3.10 \(\int (f+g x)^3 (d-c^2 d x^2)^{5/2} (a+b \cos ^{-1}(c x)) \, dx\)

Optimal. Leaf size=1281 \[ \text{result too large to display} \]

[Out]

(-3*b*d^2*f^2*g*x*Sqrt[d - c^2*d*x^2])/(7*c*Sqrt[1 - c^2*x^2]) - (2*b*d^2*g^3*x*Sqrt[d - c^2*d*x^2])/(63*c^3*S
qrt[1 - c^2*x^2]) + (25*b*c*d^2*f^3*x^2*Sqrt[d - c^2*d*x^2])/(96*Sqrt[1 - c^2*x^2]) - (15*b*d^2*f*g^2*x^2*Sqrt
[d - c^2*d*x^2])/(256*c*Sqrt[1 - c^2*x^2]) + (3*b*c*d^2*f^2*g*x^3*Sqrt[d - c^2*d*x^2])/(7*Sqrt[1 - c^2*x^2]) -
(b*d^2*g^3*x^3*Sqrt[d - c^2*d*x^2])/(189*c*Sqrt[1 - c^2*x^2]) - (5*b*c^3*d^2*f^3*x^4*Sqrt[d - c^2*d*x^2])/(96
*Sqrt[1 - c^2*x^2]) + (59*b*c*d^2*f*g^2*x^4*Sqrt[d - c^2*d*x^2])/(256*Sqrt[1 - c^2*x^2]) - (9*b*c^3*d^2*f^2*g*
x^5*Sqrt[d - c^2*d*x^2])/(35*Sqrt[1 - c^2*x^2]) + (b*c*d^2*g^3*x^5*Sqrt[d - c^2*d*x^2])/(21*Sqrt[1 - c^2*x^2])
- (17*b*c^3*d^2*f*g^2*x^6*Sqrt[d - c^2*d*x^2])/(96*Sqrt[1 - c^2*x^2]) + (3*b*c^5*d^2*f^2*g*x^7*Sqrt[d - c^2*d
*x^2])/(49*Sqrt[1 - c^2*x^2]) - (19*b*c^3*d^2*g^3*x^7*Sqrt[d - c^2*d*x^2])/(441*Sqrt[1 - c^2*x^2]) + (3*b*c^5*
d^2*f*g^2*x^8*Sqrt[d - c^2*d*x^2])/(64*Sqrt[1 - c^2*x^2]) + (b*c^5*d^2*g^3*x^9*Sqrt[d - c^2*d*x^2])/(81*Sqrt[1
- c^2*x^2]) - (b*d^2*f^3*(1 - c^2*x^2)^(5/2)*Sqrt[d - c^2*d*x^2])/(36*c) + (5*d^2*f^3*x*Sqrt[d - c^2*d*x^2]*(
a + b*ArcCos[c*x]))/16 - (15*d^2*f*g^2*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x]))/(128*c^2) + (15*d^2*f*g^2*x^
3*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x]))/64 + (5*d^2*f^3*x*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c
*x]))/24 + (5*d^2*f*g^2*x^3*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x]))/16 + (d^2*f^3*x*(1 - c^2*x^
2)^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x]))/6 + (3*d^2*f*g^2*x^3*(1 - c^2*x^2)^2*Sqrt[d - c^2*d*x^2]*(a + b*
ArcCos[c*x]))/8 - (3*d^2*f^2*g*(1 - c^2*x^2)^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x]))/(7*c^2) - (d^2*g^3*(1
- c^2*x^2)^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x]))/(7*c^4) + (d^2*g^3*(1 - c^2*x^2)^4*Sqrt[d - c^2*d*x^2]*(
a + b*ArcCos[c*x]))/(9*c^4) - (5*d^2*f^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x])^2)/(32*b*c*Sqrt[1 - c^2*x^2])
- (15*d^2*f*g^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x])^2)/(256*b*c^3*Sqrt[1 - c^2*x^2])

