3.6 \(\int (f+g x)^3 (d-c^2 d x^2)^{3/2} (a+b \cos ^{-1}(c x)) \, dx\)

Optimal. Leaf size=959 \[ -\frac{b c^3 d g^3 \sqrt{d-c^2 d x^2} x^7}{49 \sqrt{1-c^2 x^2}}-\frac{b c^3 d f g^2 \sqrt{d-c^2 d x^2} x^6}{12 \sqrt{1-c^2 x^2}}+\frac{8 b c d g^3 \sqrt{d-c^2 d x^2} x^5}{175 \sqrt{1-c^2 x^2}}-\frac{3 b c^3 d f^2 g \sqrt{d-c^2 d x^2} x^5}{25 \sqrt{1-c^2 x^2}}-\frac{b c^3 d f^3 \sqrt{d-c^2 d x^2} x^4}{16 \sqrt{1-c^2 x^2}}+\frac{7 b c d f g^2 \sqrt{d-c^2 d x^2} x^4}{32 \sqrt{1-c^2 x^2}}+\frac{3}{8} d f g^2 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right ) x^3+\frac{1}{2} d f g^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right ) x^3-\frac{b d g^3 \sqrt{d-c^2 d x^2} x^3}{105 c \sqrt{1-c^2 x^2}}+\frac{2 b c d f^2 g \sqrt{d-c^2 d x^2} x^3}{5 \sqrt{1-c^2 x^2}}+\frac{5 b c d f^3 \sqrt{d-c^2 d x^2} x^2}{16 \sqrt{1-c^2 x^2}}-\frac{3 b d f g^2 \sqrt{d-c^2 d x^2} x^2}{32 c \sqrt{1-c^2 x^2}}+\frac{3}{8} d f^3 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right ) x-\frac{3 d f g^2 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right ) x}{16 c^2}+\frac{1}{4} d f^3 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right ) x-\frac{2 b d g^3 \sqrt{d-c^2 d x^2} x}{35 c^3 \sqrt{1-c^2 x^2}}-\frac{3 b d f^2 g \sqrt{d-c^2 d x^2} x}{5 c \sqrt{1-c^2 x^2}}-\frac{3 d f^3 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{16 b c \sqrt{1-c^2 x^2}}-\frac{3 d f g^2 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{32 b c^3 \sqrt{1-c^2 x^2}}+\frac{d g^3 \left (1-c^2 x^2\right )^3 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{7 c^4}-\frac{d g^3 \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{5 c^4}-\frac{3 d f^2 g \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{5 c^2} \]

[Out]

(-3*b*d*f^2*g*x*Sqrt[d - c^2*d*x^2])/(5*c*Sqrt[1 - c^2*x^2]) - (2*b*d*g^3*x*Sqrt[d - c^2*d*x^2])/(35*c^3*Sqrt[
1 - c^2*x^2]) + (5*b*c*d*f^3*x^2*Sqrt[d - c^2*d*x^2])/(16*Sqrt[1 - c^2*x^2]) - (3*b*d*f*g^2*x^2*Sqrt[d - c^2*d
*x^2])/(32*c*Sqrt[1 - c^2*x^2]) + (2*b*c*d*f^2*g*x^3*Sqrt[d - c^2*d*x^2])/(5*Sqrt[1 - c^2*x^2]) - (b*d*g^3*x^3
*Sqrt[d - c^2*d*x^2])/(105*c*Sqrt[1 - c^2*x^2]) - (b*c^3*d*f^3*x^4*Sqrt[d - c^2*d*x^2])/(16*Sqrt[1 - c^2*x^2])
+ (7*b*c*d*f*g^2*x^4*Sqrt[d - c^2*d*x^2])/(32*Sqrt[1 - c^2*x^2]) - (3*b*c^3*d*f^2*g*x^5*Sqrt[d - c^2*d*x^2])/
(25*Sqrt[1 - c^2*x^2]) + (8*b*c*d*g^3*x^5*Sqrt[d - c^2*d*x^2])/(175*Sqrt[1 - c^2*x^2]) - (b*c^3*d*f*g^2*x^6*Sq
rt[d - c^2*d*x^2])/(12*Sqrt[1 - c^2*x^2]) - (b*c^3*d*g^3*x^7*Sqrt[d - c^2*d*x^2])/(49*Sqrt[1 - c^2*x^2]) + (3*
d*f^3*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x]))/8 - (3*d*f*g^2*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x]))/(16
*c^2) + (3*d*f*g^2*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x]))/8 + (d*f^3*x*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]
*(a + b*ArcCos[c*x]))/4 + (d*f*g^2*x^3*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x]))/2 - (3*d*f^2*g*(
1 - c^2*x^2)^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x]))/(5*c^2) - (d*g^3*(1 - c^2*x^2)^2*Sqrt[d - c^2*d*x^2]*(
a + b*ArcCos[c*x]))/(5*c^4) + (d*g^3*(1 - c^2*x^2)^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x]))/(7*c^4) - (3*d*f
^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x])^2)/(16*b*c*Sqrt[1 - c^2*x^2]) - (3*d*f*g^2*Sqrt[d - c^2*d*x^2]*(a +
b*ArcCos[c*x])^2)/(32*b*c^3*Sqrt[1 - c^2*x^2])

