3.255 \(\int \frac{x^4 (a+b \sin ^{-1}(c x))^2}{(d-c^2 d x^2)^{5/2}} \, dx\)

Optimal. Leaf size=421 \[ \frac{4 i b^2 \sqrt{1-c^2 x^2} \text{PolyLog}\left (2,-e^{2 i \sin ^{-1}(c x)}\right )}{3 c^5 d^2 \sqrt{d-c^2 d x^2}}-\frac{b x^2 \left (a+b \sin ^{-1}(c x)\right )}{3 c^3 d^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}}-\frac{x \left (a+b \sin ^{-1}(c x)\right )^2}{c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{\sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c^5 d^2 \sqrt{d-c^2 d x^2}}+\frac{4 i \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^5 d^2 \sqrt{d-c^2 d x^2}}-\frac{8 b \sqrt{1-c^2 x^2} \log \left (1+e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 c^5 d^2 \sqrt{d-c^2 d x^2}}+\frac{x^3 \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}+\frac{b^2 x}{3 c^4 d^2 \sqrt{d-c^2 d x^2}}-\frac{b^2 \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{3 c^5 d^2 \sqrt{d-c^2 d x^2}} \]

[Out]

(b^2*x)/(3*c^4*d^2*Sqrt[d - c^2*d*x^2]) - (b^2*Sqrt[1 - c^2*x^2]*ArcSin[c*x])/(3*c^5*d^2*Sqrt[d - c^2*d*x^2])
- (b*x^2*(a + b*ArcSin[c*x]))/(3*c^3*d^2*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2]) + (x^3*(a + b*ArcSin[c*x])^2)/
(3*c^2*d*(d - c^2*d*x^2)^(3/2)) - (x*(a + b*ArcSin[c*x])^2)/(c^4*d^2*Sqrt[d - c^2*d*x^2]) + (((4*I)/3)*Sqrt[1
- c^2*x^2]*(a + b*ArcSin[c*x])^2)/(c^5*d^2*Sqrt[d - c^2*d*x^2]) + (Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^3)/(3
*b*c^5*d^2*Sqrt[d - c^2*d*x^2]) - (8*b*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Log[1 + E^((2*I)*ArcSin[c*x])])/(
3*c^5*d^2*Sqrt[d - c^2*d*x^2]) + (((4*I)/3)*b^2*Sqrt[1 - c^2*x^2]*PolyLog[2, -E^((2*I)*ArcSin[c*x])])/(c^5*d^2
*Sqrt[d - c^2*d*x^2])

________________________________________________________________________________________

Rubi [A]  time = 0.725995, antiderivative size = 421, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 10, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.345, Rules used = {4703, 4643, 4641, 4675, 3719, 2190, 2279, 2391, 288, 216} \[ \frac{4 i b^2 \sqrt{1-c^2 x^2} \text{PolyLog}\left (2,-e^{2 i \sin ^{-1}(c x)}\right )}{3 c^5 d^2 \sqrt{d-c^2 d x^2}}-\frac{b x^2 \left (a+b \sin ^{-1}(c x)\right )}{3 c^3 d^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}}-\frac{x \left (a+b \sin ^{-1}(c x)\right )^2}{c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{\sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c^5 d^2 \sqrt{d-c^2 d x^2}}+\frac{4 i \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^5 d^2 \sqrt{d-c^2 d x^2}}-\frac{8 b \sqrt{1-c^2 x^2} \log \left (1+e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 c^5 d^2 \sqrt{d-c^2 d x^2}}+\frac{x^3 \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}+\frac{b^2 x}{3 c^4 d^2 \sqrt{d-c^2 d x^2}}-\frac{b^2 \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{3 c^5 d^2 \sqrt{d-c^2 d x^2}} \]

Antiderivative was successfully verified.

