\(\int \frac {x^4 (a+b \arcsin (c x))}{(d+e x^2)^3} \, dx\) [646]

Optimal result

Integrand size = 21, antiderivative size = 1082 \[ \int \frac {x^4 (a+b \arcsin (c x))}{\left (d+e x^2\right )^3} \, dx=\frac {b c \sqrt {-d} \sqrt {1-c^2 x^2}}{16 e^2 \left (c^2 d+e\right ) \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {b c \sqrt {-d} \sqrt {1-c^2 x^2}}{16 e^2 \left (c^2 d+e\right ) \left (\sqrt {-d}+\sqrt {e} x\right )}-\frac {\sqrt {-d} (a+b \arcsin (c x))}{16 e^{5/2} \left (\sqrt {-d}-\sqrt {e} x\right )^2}+\frac {5 (a+b \arcsin (c x))}{16 e^{5/2} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {\sqrt {-d} (a+b \arcsin (c x))}{16 e^{5/2} \left (\sqrt {-d}+\sqrt {e} x\right )^2}-\frac {5 (a+b \arcsin (c x))}{16 e^{5/2} \left (\sqrt {-d}+\sqrt {e} x\right )}+\frac {b c^3 d \text {arctanh}\left (\frac {\sqrt {e}-c^2 \sqrt {-d} x}{\sqrt {c^2 d+e} \sqrt {1-c^2 x^2}}\right )}{16 e^{5/2} \left (c^2 d+e\right )^{3/2}}-\frac {5 b c \text {arctanh}\left (\frac {\sqrt {e}-c^2 \sqrt {-d} x}{\sqrt {c^2 d+e} \sqrt {1-c^2 x^2}}\right )}{16 e^{5/2} \sqrt {c^2 d+e}}+\frac {b c^3 d \text {arctanh}\left (\frac {\sqrt {e}+c^2 \sqrt {-d} x}{\sqrt {c^2 d+e} \sqrt {1-c^2 x^2}}\right )}{16 e^{5/2} \left (c^2 d+e\right )^{3/2}}-\frac {5 b c \text {arctanh}\left (\frac {\sqrt {e}+c^2 \sqrt {-d} x}{\sqrt {c^2 d+e} \sqrt {1-c^2 x^2}}\right )}{16 e^{5/2} \sqrt {c^2 d+e}}+\frac {3 (a+b \arcsin (c x)) \log \left (1-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{16 \sqrt {-d} e^{5/2}}-\frac {3 (a+b \arcsin (c x)) \log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{16 \sqrt {-d} e^{5/2}}+\frac {3 (a+b \arcsin (c x)) \log \left (1-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{16 \sqrt {-d} e^{5/2}}-\frac {3 (a+b \arcsin (c x)) \log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{16 \sqrt {-d} e^{5/2}}+\frac {3 i b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{16 \sqrt {-d} e^{5/2}}-\frac {3 i b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{16 \sqrt {-d} e^{5/2}}+\frac {3 i b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{16 \sqrt {-d} e^{5/2}}-\frac {3 i b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{16 \sqrt {-d} e^{5/2}} \]

[Out]

1/16*b*c^3*d*arctanh((-c^2*x*(-d)^(1/2)+e^(1/2))/(c^2*d+e)^(1/2)/(-c^2*x^2+1)^(1/2))/e^(5/2)/(c^2*d+e)^(3/2)+1
/16*b*c^3*d*arctanh((c^2*x*(-d)^(1/2)+e^(1/2))/(c^2*d+e)^(1/2)/(-c^2*x^2+1)^(1/2))/e^(5/2)/(c^2*d+e)^(3/2)+3/1
6*(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)))/e^(5/2)/(-d)^(1/
2)-3/16*(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)))/e^(5/2)/(-
d)^(1/2)+3/16*(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)))/e^(5
/2)/(-d)^(1/2)-3/16*(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))
)/e^(5/2)/(-d)^(1/2)-3/16*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)))/e
^(5/2)/(-d)^(1/2)-3/16*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)))/e^(5
/2)/(-d)^(1/2)+3/16*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)))/e^(5/2
)/(-d)^(1/2)+3/16*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)))/e^(5/2)/
(-d)^(1/2)-1/16*(a+b*arcsin(c*x))*(-d)^(1/2)/e^(5/2)/((-d)^(1/2)-x*e^(1/2))^2+5/16*(a+b*arcsin(c*x))/e^(5/2)/(
(-d)^(1/2)-x*e^(1/2))+1/16*(a+b*arcsin(c*x))*(-d)^(1/2)/e^(5/2)/((-d)^(1/2)+x*e^(1/2))^2-5/16*(a+b*arcsin(c*x)
)/e^(5/2)/((-d)^(1/2)+x*e^(1/2))-5/16*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))/e^(5/2)/(c^2*d+e)^(1/2)-5/16*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))/e^
(5/2)/(c^2*d+e)^(1/2)+1/16*b*c*(-d)^(1/2)*(-c^2*x^2+1)^(1/2)/e^2/(c^2*d+e)/((-d)^(1/2)-x*e^(1/2))+1/16*b*c*(-d
)^(1/2)*(-c^2*x^2+1)^(1/2)/e^2/(c^2*d+e)/((-d)^(1/2)+x*e^(1/2))