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Rubi [A]  time = 1.17746, antiderivative size = 1281, normalized size of antiderivative = 1., number of steps used = 30, number of rules used = 18, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.581, Rules used = {4778, 4764, 4650, 4648, 4642, 30, 14, 261, 4678, 194, 4700, 4698, 4708, 266, 43, 4690, 12, 373} \[ \frac{b c^5 d^2 g^3 \sqrt{d-c^2 d x^2} x^9}{81 \sqrt{1-c^2 x^2}}+\frac{3 b c^5 d^2 f g^2 \sqrt{d-c^2 d x^2} x^8}{64 \sqrt{1-c^2 x^2}}-\frac{19 b c^3 d^2 g^3 \sqrt{d-c^2 d x^2} x^7}{441 \sqrt{1-c^2 x^2}}+\frac{3 b c^5 d^2 f^2 g \sqrt{d-c^2 d x^2} x^7}{49 \sqrt{1-c^2 x^2}}-\frac{17 b c^3 d^2 f g^2 \sqrt{d-c^2 d x^2} x^6}{96 \sqrt{1-c^2 x^2}}+\frac{b c d^2 g^3 \sqrt{d-c^2 d x^2} x^5}{21 \sqrt{1-c^2 x^2}}-\frac{9 b c^3 d^2 f^2 g \sqrt{d-c^2 d x^2} x^5}{35 \sqrt{1-c^2 x^2}}-\frac{5 b c^3 d^2 f^3 \sqrt{d-c^2 d x^2} x^4}{96 \sqrt{1-c^2 x^2}}+\frac{59 b c d^2 f g^2 \sqrt{d-c^2 d x^2} x^4}{256 \sqrt{1-c^2 x^2}}+\frac{15}{64} d^2 f g^2 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right ) x^3+\frac{3}{8} d^2 f g^2 \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right ) x^3+\frac{5}{16} d^2 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^2 g^3 \sqrt{d-c^2 d x^2} x^3}{189 c \sqrt{1-c^2 x^2}}+\frac{3 b c d^2 f^2 g \sqrt{d-c^2 d x^2} x^3}{7 \sqrt{1-c^2 x^2}}+\frac{25 b c d^2 f^3 \sqrt{d-c^2 d x^2} x^2}{96 \sqrt{1-c^2 x^2}}-\frac{15 b d^2 f g^2 \sqrt{d-c^2 d x^2} x^2}{256 c \sqrt{1-c^2 x^2}}+\frac{5}{16} d^2 f^3 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right ) x-\frac{15 d^2 f g^2 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right ) x}{128 c^2}+\frac{1}{6} d^2 f^3 \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right ) x+\frac{5}{24} d^2 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^2 g^3 \sqrt{d-c^2 d x^2} x}{63 c^3 \sqrt{1-c^2 x^2}}-\frac{3 b d^2 f^2 g \sqrt{d-c^2 d x^2} x}{7 c \sqrt{1-c^2 x^2}}-\frac{5 d^2 f^3 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{32 b c \sqrt{1-c^2 x^2}}-\frac{15 d^2 f g^2 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{256 b c^3 \sqrt{1-c^2 x^2}}+\frac{d^2 g^3 \left (1-c^2 x^2\right )^4 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{9 c^4}-\frac{d^2 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^2 f^2 g \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^2}-\frac{b d^2 f^3 \left (1-c^2 x^2\right )^{5/2} \sqrt{d-c^2 d x^2}}{36 c} \]

Antiderivative was successfully verified.

[In]

Int[(f + g*x)^3*(d - c^2*d*x^2)^(5/2)*(a + b*ArcCos[c*x]),x]

[Out]

(-3*b*d^2*f^2*g*x*Sqrt[d - c^2*d*x^2])/(7*c*Sqrt[1 - c^2*x^2]) - (2*b*d^2*g^3*x*Sqrt[d - c^2*d*x^2])/(63*c^3*S
qrt[1 - c^2*x^2]) + (25*b*c*d^2*f^3*x^2*Sqrt[d - c^2*d*x^2])/(96*Sqrt[1 - c^2*x^2]) - (15*b*d^2*f*g^2*x^2*Sqrt
[d - c^2*d*x^2])/(256*c*Sqrt[1 - c^2*x^2]) + (3*b*c*d^2*f^2*g*x^3*Sqrt[d - c^2*d*x^2])/(7*Sqrt[1 - c^2*x^2]) -
(b*d^2*g^3*x^3*Sqrt[d - c^2*d*x^2])/(189*c*Sqrt[1 - c^2*x^2]) - (5*b*c^3*d^2*f^3*x^4*Sqrt[d - c^2*d*x^2])/(96
*Sqrt[1 - c^2*x^2]) + (59*b*c*d^2*f*g^2*x^4*Sqrt[d - c^2*d*x^2])/(256*Sqrt[1 - c^2*x^2]) - (9*b*c^3*d^2*f^2*g*
x^5*Sqrt[d - c^2*d*x^2])/(35*Sqrt[1 - c^2*x^2]) + (b*c*d^2*g^3*x^5*Sqrt[d - c^2*d*x^2])/(21*Sqrt[1 - c^2*x^2])
- (17*b*c^3*d^2*f*g^2*x^6*Sqrt[d - c^2*d*x^2])/(96*Sqrt[1 - c^2*x^2]) + (3*b*c^5*d^2*f^2*g*x^7*Sqrt[d - c^2*d
*x^2])/(49*Sqrt[1 - c^2*x^2]) - (19*b*c^3*d^2*g^3*x^7*Sqrt[d - c^2*d*x^2])/(441*Sqrt[1 - c^2*x^2]) + (3*b*c^5*
d^2*f*g^2*x^8*Sqrt[d - c^2*d*x^2])/(64*Sqrt[1 - c^2*x^2]) + (b*c^5*d^2*g^3*x^9*Sqrt[d - c^2*d*x^2])/(81*Sqrt[1
- c^2*x^2]) - (b*d^2*f^3*(1 - c^2*x^2)^(5/2)*Sqrt[d - c^2*d*x^2])/(36*c) + (5*d^2*f^3*x*Sqrt[d - c^2*d*x^2]*(
a + b*ArcCos[c*x]))/16 - (15*d^2*f*g^2*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x]))/(128*c^2) + (15*d^2*f*g^2*x^
3*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x]))/64 + (5*d^2*f^3*x*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c
*x]))/24 + (5*d^2*f*g^2*x^3*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x]))/16 + (d^2*f^3*x*(1 - c^2*x^
2)^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x]))/6 + (3*d^2*f*g^2*x^3*(1 - c^2*x^2)^2*Sqrt[d - c^2*d*x^2]*(a + b*
ArcCos[c*x]))/8 - (3*d^2*f^2*g*(1 - c^2*x^2)^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x]))/(7*c^2) - (d^2*g^3*(1
- c^2*x^2)^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x]))/(7*c^4) + (d^2*g^3*(1 - c^2*x^2)^4*Sqrt[d - c^2*d*x^2]*(
a + b*ArcCos[c*x]))/(9*c^4) - (5*d^2*f^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x])^2)/(32*b*c*Sqrt[1 - c^2*x^2])
- (15*d^2*f*g^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x])^2)/(256*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 261