[In]

Int[(x^4*(a + b*ArcSin[c*x])^2)/(d - c^2*d*x^2)^(5/2),x]

[Out]

(b^2*x)/(3*c^4*d^2*Sqrt[d - c^2*d*x^2]) - (b^2*Sqrt[1 - c^2*x^2]*ArcSin[c*x])/(3*c^5*d^2*Sqrt[d - c^2*d*x^2])
- (b*x^2*(a + b*ArcSin[c*x]))/(3*c^3*d^2*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2]) + (x^3*(a + b*ArcSin[c*x])^2)/
(3*c^2*d*(d - c^2*d*x^2)^(3/2)) - (x*(a + b*ArcSin[c*x])^2)/(c^4*d^2*Sqrt[d - c^2*d*x^2]) + (((4*I)/3)*Sqrt[1
- c^2*x^2]*(a + b*ArcSin[c*x])^2)/(c^5*d^2*Sqrt[d - c^2*d*x^2]) + (Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^3)/(3
*b*c^5*d^2*Sqrt[d - c^2*d*x^2]) - (8*b*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Log[1 + E^((2*I)*ArcSin[c*x])])/(
3*c^5*d^2*Sqrt[d - c^2*d*x^2]) + (((4*I)/3)*b^2*Sqrt[1 - c^2*x^2]*PolyLog[2, -E^((2*I)*ArcSin[c*x])])/(c^5*d^2
*Sqrt[d - c^2*d*x^2])

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 4643

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Dist[Sqrt[1 - c^2*x^2]/Sq
rt[d + e*x^2], Int[(a + b*ArcSin[c*x])^n/Sqrt[1 - c^2*x^2], x], x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^2*
d + e, 0] &&  !GtQ[d, 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 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 288

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^
n)^(p + 1))/(b*n*(p + 1)), x] - Dist[(c^n*(m - n + 1))/(b*n*(p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x
], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] &&  !ILtQ[(m + n*(p + 1) + 1)/n, 0]
&& IntBinomialQ[a, b, c, n, m, p, x]