Rubi [A] (verified)

Time = 2.41 (sec) , antiderivative size = 1082, normalized size of antiderivative = 1.00, number of steps used = 80, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.524, Rules used = {4817, 4757, 4827, 745, 739, 212, 4825, 4617, 2221, 2317, 2438} \[ \int \frac {x^4 (a+b \arcsin (c x))}{\left (d+e x^2\right )^3} \, dx=\frac {b d \text {arctanh}\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^{5/2} \left (d c^2+e\right )^{3/2}}+\frac {b d \text {arctanh}\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^{5/2} \left (d c^2+e\right )^{3/2}}-\frac {5 b \text {arctanh}\left (\frac {\sqrt {e}-c^2 \sqrt {-d} x}{\sqrt {d c^2+e} \sqrt {1-c^2 x^2}}\right ) c}{16 e^{5/2} \sqrt {d c^2+e}}-\frac {5 b \text {arctanh}\left (\frac {\sqrt {-d} x c^2+\sqrt {e}}{\sqrt {d c^2+e} \sqrt {1-c^2 x^2}}\right ) c}{16 e^{5/2} \sqrt {d c^2+e}}+\frac {b \sqrt {-d} \sqrt {1-c^2 x^2} c}{16 e^2 \left (d c^2+e\right ) \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {b \sqrt {-d} \sqrt {1-c^2 x^2} c}{16 e^2 \left (d c^2+e\right ) \left (\sqrt {e} x+\sqrt {-d}\right )}+\frac {5 (a+b \arcsin (c x))}{16 e^{5/2} \left (\sqrt {-d}-\sqrt {e} x\right )}-\frac {5 (a+b \arcsin (c x))}{16 e^{5/2} \left (\sqrt {e} x+\sqrt {-d}\right )}-\frac {\sqrt {-d} (a+b \arcsin (c x))}{16 e^{5/2} \left (\sqrt {-d}-\sqrt {e} x\right )^2}+\frac {\sqrt {-d} (a+b \arcsin (c x))}{16 e^{5/2} \left (\sqrt {e} x+\sqrt {-d}\right )^2}+\frac {3 (a+b \arcsin (c x)) \log \left (1-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {d c^2+e}}\right )}{16 \sqrt {-d} e^{5/2}}-\frac {3 (a+b \arcsin (c x)) \log \left (\frac {e^{i \arcsin (c x)} \sqrt {e}}{i c \sqrt {-d}-\sqrt {d c^2+e}}+1\right )}{16 \sqrt {-d} e^{5/2}}+\frac {3 (a+b \arcsin (c x)) \log \left (1-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i \sqrt {-d} c+\sqrt {d c^2+e}}\right )}{16 \sqrt {-d} e^{5/2}}-\frac {3 (a+b \arcsin (c x)) \log \left (\frac {e^{i \arcsin (c x)} \sqrt {e}}{i \sqrt {-d} c+\sqrt {d c^2+e}}+1\right )}{16 \sqrt {-d} e^{5/2}}+\frac {3 i b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {d c^2+e}}\right )}{16 \sqrt {-d} e^{5/2}}-\frac {3 i b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {d c^2+e}}\right )}{16 \sqrt {-d} e^{5/2}}+\frac {3 i b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i \sqrt {-d} c+\sqrt {d c^2+e}}\right )}{16 \sqrt {-d} e^{5/2}}-\frac {3 i b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{i \sqrt {-d} c+\sqrt {d c^2+e}}\right )}{16 \sqrt {-d} e^{5/2}} \]

