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

Optimal. Leaf size=738 \[ -\frac{2 i b^2 g \sqrt{1-c^2 x^2} \left (3 c^2 f^2+g^2\right ) \text{PolyLog}\left (2,-i e^{i \sin ^{-1}(c x)}\right )}{c^4 d \sqrt{d-c^2 d x^2}}+\frac{2 i b^2 g \sqrt{1-c^2 x^2} \left (3 c^2 f^2+g^2\right ) \text{PolyLog}\left (2,i e^{i \sin ^{-1}(c x)}\right )}{c^4 d \sqrt{d-c^2 d x^2}}-\frac{i b^2 f \sqrt{1-c^2 x^2} \left (c^2 f^2+3 g^2\right ) \text{PolyLog}\left (2,-e^{2 i \sin ^{-1}(c x)}\right )}{c^3 d \sqrt{d-c^2 d x^2}}+\frac{g \left (3 c^2 f^2+g^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c^4 d \sqrt{d-c^2 d x^2}}+\frac{f x \left (\frac{3 g^2}{c^2}+f^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{d \sqrt{d-c^2 d x^2}}-\frac{i f \sqrt{1-c^2 x^2} \left (c^2 f^2+3 g^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c^3 d \sqrt{d-c^2 d x^2}}+\frac{2 b f \sqrt{1-c^2 x^2} \left (c^2 f^2+3 g^2\right ) \log \left (1+e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{c^3 d \sqrt{d-c^2 d x^2}}+\frac{4 i b g \sqrt{1-c^2 x^2} \left (3 c^2 f^2+g^2\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{c^4 d \sqrt{d-c^2 d x^2}}-\frac{f g^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{b c^3 d \sqrt{d-c^2 d x^2}}-\frac{2 a b g^3 x \sqrt{1-c^2 x^2}}{c^3 d \sqrt{d-c^2 d x^2}}+\frac{g^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c^4 d \sqrt{d-c^2 d x^2}}-\frac{2 b^2 g^3 \left (1-c^2 x^2\right )}{c^4 d \sqrt{d-c^2 d x^2}}-\frac{2 b^2 g^3 x \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{c^3 d \sqrt{d-c^2 d x^2}} \]

[Out]

(-2*a*b*g^3*x*Sqrt[1 - c^2*x^2])/(c^3*d*Sqrt[d - c^2*d*x^2]) - (2*b^2*g^3*(1 - c^2*x^2))/(c^4*d*Sqrt[d - c^2*d
*x^2]) - (2*b^2*g^3*x*Sqrt[1 - c^2*x^2]*ArcSin[c*x])/(c^3*d*Sqrt[d - c^2*d*x^2]) + (g*(3*c^2*f^2 + g^2)*(a + b
*ArcSin[c*x])^2)/(c^4*d*Sqrt[d - c^2*d*x^2]) + (f*(f^2 + (3*g^2)/c^2)*x*(a + b*ArcSin[c*x])^2)/(d*Sqrt[d - c^2
*d*x^2]) - (I*f*(c^2*f^2 + 3*g^2)*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/(c^3*d*Sqrt[d - c^2*d*x^2]) + (g^3*
(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/(c^4*d*Sqrt[d - c^2*d*x^2]) - (f*g^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]
)^3)/(b*c^3*d*Sqrt[d - c^2*d*x^2]) + ((4*I)*b*g*(3*c^2*f^2 + g^2)*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*ArcTan
[E^(I*ArcSin[c*x])])/(c^4*d*Sqrt[d - c^2*d*x^2]) + (2*b*f*(c^2*f^2 + 3*g^2)*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*
x])*Log[1 + E^((2*I)*ArcSin[c*x])])/(c^3*d*Sqrt[d - c^2*d*x^2]) - ((2*I)*b^2*g*(3*c^2*f^2 + g^2)*Sqrt[1 - c^2*
x^2]*PolyLog[2, (-I)*E^(I*ArcSin[c*x])])/(c^4*d*Sqrt[d - c^2*d*x^2]) + ((2*I)*b^2*g*(3*c^2*f^2 + g^2)*Sqrt[1 -
c^2*x^2]*PolyLog[2, I*E^(I*ArcSin[c*x])])/(c^4*d*Sqrt[d - c^2*d*x^2]) - (I*b^2*f*(c^2*f^2 + 3*g^2)*Sqrt[1 - c
^2*x^2]*PolyLog[2, -E^((2*I)*ArcSin[c*x])])/(c^3*d*Sqrt[d - c^2*d*x^2])

