3.641 \(\int \frac{x^5 (a+b \sin ^{-1}(c x))}{(d+e x^2)^3} \, dx\)

Optimal. Leaf size=705 \[ -\frac{i b \text{PolyLog}\left (2,-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{-\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{2 e^3}-\frac{i b \text{PolyLog}\left (2,\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{-\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{2 e^3}-\frac{i b \text{PolyLog}\left (2,-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{2 e^3}-\frac{i b \text{PolyLog}\left (2,\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{2 e^3}+\frac{\left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{-\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{2 e^3}+\frac{\left (a+b \sin ^{-1}(c x)\right ) \log \left (1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{-\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{2 e^3}+\frac{\left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{2 e^3}+\frac{\left (a+b \sin ^{-1}(c x)\right ) \log \left (1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{2 e^3}-\frac{d^2 \left (a+b \sin ^{-1}(c x)\right )}{4 e^3 \left (d+e x^2\right )^2}+\frac{d \left (a+b \sin ^{-1}(c x)\right )}{e^3 \left (d+e x^2\right )}-\frac{i \left (a+b \sin ^{-1}(c x)\right )^2}{2 b e^3}+\frac{b c d x \sqrt{1-c^2 x^2}}{8 e^2 \left (c^2 d+e\right ) \left (d+e x^2\right )}+\frac{b c \sqrt{d} \left (2 c^2 d+e\right ) \tan ^{-1}\left (\frac{x \sqrt{c^2 d+e}}{\sqrt{d} \sqrt{1-c^2 x^2}}\right )}{8 e^3 \left (c^2 d+e\right )^{3/2}}-\frac{b c \sqrt{d} \tan ^{-1}\left (\frac{x \sqrt{c^2 d+e}}{\sqrt{d} \sqrt{1-c^2 x^2}}\right )}{e^3 \sqrt{c^2 d+e}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 1.09456, antiderivative size = 705, normalized size of antiderivative = 1., number of steps used = 27, number of rules used = 10, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.476, Rules used = {4733, 4729, 382, 377, 205, 4741, 4521, 2190, 2279, 2391} \[ -\frac{i b \text{PolyLog}\left (2,-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{-\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{2 e^3}-\frac{i b \text{PolyLog}\left (2,\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{-\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{2 e^3}-\frac{i b \text{PolyLog}\left (2,-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{2 e^3}-\frac{i b \text{PolyLog}\left (2,\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{2 e^3}+\frac{\left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{-\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{2 e^3}+\frac{\left (a+b \sin ^{-1}(c x)\right ) \log \left (1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{-\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{2 e^3}+\frac{\left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{2 e^3}+\frac{\left (a+b \sin ^{-1}(c x)\right ) \log \left (1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{2 e^3}-\frac{d^2 \left (a+b \sin ^{-1}(c x)\right )}{4 e^3 \left (d+e x^2\right )^2}+\frac{d \left (a+b \sin ^{-1}(c x)\right )}{e^3 \left (d+e x^2\right )}-\frac{i \left (a+b \sin ^{-1}(c x)\right )^2}{2 b e^3}+\frac{b c d x \sqrt{1-c^2 x^2}}{8 e^2 \left (c^2 d+e\right ) \left (d+e x^2\right )}+\frac{b c \sqrt{d} \left (2 c^2 d+e\right ) \tan ^{-1}\left (\frac{x \sqrt{c^2 d+e}}{\sqrt{d} \sqrt{1-c^2 x^2}}\right )}{8 e^3 \left (c^2 d+e\right )^{3/2}}-\frac{b c \sqrt{d} \tan ^{-1}\left (\frac{x \sqrt{c^2 d+e}}{\sqrt{d} \sqrt{1-c^2 x^2}}\right )}{e^3 \sqrt{c^2 d+e}} \]

Antiderivative was successfully verified.