[In]

Int[(x^4*(a + b*ArcSin[c*x]))/(d + e*x^2)^3,x]

[Out]

(b*c*Sqrt[-d]*Sqrt[1 - c^2*x^2])/(16*e^2*(c^2*d + e)*(Sqrt[-d] - Sqrt[e]*x)) + (b*c*Sqrt[-d]*Sqrt[1 - c^2*x^2]
)/(16*e^2*(c^2*d + e)*(Sqrt[-d] + Sqrt[e]*x)) - (Sqrt[-d]*(a + b*ArcSin[c*x]))/(16*e^(5/2)*(Sqrt[-d] - Sqrt[e]
*x)^2) + (5*(a + b*ArcSin[c*x]))/(16*e^(5/2)*(Sqrt[-d] - Sqrt[e]*x)) + (Sqrt[-d]*(a + b*ArcSin[c*x]))/(16*e^(5
/2)*(Sqrt[-d] + Sqrt[e]*x)^2) - (5*(a + b*ArcSin[c*x]))/(16*e^(5/2)*(Sqrt[-d] + Sqrt[e]*x)) + (b*c^3*d*ArcTanh
[(Sqrt[e] - c^2*Sqrt[-d]*x)/(Sqrt[c^2*d + e]*Sqrt[1 - c^2*x^2])])/(16*e^(5/2)*(c^2*d + e)^(3/2)) - (5*b*c*ArcT
anh[(Sqrt[e] - c^2*Sqrt[-d]*x)/(Sqrt[c^2*d + e]*Sqrt[1 - c^2*x^2])])/(16*e^(5/2)*Sqrt[c^2*d + e]) + (b*c^3*d*A
rcTanh[(Sqrt[e] + c^2*Sqrt[-d]*x)/(Sqrt[c^2*d + e]*Sqrt[1 - c^2*x^2])])/(16*e^(5/2)*(c^2*d + e)^(3/2)) - (5*b*
c*ArcTanh[(Sqrt[e] + c^2*Sqrt[-d]*x)/(Sqrt[c^2*d + e]*Sqrt[1 - c^2*x^2])])/(16*e^(5/2)*Sqrt[c^2*d + 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])])/(16*Sqrt[-d]*e^(5/2)
) - (3*(a + b*ArcSin[c*x])*Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/(16*Sqrt[-d]
*e^(5/2)) + (3*(a + b*ArcSin[c*x])*Log[1 - (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])])/(16*
Sqrt[-d]*e^(5/2)) - (3*(a + b*ArcSin[c*x])*Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e]
)])/(16*Sqrt[-d]*e^(5/2)) + (((3*I)/16)*b*PolyLog[2, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d
+ e]))])/(Sqrt[-d]*e^(5/2)) - (((3*I)/16)*b*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d
+ e])])/(Sqrt[-d]*e^(5/2)) + (((3*I)/16)*b*PolyLog[2, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d
+ e]))])/(Sqrt[-d]*e^(5/2)) - (((3*I)/16)*b*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d
+ e])])/(Sqrt[-d]*e^(5/2))

Rule 212

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

Rule 739

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 745

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 2221

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/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], 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 2317

