3.13 \(\int \frac{(d-c^2 d x^2)^{5/2} (a+b \cos ^{-1}(c x))}{f+g x} \, dx\)

Optimal. Leaf size=1637 \[ \text{result too large to display} \]

[Out]

(a*d^2*(c^2*f^2 - g^2)^2*Sqrt[d - c^2*d*x^2])/g^5 - (2*b*c*d^2*x*Sqrt[d - c^2*d*x^2])/(15*g*Sqrt[1 - c^2*x^2])
- (b*c*d^2*(c^2*f^2 - 2*g^2)*x*Sqrt[d - c^2*d*x^2])/(3*g^3*Sqrt[1 - c^2*x^2]) + (b*c*d^2*(c^2*f^2 - g^2)^2*x*
Sqrt[d - c^2*d*x^2])/(g^5*Sqrt[1 - c^2*x^2]) + (b*c^3*d^2*f*x^2*Sqrt[d - c^2*d*x^2])/(16*g^2*Sqrt[1 - c^2*x^2]
) - (b*c^3*d^2*f*(c^2*f^2 - 2*g^2)*x^2*Sqrt[d - c^2*d*x^2])/(4*g^4*Sqrt[1 - c^2*x^2]) - (b*c^3*d^2*x^3*Sqrt[d
- c^2*d*x^2])/(45*g*Sqrt[1 - c^2*x^2]) + (b*c^3*d^2*(c^2*f^2 - 2*g^2)*x^3*Sqrt[d - c^2*d*x^2])/(9*g^3*Sqrt[1 -
c^2*x^2]) - (b*c^5*d^2*f*x^4*Sqrt[d - c^2*d*x^2])/(16*g^2*Sqrt[1 - c^2*x^2]) + (b*c^5*d^2*x^5*Sqrt[d - c^2*d*
x^2])/(25*g*Sqrt[1 - c^2*x^2]) + (b*d^2*(c^2*f^2 - g^2)^2*Sqrt[d - c^2*d*x^2]*ArcCos[c*x])/g^5 + (c^2*d^2*f*x*
Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x]))/(8*g^2) - (c^2*d^2*f*(c^2*f^2 - 2*g^2)*x*Sqrt[d - c^2*d*x^2]*(a + b*A
rcCos[c*x]))/(2*g^4) - (c^4*d^2*f*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x]))/(4*g^2) - (d^2*(1 - c^2*x^2)*Sq
rt[d - c^2*d*x^2]*(a + b*ArcCos[c*x]))/(3*g) - (d^2*(c^2*f^2 - 2*g^2)*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]*(a + b
*ArcCos[c*x]))/(3*g^3) + (d^2*(1 - c^2*x^2)^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x]))/(5*g) + (c*d^2*f*Sqrt[d
- c^2*d*x^2]*(a + b*ArcCos[c*x])^2)/(16*b*g^2*Sqrt[1 - c^2*x^2]) + (c*d^2*f*(c^2*f^2 - 2*g^2)*Sqrt[d - c^2*d*
x^2]*(a + b*ArcCos[c*x])^2)/(4*b*g^4*Sqrt[1 - c^2*x^2]) - (c*d^2*(c^2*f^2 - g^2)^2*x*Sqrt[d - c^2*d*x^2]*(a +
b*ArcCos[c*x])^2)/(2*b*g^5*Sqrt[1 - c^2*x^2]) - (d^2*(c^2*f^2 - g^2)^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x])
^2)/(2*b*c*g^6*(f + g*x)*Sqrt[1 - c^2*x^2]) - (d^2*(c^2*f^2 - g^2)^2*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2]*(a
+ b*ArcCos[c*x])^2)/(2*b*c*g^4*(f + g*x)) - (a*d^2*(c^2*f^2 - g^2)^(5/2)*Sqrt[d - c^2*d*x^2]*ArcTan[(g + c^2*f
*x)/(Sqrt[c^2*f^2 - g^2]*Sqrt[1 - c^2*x^2])])/(g^6*Sqrt[1 - c^2*x^2]) - (I*b*d^2*(c^2*f^2 - g^2)^(5/2)*Sqrt[d
- c^2*d*x^2]*ArcCos[c*x]*Log[1 + (E^(I*ArcCos[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/(g^6*Sqrt[1 - c^2*x^2]) +
(I*b*d^2*(c^2*f^2 - g^2)^(5/2)*Sqrt[d - c^2*d*x^2]*ArcCos[c*x]*Log[1 + (E^(I*ArcCos[c*x])*g)/(c*f + Sqrt[c^2*
f^2 - g^2])])/(g^6*Sqrt[1 - c^2*x^2]) - (b*d^2*(c^2*f^2 - g^2)^(5/2)*Sqrt[d - c^2*d*x^2]*PolyLog[2, -((E^(I*Ar
cCos[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2]))])/(g^6*Sqrt[1 - c^2*x^2]) + (b*d^2*(c^2*f^2 - g^2)^(5/2)*Sqrt[d - c
^2*d*x^2]*PolyLog[2, -((E^(I*ArcCos[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2]))])/(g^6*Sqrt[1 - c^2*x^2])

________________________________________________________________________________________

Rubi [A]  time = 2.73, antiderivative size = 1637, normalized size of antiderivative = 1., number of steps used = 37, number of rules used = 28, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.903, Rules used = {4778, 4768, 4648, 4642, 30, 4678, 4698, 4708, 266, 43, 4690, 12, 4766, 683, 4758, 6742, 725, 204, 1654, 4800, 4798, 8, 4774, 3321, 2264, 2190, 2279, 2391} \[ \frac{b d^2 x^5 \sqrt{d-c^2 d x^2} c^5}{25 g \sqrt{1-c^2 x^2}}-\frac{b d^2 f x^4 \sqrt{d-c^2 d x^2} c^5}{16 g^2 \sqrt{1-c^2 x^2}}-\frac{d^2 f x^3 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right ) c^4}{4 g^2}+\frac{b d^2 \left (c^2 f^2-2 g^2\right ) x^3 \sqrt{d-c^2 d x^2} c^3}{9 g^3 \sqrt{1-c^2 x^2}}-\frac{b d^2 x^3 \sqrt{d-c^2 d x^2} c^3}{45 g \sqrt{1-c^2 x^2}}-\frac{b d^2 f \left (c^2 f^2-2 g^2\right ) x^2 \sqrt{d-c^2 d x^2} c^3}{4 g^4 \sqrt{1-c^2 x^2}}+\frac{b d^2 f x^2 \sqrt{d-c^2 d x^2} c^3}{16 g^2 \sqrt{1-c^2 x^2}}-\frac{d^2 f \left (c^2 f^2-2 g^2\right ) x \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right ) c^2}{2 g^4}+\frac{d^2 f x \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right ) c^2}{8 g^2}+\frac{d^2 f \left (c^2 f^2-2 g^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2 c}{4 b g^4 \sqrt{1-c^2 x^2}}-\frac{d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2 c}{2 b g^5 \sqrt{1-c^2 x^2}}+\frac{d^2 f \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2 c}{16 b g^2 \sqrt{1-c^2 x^2}}+\frac{b d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt{d-c^2 d x^2} c}{g^5 \sqrt{1-c^2 x^2}}-\frac{b d^2 \left (c^2 f^2-2 g^2\right ) x \sqrt{d-c^2 d x^2} c}{3 g^3 \sqrt{1-c^2 x^2}}-\frac{2 b d^2 x \sqrt{d-c^2 d x^2} c}{15 g \sqrt{1-c^2 x^2}}+\frac{b d^2 \left (c^2 f^2-g^2\right )^2 \sqrt{d-c^2 d x^2} \cos ^{-1}(c x)}{g^5}+\frac{d^2 \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{5 g}-\frac{d^2 \left (c^2 f^2-2 g^2\right ) \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{3 g^3}-\frac{d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{3 g}-\frac{a d^2 \left (c^2 f^2-g^2\right )^{5/2} \sqrt{d-c^2 d x^2} \tan ^{-1}\left (\frac{f x c^2+g}{\sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}\right )}{g^6 \sqrt{1-c^2 x^2}}-\frac{i b d^2 \left (c^2 f^2-g^2\right )^{5/2} \sqrt{d-c^2 d x^2} \cos ^{-1}(c x) \log \left (\frac{e^{i \cos ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}+1\right )}{g^6 \sqrt{1-c^2 x^2}}+\frac{i b d^2 \left (c^2 f^2-g^2\right )^{5/2} \sqrt{d-c^2 d x^2} \cos ^{-1}(c x) \log \left (\frac{e^{i \cos ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}+1\right )}{g^6 \sqrt{1-c^2 x^2}}-\frac{b d^2 \left (c^2 f^2-g^2\right )^{5/2} \sqrt{d-c^2 d x^2} \text{PolyLog}\left (2,-\frac{e^{i \cos ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right )}{g^6 \sqrt{1-c^2 x^2}}+\frac{b d^2 \left (c^2 f^2-g^2\right )^{5/2} \sqrt{d-c^2 d x^2} \text{PolyLog}\left (2,-\frac{e^{i \cos ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right )}{g^6 \sqrt{1-c^2 x^2}}+\frac{a d^2 \left (c^2 f^2-g^2\right )^2 \sqrt{d-c^2 d x^2}}{g^5}-\frac{d^2 \left (c^2 f^2-g^2\right )^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b g^4 (f+g x) c}-\frac{d^2 \left (c^2 f^2-g^2\right )^3 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b g^6 (f+g x) \sqrt{1-c^2 x^2} c} \]

Antiderivative was successfully verified.