________________________________________________________________________________________

Rubi [A]  time = 0.96175, antiderivative size = 959, normalized size of antiderivative = 1., number of steps used = 24, number of rules used = 17, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.548, Rules used = {4778, 4764, 4650, 4648, 4642, 30, 14, 4678, 194, 4700, 4698, 4708, 266, 43, 4690, 12, 373} \[ -\frac{b c^3 d g^3 \sqrt{d-c^2 d x^2} x^7}{49 \sqrt{1-c^2 x^2}}-\frac{b c^3 d f g^2 \sqrt{d-c^2 d x^2} x^6}{12 \sqrt{1-c^2 x^2}}+\frac{8 b c d g^3 \sqrt{d-c^2 d x^2} x^5}{175 \sqrt{1-c^2 x^2}}-\frac{3 b c^3 d f^2 g \sqrt{d-c^2 d x^2} x^5}{25 \sqrt{1-c^2 x^2}}-\frac{b c^3 d f^3 \sqrt{d-c^2 d x^2} x^4}{16 \sqrt{1-c^2 x^2}}+\frac{7 b c d f g^2 \sqrt{d-c^2 d x^2} x^4}{32 \sqrt{1-c^2 x^2}}+\frac{3}{8} d f g^2 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right ) x^3+\frac{1}{2} d f g^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right ) x^3-\frac{b d g^3 \sqrt{d-c^2 d x^2} x^3}{105 c \sqrt{1-c^2 x^2}}+\frac{2 b c d f^2 g \sqrt{d-c^2 d x^2} x^3}{5 \sqrt{1-c^2 x^2}}+\frac{5 b c d f^3 \sqrt{d-c^2 d x^2} x^2}{16 \sqrt{1-c^2 x^2}}-\frac{3 b d f g^2 \sqrt{d-c^2 d x^2} x^2}{32 c \sqrt{1-c^2 x^2}}+\frac{3}{8} d f^3 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right ) x-\frac{3 d f g^2 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right ) x}{16 c^2}+\frac{1}{4} d f^3 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right ) x-\frac{2 b d g^3 \sqrt{d-c^2 d x^2} x}{35 c^3 \sqrt{1-c^2 x^2}}-\frac{3 b d f^2 g \sqrt{d-c^2 d x^2} x}{5 c \sqrt{1-c^2 x^2}}-\frac{3 d f^3 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{16 b c \sqrt{1-c^2 x^2}}-\frac{3 d f g^2 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{32 b c^3 \sqrt{1-c^2 x^2}}+\frac{d g^3 \left (1-c^2 x^2\right )^3 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{7 c^4}-\frac{d g^3 \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{5 c^4}-\frac{3 d f^2 g \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{5 c^2} \]

Antiderivative was successfully verified.

[In]

Int[(f + g*x)^3*(d - c^2*d*x^2)^(3/2)*(a + b*ArcCos[c*x]),x]

[Out]

(-3*b*d*f^2*g*x*Sqrt[d - c^2*d*x^2])/(5*c*Sqrt[1 - c^2*x^2]) - (2*b*d*g^3*x*Sqrt[d - c^2*d*x^2])/(35*c^3*Sqrt[
1 - c^2*x^2]) + (5*b*c*d*f^3*x^2*Sqrt[d - c^2*d*x^2])/(16*Sqrt[1 - c^2*x^2]) - (3*b*d*f*g^2*x^2*Sqrt[d - c^2*d
*x^2])/(32*c*Sqrt[1 - c^2*x^2]) + (2*b*c*d*f^2*g*x^3*Sqrt[d - c^2*d*x^2])/(5*Sqrt[1 - c^2*x^2]) - (b*d*g^3*x^3
*Sqrt[d - c^2*d*x^2])/(105*c*Sqrt[1 - c^2*x^2]) - (b*c^3*d*f^3*x^4*Sqrt[d - c^2*d*x^2])/(16*Sqrt[1 - c^2*x^2])
+ (7*b*c*d*f*g^2*x^4*Sqrt[d - c^2*d*x^2])/(32*Sqrt[1 - c^2*x^2]) - (3*b*c^3*d*f^2*g*x^5*Sqrt[d - c^2*d*x^2])/
(25*Sqrt[1 - c^2*x^2]) + (8*b*c*d*g^3*x^5*Sqrt[d - c^2*d*x^2])/(175*Sqrt[1 - c^2*x^2]) - (b*c^3*d*f*g^2*x^6*Sq
rt[d - c^2*d*x^2])/(12*Sqrt[1 - c^2*x^2]) - (b*c^3*d*g^3*x^7*Sqrt[d - c^2*d*x^2])/(49*Sqrt[1 - c^2*x^2]) + (3*
d*f^3*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x]))/8 - (3*d*f*g^2*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x]))/(16
*c^2) + (3*d*f*g^2*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x]))/8 + (d*f^3*x*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]
*(a + b*ArcCos[c*x]))/4 + (d*f*g^2*x^3*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x]))/2 - (3*d*f^2*g*(
1 - c^2*x^2)^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x]))/(5*c^2) - (d*g^3*(1 - c^2*x^2)^2*Sqrt[d - c^2*d*x^2]*(
a + b*ArcCos[c*x]))/(5*c^4) + (d*g^3*(1 - c^2*x^2)^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x]))/(7*c^4) - (3*d*f
^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x])^2)/(16*b*c*Sqrt[1 - c^2*x^2]) - (3*d*f*g^2*Sqrt[d - c^2*d*x^2]*(a +
b*ArcCos[c*x])^2)/(32*b*c^3*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 4764

