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

Optimal. Leaf size=1092 \[ \text{result too large to display} \]

[Out]

(b*c*Sqrt[1 - c^2*x^2])/(16*Sqrt[-d]*e*(c^2*d + e)*(Sqrt[-d] - Sqrt[e]*x)) + (b*c*Sqrt[1 - c^2*x^2])/(16*Sqrt[
-d]*e*(c^2*d + e)*(Sqrt[-d] + Sqrt[e]*x)) - (a + b*ArcSin[c*x])/(16*Sqrt[-d]*e^(3/2)*(Sqrt[-d] - Sqrt[e]*x)^2)
- (a + b*ArcSin[c*x])/(16*d*e^(3/2)*(Sqrt[-d] - Sqrt[e]*x)) + (a + b*ArcSin[c*x])/(16*Sqrt[-d]*e^(3/2)*(Sqrt[
-d] + Sqrt[e]*x)^2) + (a + b*ArcSin[c*x])/(16*d*e^(3/2)*(Sqrt[-d] + Sqrt[e]*x)) - (b*c^3*ArcTanh[(Sqrt[e] - c^
2*Sqrt[-d]*x)/(Sqrt[c^2*d + e]*Sqrt[1 - c^2*x^2])])/(16*e^(3/2)*(c^2*d + e)^(3/2)) + (b*c*ArcTanh[(Sqrt[e] - c
^2*Sqrt[-d]*x)/(Sqrt[c^2*d + e]*Sqrt[1 - c^2*x^2])])/(16*d*e^(3/2)*Sqrt[c^2*d + e]) - (b*c^3*ArcTanh[(Sqrt[e]
+ c^2*Sqrt[-d]*x)/(Sqrt[c^2*d + e]*Sqrt[1 - c^2*x^2])])/(16*e^(3/2)*(c^2*d + e)^(3/2)) + (b*c*ArcTanh[(Sqrt[e]
+ c^2*Sqrt[-d]*x)/(Sqrt[c^2*d + e]*Sqrt[1 - c^2*x^2])])/(16*d*e^(3/2)*Sqrt[c^2*d + e]) - ((a + b*ArcSin[c*x])
*Log[1 - (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/(16*(-d)^(3/2)*e^(3/2)) + ((a + b*ArcS
in[c*x])*Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/(16*(-d)^(3/2)*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])])/(16*(-d)^(3/2)*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])])/(16*(-d)^(3/2)
*e^(3/2)) - ((I/16)*b*PolyLog[2, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e]))])/((-d)^(3/2)
*e^(3/2)) + ((I/16)*b*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/((-d)^(3/2)*e^
(3/2)) - ((I/16)*b*PolyLog[2, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e]))])/((-d)^(3/2)*e^
(3/2)) + ((I/16)*b*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])])/((-d)^(3/2)*e^(3/
2))

