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

Optimal. Leaf size=1092 \[ -\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 {Li}_2\left (-\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 {Li}_2\left (\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 {Li}_2\left (-\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 {Li}_2\left (\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}} \]

[Out]

-1/16*b*c^3*arctanh((-c^2*x*(-d)^(1/2)+e^(1/2))/(c^2*d+e)^(1/2)/(-c^2*x^2+1)^(1/2))/e^(3/2)/(c^2*d+e)^(3/2)-1/
16*b*c^3*arctanh((c^2*x*(-d)^(1/2)+e^(1/2))/(c^2*d+e)^(1/2)/(-c^2*x^2+1)^(1/2))/e^(3/2)/(c^2*d+e)^(3/2)-1/16*(
a+b*arcsin(c*x))*ln(1-(I*c*x+(-c^2*x^2+1)^(1/2))*e^(1/2)/(I*c*(-d)^(1/2)-(c^2*d+e)^(1/2)))/(-d)^(3/2)/e^(3/2)+
1/16*(a+b*arcsin(c*x))*ln(1+(I*c*x+(-c^2*x^2+1)^(1/2))*e^(1/2)/(I*c*(-d)^(1/2)-(c^2*d+e)^(1/2)))/(-d)^(3/2)/e^
(3/2)-1/16*(a+b*arcsin(c*x))*ln(1-(I*c*x+(-c^2*x^2+1)^(1/2))*e^(1/2)/(I*c*(-d)^(1/2)+(c^2*d+e)^(1/2)))/(-d)^(3
/2)/e^(3/2)+1/16*(a+b*arcsin(c*x))*ln(1+(I*c*x+(-c^2*x^2+1)^(1/2))*e^(1/2)/(I*c*(-d)^(1/2)+(c^2*d+e)^(1/2)))/(
-d)^(3/2)/e^(3/2)+1/16*I*b*polylog(2,(I*c*x+(-c^2*x^2+1)^(1/2))*e^(1/2)/(I*c*(-d)^(1/2)-(c^2*d+e)^(1/2)))/(-d)
^(3/2)/e^(3/2)-1/16*I*b*polylog(2,-(I*c*x+(-c^2*x^2+1)^(1/2))*e^(1/2)/(I*c*(-d)^(1/2)+(c^2*d+e)^(1/2)))/(-d)^(
3/2)/e^(3/2)+1/16*I*b*polylog(2,(I*c*x+(-c^2*x^2+1)^(1/2))*e^(1/2)/(I*c*(-d)^(1/2)+(c^2*d+e)^(1/2)))/(-d)^(3/2
)/e^(3/2)-1/16*I*b*polylog(2,-(I*c*x+(-c^2*x^2+1)^(1/2))*e^(1/2)/(I*c*(-d)^(1/2)-(c^2*d+e)^(1/2)))/(-d)^(3/2)/
e^(3/2)+1/16*(-a-b*arcsin(c*x))/e^(3/2)/(-d)^(1/2)/((-d)^(1/2)-x*e^(1/2))^2+1/16*(-a-b*arcsin(c*x))/d/e^(3/2)/
((-d)^(1/2)-x*e^(1/2))+1/16*(a+b*arcsin(c*x))/e^(3/2)/(-d)^(1/2)/((-d)^(1/2)+x*e^(1/2))^2+1/16*(a+b*arcsin(c*x
))/d/e^(3/2)/((-d)^(1/2)+x*e^(1/2))+1/16*b*c*arctanh((-c^2*x*(-d)^(1/2)+e^(1/2))/(c^2*d+e)^(1/2)/(-c^2*x^2+1)^
(1/2))/d/e^(3/2)/(c^2*d+e)^(1/2)+1/16*b*c*arctanh((c^2*x*(-d)^(1/2)+e^(1/2))/(c^2*d+e)^(1/2)/(-c^2*x^2+1)^(1/2
))/d/e^(3/2)/(c^2*d+e)^(1/2)+1/16*b*c*(-c^2*x^2+1)^(1/2)/e/(c^2*d+e)/(-d)^(1/2)/((-d)^(1/2)-x*e^(1/2))+1/16*b*
c*(-c^2*x^2+1)^(1/2)/e/(c^2*d+e)/(-d)^(1/2)/((-d)^(1/2)+x*e^(1/2))

________________________________________________________________________________________

Rubi [A]  time = 2.61, antiderivative size = 1092, normalized size of antiderivative = 1.00, 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 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 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 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 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 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 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 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 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 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]

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*}

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Mathematica [A]  time = 6.08, 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 {Li}_2\left (\frac {\sqrt {e} e^{i \sin ^{-1}(c x)}}{\sqrt {d c^2+e}-c \sqrt {d}}\right )+2 \text {Li}_2\left (-\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 {Li}_2\left (\frac {\sqrt {e} e^{i \sin ^{-1}(c x)}}{c \sqrt {d}-\sqrt {d c^2+e}}\right )+2 \text {Li}_2\left (\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]

-1/4*(a*x)/(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]*Sq
rt[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*Sq
rt[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*Sqr
t[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*Poly
Log[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]*(Arc
Sin[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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fricas [F]  time = 0.63, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b x^{2} \arcsin \left (c x\right ) + a x^{2}}{e^{3} x^{6} + 3 \, d e^{2} x^{4} + 3 \, d^{2} e x^{2} + d^{3}}, x\right ) \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \arcsin \left (c x\right ) + a\right )} x^{2}}{{\left (e x^{2} + d\right )}^{3}}\,{d x} \]

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)

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maple [C]  time = 2.18, size = 2259, normalized size = 2.07 \[ \text {result too large to display} \]

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]

1/8*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*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)-1/4*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)/e^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+1/8*c^6*b/(c^2*d+e)/(c^2*e*x^2+c
^2*d)^2*arcsin(c*x)*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)*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)/e^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))/(c^
2*d+e)^2/e^2+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))+1/16*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))+1/16*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((_R1-I*c*x-(-c^2*x^2+1)^(1/2))/_R1)),_R1=Roo
tOf(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
))+1/8*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*b*(-(2*c^2*d-2*(c^2*d*(c^2*d+e))^(1/2)+e)*e)^(1/2)*ar
ctan(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)-1/4*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^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)-1/4*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^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^5*b/e*d/(c^2*d+e)/(c^2*e*x^2+c^2*d)^2*(-c^2*x^2+1)^(1/2)-1/8*c^4*a/(c^2*e*x^2+c^
2*d)^2/e*x+1/8*c^4*a/(c^2*e*x^2+c^2*d)^2/d*x^3+1/8*a/d/e/(d*e)^(1/2)*arctan(e*x/(d*e)^(1/2))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{8} \, a {\left (\frac {e x^{3} - d x}{d e^{3} x^{4} + 2 \, d^{2} e^{2} x^{2} + d^{3} e} + \frac {\arctan \left (\frac {e x}{\sqrt {d e}}\right )}{\sqrt {d e} d e}\right )} + b \int \frac {x^{2} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )}{e^{3} x^{6} + 3 \, d e^{2} x^{4} + 3 \, d^{2} e x^{2} + d^{3}}\,{d x} \]

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]

1/8*a*((e*x^3 - d*x)/(d*e^3*x^4 + 2*d^2*e^2*x^2 + d^3*e) + arctan(e*x/sqrt(d*e))/(sqrt(d*e)*d*e)) + b*integrat
e(x^2*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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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^2\,\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}{{\left (e\,x^2+d\right )}^3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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