Rule 216

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

Rubi steps

\begin{align*} \int \frac{x^4 \left (a+b \sin ^{-1}(c x)\right )^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx &=\frac{x^3 \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}-\frac{\int \frac{x^2 \left (a+b \sin ^{-1}(c x)\right )^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx}{c^2 d}-\frac{\left (2 b \sqrt{1-c^2 x^2}\right ) \int \frac{x^3 \left (a+b \sin ^{-1}(c x)\right )}{\left (1-c^2 x^2\right )^2} \, dx}{3 c d^2 \sqrt{d-c^2 d x^2}}\\ &=-\frac{b x^2 \left (a+b \sin ^{-1}(c x)\right )}{3 c^3 d^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}}+\frac{x^3 \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}-\frac{x \left (a+b \sin ^{-1}(c x)\right )^2}{c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{\int \frac{\left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt{d-c^2 d x^2}} \, dx}{c^4 d^2}+\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}{3 c^3 d^2 \sqrt{d-c^2 d x^2}}+\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^3 d^2 \sqrt{d-c^2 d x^2}}+\frac{\left (b^2 \sqrt{1-c^2 x^2}\right ) \int \frac{x^2}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{3 c^2 d^2 \sqrt{d-c^2 d x^2}}\\ &=\frac{b^2 x}{3 c^4 d^2 \sqrt{d-c^2 d x^2}}-\frac{b x^2 \left (a+b \sin ^{-1}(c x)\right )}{3 c^3 d^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}}+\frac{x^3 \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}-\frac{x \left (a+b \sin ^{-1}(c x)\right )^2}{c^4 d^2 \sqrt{d-c^2 d x^2}}+\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 )}{3 c^5 d^2 \sqrt{d-c^2 d x^2}}+\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^5 d^2 \sqrt{d-c^2 d x^2}}+\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^4 d^2 \sqrt{d-c^2 d x^2}}-\frac{\left (b^2 \sqrt{1-c^2 x^2}\right ) \int \frac{1}{\sqrt{1-c^2 x^2}} \, dx}{3 c^4 d^2 \sqrt{d-c^2 d x^2}}\\ &=\frac{b^2 x}{3 c^4 d^2 \sqrt{d-c^2 d x^2}}-\frac{b^2 \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{3 c^5 d^2 \sqrt{d-c^2 d x^2}}-\frac{b x^2 \left (a+b \sin ^{-1}(c x)\right )}{3 c^3 d^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}}+\frac{x^3 \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}-\frac{x \left (a+b \sin ^{-1}(c x)\right )^2}{c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{4 i \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^5 d^2 \sqrt{d-c^2 d x^2}}+\frac{\sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c^5 d^2 \sqrt{d-c^2 d x^2}}-\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 )}{3 c^5 d^2 \sqrt{d-c^2 d x^2}}-\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^5 d^2 \sqrt{d-c^2 d x^2}}\\ &=\frac{b^2 x}{3 c^4 d^2 \sqrt{d-c^2 d x^2}}-\frac{b^2 \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{3 c^5 d^2 \sqrt{d-c^2 d x^2}}-\frac{b x^2 \left (a+b \sin ^{-1}(c x)\right )}{3 c^3 d^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}}+\frac{x^3 \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}-\frac{x \left (a+b \sin ^{-1}(c x)\right )^2}{c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{4 i \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^5 d^2 \sqrt{d-c^2 d x^2}}+\frac{\sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c^5 d^2 \sqrt{d-c^2 d x^2}}-\frac{8 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 )}{3 c^5 d^2 \sqrt{d-c^2 d x^2}}+\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 )}{3 c^5 d^2 \sqrt{d-c^2 d x^2}}+\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^5 d^2 \sqrt{d-c^2 d x^2}}\\ &=\frac{b^2 x}{3 c^4 d^2 \sqrt{d-c^2 d x^2}}-\frac{b^2 \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{3 c^5 d^2 \sqrt{d-c^2 d x^2}}-\frac{b x^2 \left (a+b \sin ^{-1}(c x)\right )}{3 c^3 d^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}}+\frac{x^3 \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}-\frac{x \left (a+b \sin ^{-1}(c x)\right )^2}{c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{4 i \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^5 d^2 \sqrt{d-c^2 d x^2}}+\frac{\sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c^5 d^2 \sqrt{d-c^2 d x^2}}-\frac{8 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 )}{3 c^5 d^2 \sqrt{d-c^2 d x^2}}-\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 )}{3 c^5 d^2 \sqrt{d-c^2 d x^2}}-\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^5 d^2 \sqrt{d-c^2 d x^2}}\\ &=\frac{b^2 x}{3 c^4 d^2 \sqrt{d-c^2 d x^2}}-\frac{b^2 \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{3 c^5 d^2 \sqrt{d-c^2 d x^2}}-\frac{b x^2 \left (a+b \sin ^{-1}(c x)\right )}{3 c^3 d^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}}+\frac{x^3 \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}-\frac{x \left (a+b \sin ^{-1}(c x)\right )^2}{c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{4 i \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^5 d^2 \sqrt{d-c^2 d x^2}}+\frac{\sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c^5 d^2 \sqrt{d-c^2 d x^2}}-\frac{8 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 )}{3 c^5 d^2 \sqrt{d-c^2 d x^2}}+\frac{4 i b^2 \sqrt{1-c^2 x^2} \text{Li}_2\left (-e^{2 i \sin ^{-1}(c x)}\right )}{3 c^5 d^2 \sqrt{d-c^2 d x^2}}\\ \end{align*}