________________________________________________________________________________________

Rubi [A]  time = 2.61063, antiderivative size = 1092, normalized size of antiderivative = 1., number of steps used = 62, number of rules used = 11, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.524, Rules used = {4733, 4667, 4743, 731, 725, 206, 4741, 4521, 2190, 2279, 2391} \[ -\frac{b \tanh ^{-1}\left (\frac{\sqrt{e}-c^2 \sqrt{-d} x}{\sqrt{d c^2+e} \sqrt{1-c^2 x^2}}\right ) c^3}{16 e^{3/2} \left (d c^2+e\right )^{3/2}}-\frac{b \tanh ^{-1}\left (\frac{\sqrt{-d} x c^2+\sqrt{e}}{\sqrt{d c^2+e} \sqrt{1-c^2 x^2}}\right ) c^3}{16 e^{3/2} \left (d c^2+e\right )^{3/2}}+\frac{b \tanh ^{-1}\left (\frac{\sqrt{e}-c^2 \sqrt{-d} x}{\sqrt{d c^2+e} \sqrt{1-c^2 x^2}}\right ) c}{16 d e^{3/2} \sqrt{d c^2+e}}+\frac{b \tanh ^{-1}\left (\frac{\sqrt{-d} x c^2+\sqrt{e}}{\sqrt{d c^2+e} \sqrt{1-c^2 x^2}}\right ) c}{16 d e^{3/2} \sqrt{d c^2+e}}+\frac{b \sqrt{1-c^2 x^2} c}{16 \sqrt{-d} e \left (d c^2+e\right ) \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{b \sqrt{1-c^2 x^2} c}{16 \sqrt{-d} e \left (d c^2+e\right ) \left (\sqrt{e} x+\sqrt{-d}\right )}-\frac{a+b \sin ^{-1}(c x)}{16 d e^{3/2} \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{a+b \sin ^{-1}(c x)}{16 d e^{3/2} \left (\sqrt{e} x+\sqrt{-d}\right )}-\frac{a+b \sin ^{-1}(c x)}{16 \sqrt{-d} e^{3/2} \left (\sqrt{-d}-\sqrt{e} x\right )^2}+\frac{a+b \sin ^{-1}(c x)}{16 \sqrt{-d} e^{3/2} \left (\sqrt{e} x+\sqrt{-d}\right )^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{d c^2+e}}\right )}{16 (-d)^{3/2} e^{3/2}}+\frac{\left (a+b \sin ^{-1}(c x)\right ) \log \left (\frac{e^{i \sin ^{-1}(c x)} \sqrt{e}}{i c \sqrt{-d}-\sqrt{d c^2+e}}+1\right )}{16 (-d)^{3/2} e^{3/2}}-\frac{\left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i \sqrt{-d} c+\sqrt{d c^2+e}}\right )}{16 (-d)^{3/2} e^{3/2}}+\frac{\left (a+b \sin ^{-1}(c x)\right ) \log \left (\frac{e^{i \sin ^{-1}(c x)} \sqrt{e}}{i \sqrt{-d} c+\sqrt{d c^2+e}}+1\right )}{16 (-d)^{3/2} e^{3/2}}-\frac{i b \text{PolyLog}\left (2,-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}-\sqrt{d c^2+e}}\right )}{16 (-d)^{3/2} e^{3/2}}+\frac{i b \text{PolyLog}\left (2,\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}-\sqrt{d c^2+e}}\right )}{16 (-d)^{3/2} e^{3/2}}-\frac{i b \text{PolyLog}\left (2,-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i \sqrt{-d} c+\sqrt{d c^2+e}}\right )}{16 (-d)^{3/2} e^{3/2}}+\frac{i b \text{PolyLog}\left (2,\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i \sqrt{-d} c+\sqrt{d c^2+e}}\right )}{16 (-d)^{3/2} e^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*c*Sqrt[1 - c^2*x^2])/(16*Sqrt[-d]*e*(c^2*d + e)*(Sqrt[-d] - Sqrt[e]*x)) + (b*c*Sqrt[1 - c^2*x^2])/(16*Sqrt[
-d]*e*(c^2*d + e)*(Sqrt[-d] + Sqrt[e]*x)) - (a + b*ArcSin[c*x])/(16*Sqrt[-d]*e^(3/2)*(Sqrt[-d] - Sqrt[e]*x)^2)
- (a + b*ArcSin[c*x])/(16*d*e^(3/2)*(Sqrt[-d] - Sqrt[e]*x)) + (a + b*ArcSin[c*x])/(16*Sqrt[-d]*e^(3/2)*(Sqrt[
-d] + Sqrt[e]*x)^2) + (a + b*ArcSin[c*x])/(16*d*e^(3/2)*(Sqrt[-d] + Sqrt[e]*x)) - (b*c^3*ArcTanh[(Sqrt[e] - c^
2*Sqrt[-d]*x)/(Sqrt[c^2*d + e]*Sqrt[1 - c^2*x^2])])/(16*e^(3/2)*(c^2*d + e)^(3/2)) + (b*c*ArcTanh[(Sqrt[e] - c
^2*Sqrt[-d]*x)/(Sqrt[c^2*d + e]*Sqrt[1 - c^2*x^2])])/(16*d*e^(3/2)*Sqrt[c^2*d + e]) - (b*c^3*ArcTanh[(Sqrt[e]
+ c^2*Sqrt[-d]*x)/(Sqrt[c^2*d + e]*Sqrt[1 - c^2*x^2])])/(16*e^(3/2)*(c^2*d + e)^(3/2)) + (b*c*ArcTanh[(Sqrt[e]
+ c^2*Sqrt[-d]*x)/(Sqrt[c^2*d + e]*Sqrt[1 - c^2*x^2])])/(16*d*e^(3/2)*Sqrt[c^2*d + e]) - ((a + b*ArcSin[c*x])
*Log[1 - (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/(16*(-d)^(3/2)*e^(3/2)) + ((a + b*ArcS
in[c*x])*Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/(16*(-d)^(3/2)*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])])/(16*(-d)^(3/2)*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])])/(16*(-d)^(3/2)
*e^(3/2)) - ((I/16)*b*PolyLog[2, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e]))])/((-d)^(3/2)
*e^(3/2)) + ((I/16)*b*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/((-d)^(3/2)*e^
(3/2)) - ((I/16)*b*PolyLog[2, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e]))])/((-d)^(3/2)*e^
(3/2)) + ((I/16)*b*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])])/((-d)^(3/2)*e^(3/
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 4667

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

Rule 4743

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

Rule 731

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^(m + 1)*(a + c*x^2)^(p
+ 1))/((m + 1)*(c*d^2 + a*e^2)), x] + Dist[(c*d)/(c*d^2 + a*e^2), Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x]
/; FreeQ[{a, c, d, e, m, p}, x] && NeQ[c*d^2 + a*e^2, 0] && EqQ[m + 2*p + 3, 0]

Rule 725

Int[1/(((d_) + (e_.)*(x_))*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> -Subst[Int[1/(c*d^2 + a*e^2 - x^2), x], x,
(a*e - c*d*x)/Sqrt[a + c*x^2]] /; FreeQ[{a, c, d, e}, x]