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

Rule 4650

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(x*(d + e*x^2)^p*(
a + b*ArcCos[c*x])^n)/(2*p + 1), x] + (Dist[(2*d*p)/(2*p + 1), Int[(d + e*x^2)^(p - 1)*(a + b*ArcCos[c*x])^n,
x], x] + Dist[(b*c*n*d^IntPart[p]*(d + e*x^2)^FracPart[p])/((2*p + 1)*(1 - c^2*x^2)^FracPart[p]), Int[x*(1 - c
^2*x^2)^(p - 1/2)*(a + b*ArcCos[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && Gt
Q[n, 0] && GtQ[p, 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 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

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 194

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^n)^p, x], x] /; FreeQ[{a, b}, x]
&& IGtQ[n, 0] && IGtQ[p, 0]

Rule 4700

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[
((f*x)^(m + 1)*(d + e*x^2)^p*(a + b*ArcCos[c*x])^n)/(f*(m + 2*p + 1)), x] + (Dist[(2*d*p)/(m + 2*p + 1), Int[(
f*x)^m*(d + e*x^2)^(p - 1)*(a + b*ArcCos[c*x])^n, x], x] + Dist[(b*c*n*d^IntPart[p]*(d + e*x^2)^FracPart[p])/(
f*(m + 2*p + 1)*(1 - c^2*x^2)^FracPart[p]), Int[(f*x)^(m + 1)*(1 - c^2*x^2)^(p - 1/2)*(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] && GtQ[p, 0] &&  !LtQ[m, -1]
&& (RationalQ[m] || EqQ[n, 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 373

Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^n
)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[p, 0] && IGtQ[q, 0]