Mathematica [A]  time = 1.48498, size = 374, normalized size = 0.89 \[ \frac{b^2 \sqrt{d} \left (4 i \left (1-c^2 x^2\right )^{3/2} \text{PolyLog}\left (2,-e^{2 i \sin ^{-1}(c x)}\right )-c^3 x^3+4 c^3 x^3 \sin ^{-1}(c x)^2+\left (1-c^2 x^2\right )^{3/2} \sin ^{-1}(c x)^3+4 i \left (1-c^2 x^2\right )^{3/2} \sin ^{-1}(c x)^2-\sqrt{1-c^2 x^2} \sin ^{-1}(c x)-8 \left (1-c^2 x^2\right )^{3/2} \sin ^{-1}(c x) \log \left (1+e^{2 i \sin ^{-1}(c x)}\right )+c x-3 c x \sin ^{-1}(c x)^2\right )+a^2 c \sqrt{d} x \left (4 c^2 x^2-3\right )+3 a^2 \left (c^2 x^2-1\right ) \sqrt{d-c^2 d x^2} \tan ^{-1}\left (\frac{c x \sqrt{d-c^2 d x^2}}{\sqrt{d} \left (c^2 x^2-1\right )}\right )-a b \sqrt{d} \left (\sqrt{1-c^2 x^2}+\left (1-c^2 x^2\right )^{3/2} \left (4 \log \left (1-c^2 x^2\right )-3 \sin ^{-1}(c x)^2\right )+2 \sin ^{-1}(c x) \sin \left (3 \sin ^{-1}(c x)\right )\right )}{3 c^5 d^{5/2} \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(x^4*(a + b*ArcSin[c*x])^2)/(d - c^2*d*x^2)^(5/2),x]

[Out]

(a^2*c*Sqrt[d]*x*(-3 + 4*c^2*x^2) + 3*a^2*(-1 + c^2*x^2)*Sqrt[d - c^2*d*x^2]*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/
(Sqrt[d]*(-1 + c^2*x^2))] + b^2*Sqrt[d]*(c*x - c^3*x^3 - Sqrt[1 - c^2*x^2]*ArcSin[c*x] - 3*c*x*ArcSin[c*x]^2 +
4*c^3*x^3*ArcSin[c*x]^2 + (4*I)*(1 - c^2*x^2)^(3/2)*ArcSin[c*x]^2 + (1 - c^2*x^2)^(3/2)*ArcSin[c*x]^3 - 8*(1
- c^2*x^2)^(3/2)*ArcSin[c*x]*Log[1 + E^((2*I)*ArcSin[c*x])] + (4*I)*(1 - c^2*x^2)^(3/2)*PolyLog[2, -E^((2*I)*A
rcSin[c*x])]) - a*b*Sqrt[d]*(Sqrt[1 - c^2*x^2] + (1 - c^2*x^2)^(3/2)*(-3*ArcSin[c*x]^2 + 4*Log[1 - c^2*x^2]) +
2*ArcSin[c*x]*Sin[3*ArcSin[c*x]]))/(3*c^5*d^(5/2)*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2])

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Maple [B]  time = 0.556, size = 3907, normalized size = 9.3 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(a+b*arcsin(c*x))^2/(-c^2*d*x^2+d)^(5/2),x)

[Out]