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -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 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 )^{5/2} \left (a+b \cos ^{-1}(c x)\right ) \, dx &=\frac{\left (d^2 \sqrt{d-c^2 d x^2}\right ) \int (f+g x)^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \cos ^{-1}(c x)\right ) \, dx}{\sqrt{1-c^2 x^2}}\\ &=\frac{\left (d^2 \sqrt{d-c^2 d x^2}\right ) \int \left (f^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \cos ^{-1}(c x)\right )+3 f^2 g x \left (1-c^2 x^2\right )^{5/2} \left (a+b \cos ^{-1}(c x)\right )+3 f g^2 x^2 \left (1-c^2 x^2\right )^{5/2} \left (a+b \cos ^{-1}(c x)\right )+g^3 x^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \cos ^{-1}(c x)\right )\right ) \, dx}{\sqrt{1-c^2 x^2}}\\ &=\frac{\left (d^2 f^3 \sqrt{d-c^2 d x^2}\right ) \int \left (1-c^2 x^2\right )^{5/2} \left (a+b \cos ^{-1}(c x)\right ) \, dx}{\sqrt{1-c^2 x^2}}+\frac{\left (3 d^2 f^2 g \sqrt{d-c^2 d x^2}\right ) \int x \left (1-c^2 x^2\right )^{5/2} \left (a+b \cos ^{-1}(c x)\right ) \, dx}{\sqrt{1-c^2 x^2}}+\frac{\left (3 d^2 f g^2 \sqrt{d-c^2 d x^2}\right ) \int x^2 \left (1-c^2 x^2\right )^{5/2} \left (a+b \cos ^{-1}(c x)\right ) \, dx}{\sqrt{1-c^2 x^2}}+\frac{\left (d^2 g^3 \sqrt{d-c^2 d x^2}\right ) \int x^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \cos ^{-1}(c x)\right ) \, dx}{\sqrt{1-c^2 x^2}}\\ &=\frac{1}{6} d^2 f^3 x \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )+\frac{3}{8} d^2 f g^2 x^3 \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )-\frac{3 d^2 f^2 g \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^2}-\frac{d^2 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^2 g^3 \left (1-c^2 x^2\right )^4 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{9 c^4}+\frac{\left (5 d^2 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}{6 \sqrt{1-c^2 x^2}}+\frac{\left (b c d^2 f^3 \sqrt{d-c^2 d x^2}\right ) \int x \left (1-c^2 x^2\right )^2 \, dx}{6 \sqrt{1-c^2 x^2}}-\frac{\left (3 b d^2 f^2 g \sqrt{d-c^2 d x^2}\right ) \int \left (1-c^2 x^2\right )^3 \, dx}{7 c \sqrt{1-c^2 x^2}}+\frac{\left (15 d^2 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}{8 \sqrt{1-c^2 x^2}}+\frac{\left (3 b c d^2 f g^2 \sqrt{d-c^2 d x^2}\right ) \int x^3 \left (1-c^2 x^2\right )^2 \, dx}{8 \sqrt{1-c^2 x^2}}+\frac{\left (b c d^2 g^3 \sqrt{d-c^2 d x^2}\right ) \int \frac{\left (-2-7 c^2 x^2\right ) \left (1-c^2 x^2\right )^3}{63 c^4} \, dx}{\sqrt{1-c^2 x^2}}\\ &=-\frac{b d^2 f^3 \left (1-c^2 x^2\right )^{5/2} \sqrt{d-c^2 d x^2}}{36 c}+\frac{5}{24} d^2 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{5}{16} d^2 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{1}{6} d^2 f^3 x \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )+\frac{3}{8} d^2 f g^2 x^3 \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )-\frac{3 d^2 f^2 g \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^2}-\frac{d^2 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^2 g^3 \left (1-c^2 x^2\right )^4 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{9 c^4}+\frac{\left (5 d^2 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}{8 \sqrt{1-c^2 x^2}}+\frac{\left (5 b c d^2 f^3 \sqrt{d-c^2 d x^2}\right ) \int x \left (1-c^2 x^2\right ) \, dx}{24 \sqrt{1-c^2 x^2}}-\frac{\left (3 b d^2 f^2 g \sqrt{d-c^2 d x^2}\right ) \int \left (1-3 c^2 x^2+3 c^4 x^4-c^6 x^6\right ) \, dx}{7 c \sqrt{1-c^2 x^2}}+\frac{\left (15 d^2 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}{16 \sqrt{1-c^2 x^2}}+\frac{\left (3 b c d^2 f g^2 \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int x \left (1-c^2 x\right )^2 \, dx,x,x^2\right )}{16 \sqrt{1-c^2 x^2}}+\frac{\left (5 b c d^2 f g^2 \sqrt{d-c^2 d x^2}\right ) \int x^3 \left (1-c^2 x^2\right ) \, dx}{16 \sqrt{1-c^2 x^2}}+\frac{\left (b d^2 g^3 \sqrt{d-c^2 d x^2}\right ) \int \left (-2-7 c^2 x^2\right ) \left (1-c^2 x^2\right )^3 \, dx}{63 c^3 \sqrt{1-c^2 x^2}}\\ &=-\frac{3 b d^2 f^2 g x \sqrt{d-c^2 d x^2}}{7 c \sqrt{1-c^2 x^2}}+\frac{3 b c d^2 f^2 g x^3 \sqrt{d-c^2 d x^2}}{7 \sqrt{1-c^2 x^2}}-\frac{9 b c^3 d^2 f^2 g x^5 \sqrt{d-c^2 d x^2}}{35 \sqrt{1-c^2 x^2}}+\frac{3 b c^5 d^2 f^2 g x^7 \sqrt{d-c^2 d x^2}}{49 \sqrt{1-c^2 x^2}}-\frac{b d^2 f^3 \left (1-c^2 x^2\right )^{5/2} \sqrt{d-c^2 d x^2}}{36 c}+\frac{5}{16} d^2 