[In]

Int[(x^5*(a + b*ArcSin[c*x]))/(d + e*x^2)^3,x]

[Out]

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

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 4729

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

Rule 382

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> -Simp[(b*x*(a + b*x^n)^(p + 1)*(
c + d*x^n)^(q + 1))/(a*n*(p + 1)*(b*c - a*d)), x] + Dist[(b*c + n*(p + 1)*(b*c - a*d))/(a*n*(p + 1)*(b*c - a*d
)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, n, q}, x] && NeQ[b*c - a*d, 0] && EqQ[
n*(p + q + 2) + 1, 0] && (LtQ[p, -1] ||  !LtQ[q, -1]) && NeQ[p, -1]

Rule 377

Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Subst[Int[1/(c - (b*c - a*d)*x^n), x]
, x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/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^5 \left (a+b \sin ^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx &=\int \left (\frac{d^2 x \left (a+b \sin ^{-1}(c x)\right )}{e^2 \left (d+e x^2\right )^3}-\frac{2 d x \left (a+b \sin ^{-1}(c x)\right )}{e^2 \left (d+e x^2\right )^2}+\frac{x \left (a+b \sin ^{-1}(c x)\right )}{e^2 \left (d+e x^2\right )}\right ) \, dx\\ &=\frac{\int \frac{x \left (a+b \sin ^{-1}(c x)\right )}{d+e x^2} \, dx}{e^2}-\frac{(2 d) \int \frac{x \left (a+b \sin ^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx}{e^2}+\frac{d^2 \int \frac{x \left (a+b \sin ^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx}{e^2}\\ &=-\frac{d^2 \left (a+b \sin ^{-1}(c x)\right )}{4 e^3 \left (d+e x^2\right )^2}+\frac{d \left (a+b \sin ^{-1}(c x)\right )}{e^3 \left (d+e x^2\right )}-\frac{(b c d) \int \frac{1}{\sqrt{1-c^2 x^2} \left (d+e x^2\right )} \, dx}{e^3}+\frac{\left (b c d^2\right ) \int \frac{1}{\sqrt{1-c^2 x^2} \left (d+e x^2\right )^2} \, dx}{4 e^3}+\frac{\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^2}\\ &=\frac{b c d x \sqrt{1-c^2 x^2}}{8 e^2 \left (c^2 d+e\right ) \left (d+e x^2\right )}-\frac{d^2 \left (a+b \sin ^{-1}(c x)\right )}{4 e^3 \left (d+e x^2\right )^2}+\frac{d \left (a+b \sin ^{-1}(c x)\right )}{e^3 \left (d+e x^2\right )}-\frac{(b c d) \operatorname{Subst}\left (\int \frac{1}{d-\left (-c^2 d-e\right ) x^2} \, dx,x,\frac{x}{\sqrt{1-c^2 x^2}}\right )}{e^3}-\frac{\int \frac{a+b \sin ^{-1}(c x)}{\sqrt{-d}-\sqrt{e} x} \, dx}{2 e^{5/2}}+\frac{\int \frac{a+b \sin ^{-1}(c x)}{\sqrt{-d}+\sqrt{e} x} \, dx}{2 e^{5/2}}+\frac{\left (b c d \left (2 c^2 d+e\right )\right ) \int \frac{1}{\sqrt{1-c^2 x^2} \left (d+e x^2\right )} \, dx}{8 e^3 \left (c^2 d+e\right )}\\ &=\frac{b c d x \sqrt{1-c^2 x^2}}{8 e^2 \left (c^2 d+e\right ) \left (d+e x^2\right )}-\frac{d^2 \left (a+b \sin ^{-1}(c x)\right )}{4 e^3 \left (d+e x^2\right )^2}+\frac{d \left (a+b \sin ^{-1}(c x)\right )}{e^3 \left (d+e x^2\right )}-\frac{b c \sqrt{d} \tan ^{-1}\left (\frac{\sqrt{c^2 d+e} x}{\sqrt{d} \sqrt{1-c^2 x^2}}\right )}{e^3 \sqrt{c^2 d+e}}-\frac{\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^{5/2}}+\frac{\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^{5/2}}+\frac{\left (b c d \left (2 c^2 d+e\right )\right ) \operatorname{Subst}\left (\int \frac{1}{d-\left (-c^2 d-e\right ) x^2} \, dx,x,\frac{x}{\sqrt{1-c^2 x^2}}\right )}{8 e^3 \left (c^2 d+e\right )}\\ &=\frac{b c d x \sqrt{1-c^2 x^2}}{8 e^2 \left (c^2 d+e\right ) \left (d+e x^2\right )}-\frac{d^2 \left (a+b \sin ^{-1}(c x)\right )}{4 e^3 \left (d+e x^2\right )^2}+\frac{d \left (a+b \sin ^{-1}(c x)\right )}{e^3 \left (d+e x^2\right )}-\frac{i \left (a+b \sin ^{-1}(c x)\right )^2}{2 b e^3}-\frac{b c \sqrt{d} \tan ^{-1}\left (\frac{\sqrt{c^2 d+e} x}{\sqrt{d} \sqrt{1-c^2 x^2}}\right )}{e^3 \sqrt{c^2 d+e}}+\frac{b c \sqrt{d} \left (2 c^2 d+e\right ) \tan ^{-1}\left (\frac{\sqrt{c^2 d+e} x}{\sqrt{d} \sqrt{1-c^2 x^2}}\right )}{8 e^3 \left (c^2 d+e\right )^{3/2}}-\frac{i \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^{5/2}}-\frac{i \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^{5/2}}+\frac{i \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^{5/2}}+\frac{i \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^{5/2}}\\ &=\frac{b c d x \sqrt{1-c^2 x^2}}{8 e^2 \left (c^2 d+e\right ) \left (d+e x^2\right )}-\frac{d^2 \left (a+b \sin ^{-1}(c x)\right )}{4 e^3 \left (d+e x^2\right )^2}+\frac{d \left (a+b \sin ^{-1}(c x)\right )}{e^3 \left (d+e x^2\right )}-\frac{i \left (a+b \sin ^{-1}(c x)\right )^2}{2 b e^3}-\frac{b c \sqrt{d} \tan ^{-1}\left (\frac{\sqrt{c^2 d+e} x}{\sqrt{d} \sqrt{1-c^2 x^2}}\right )}{e^3 \sqrt{c^2 d+e}}+\frac{b c \sqrt{d} \left (2 c^2 d+e\right ) \tan ^{-1}\left (\frac{\sqrt{c^2 d+e} x}{\sqrt{d} \sqrt{1-c^2 x^2}}\right )}{8 e^3 \left (c^2 d+e\right )^{3/2}}+\frac{\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^3}+\frac{\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^3}+\frac{\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^3}+\frac{\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^3}-\frac{b \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^3}-\frac{b \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^3}-\frac{b \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^3}-\frac{b \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^3}\\ &=\frac{b c d x \sqrt{1-c^2 x^2}}{8 e^2 \left (c^2 d+e\right ) \left (d+e x^2\right )}-\frac{d^2 \left (a+b \sin ^{-1}(c x)\right )}{4 e^3 \left (d+e x^2\right )^2}+\frac{d \left (a+b \sin ^{-1}(c x)\right )}{e^3 \left (d+e x^2\right )}-\frac{i \left (a+b \sin ^{-1}(c x)\right )^2}{2 b e^3}-\frac{b c \sqrt{d} \tan ^{-1}\left (\frac{\sqrt{c^2 d+e} x}{\sqrt{d} \sqrt{1-c^2 x^2}}\right )}{e^3 \sqrt{c^2 d+e}}+\frac{b c \sqrt{d} \left (2 c^2 d+e\right ) \tan ^{-1}\left (\frac{\sqrt{c^2 d+e} x}{\sqrt{d} \sqrt{1-c^2 x^2}}\right )}{8 e^3 \left (c^2 d+e\right )^{3/2}}+\frac{\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^3}+\frac{\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^3}+\frac{\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^3}+\frac{\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^3}+\frac{(i b) \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^3}+\frac{(i b) \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^3}+\frac{(i b) \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^3}+\frac{(i b) \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^3}\\ &=\frac{b c d x \sqrt{1-c^2 x^2}}{8 e^2 \left (c^2 d+e\right ) \left (d+e x^2\right )}-\frac{d^2 \left (a+b \sin ^{-1}(c x)\right )}{4 e^3 \left (d+e x^2\right )^2}+\frac{d \left (a+b \sin ^{-1}(c x)\right )}{e^3 \left (d+e x^2\right )}-\frac{i \left (a+b \sin ^{-1}(c x)\right )^2}{2 b e^3}-\frac{b c \sqrt{d} \tan ^{-1}\left (\frac{\sqrt{c^2 d+e} x}{\sqrt{d} \sqrt{1-c^2 x^2}}\right )}{e^3 \sqrt{c^2 d+e}}+\frac{b c \sqrt{d} \left (2 c^2 d+e\right ) \tan ^{-1}\left (\frac{\sqrt{c^2 d+e} x}{\sqrt{d} \sqrt{1-c^2 x^2}}\right )}{8 e^3 \left (c^2 d+e\right )^{3/2}}+\frac{\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^3}+\frac{\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^3}+\frac{\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^3}+\frac{\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^3}-\frac{i b \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^3}-\frac{i b \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^3}-\frac{i b \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^3}-\frac{i b \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^3}\\ \end{align*}