Rubi steps

\begin{align*} \int (f+g x)^3 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \cos ^{-1}(c x)\right ) \, dx &=\frac{\left (d \sqrt{d-c^2 d x^2}\right ) \int (f+g x)^3 \left (1-c^2 x^2\right )^{3/2} \left (a+b \cos ^{-1}(c x)\right ) \, dx}{\sqrt{1-c^2 x^2}}\\ &=\frac{\left (d \sqrt{d-c^2 d x^2}\right ) \int \left (f^3 \left (1-c^2 x^2\right )^{3/2} \left (a+b \cos ^{-1}(c x)\right )+3 f^2 g x \left (1-c^2 x^2\right )^{3/2} \left (a+b \cos ^{-1}(c x)\right )+3 f g^2 x^2 \left (1-c^2 x^2\right )^{3/2} \left (a+b \cos ^{-1}(c x)\right )+g^3 x^3 \left (1-c^2 x^2\right )^{3/2} \left (a+b \cos ^{-1}(c x)\right )\right ) \, dx}{\sqrt{1-c^2 x^2}}\\ &=\frac{\left (d f^3 \sqrt{d-c^2 d x^2}\right ) \int \left (1-c^2 x^2\right )^{3/2} \left (a+b \cos ^{-1}(c x)\right ) \, dx}{\sqrt{1-c^2 x^2}}+\frac{\left (3 d f^2 g \sqrt{d-c^2 d x^2}\right ) \int x \left (1-c^2 x^2\right )^{3/2} \left (a+b \cos ^{-1}(c x)\right ) \, dx}{\sqrt{1-c^2 x^2}}+\frac{\left (3 d f g^2 \sqrt{d-c^2 d x^2}\right ) \int x^2 \left (1-c^2 x^2\right )^{3/2} \left (a+b \cos ^{-1}(c x)\right ) \, dx}{\sqrt{1-c^2 x^2}}+\frac{\left (d g^3 \sqrt{d-c^2 d x^2}\right ) \int x^3 \left (1-c^2 x^2\right )^{3/2} \left (a+b \cos ^{-1}(c x)\right ) \, dx}{\sqrt{1-c^2 x^2}}\\ &=\frac{1}{4} d f^3 x \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )+\frac{1}{2} d f g^2 x^3 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )-\frac{3 d f^2 g \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{5 c^2}-\frac{d g^3 \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{5 c^4}+\frac{d g^3 \left (1-c^2 x^2\right )^3 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{7 c^4}+\frac{\left (3 d f^3 \sqrt{d-c^2 d x^2}\right ) \int \sqrt{1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right ) \, dx}{4 \sqrt{1-c^2 x^2}}+\frac{\left (b c d f^3 \sqrt{d-c^2 d x^2}\right ) \int x \left (1-c^2 x^2\right ) \, dx}{4 \sqrt{1-c^2 x^2}}-\frac{\left (3 b d f^2 g \sqrt{d-c^2 d x^2}\right ) \int \left (1-c^2 x^2\right )^2 \, dx}{5 c \sqrt{1-c^2 x^2}}+\frac{\left (3 d f g^2 \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}{2 \sqrt{1-c^2 x^2}}+\frac{\left (b c d f g^2 \sqrt{d-c^2 d x^2}\right ) \int x^3 \left (1-c^2 x^2\right ) \, dx}{2 \sqrt{1-c^2 x^2}}+\frac{\left (b c d g^3 \sqrt{d-c^2 d x^2}\right ) \int \frac{\left (-2-5 c^2 x^2\right ) \left (1-c^2 x^2\right )^2}{35 c^4} \, dx}{\sqrt{1-c^2 x^2}}\\ &=\frac{3}{8} d f^3 x \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )+\frac{3}{8} d f g^2 x^3 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )+\frac{1}{4} d f^3 x \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )+\frac{1}{2} d f g^2 x^3 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )-\frac{3 d f^2 g \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{5 c^2}-\frac{d g^3 \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{5 c^4}+\frac{d g^3 \left (1-c^2 x^2\right )^3 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{7 c^4}+\frac{\left (3 d f^3 \sqrt{d-c^2 d x^2}\right ) \int \frac{a+b \cos ^{-1}(c x)}{\sqrt{1-c^2 x^2}} \, dx}{8 \sqrt{1-c^2 x^2}}+\frac{\left (b c d f^3 \sqrt{d-c^2 d x^2}\right ) \int \left (x-c^2 x^3\right ) \, dx}{4 \sqrt{1-c^2 x^2}}+\frac{\left (3 b c d f^3 \sqrt{d-c^2 d x^2}\right ) \int x \, dx}{8 \sqrt{1-c^2 x^2}}-\frac{\left (3 b d f^2 g \sqrt{d-c^2 d x^2}\right ) \int \left (1-2 c^2 x^2+c^4 x^4\right ) \, dx}{5 c \sqrt{1-c^2 x^2}}+\frac{\left (3 d f g^2 \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}{8 \sqrt{1-c^2 x^2}}+\frac{\left (3 b c d f g^2 \sqrt{d-c^2 d x^2}\right ) \int x^3 \, dx}{8 \sqrt{1-c^2 x^2}}+\frac{\left (b c d f g^2 \sqrt{d-c^2 d x^2}\right ) \int \left (x^3-c^2 x^5\right ) \, dx}{2 \sqrt{1-c^2 x^2}}+\frac{\left (b d g^3 \sqrt{d-c^2 d x^2}\right ) \int \left (-2-5 c^2 x^2\right ) \left (1-c^2 x^2\right )^2 \, dx}{35 c^3 \sqrt{1-c^2 x^2}}\\ &=-\frac{3 b d f^2 g x \sqrt{d-c^2 d x^2}}{5 c \sqrt{1-c^2 x^2}}+\frac{5 b c d f^3 x^2 \sqrt{d-c^2 d x^2}}{16 \sqrt{1-c^2 x^2}}+\frac{2 b c d f^2 g x^3 \sqrt{d-c^2 d x^2}}{5 \sqrt{1-c^2 x^2}}-\frac{b c^3 d f^3 x^4 \sqrt{d-c^2 d x^2}}{16 \sqrt{1-c^2 x^2}}+\frac{7 b c d f g^2 x^4 \sqrt{d-c^2 d x^2}}{32 \sqrt{1-c^2 x^2}}-\frac{3 b c^3 d f^2 g x^5 \sqrt{d-c^2 d x^2}}{25 \sqrt{1-c^2 x^2}}-\frac{b c^3 d f g^2 x^6 \sqrt{d-c^2 d x^2}}{12 \sqrt{1-c^2 x^2}}+\frac{3}{8} d f^3 x \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )-\frac{3 d f g^2 x \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{16 c^2}+\frac{3}{8} d f g^2 x^3 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )+\frac{1}{4} d f^3 x \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )+\frac{1}{2} d f g^2 