f^3 x \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )+\frac{15}{64} d^2 f g^2 x^3 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )+\frac{5}{24} d^2 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{5}{16} d^2 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{1}{6} d^2 f^3 x \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )+\frac{3}{8} d^2 f g^2 x^3 \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )-\frac{3 d^2 f^2 g \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^2}-\frac{d^2 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^2 g^3 \left (1-c^2 x^2\right )^4 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{9 c^4}+\frac{\left (5 d^2 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}{16 \sqrt{1-c^2 x^2}}+\frac{\left (5 b c d^2 f^3 \sqrt{d-c^2 d x^2}\right ) \int \left (x-c^2 x^3\right ) \, dx}{24 \sqrt{1-c^2 x^2}}+\frac{\left (5 b c d^2 f^3 \sqrt{d-c^2 d x^2}\right ) \int x \, dx}{16 \sqrt{1-c^2 x^2}}+\frac{\left (15 d^2 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}{64 \sqrt{1-c^2 x^2}}+\frac{\left (3 b c d^2 f g^2 \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \left (x-2 c^2 x^2+c^4 x^3\right ) \, dx,x,x^2\right )}{16 \sqrt{1-c^2 x^2}}+\frac{\left (15 b c d^2 f g^2 \sqrt{d-c^2 d x^2}\right ) \int x^3 \, dx}{64 \sqrt{1-c^2 x^2}}+\frac{\left (5 b c d^2 f g^2 \sqrt{d-c^2 d x^2}\right ) \int \left (x^3-c^2 x^5\right ) \, dx}{16 \sqrt{1-c^2 x^2}}+\frac{\left (b d^2 g^3 \sqrt{d-c^2 d x^2}\right ) \int \left (-2-c^2 x^2+15 c^4 x^4-19 c^6 x^6+7 c^8 x^8\right ) \, dx}{63 c^3 \sqrt{1-c^2 x^2}}\\ &=-\frac{3 b d^2 f^2 g x \sqrt{d-c^2 d x^2}}{7 c \sqrt{1-c^2 x^2}}-\frac{2 b d^2 g^3 x \sqrt{d-c^2 d x^2}}{63 c^3 \sqrt{1-c^2 x^2}}+\frac{25 b c d^2 f^3 x^2 \sqrt{d-c^2 d x^2}}{96 \sqrt{1-c^2 x^2}}+\frac{3 b c d^2 f^2 g x^3 \sqrt{d-c^2 d x^2}}{7 \sqrt{1-c^2 x^2}}-\frac{b d^2 g^3 x^3 \sqrt{d-c^2 d x^2}}{189 c \sqrt{1-c^2 x^2}}-\frac{5 b c^3 d^2 f^3 x^4 \sqrt{d-c^2 d x^2}}{96 \sqrt{1-c^2 x^2}}+\frac{59 b c d^2 f g^2 x^4 \sqrt{d-c^2 d x^2}}{256 \sqrt{1-c^2 x^2}}-\frac{9 b c^3 d^2 f^2 g x^5 \sqrt{d-c^2 d x^2}}{35 \sqrt{1-c^2 x^2}}+\frac{b c d^2 g^3 x^5 \sqrt{d-c^2 d x^2}}{21 \sqrt{1-c^2 x^2}}-\frac{17 b c^3 d^2 f g^2 x^6 \sqrt{d-c^2 d x^2}}{96 \sqrt{1-c^2 x^2}}+\frac{3 b c^5 d^2 f^2 g x^7 \sqrt{d-c^2 d x^2}}{49 \sqrt{1-c^2 x^2}}-\frac{19 b c^3 d^2 g^3 x^7 \sqrt{d-c^2 d x^2}}{441 \sqrt{1-c^2 x^2}}+\frac{3 b c^5 d^2 f g^2 x^8 \sqrt{d-c^2 d x^2}}{64 \sqrt{1-c^2 x^2}}+\frac{b c^5 d^2 g^3 x^9 \sqrt{d-c^2 d x^2}}{81 \sqrt{1-c^2 x^2}}-\frac{b d^2 f^3 \left (1-c^2 x^2\right )^{5/2} \sqrt{d-c^2 d x^2}}{36 c}+\frac{5}{16} d^2 f^3 x \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )-\frac{15 d^2 f g^2 x \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{128 c^2}+\frac{15}{64} d^2 f g^2 x^3 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )+\frac{5}{24} d^2 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{5}{16} d^2 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{1}{6} d^2 f^3 x \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )+\frac{3}{8} d^2 f g^2 x^3 \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )-\frac{3 d^2 f^2 g \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^2}-\frac{d^2 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^2 g^3 \left (1-c^2 x^2\right )^4 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{9 c^4}-\frac{5 d^2 f^3 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{32 b c \sqrt{1-c^2 x^2}}+\frac{\left (15 d^2 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}{128 c^2 \sqrt{1-c^2 x^2}}-\frac{\left (15 b d^2 f g^2 \sqrt{d-c^2 d x^2}\right ) \int x \, dx}{128 c \sqrt{1-c^2 x^2}}\\ &=-\frac{3 b d^2 f^2 g x \sqrt{d-c^2 d x^2}}{7 c \sqrt{1-c^2 x^2}}-\frac{2 b d^2 g^3 x \sqrt{d-c^2 d x^2}}{63 c^3 \sqrt{1-c^2 x^2}}+\frac{25 b c d^2 f^3 x^2 \sqrt{d-c^2 d x^2}}{96 \sqrt{1-c^2 x^2}}-\frac{15 b d^2 f g^2 x^2 \sqrt{d-c^2 d x^2}}{256 c \sqrt{1-c^2 x^2}}+\frac{3 b c d^2 f^2 g x^3 \sqrt{d-c^2 d x^2}}{7 \sqrt{1-c^2 x^2}}-\frac{b d^2 g^3 x^3 \sqrt{d-c^2 d x^2}}{189 c \sqrt{1-c^2 x^2}}-\frac{5 b c^3 d^2 f^3 x^4 \sqrt{d-c^2 d x^2}}{96 \sqrt{1-c^2 x^2}}+\frac{59 b c d^2 f g^2 x^4 \sqrt{d-c^2 d x^2}}{256 \sqrt{1-c^2 x^2}}-\frac{9 b c^3 d^2 f^2 g x^5 \sqrt{d-c^2 d x^2}}{35 \sqrt{1-c^2 x^2}}+\frac{b c d^2 g^3 x^5 \sqrt{d-c^2 d x^2}}{21 \sqrt{1-c^2 x^2}}-\frac{17 b c^3 d^2 f g^2 x^6 \sqrt{d-c^2 d x^2}}{96 \sqrt{1-c^2 x^2}}+\frac{3 b c^5 d^2 f^2 g x^7 \sqrt{d-c^2 d x^2}}{49 \sqrt{1-c^2 x^2}}-\frac{19 b c^3 d^2 g^3 x^7 \sqrt{d-c^2 d x^2}}{441 \sqrt{1-c^2 x^2}}+\frac{3 b c^5 d^2 f g^2 x^8 \sqrt{d-c^2 d x^2}}{64 \sqrt{1-c^2 x^2}}+\frac{b c^5 d^2 g^3 x^9 \sqrt{d-c^2 d x^2}}{81 \sqrt{1-c^2 x^2}}-\frac{b d^2 f^3 \left (1-c^2 x^2\right )^{5/2} \sqrt{d-c^2 d x^2}}{36 c}+\frac{5}{16} d^2 f^3 x \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )-\frac{15 d^2 f g^2 x \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{128 c^2}+\frac{15}{64} d^2 f g^2 x^3 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )+\frac{5}{24} d^2 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{5}{16} d^2 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{1}{6} d^2 f^3 x \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )+\frac{3}{8} d^2 f g^2 x^3 \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )-\frac{3 d^2 f^2 g \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^2}-\frac{d^2 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^2 g^3 \left (1-c^2 x^2\right )^4 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{9 c^4}-\frac{5 d^2 f^3 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{32 b c \sqrt{1-c^2 x^2}}-\frac{15 d^2 f g^2 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{256 b c^3 \sqrt{1-c^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 7.46235, size = 1144, normalized size = 0.89 \[ \frac{d^2 \left (-3175200 b c f \left (8 c^2 f^2+3 g^2\right ) \sqrt{d-c^2 d x^2} \cos ^{-1}(c x)^2+504 b \sqrt{d-c^2 d x^2} \left (503424 f^2 g x^2 \sqrt{1-c^2 x^2} c^4+75600 f^3 \sin \left (2 \cos ^{-1}(c x)\right ) c^3-15120 f^3 \sin \left (4 \cos ^{-1}(c x)\right ) c^3+1680 f^3 \sin \left (6 \cos ^{-1}(c x)\right ) c^3-40320 f^2 g \sin \left (3 \cos ^{-1}(c x)\right ) c^2-24192 f^2 g \sin \left (5 \cos ^{-1}(c x)\right ) c^2-120576 g^3 x^2 \sqrt{1-c^2 x^2} c^2-261504 f^2 g \sqrt{1-c^2 x^2} c^2+15120 f g^2 \sin \left (2 \cos ^{-1}(c x)\right ) c+7560 f g^2 \sin \left (4 \cos ^{-1}(c x)\right ) c-5040 f g^2 \sin \left (6 \cos ^{-1}(c x)\right ) c+945 f g^2 \sin \left (8 \cos ^{-1}(c x)\right ) c-41472 g \left (3 c^2 f^2-2 g^2\right ) \left (1-c^2 x^2\right )^{3/2} \cos \left (2 \cos ^{-1}(c x)\right )-5760 g \left (3 c^2 f^2-2 g^2\right ) \left (1-c^2 x^2\right )^{3/2} \cos \left (4 \cos ^{-1}(c x)\right )+6720 g^3 \sin \left (3 \cos ^{-1}(c x)\right )+6048 g^3 \sin \left (5 \cos ^{-1}(c x)\right )+900 g^3 \sin \left (7 \cos ^{-1}(c x)\right )+140 g^3 \sin \left (9 \cos ^{-1}(c x)\right )+62616 g^3 \sqrt{1-c^2 x^2}\right ) \cos ^{-1}(c x)-6350400 a c \sqrt{d} f \left (8 c^2 f^2+3 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 )+\sqrt{d-c^2 d x^2} \left (18063360 a g^3 x^8 \sqrt{1-c^2 x^2} c^8+60963840 a f g^2 x^7 \sqrt{1-c^2 x^2} c^8+69672960 a f^2 g x^6 \sqrt{1-c^2 x^2} c^8+27095040 a f^3 x^5 \sqrt{1-c^2 x^2} c^8-49029120 a g^3 x^6 \sqrt{1-c^2 x^2} c^6-172730880 a f g^2 x^5 \sqrt{1-c^2 x^2} c^6-209018880 a f^2 g x^4 \sqrt{1-c^2 x^2} c^6-88058880 a f^3 x^3 \sqrt{1-c^2 x^2} c^6+38707200 a g^3 x^4 \sqrt{1-c^2 x^2} c^4+149869440 a f g^2 x^3 \sqrt{1-c^2 x^2} c^4+209018880 a f^2 g x^2 \sqrt{1-c^2 x^2} c^4+111767040 a f^3 x \sqrt{1-c^2 x^2} c^4-38102400 b f^2 g x c^3-1905120 b f^3 \cos \left (4 \cos ^{-1}(c x)\right ) c^3+141120 b f^3 \cos \left (6 \cos ^{-1}(c x)\right ) c^3-1524096 b f^2 g \cos \left (5 \cos ^{-1}(c x)\right ) c^2+155520 b f^2 g \cos \left (7 \cos ^{-1}(c x)\right ) c^2-2580480 a g^3 x^2 \sqrt{1-c^2 x^2} c^2-69672960 a f^2 g \sqrt{1-c^2 x^2} c^2-19051200 a f g^2 x \sqrt{1-c^2 x^2} c^2-3810240 b g^3 x c+3810240 b f \left (5 c^2 f^2+g^2\right ) \cos \left (2 \cos ^{-1}(c x)\right ) c+952560 b f g^2 \cos \left (4 \cos ^{-1}(c x)\right ) c-423360 b f g^2 \cos \left (6 \cos ^{-1}(c x)\right ) c+59535 b f g^2 \cos \left (8 \cos ^{-1}(c x)\right ) c+282240 b g \left (27 c^2 f^2+2 g^2\right ) \cos \left (3 \cos ^{-1}(c x)\right )-38880 b g^3 \cos \left (7 \cos ^{-1}(c x)\right )+7840 b g^3 \cos \left (9 \cos ^{-1}(c x)\right )-5160960 a g^3 \sqrt{1-c^2 x^2}\right )\right )}{162570240 c^4 \sqrt{1-c^2 x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(f + g*x)^3*(d - c^2*d*x^2)^(5/2)*(a + b*ArcCos[c*x]),x]

