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

Optimal. Leaf size=559 \[ \frac{i b d \text{PolyLog}\left (2,-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{-\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{2 e^2}+\frac{i b d \text{PolyLog}\left (2,\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{-\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{2 e^2}+\frac{i b d \text{PolyLog}\left (2,-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{2 e^2}+\frac{i b d \text{PolyLog}\left (2,\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{2 e^2}-\frac{d \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{-\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{2 e^2}-\frac{d \left (a+b \sin ^{-1}(c x)\right ) \log \left (1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{-\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{2 e^2}-\frac{d \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{2 e^2}-\frac{d \left (a+b \sin ^{-1}(c x)\right ) \log \left (1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{2 e^2}+\frac{i d \left (a+b \sin ^{-1}(c x)\right )^2}{2 b e^2}+\frac{x^2 \left (a+b \sin ^{-1}(c x)\right )}{2 e}+\frac{b x \sqrt{1-c^2 x^2}}{4 c e}-\frac{b \sin ^{-1}(c x)}{4 c^2 e} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.910711, antiderivative size = 559, normalized size of antiderivative = 1., number of steps used = 23, number of rules used = 9, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {4733, 4627, 321, 216, 4741, 4521, 2190, 2279, 2391} \[ \frac{i b d \text{PolyLog}\left (2,-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{-\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{2 e^2}+\frac{i b d \text{PolyLog}\left (2,\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{-\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{2 e^2}+\frac{i b d \text{PolyLog}\left (2,-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{2 e^2}+\frac{i b d \text{PolyLog}\left (2,\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{2 e^2}-\frac{d \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{-\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{2 e^2}-\frac{d \left (a+b \sin ^{-1}(c x)\right ) \log \left (1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{-\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{2 e^2}-\frac{d \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{2 e^2}-\frac{d \left (a+b \sin ^{-1}(c x)\right ) \log \left (1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{2 e^2}+\frac{i d \left (a+b \sin ^{-1}(c x)\right )^2}{2 b e^2}+\frac{x^2 \left (a+b \sin ^{-1}(c x)\right )}{2 e}+\frac{b x \sqrt{1-c^2 x^2}}{4 c e}-\frac{b \sin ^{-1}(c x)}{4 c^2 e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 4733

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Int[
ExpandIntegrand[(a + b*ArcSin[c*x])^n, (f*x)^m*(d + e*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[c^
2*d + e, 0] && IGtQ[n, 0] && IntegerQ[p] && IntegerQ[m]

Rule 4627

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

Rule 321

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*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 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]

Rule 4741

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

Rule 4521

Int[(Cos[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sin[(c_.) + (d_.)*(x_)]), x_Symbol] :>
-Simp[(I*(e + f*x)^(m + 1))/(b*f*(m + 1)), x] + (Dist[I, Int[((e + f*x)^m*E^(I*(c + d*x)))/(I*a - Rt[-a^2 + b^
2, 2] + b*E^(I*(c + d*x))), x], x] + Dist[I, Int[((e + f*x)^m*E^(I*(c + d*x)))/(I*a + Rt[-a^2 + b^2, 2] + b*E^
(I*(c + d*x))), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && NegQ[a^2 - b^2]