[In]

Int[((d - c^2*d*x^2)^(5/2)*(a + b*ArcCos[c*x]))/(f + g*x),x]

[Out]

(a*d^2*(c^2*f^2 - g^2)^2*Sqrt[d - c^2*d*x^2])/g^5 - (2*b*c*d^2*x*Sqrt[d - c^2*d*x^2])/(15*g*Sqrt[1 - c^2*x^2])
- (b*c*d^2*(c^2*f^2 - 2*g^2)*x*Sqrt[d - c^2*d*x^2])/(3*g^3*Sqrt[1 - c^2*x^2]) + (b*c*d^2*(c^2*f^2 - g^2)^2*x*
Sqrt[d - c^2*d*x^2])/(g^5*Sqrt[1 - c^2*x^2]) + (b*c^3*d^2*f*x^2*Sqrt[d - c^2*d*x^2])/(16*g^2*Sqrt[1 - c^2*x^2]
) - (b*c^3*d^2*f*(c^2*f^2 - 2*g^2)*x^2*Sqrt[d - c^2*d*x^2])/(4*g^4*Sqrt[1 - c^2*x^2]) - (b*c^3*d^2*x^3*Sqrt[d
- c^2*d*x^2])/(45*g*Sqrt[1 - c^2*x^2]) + (b*c^3*d^2*(c^2*f^2 - 2*g^2)*x^3*Sqrt[d - c^2*d*x^2])/(9*g^3*Sqrt[1 -
c^2*x^2]) - (b*c^5*d^2*f*x^4*Sqrt[d - c^2*d*x^2])/(16*g^2*Sqrt[1 - c^2*x^2]) + (b*c^5*d^2*x^5*Sqrt[d - c^2*d*
x^2])/(25*g*Sqrt[1 - c^2*x^2]) + (b*d^2*(c^2*f^2 - g^2)^2*Sqrt[d - c^2*d*x^2]*ArcCos[c*x])/g^5 + (c^2*d^2*f*x*
Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x]))/(8*g^2) - (c^2*d^2*f*(c^2*f^2 - 2*g^2)*x*Sqrt[d - c^2*d*x^2]*(a + b*A
rcCos[c*x]))/(2*g^4) - (c^4*d^2*f*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x]))/(4*g^2) - (d^2*(1 - c^2*x^2)*Sq
rt[d - c^2*d*x^2]*(a + b*ArcCos[c*x]))/(3*g) - (d^2*(c^2*f^2 - 2*g^2)*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]*(a + b
*ArcCos[c*x]))/(3*g^3) + (d^2*(1 - c^2*x^2)^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x]))/(5*g) + (c*d^2*f*Sqrt[d
- c^2*d*x^2]*(a + b*ArcCos[c*x])^2)/(16*b*g^2*Sqrt[1 - c^2*x^2]) + (c*d^2*f*(c^2*f^2 - 2*g^2)*Sqrt[d - c^2*d*
x^2]*(a + b*ArcCos[c*x])^2)/(4*b*g^4*Sqrt[1 - c^2*x^2]) - (c*d^2*(c^2*f^2 - g^2)^2*x*Sqrt[d - c^2*d*x^2]*(a +
b*ArcCos[c*x])^2)/(2*b*g^5*Sqrt[1 - c^2*x^2]) - (d^2*(c^2*f^2 - g^2)^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x])
^2)/(2*b*c*g^6*(f + g*x)*Sqrt[1 - c^2*x^2]) - (d^2*(c^2*f^2 - g^2)^2*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2]*(a
+ b*ArcCos[c*x])^2)/(2*b*c*g^4*(f + g*x)) - (a*d^2*(c^2*f^2 - g^2)^(5/2)*Sqrt[d - c^2*d*x^2]*ArcTan[(g + c^2*f
*x)/(Sqrt[c^2*f^2 - g^2]*Sqrt[1 - c^2*x^2])])/(g^6*Sqrt[1 - c^2*x^2]) - (I*b*d^2*(c^2*f^2 - g^2)^(5/2)*Sqrt[d
- c^2*d*x^2]*ArcCos[c*x]*Log[1 + (E^(I*ArcCos[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/(g^6*Sqrt[1 - c^2*x^2]) +
(I*b*d^2*(c^2*f^2 - g^2)^(5/2)*Sqrt[d - c^2*d*x^2]*ArcCos[c*x]*Log[1 + (E^(I*ArcCos[c*x])*g)/(c*f + Sqrt[c^2*
f^2 - g^2])])/(g^6*Sqrt[1 - c^2*x^2]) - (b*d^2*(c^2*f^2 - g^2)^(5/2)*Sqrt[d - c^2*d*x^2]*PolyLog[2, -((E^(I*Ar
cCos[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2]))])/(g^6*Sqrt[1 - c^2*x^2]) + (b*d^2*(c^2*f^2 - g^2)^(5/2)*Sqrt[d - c
^2*d*x^2]*PolyLog[2, -((E^(I*ArcCos[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2]))])/(g^6*Sqrt[1 - c^2*x^2])

Rule 4778

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :
> Dist[(d^IntPart[p]*(d + e*x^2)^FracPart[p])/(1 - c^2*x^2)^FracPart[p], Int[(f + g*x)^m*(1 - c^2*x^2)^p*(a +
b*ArcCos[c*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && EqQ[c^2*d + e, 0] && IntegerQ[m] && IntegerQ
[p - 1/2] &&  !GtQ[d, 0]

Rule 4768

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :
> Int[ExpandIntegrand[Sqrt[d + e*x^2]*(a + b*ArcCos[c*x])^n, (f + g*x)^m*(d + e*x^2)^(p - 1/2), x], x] /; Free
Q[{a, b, c, d, e, f, g}, x] && EqQ[c^2*d + e, 0] && IntegerQ[m] && IGtQ[p + 1/2, 0] && GtQ[d, 0] && IGtQ[n, 0]

Rule 4648

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(x*Sqrt[d + e*x^2]*(
a + b*ArcCos[c*x])^n)/2, x] + (Dist[Sqrt[d + e*x^2]/(2*Sqrt[1 - c^2*x^2]), Int[(a + b*ArcCos[c*x])^n/Sqrt[1 -
c^2*x^2], x], x] + Dist[(b*c*n*Sqrt[d + e*x^2])/(2*Sqrt[1 - c^2*x^2]), Int[x*(a + b*ArcCos[c*x])^(n - 1), x],
x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0]

Rule 4642

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> -Simp[(a + b*ArcCos[c*x])
^(n + 1)/(b*c*Sqrt[d]*(n + 1)), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^2*d + e, 0] && GtQ[d, 0] && NeQ[n,
-1]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 4678

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[((d + e*x^2)^
(p + 1)*(a + b*ArcCos[c*x])^n)/(2*e*(p + 1)), x] - Dist[(b*n*d^IntPart[p]*(d + e*x^2)^FracPart[p])/(2*c*(p + 1
)*(1 - c^2*x^2)^FracPart[p]), Int[(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcCos[c*x])^(n - 1), x], x] /; FreeQ[{a, b,
c, d, e, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && NeQ[p, -1]

Rule 4698

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[((
f*x)^(m + 1)*Sqrt[d + e*x^2]*(a + b*ArcCos[c*x])^n)/(f*(m + 2)), x] + (Dist[Sqrt[d + e*x^2]/((m + 2)*Sqrt[1 -
c^2*x^2]), Int[((f*x)^m*(a + b*ArcCos[c*x])^n)/Sqrt[1 - c^2*x^2], x], x] + Dist[(b*c*n*Sqrt[d + e*x^2])/(f*(m
+ 2)*Sqrt[1 - c^2*x^2]), Int[(f*x)^(m + 1)*(a + b*ArcCos[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f, m}
, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] &&  !LtQ[m, -1] && (RationalQ[m] || EqQ[n, 1])

Rule 4708

Int[(((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[
(f*(f*x)^(m - 1)*Sqrt[d + e*x^2]*(a + b*ArcCos[c*x])^n)/(e*m), x] + (Dist[(f^2*(m - 1))/(c^2*m), Int[((f*x)^(m
- 2)*(a + b*ArcCos[c*x])^n)/Sqrt[d + e*x^2], x], x] - Dist[(b*f*n*Sqrt[1 - c^2*x^2])/(c*m*Sqrt[d + e*x^2]), I
nt[(f*x)^(m - 1)*(a + b*ArcCos[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] &&
GtQ[n, 0] && GtQ[m, 1] && IntegerQ[m]

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 4690

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))*(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> With[{u = IntHide[x^
m*(1 - c^2*x^2)^p, x]}, Dist[d^p*(a + b*ArcCos[c*x]), u, x] + Dist[b*c*d^p, Int[SimplifyIntegrand[u/Sqrt[1 - c
^2*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IntegerQ[p - 1/2] && (IGtQ[(m + 1)/2
, 0] || ILtQ[(m + 2*p + 3)/2, 0]) && NeQ[p, -2^(-1)] && GtQ[d, 0]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 4766

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_)*Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :>
-Simp[((f + g*x)^m*(d + e*x^2)*(a + b*ArcCos[c*x])^(n + 1))/(b*c*Sqrt[d]*(n + 1)), x] + Dist[1/(b*c*Sqrt[d]*(n
+ 1)), Int[(d*g*m + 2*e*f*x + e*g*(m + 2)*x^2)*(f + g*x)^(m - 1)*(a + b*ArcCos[c*x])^(n + 1), x], x] /; FreeQ
[{a, b, c, d, e, f, g}, x] && EqQ[c^2*d + e, 0] && ILtQ[m, 0] && GtQ[d, 0] && IGtQ[n, 0]

Rule 683

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[2*c*d - b*e,
0] && IGtQ[p, 0] &&  !(EqQ[m, 3] && NeQ[p, 1])

Rule 4758

Int[(((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_)*((f_.) + (g_.)*(x_) + (h_.)*(x_)^2)^(p_.))/((d_) + (e_.)*(x_))^2,
x_Symbol] :> With[{u = IntHide[(f + g*x + h*x^2)^p/(d + e*x)^2, x]}, Dist[(a + b*ArcCos[c*x])^n, u, x] + Dist
[b*c*n, Int[SimplifyIntegrand[(u*(a + b*ArcCos[c*x])^(n - 1))/Sqrt[1 - c^2*x^2], x], x], x]] /; FreeQ[{a, b, c
, d, e, f, g, h}, x] && IGtQ[n, 0] && IGtQ[p, 0] && EqQ[e*g - 2*d*h, 0]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rule 725

