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

Optimal. Leaf size=295 \[ -\frac{i b^2 \sqrt{1-c^2 x^2} \text{PolyLog}\left (2,-e^{2 i \sin ^{-1}(c x)}\right )}{c^3 d e \sqrt{c d x+d} \sqrt{e-c e x}}-\frac{\sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c^3 d e \sqrt{c d x+d} \sqrt{e-c e x}}-\frac{i \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{c^3 d e \sqrt{c d x+d} \sqrt{e-c e x}}+\frac{2 b \sqrt{1-c^2 x^2} \log \left (1+e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{c^3 d e \sqrt{c d x+d} \sqrt{e-c e x}}+\frac{x \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 d e \sqrt{c d x+d} \sqrt{e-c e x}} \]

[Out]

(x*(a + b*ArcSin[c*x])^2)/(c^2*d*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]) - (I*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])
^2)/(c^3*d*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]) - (Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^3)/(3*b*c^3*d*e*Sqrt[d
+ c*d*x]*Sqrt[e - c*e*x]) + (2*b*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Log[1 + E^((2*I)*ArcSin[c*x])])/(c^3*d*
e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]) - (I*b^2*Sqrt[1 - c^2*x^2]*PolyLog[2, -E^((2*I)*ArcSin[c*x])])/(c^3*d*e*Sqr
t[d + c*d*x]*Sqrt[e - c*e*x])

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Rubi [A]  time = 0.74251, antiderivative size = 295, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.229, Rules used = {4739, 4703, 4641, 4675, 3719, 2190, 2279, 2391} \[ -\frac{i b^2 \sqrt{1-c^2 x^2} \text{PolyLog}\left (2,-e^{2 i \sin ^{-1}(c x)}\right )}{c^3 d e \sqrt{c d x+d} \sqrt{e-c e x}}-\frac{\sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c^3 d e \sqrt{c d x+d} \sqrt{e-c e x}}-\frac{i \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{c^3 d e \sqrt{c d x+d} \sqrt{e-c e x}}+\frac{2 b \sqrt{1-c^2 x^2} \log \left (1+e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{c^3 d e \sqrt{c d x+d} \sqrt{e-c e x}}+\frac{x \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 d e \sqrt{c d x+d} \sqrt{e-c e x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(a + b*ArcSin[c*x])^2)/(c^2*d*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]) - (I*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])
^2)/(c^3*d*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]) - (Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^3)/(3*b*c^3*d*e*Sqrt[d
+ c*d*x]*Sqrt[e - c*e*x]) + (2*b*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Log[1 + E^((2*I)*ArcSin[c*x])])/(c^3*d*
e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]) - (I*b^2*Sqrt[1 - c^2*x^2]*PolyLog[2, -E^((2*I)*ArcSin[c*x])])/(c^3*d*e*Sqr
t[d + c*d*x]*Sqrt[e - c*e*x])

Rule 4739

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((h_.)*(x_))^(m_.)*((d_) + (e_.)*(x_))^(p_)*((f_) + (g_.)*(x_))^(
q_), x_Symbol] :> Dist[((-((d^2*g)/e))^IntPart[q]*(d + e*x)^FracPart[q]*(f + g*x)^FracPart[q])/(1 - c^2*x^2)^F
racPart[q], Int[(h*x)^m*(d + e*x)^(p - q)*(1 - c^2*x^2)^q*(a + b*ArcSin[c*x])^n, x], x] /; FreeQ[{a, b, c, d,
e, f, g, h, m, n}, x] && EqQ[e*f + d*g, 0] && EqQ[c^2*d^2 - e^2, 0] && HalfIntegerQ[p, q] && GeQ[p - q, 0]

Rule 4703

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

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 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]