________________________________________________________________________________________

Rubi [A]  time = 1.19116, antiderivative size = 738, normalized size of antiderivative = 1., number of steps used = 23, number of rules used = 15, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.454, Rules used = {4777, 4775, 4763, 4651, 4675, 3719, 2190, 2279, 2391, 4677, 4657, 4181, 4641, 4619, 261} \[ -\frac{2 i b^2 g \sqrt{1-c^2 x^2} \left (3 c^2 f^2+g^2\right ) \text{PolyLog}\left (2,-i e^{i \sin ^{-1}(c x)}\right )}{c^4 d \sqrt{d-c^2 d x^2}}+\frac{2 i b^2 g \sqrt{1-c^2 x^2} \left (3 c^2 f^2+g^2\right ) \text{PolyLog}\left (2,i e^{i \sin ^{-1}(c x)}\right )}{c^4 d \sqrt{d-c^2 d x^2}}-\frac{i b^2 f \sqrt{1-c^2 x^2} \left (c^2 f^2+3 g^2\right ) \text{PolyLog}\left (2,-e^{2 i \sin ^{-1}(c x)}\right )}{c^3 d \sqrt{d-c^2 d x^2}}+\frac{g \left (3 c^2 f^2+g^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c^4 d \sqrt{d-c^2 d x^2}}+\frac{f x \left (\frac{3 g^2}{c^2}+f^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{d \sqrt{d-c^2 d x^2}}-\frac{i f \sqrt{1-c^2 x^2} \left (c^2 f^2+3 g^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c^3 d \sqrt{d-c^2 d x^2}}+\frac{2 b f \sqrt{1-c^2 x^2} \left (c^2 f^2+3 g^2\right ) \log \left (1+e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{c^3 d \sqrt{d-c^2 d x^2}}+\frac{4 i b g \sqrt{1-c^2 x^2} \left (3 c^2 f^2+g^2\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{c^4 d \sqrt{d-c^2 d x^2}}-\frac{f g^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{b c^3 d \sqrt{d-c^2 d x^2}}-\frac{2 a b g^3 x \sqrt{1-c^2 x^2}}{c^3 d \sqrt{d-c^2 d x^2}}+\frac{g^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c^4 d \sqrt{d-c^2 d x^2}}-\frac{2 b^2 g^3 \left (1-c^2 x^2\right )}{c^4 d \sqrt{d-c^2 d x^2}}-\frac{2 b^2 g^3 x \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{c^3 d \sqrt{d-c^2 d x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*a*b*g^3*x*Sqrt[1 - c^2*x^2])/(c^3*d*Sqrt[d - c^2*d*x^2]) - (2*b^2*g^3*(1 - c^2*x^2))/(c^4*d*Sqrt[d - c^2*d
*x^2]) - (2*b^2*g^3*x*Sqrt[1 - c^2*x^2]*ArcSin[c*x])/(c^3*d*Sqrt[d - c^2*d*x^2]) + (g*(3*c^2*f^2 + g^2)*(a + b
*ArcSin[c*x])^2)/(c^4*d*Sqrt[d - c^2*d*x^2]) + (f*(f^2 + (3*g^2)/c^2)*x*(a + b*ArcSin[c*x])^2)/(d*Sqrt[d - c^2
*d*x^2]) - (I*f*(c^2*f^2 + 3*g^2)*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/(c^3*d*Sqrt[d - c^2*d*x^2]) + (g^3*
(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/(c^4*d*Sqrt[d - c^2*d*x^2]) - (f*g^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]
)^3)/(b*c^3*d*Sqrt[d - c^2*d*x^2]) + ((4*I)*b*g*(3*c^2*f^2 + g^2)*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*ArcTan
[E^(I*ArcSin[c*x])])/(c^4*d*Sqrt[d - c^2*d*x^2]) + (2*b*f*(c^2*f^2 + 3*g^2)*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*
x])*Log[1 + E^((2*I)*ArcSin[c*x])])/(c^3*d*Sqrt[d - c^2*d*x^2]) - ((2*I)*b^2*g*(3*c^2*f^2 + g^2)*Sqrt[1 - c^2*
x^2]*PolyLog[2, (-I)*E^(I*ArcSin[c*x])])/(c^4*d*Sqrt[d - c^2*d*x^2]) + ((2*I)*b^2*g*(3*c^2*f^2 + g^2)*Sqrt[1 -
c^2*x^2]*PolyLog[2, I*E^(I*ArcSin[c*x])])/(c^4*d*Sqrt[d - c^2*d*x^2]) - (I*b^2*f*(c^2*f^2 + 3*g^2)*Sqrt[1 - c
^2*x^2]*PolyLog[2, -E^((2*I)*ArcSin[c*x])])/(c^3*d*Sqrt[d - c^2*d*x^2])