-a^2/c^4/d^2*x/(-c^2*d*x^2+d)^(1/2)+1/3*a^2*x^3/c^2/d/(-c^2*d*x^2+d)^(3/2)+17*b^2*(-d*(c^2*x^2-1))^(1/2)/d^3/(
24*c^8*x^8-87*c^6*x^6+118*c^4*x^4-71*c^2*x^2+16)*x^5+13*b^2*(-d*(c^2*x^2-1))^(1/2)/d^3/(24*c^8*x^8-87*c^6*x^6+
118*c^4*x^4-71*c^2*x^2+16)/c^3*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*x^2+64*I*a*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(24*c^8*
x^8-87*c^6*x^6+118*c^4*x^4-71*c^2*x^2+16)*c*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*x^6-168*I*a*b*(-d*(c^2*x^2-1))^(1/2
)/d^3/(24*c^8*x^8-87*c^6*x^6+118*c^4*x^4-71*c^2*x^2+16)/c*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*x^4-40/3*I*b^2*(-d*(c
^2*x^2-1))^(1/2)/d^3/(24*c^8*x^8-87*c^6*x^6+118*c^4*x^4-71*c^2*x^2+16)/c^2*arcsin(c*x)*x^3-55/3*I*b^2*(-d*(c^2
*x^2-1))^(1/2)/d^3/(24*c^8*x^8-87*c^6*x^6+118*c^4*x^4-71*c^2*x^2+16)/c^3*x^2*(-c^2*x^2+1)^(1/2)-64/3*I*b^2*(-d
*(c^2*x^2-1))^(1/2)/d^3/(24*c^8*x^8-87*c^6*x^6+118*c^4*x^4-71*c^2*x^2+16)/c^5*arcsin(c*x)^2*(-c^2*x^2+1)^(1/2)
-16/3*I*b^2*(-d*(c^2*x^2-1))^(1/2)/d^3/(24*c^8*x^8-87*c^6*x^6+118*c^4*x^4-71*c^2*x^2+16)*c^2*arcsin(c*x)*x^7+8
/3*b^2*(-c^2*x^2+1)^(1/2)*(-d*(c^2*x^2-1))^(1/2)/c^5/d^3/(c^2*x^2-1)*arcsin(c*x)*ln(1+(I*c*x+(-c^2*x^2+1)^(1/2
))^2)-8*b^2*(-d*(c^2*x^2-1))^(1/2)/d^3/(24*c^8*x^8-87*c^6*x^6+118*c^4*x^4-71*c^2*x^2+16)/c*arcsin(c*x)*(-c^2*x
^2+1)^(1/2)*x^4+21*I*b^2*(-d*(c^2*x^2-1))^(1/2)/d^3/(24*c^8*x^8-87*c^6*x^6+118*c^4*x^4-71*c^2*x^2+16)/c*(-c^2*
x^2+1)^(1/2)*x^4+362/3*a*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(24*c^8*x^8-87*c^6*x^6+118*c^4*x^4-71*c^2*x^2+16)/c^2*ar
csin(c*x)*x^3+13*a*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(24*c^8*x^8-87*c^6*x^6+118*c^4*x^4-71*c^2*x^2+16)/c^3*x^2*(-c^
2*x^2+1)^(1/2)-32*a*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(24*c^8*x^8-87*c^6*x^6+118*c^4*x^4-71*c^2*x^2+16)/c^4*arcsin(
c*x)*x+8/3*a*b*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/c^5/d^3/(c^2*x^2-1)*ln(1+(I*c*x+(-c^2*x^2+1)^(1/2))^2
)-a*b*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/c^5/d^3/(c^2*x^2-1)*arcsin(c*x)^2-16/3*I*a*b*(-d*(c^2*x^2-1))^
(1/2)/d^3/(24*c^8*x^8-87*c^6*x^6+118*c^4*x^4-71*c^2*x^2+16)*c^2*x^7-40/3*I*a*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(24*
c^8*x^8-87*c^6*x^6+118*c^4*x^4-71*c^2*x^2+16)/c^2*x^3+4*I*a*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(24*c^8*x^8-87*c^6*x^
6+118*c^4*x^4-71*c^2*x^2+16)/c^4*x-16/3*I*a*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(24*c^8*x^8-87*c^6*x^6+118*c^4*x^4-71
*c^2*x^2+16)*(-c^2*x^2+1)*x^5+64*a*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(24*c^8*x^8-87*c^6*x^6+118*c^4*x^4-71*c^2*x^2+
16)*c^2*arcsin(c*x)*x^7-8*a*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(24*c^8*x^8-87*c^6*x^6+118*c^4*x^4-71*c^2*x^2+16)/c*(
-c^2*x^2+1)^(1/2)*x^4+4*I*b^2*(-d*(c^2*x^2-1))^(1/2)/d^3/(24*c^8*x^8-87*c^6*x^6+118*c^4*x^4-71*c^2*x^2+16)/c^4
*arcsin(c*x)*x-8/3*I*b^2*(-c^2*x^2+1)^(1/2)*(-d*(c^2*x^2-1))^(1/2)/c^5/d^3/(c^2*x^2-1)*arcsin(c*x)^2-4/3*I*b^2
*(-c^2*x^2+1)^(1/2)*(-d*(c^2*x^2-1))^(1/2)/c^5/d^3/(c^2*x^2-1)*polylog(2,-(I*c*x+(-c^2*x^2+1)^(1/2))^2)-16/3*I
*b^2*(-d*(c^2*x^2-1))^(1/2)/d^3/(24*c^8*x^8-87*c^6*x^6+118*c^4*x^4-71*c^2*x^2+16)*arcsin(c*x)*(-c^2*x^2+1)*x^5
-8*I*b^2*(-d*(c^2*x^2-1))^(1/2)/d^3/(24*c^8*x^8-87*c^6*x^6+118*c^4*x^4-71*c^2*x^2+16)*c*(-c^2*x^2+1)^(1/2)*x^6