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

Mathematica [A]  time = 0.35845, size = 454, normalized size = 0.81 \[ \frac{b \left (i c^2 d \left (2 \text{PolyLog}\left (2,\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}-c \sqrt{d}}\right )+2 \text{PolyLog}\left (2,-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+c \sqrt{d}}\right )+\sin ^{-1}(c x) \left (\sin ^{-1}(c x)+2 i \left (\log \left (1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{c \sqrt{d}-\sqrt{c^2 d+e}}\right )+\log \left (1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+c \sqrt{d}}\right )\right )\right )\right )+i c^2 d \left (2 \text{PolyLog}\left (2,\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{c \sqrt{d}-\sqrt{c^2 d+e}}\right )+2 \text{PolyLog}\left (2,\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+c \sqrt{d}}\right )+\sin ^{-1}(c x) \left (\sin ^{-1}(c x)+2 i \left (\log \left (1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}-c \sqrt{d}}\right )+\log \left (1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+c \sqrt{d}}\right )\right )\right )\right )+e \left (c x \sqrt{1-c^2 x^2}+\left (2 c^2 x^2-1\right ) \sin ^{-1}(c x)\right )\right )-2 a c^2 d \log \left (d+e x^2\right )+2 a c^2 e x^2}{4 c^2 e^2} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(2*a*c^2*e*x^2 - 2*a*c^2*d*Log[d + e*x^2] + b*(e*(c*x*Sqrt[1 - c^2*x^2] + (-1 + 2*c^2*x^2)*ArcSin[c*x]) + I*c^
2*d*(ArcSin[c*x]*(ArcSin[c*x] + (2*I)*(Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] - Sqrt[c^2*d + e])] + Lo
g[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] + Sqrt[c^2*d + e])])) + 2*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/
(-(c*Sqrt[d]) + Sqrt[c^2*d + e])] + 2*PolyLog[2, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] + Sqrt[c^2*d + e]))]
) + I*c^2*d*(ArcSin[c*x]*(ArcSin[c*x] + (2*I)*(Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(-(c*Sqrt[d]) + Sqrt[c^2*d
+ e])] + Log[1 - (Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] + Sqrt[c^2*d + e])])) + 2*PolyLog[2, (Sqrt[e]*E^(I*Arc
Sin[c*x]))/(c*Sqrt[d] - Sqrt[c^2*d + e])] + 2*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] + Sqrt[c^2*d +
e])])))/(4*c^2*e^2)