x^3 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )-\frac{3 d f^2 g \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{5 c^2}-\frac{d g^3 \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{5 c^4}+\frac{d g^3 \left (1-c^2 x^2\right )^3 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{7 c^4}-\frac{3 d f^3 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{16 b c \sqrt{1-c^2 x^2}}+\frac{\left (3 d f g^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{a+b \cos ^{-1}(c x)}{\sqrt{1-c^2 x^2}} \, dx}{16 c^2 \sqrt{1-c^2 x^2}}-\frac{\left (3 b d f g^2 \sqrt{d-c^2 d x^2}\right ) \int x \, dx}{16 c \sqrt{1-c^2 x^2}}+\frac{\left (b d g^3 \sqrt{d-c^2 d x^2}\right ) \int \left (-2-c^2 x^2+8 c^4 x^4-5 c^6 x^6\right ) \, dx}{35 c^3 \sqrt{1-c^2 x^2}}\\ &=-\frac{3 b d f^2 g x \sqrt{d-c^2 d x^2}}{5 c \sqrt{1-c^2 x^2}}-\frac{2 b d g^3 x \sqrt{d-c^2 d x^2}}{35 c^3 \sqrt{1-c^2 x^2}}+\frac{5 b c d f^3 x^2 \sqrt{d-c^2 d x^2}}{16 \sqrt{1-c^2 x^2}}-\frac{3 b d f g^2 x^2 \sqrt{d-c^2 d x^2}}{32 c \sqrt{1-c^2 x^2}}+\frac{2 b c d f^2 g x^3 \sqrt{d-c^2 d x^2}}{5 \sqrt{1-c^2 x^2}}-\frac{b d g^3 x^3 \sqrt{d-c^2 d x^2}}{105 c \sqrt{1-c^2 x^2}}-\frac{b c^3 d f^3 x^4 \sqrt{d-c^2 d x^2}}{16 \sqrt{1-c^2 x^2}}+\frac{7 b c d f g^2 x^4 \sqrt{d-c^2 d x^2}}{32 \sqrt{1-c^2 x^2}}-\frac{3 b c^3 d f^2 g x^5 \sqrt{d-c^2 d x^2}}{25 \sqrt{1-c^2 x^2}}+\frac{8 b c d g^3 x^5 \sqrt{d-c^2 d x^2}}{175 \sqrt{1-c^2 x^2}}-\frac{b c^3 d f g^2 x^6 \sqrt{d-c^2 d x^2}}{12 \sqrt{1-c^2 x^2}}-\frac{b c^3 d g^3 x^7 \sqrt{d-c^2 d x^2}}{49 \sqrt{1-c^2 x^2}}+\frac{3}{8} d f^3 x \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )-\frac{3 d f g^2 x \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{16 c^2}+\frac{3}{8} d f g^2 x^3 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )+\frac{1}{4} d f^3 x \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )+\frac{1}{2} d f g^2 x^3 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )-\frac{3 d f^2 g \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{5 c^2}-\frac{d g^3 \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{5 c^4}+\frac{d g^3 \left (1-c^2 x^2\right )^3 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{7 c^4}-\frac{3 d f^3 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{16 b c \sqrt{1-c^2 x^2}}-\frac{3 d f g^2 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{32 b c^3 \sqrt{1-c^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 4.57178, size = 910, normalized size = 0.95 \[ \frac{-88200 b c d f \left (2 c^2 f^2+g^2\right ) \sqrt{d-c^2 d x^2} \cos ^{-1}(c x)^2+140 b d \sqrt{d-c^2 d x^2} \left (6720 f^2 g x^2 \sqrt{1-c^2 x^2} c^4+1680 f^3 \sin \left (2 \cos ^{-1}(c x)\right ) c^3-210 f^3 \sin \left (4 \cos ^{-1}(c x)\right ) c^3-420 f^2 g \sin \left (3 \cos ^{-1}(c x)\right ) c^2-252 f^2 g \sin \left (5 \cos ^{-1}(c x)\right ) c^2-1256 g^3 x^2 \sqrt{1-c^2 x^2} c^2-4200 f^2 g \sqrt{1-c^2 x^2} c^2+315 f g^2 \sin \left (2 \cos ^{-1}(c x)\right ) c+315 f g^2 \sin \left (4 \cos ^{-1}(c x)\right ) c-105 f g^2 \sin \left (6 \cos ^{-1}(c x)\right ) c+864 g^3 \left (1-c^2 x^2\right )^{3/2} \cos \left (2 \cos ^{-1}(c x)\right )+120 g^3 \left (1-c^2 x^2\right )^{3/2} \cos \left (4 \cos ^{-1}(c x)\right )+140 g^3 \sin \left (3 \cos ^{-1}(c x)\right )+84 g^3 \sin \left (5 \cos ^{-1}(c x)\right )+416 g^3 \sqrt{1-c^2 x^2}\right ) \cos ^{-1}(c x)-176400 a c d^{3/2} f \left (2 c^2 f^2+g^2\right ) \sqrt{1-c^2 x^2} \tan ^{-1}\left (\frac{c x \sqrt{d-c^2 d x^2}}{\sqrt{d} \left (c^2 x^2-1\right )}\right )-d \sqrt{d-c^2 d x^2} \left (134400 a g^3 x^6 \sqrt{1-c^2 x^2} c^6+470400 a f g^2 x^5 \sqrt{1-c^2 x^2} c^6+564480 a f^2 g x^4 \sqrt{1-c^2 x^2} c^6+235200 a f^3 x^3 \sqrt{1-c^2 x^2} c^6-215040 a g^3 x^4 \sqrt{1-c^2 x^2} c^4-823200 a f g^2 x^3 \sqrt{1-c^2 x^2} c^4-1128960 a f^2 g x^2 \sqrt{1-c^2 x^2} c^4-588000 a f^3 x \sqrt{1-c^2 x^2} c^4+352800 b f^2 g x c^3+7350 b f^3 \cos \left (4 \cos ^{-1}(c x)\right ) c^3+7056 b f^2 g \cos \left (5 \cos ^{-1}(c x)\right ) c^2+26880 a g^3 x^2 \sqrt{1-c^2 x^2} c^2+564480 a f^2 g \sqrt{1-c^2 x^2} c^2+176400 a f g^2 x \sqrt{1-c^2 x^2} c^2+44100 b g^3 x c-7350 b f \left (16 c^2 f^2+3 g^2\right ) \cos \left (2 \cos ^{-1}(c x)\right ) c-11025 b f g^2 \cos \left (4 \cos ^{-1}(c x)\right ) c+2450 b f g^2 \cos \left (6 \cos ^{-1}(c x)\right ) c-4900 b g \left (12 c^2 f^2+g^2\right ) \cos \left (3 \cos ^{-1}(c x)\right )-588 b g^3 \cos \left (5 \cos ^{-1}(c x)\right )+300 b g^3 \cos \left (7 \cos ^{-1}(c x)\right )+53760 a g^3 \sqrt{1-c^2 x^2}\right )}{940800 c^4 \sqrt{1-c^2 x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(f + g*x)^3*(d - c^2*d*x^2)^(3/2)*(a + b*ArcCos[c*x]),x]