Int[1/(((d_) + (e_.)*(x_))*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> -Subst[Int[1/(c*d^2 + a*e^2 - x^2), x], x,
(a*e - c*d*x)/Sqrt[a + c*x^2]] /; FreeQ[{a, c, d, e}, x]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 1654

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], f = Coeff
[Pq, x, Expon[Pq, x]]}, Simp[(f*(d + e*x)^(m + q - 1)*(a + c*x^2)^(p + 1))/(c*e^(q - 1)*(m + q + 2*p + 1)), x]
+ Dist[1/(c*e^q*(m + q + 2*p + 1)), Int[(d + e*x)^m*(a + c*x^2)^p*ExpandToSum[c*e^q*(m + q + 2*p + 1)*Pq - c*
f*(m + q + 2*p + 1)*(d + e*x)^q - f*(d + e*x)^(q - 2)*(a*e^2*(m + q - 1) - c*d^2*(m + q + 2*p + 1) - 2*c*d*e*(
m + q + p)*x), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, c, d, e, m, p}, x] && PolyQ[Pq
, x] && NeQ[c*d^2 + a*e^2, 0] &&  !(EqQ[d, 0] && True) &&  !(IGtQ[m, 0] && RationalQ[a, c, d, e] && (IntegerQ[
p] || ILtQ[p + 1/2, 0]))

Rule 4800

Int[(ArcCos[(c_.)*(x_)]*(b_.) + (a_))^(n_.)*(RFx_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Int[ExpandIntegran
d[(d + e*x^2)^p, RFx*(a + b*ArcCos[c*x])^n, x], x] /; FreeQ[{a, b, c, d, e}, x] && RationalFunctionQ[RFx, x] &
& IGtQ[n, 0] && EqQ[c^2*d + e, 0] && IntegerQ[p - 1/2]

Rule 4798

Int[ArcCos[(c_.)*(x_)]^(n_.)*(RFx_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> With[{u = ExpandIntegrand[(d + e*
x^2)^p*ArcCos[c*x]^n, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{c, d, e}, x] && RationalFunctionQ[RFx, x] && I
GtQ[n, 0] && EqQ[c^2*d + e, 0] && IntegerQ[p - 1/2]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 4774

Int[(((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol]
:> -Dist[(c^(m + 1)*Sqrt[d])^(-1), Subst[Int[(a + b*x)^n*(c*f + g*Cos[x])^m, x], x, ArcCos[c*x]], x] /; FreeQ[
{a, b, c, d, e, f, g, n}, x] && EqQ[c^2*d + e, 0] && IntegerQ[m] && GtQ[d, 0] && (GtQ[m, 0] || IGtQ[n, 0])

Rule 3321

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + Pi*(k_.) + (f_.)*(x_)]), x_Symbol] :> Dist[2, Int[((c
+ d*x)^m*E^(I*Pi*(k - 1/2))*E^(I*(e + f*x)))/(b + 2*a*E^(I*Pi*(k - 1/2))*E^(I*(e + f*x)) - b*E^(2*I*k*Pi)*E^(
2*I*(e + f*x))), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IntegerQ[2*k] && NeQ[a^2 - b^2, 0] && IGtQ[m, 0]

Rule 2264

Int[((F_)^(u_)*((f_.) + (g_.)*(x_))^(m_.))/((a_.) + (b_.)*(F_)^(u_) + (c_.)*(F_)^(v_)), x_Symbol] :> With[{q =
Rt[b^2 - 4*a*c, 2]}, Dist[(2*c)/q, Int[((f + g*x)^m*F^u)/(b - q + 2*c*F^u), x], x] - Dist[(2*c)/q, Int[((f +
g*x)^m*F^u)/(b + q + 2*c*F^u), x], x]] /; FreeQ[{F, a, b, c, f, g}, x] && EqQ[v, 2*u] && LinearQ[u, x] && NeQ[
b^2 - 4*a*c, 0] && IGtQ[m, 0]

Rule 2190

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), x]
- Dist[(d*m)/(b*f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*Log[1 + (b*(F^(g*(e + f*x)))^n)/a], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2279