Rule 206

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

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^2 \left (a+b \sin ^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx &=\int \left (-\frac{d \left (a+b \sin ^{-1}(c x)\right )}{e \left (d+e x^2\right )^3}+\frac{a+b \sin ^{-1}(c x)}{e \left (d+e x^2\right )^2}\right ) \, dx\\ &=\frac{\int \frac{a+b \sin ^{-1}(c x)}{\left (d+e x^2\right )^2} \, dx}{e}-\frac{d \int \frac{a+b \sin ^{-1}(c x)}{\left (d+e x^2\right )^3} \, dx}{e}\\ &=\frac{\int \left (-\frac{e \left (a+b \sin ^{-1}(c x)\right )}{4 d \left (\sqrt{-d} \sqrt{e}-e x\right )^2}-\frac{e \left (a+b \sin ^{-1}(c x)\right )}{4 d \left (\sqrt{-d} \sqrt{e}+e x\right )^2}-\frac{e \left (a+b \sin ^{-1}(c x)\right )}{2 d \left (-d e-e^2 x^2\right )}\right ) \, dx}{e}-\frac{d \int \left (-\frac{e^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{8 (-d)^{3/2} \left (\sqrt{-d} \sqrt{e}-e x\right )^3}-\frac{3 e \left (a+b \sin ^{-1}(c x)\right )}{16 d^2 \left (\sqrt{-d} \sqrt{e}-e x\right )^2}-\frac{e^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{8 (-d)^{3/2} \left (\sqrt{-d} \sqrt{e}+e x\right )^3}-\frac{3 e \left (a+b \sin ^{-1}(c x)\right )}{16 d^2 \left (\sqrt{-d} \sqrt{e}+e x\right )^2}-\frac{3 e \left (a+b \sin ^{-1}(c x)\right )}{8 d^2 \left (-d e-e^2 x^2\right )}\right ) \, dx}{e}\\ &=\frac{3 \int \frac{a+b \sin ^{-1}(c x)}{\left (\sqrt{-d} \sqrt{e}-e x\right )^2} \, dx}{16 d}+\frac{3 \int \frac{a+b \sin ^{-1}(c x)}{\left (\sqrt{-d} \sqrt{e}+e x\right )^2} \, dx}{16 d}-\frac{\int \frac{a+b \sin ^{-1}(c x)}{\left (\sqrt{-d} \sqrt{e}-e x\right )^2} \, dx}{4 d}-\frac{\int \frac{a+b \sin ^{-1}(c x)}{\left (\sqrt{-d} \sqrt{e}+e x\right )^2} \, dx}{4 d}+\frac{3 \int \frac{a+b \sin ^{-1}(c x)}{-d e-e^2 x^2} \, dx}{8 d}-\frac{\int \frac{a+b \sin ^{-1}(c x)}{-d e-e^2 x^2} \, dx}{2 d}-\frac{\sqrt{e} \int \frac{a+b \sin ^{-1}(c x)}{\left (\sqrt{-d} \sqrt{e}-e x\right )^3} \, dx}{8 \sqrt{-d}}-\frac{\sqrt{e} \int \frac{a+b \sin ^{-1}(c x)}{\left (\sqrt{-d} \sqrt{e}+e x\right )^3} \, dx}{8 \sqrt{-d}}\\ &=-\frac{a+b \sin ^{-1}(c x)}{16 \sqrt{-d} e^{3/2} \left (\sqrt{-d}-\sqrt{e} x\right )^2}-\frac{a+b \sin ^{-1}(c x)}{16 d e^{3/2} \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{a+b \sin ^{-1}(c x)}{16 \sqrt{-d} e^{3/2} \left (\sqrt{-d}+\sqrt{e} x\right )^2}+\frac{a+b \sin ^{-1}(c x)}{16 d e^{3/2} \left (\sqrt{-d}+\sqrt{e} x\right )}+\frac{3 \int \left (-\frac{\sqrt{-d} \left (a+b \sin ^{-1}(c x)\right )}{2 d e \left (\sqrt{-d}-\sqrt{e} x\right )}-\frac{\sqrt{-d} \left (a+b \sin ^{-1}(c x)\right )}{2 d e \left (\sqrt{-d}+\sqrt{e} x\right )}\right ) \, dx}{8 d}-\frac{\int \left (-\frac{\sqrt{-d} \left (a+b \sin ^{-1}(c x)\right )}{2 d e \left (\sqrt{-d}-\sqrt{e} x\right )}-\frac{\sqrt{-d} \left (a+b \sin ^{-1}(c x)\right )}{2 d e \left (\sqrt{-d}+\sqrt{e} x\right )}\right ) \, dx}{2 d}-\frac{(3 b c) \int \frac{1}{\left (\sqrt{-d} \sqrt{e}-e x\right ) \sqrt{1-c^2 x^2}} \, dx}{16 d e}+\frac{(3 b c) \int \frac{1}{\left (\sqrt{-d} \sqrt{e}+e x\right ) \sqrt{1-c^2 x^2}} \, dx}{16 d e}+\frac{(b c) \int \frac{1}{\left (\sqrt{-d} \sqrt{e}-e x\right ) \sqrt{1-c^2 x^2}} \, dx}{4 d e}-\frac{(b c) \int \frac{1}{\left (\sqrt{-d} \sqrt{e}+e x\right ) \sqrt{1-c^2 x^2}} \, dx}{4 d e}+\frac{(b c) \int \frac{1}{\left (\sqrt{-d} \sqrt{e}-e x\right )^2 \sqrt{1-c^2 x^2}} \, dx}{16 \sqrt{-d} \sqrt{e}}-\frac{(b c) \int \frac{1}{\left (\sqrt{-d} \sqrt{e}+e x\right )^2 \sqrt{1-c^2 x^2}} \, dx}{16 \sqrt{-d} \sqrt{e}}\\ &=\frac{b c \sqrt{1-c^2 x^2}}{16 \sqrt{-d} e \left (c^2 d+e\right ) \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{b c \sqrt{1-c^2 x^2}}{16 \sqrt{-d} e \left (c^2 d+e\right ) \left (\sqrt{-d}+\sqrt{e} x\right )}-\frac{a+b \sin ^{-1}(c x)}{16 \sqrt{-d} e^{3/2} \left (\sqrt{-d}-\sqrt{e} x\right )^2}-\frac{a+b \sin ^{-1}(c x)}{16 d e^{3/2} \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{a+b \sin ^{-1}(c x)}{16 \sqrt{-d} e^{3/2} \left (\sqrt{-d}+\sqrt{e} x\right )^2}+\frac{a+b \sin ^{-1}(c x)}{16 d e^{3/2} \left (\sqrt{-d}+\sqrt{e} x\right )}-\frac{3 \int \frac{a+b \sin ^{-1}(c x)}{\sqrt{-d}-\sqrt{e} x} \, dx}{16 (-d)^{3/2} e}-\frac{3 \int \frac{a+b \sin ^{-1}(c x)}{\sqrt{-d}+\sqrt{e} x} \, dx}{16 (-d)^{3/2} e}+\frac{\int \frac{a+b \sin ^{-1}(c x)}{\sqrt{-d}-\sqrt{e} x} \, dx}{4 (-d)^{3/2} e}+\frac{\int \frac{a+b \sin ^{-1}(c x)}{\sqrt{-d}+\sqrt{e} x} \, dx}{4 (-d)^{3/2} e}+\frac{(3 b c) \operatorname{Subst}\left (\int \frac{1}{c^2 d e+e^2-x^2} \, dx,x,\frac{-e+c^2 \sqrt{-d} \sqrt{e} x}{\sqrt{1-c^2 x^2}}\right )}{16 d e}-\frac{(3 b c) \operatorname{Subst}\left (\int \frac{1}{c^2 d e+e^2-x^2} \, dx,x,\frac{e+c^2 \sqrt{-d} \sqrt{e} x}{\sqrt{1-c^2 x^2}}\right )}{16 d e}-\frac{(b c) \operatorname{Subst}\left (\int \frac{1}{c^2 d e+e^2-x^2} \, dx,x,\frac{-e+c^2 \sqrt{-d} \sqrt{e} x}{\sqrt{1-c^2 x^2}}\right )}{4 d e}+\frac{(b c) \operatorname{Subst}\left (\int \frac{1}{c^2 d e+e^2-x^2} \, dx,x,\frac{e+c^2 \sqrt{-d} \sqrt{e} x}{\sqrt{1-c^2 x^2}}\right )}{4 d e}-\frac{\left (b c^3\right ) \int \frac{1}{\left (\sqrt{-d} \sqrt{e}-e x\right ) \sqrt{1-c^2 x^2}} \, dx}{16 e \left (c^2 d+e\right )}+\frac{\left (b c^3\right ) \int \frac{1}{\left (\sqrt{-d} \sqrt{e}+e x\right ) \sqrt{1-c^2 x^2}} \, dx}{16 e \left (c^2 d+e\right )}\\ &=\frac{b c \sqrt{1-c^2 x^2}}{16 \sqrt{-d} e \left (c^2 d+e\right ) \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{b c \sqrt{1-c^2 x^2}}{16 \sqrt{-d} e \left (c^2 d+e\right ) \left (\sqrt{-d}+\sqrt{e} x\right )}-\frac{a+b \sin ^{-1}(c x)}{16 \sqrt{-d} e^{3/2} \left (\sqrt{-d}-\sqrt{e} x\right )^2}-\frac{a+b \sin ^{-1}(c x)}{16 d e^{3/2} \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{a+b \sin ^{-1}(c x)}{16 \sqrt{-d} e^{3/2} \left (\sqrt{-d}+\sqrt{e} x\right )^2}+\frac{a+b \sin ^{-1}(c x)}{16 d e^{3/2} \left (\sqrt{-d}+\sqrt{e} x\right )}+\frac{b c \tanh ^{-1}\left (\frac{\sqrt{e}-c^2 \sqrt{-d} x}{\sqrt{c^2 d+e} \sqrt{1-c^2 x^2}}\right )}{16 d e^{3/2} \sqrt{c^2 d+e}}+\frac{b c \tanh ^{-1}\left (\frac{\sqrt{e}+c^2 \sqrt{-d} x}{\sqrt{c^2 d+e} \sqrt{1-c^2 x^2}}\right )}{16 d e^{3/2} \sqrt{c^2 d+e}}-\frac{3 \operatorname{Subst}\left (\int \frac{(a+b x) \cos (x)}{c \sqrt{-d}-\sqrt{e} \sin (x)} \, dx,x,\sin ^{-1}(c x)\right )}{16 (-d)^{3/2} e}-\frac{3 \operatorname{Subst}\left (\int \frac{(a+b x) \cos (x)}{c \sqrt{-d}+\sqrt{e} \sin (x)} \, dx,x,\sin ^{-1}(c x)\right )}{16 (-d)^{3/2} 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 )}{4 (-d)^{3/2} 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 )}{4 (-d)^{3/2} e}+\frac{\left (b c^3\right ) \operatorname{Subst}\left (\int \frac{1}{c^2 d e+e^2-x^2} \, dx,x,\frac{-e+c^2 \sqrt{-d} \sqrt{e} x}{\sqrt{1-c^2 x^2}}\right )}{16 e \left (c^2 d+e\right )}-\frac{\left (b c^3\right ) \operatorname{Subst}\left (\int \frac{1}{c^2 d e+e^2-x^2} \, dx,x,\frac{e+c^2 \sqrt{-d} \sqrt{e} x}{\sqrt{1-c^2 x^2}}\right )}{16 e \left (c^2 d+e\right )}\\ &=\frac{b c \sqrt{1-c^2 x^2}}{16 \sqrt{-d} e \left (c^2 d+e\right ) \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{b c \sqrt{1-c^2 x^2}}{16 \sqrt{-d} e \left (c^2 d+e\right ) \left (\sqrt{-d}+\sqrt{e} x\right )}-\frac{a+b \sin ^{-1}(c x)}{16 \sqrt{-d} e^{3/2} \left (\sqrt{-d}-\sqrt{e} x\right )^2}-\frac{a+b \sin ^{-1}(c x)}{16 d e^{3/2} \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{a+b \sin ^{-1}(c x)}{16 \sqrt{-d} e^{3/2} \left (\sqrt{-d}+\sqrt{e} x\right )^2}+\frac{a+b \sin ^{-1}(c x)}{16 d e^{3/2} \left (\sqrt{-d}+\sqrt{e} x\right )}-\frac{b c^3 \tanh ^{-1}\left (\frac{\sqrt{e}-c^2 \sqrt{-d} x}{\sqrt{c^2 d+e} \sqrt{1-c^2 x^2}}\right )}{16 e^{3/2} \left (c^2 d+e\right )^{3/2}}+\frac{b c \tanh ^{-1}\left (\frac{\sqrt{e}-c^2 \sqrt{-d} x}{\sqrt{c^2 d+e} \sqrt{1-c^2 x^2}}\right )}{16 d e^{3/2} \sqrt{c^2 d+e}}-\frac{b c^3 \tanh ^{-1}\left (\frac{\sqrt{e}+c^2 \sqrt{-d} x}{\sqrt{c^2 d+e} \sqrt{1-c^2 x^2}}\right )}{16 e^{3/2} \left (c^2 d+e\right )^{3/2}}+\frac{b c \tanh ^{-1}\left (\frac{\sqrt{e}+c^2 \sqrt{-d} x}{\sqrt{c^2 d+e} \sqrt{1-c^2 x^2}}\right )}{16 d e^{3/2} \sqrt{c^2 d+e}}-\frac{(3 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 )}{16 (-d)^{3/2} e}-\frac{(3 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 )}{16 (-d)^{3/2} e}-\frac{(3 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 )}{16 (-d)^{3/2} e}-\frac{(3 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 )}{16 (-d)^{3/2} e}+\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 )}{4 (-d)^{3/2} e}+\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 )}{4 (-d)^{3/2} e}+\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 )}{4 (-d)^{3/2} e}+\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 )}{4 (-d)^{3/2} e}\\ &=\frac{b c \sqrt{1-c^2 x^2}}{16 \sqrt{-d} e \left (c^2 d+e\right ) \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{b c \sqrt{1-c^2 x^2}}{16 \sqrt{-d} e \left (c^2 d+e\right ) \left (\sqrt{-d}+\sqrt{e} x\right )}-\frac{a+b \sin ^{-1}(c x)}{16 \sqrt{-d} e^{3/2} \left (\sqrt{-d}-\sqrt{e} x\right )^2}-\frac{a+b \sin ^{-1}(c x)}{16 d e^{3/2} \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{a+b \sin ^{-1}(c x)}{16 \sqrt{-d} e^{3/2} \left (\sqrt{-d}+\sqrt{e} x\right )^2}+\frac{a+b \sin ^{-1}(c x)}{16 d e^{3/2} \left (\sqrt{-d}+\sqrt{e} x\right )}-\frac{b c^3 \tanh ^{-1}\left (\frac{\sqrt{e}-c^2 \sqrt{-d} x}{\sqrt{c^2 d+e} \sqrt{1-c^2 x^2}}\right )}{16 e^{3/2} \left (c^2 d+e\right )^{3/2}}+\frac{b c \tanh ^{-1}\left (\frac{\sqrt{e}-c^2 \sqrt{-d} x}{\sqrt{c^2 d+e} \sqrt{1-c^2 x^2}}\right )}{16 d e^{3/2} \sqrt{c^2 d+e}}-\frac{b c^3 \tanh ^{-1}\left (\frac{\sqrt{e}+c^2 \sqrt{-d} x}{\sqrt{c^2 d+e} \sqrt{1-c^2 x^2}}\right )}{16 e^{3/2} \left (c^2 d+e\right )^{3/2}}+\frac{b c \tanh ^{-1}\left (\frac{\sqrt{e}+c^2 \sqrt{-d} x}{\sqrt{c^2 d+e} \sqrt{1-c^2 x^2}}\right )}{16 d e^{3/2} \sqrt{c^2 d+e}}-\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 )}{16 (-d)^{3/2} e^{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 )}{16 (-d)^{3/2} e^{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 )}{16 (-d)^{3/2} e^{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 )}{16 (-d)^{3/2} e^{3/2}}-\frac{(3 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 )}{16 (-d)^{3/2} e^{3/2}}+\frac{(3 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 )}{16 (-d)^{3/2} e^{3/2}}-\frac{(3 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 )}{16 (-d)^{3/2} e^{3/2}}+\frac{(3 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 )}{16 (-d)^{3/2} e^{3/2}}+\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 )}{4 (-d)^{3/2} e^{3/2}}-\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 )}{4 (-d)^{3/2} e^{3/2}}+\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 )}{4 (-d)^{3/2} e^{3/2}}-\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 )}{4 (-d)^{3/2} e^{3/2}}\\ &=\frac{b c \sqrt{1-c^2 x^2}}{16 \sqrt{-d} e \left (c^2 d+e\right ) \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{b c \sqrt{1-c^2 x^2}}{16 \sqrt{-d} e \left (c^2 d+e\right ) \left (\sqrt{-d}+\sqrt{e} x\right )}-\frac{a+b \sin ^{-1}(c x)}{16 \sqrt{-d} e^{3/2} \left (\sqrt{-d}-\sqrt{e} x\right )^2}-\frac{a+b \sin ^{-1}(c x)}{16 d e^{3/2} \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{a+b \sin ^{-1}(c x)}{16 \sqrt{-d} e^{3/2} \left (\sqrt{-d}+\sqrt{e} x\right )^2}+\frac{a+b \sin ^{-1}(c x)}{16 d e^{3/2} \left (\sqrt{-d}+\sqrt{e} x\right )}-\frac{b c^3 \tanh ^{-1}\left (\frac{\sqrt{e}-c^2 \sqrt{-d} x}{\sqrt{c^2 d+e} \sqrt{1-c^2 x^2}}\right )}{16 e^{3/2} \left (c^2 d+e\right )^{3/2}}+\frac{b c \tanh ^{-1}\left (\frac{\sqrt{e}-c^2 \sqrt{-d} x}{\sqrt{c^2 d+e} \sqrt{1-c^2 x^2}}\right )}{16 d e^{3/2} \sqrt{c^2 d+e}}-\frac{b c^3 \tanh ^{-1}\left (\frac{\sqrt{e}+c^2 \sqrt{-d} x}{\sqrt{c^2 d+e} \sqrt{1-c^2 x^2}}\right )}{16 e^{3/2} \left (c^2 d+e\right )^{3/2}}+\frac{b c \tanh ^{-1}\left (\frac{\sqrt{e}+c^2 \sqrt{-d} x}{\sqrt{c^2 d+e} \sqrt{1-c^2 x^2}}\right )}{16 d e^{3/2} \sqrt{c^2 d+e}}-\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 )}{16 (-d)^{3/2} e^{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 )}{16 (-d)^{3/2} e^{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 )}{16 (-d)^{3/2} e^{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 )}{16 (-d)^{3/2} e^{3/2}}+\frac{(3 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 )}{16 (-d)^{3/2} e^{3/2}}-\frac{(3 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 )}{16 (-d)^{3/2} e^{3/2}}+\frac{(3 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 )}{16 (-d)^{3/2} e^{3/2}}-\frac{(3 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 )}{16 (-d)^{3/2} e^{3/2}}-\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 )}{4 (-d)^{3/2} e^{3/2}}+\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 )}{4 (-d)^{3/2} e^{3/2}}-\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 )}{4 (-d)^{3/2} e^{3/2}}+\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 )}{4 (-d)^{3/2} e^{3/2}}\\ &=\frac{b c \sqrt{1-c^2 x^2}}{16 \sqrt{-d} e \left (c^2 d+e\right ) \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{b c \sqrt{1-c^2 x^2}}{16 \sqrt{-d} e \left (c^2 d+e\right ) \left (\sqrt{-d}+\sqrt{e} x\right )}-\frac{a+b \sin ^{-1}(c x)}{16 \sqrt{-d} e^{3/2} \left (\sqrt{-d}-\sqrt{e} x\right )^2}-\frac{a+b \sin ^{-1}(c x)}{16 d e^{3/2} \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{a+b \sin ^{-1}(c x)}{16 \sqrt{-d} e^{3/2} \left (\sqrt{-d}+\sqrt{e} x\right )^2}+\frac{a+b \sin ^{-1}(c x)}{16 d e^{3/2} \left (\sqrt{-d}+\sqrt{e} x\right )}-\frac{b c^3 \tanh ^{-1}\left (\frac{\sqrt{e}-c^2 \sqrt{-d} x}{\sqrt{c^2 d+e} \sqrt{1-c^2 x^2}}\right )}{16 e^{3/2} \left (c^2 d+e\right )^{3/2}}+\frac{b c \tanh ^{-1}\left (\frac{\sqrt{e}-c^2 \sqrt{-d} x}{\sqrt{c^2 d+e} \sqrt{1-c^2 x^2}}\right )}{16 d e^{3/2} \sqrt{c^2 d+e}}-\frac{b c^3 \tanh ^{-1}\left (\frac{\sqrt{e}+c^2 \sqrt{-d} x}{\sqrt{c^2 d+e} \sqrt{1-c^2 x^2}}\right )}{16 e^{3/2} \left (c^2 d+e\right )^{3/2}}+\frac{b c \tanh ^{-1}\left (\frac{\sqrt{e}+c^2 \sqrt{-d} x}{\sqrt{c^2 d+e} \sqrt{1-c^2 x^2}}\right )}{16 d e^{3/2} \sqrt{c^2 d+e}}-\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 )}{16 (-d)^{3/2} e^{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 )}{16 (-d)^{3/2} e^{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 )}{16 (-d)^{3/2} e^{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 )}{16 (-d)^{3/2} e^{3/2}}-\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 )}{16 (-d)^{3/2} e^{3/2}}+\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 )}{16 (-d)^{3/2} e^{3/2}}-\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 )}{16 (-d)^{3/2} e^{3/2}}+\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 )}{16 (-d)^{3/2} e^{3/2}}\\ \end{align*}