[Out]

(-88200*b*c*d*f*(2*c^2*f^2 + g^2)*Sqrt[d - c^2*d*x^2]*ArcCos[c*x]^2 - 176400*a*c*d^(3/2)*f*(2*c^2*f^2 + g^2)*S
qrt[1 - c^2*x^2]*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] - d*Sqrt[d - c^2*d*x^2]*(352800*b*
c^3*f^2*g*x + 44100*b*c*g^3*x + 564480*a*c^2*f^2*g*Sqrt[1 - c^2*x^2] + 53760*a*g^3*Sqrt[1 - c^2*x^2] - 588000*
a*c^4*f^3*x*Sqrt[1 - c^2*x^2] + 176400*a*c^2*f*g^2*x*Sqrt[1 - c^2*x^2] - 1128960*a*c^4*f^2*g*x^2*Sqrt[1 - c^2*
x^2] + 26880*a*c^2*g^3*x^2*Sqrt[1 - c^2*x^2] + 235200*a*c^6*f^3*x^3*Sqrt[1 - c^2*x^2] - 823200*a*c^4*f*g^2*x^3
*Sqrt[1 - c^2*x^2] + 564480*a*c^6*f^2*g*x^4*Sqrt[1 - c^2*x^2] - 215040*a*c^4*g^3*x^4*Sqrt[1 - c^2*x^2] + 47040
0*a*c^6*f*g^2*x^5*Sqrt[1 - c^2*x^2] + 134400*a*c^6*g^3*x^6*Sqrt[1 - c^2*x^2] - 7350*b*c*f*(16*c^2*f^2 + 3*g^2)
*Cos[2*ArcCos[c*x]] - 4900*b*g*(12*c^2*f^2 + g^2)*Cos[3*ArcCos[c*x]] + 7350*b*c^3*f^3*Cos[4*ArcCos[c*x]] - 110
25*b*c*f*g^2*Cos[4*ArcCos[c*x]] + 7056*b*c^2*f^2*g*Cos[5*ArcCos[c*x]] - 588*b*g^3*Cos[5*ArcCos[c*x]] + 2450*b*
c*f*g^2*Cos[6*ArcCos[c*x]] + 300*b*g^3*Cos[7*ArcCos[c*x]]) + 140*b*d*Sqrt[d - c^2*d*x^2]*ArcCos[c*x]*(-4200*c^
2*f^2*g*Sqrt[1 - c^2*x^2] + 416*g^3*Sqrt[1 - c^2*x^2] + 6720*c^4*f^2*g*x^2*Sqrt[1 - c^2*x^2] - 1256*c^2*g^3*x^
2*Sqrt[1 - c^2*x^2] + 864*g^3*(1 - c^2*x^2)^(3/2)*Cos[2*ArcCos[c*x]] + 120*g^3*(1 - c^2*x^2)^(3/2)*Cos[4*ArcCo
s[c*x]] + 1680*c^3*f^3*Sin[2*ArcCos[c*x]] + 315*c*f*g^2*Sin[2*ArcCos[c*x]] - 420*c^2*f^2*g*Sin[3*ArcCos[c*x]]
+ 140*g^3*Sin[3*ArcCos[c*x]] - 210*c^3*f^3*Sin[4*ArcCos[c*x]] + 315*c*f*g^2*Sin[4*ArcCos[c*x]] - 252*c^2*f^2*g
*Sin[5*ArcCos[c*x]] + 84*g^3*Sin[5*ArcCos[c*x]] - 105*c*f*g^2*Sin[6*ArcCos[c*x]]))/(940800*c^4*Sqrt[1 - c^2*x^
2])