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rubi steps

\begin{align*} \int \frac{\left (d-c^2 d x^2\right )^{5/2} \left (a+b \cos ^{-1}(c x)\right )}{f+g x} \, dx &=\frac{\left (d^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{\left (1-c^2 x^2\right )^{5/2} \left (a+b \cos ^{-1}(c x)\right )}{f+g x} \, dx}{\sqrt{1-c^2 x^2}}\\ &=\frac{\left (d^2 \sqrt{d-c^2 d x^2}\right ) \int \left (-\frac{c^2 f \left (c^2 f^2-2 g^2\right ) \sqrt{1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right )}{g^4}+\frac{c^2 \left (c^2 f^2-2 g^2\right ) x \sqrt{1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right )}{g^3}-\frac{c^4 f x^2 \sqrt{1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right )}{g^2}+\frac{c^4 x^3 \sqrt{1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right )}{g}+\frac{\left (-c^2 f^2+g^2\right )^2 \sqrt{1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right )}{g^4 (f+g x)}\right ) \, dx}{\sqrt{1-c^2 x^2}}\\ &=-\frac{\left (c^4 d^2 f \sqrt{d-c^2 d x^2}\right ) \int x^2 \sqrt{1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right ) \, dx}{g^2 \sqrt{1-c^2 x^2}}+\frac{\left (c^4 d^2 \sqrt{d-c^2 d x^2}\right ) \int x^3 \sqrt{1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right ) \, dx}{g \sqrt{1-c^2 x^2}}-\frac{\left (c^2 d^2 f \left (c^2 f^2-2 g^2\right ) \sqrt{d-c^2 d x^2}\right ) \int \sqrt{1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right ) \, dx}{g^4 \sqrt{1-c^2 x^2}}+\frac{\left (c^2 d^2 \left (c^2 f^2-2 g^2\right ) \sqrt{d-c^2 d x^2}\right ) \int x \sqrt{1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right ) \, dx}{g^3 \sqrt{1-c^2 x^2}}+\frac{\left (d^2 \left (c^2 f^2-g^2\right )^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{\sqrt{1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right )}{f+g x} \, dx}{g^4 \sqrt{1-c^2 x^2}}\\ &=-\frac{c^2 d^2 f \left (c^2 f^2-2 g^2\right ) x \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{2 g^4}-\frac{c^4 d^2 f x^3 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{4 g^2}-\frac{d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{3 g}-\frac{d^2 \left (c^2 f^2-2 g^2\right ) \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{3 g^3}+\frac{d^2 \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{5 g}-\frac{d^2 \left (c^2 f^2-g^2\right )^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b c g^4 (f+g x)}-\frac{\left (c^4 d^2 f \sqrt{d-c^2 d x^2}\right ) \int \frac{x^2 \left (a+b \cos ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}} \, dx}{4 g^2 \sqrt{1-c^2 x^2}}-\frac{\left (b c^5 d^2 f \sqrt{d-c^2 d x^2}\right ) \int x^3 \, dx}{4 g^2 \sqrt{1-c^2 x^2}}+\frac{\left (b c^5 d^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{-2-c^2 x^2+3 c^4 x^4}{15 c^4} \, dx}{g \sqrt{1-c^2 x^2}}-\frac{\left (c^2 d^2 f \left (c^2 f^2-2 g^2\right ) \sqrt{d-c^2 d x^2}\right ) \int \frac{a+b \cos ^{-1}(c x)}{\sqrt{1-c^2 x^2}} \, dx}{2 g^4 \sqrt{1-c^2 x^2}}-\frac{\left (b c^3 d^2 f \left (c^2 f^2-2 g^2\right ) \sqrt{d-c^2 d x^2}\right ) \int x \, dx}{2 g^4 \sqrt{1-c^2 x^2}}-\frac{\left (b c d^2 \left (c^2 f^2-2 g^2\right ) \sqrt{d-c^2 d x^2}\right ) \int \left (1-c^2 x^2\right ) \, dx}{3 g^3 \sqrt{1-c^2 x^2}}+\frac{\left (d^2 \left (c^2 f^2-g^2\right )^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{\left (-g-2 c^2 f x-c^2 g x^2\right ) \left (a+b \cos ^{-1}(c x)\right )^2}{(f+g x)^2} \, dx}{2 b c g^4 \sqrt{1-c^2 x^2}}\\ &=-\frac{b c d^2 \left (c^2 f^2-2 g^2\right ) x \sqrt{d-c^2 d x^2}}{3 g^3 \sqrt{1-c^2 x^2}}-\frac{b c^3 d^2 f \left (c^2 f^2-2 g^2\right ) x^2 \sqrt{d-c^2 d x^2}}{4 g^4 \sqrt{1-c^2 x^2}}+\frac{b c^3 d^2 \left (c^2 f^2-2 g^2\right ) x^3 \sqrt{d-c^2 d x^2}}{9 g^3 \sqrt{1-c^2 x^2}}-\frac{b c^5 d^2 f x^4 \sqrt{d-c^2 d x^2}}{16 g^2 \sqrt{1-c^2 x^2}}+\frac{c^2 d^2 f x \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{8 g^2}-\frac{c^2 d^2 f \left (c^2 f^2-2 g^2\right ) x \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{2 g^4}-\frac{c^4 d^2 f x^3 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{4 g^2}-\frac{d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{3 g}-\frac{d^2 \left (c^2 f^2-2 g^2\right ) \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{3 g^3}+\frac{d^2 \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{5 g}+\frac{c d^2 f \left (c^2 f^2-2 g^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{4 b g^4 \sqrt{1-c^2 x^2}}-\frac{c d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b g^5 \sqrt{1-c^2 x^2}}-\frac{d^2 \left (c^2 f^2-g^2\right )^3 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b c g^6 (f+g x) \sqrt{1-c^2 x^2}}-\frac{d^2 \left (c^2 f^2-g^2\right )^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b c g^4 (f+g x)}-\frac{\left (c^2 d^2 f \sqrt{d-c^2 d x^2}\right ) \int \frac{a+b \cos ^{-1}(c x)}{\sqrt{1-c^2 x^2}} \, dx}{8 g^2 \sqrt{1-c^2 x^2}}+\frac{\left (b c^3 d^2 f \sqrt{d-c^2 d x^2}\right ) \int x \, dx}{8 g^2 \sqrt{1-c^2 x^2}}+\frac{\left (b c d^2 \sqrt{d-c^2 d x^2}\right ) \int \left (-2-c^2 x^2+3 c^4 x^4\right ) \, dx}{15 g \sqrt{1-c^2 x^2}}+\frac{\left (d^2 \left (c^2 f^2-g^2\right )^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{\left (\frac{1}{f+g x}-\frac{c^2 \left (g x+\frac{f^2}{f+g x}\right )}{g^2}\right ) \left (a+b \cos ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}} \, dx}{g^4 \sqrt{1-c^2 x^2}}\\ &=-\frac{2 b c d^2 x \sqrt{d-c^2 d x^2}}{15 g \sqrt{1-c^2 x^2}}-\frac{b c d^2 \left (c^2 f^2-2 g^2\right ) x \sqrt{d-c^2 d x^2}}{3 g^3 \sqrt{1-c^2 x^2}}+\frac{b c^3 d^2 f x^2 \sqrt{d-c^2 d x^2}}{16 g^2 \sqrt{1-c^2 x^2}}-\frac{b c^3 d^2 f \left (c^2 f^2-2 g^2\right ) x^2 \sqrt{d-c^2 d x^2}}{4 g^4 \sqrt{1-c^2 x^2}}-\frac{b c^3 d^2 x^3 \sqrt{d-c^2 d x^2}}{45 g \sqrt{1-c^2 x^2}}+\frac{b c^3 d^2 \left (c^2 f^2-2 g^2\right ) x^3 \sqrt{d-c^2 d x^2}}{9 g^3 \sqrt{1-c^2 x^2}}-\frac{b c^5 d^2 f x^4 \sqrt{d-c^2 d x^2}}{16 g^2 \sqrt{1-c^2 x^2}}+\frac{b c^5 d^2 x^5 \sqrt{d-c^2 d x^2}}{25 g \sqrt{1-c^2 x^2}}+\frac{c^2 d^2 f x \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{8 g^2}-\frac{c^2 d^2 f \left (c^2 f^2-2 g^2\right ) x \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{2 g^4}-\frac{c^4 d^2 f x^3 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{4 g^2}-\frac{d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{3 g}-\frac{d^2 \left (c^2 f^2-2 g^2\right ) \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{3 g^3}+\frac{d^2 \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{5 g}+\frac{c d^2 f \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{16 b g^2 \sqrt{1-c^2 x^2}}+\frac{c d^2 f \left (c^2 f^2-2 g^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{4 b g^4 \sqrt{1-c^2 x^2}}-\frac{c d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b g^5 \sqrt{1-c^2 x^2}}-\frac{d^2 \left (c^2 f^2-g^2\right )^3 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b c g^6 (f+g x) \sqrt{1-c^2 x^2}}-\frac{d^2 \left (c^2 f^2-g^2\right )^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b c g^4 (f+g x)}+\frac{\left (d^2 \left (c^2 f^2-g^2\right )^2 \sqrt{d-c^2 d x^2}\right ) \int \left (-\frac{a \left (c^2 f^2-g^2+c^2 f g x+c^2 g^2 x^2\right )}{g^2 (f+g x) \sqrt{1-c^2 x^2}}-\frac{b \left (c^2 f^2-g^2+c^2 f g x+c^2 g^2 x^2\right ) \cos ^{-1}(c x)}{g^2 (f+g x) \sqrt{1-c^2 x^2}}\right ) \, dx}{g^4 \sqrt{1-c^2 x^2}}\\ &=-\frac{2 b c d^2 x \sqrt{d-c^2 d x^2}}{15 g \sqrt{1-c^2 