Mathematica [A]  time = 6.02878, size = 1064, normalized size = 0.97 \[ \frac{a x}{8 d e \left (e x^2+d\right )}-\frac{a x}{4 e \left (e x^2+d\right )^2}+\frac{a \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{8 d^{3/2} e^{3/2}}+b \left (\frac{i \left (\frac{\sin ^{-1}(c x)}{i \sqrt{e} x+\sqrt{d}}-\frac{c \tan ^{-1}\left (\frac{\sqrt{d} x c^2+i \sqrt{e}}{\sqrt{d c^2+e} \sqrt{1-c^2 x^2}}\right )}{\sqrt{d c^2+e}}\right )}{16 d e^{3/2}}-\frac{-\frac{\sin ^{-1}(c x)}{\sqrt{e} x+i \sqrt{d}}-\frac{c \tanh ^{-1}\left (\frac{i \sqrt{d} x c^2+\sqrt{e}}{\sqrt{d c^2+e} \sqrt{1-c^2 x^2}}\right )}{\sqrt{d c^2+e}}}{16 d e^{3/2}}-\frac{i \left (-\frac{i \sqrt{d} \left (\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 )+\log (4)\right ) c^3}{\sqrt{e} \left (d c^2+e\right )^{3/2}}-\frac{\sqrt{1-c^2 x^2} c}{\left (d c^2+e\right ) \left (\sqrt{e} x-i \sqrt{d}\right )}-\frac{\sin ^{-1}(c x)}{\sqrt{e} \left (\sqrt{e} x-i \sqrt{d}\right )^2}\right )}{16 \sqrt{d} e}+\frac{i \left (\frac{i \sqrt{d} \left (\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 )+\log (4)\right ) c^3}{\sqrt{e} \left (d c^2+e\right )^{3/2}}-\frac{\sqrt{1-c^2 x^2} c}{\left (d c^2+e\right ) \left (\sqrt{e} x+i \sqrt{d}\right )}-\frac{\sin ^{-1}(c x)}{\sqrt{e} \left (\sqrt{e} x+i \sqrt{d}\right )^2}\right )}{16 \sqrt{d} e}-\frac{\sin ^{-1}(c x) \left (\sin ^{-1}(c x)+2 i \left (\log \left (\frac{e^{i \sin ^{-1}(c x)} \sqrt{e}}{c \sqrt{d}-\sqrt{d c^2+e}}+1\right )+\log \left (\frac{e^{i \sin ^{-1}(c x)} \sqrt{e}}{\sqrt{d} c+\sqrt{d c^2+e}}+1\right )\right )\right )+2 \text{PolyLog}\left (2,\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{d c^2+e}-c \sqrt{d}}\right )+2 \text{PolyLog}\left (2,-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{d} c+\sqrt{d c^2+e}}\right )}{32 d^{3/2} e^{3/2}}+\frac{\sin ^{-1}(c x) \left (\sin ^{-1}(c x)+2 i \left (\log \left (\frac{e^{i \sin ^{-1}(c x)} \sqrt{e}}{\sqrt{d c^2+e}-c \sqrt{d}}+1\right )+\log \left (1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{d} c+\sqrt{d c^2+e}}\right )\right )\right )+2 \text{PolyLog}\left (2,\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{c \sqrt{d}-\sqrt{d c^2+e}}\right )+2 \text{PolyLog}\left (2,\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{d} c+\sqrt{d c^2+e}}\right )}{32 d^{3/2} e^{3/2}}\right ) \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-(a*x)/(4*e*(d + e*x^2)^2) + (a*x)/(8*d*e*(d + e*x^2)) + (a*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(8*d^(3/2)*e^(3/2)) +
b*(((I/16)*(ArcSin[c*x]/(Sqrt[d] + I*Sqrt[e]*x) - (c*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]))/(d*e^(3/2)) - (-(ArcSin[c*x]/(I*Sqrt[d] + Sqrt[e]*x)) - (c*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])/(16*d*e^(3/2)) - ((I/16)*(-((c*Sqrt
[1 - c^2*x^2])/((c^2*d + e)*((-I)*Sqrt[d] + Sqrt[e]*x))) - ArcSin[c*x]/(Sqrt[e]*((-I)*Sqrt[d] + Sqrt[e]*x)^2)
- (I*c^3*Sqrt[d]*(Log[4] + 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))]))/(Sqrt[e]*(c^2*d + e)^(3/2))))/(Sqrt[d]*e) + ((I/16)*(-((c*Sqrt[1 - c^2
*x^2])/((c^2*d + e)*(I*Sqrt[d] + Sqrt[e]*x))) - ArcSin[c*x]/(Sqrt[e]*(I*Sqrt[d] + Sqrt[e]*x)^2) + (I*c^3*Sqrt[
d]*(Log[4] + 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))]))/(Sqrt[e]*(c^2*d + e)^(3/2))))/(Sqrt[d]*e) - (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*ArcSin[c*x]))/(-(c*Sqrt[d]) + Sqrt[c^2*d + e])] + 2*PolyLo
g[2, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] + Sqrt[c^2*d + e]))])/(32*d^(3/2)*e^(3/2)) + (ArcSin[c*x]*(ArcSi
n[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*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])])/(32*d^(3/2)*e^(3/2)))