Mathematica [A]  time = 6.48581, size = 973, normalized size = 1.38 \[ \frac{-\frac{4 a d^2}{\left (e x^2+d\right )^2}+\frac{16 a d}{e x^2+d}+8 a \log \left (e x^2+d\right )+b \left (\frac{i d^{3/2} \log \left (\frac{e \sqrt{d c^2+e} \left (-i \sqrt{d} x c^2+\sqrt{e}+\sqrt{d c^2+e} \sqrt{1-c^2 x^2}\right )}{c^3 \left (d+i \sqrt{e} x \sqrt{d}\right )}\right ) c^3}{\left (d c^2+e\right )^{3/2}}-\frac{i d^{3/2} \log \left (\frac{e \sqrt{d c^2+e} \left (i \sqrt{d} x c^2+\sqrt{e}+\sqrt{d c^2+e} \sqrt{1-c^2 x^2}\right )}{c^3 \left (d-i \sqrt{d} \sqrt{e} x\right )}\right ) c^3}{\left (d c^2+e\right )^{3/2}}-\frac{7 \sqrt{d} \tan ^{-1}\left (\frac{\sqrt{d} x c^2+i \sqrt{e}}{\sqrt{d c^2+e} \sqrt{1-c^2 x^2}}\right ) c}{\sqrt{d c^2+e}}+\frac{7 i \sqrt{d} \tanh ^{-1}\left (\frac{i \sqrt{d} x c^2+\sqrt{e}}{\sqrt{d c^2+e} \sqrt{1-c^2 x^2}}\right ) c}{\sqrt{d c^2+e}}+\frac{d \sqrt{e} \sqrt{1-c^2 x^2} c}{\left (d c^2+e\right ) \left (\sqrt{e} x-i \sqrt{d}\right )}+\frac{d \sqrt{e} \sqrt{1-c^2 x^2} c}{\left (d c^2+e\right ) \left (\sqrt{e} x+i \sqrt{d}\right )}-8 i \sin ^{-1}(c x)^2+\frac{7 \sqrt{d} \sin ^{-1}(c x)}{\sqrt{d}-i \sqrt{e} x}+\frac{7 \sqrt{d} \sin ^{-1}(c x)}{i \sqrt{e} x+\sqrt{d}}-\frac{d \sin ^{-1}(c x)}{\left (i \sqrt{e} x+\sqrt{d}\right )^2}+\frac{d \sin ^{-1}(c x)}{\left (\sqrt{e} x+i \sqrt{d}\right )^2}+8 \sin ^{-1}(c x) \log \left (\frac{e^{i \sin ^{-1}(c x)} \sqrt{e}}{c \sqrt{d}-\sqrt{d c^2+e}}+1\right )+8 \sin ^{-1}(c x) \log \left (\frac{e^{i \sin ^{-1}(c x)} \sqrt{e}}{\sqrt{d c^2+e}-c \sqrt{d}}+1\right )+8 \sin ^{-1}(c x) \log \left (1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{d} c+\sqrt{d c^2+e}}\right )+8 \sin ^{-1}(c x) \log \left (\frac{e^{i \sin ^{-1}(c x)} \sqrt{e}}{\sqrt{d} c+\sqrt{d c^2+e}}+1\right )-8 i \text{PolyLog}\left (2,\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{c \sqrt{d}-\sqrt{d c^2+e}}\right )-8 i \text{PolyLog}\left (2,\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{d c^2+e}-c \sqrt{d}}\right )-8 i \text{PolyLog}\left (2,-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{d} c+\sqrt{d c^2+e}}\right )-8 i \text{PolyLog}\left (2,\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{d} c+\sqrt{d c^2+e}}\right )\right )}{16 e^3} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(x^5*(a + b*ArcSin[c*x]))/(d + e*x^2)^3,x]

[Out]