Rule 4777

Int[((a_.) + ArcSin[(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*ArcSin[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 4775

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :
> Int[ExpandIntegrand[(a + b*ArcSin[c*x])^n/Sqrt[d + e*x^2], (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] && ILtQ[p + 1/2, 0] && GtQ[d, 0] && IGtQ[n, 0]

Rule 4763

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :
> Int[ExpandIntegrand[(d + e*x^2)^p*(a + b*ArcSin[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 4651

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[(x*(a + b*ArcSin[c
*x])^n)/(d*Sqrt[d + e*x^2]), x] - Dist[(b*c*n)/Sqrt[d], Int[(x*(a + b*ArcSin[c*x])^(n - 1))/(d + e*x^2), x], x
] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && GtQ[d, 0]

Rule 4675

Int[(((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*(x_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> -Dist[e^(-1), Subst[In
t[(a + b*x)^n*Tan[x], x], x, ArcSin[c*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0]

Rule 3719

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[(I*(c + d*x)^(m + 1))/(d*(m + 1)), x
] - Dist[2*I, Int[((c + d*x)^m*E^(2*I*(e + f*x)))/(1 + E^(2*I*(e + f*x))), x], x] /; FreeQ[{c, d, e, f}, x] &&
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]

Rule 4677

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[((d + e*x^2)^
(p + 1)*(a + b*ArcSin[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*ArcSin[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 4657

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

Rule 4181

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

Rule 4641

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(a + b*ArcSin[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 4619

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

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]