x^2}}-\frac{b c d^2 \left (c^2 f^2-2 g^2\right ) x \sqrt{d-c^2 d x^2}}{3 g^3 \sqrt{1-c^2 x^2}}+\frac{b c^3 d^2 f x^2 \sqrt{d-c^2 d x^2}}{16 g^2 \sqrt{1-c^2 x^2}}-\frac{b c^3 d^2 f \left (c^2 f^2-2 g^2\right ) x^2 \sqrt{d-c^2 d x^2}}{4 g^4 \sqrt{1-c^2 x^2}}-\frac{b c^3 d^2 x^3 \sqrt{d-c^2 d x^2}}{45 g \sqrt{1-c^2 x^2}}+\frac{b c^3 d^2 \left (c^2 f^2-2 g^2\right ) x^3 \sqrt{d-c^2 d x^2}}{9 g^3 \sqrt{1-c^2 x^2}}-\frac{b c^5 d^2 f x^4 \sqrt{d-c^2 d x^2}}{16 g^2 \sqrt{1-c^2 x^2}}+\frac{b c^5 d^2 x^5 \sqrt{d-c^2 d x^2}}{25 g \sqrt{1-c^2 x^2}}+\frac{c^2 d^2 f x \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{8 g^2}-\frac{c^2 d^2 f \left (c^2 f^2-2 g^2\right ) x \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{2 g^4}-\frac{c^4 d^2 f x^3 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{4 g^2}-\frac{d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{3 g}-\frac{d^2 \left (c^2 f^2-2 g^2\right ) \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{3 g^3}+\frac{d^2 \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{5 g}+\frac{c d^2 f \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{16 b g^2 \sqrt{1-c^2 x^2}}+\frac{c d^2 f \left (c^2 f^2-2 g^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{4 b g^4 \sqrt{1-c^2 x^2}}-\frac{c d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b g^5 \sqrt{1-c^2 x^2}}-\frac{d^2 \left (c^2 f^2-g^2\right )^3 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b c g^6 (f+g x) \sqrt{1-c^2 x^2}}-\frac{d^2 \left (c^2 f^2-g^2\right )^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b c g^4 (f+g x)}-\frac{\left (a d^2 \left (c^2 f^2-g^2\right )^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{c^2 f^2-g^2+c^2 f g x+c^2 g^2 x^2}{(f+g x) \sqrt{1-c^2 x^2}} \, dx}{g^6 \sqrt{1-c^2 x^2}}-\frac{\left (b d^2 \left (c^2 f^2-g^2\right )^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{\left (c^2 f^2-g^2+c^2 f g x+c^2 g^2 x^2\right ) \cos ^{-1}(c x)}{(f+g x) \sqrt{1-c^2 x^2}} \, dx}{g^6 \sqrt{1-c^2 x^2}}\\ &=\frac{a d^2 \left (c^2 f^2-g^2\right )^2 \sqrt{d-c^2 d x^2}}{g^5}-\frac{2 b c d^2 x \sqrt{d-c^2 d x^2}}{15 g \sqrt{1-c^2 x^2}}-\frac{b c d^2 \left (c^2 f^2-2 g^2\right ) x \sqrt{d-c^2 d x^2}}{3 g^3 \sqrt{1-c^2 x^2}}+\frac{b c^3 d^2 f x^2 \sqrt{d-c^2 d x^2}}{16 g^2 \sqrt{1-c^2 x^2}}-\frac{b c^3 d^2 f \left (c^2 f^2-2 g^2\right ) x^2 \sqrt{d-c^2 d x^2}}{4 g^4 \sqrt{1-c^2 x^2}}-\frac{b c^3 d^2 x^3 \sqrt{d-c^2 d x^2}}{45 g \sqrt{1-c^2 x^2}}+\frac{b c^3 d^2 \left (c^2 f^2-2 g^2\right ) x^3 \sqrt{d-c^2 d x^2}}{9 g^3 \sqrt{1-c^2 x^2}}-\frac{b c^5 d^2 f x^4 \sqrt{d-c^2 d x^2}}{16 g^2 \sqrt{1-c^2 x^2}}+\frac{b c^5 d^2 x^5 \sqrt{d-c^2 d x^2}}{25 g \sqrt{1-c^2 x^2}}+\frac{c^2 d^2 f x \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{8 g^2}-\frac{c^2 d^2 f \left (c^2 f^2-2 g^2\right ) x \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{2 g^4}-\frac{c^4 d^2 f x^3 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{4 g^2}-\frac{d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{3 g}-\frac{d^2 \left (c^2 f^2-2 g^2\right ) \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{3 g^3}+\frac{d^2 \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{5 g}+\frac{c d^2 f \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{16 b g^2 \sqrt{1-c^2 x^2}}+\frac{c d^2 f \left (c^2 f^2-2 g^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{4 b g^4 \sqrt{1-c^2 x^2}}-\frac{c d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b g^5 \sqrt{1-c^2 x^2}}-\frac{d^2 \left (c^2 f^2-g^2\right )^3 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b c g^6 (f+g x) \sqrt{1-c^2 x^2}}-\frac{d^2 \left (c^2 f^2-g^2\right )^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b c g^4 (f+g x)}-\frac{\left (a d^2 \left (c^2 f^2-g^2\right )^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{c^2 g^2 \left (c^2 f^2-g^2\right )}{(f+g x) \sqrt{1-c^2 x^2}} \, dx}{c^2 g^8 \sqrt{1-c^2 x^2}}-\frac{\left (b d^2 \left (c^2 f^2-g^2\right )^2 \sqrt{d-c^2 d x^2}\right ) \int \left (\frac{c^2 g x \cos ^{-1}(c x)}{\sqrt{1-c^2 x^2}}+\frac{\left (c^2 f^2-g^2\right ) \cos ^{-1}(c x)}{(f+g x) \sqrt{1-c^2 x^2}}\right ) \, dx}{g^6 \sqrt{1-c^2 x^2}}\\ &=\frac{a d^2 \left (c^2 f^2-g^2\right )^2 \sqrt{d-c^2 d x^2}}{g^5}-\frac{2 b c d^2 x \sqrt{d-c^2 d x^2}}{15 g \sqrt{1-c^2 x^2}}-\frac{b c d^2 \left (c^2 f^2-2 g^2\right ) x \sqrt{d-c^2 d x^2}}{3 g^3 \sqrt{1-c^2 x^2}}+\frac{b c^3 d^2 f x^2 \sqrt{d-c^2 d x^2}}{16 g^2 \sqrt{1-c^2 x^2}}-\frac{b c^3 d^2 f \left (c^2 f^2-2 g^2\right ) x^2 \sqrt{d-c^2 d x^2}}{4 g^4 \sqrt{1-c^2 x^2}}-\frac{b c^3 d^2 x^3 \sqrt{d-c^2 d x^2}}{45 g \sqrt{1-c^2 x^2}}+\frac{b c^3 d^2 \left (c^2 f^2-2 g^2\right ) x^3 \sqrt{d-c^2 d x^2}}{9 g^3 \sqrt{1-c^2 x^2}}-\frac{b c^5 d^2 f x^4 \sqrt{d-c^2 d x^2}}{16 g^2 \sqrt{1-c^2 x^2}}+\frac{b c^5 d^2 x^5 \sqrt{d-c^2 d x^2}}{25 g \sqrt{1-c^2 x^2}}+\frac{c^2 d^2 f x \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{8 g^2}-\frac{c^2 d^2 f \left (c^2 f^2-2 g^2\right ) x \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{2 g^4}-\frac{c^4 d^2 f x^3 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{4 g^2}-\frac{d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{3 g}-\frac{d^2 \left (c^2 f^2-2 g^2\right ) \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{3 g^3}+\frac{d^2 \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{5 g}+\frac{c d^2 f \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{16 b g^2 \sqrt{1-c^2 x^2}}+\frac{c d^2 f \left (c^2 f^2-2 g^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{4 b g^4 \sqrt{1-c^2 x^2}}-\frac{c d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b g^5 \sqrt{1-c^2 x^2}}-\frac{d^2 \left (c^2 f^2-g^2\right )^3 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b c g^6 (f+g x) \sqrt{1-c^2 x^2}}-\frac{d^2 \left (c^2 f^2-g^2\right )^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b c g^4 (f+g x)}-\frac{\left (b c^2 d^2 \left (c^2 f^2-g^2\right )^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{x \cos ^{-1}(c x)}{\sqrt{1-c^2 x^2}} \, dx}{g^5 \sqrt{1-c^2 x^2}}-\frac{\left (a d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{1}{(f+g x) \sqrt{1-c^2 x^2}} \, dx}{g^6 \sqrt{1-c^2 x^2}}-\frac{\left (b d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{\cos ^{-1}(c x)}{(f+g x) \sqrt{1-c^2 x^2}} \, dx}{g^6 \sqrt{1-c^2 x^2}}\\ &=\frac{a d^2 \left (c^2 f^2-g^2\right )^2 \sqrt{d-c^2 d x^2}}{g^5}-\frac{2 b c d^2 x \sqrt{d-c^2 d x^2}}{15 g \sqrt{1-c^2 x^2}}-\frac{b c d^2 \left (c^2 f^2-2 g^2\right ) x \sqrt{d-c^2 d x^2}}{3 g^3 \sqrt{1-c^2 x^2}}+\frac{b c^3 d^2 f x^2 \sqrt{d-c^2 d x^2}}{16 g^2 \sqrt{1-c^2 x^2}}-\frac{b c^3 d^2 f \left (c^2 f^2-2 g^2\right ) x^2 \sqrt{d-c^2 d x^2}}{4 g^4 \sqrt{1-c^2 x^2}}-\frac{b c^3 d^2 x^3 \sqrt{d-c^2 d x^2}}{45 g \sqrt{1-c^2 x^2}}+\frac{b c^3 d^2 \left (c^2 f^2-2 g^2\right ) x^3 \sqrt{d-c^2 d x^2}}{9 g^3 \sqrt{1-c^2 x^2}}-\frac{b c^5 d^2 f x^4 \sqrt{d-c^2 d x^2}}{16 g^2 \sqrt{1-c^2 x^2}}+\frac{b c^5 d^2 x^5 \sqrt{d-c^2 d x^2}}{25 g \sqrt{1-c^2 x^2}}+\frac{b d^2 \left (c^2 f^2-g^2\right )^2 \sqrt{d-c^2 d x^2} \cos ^{-1}(c x)}{g^5}+\frac{c^2 d^2 f x \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{8 g^2}-\frac{c^2 d^2 f \left (c^2 f^2-2 g^2\right ) x \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{2 g^4}-\frac{c^4 d^2 f x^3 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{4 g^2}-\frac{d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{3 