[Out]

(d^2*(-3175200*b*c*f*(8*c^2*f^2 + 3*g^2)*Sqrt[d - c^2*d*x^2]*ArcCos[c*x]^2 - 6350400*a*c*Sqrt[d]*f*(8*c^2*f^2
+ 3*g^2)*Sqrt[1 - c^2*x^2]*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] + Sqrt[d - c^2*d*x^2]*(-
38102400*b*c^3*f^2*g*x - 3810240*b*c*g^3*x - 69672960*a*c^2*f^2*g*Sqrt[1 - c^2*x^2] - 5160960*a*g^3*Sqrt[1 - c
^2*x^2] + 111767040*a*c^4*f^3*x*Sqrt[1 - c^2*x^2] - 19051200*a*c^2*f*g^2*x*Sqrt[1 - c^2*x^2] + 209018880*a*c^4
*f^2*g*x^2*Sqrt[1 - c^2*x^2] - 2580480*a*c^2*g^3*x^2*Sqrt[1 - c^2*x^2] - 88058880*a*c^6*f^3*x^3*Sqrt[1 - c^2*x
^2] + 149869440*a*c^4*f*g^2*x^3*Sqrt[1 - c^2*x^2] - 209018880*a*c^6*f^2*g*x^4*Sqrt[1 - c^2*x^2] + 38707200*a*c
^4*g^3*x^4*Sqrt[1 - c^2*x^2] + 27095040*a*c^8*f^3*x^5*Sqrt[1 - c^2*x^2] - 172730880*a*c^6*f*g^2*x^5*Sqrt[1 - c
^2*x^2] + 69672960*a*c^8*f^2*g*x^6*Sqrt[1 - c^2*x^2] - 49029120*a*c^6*g^3*x^6*Sqrt[1 - c^2*x^2] + 60963840*a*c
^8*f*g^2*x^7*Sqrt[1 - c^2*x^2] + 18063360*a*c^8*g^3*x^8*Sqrt[1 - c^2*x^2] + 3810240*b*c*f*(5*c^2*f^2 + g^2)*Co
s[2*ArcCos[c*x]] + 282240*b*g*(27*c^2*f^2 + 2*g^2)*Cos[3*ArcCos[c*x]] - 1905120*b*c^3*f^3*Cos[4*ArcCos[c*x]] +
952560*b*c*f*g^2*Cos[4*ArcCos[c*x]] - 1524096*b*c^2*f^2*g*Cos[5*ArcCos[c*x]] + 141120*b*c^3*f^3*Cos[6*ArcCos[
c*x]] - 423360*b*c*f*g^2*Cos[6*ArcCos[c*x]] + 155520*b*c^2*f^2*g*Cos[7*ArcCos[c*x]] - 38880*b*g^3*Cos[7*ArcCos
[c*x]] + 59535*b*c*f*g^2*Cos[8*ArcCos[c*x]] + 7840*b*g^3*Cos[9*ArcCos[c*x]]) + 504*b*Sqrt[d - c^2*d*x^2]*ArcCo
s[c*x]*(-261504*c^2*f^2*g*Sqrt[1 - c^2*x^2] + 62616*g^3*Sqrt[1 - c^2*x^2] + 503424*c^4*f^2*g*x^2*Sqrt[1 - c^2*
x^2] - 120576*c^2*g^3*x^2*Sqrt[1 - c^2*x^2] - 41472*g*(3*c^2*f^2 - 2*g^2)*(1 - c^2*x^2)^(3/2)*Cos[2*ArcCos[c*x
]] - 5760*g*(3*c^2*f^2 - 2*g^2)*(1 - c^2*x^2)^(3/2)*Cos[4*ArcCos[c*x]] + 75600*c^3*f^3*Sin[2*ArcCos[c*x]] + 15
120*c*f*g^2*Sin[2*ArcCos[c*x]] - 40320*c^2*f^2*g*Sin[3*ArcCos[c*x]] + 6720*g^3*Sin[3*ArcCos[c*x]] - 15120*c^3*
f^3*Sin[4*ArcCos[c*x]] + 7560*c*f*g^2*Sin[4*ArcCos[c*x]] - 24192*c^2*f^2*g*Sin[5*ArcCos[c*x]] + 6048*g^3*Sin[5
*ArcCos[c*x]] + 1680*c^3*f^3*Sin[6*ArcCos[c*x]] - 5040*c*f*g^2*Sin[6*ArcCos[c*x]] + 900*g^3*Sin[7*ArcCos[c*x]]
+ 945*c*f*g^2*Sin[8*ArcCos[c*x]] + 140*g^3*Sin[9*ArcCos[c*x]])))/(162570240*c^4*Sqrt[1 - c^2*x^2])