Rubi steps

\begin{align*} \int \frac{(f+g x)^3 \left (a+b \sin ^{-1}(c x)\right )^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx &=\frac{\sqrt{1-c^2 x^2} \int \frac{(f+g x)^3 \left (a+b \sin ^{-1}(c x)\right )^2}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{d \sqrt{d-c^2 d x^2}}\\ &=\frac{\sqrt{1-c^2 x^2} \int \left (\frac{\left (c^2 f^3+3 f g^2+g \left (3 c^2 f^2+g^2\right ) x\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 \left (1-c^2 x^2\right )^{3/2}}-\frac{3 f g^2 \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 \sqrt{1-c^2 x^2}}-\frac{g^3 x \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 \sqrt{1-c^2 x^2}}\right ) \, dx}{d \sqrt{d-c^2 d x^2}}\\ &=\frac{\sqrt{1-c^2 x^2} \int \frac{\left (c^2 f^3+3 f g^2+g \left (3 c^2 f^2+g^2\right ) x\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{c^2 d \sqrt{d-c^2 d x^2}}-\frac{\left (3 f g^2 \sqrt{1-c^2 x^2}\right ) \int \frac{\left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt{1-c^2 x^2}} \, dx}{c^2 d \sqrt{d-c^2 d x^2}}-\frac{\left (g^3 \sqrt{1-c^2 x^2}\right ) \int \frac{x \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt{1-c^2 x^2}} \, dx}{c^2 d \sqrt{d-c^2 d x^2}}\\ &=\frac{g^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c^4 d \sqrt{d-c^2 d x^2}}-\frac{f g^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{b c^3 d \sqrt{d-c^2 d x^2}}+\frac{\sqrt{1-c^2 x^2} \int \left (\frac{c^2 f^3 \left (1+\frac{3 g^2}{c^2 f^2}\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{\left (1-c^2 x^2\right )^{3/2}}+\frac{g \left (3 c^2 f^2+g^2\right ) x \left (a+b \sin ^{-1}(c x)\right )^2}{\left (1-c^2 x^2\right )^{3/2}}\right ) \, dx}{c^2 d \sqrt{d-c^2 d x^2}}-\frac{\left (2 b g^3 \sqrt{1-c^2 x^2}\right ) \int \left (a+b \sin ^{-1}(c x)\right ) \, dx}{c^3 d \sqrt{d-c^2 d x^2}}\\ &=-\frac{2 a b g^3 x \sqrt{1-c^2 x^2}}{c^3 d \sqrt{d-c^2 d x^2}}+\frac{g^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c^4 d \sqrt{d-c^2 d x^2}}-\frac{f g^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{b c^3 d \sqrt{d-c^2 d x^2}}-\frac{\left (2 b^2 g^3 \sqrt{1-c^2 x^2}\right ) \int \sin ^{-1}(c x) \, dx}{c^3 d \sqrt{d-c^2 d x^2}}+\frac{\left (g \left (3 c^2 f^2+g^2\right ) \sqrt{1-c^2 x^2}\right ) \int \frac{x \left (a+b \sin ^{-1}(c x)\right )^2}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{c^2 d \sqrt{d-c^2 d x^2}}+\frac{\left (f \left (c^2 f^2+3 g^2\right ) \sqrt{1-c^2 x^2}\right ) \int \frac{\left (a+b \sin ^{-1}(c x)\right )^2}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{c^2 d \sqrt{d-c^2 d x^2}}\\ &=-\frac{2 a b g^3 x \sqrt{1-c^2 x^2}}{c^3 d \sqrt{d-c^2 d x^2}}-\frac{2 b^2 g^3 x \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{c^3 d \sqrt{d-c^2 d x^2}}+\frac{g \left (3 c^2 f^2+g^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c^4 d \sqrt{d-c^2 d x^2}}+\frac{f \left (c^2 f^2+3 g^2\right ) x \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 d \sqrt{d-c^2 d x^2}}+\frac{g^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c^4 d \sqrt{d-c^2 d x^2}}-\frac{f g^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{b c^3 d \sqrt{d-c^2 d x^2}}+\frac{\left (2 b^2 g^3 \sqrt{1-c^2 x^2}\right ) \int \frac{x}{\sqrt{1-c^2 x^2}} \, dx}{c^2 d \sqrt{d-c^2 d x^2}}-\frac{\left (2 b g \left (3 c^2 f^2+g^2\right ) \sqrt{1-c^2 x^2}\right ) \int \frac{a+b \sin ^{-1}(c x)}{1-c^2 x^2} \, dx}{c^3 d \sqrt{d-c^2 d x^2}}-\frac{\left (2 b f \left (c^2 f^2+3 g^2\right ) \sqrt{1-c^2 x^2}\right ) \int \frac{x \left (a+b \sin ^{-1}(c x)\right )}{1-c^2 x^2} \, dx}{c d \sqrt{d-c^2 