________________________________________________________________________________________

Maple [C]  time = 0.512, size = 2854, normalized size = 5.1 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*b*arcsin(c*x)/c^2/e+1/2*a/e*x^2+I*c^4*b*d^3*polylog(2,e*(I*c*x+(-c^2*x^2+1)^(1/2))^2/(2*c^2*d+2*(c^2*d*(c
^2*d+e))^(1/2)+e))/e^4/(c^2*d+e)*(c^2*d*(c^2*d+e))^(1/2)+3*I*c^2*b*d^2*arcsin(c*x)^2/e^3/(c^2*d+e)*(c^2*d*(c^2
*d+e))^(1/2)+2*I*c^4*b*d^3*arcsin(c*x)^2/e^4/(c^2*d+e)*(c^2*d*(c^2*d+e))^(1/2)+3/2*I*c^2*b*d^2*polylog(2,e*(I*
c*x+(-c^2*x^2+1)^(1/2))^2/(2*c^2*d+2*(c^2*d*(c^2*d+e))^(1/2)+e))/e^3/(c^2*d+e)*(c^2*d*(c^2*d+e))^(1/2)+1/4*b*x
*(-c^2*x^2+1)^(1/2)/c/e-1/2*b/e^2*ln(1-e*(I*c*x+(-c^2*x^2+1)^(1/2))^2/(2*c^2*d+2*(c^2*d*(c^2*d+e))^(1/2)+e))*a
rcsin(c*x)*d-2*c^4*b/e^4*d^3/(c^2*d+e)*ln(1-e*(I*c*x+(-c^2*x^2+1)^(1/2))^2/(2*c^2*d+2*(c^2*d*(c^2*d+e))^(1/2)+
e))*arcsin(c*x)*(c^2*d*(c^2*d+e))^(1/2)-3*c^2*b/e^3*d^2/(c^2*d+e)*ln(1-e*(I*c*x+(-c^2*x^2+1)^(1/2))^2/(2*c^2*d
+2*(c^2*d*(c^2*d+e))^(1/2)+e))*arcsin(c*x)*(c^2*d*(c^2*d+e))^(1/2)-2*c^4*b/e^4*d^3*ln(1-e*(I*c*x+(-c^2*x^2+1)^
(1/2))^2/(2*c^2*d+2*(c^2*d*(c^2*d+e))^(1/2)+e))*arcsin(c*x)-2*c^2*b/e^3*d^2*ln(1-e*(I*c*x+(-c^2*x^2+1)^(1/2))^
2/(2*c^2*d+2*(c^2*d*(c^2*d+e))^(1/2)+e))*arcsin(c*x)+I*c^4*b*polylog(2,e*(I*c*x+(-c^2*x^2+1)^(1/2))^2/(2*c^2*d
+2*(c^2*d*(c^2*d+e))^(1/2)+e))*d^3/e^4+I*c^2*b*polylog(2,e*(I*c*x+(-c^2*x^2+1)^(1/2))^2/(2*c^2*d+2*(c^2*d*(c^2
*d+e))^(1/2)+e))*d^2/e^3+1/2*b*arcsin(c*x)/e*x^2-1/2*a*d/e^2*ln(c^2*e*x^2+c^2*d)-3/2*b/e^2/(c^2*d+e)*ln(1-e*(I
*c*x+(-c^2*x^2+1)^(1/2))^2/(2*c^2*d+2*(c^2*d*(c^2*d+e))^(1/2)+e))*arcsin(c*x)*(c^2*d*(c^2*d+e))^(1/2)*d+2*c^6*
b/e^4*d^4/(c^2*d+e)*ln(1-e*(I*c*x+(-c^2*x^2+1)^(1/2))^2/(2*c^2*d+2*(c^2*d*(c^2*d+e))^(1/2)+e))*arcsin(c*x)+4*c
^4*b/e^3*d^3/(c^2*d+e)*ln(1-e*(I*c*x+(-c^2*x^2+1)^(1/2))^2/(2*c^2*d+2*(c^2*d*(c^2*d+e))^(1/2)+e))*arcsin(c*x)+
5/2*c^2*b/e^2/(c^2*d+e)*ln(1-e*(I*c*x+(-c^2*x^2+1)^(1/2))^2/(2*c^2*d+2*(c^2*d*(c^2*d+e))^(1/2)+e))*arcsin(c*x)
*d^2+2*c^2*b/e^4*d^2*ln(1-e*(I*c*x+(-c^2*x^2+1)^(1/2))^2/(2*c^2*d+2*(c^2*d*(c^2*d+e))^(1/2)+e))*arcsin(c*x)*(c
^2*d*(c^2*d+e))^(1/2)+1/4/c^2*b*(c^2*d*(c^2*d+e))^(1/2)/e/(c^2*d+e)*arcsin(c*x)*ln(1-e*(I*c*x+(-c^2*x^2+1)^(1/
2))^2/(2*c^2*d-2*(c^2*d*(c^2*d+e))^(1/2)+e))-1/4/c^2*b/e/(c^2*d+e)*ln(1-e*(I*c*x+(-c^2*x^2+1)^(1/2))^2/(2*c^2*
d+2*(c^2*d*(c^2*d+e))^(1/2)+e))*arcsin(c*x)*(c^2*d*(c^2*d+e))^(1/2)+I*b*(c^2*d*(c^2*d+e))^(1/2)/e^2*d/(c^2*d+e
)*arcsin(c*x)^2+1/2*b*(c^2*d*(c^2*d+e))^(1/2)/e^2*d/(c^2*d+e)*arcsin(c*x)*ln(1-e*(I*c*x+(-c^2*x^2+1)^(1/2))^2/
(2*c^2*d-2*(c^2*d*(c^2*d+e))^(1/2)+e))-4*I*c^4*b*d^3*arcsin(c*x)^2/e^3/(c^2*d+e)-1/8*I/c^2*b*(c^2*d*(c^2*d+e))