________________________________________________________________________________________

Maple [B]  time = 0.811, size = 1734, normalized size = 1.8 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((g*x+f)^3*(-c^2*d*x^2+d)^(3/2)*(a+b*arccos(c*x)),x)

[Out]

3/5*b*(-d*(c^2*x^2-1))^(1/2)*g*d/c^2/(c^2*x^2-1)*arccos(c*x)*f^2-1/4*b*(-d*(c^2*x^2-1))^(1/2)*f^3*d*c^4/(c^2*x
^2-1)*arccos(c*x)*x^5+7/8*b*(-d*(c^2*x^2-1))^(1/2)*f^3*d*c^2/(c^2*x^2-1)*arccos(c*x)*x^3-17/16*b*(-d*(c^2*x^2-
1))^(1/2)*f*g^2*d/(c^2*x^2-1)*arccos(c*x)*x^3-9/5*b*(-d*(c^2*x^2-1))^(1/2)*g*d/(c^2*x^2-1)*arccos(c*x)*x^2*f^2
+3/16*b*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/c/(c^2*x^2-1)*arccos(c*x)^2*f^3*d-1/7*b*(-d*(c^2*x^2-1))^(1/
2)*g^3*d*c^4/(c^2*x^2-1)*arccos(c*x)*x^8+13/35*b*(-d*(c^2*x^2-1))^(1/2)*g^3*d*c^2/(c^2*x^2-1)*arccos(c*x)*x^6-
1/35*b*(-d*(c^2*x^2-1))^(1/2)*g^3*d/c^2/(c^2*x^2-1)*arccos(c*x)*x^2+3/8*a*f^3*d*x*(-c^2*d*x^2+d)^(1/2)+3/8*a*f
^3*d^2/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2))-2/35*a*g^3/d/c^4*(-c^2*d*x^2+d)^(5/2)-9/35*b
*(-d*(c^2*x^2-1))^(1/2)*g^3*d/(c^2*x^2-1)*arccos(c*x)*x^4-5/8*b*(-d*(c^2*x^2-1))^(1/2)*f^3*d/(c^2*x^2-1)*arcco
s(c*x)*x+2/35*b*(-d*(c^2*x^2-1))^(1/2)*g^3*d/c^4/(c^2*x^2-1)*arccos(c*x)-1/7*a*g^3*x^2*(-c^2*d*x^2+d)^(5/2)/c^
2/d+1/8*a*f*g^2/c^2*x*(-c^2*d*x^2+d)^(3/2)-3/5*a*f^2*g/c^2/d*(-c^2*d*x^2+d)^(5/2)+1/4*a*f^3*x*(-c^2*d*x^2+d)^(
3/2)-1/2*b*(-d*(c^2*x^2-1))^(1/2)*f*g^2*d*c^4/(c^2*x^2-1)*arccos(c*x)*x^7+11/8*b*(-d*(c^2*x^2-1))^(1/2)*f*g^2*
d*c^2/(c^2*x^2-1)*arccos(c*x)*x^5+3/16*b*(-d*(c^2*x^2-1))^(1/2)*f*g^2*d/c^2/(c^2*x^2-1)*arccos(c*x)*x-3/5*b*(-
d*(c^2*x^2-1))^(1/2)*g*d*c^4/(c^2*x^2-1)*arccos(c*x)*x^6*f^2+9/5*b*(-d*(c^2*x^2-1))^(1/2)*g*d*c^2/(c^2*x^2-1)*
arccos(c*x)*x^4*f^2+3/32*b*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/c^3/(c^2*x^2-1)*arccos(c*x)^2*f*d*g^2-7/3
2*b*(-d*(c^2*x^2-1))^(1/2)*f*g^2*d*c/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x^4+3/32*b*(-d*(c^2*x^2-1))^(1/2)*f*g^2*d/
c/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x^2+3/25*b*(-d*(c^2*x^2-1))^(1/2)*g*d*c^3/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x^5*
f^2-2/5*b*(-d*(c^2*x^2-1))^(1/2)*g*d*c/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x^3*f^2+3/5*b*(-d*(c^2*x^2-1))^(1/2)*g*d
/c/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x*f^2+1/12*b*(-d*(c^2*x^2-1))^(1/2)*f*g^2*d*c^3/(c^2*x^2-1)*(-c^2*x^2+1)^(1/
2)*x^6-1/2*a*f*g^2*x*(-c^2*d*x^2+d)^(5/2)/c^2/d+3/16*a*f*g^2/c^2*d*x*(-c^2*d*x^2+d)^(1/2)+1/49*b*(-d*(c^2*x^2-
1))^(1/2)*g^3*d*c^3/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x^7-8/175*b*(-d*(c^2*x^2-1))^(1/2)*g^3*d*c/(c^2*x^2-1)*(-c^
2*x^2+1)^(1/2)*x^5+1/105*b*(-d*(c^2*x^2-1))^(1/2)*g^3*d/c/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x^3+2/35*b*(-d*(c^2*x
^2-1))^(1/2)*g^3*d/c^3/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x+7/768*b*(-d*(c^2*x^2-1))^(1/2)*f*g^2*d/c^3/(c^2*x^2-1)
*(-c^2*x^2+1)^(1/2)+1/16*b*(-d*(c^2*x^2-1))^(1/2)*f^3*d*c^3/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x^4-5/16*b*(-d*(c^2
*x^2-1))^(1/2)*f^3*d*c/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x^2+3/16*a*f*g^2/c^2*d^2/(c^2*d)^(1/2)*arctan((c^2*d)^(1
/2)*x/(-c^2*d*x^2+d)^(1/2))+17/128*b*(-d*(c^2*x^2-1))^(1/2)*f^3*d/c/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)