g}-\frac{d^2 \left (c^2 f^2-2 g^2\right ) \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{3 g^3}+\frac{d^2 \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{5 g}+\frac{c d^2 f \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{16 b g^2 \sqrt{1-c^2 x^2}}+\frac{c d^2 f \left (c^2 f^2-2 g^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{4 b g^4 \sqrt{1-c^2 x^2}}-\frac{c d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b g^5 \sqrt{1-c^2 x^2}}-\frac{d^2 \left (c^2 f^2-g^2\right )^3 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b c g^6 (f+g x) \sqrt{1-c^2 x^2}}-\frac{d^2 \left (c^2 f^2-g^2\right )^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b c g^4 (f+g x)}+\frac{\left (b c d^2 \left (c^2 f^2-g^2\right )^2 \sqrt{d-c^2 d x^2}\right ) \int 1 \, dx}{g^5 \sqrt{1-c^2 x^2}}+\frac{\left (a d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^2 \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{-c^2 f^2+g^2-x^2} \, dx,x,\frac{g+c^2 f x}{\sqrt{1-c^2 x^2}}\right )}{g^6 \sqrt{1-c^2 x^2}}+\frac{\left (b d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^2 \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{x}{c f+g \cos (x)} \, dx,x,\cos ^{-1}(c x)\right )}{g^6 \sqrt{1-c^2 x^2}}\\ &=\frac{a d^2 \left (c^2 f^2-g^2\right )^2 \sqrt{d-c^2 d x^2}}{g^5}-\frac{2 b c d^2 x \sqrt{d-c^2 d x^2}}{15 g \sqrt{1-c^2 x^2}}-\frac{b c d^2 \left (c^2 f^2-2 g^2\right ) x \sqrt{d-c^2 d x^2}}{3 g^3 \sqrt{1-c^2 x^2}}+\frac{b c d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt{d-c^2 d x^2}}{g^5 \sqrt{1-c^2 x^2}}+\frac{b c^3 d^2 f x^2 \sqrt{d-c^2 d x^2}}{16 g^2 \sqrt{1-c^2 x^2}}-\frac{b c^3 d^2 f \left (c^2 f^2-2 g^2\right ) x^2 \sqrt{d-c^2 d x^2}}{4 g^4 \sqrt{1-c^2 x^2}}-\frac{b c^3 d^2 x^3 \sqrt{d-c^2 d x^2}}{45 g \sqrt{1-c^2 x^2}}+\frac{b c^3 d^2 \left (c^2 f^2-2 g^2\right ) x^3 \sqrt{d-c^2 d x^2}}{9 g^3 \sqrt{1-c^2 x^2}}-\frac{b c^5 d^2 f x^4 \sqrt{d-c^2 d x^2}}{16 g^2 \sqrt{1-c^2 x^2}}+\frac{b c^5 d^2 x^5 \sqrt{d-c^2 d x^2}}{25 g \sqrt{1-c^2 x^2}}+\frac{b d^2 \left (c^2 f^2-g^2\right )^2 \sqrt{d-c^2 d x^2} \cos ^{-1}(c x)}{g^5}+\frac{c^2 d^2 f x \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{8 g^2}-\frac{c^2 d^2 f \left (c^2 f^2-2 g^2\right ) x \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{2 g^4}-\frac{c^4 d^2 f x^3 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{4 g^2}-\frac{d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{3 g}-\frac{d^2 \left (c^2 f^2-2 g^2\right ) \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{3 g^3}+\frac{d^2 \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{5 g}+\frac{c d^2 f \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{16 b g^2 \sqrt{1-c^2 x^2}}+\frac{c d^2 f \left (c^2 f^2-2 g^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{4 b g^4 \sqrt{1-c^2 x^2}}-\frac{c d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b g^5 \sqrt{1-c^2 x^2}}-\frac{d^2 \left (c^2 f^2-g^2\right )^3 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b c g^6 (f+g x) \sqrt{1-c^2 x^2}}-\frac{d^2 \left (c^2 f^2-g^2\right )^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b c g^4 (f+g x)}-\frac{a d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt{d-c^2 d x^2} \tan ^{-1}\left (\frac{g+c^2 f x}{\sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}\right )}{g^6 \sqrt{1-c^2 x^2}}+\frac{\left (2 b d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^2 \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{e^{i x} x}{2 c e^{i x} f+g+e^{2 i x} g} \, dx,x,\cos ^{-1}(c x)\right )}{g^6 \sqrt{1-c^2 x^2}}\\ &=\frac{a d^2 \left (c^2 f^2-g^2\right )^2 \sqrt{d-c^2 d x^2}}{g^5}-\frac{2 b c d^2 x \sqrt{d-c^2 d x^2}}{15 g \sqrt{1-c^2 x^2}}-\frac{b c d^2 \left (c^2 f^2-2 g^2\right ) x \sqrt{d-c^2 d x^2}}{3 g^3 \sqrt{1-c^2 x^2}}+\frac{b c d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt{d-c^2 d x^2}}{g^5 \sqrt{1-c^2 x^2}}+\frac{b c^3 d^2 f x^2 \sqrt{d-c^2 d x^2}}{16 g^2 \sqrt{1-c^2 x^2}}-\frac{b c^3 d^2 f \left (c^2 f^2-2 g^2\right ) x^2 \sqrt{d-c^2 d x^2}}{4 g^4 \sqrt{1-c^2 x^2}}-\frac{b c^3 d^2 x^3 \sqrt{d-c^2 d x^2}}{45 g \sqrt{1-c^2 x^2}}+\frac{b c^3 d^2 \left (c^2 f^2-2 g^2\right ) x^3 \sqrt{d-c^2 d x^2}}{9 g^3 \sqrt{1-c^2 x^2}}-\frac{b c^5 d^2 f x^4 \sqrt{d-c^2 d x^2}}{16 g^2 \sqrt{1-c^2 x^2}}+\frac{b c^5 d^2 x^5 \sqrt{d-c^2 d x^2}}{25 g \sqrt{1-c^2 x^2}}+\frac{b d^2 \left (c^2 f^2-g^2\right )^2 \sqrt{d-c^2 d x^2} \cos ^{-1}(c x)}{g^5}+\frac{c^2 d^2 f x \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{8 g^2}-\frac{c^2 d^2 f \left (c^2 f^2-2 g^2\right ) x \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{2 g^4}-\frac{c^4 d^2 f x^3 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{4 g^2}-\frac{d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{3 g}-\frac{d^2 \left (c^2 f^2-2 g^2\right ) \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{3 g^3}+\frac{d^2 \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{5 g}+\frac{c d^2 f \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{16 b g^2 \sqrt{1-c^2 x^2}}+\frac{c d^2 f \left (c^2 f^2-2 g^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{4 b g^4 \sqrt{1-c^2 x^2}}-\frac{c d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b g^5 \sqrt{1-c^2 x^2}}-\frac{d^2 \left (c^2 f^2-g^2\right )^3 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b c g^6 (f+g x) \sqrt{1-c^2 x^2}}-\frac{d^2 \left (c^2 f^2-g^2\right )^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b c g^4 (f+g x)}-\frac{a d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt{d-c^2 d x^2} \tan ^{-1}\left (\frac{g+c^2 f x}{\sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}\right )}{g^6 \sqrt{1-c^2 x^2}}+\frac{\left (2 b d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{e^{i x} x}{2 c f+2 e^{i x} g-2 \sqrt{c^2 f^2-g^2}} \, dx,x,\cos ^{-1}(c x)\right )}{g^5 \sqrt{1-c^2 x^2}}-\frac{\left (2 b d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{e^{i x} x}{2 c f+2 e^{i x} g+2 \sqrt{c^2 f^2-g^2}} \, dx,x,\cos ^{-1}(c x)\right )}{g^5 \sqrt{1-c^2 x^2}}\\ &=\frac{a d^2 \left (c^2 f^2-g^2\right )^2 \sqrt{d-c^2 d x^2}}{g^5}-\frac{2 b c d^2 x \sqrt{d-c^2 d x^2}}{15 g \sqrt{1-c^2 x^2}}-\frac{b c d^2 \left (c^2 f^2-2 g^2\right ) x \sqrt{d-c^2 d x^2}}{3 g^3 \sqrt{1-c^2 x^2}}+\frac{b c d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt{d-c^2 d x^2}}{g^5 \sqrt{1-c^2 x^2}}+\frac{b c^3 d^2 f x^2 \sqrt{d-c^2 d x^2}}{16 g^2 \sqrt{1-c^2 x^2}}-\frac{b c^3 d^2 f \left (c^2 f^2-2 g^2\right ) x^2 \sqrt{d-c^2 d x^2}}{4 g^4 \sqrt{1-c^2 x^2}}-\frac{b c^3 d^2 x^3 \sqrt{d-c^2 d x^2}}{45 g \sqrt{1-c^2 x^2}}+\frac{b c^3 d^2 \left (c^2 f^2-2 g^2\right ) x^3 \sqrt{d-c^2 d x^2}}{9 g^3 \sqrt{1-c^2 x^2}}-\frac{b c^5 d^2 f x^4 \sqrt{d-c^2 d x^2}}{16 g^2 \sqrt{1-c^2 x^2}}+\frac{b c^5 d^2 x^5 \sqrt{d-c^2 d x^2}}{25 g \sqrt{1-c^2 x^2}}+\frac{b d^2 \left (c^2 f^2-g^2\right )^2 \sqrt{d-c^2 d x^2} \cos ^{-1}(c x)}{g^5}+\frac{c^2 d^2 f x \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{8 g^2}-\frac{c^2 d^2 f \left (c^2 f^2-2 g^2\right ) x \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{2 g^4}-\frac{c^4 d^2 f x^3 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{4 g^2}-\frac{d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{3 g}-\frac{d^2 \left (c^2 f^2-2 g^2\right ) \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{3 g^3}+\frac{d^2 \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{5 