Rubi steps

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

Mathematica [B]  time = 2.53003, size = 636, normalized size = 2.16 \[ \frac{b^2 \sqrt{d} e \left (-6 i \sqrt{1-c^2 x^2} \text{PolyLog}\left (2,-i e^{i \sin ^{-1}(c x)}\right )-6 i \sqrt{1-c^2 x^2} \text{PolyLog}\left (2,i e^{i \sin ^{-1}(c x)}\right )-\sqrt{1-c^2 x^2} \sin ^{-1}(c x)^3-3 i \sqrt{1-c^2 x^2} \sin ^{-1}(c x)^2+6 i \pi \sqrt{1-c^2 x^2} \sin ^{-1}(c x)+6 \sqrt{1-c^2 x^2} \sin ^{-1}(c x) \log \left (1-i e^{i \sin ^{-1}(c x)}\right )+6 \sqrt{1-c^2 x^2} \sin ^{-1}(c x) \log \left (1+i e^{i \sin ^{-1}(c x)}\right )+12 \pi \sqrt{1-c^2 x^2} \log \left (1+e^{-i \sin ^{-1}(c x)}\right )+3 \pi \sqrt{1-c^2 x^2} \log \left (1-i e^{i \sin ^{-1}(c x)}\right )-3 \pi \sqrt{1-c^2 x^2} \log \left (1+i e^{i \sin ^{-1}(c x)}\right )-3 \pi \sqrt{1-c^2 x^2} \log \left (\sin \left (\frac{1}{4} \left (2 \sin ^{-1}(c x)+\pi \right )\right )\right )-12 \pi \sqrt{1-c^2 x^2} \log \left (\cos \left (\frac{1}{2} \sin ^{-1}(c x)\right )\right )+3 \pi \sqrt{1-c^2 x^2} \log \left (-\cos \left (\frac{1}{4} \left (2 \sin ^{-1}(c x)+\pi \right )\right )\right )+3 c x \sin ^{-1}(c x)^2\right )+3 a^2 \sqrt{e} \sqrt{c d x+d} \sqrt{e-c e x} \tan ^{-1}\left (\frac{c x \sqrt{c d x+d} \sqrt{e-c e x}}{\sqrt{d} \sqrt{e} \left (c^2 x^2-1\right )}\right )+3 a^2 c \sqrt{d} e x+3 a b \sqrt{d} e \left (\sqrt{1-c^2 x^2} \left (2 \left (\log \left (\cos \left (\frac{1}{2} \sin ^{-1}(c x)\right )-\sin \left (\frac{1}{2} \sin ^{-1}(c x)\right )\right )+\log \left (\sin \left (\frac{1}{2} \sin ^{-1}(c x)\right )+\cos \left (\frac{1}{2} \sin ^{-1}(c x)\right )\right )\right )-\sin ^{-1}(c x)^2\right )+2 c x \sin ^{-1}(c x)\right )}{3 c^3 d^{3/2} e^2 \sqrt{c d x+d} \sqrt{e-c e x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*a^2*c*Sqrt[d]*e*x + 3*a^2*Sqrt[e]*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*ArcTan[(c*x*Sqrt[d + c*d*x]*Sqrt[e - c*e*
x])/(Sqrt[d]*Sqrt[e]*(-1 + c^2*x^2))] + 3*a*b*Sqrt[d]*e*(2*c*x*ArcSin[c*x] + Sqrt[1 - c^2*x^2]*(-ArcSin[c*x]^2
+ 2*(Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]] + Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]]))) + b^2*Sqr
t[d]*e*((6*I)*Pi*Sqrt[1 - c^2*x^2]*ArcSin[c*x] + 3*c*x*ArcSin[c*x]^2 - (3*I)*Sqrt[1 - c^2*x^2]*ArcSin[c*x]^2 -
Sqrt[1 - c^2*x^2]*ArcSin[c*x]^3 + 12*Pi*Sqrt[1 - c^2*x^2]*Log[1 + E^((-I)*ArcSin[c*x])] + 3*Pi*Sqrt[1 - c^2*x
^2]*Log[1 - I*E^(I*ArcSin[c*x])] + 6*Sqrt[1 - c^2*x^2]*ArcSin[c*x]*Log[1 - I*E^(I*ArcSin[c*x])] - 3*Pi*Sqrt[1
- c^2*x^2]*Log[1 + I*E^(I*ArcSin[c*x])] + 6*Sqrt[1 - c^2*x^2]*ArcSin[c*x]*Log[1 + I*E^(I*ArcSin[c*x])] - 12*Pi
*Sqrt[1 - c^2*x^2]*Log[Cos[ArcSin[c*x]/2]] + 3*Pi*Sqrt[1 - c^2*x^2]*Log[-Cos[(Pi + 2*ArcSin[c*x])/4]] - 3*Pi*S
qrt[1 - c^2*x^2]*Log[Sin[(Pi + 2*ArcSin[c*x])/4]] - (6*I)*Sqrt[1 - c^2*x^2]*PolyLog[2, (-I)*E^(I*ArcSin[c*x])]
- (6*I)*Sqrt[1 - c^2*x^2]*PolyLog[2, I*E^(I*ArcSin[c*x])]))/(3*c^3*d^(3/2)*e^2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x
])

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Maple [F]  time = 0.338, size = 0, normalized size = 0. \begin{align*} \int{{x}^{2} \left ( a+b\arcsin \left ( cx \right ) \right ) ^{2} \left ( cdx+d \right ) ^{-{\frac{3}{2}}} \left ( -cex+e \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(a+b*arcsin(c*x))^2/(c*d*x+d)^(3/2)/(-c*e*x+e)^(3/2),x)

[Out]

int(x^2*(a+b*arcsin(c*x))^2/(c*d*x+d)^(3/2)/(-c*e*x+e)^(3/2),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(x^2*(a+b*arcsin(c*x))^2/(c*d*x+d)^(3/2)/(-c*e*x+e)^(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 (b^{2} x^{2} \arcsin \left (c x\right )^{2} + 2 \, a b x^{2} \arcsin \left (c x\right ) + a^{2} x^{2}\right )} \sqrt{c d x + d} \sqrt{-c e x + e}}{c^{4} d^{2} e^{2} x^{4} - 2 \, c^{2} d^{2} e^{2} x^{2} + d^{2} e^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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