d x^2}}\\ &=-\frac{2 a b g^3 x \sqrt{1-c^2 x^2}}{c^3 d \sqrt{d-c^2 d x^2}}-\frac{2 b^2 g^3 \left (1-c^2 x^2\right )}{c^4 d \sqrt{d-c^2 d x^2}}-\frac{2 b^2 g^3 x \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{c^3 d \sqrt{d-c^2 d x^2}}+\frac{g \left (3 c^2 f^2+g^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c^4 d \sqrt{d-c^2 d x^2}}+\frac{f \left (c^2 f^2+3 g^2\right ) x \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 d \sqrt{d-c^2 d x^2}}+\frac{g^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c^4 d \sqrt{d-c^2 d x^2}}-\frac{f g^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{b c^3 d \sqrt{d-c^2 d x^2}}-\frac{\left (2 b g \left (3 c^2 f^2+g^2\right ) \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int (a+b x) \sec (x) \, dx,x,\sin ^{-1}(c x)\right )}{c^4 d \sqrt{d-c^2 d x^2}}-\frac{\left (2 b f \left (c^2 f^2+3 g^2\right ) \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int (a+b x) \tan (x) \, dx,x,\sin ^{-1}(c x)\right )}{c^3 d \sqrt{d-c^2 d x^2}}\\ &=-\frac{2 a b g^3 x \sqrt{1-c^2 x^2}}{c^3 d \sqrt{d-c^2 d x^2}}-\frac{2 b^2 g^3 \left (1-c^2 x^2\right )}{c^4 d \sqrt{d-c^2 d x^2}}-\frac{2 b^2 g^3 x \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{c^3 d \sqrt{d-c^2 d x^2}}+\frac{g \left (3 c^2 f^2+g^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c^4 d \sqrt{d-c^2 d x^2}}+\frac{f \left (c^2 f^2+3 g^2\right ) x \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 d \sqrt{d-c^2 d x^2}}-\frac{i f \left (c^2 f^2+3 g^2\right ) \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{c^3 d \sqrt{d-c^2 d x^2}}+\frac{g^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c^4 d \sqrt{d-c^2 d x^2}}-\frac{f g^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{b c^3 d \sqrt{d-c^2 d x^2}}+\frac{4 i b g \left (3 c^2 f^2+g^2\right ) \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{c^4 d \sqrt{d-c^2 d x^2}}+\frac{\left (2 b^2 g \left (3 c^2 f^2+g^2\right ) \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \log \left (1-i e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{c^4 d \sqrt{d-c^2 d x^2}}-\frac{\left (2 b^2 g \left (3 c^2 f^2+g^2\right ) \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \log \left (1+i e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{c^4 d \sqrt{d-c^2 d x^2}}+\frac{\left (4 i b f \left (c^2 f^2+3 g^2\right ) \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{e^{2 i x} (a+b x)}{1+e^{2 i x}} \, dx,x,\sin ^{-1}(c x)\right )}{c^3 d \sqrt{d-c^2 d x^2}}\\ &=-\frac{2 a b g^3 x \sqrt{1-c^2 x^2}}{c^3 d \sqrt{d-c^2 d x^2}}-\frac{2 b^2 g^3 \left (1-c^2 x^2\right )}{c^4 d \sqrt{d-c^2 d x^2}}-\frac{2 b^2 g^3 x \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{c^3 d \sqrt{d-c^2 d x^2}}+\frac{g \left (3 c^2 f^2+g^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c^4 d \sqrt{d-c^2 d x^2}}+\frac{f \left (c^2 f^2+3 g^2\right ) x \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 d \sqrt{d-c^2 d x^2}}-\frac{i f \left (c^2 f^2+3 g^2\right ) \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{c^3 d \sqrt{d-c^2 d x^2}}+\frac{g^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c^4 d \sqrt{d-c^2 d x^2}}-\frac{f g^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{b c^3 d \sqrt{d-c^2 d x^2}}+\frac{4 i b g \left (3 c^2 f^2+g^2\right ) \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{c^4 d \sqrt{d-c^2 d x^2}}+\frac{2 b f \left (c^2 f^2+3 g^2\right ) \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1+e^{2 i \sin ^{-1}(c x)}\right )}{c^3 d \sqrt{d-c^2 d x^2}}-\frac{\left (2 i b^2 g \left (3 c^2 f^2+g^2\right ) \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\log (1-i x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{c^4 d \sqrt{d-c^2 d x^2}}+\frac{\left (2 i b^2 g \left (3 c^2 f^2+g^2\right ) \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\log (1+i x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{c^4 d \sqrt{d-c^2 d x^2}}-\frac{\left (2 b^2 f \left (c^2 f^2+3 g^2\right ) \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \log \left (1+e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{c^3 d \sqrt{d-c^2 d x^2}}\\ &=-\frac{2 a b g^3 x \sqrt{1-c^2 x^2}}{c^3 d \sqrt{d-c^2 d x^2}}-\frac{2 b^2 g^3 \left (1-c^2 x^2\right )}{c^4 d \sqrt{d-c^2 d x^2}}-\frac{2 b^2 g^3 x \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{c^3 d \sqrt{d-c^2 d x^2}}+\frac{g \left (3 c^2 f^2+g^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c^4 d \sqrt{d-c^2 d x^2}}+\frac{f \left (c^2 f^2+3 g^2\right ) x \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 d \sqrt{d-c^2 d x^2}}-\frac{i f \left (c^2 f^2+3 g^2\right ) \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{c^3 d \sqrt{d-c^2 d x^2}}+\frac{g^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c^4 d \sqrt{d-c^2 d x^2}}-\frac{f g^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{b c^3 d \sqrt{d-c^2 d x^2}}+\frac{4 i b g \left (3 c^2 f^2+g^2\right ) \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{c^4 d \sqrt{d-c^2 d x^2}}+\frac{2 b f \left (c^2 f^2+3 g^2\right ) \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1+e^{2 i \sin ^{-1}(c x)}\right )}{c^3 d \sqrt{d-c^2 d x^2}}-\frac{2 i b^2 g \left (3 c^2 f^2+g^2\right ) \sqrt{1-c^2 x^2} \text{Li}_2\left (-i e^{i \sin ^{-1}(c x)}\right )}{c^4 d \sqrt{d-c^2 d x^2}}+\frac{2 i b^2 g \left (3 c^2 f^2+g^2\right ) \sqrt{1-c^2 x^2} \text{Li}_2\left (i e^{i \sin ^{-1}(c x)}\right )}{c^4 d \sqrt{d-c^2 d x^2}}+\frac{\left (i b^2 f \left (c^2 f^2+3 g^2\right ) \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{2 i \sin ^{-1}(c x)}\right )}{c^3 d \sqrt{d-c^2 d x^2}}\\ &=-\frac{2 a b g^3 x \sqrt{1-c^2 x^2}}{c^3 d \sqrt{d-c^2 d x^2}}-\frac{2 b^2 g^3 \left (1-c^2 x^2\right )}{c^4 d \sqrt{d-c^2 d x^2}}-\frac{2 b^2 g^3 x \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{c^3 d \sqrt{d-c^2 d x^2}}+\frac{g \left (3 c^2 f^2+g^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c^4 d \sqrt{d-c^2 d x^2}}+\frac{f \left (c^2 f^2+3 g^2\right ) x \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 d \sqrt{d-c^2 d x^2}}-\frac{i f \left (c^2 f^2+3 g^2\right ) \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{c^3 d \sqrt{d-c^2 d x^2}}+\frac{g^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c^4 d \sqrt{d-c^2 d x^2}}-\frac{f g^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{b c^3 d \sqrt{d-c^2 d x^2}}+\frac{4 i b g \left (3 c^2 f^2+g^2\right ) \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{c^4 d \sqrt{d-c^2 d x^2}}+\frac{2 b f \left (c^2 f^2+3 g^2\right ) \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1+e^{2 i \sin ^{-1}(c x)}\right )}{c^3 d \sqrt{d-c^2 d x^2}}-\frac{2 i b^2 g \left (3 c^2 f^2+g^2\right ) \sqrt{1-c^2 x^2} \text{Li}_2\left (-i e^{i \sin ^{-1}(c x)}\right )}{c^4 d \sqrt{d-c^2 d x^2}}+\frac{2 i b^2 g \left (3 c^2 f^2+g^2\right ) \sqrt{1-c^2 x^2} \text{Li}_2\left (i e^{i \sin ^{-1}(c x)}\right )}{c^4 d \sqrt{d-c^2 d x^2}}-\frac{i b^2 f \left (c^2 f^2+3 g^2\right ) \sqrt{1-c^2 x^2} \text{Li}_2\left (-e^{2 i \sin ^{-1}(c x)}\right )}{c^3 d \sqrt{d-c^2 d x^2}}\\ \end{align*}