g}+\frac{c d^2 f \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{16 b g^2 \sqrt{1-c^2 x^2}}+\frac{c d^2 f \left (c^2 f^2-2 g^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{4 b g^4 \sqrt{1-c^2 x^2}}-\frac{c d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b g^5 \sqrt{1-c^2 x^2}}-\frac{d^2 \left (c^2 f^2-g^2\right )^3 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b c g^6 (f+g x) \sqrt{1-c^2 x^2}}-\frac{d^2 \left (c^2 f^2-g^2\right )^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b c g^4 (f+g x)}-\frac{a d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt{d-c^2 d x^2} \tan ^{-1}\left (\frac{g+c^2 f x}{\sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}\right )}{g^6 \sqrt{1-c^2 x^2}}-\frac{i b d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt{d-c^2 d x^2} \cos ^{-1}(c x) \log \left (1+\frac{e^{i \cos ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right )}{g^6 \sqrt{1-c^2 x^2}}+\frac{i b d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt{d-c^2 d x^2} \cos ^{-1}(c x) \log \left (1+\frac{e^{i \cos ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right )}{g^6 \sqrt{1-c^2 x^2}}+\frac{\left (i b d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \log \left (1+\frac{2 e^{i x} g}{2 c f-2 \sqrt{c^2 f^2-g^2}}\right ) \, dx,x,\cos ^{-1}(c x)\right )}{g^6 \sqrt{1-c^2 x^2}}-\frac{\left (i b d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \log \left (1+\frac{2 e^{i x} g}{2 c f+2 \sqrt{c^2 f^2-g^2}}\right ) \, dx,x,\cos ^{-1}(c x)\right )}{g^6 \sqrt{1-c^2 x^2}}\\ &=\frac{a d^2 \left (c^2 f^2-g^2\right )^2 \sqrt{d-c^2 d x^2}}{g^5}-\frac{2 b c d^2 x \sqrt{d-c^2 d x^2}}{15 g \sqrt{1-c^2 x^2}}-\frac{b c d^2 \left (c^2 f^2-2 g^2\right ) x \sqrt{d-c^2 d x^2}}{3 g^3 \sqrt{1-c^2 x^2}}+\frac{b c d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt{d-c^2 d x^2}}{g^5 \sqrt{1-c^2 x^2}}+\frac{b c^3 d^2 f x^2 \sqrt{d-c^2 d x^2}}{16 g^2 \sqrt{1-c^2 x^2}}-\frac{b c^3 d^2 f \left (c^2 f^2-2 g^2\right ) x^2 \sqrt{d-c^2 d x^2}}{4 g^4 \sqrt{1-c^2 x^2}}-\frac{b c^3 d^2 x^3 \sqrt{d-c^2 d x^2}}{45 g \sqrt{1-c^2 x^2}}+\frac{b c^3 d^2 \left (c^2 f^2-2 g^2\right ) x^3 \sqrt{d-c^2 d x^2}}{9 g^3 \sqrt{1-c^2 x^2}}-\frac{b c^5 d^2 f x^4 \sqrt{d-c^2 d x^2}}{16 g^2 \sqrt{1-c^2 x^2}}+\frac{b c^5 d^2 x^5 \sqrt{d-c^2 d x^2}}{25 g \sqrt{1-c^2 x^2}}+\frac{b d^2 \left (c^2 f^2-g^2\right )^2 \sqrt{d-c^2 d x^2} \cos ^{-1}(c x)}{g^5}+\frac{c^2 d^2 f x \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{8 g^2}-\frac{c^2 d^2 f \left (c^2 f^2-2 g^2\right ) x \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{2 g^4}-\frac{c^4 d^2 f x^3 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{4 g^2}-\frac{d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{3 g}-\frac{d^2 \left (c^2 f^2-2 g^2\right ) \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{3 g^3}+\frac{d^2 \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{5 g}+\frac{c d^2 f \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{16 b g^2 \sqrt{1-c^2 x^2}}+\frac{c d^2 f \left (c^2 f^2-2 g^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{4 b g^4 \sqrt{1-c^2 x^2}}-\frac{c d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b g^5 \sqrt{1-c^2 x^2}}-\frac{d^2 \left (c^2 f^2-g^2\right )^3 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b c g^6 (f+g x) \sqrt{1-c^2 x^2}}-\frac{d^2 \left (c^2 f^2-g^2\right )^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b c g^4 (f+g x)}-\frac{a d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt{d-c^2 d x^2} \tan ^{-1}\left (\frac{g+c^2 f x}{\sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}\right )}{g^6 \sqrt{1-c^2 x^2}}-\frac{i b d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt{d-c^2 d x^2} \cos ^{-1}(c x) \log \left (1+\frac{e^{i \cos ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right )}{g^6 \sqrt{1-c^2 x^2}}+\frac{i b d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt{d-c^2 d x^2} \cos ^{-1}(c x) \log \left (1+\frac{e^{i \cos ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right )}{g^6 \sqrt{1-c^2 x^2}}+\frac{\left (b d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{2 g x}{2 c f-2 \sqrt{c^2 f^2-g^2}}\right )}{x} \, dx,x,e^{i \cos ^{-1}(c x)}\right )}{g^6 \sqrt{1-c^2 x^2}}-\frac{\left (b d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{2 g x}{2 c f+2 \sqrt{c^2 f^2-g^2}}\right )}{x} \, dx,x,e^{i \cos ^{-1}(c x)}\right )}{g^6 \sqrt{1-c^2 x^2}}\\ &=\frac{a d^2 \left (c^2 f^2-g^2\right )^2 \sqrt{d-c^2 d x^2}}{g^5}-\frac{2 b c d^2 x \sqrt{d-c^2 d x^2}}{15 g \sqrt{1-c^2 x^2}}-\frac{b c d^2 \left (c^2 f^2-2 g^2\right ) x \sqrt{d-c^2 d x^2}}{3 g^3 \sqrt{1-c^2 x^2}}+\frac{b c d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt{d-c^2 d x^2}}{g^5 \sqrt{1-c^2 x^2}}+\frac{b c^3 d^2 f x^2 \sqrt{d-c^2 d x^2}}{16 g^2 \sqrt{1-c^2 x^2}}-\frac{b c^3 d^2 f \left (c^2 f^2-2 g^2\right ) x^2 \sqrt{d-c^2 d x^2}}{4 g^4 \sqrt{1-c^2 x^2}}-\frac{b c^3 d^2 x^3 \sqrt{d-c^2 d x^2}}{45 g \sqrt{1-c^2 x^2}}+\frac{b c^3 d^2 \left (c^2 f^2-2 g^2\right ) x^3 \sqrt{d-c^2 d x^2}}{9 g^3 \sqrt{1-c^2 x^2}}-\frac{b c^5 d^2 f x^4 \sqrt{d-c^2 d x^2}}{16 g^2 \sqrt{1-c^2 x^2}}+\frac{b c^5 d^2 x^5 \sqrt{d-c^2 d x^2}}{25 g \sqrt{1-c^2 x^2}}+\frac{b d^2 \left (c^2 f^2-g^2\right )^2 \sqrt{d-c^2 d x^2} \cos ^{-1}(c x)}{g^5}+\frac{c^2 d^2 f x \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{8 g^2}-\frac{c^2 d^2 f \left (c^2 f^2-2 g^2\right ) x \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{2 g^4}-\frac{c^4 d^2 f x^3 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{4 g^2}-\frac{d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{3 g}-\frac{d^2 \left (c^2 f^2-2 g^2\right ) \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{3 g^3}+\frac{d^2 \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{5 g}+\frac{c d^2 f \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{16 b g^2 \sqrt{1-c^2 x^2}}+\frac{c d^2 f \left (c^2 f^2-2 g^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{4 b g^4 \sqrt{1-c^2 x^2}}-\frac{c d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b g^5 \sqrt{1-c^2 x^2}}-\frac{d^2 \left (c^2 f^2-g^2\right )^3 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b c g^6 (f+g x) \sqrt{1-c^2 x^2}}-\frac{d^2 \left (c^2 f^2-g^2\right )^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b c g^4 (f+g x)}-\frac{a d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt{d-c^2 d x^2} \tan ^{-1}\left (\frac{g+c^2 f x}{\sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}\right )}{g^6 \sqrt{1-c^2 x^2}}-\frac{i b d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt{d-c^2 d x^2} \cos ^{-1}(c x) \log \left (1+\frac{e^{i \cos ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right )}{g^6 \sqrt{1-c^2 x^2}}+\frac{i b d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt{d-c^2 d x^2} \cos ^{-1}(c x) \log \left (1+\frac{e^{i \cos ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right )}{g^6 \sqrt{1-c^2 x^2}}-\frac{b d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt{d-c^2 d x^2} \text{Li}_2\left (-\frac{e^{i \cos ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right )}{g^6 \sqrt{1-c^2 x^2}}+\frac{b d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt{d-c^2 d x^2} \text{Li}_2\left (-\frac{e^{i \cos ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right )}{g^6 \sqrt{1-c^2 x^2}}\\ \end{align*}