+220/3*I*b^2*(-d*(c^2*x^2-1))^(1/2)/d^3/(24*c^8*x^8-87*c^6*x^6+118*c^4*x^4-71*c^2*x^2+16)/c^3*arcsin(c*x)^2*(-
c^2*x^2+1)^(1/2)*x^2-4*I*b^2*(-d*(c^2*x^2-1))^(1/2)/d^3/(24*c^8*x^8-87*c^6*x^6+118*c^4*x^4-71*c^2*x^2+16)/c^4*
arcsin(c*x)*(-c^2*x^2+1)*x+32*I*b^2*(-d*(c^2*x^2-1))^(1/2)/d^3/(24*c^8*x^8-87*c^6*x^6+118*c^4*x^4-71*c^2*x^2+1
6)*c*arcsin(c*x)^2*(-c^2*x^2+1)^(1/2)*x^6+28/3*I*a*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(24*c^8*x^8-87*c^6*x^6+118*c^4
*x^4-71*c^2*x^2+16)/c^2*(-c^2*x^2+1)*x^3-4*I*a*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(24*c^8*x^8-87*c^6*x^6+118*c^4*x^4
-71*c^2*x^2+16)/c^4*(-c^2*x^2+1)*x-128/3*I*a*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(24*c^8*x^8-87*c^6*x^6+118*c^4*x^4-7
1*c^2*x^2+16)/c^5*arcsin(c*x)*(-c^2*x^2+1)^(1/2)-16/3*I*a*b*(-c^2*x^2+1)^(1/2)*(-d*(c^2*x^2-1))^(1/2)/c^5/d^3/
(c^2*x^2-1)*arcsin(c*x)-84*I*b^2*(-d*(c^2*x^2-1))^(1/2)/d^3/(24*c^8*x^8-87*c^6*x^6+118*c^4*x^4-71*c^2*x^2+16)/
c*arcsin(c*x)^2*(-c^2*x^2+1)^(1/2)*x^4+28/3*I*b^2*(-d*(c^2*x^2-1))^(1/2)/d^3/(24*c^8*x^8-87*c^6*x^6+118*c^4*x^
4-71*c^2*x^2+16)/c^2*arcsin(c*x)*(-c^2*x^2+1)*x^3+440/3*I*a*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(24*c^8*x^8-87*c^6*x^
6+118*c^4*x^4-71*c^2*x^2+16)/c^3*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*x^2+4*b^2*(-d*(c^2*x^2-1))^(1/2)/d^3/(24*c^8*x
^8-87*c^6*x^6+118*c^4*x^4-71*c^2*x^2+16)/c^4*x+4/3*b^2*(-d*(c^2*x^2-1))^(1/2)/d^3/(24*c^8*x^8-87*c^6*x^6+118*c
^4*x^4-71*c^2*x^2+16)*(-c^2*x^2+1)*x^5-76*b^2*(-d*(c^2*x^2-1))^(1/2)/d^3/(24*c^8*x^8-87*c^6*x^6+118*c^4*x^4-71
*c^2*x^2+16)*arcsin(c*x)^2*x^5-20/3*b^2*(-d*(c^2*x^2-1))^(1/2)/d^3/(24*c^8*x^8-87*c^6*x^6+118*c^4*x^4-71*c^2*x
^2+16)*c^2*x^7-43/3*b^2*(-d*(c^2*x^2-1))^(1/2)/d^3/(24*c^8*x^8-87*c^6*x^6+118*c^4*x^4-71*c^2*x^2+16)/c^2*x^3+a
^2/c^4/d^2/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2))-8/3*b^2*(-d*(c^2*x^2-1))^(1/2)/d^3/(24*c
^8*x^8-87*c^6*x^6+118*c^4*x^4-71*c^2*x^2+16)/c^2*(-c^2*x^2+1)*x^3+4/3*b^2*(-d*(c^2*x^2-1))^(1/2)/d^3/(24*c^8*x
^8-87*c^6*x^6+118*c^4*x^4-71*c^2*x^2+16)/c^4*(-c^2*x^2+1)*x-16/3*b^2*(-d*(c^2*x^2-1))^(1/2)/d^3/(24*c^8*x^8-87
*c^6*x^6+118*c^4*x^4-71*c^2*x^2+16)/c^5*arcsin(c*x)*(-c^2*x^2+1)^(1/2)+32*b^2*(-d*(c^2*x^2-1))^(1/2)/d^3/(24*c
^8*x^8-87*c^6*x^6+118*c^4*x^4-71*c^2*x^2+16)*c^2*arcsin(c*x)^2*x^7+181/3*b^2*(-d*(c^2*x^2-1))^(1/2)/d^3/(24*c^
8*x^8-87*c^6*x^6+118*c^4*x^4-71*c^2*x^2+16)/c^2*arcsin(c*x)^2*x^3-16*b^2*(-d*(c^2*x^2-1))^(1/2)/d^3/(24*c^8*x^
8-87*c^6*x^6+118*c^4*x^4-71*c^2*x^2+16)/c^4*arcsin(c*x)^2*x-1/3*b^2*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/
c^5/d^3/(c^2*x^2-1)*arcsin(c*x)^3+16/3*I*b^2*(-d*(c^2*x^2-1))^(1/2)/d^3/(24*c^8*x^8-87*c^6*x^6+118*c^4*x^4-71*
c^2*x^2+16)/c^5*(-c^2*x^2+1)^(1/2)+44/3*I*b^2*(-d*(c^2*x^2-1))^(1/2)/d^3/(24*c^8*x^8-87*c^6*x^6+118*c^4*x^4-71
*c^2*x^2+16)*arcsin(c*x)*x^5-16/3*a*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(24*c^8*x^8-87*c^6*x^6+118*c^4*x^4-71*c^2*x^2
+16)/c^5*(-c^2*x^2+1)^(1/2)-152*a*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(24*c^8*x^8-87*c^6*x^6+118*c^4*x^4-71*c^2*x^2+1
6)*arcsin(c*x)*x^5+44/3*I*a*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(24*c^8*x^8-87*c^6*x^6+118*c^4*x^4-71*c^2*x^2+16)*x^5