Mathematica [A]  time = 3.32526, size = 325, normalized size = 0.44 \[ \frac{\sqrt{1-c^2 x^2} \left (-(c f+g)^3 \left (-\tan \left (\frac{1}{4} \left (2 \sin ^{-1}(c x)+\pi \right )\right ) \left (a+b \sin ^{-1}(c x)\right )^2+i \left (4 b^2 \text{PolyLog}\left (2,-i e^{i \sin ^{-1}(c x)}\right )+\left (a+b \sin ^{-1}(c x)\right ) \left (a+b \sin ^{-1}(c x)+4 i b \log \left (1+i e^{i \sin ^{-1}(c x)}\right )\right )\right )\right )+(c f-g)^3 \left (-\cot \left (\frac{1}{4} \left (2 \sin ^{-1}(c x)+\pi \right )\right ) \left (a+b \sin ^{-1}(c x)\right )^2+i \left (4 b^2 \text{PolyLog}\left (2,-i e^{-i \sin ^{-1}(c x)}\right )+\left (a+b \sin ^{-1}(c x)\right ) \left (a+b \sin ^{-1}(c x)-4 i b \log \left (1+i e^{-i \sin ^{-1}(c x)}\right )\right )\right )\right )+2 g^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2-4 b g^3 \left (a c x+b \sqrt{1-c^2 x^2}+b c x \sin ^{-1}(c x)\right )-\frac{2 c f g^2 \left (a+b \sin ^{-1}(c x)\right )^3}{b}\right )}{2 c^4 d \sqrt{d-c^2 d x^2}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(Sqrt[1 - c^2*x^2]*(2*g^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2 - (2*c*f*g^2*(a + b*ArcSin[c*x])^3)/b - 4*b*
g^3*(a*c*x + b*Sqrt[1 - c^2*x^2] + b*c*x*ArcSin[c*x]) + (c*f - g)^3*(-((a + b*ArcSin[c*x])^2*Cot[(Pi + 2*ArcSi
n[c*x])/4]) + I*((a + b*ArcSin[c*x])*(a + b*ArcSin[c*x] - (4*I)*b*Log[1 + I/E^(I*ArcSin[c*x])]) + 4*b^2*PolyLo
g[2, (-I)/E^(I*ArcSin[c*x])])) - (c*f + g)^3*(I*((a + b*ArcSin[c*x])*(a + b*ArcSin[c*x] + (4*I)*b*Log[1 + I*E^
(I*ArcSin[c*x])]) + 4*b^2*PolyLog[2, (-I)*E^(I*ArcSin[c*x])]) - (a + b*ArcSin[c*x])^2*Tan[(Pi + 2*ArcSin[c*x])
/4])))/(2*c^4*d*Sqrt[d - c^2*d*x^2])