Mathematica [B]  time = 19.6051, size = 6216, normalized size = 3.8 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[((d - c^2*d*x^2)^(5/2)*(a + b*ArcCos[c*x]))/(f + g*x),x]

[Out]

Result too large to show

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Maple [B]  time = 0.531, size = 4692, normalized size = 2.9 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-c^2*d*x^2+d)^(5/2)*(a+b*arccos(c*x))/(g*x+f),x)

[Out]

-5/4*b*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/(c^2*x^2-1)*arccos(c*x)^2*f^3*d^2*c^3/g^4-8/3*b*(-d*(c^2*x^2-
1))^(1/2)*d^2/(c^2*x^2-1)/g^3*arccos(c*x)*x^2*c^4*f^2-1/2*b*(-d*(c^2*x^2-1))^(1/2)*f^3*d^2*c^6/(c^2*x^2-1)/g^4
*arccos(c*x)*x^3+1/2*b*(-d*(c^2*x^2-1))^(1/2)*f^3*d^2*c^4/(c^2*x^2-1)/g^4*arccos(c*x)*x+b*(-d*(c^2*x^2-1))^(1/
2)*d^2/(c^2*x^2-1)/g^5*arccos(c*x)*x^2*c^6*f^4+1/2*b*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/(c^2*x^2-1)*arc
cos(c*x)^2*f^5*d^2*c^5/g^6+1/5*a/g*(-d*c^2*(x+f/g)^2+2*c^2*d*f/g*(x+f/g)-d*(c^2*f^2-g^2)/g^2)^(5/2)+1/3*a/g*d*
(-d*c^2*(x+f/g)^2+2*c^2*d*f/g*(x+f/g)-d*(c^2*f^2-g^2)/g^2)^(3/2)+a/g*d^2*(-d*c^2*(x+f/g)^2+2*c^2*d*f/g*(x+f/g)
-d*(c^2*f^2-g^2)/g^2)^(1/2)-a/g*d^3/(-d*(c^2*f^2-g^2)/g^2)^(1/2)*ln((-2*d*(c^2*f^2-g^2)/g^2+2*c^2*d*f/g*(x+f/g
)+2*(-d*(c^2*f^2-g^2)/g^2)^(1/2)*(-d*c^2*(x+f/g)^2+2*c^2*d*f/g*(x+f/g)-d*(c^2*f^2-g^2)/g^2)^(1/2))/(x+f/g))+15
/16*b*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/(c^2*x^2-1)*arccos(c*x)^2*f*d^2*c/g^2-1/4*b*(-d*(c^2*x^2-1))^(
1/2)*f*d^2*c^6/(c^2*x^2-1)/g^2*arccos(c*x)*x^5+11/8*b*(-d*(c^2*x^2-1))^(1/2)*f*d^2*c^4/(c^2*x^2-1)/g^2*arccos(
c*x)*x^3-9/8*b*(-d*(c^2*x^2-1))^(1/2)*f*d^2*c^2/(c^2*x^2-1)/g^2*arccos(c*x)*x+1/3*b*(-d*(c^2*x^2-1))^(1/2)*d^2
/(c^2*x^2-1)/g^3*arccos(c*x)*x^4*c^6*f^2+a/g^6*d^3*c^6*f^5/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-d*c^2*(x+f/g
)^2+2*c^2*d*f/g*(x+f/g)-d*(c^2*f^2-g^2)/g^2)^(1/2))+a/g^7*d^3/(-d*(c^2*f^2-g^2)/g^2)^(1/2)*ln((-2*d*(c^2*f^2-g
^2)/g^2+2*c^2*d*f/g*(x+f/g)+2*(-d*(c^2*f^2-g^2)/g^2)^(1/2)*(-d*c^2*(x+f/g)^2+2*c^2*d*f/g*(x+f/g)-d*(c^2*f^2-g^
2)/g^2)^(1/2))/(x+f/g))*c^6*f^6-3*a/g^5*d^3/(-d*(c^2*f^2-g^2)/g^2)^(1/2)*ln((-2*d*(c^2*f^2-g^2)/g^2+2*c^2*d*f/
g*(x+f/g)+2*(-d*(c^2*f^2-g^2)/g^2)^(1/2)*(-d*c^2*(x+f/g)^2+2*c^2*d*f/g*(x+f/g)-d*(c^2*f^2-g^2)/g^2)^(1/2))/(x+
f/g))*c^4*f^4+3*a/g^3*d^3/(-d*(c^2*f^2-g^2)/g^2)^(1/2)*ln((-2*d*(c^2*f^2-g^2)/g^2+2*c^2*d*f/g*(x+f/g)+2*(-d*(c
^2*f^2-g^2)/g^2)^(1/2)*(-d*c^2*(x+f/g)^2+2*c^2*d*f/g*(x+f/g)-d*(c^2*f^2-g^2)/g^2)^(1/2))/(x+f/g))*c^2*f^2+15/8
*a/g^2*c^2*d^3*f/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-d*c^2*(x+f/g)^2+2*c^2*d*f/g*(x+f/g)-d*(c^2*f^2-g^2)/g^
2)^(1/2))-1/2*a/g^4*d^2*c^4*f^3*(-d*c^2*(x+f/g)^2+2*c^2*d*f/g*(x+f/g)-d*(c^2*f^2-g^2)/g^2)^(1/2)*x-5/2*a/g^4*d
^3*c^4*f^3/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-d*c^2*(x+f/g)^2+2*c^2*d*f/g*(x+f/g)-d*(c^2*f^2-g^2)/g^2)^(1/
2))+1/4*a/g^2*c^2*d*f*(-d*c^2*(x+f/g)^2+2*c^2*d*f/g*(x+f/g)-d*(c^2*f^2-g^2)/g^2)^(3/2)*x+7/8*a/g^2*c^2*d^2*f*(
-d*c^2*(x+f/g)^2+2*c^2*d*f/g*(x+f/g)-d*(c^2*f^2-g^2)/g^2)^(1/2)*x-1/25*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(c^2*x^2-1
)/g*(-c^2*x^2+1)^(1/2)*x^5*c^5+33/128*b*(-d*(c^2*x^2-1))^(1/2)*f*d^2*c/(c^2*x^2-1)/g^2*(-c^2*x^2+1)^(1/2)+11/4
5*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(c^2*x^2-1)/g*(-c^2*x^2+1)^(1/2)*x^3*c^3-23/15*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(c^
2*x^2-1)/g*(-c^2*x^2+1)^(1/2)*x*c-1/8*b*(-d*(c^2*x^2-1))^(1/2)*f^3*d^2*c^3/(c^2*x^2-1)/g^4*(-c^2*x^2+1)^(1/2)-
2*I*b*d^2*(-d*(c^2*x^2-1))^(1/2)*(c^2*f^2-g^2)^(1/2)*(-c^2*x^2+1)^(1/2)/(c^2*x^2-1)/g^4*arccos(c*x)*ln((-(c*x+
I*(-c^2*x^2+1)^(1/2))*g-c*f+(c^2*f^2-g^2)^(1/2))/(-c*f+(c^2*f^2-g^2)^(1/2)))*c^2*f^2+2*I*b*d^2*(-d*(c^2*x^2-1)
)^(1/2)*(c^2*f^2-g^2)^(1/2)*(-c^2*x^2+1)^(1/2)/(c^2*x^2-1)/g^4*arccos(c*x)*ln(((c*x+I*(-c^2*x^2+1)^(1/2))*g+c*
f+(c^2*f^2-g^2)^(1/2))/(c*f+(c^2*f^2-g^2)^(1/2)))*c^2*f^2+34/15*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(c^2*x^2-1)/g*arc
cos(c*x)*x^2*c^2+7/3*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(c^2*x^2-1)/g^3*arccos(c*x)*c^2*f^2-b*(-d*(c^2*x^2-1))^(1/2)
*d^2/(c^2*x^2-1)/g^5*arccos(c*x)*c^4*f^4+b*d^2*(-d*(c^2*x^2-1))^(1/2)*(c^2*f^2-g^2)^(1/2)*(-c^2*x^2+1)^(1/2)/(
c^2*x^2-1)/g^2*dilog(-1/(-c*f+(c^2*f^2-g^2)^(1/2))*(c*x+I*(-c^2*x^2+1)^(1/2))*g-1/(-c*f+(c^2*f^2-g^2)^(1/2))*c
*f+1/(-c*f+(c^2*f^2-g^2)^(1/2))*(c^2*f^2-g^2)^(1/2))-b*d^2*(-d*(c^2*x^2-1))^(1/2)*(c^2*f^2-g^2)^(1/2)*(-c^2*x^
2+1)^(1/2)/(c^2*x^2-1)/g^2*dilog((c*x+I*(-c^2*x^2+1)^(1/2))*g/(c*f+(c^2*f^2-g^2)^(1/2))+1/(c*f+(c^2*f^2-g^2)^(
1/2))*c*f+1/(c*f+(c^2*f^2-g^2)^(1/2))*(c^2*f^2-g^2)^(1/2))+1/5*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(c^2*x^2-1)/g*arcc
os(c*x)*x^6*c^6-14/15*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(c^2*x^2-1)/g*arccos(c*x)*x^4*c^4-1/3*a/g^3*d*(-d*c^2*(x+f/
g)^2+2*c^2*d*f/g*(x+f/g)-d*(c^2*f^2-g^2)/g^2)^(3/2)*c^2*f^2+a/g^5*d^2*(-d*c^2*(x+f/g)^2+2*c^2*d*f/g*(x+f/g)-d*
(c^2*f^2-g^2)/g^2)^(1/2)*c^4*f^4-2*a/g^3*d^2*(-d*c^2*(x+f/g)^2+2*c^2*d*f/g*(x+f/g)-d*(c^2*f^2-g^2)/g^2)^(1/2)*
c^2*f^2+I*b*d^2*(-d*(c^2*x^2-1))^(1/2)*(c^2*f^2-g^2)^(1/2)*(-c^2*x^2+1)^(1/2)/(c^2*x^2-1)/g^6*arccos(c*x)*ln((
-(c*x+I*(-c^2*x^2+1)^(1/2))*g-c*f+(c^2*f^2-g^2)^(1/2))/(-c*f+(c^2*f^2-g^2)^(1/2)))*c^4*f^4-I*b*d^2*(-d*(c^2*x^
2-1))^(1/2)*(c^2*f^2-g^2)^(1/2)*(-c^2*x^2+1)^(1/2)/(c^2*x^2-1)/g^6*arccos(c*x)*ln(((c*x+I*(-c^2*x^2+1)^(1/2))*
g+c*f+(c^2*f^2-g^2)^(1/2))/(c*f+(c^2*f^2-g^2)^(1/2)))*c^4*f^4+2*b*d^2*(-d*(c^2*x^2-1))^(1/2)*(c^2*f^2-g^2)^(1/
2)*(-c^2*x^2+1)^(1/2)/(c^2*x^2-1)/g^4*dilog((c*x+I*(-c^2*x^2+1)^(1/2))*g/(c*f+(c^2*f^2-g^2)^(1/2))+1/(c*f+(c^2
*f^2-g^2)^(1/2))*c*f+1/(c*f+(c^2*f^2-g^2)^(1/2))*(c^2*f^2-g^2)^(1/2))*c^2*f^2+b*d^2*(-d*(c^2*x^2-1))^(1/2)*(c^
2*f^2-g^2)^(1/2)*(-c^2*x^2+1)^(1/2)/(c^2*x^2-1)/g^6*dilog(-1/(-c*f+(c^2*f^2-g^2)^(1/2))*(c*x+I*(-c^2*x^2+1)^(1
/2))*g-1/(-c*f+(c^2*f^2-g^2)^(1/2))*c*f+1/(-c*f+(c^2*f^2-g^2)^(1/2))*(c^2*f^2-g^2)^(1/2))*c^4*f^4-b*d^2*(-d*(c
^2*x^2-1))^(1/2)*(c^2*f^2-g^2)^(1/2)*(-c^2*x^2+1)^(1/2)/(c^2*x^2-1)/g^6*dilog((c*x+I*(-c^2*x^2+1)^(1/2))*g/(c*
f+(c^2*f^2-g^2)^(1/2))+1/(c*f+(c^2*f^2-g^2)^(1/2))*c*f+1/(c*f+(c^2*f^2-g^2)^(1/2))*(c^2*f^2-g^2)^(1/2))*c^4*f^
4+I*b*d^2*(-d*(c^2*x^2-1))^(1/2)*(c^2*f^2-g^2)^(1/2)*(-c^2*x^2+1)^(1/2)/(c^2*x^2-1)/g^2*arccos(c*x)*ln((-(c*x+
I*(-c^2*x^2+1)^(1/2))*g-c*f+(c^2*f^2-g^2)^(1/2))/(-c*f+(c^2*f^2-g^2)^(1/2)))-2*b*d^2*(-d*(c^2*x^2-1))^(1/2)*(c
^2*f^2-g^2)^(1/2)*(-c^2*x^2+1)^(1/2)/(c^2*x^2-1)/g^4*dilog(-1/(-c*f+(c^2*f^2-g^2)^(1/2))*(c*x+I*(-c^2*x^2+1)^(
1/2))*g-1/(-c*f+(c^2*f^2-g^2)^(1/2))*c*f+1/(-c*f+(c^2*f^2-g^2)^(1/2))*(c^2*f^2-g^2)^(1/2))*c^2*f^2-I*b*d^2*(-d
*(c^2*x^2-1))^(1/2)*(c^2*f^2-g^2)^(1/2)*(-c^2*x^2+1)^(1/2)/(c^2*x^2-1)/g^2*arccos(c*x)*ln(((c*x+I*(-c^2*x^2+1)
^(1/2))*g+c*f+(c^2*f^2-g^2)^(1/2))/(c*f+(c^2*f^2-g^2)^(1/2)))-23/15*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(c^2*x^2-1)/g
*arccos(c*x)-9/16*b*(-d*(c^2*x^2-1))^(1/2)*f*d^2*c^3/(c^2*x^2-1)/g^2*(-c^2*x^2+1)^(1/2)*x^2-1/9*b*(-d*(c^2*x^2
-1))^(1/2)*d^2/(c^2*x^2-1)/g^3*(-c^2*x^2+1)^(1/2)*x^3*c^5*f^2+7/3*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(c^2*x^2-1)/g^3
*(-c^2*x^2+1)^(1/2)*x*c^3*f^2+1/4*b*(-d*(c^2*x^2-1))^(1/2)*f^3*d^2*c^5/(c^2*x^2-1)/g^4*(-c^2*x^2+1)^(1/2)*x^2-
b*(-d*(c^2*x^2-1))^(1/2)*d^2/(c^2*x^2-1)/g^5*(-c^2*x^2+1)^(1/2)*x*c^5*f^4+1/16*b*(-d*(c^2*x^2-1))^(1/2)*f*d^2*
c^5/(c^2*x^2-1)/g^2*(-c^2*x^2+1)^(1/2)*x^4

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c^2*d*x^2+d)^(5/2)*(a+b*arccos(c*x))/(g*x+f),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (a c^{4} d^{2} x^{4} - 2 \, a c^{2} d^{2} x^{2} + a d^{2} +{\left (b c^{4} d^{2} x^{4} - 2 \, b c^{2} d^{2} x^{2} + b d^{2}\right )} \arccos \left (c x\right )\right )} \sqrt{-c^{2} d x^{2} + d}}{g x + f}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c^2*d*x^2+d)^(5/2)*(a+b*arccos(c*x))/(g*x+f),x, algorithm="fricas")

[Out]

integral((a*c^4*d^2*x^4 - 2*a*c^2*d^2*x^2 + a*d^2 + (b*c^4*d^2*x^4 - 2*b*c^2*d^2*x^2 + b*d^2)*arccos(c*x))*sqr
t(-c^2*d*x^2 + d)/(g*x + f), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c**2*d*x**2+d)**(5/2)*(a+b*acos(c*x))/(g*x+f),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (-c^{2} d x^{2} + d\right )}^{\frac{5}{2}}{\left (b \arccos \left (c x\right ) + a\right )}}{g x + f}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c^2*d*x^2+d)^(5/2)*(a+b*arccos(c*x))/(g*x+f),x, algorithm="giac")

[Out]

integrate((-c^2*d*x^2 + d)^(5/2)*(b*arccos(c*x) + a)/(g*x + f), x)