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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((g*x+f)^3*(-c^2*d*x^2+d)^(3/2)*(a+b*arccos(c*x)),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 (-{\left (a c^{2} d g^{3} x^{5} + 3 \, a c^{2} d f g^{2} x^{4} - 3 \, a d f^{2} g x - a d f^{3} +{\left (3 \, a c^{2} d f^{2} g - a d g^{3}\right )} x^{3} +{\left (a c^{2} d f^{3} - 3 \, a d f g^{2}\right )} x^{2} +{\left (b c^{2} d g^{3} x^{5} + 3 \, b c^{2} d f g^{2} x^{4} - 3 \, b d f^{2} g x - b d f^{3} +{\left (3 \, b c^{2} d f^{2} g - b d g^{3}\right )} x^{3} +{\left (b c^{2} d f^{3} - 3 \, b d f g^{2}\right )} x^{2}\right )} \arccos \left (c x\right )\right )} \sqrt{-c^{2} d x^{2} + d}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)^3*(-c^2*d*x^2+d)^(3/2)*(a+b*arccos(c*x)),x, algorithm="fricas")

[Out]

integral(-(a*c^2*d*g^3*x^5 + 3*a*c^2*d*f*g^2*x^4 - 3*a*d*f^2*g*x - a*d*f^3 + (3*a*c^2*d*f^2*g - a*d*g^3)*x^3 +
(a*c^2*d*f^3 - 3*a*d*f*g^2)*x^2 + (b*c^2*d*g^3*x^5 + 3*b*c^2*d*f*g^2*x^4 - 3*b*d*f^2*g*x - b*d*f^3 + (3*b*c^2
*d*f^2*g - b*d*g^3)*x^3 + (b*c^2*d*f^3 - 3*b*d*f*g^2)*x^2)*arccos(c*x))*sqrt(-c^2*d*x^2 + d), 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((g*x+f)**3*(-c**2*d*x**2+d)**(3/2)*(a+b*acos(c*x)),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (-c^{2} d x^{2} + d\right )}^{\frac{3}{2}}{\left (g x + f\right )}^{3}{\left (b \arccos \left (c x\right ) + a\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)^3*(-c^2*d*x^2+d)^(3/2)*(a+b*arccos(c*x)),x, algorithm="giac")

[Out]

integrate((-c^2*d*x^2 + d)^(3/2)*(g*x + f)^3*(b*arccos(c*x) + a), x)