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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^4*(a+b*arcsin(c*x))^2/(-c^2*d*x^2+d)^(5/2),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^{4} \arcsin \left (c x\right )^{2} + 2 \, a b x^{4} \arcsin \left (c x\right ) + a^{2} x^{4}\right )} \sqrt{-c^{2} d x^{2} + d}}{c^{6} d^{3} x^{6} - 3 \, c^{4} d^{3} x^{4} + 3 \, c^{2} d^{3} x^{2} - d^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(a+b*arcsin(c*x))^2/(-c^2*d*x^2+d)^(5/2),x, algorithm="fricas")

[Out]

integral(-(b^2*x^4*arcsin(c*x)^2 + 2*a*b*x^4*arcsin(c*x) + a^2*x^4)*sqrt(-c^2*d*x^2 + d)/(c^6*d^3*x^6 - 3*c^4*
d^3*x^4 + 3*c^2*d^3*x^2 - d^3), 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**4*(a+b*asin(c*x))**2/(-c**2*d*x**2+d)**(5/2),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^{4}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(a+b*arcsin(c*x))^2/(-c^2*d*x^2+d)^(5/2),x, algorithm="giac")

[Out]

integrate((b*arcsin(c*x) + a)^2*x^4/(-c^2*d*x^2 + d)^(5/2), x)