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Maple [A]  time = 0.959, size = 2236, 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)^(5/2)*(a+b*arccos(c*x)),x)

[Out]

5/16*a*f^3*d^2*x*(-c^2*d*x^2+d)^(1/2)+5/16*a*f^3*d^3/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2)
)-23/16*b*(-d*(c^2*x^2-1))^(1/2)*f*g^2*d^2*c^4/(c^2*x^2-1)*arccos(c*x)*x^7+3/7*b*(-d*(c^2*x^2-1))^(1/2)*g*d^2*
c^6/(c^2*x^2-1)*arccos(c*x)*x^8*f^2-12/7*b*(-d*(c^2*x^2-1))^(1/2)*g*d^2*c^4/(c^2*x^2-1)*arccos(c*x)*x^6*f^2+18
/7*b*(-d*(c^2*x^2-1))^(1/2)*g*d^2*c^2/(c^2*x^2-1)*arccos(c*x)*x^4*f^2-2/63*a*g^3/d/c^4*(-c^2*d*x^2+d)^(7/2)+3/
7*b*(-d*(c^2*x^2-1))^(1/2)*g*d^2/c^2/(c^2*x^2-1)*arccos(c*x)*f^2+1/6*b*(-d*(c^2*x^2-1))^(1/2)*f^3*d^2*c^6/(c^2
*x^2-1)*arccos(c*x)*x^7-17/24*b*(-d*(c^2*x^2-1))^(1/2)*f^3*d^2*c^4/(c^2*x^2-1)*arccos(c*x)*x^5+59/48*b*(-d*(c^
2*x^2-1))^(1/2)*f^3*d^2*c^2/(c^2*x^2-1)*arccos(c*x)*x^3-1/36*b*(-d*(c^2*x^2-1))^(1/2)*f^3*d^2*c^5/(c^2*x^2-1)*
(-c^2*x^2+1)^(1/2)*x^6+13/96*b*(-d*(c^2*x^2-1))^(1/2)*f^3*d^2*c^3/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x^4-11/32*b*(
-d*(c^2*x^2-1))^(1/2)*f^3*d^2*c/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x^2-1/81*b*(-d*(c^2*x^2-1))^(1/2)*g^3*d^2*c^5/(
c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x^9+19/441*b*(-d*(c^2*x^2-1))^(1/2)*g^3*d^2*c^3/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x
^7-1/21*b*(-d*(c^2*x^2-1))^(1/2)*g^3*d^2*c/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x^5+1/189*b*(-d*(c^2*x^2-1))^(1/2)*g
^3*d^2/c/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x^3+2/63*b*(-d*(c^2*x^2-1))^(1/2)*g^3*d^2/c^3/(c^2*x^2-1)*(-c^2*x^2+1)
^(1/2)*x+359/24576*b*(-d*(c^2*x^2-1))^(1/2)*f*g^2*d^2/c^3/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)-133/128*b*(-d*(c^2*x^
2-1))^(1/2)*f*g^2*d^2/(c^2*x^2-1)*arccos(c*x)*x^3-12/7*b*(-d*(c^2*x^2-1))^(1/2)*g*d^2/(c^2*x^2-1)*arccos(c*x)*
x^2*f^2+5/32*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^2+1/9*b*(-d*(c^2*x^
2-1))^(1/2)*g^3*d^2*c^6/(c^2*x^2-1)*arccos(c*x)*x^10-26/63*b*(-d*(c^2*x^2-1))^(1/2)*g^3*d^2*c^4/(c^2*x^2-1)*ar
ccos(c*x)*x^8+34/63*b*(-d*(c^2*x^2-1))^(1/2)*g^3*d^2*c^2/(c^2*x^2-1)*arccos(c*x)*x^6-1/63*b*(-d*(c^2*x^2-1))^(
1/2)*g^3*d^2/c^2/(c^2*x^2-1)*arccos(c*x)*x^2+3/7*b*(-d*(c^2*x^2-1))^(1/2)*g*d^2/c/(c^2*x^2-1)*(-c^2*x^2+1)^(1/
2)*x*f^2-3/64*b*(-d*(c^2*x^2-1))^(1/2)*f*g^2*d^2*c^5/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x^8+17/96*b*(-d*(c^2*x^2-1
))^(1/2)*f*g^2*d^2*c^3/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x^6-3/49*b*(-d*(c^2*x^2-1))^(1/2)*g*d^2*c^5/(c^2*x^2-1)*
(-c^2*x^2+1)^(1/2)*x^7*f^2-3/7*b*(-d*(c^2*x^2-1))^(1/2)*g*d^2*c/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x^3*f^2-59/256*
b*(-d*(c^2*x^2-1))^(1/2)*f*g^2*d^2*c/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x^4+15/256*b*(-d*(c^2*x^2-1))^(1/2)*f*g^2*
d^2/c/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x^2+15/256*b*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/c^3/(c^2*x^2-1)*ar
ccos(c*x)^2*f*d^2*g^2+3/8*b*(-d*(c^2*x^2-1))^(1/2)*f*g^2*d^2*c^6/(c^2*x^2-1)*arccos(c*x)*x^9+127/64*b*(-d*(c^2
*x^2-1))^(1/2)*f*g^2*d^2*c^2/(c^2*x^2-1)*arccos(c*x)*x^5+15/128*b*(-d*(c^2*x^2-1))^(1/2)*f*g^2*d^2/c^2/(c^2*x^
2-1)*arccos(c*x)*x+9/35*b*(-d*(c^2*x^2-1))^(1/2)*g*d^2*c^3/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x^5*f^2+5/24*a*f^3*d
*x*(-c^2*d*x^2+d)^(3/2)-1/9*a*g^3*x^2*(-c^2*d*x^2+d)^(7/2)/c^2/d+1/16*a*f*g^2/c^2*x*(-c^2*d*x^2+d)^(5/2)-3/7*a
*f^2*g*(-c^2*d*x^2+d)^(7/2)/c^2/d+1/6*a*f^3*x*(-c^2*d*x^2+d)^(5/2)-3/8*a*f*g^2*x*(-c^2*d*x^2+d)^(7/2)/c^2/d+5/
64*a*f*g^2/c^2*d*x*(-c^2*d*x^2+d)^(3/2)+15/128*a*f*g^2/c^2*d^2*x*(-c^2*d*x^2+d)^(1/2)+15/128*a*f*g^2/c^2*d^3/(
c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2))+299/2304*b*(-d*(c^2*x^2-1))^(1/2)*f^3*d^2/c/(c^2*x^2
-1)*(-c^2*x^2+1)^(1/2)+2/63*b*(-d*(c^2*x^2-1))^(1/2)*g^3*d^2/c^4/(c^2*x^2-1)*arccos(c*x)-16/63*b*(-d*(c^2*x^2-
1))^(1/2)*g^3*d^2/(c^2*x^2-1)*arccos(c*x)*x^4-11/16*b*(-d*(c^2*x^2-1))^(1/2)*f^3*d^2/(c^2*x^2-1)*arccos(c*x)*x