^(1/2)/e/(c^2*d+e)*polylog(2,e*(I*c*x+(-c^2*x^2+1)^(1/2))^2/(2*c^2*d-2*(c^2*d*(c^2*d+e))^(1/2)+e))-2*I*c^4*b*d
^3*polylog(2,e*(I*c*x+(-c^2*x^2+1)^(1/2))^2/(2*c^2*d+2*(c^2*d*(c^2*d+e))^(1/2)+e))/e^3/(c^2*d+e)-5/4*I*c^2*b*p
olylog(2,e*(I*c*x+(-c^2*x^2+1)^(1/2))^2/(2*c^2*d+2*(c^2*d*(c^2*d+e))^(1/2)+e))/e^2/(c^2*d+e)*d^2-5/2*I*c^2*b*d
^2*arcsin(c*x)^2/e^2/(c^2*d+e)-I*c^2*b*polylog(2,e*(I*c*x+(-c^2*x^2+1)^(1/2))^2/(2*c^2*d+2*(c^2*d*(c^2*d+e))^(
1/2)+e))*d^2/e^4*(c^2*d*(c^2*d+e))^(1/2)-1/4*I*b*(c^2*d*(c^2*d+e))^(1/2)/e^2*d/(c^2*d+e)*polylog(2,e*(I*c*x+(-
c^2*x^2+1)^(1/2))^2/(2*c^2*d-2*(c^2*d*(c^2*d+e))^(1/2)+e))+3/4*I*b*polylog(2,e*(I*c*x+(-c^2*x^2+1)^(1/2))^2/(2
*c^2*d+2*(c^2*d*(c^2*d+e))^(1/2)+e))/e^2/(c^2*d+e)*(c^2*d*(c^2*d+e))^(1/2)*d-2*I*c^6*b*d^4*arcsin(c*x)^2/e^4/(
c^2*d+e)-I*c^6*b*d^4*polylog(2,e*(I*c*x+(-c^2*x^2+1)^(1/2))^2/(2*c^2*d+2*(c^2*d*(c^2*d+e))^(1/2)+e))/e^4/(c^2*
d+e)-2*I*c^2*b*arcsin(c*x)^2*d^2/e^4*(c^2*d*(c^2*d+e))^(1/2)+1/8*I/c^2*b*polylog(2,e*(I*c*x+(-c^2*x^2+1)^(1/2)
)^2/(2*c^2*d+2*(c^2*d*(c^2*d+e))^(1/2)+e))/e/(c^2*d+e)*(c^2*d*(c^2*d+e))^(1/2)+1/2*b/e/(c^2*d+e)*ln(1-e*(I*c*x
+(-c^2*x^2+1)^(1/2))^2/(2*c^2*d+2*(c^2*d*(c^2*d+e))^(1/2)+e))*arcsin(c*x)*d+b/e^3*ln(1-e*(I*c*x+(-c^2*x^2+1)^(
1/2))^2/(2*c^2*d+2*(c^2*d*(c^2*d+e))^(1/2)+e))*arcsin(c*x)*d*(c^2*d*(c^2*d+e))^(1/2)+2*I*c^2*b*arcsin(c*x)^2*d
^2/e^3+2*I*c^4*b*arcsin(c*x)^2*d^3/e^4-1/2*I*b*polylog(2,e*(I*c*x+(-c^2*x^2+1)^(1/2))^2/(2*c^2*d+2*(c^2*d*(c^2
*d+e))^(1/2)+e))*d/e^3*(c^2*d*(c^2*d+e))^(1/2)-I*b*arcsin(c*x)^2*d/e^3*(c^2*d*(c^2*d+e))^(1/2)-1/2*I*b*arcsin(
c*x)^2/e/(c^2*d+e)*d-1/4*I*b*polylog(2,e*(I*c*x+(-c^2*x^2+1)^(1/2))^2/(2*c^2*d+2*(c^2*d*(c^2*d+e))^(1/2)+e))/e
/(c^2*d+e)*d+I*b*d*arcsin(c*x)^2/e^2+1/2*I*b*d/e^2*sum((_R1^2*e-4*c^2*d-2*e)/(_R1^2*e-2*c^2*d-e)*(I*arcsin(c*x
)*ln((_R1-I*c*x-(-c^2*x^2+1)^(1/2))/_R1)+dilog((_R1-I*c*x-(-c^2*x^2+1)^(1/2))/_R1)),_R1=RootOf(e*_Z^4+(-4*c^2*
d-2*e)*_Z^2+e))+1/4*I*b*polylog(2,e*(I*c*x+(-c^2*x^2+1)^(1/2))^2/(2*c^2*d+2*(c^2*d*(c^2*d+e))^(1/2)+e))*d/e^2

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{2} \, a{\left (\frac{x^{2}}{e} - \frac{d \log \left (e x^{2} + d\right )}{e^{2}}\right )} + b \int \frac{x^{3} \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right )}{e x^{2} + d}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*a*(x^2/e - d*log(e*x^2 + d)/e^2) + b*integrate(x^3*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))/(e*x^2 + d),
x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(a+b*asin(c*x))/(e*x**2+d),x)

[Out]

Integral(x**3*(a + b*asin(c*x))/(d + e*x**2), x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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