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Maple [C]  time = 1.247, size = 2259, normalized size = 2.1 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*c*b*(-(2*c^2*d-2*(c^2*d*(c^2*d+e))^(1/2)+e)*e)^(1/2)*arctan(e*(I*c*x+(-c^2*x^2+1)^(1/2))/((-2*c^2*d+2*(c^2
*d*(c^2*d+e))^(1/2)-e)*e)^(1/2))/e^3/d/(c^2*d+e)*(c^2*d*(c^2*d+e))^(1/2)+1/8*c*b*((2*c^2*d+2*(c^2*d*(c^2*d+e))
^(1/2)+e)*e)^(1/2)*arctanh(e*(I*c*x+(-c^2*x^2+1)^(1/2))/((2*c^2*d+2*(c^2*d*(c^2*d+e))^(1/2)+e)*e)^(1/2))/(c^2*
d+e)^2/d/e^2*(c^2*d*(c^2*d+e))^(1/2)+1/8*c^4*b*e/d/(c^2*d+e)/(c^2*e*x^2+c^2*d)^2*arcsin(c*x)*x^3-1/8*c^6*b/e/(
c^2*d+e)/(c^2*e*x^2+c^2*d)^2*arcsin(c*x)*x*d-1/4*c*b*((2*c^2*d+2*(c^2*d*(c^2*d+e))^(1/2)+e)*e)^(1/2)*arctanh(e
*(I*c*x+(-c^2*x^2+1)^(1/2))/((2*c^2*d+2*(c^2*d*(c^2*d+e))^(1/2)+e)*e)^(1/2))/e^3/d/(c^2*d+e)*(c^2*d*(c^2*d+e))
^(1/2)-1/8*c*b*(-(2*c^2*d-2*(c^2*d*(c^2*d+e))^(1/2)+e)*e)^(1/2)*arctan(e*(I*c*x+(-c^2*x^2+1)^(1/2))/((-2*c^2*d
+2*(c^2*d*(c^2*d+e))^(1/2)-e)*e)^(1/2))/(c^2*d+e)^2/d/e^2*(c^2*d*(c^2*d+e))^(1/2)-1/4*c^5*b*((2*c^2*d+2*(c^2*d
*(c^2*d+e))^(1/2)+e)*e)^(1/2)*arctanh(e*(I*c*x+(-c^2*x^2+1)^(1/2))/((2*c^2*d+2*(c^2*d*(c^2*d+e))^(1/2)+e)*e)^(
1/2))/e^3/(c^2*d+e)^2*d-1/4*c^5*b*(-(2*c^2*d-2*(c^2*d*(c^2*d+e))^(1/2)+e)*e)^(1/2)*arctan(e*(I*c*x+(-c^2*x^2+1
)^(1/2))/((-2*c^2*d+2*(c^2*d*(c^2*d+e))^(1/2)-e)*e)^(1/2))/e^3/(c^2*d+e)^2*d+1/8*c*b*(-(2*c^2*d-2*(c^2*d*(c^2*
d+e))^(1/2)+e)*e)^(1/2)*arctan(e*(I*c*x+(-c^2*x^2+1)^(1/2))/((-2*c^2*d+2*(c^2*d*(c^2*d+e))^(1/2)-e)*e)^(1/2))/
e^2/d/(c^2*d+e)+1/8*c*b*((2*c^2*d+2*(c^2*d*(c^2*d+e))^(1/2)+e)*e)^(1/2)*arctanh(e*(I*c*x+(-c^2*x^2+1)^(1/2))/(
(2*c^2*d+2*(c^2*d*(c^2*d+e))^(1/2)+e)*e)^(1/2))/e^2/d/(c^2*d+e)-1/8*c^5*b/e*d/(c^2*d+e)/(c^2*e*x^2+c^2*d)^2*(-
c^2*x^2+1)^(1/2)-1/4*c^3*b*(-(2*c^2*d-2*(c^2*d*(c^2*d+e))^(1/2)+e)*e)^(1/2)*arctan(e*(I*c*x+(-c^2*x^2+1)^(1/2)
)/((-2*c^2*d+2*(c^2*d*(c^2*d+e))^(1/2)-e)*e)^(1/2))/e^3/(c^2*d+e)^2*(c^2*d*(c^2*d+e))^(1/2)+1/4*c^3*b*((2*c^2*
d+2*(c^2*d*(c^2*d+e))^(1/2)+e)*e)^(1/2)*arctanh(e*(I*c*x+(-c^2*x^2+1)^(1/2))/((2*c^2*d+2*(c^2*d*(c^2*d+e))^(1/
2)+e)*e)^(1/2))/e^3/(c^2*d+e)^2*(c^2*d*(c^2*d+e))^(1/2)+1/16*c^3*b/e/(c^2*d+e)*sum(_R1/(_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))+1/8*a/d/e/(d*e)^(1/2)*arctan(e*x/(d*e)^(1/2))-1/8*c^4*a/(c^2*e*x^2+c^2*d)^2/e*x+1/16
*c*b/d/(c^2*d+e)*sum(1/_R1/(_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((_R
1-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))+1/16*c*b/d/(c^2*d+e)*sum(_R1/(_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))+1/16*c^3*b/e/(c^2*d+e)*sum(1/_R1/(_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))+1/8*c^4*a/(c^2*e*x^2+c^2*d)^2/d*x^3+1/4*c^3*b*((2*c^2*d+2*(c^2*d*(c^2*d+e))^(1/2)+e)*e)^(1/2)*
arctanh(e*(I*c*x+(-c^2*x^2+1)^(1/2))/((2*c^2*d+2*(c^2*d*(c^2*d+e))^(1/2)+e)*e)^(1/2))/e^3/(c^2*d+e)+1/8*c^6*b/
(c^2*d+e)/(c^2*e*x^2+c^2*d)^2*arcsin(c*x)*x^3-1/8*c^5*b/(c^2*d+e)/(c^2*e*x^2+c^2*d)^2*(-c^2*x^2+1)^(1/2)*x^2-1
/8*c^4*b/(c^2*d+e)/(c^2*e*x^2+c^2*d)^2*arcsin(c*x)*x-1/4*c^3*b*(-(2*c^2*d-2*(c^2*d*(c^2*d+e))^(1/2)+e)*e)^(1/2
)*arctan(e*(I*c*x+(-c^2*x^2+1)^(1/2))/((-2*c^2*d+2*(c^2*d*(c^2*d+e))^(1/2)-e)*e)^(1/2))/(c^2*d+e)^2/e^2-1/4*c^
3*b*((2*c^2*d+2*(c^2*d*(c^2*d+e))^(1/2)+e)*e)^(1/2)*arctanh(e*(I*c*x+(-c^2*x^2+1)^(1/2))/((2*c^2*d+2*(c^2*d*(c
^2*d+e))^(1/2)+e)*e)^(1/2))/(c^2*d+e)^2/e^2+1/4*c^3*b*(-(2*c^2*d-2*(c^2*d*(c^2*d+e))^(1/2)+e)*e)^(1/2)*arctan(
e*(I*c*x+(-c^2*x^2+1)^(1/2))/((-2*c^2*d+2*(c^2*d*(c^2*d+e))^(1/2)-e)*e)^(1/2))/e^3/(c^2*d+e)

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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))/(e*x^2+d)^3,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{b x^{2} \arcsin \left (c x\right ) + a x^{2}}{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^2*(a+b*arcsin(c*x))/(e*x^2+d)^3,x, algorithm="fricas")

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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