((-4*a*d^2)/(d + e*x^2)^2 + (16*a*d)/(d + e*x^2) + 8*a*Log[d + e*x^2] + b*((c*d*Sqrt[e]*Sqrt[1 - c^2*x^2])/((c
^2*d + e)*((-I)*Sqrt[d] + Sqrt[e]*x)) + (c*d*Sqrt[e]*Sqrt[1 - c^2*x^2])/((c^2*d + e)*(I*Sqrt[d] + Sqrt[e]*x))
+ (7*Sqrt[d]*ArcSin[c*x])/(Sqrt[d] - I*Sqrt[e]*x) - (d*ArcSin[c*x])/(Sqrt[d] + I*Sqrt[e]*x)^2 + (7*Sqrt[d]*Arc
Sin[c*x])/(Sqrt[d] + I*Sqrt[e]*x) + (d*ArcSin[c*x])/(I*Sqrt[d] + Sqrt[e]*x)^2 - (8*I)*ArcSin[c*x]^2 - (7*c*Sqr
t[d]*ArcTan[(I*Sqrt[e] + c^2*Sqrt[d]*x)/(Sqrt[c^2*d + e]*Sqrt[1 - c^2*x^2])])/Sqrt[c^2*d + e] + ((7*I)*c*Sqrt[
d]*ArcTanh[(Sqrt[e] + I*c^2*Sqrt[d]*x)/(Sqrt[c^2*d + e]*Sqrt[1 - c^2*x^2])])/Sqrt[c^2*d + e] + 8*ArcSin[c*x]*L
og[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] - Sqrt[c^2*d + e])] + 8*ArcSin[c*x]*Log[1 + (Sqrt[e]*E^(I*ArcSin
[c*x]))/(-(c*Sqrt[d]) + Sqrt[c^2*d + e])] + 8*ArcSin[c*x]*Log[1 - (Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] + Sqr
t[c^2*d + e])] + 8*ArcSin[c*x]*Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] + Sqrt[c^2*d + e])] + (I*c^3*d^(
3/2)*Log[(e*Sqrt[c^2*d + e]*(Sqrt[e] - I*c^2*Sqrt[d]*x + Sqrt[c^2*d + e]*Sqrt[1 - c^2*x^2]))/(c^3*(d + I*Sqrt[
d]*Sqrt[e]*x))])/(c^2*d + e)^(3/2) - (I*c^3*d^(3/2)*Log[(e*Sqrt[c^2*d + e]*(Sqrt[e] + I*c^2*Sqrt[d]*x + Sqrt[c
^2*d + e]*Sqrt[1 - c^2*x^2]))/(c^3*(d - I*Sqrt[d]*Sqrt[e]*x))])/(c^2*d + e)^(3/2) - (8*I)*PolyLog[2, (Sqrt[e]*
E^(I*ArcSin[c*x]))/(c*Sqrt[d] - Sqrt[c^2*d + e])] - (8*I)*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(-(c*Sqrt[d])
+ Sqrt[c^2*d + e])] - (8*I)*PolyLog[2, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] + Sqrt[c^2*d + e]))] - (8*I)*
PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] + Sqrt[c^2*d + e])]))/(16*e^3)

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Maple [C]  time = 1.639, size = 5124, normalized size = 7.3 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(a+b*arcsin(c*x))/(e*x^2+d)^3,x)

[Out]

result too large to display

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{4} \, a{\left (\frac{4 \, d e x^{2} + 3 \, d^{2}}{e^{5} x^{4} + 2 \, d e^{4} x^{2} + d^{2} e^{3}} + \frac{2 \, \log \left (e x^{2} + d\right )}{e^{3}}\right )} + b \int \frac{x^{5} \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right )}{e^{3} x^{6} + 3 \, d e^{2} x^{4} + 3 \, d^{2} e x^{2} + d^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(a+b*arcsin(c*x))/(e*x^2+d)^3,x, algorithm="maxima")

[Out]

1/4*a*((4*d*e*x^2 + 3*d^2)/(e^5*x^4 + 2*d*e^4*x^2 + d^2*e^3) + 2*log(e*x^2 + d)/e^3) + b*integrate(x^5*arctan2
(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))/(e^3*x^6 + 3*d*e^2*x^4 + 3*d^2*e*x^2 + d^3), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b x^{5} \arcsin \left (c x\right ) + a x^{5}}{e^{3} x^{6} + 3 \, d e^{2} x^{4} + 3 \, d^{2} e x^{2} + d^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(a+b*arcsin(c*x))/(e*x^2+d)^3,x, algorithm="fricas")

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(a+b*arcsin(c*x))/(e*x^2+d)^3,x, algorithm="giac")

[Out]

integrate((b*arcsin(c*x) + a)*x^5/(e*x^2 + d)^3, x)