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Maple [B]  time = 0.635, size = 2663, normalized size = 3.6 \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*(a+b*arcsin(c*x))^2/(-c^2*d*x^2+d)^(3/2),x)

[Out]

-6*a*b*(-d*(c^2*x^2-1))^(1/2)/c^2/d^2/(c^2*x^2-1)*arcsin(c*x)*x*f*g^2-6*b^2*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1
)^(1/2)/c^2/d^2/(c^2*x^2-1)*f^2*g*arcsin(c*x)*ln(1+I*(I*c*x+(-c^2*x^2+1)^(1/2)))+6*b^2*(-d*(c^2*x^2-1))^(1/2)*
(-c^2*x^2+1)^(1/2)/c^2/d^2/(c^2*x^2-1)*f^2*g*arcsin(c*x)*ln(1-I*(I*c*x+(-c^2*x^2+1)^(1/2)))-6*b^2*(-d*(c^2*x^2
-1))^(1/2)*(-c^2*x^2+1)^(1/2)/c^3/d^2/(c^2*x^2-1)*f*g^2*arcsin(c*x)*ln(1+(I*c*x+(-c^2*x^2+1)^(1/2))^2)+3*I*b^2
*(-d*(c^2*x^2-1))^(1/2)/c^3/d^2/(c^2*x^2-1)*arcsin(c*x)^2*(-c^2*x^2+1)^(1/2)*f*g^2+3*a^2*f*g^2*x/c^2/d/(-c^2*d
*x^2+d)^(1/2)-3*a^2*f*g^2/c^2/d/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2))-b^2*(-d*(c^2*x^2-1)
)^(1/2)/d^2/(c^2*x^2-1)*arcsin(c*x)^2*x*f^3-2*b^2*(-d*(c^2*x^2-1))^(1/2)*g^3/c^2/d^2/(c^2*x^2-1)*x^2-2*b^2*(-d
*(c^2*x^2-1))^(1/2)*g^3/c^4/d^2/(c^2*x^2-1)*arcsin(c*x)^2+2*a^2*g^3/d/c^4/(-c^2*d*x^2+d)^(1/2)+a^2*f^3/d*x/(-c
^2*d*x^2+d)^(1/2)-4*a*b*(-d*(c^2*x^2-1))^(1/2)*g^3/c^4/d^2/(c^2*x^2-1)*arcsin(c*x)-2*a*b*(-d*(c^2*x^2-1))^(1/2
)/d^2/(c^2*x^2-1)*arcsin(c*x)*x*f^3+b^2*(-d*(c^2*x^2-1))^(1/2)*g^3/c^2/d^2/(c^2*x^2-1)*arcsin(c*x)^2*x^2-3*b^2
*(-d*(c^2*x^2-1))^(1/2)/c^2/d^2/(c^2*x^2-1)*arcsin(c*x)^2*f^2*g+6*I*a*b*(-c^2*x^2+1)^(1/2)*(-d*(c^2*x^2-1))^(1
/2)/c^3/d^2/(c^2*x^2-1)*f*arcsin(c*x)*g^2+6*I*b^2*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/c^2/d^2/(c^2*x^2-1
)*f^2*g*dilog(1+I*(I*c*x+(-c^2*x^2+1)^(1/2)))-6*I*b^2*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/c^2/d^2/(c^2*x
^2-1)*f^2*g*dilog(1-I*(I*c*x+(-c^2*x^2+1)^(1/2)))+3*I*b^2*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/c^3/d^2/(c
^2*x^2-1)*f*g^2*polylog(2,-(I*c*x+(-c^2*x^2+1)^(1/2))^2)-6*a*b*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/c^3/d
^2/(c^2*x^2-1)*ln(I*c*x+(-c^2*x^2+1)^(1/2)-I)*f*g^2+6*a*b*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)*ln(I*c*x+(
-c^2*x^2+1)^(1/2)+I)/c^2/d^2/(c^2*x^2-1)*f^2*g-6*a*b*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)*ln(I*c*x+(-c^2*
x^2+1)^(1/2)+I)/c^3/d^2/(c^2*x^2-1)*f*g^2+3*a*b*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/c^3/d^2/(c^2*x^2-1)*
arcsin(c*x)^2*f*g^2+2*I*a*b*(-c^2*x^2+1)^(1/2)*(-d*(c^2*x^2-1))^(1/2)/c/d^2/(c^2*x^2-1)*f^3*arcsin(c*x)-6*a*b*
(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/c^2/d^2/(c^2*x^2-1)*ln(I*c*x+(-c^2*x^2+1)^(1/2)-I)*f^2*g+I*b^2*(-d*(
c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/c/d^2/(c^2*x^2-1)*f^3*polylog(2,-(I*c*x+(-c^2*x^2+1)^(1/2))^2)-2*b^2*(-d*
(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/c^4/d^2/(c^2*x^2-1)*g^3*arcsin(c*x)*ln(1+I*(I*c*x+(-c^2*x^2+1)^(1/2)))+2
*b^2*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/c^4/d^2/(c^2*x^2-1)*g^3*arcsin(c*x)*ln(1-I*(I*c*x+(-c^2*x^2+1)^
(1/2)))-6*a*b*(-d*(c^2*x^2-1))^(1/2)/c^2/d^2/(c^2*x^2-1)*arcsin(c*x)*f^2*g+3*a^2*f^2*g/c^2/d/(-c^2*d*x^2+d)^(1
/2)-a^2*g^3*x^2/c^2/d/(-c^2*d*x^2+d)^(1/2)+2*b^2*(-d*(c^2*x^2-1))^(1/2)*g^3/c^4/d^2/(c^2*x^2-1)-2*a*b*(-d*(c^2
*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/c/d^2/(c^2*x^2-1)*ln(I*c*x+(-c^2*x^2+1)^(1/2)-I)*f^3-2*a*b*(-d*(c^2*x^2-1))^
(1/2)*(-c^2*x^2+1)^(1/2)/c^4/d^2/(c^2*x^2-1)*ln(I*c*x+(-c^2*x^2+1)^(1/2)-I)*g^3-2*a*b*(-d*(c^2*x^2-1))^(1/2)*(
-c^2*x^2+1)^(1/2)*ln(I*c*x+(-c^2*x^2+1)^(1/2)+I)/c/d^2/(c^2*x^2-1)*f^3+2*a*b*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+
1)^(1/2)*ln(I*c*x+(-c^2*x^2+1)^(1/2)+I)/c^4/d^2/(c^2*x^2-1)*g^3+2*a*b*(-d*(c^2*x^2-1))^(1/2)*g^3/c^3/d^2/(c^2*
x^2-1)*(-c^2*x^2+1)^(1/2)*x+2*a*b*(-d*(c^2*x^2-1))^(1/2)*g^3/c^2/d^2/(c^2*x^2-1)*arcsin(c*x)*x^2+b^2*(-d*(c^2*
x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/c^3/d^2/(c^2*x^2-1)*arcsin(c*x)^3*f*g^2+2*I*b^2*(-d*(c^2*x^2-1))^(1/2)*(-c^2*
x^2+1)^(1/2)/c^4/d^2/(c^2*x^2-1)*g^3*dilog(1+I*(I*c*x+(-c^2*x^2+1)^(1/2)))-2*I*b^2*(-d*(c^2*x^2-1))^(1/2)*(-c^
2*x^2+1)^(1/2)/c^4/d^2/(c^2*x^2-1)*g^3*dilog(1-I*(I*c*x+(-c^2*x^2+1)^(1/2)))+2*b^2*(-d*(c^2*x^2-1))^(1/2)*g^3/
c^3/d^2/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*arcsin(c*x)*x+I*b^2*(-d*(c^2*x^2-1))^(1/2)/c/d^2/(c^2*x^2-1)*arcsin(c*x
)^2*(-c^2*x^2+1)^(1/2)*f^3-3*b^2*(-d*(c^2*x^2-1))^(1/2)/c^2/d^2/(c^2*x^2-1)*arcsin(c*x)^2*x*f*g^2-2*b^2*(-d*(c
^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/c/d^2/(c^2*x^2-1)*f^3*arcsin(c*x)*ln(1+(I*c*x+(-c^2*x^2+1)^(1/2))^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*(a+b*arcsin(c*x))^2/(-c^2*d*x^2+d)^(3/2),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^{2} g^{3} x^{3} + 3 \, a^{2} f g^{2} x^{2} + 3 \, a^{2} f^{2} g x + a^{2} f^{3} +{\left (b^{2} g^{3} x^{3} + 3 \, b^{2} f g^{2} x^{2} + 3 \, b^{2} f^{2} g x + b^{2} f^{3}\right )} \arcsin \left (c x\right )^{2} + 2 \,{\left (a b g^{3} x^{3} + 3 \, a b f g^{2} x^{2} + 3 \, a b f^{2} g x + a b f^{3}\right )} \arcsin \left (c x\right )\right )} \sqrt{-c^{2} d x^{2} + d}}{c^{4} d^{2} x^{4} - 2 \, c^{2} d^{2} x^{2} + d^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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