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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)^(5/2)*(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^{4} d^{2} g^{3} x^{7} + 3 \, a c^{4} d^{2} f g^{2} x^{6} + 3 \, a d^{2} f^{2} g x + a d^{2} f^{3} +{\left (3 \, a c^{4} d^{2} f^{2} g - 2 \, a c^{2} d^{2} g^{3}\right )} x^{5} +{\left (a c^{4} d^{2} f^{3} - 6 \, a c^{2} d^{2} f g^{2}\right )} x^{4} -{\left (6 \, a c^{2} d^{2} f^{2} g - a d^{2} g^{3}\right )} x^{3} -{\left (2 \, a c^{2} d^{2} f^{3} - 3 \, a d^{2} f g^{2}\right )} x^{2} +{\left (b c^{4} d^{2} g^{3} x^{7} + 3 \, b c^{4} d^{2} f g^{2} x^{6} + 3 \, b d^{2} f^{2} g x + b d^{2} f^{3} +{\left (3 \, b c^{4} d^{2} f^{2} g - 2 \, b c^{2} d^{2} g^{3}\right )} x^{5} +{\left (b c^{4} d^{2} f^{3} - 6 \, b c^{2} d^{2} f g^{2}\right )} x^{4} -{\left (6 \, b c^{2} d^{2} f^{2} g - b d^{2} g^{3}\right )} x^{3} -{\left (2 \, b c^{2} d^{2} f^{3} - 3 \, b d^{2} 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)^(5/2)*(a+b*arccos(c*x)),x, algorithm="fricas")

[Out]

integral((a*c^4*d^2*g^3*x^7 + 3*a*c^4*d^2*f*g^2*x^6 + 3*a*d^2*f^2*g*x + a*d^2*f^3 + (3*a*c^4*d^2*f^2*g - 2*a*c
^2*d^2*g^3)*x^5 + (a*c^4*d^2*f^3 - 6*a*c^2*d^2*f*g^2)*x^4 - (6*a*c^2*d^2*f^2*g - a*d^2*g^3)*x^3 - (2*a*c^2*d^2
*f^3 - 3*a*d^2*f*g^2)*x^2 + (b*c^4*d^2*g^3*x^7 + 3*b*c^4*d^2*f*g^2*x^6 + 3*b*d^2*f^2*g*x + b*d^2*f^3 + (3*b*c^
4*d^2*f^2*g - 2*b*c^2*d^2*g^3)*x^5 + (b*c^4*d^2*f^3 - 6*b*c^2*d^2*f*g^2)*x^4 - (6*b*c^2*d^2*f^2*g - b*d^2*g^3)
*x^3 - (2*b*c^2*d^2*f^3 - 3*b*d^2*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)**(5/2)*(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{5}{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)^(5/2)*(a+b*arccos(c*x)),x, algorithm="giac")

[Out]

integrate((-c^2*d*x^2 + d)^(5/2)*(g*x + f)^3*(b*arccos(c*x) + a), x)