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 2438

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 4617

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 4757

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 4817

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 4825

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 4827

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*} \text {integral}& = \int \left (\frac {d^2 (a+b \arcsin (c x))}{e^2 \left (d+e x^2\right )^3}-\frac {2 d (a+b \arcsin (c x))}{e^2 \left (d+e x^2\right )^2}+\frac {a+b \arcsin (c x)}{e^2 \left (d+e x^2\right )}\right ) \, dx \\ & = \frac {\int \frac {a+b \arcsin (c x)}{d+e x^2} \, dx}{e^2}-\frac {(2 d) \int \frac {a+b \arcsin (c x)}{\left (d+e x^2\right )^2} \, dx}{e^2}+\frac {d^2 \int \frac {a+b \arcsin (c x)}{\left (d+e x^2\right )^3} \, dx}{e^2} \\ & = \frac {\int \left (\frac {\sqrt {-d} (a+b \arcsin (c x))}{2 d \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {\sqrt {-d} (a+b \arcsin (c x))}{2 d \left (\sqrt {-d}+\sqrt {e} x\right )}\right ) \, dx}{e^2}-\frac {(2 d) \int \left (-\frac {e (a+b \arcsin (c x))}{4 d \left (\sqrt {-d} \sqrt {e}-e x\right )^2}-\frac {e (a+b \arcsin (c x))}{4 d \left (\sqrt {-d} \sqrt {e}+e x\right )^2}-\frac {e (a+b \arcsin (c x))}{2 d \left (-d e-e^2 x^2\right )}\right ) \, dx}{e^2}+\frac {d^2 \int \left (-\frac {e^{3/2} (a+b \arcsin (c x))}{8 (-d)^{3/2} \left (\sqrt {-d} \sqrt {e}-e x\right )^3}-\frac {3 e (a+b \arcsin (c x))}{16 d^2 \left (\sqrt {-d} \sqrt {e}-e x\right )^2}-\frac {e^{3/2} (a+b \arcsin (c x))}{8 (-d)^{3/2} \left (\sqrt {-d} \sqrt {e}+e x\right )^3}-\frac {3 e (a+b \arcsin (c x))}{16 d^2 \left (\sqrt {-d} \sqrt {e}+e x\right )^2}-\frac {3 e (a+b \arcsin (c x))}{8 d^2 \left (-d e-e^2 x^2\right )}\right ) \, dx}{e^2} \\ & = -\frac {\int \frac {a+b \arcsin (c x)}{\sqrt {-d}-\sqrt {e} x} \, dx}{2 \sqrt {-d} e^2}-\frac {\int \frac {a+b \arcsin (c x)}{\sqrt {-d}+\sqrt {e} x} \, dx}{2 \sqrt {-d} e^2}-\frac {3 \int \frac {a+b \arcsin (c x)}{\left (\sqrt {-d} \sqrt {e}-e x\right )^2} \, dx}{16 e}-\frac {3 \int \frac {a+b \arcsin (c x)}{\left (\sqrt {-d} \sqrt {e}+e x\right )^2} \, dx}{16 e}-\frac {3 \int \frac {a+b \arcsin (c x)}{-d e-e^2 x^2} \, dx}{8 e}+\frac {\int \frac {a+b \arcsin (c x)}{\left (\sqrt {-d} \sqrt {e}-e x\right )^2} \, dx}{2 e}+\frac {\int \frac {a+b \arcsin (c x)}{\left (\sqrt {-d} \sqrt {e}+e x\right )^2} \, dx}{2 e}+\frac {\int \frac {a+b \arcsin (c x)}{-d e-e^2 x^2} \, dx}{e}-\frac {\sqrt {-d} \int \frac {a+b \arcsin (c x)}{\left (\sqrt {-d} \sqrt {e}-e x\right )^3} \, dx}{8 \sqrt {e}}-\frac {\sqrt {-d} \int \frac {a+b \arcsin (c x)}{\left (\sqrt {-d} \sqrt {e}+e x\right )^3} \, dx}{8 \sqrt {e}} \\ & = -\frac {\sqrt {-d} (a+b \arcsin (c x))}{16 e^{5/2} \left (\sqrt {-d}-\sqrt {e} x\right )^2}+\frac {5 (a+b \arcsin (c x))}{16 e^{5/2} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {\sqrt {-d} (a+b \arcsin (c x))}{16 e^{5/2} \left (\sqrt {-d}+\sqrt {e} x\right )^2}-\frac {5 (a+b \arcsin (c x))}{16 e^{5/2} \left (\sqrt {-d}+\sqrt {e} x\right )}+\frac {(3 b c) \int \frac {1}{\left (\sqrt {-d} \sqrt {e}-e x\right ) \sqrt {1-c^2 x^2}} \, dx}{16 e^2}-\frac {(3 b c) \int \frac {1}{\left (\sqrt {-d} \sqrt {e}+e x\right ) \sqrt {1-c^2 x^2}} \, dx}{16 e^2}-\frac {(b c) \int \frac {1}{\left (\sqrt {-d} \sqrt {e}-e x\right ) \sqrt {1-c^2 x^2}} \, dx}{2 e^2}+\frac {(b c) \int \frac {1}{\left (\sqrt {-d} \sqrt {e}+e x\right ) \sqrt {1-c^2 x^2}} \, dx}{2 e^2}-\frac {\text {Subst}\left (\int \frac {(a+b x) \cos (x)}{c \sqrt {-d}-\sqrt {e} \sin (x)} \, dx,x,\arcsin (c x)\right )}{2 \sqrt {-d} e^2}-\frac {\text {Subst}\left (\int \frac {(a+b x) \cos (x)}{c \sqrt {-d}+\sqrt {e} \sin (x)} \, dx,x,\arcsin (c x)\right )}{2 \sqrt {-d} e^2}+\frac {\left (b c \sqrt {-d}\right ) \int \frac {1}{\left (\sqrt {-d} \sqrt {e}-e x\right )^2 \sqrt {1-c^2 x^2}} \, dx}{16 e^{3/2}}-\frac {\left (b c \sqrt {-d}\right ) \int \frac {1}{\left (\sqrt {-d} \sqrt {e}+e x\right )^2 \sqrt {1-c^2 x^2}} \, dx}{16 e^{3/2}}-\frac {3 \int \left (-\frac {\sqrt {-d} (a+b \arcsin (c x))}{2 d e \left (\sqrt {-d}-\sqrt {e} x\right )}-\frac {\sqrt {-d} (a+b \arcsin (c x))}{2 d e \left (\sqrt {-d}+\sqrt {e} x\right )}\right ) \, dx}{8 e}+\frac {\int \left (-\frac {\sqrt {-d} (a+b \arcsin (c x))}{2 d e \left (\sqrt {-d}-\sqrt {e} x\right )}-\frac {\sqrt {-d} (a+b \arcsin (c x))}{2 d e \left (\sqrt {-d}+\sqrt {e} x\right )}\right ) \, dx}{e} \\ & = \frac {b c \sqrt {-d} \sqrt {1-c^2 x^2}}{16 e^2 \left (c^2 d+e\right ) \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {b c \sqrt {-d} \sqrt {1-c^2 x^2}}{16 e^2 \left (c^2 d+e\right ) \left (\sqrt {-d}+\sqrt {e} x\right )}-\frac {\sqrt {-d} (a+b \arcsin (c x))}{16 e^{5/2} \left (\sqrt {-d}-\sqrt {e} x\right )^2}+\frac {5 (a+b \arcsin (c x))}{16 e^{5/2} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {\sqrt {-d} (a+b \arcsin (c x))}{16 e^{5/2} \left (\sqrt {-d}+\sqrt {e} x\right )^2}-\frac {5 (a+b \arcsin (c x))}{16 e^{5/2} \left (\sqrt {-d}+\sqrt {e} x\right )}-\frac {(3 b c) \text {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^2}+\frac {(3 b c) \text {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^2}+\frac {(b c) \text {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 )}{2 e^2}-\frac {(b c) \text {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 )}{2 e^2}-\frac {i \text {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,\arcsin (c x)\right )}{2 \sqrt {-d} e^2}-\frac {i \text {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,\arcsin (c x)\right )}{2 \sqrt {-d} e^2}-\frac {i \text {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,\arcsin (c x)\right )}{2 \sqrt {-d} e^2}-\frac {i \text {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,\arcsin (c x)\right )}{2 \sqrt {-d} e^2}-\frac {3 \int \frac {a+b \arcsin (c x)}{\sqrt {-d}-\sqrt {e} x} \, dx}{16 \sqrt {-d} e^2}-\frac {3 \int \frac {a+b \arcsin (c x)}{\sqrt {-d}+\sqrt {e} x} \, dx}{16 \sqrt {-d} e^2}+\frac {\int \frac {a+b \arcsin (c x)}{\sqrt {-d}-\sqrt {e} x} \, dx}{2 \sqrt {-d} e^2}+\frac {\int \frac {a+b \arcsin (c x)}{\sqrt {-d}+\sqrt {e} x} \, dx}{2 \sqrt {-d} e^2}+\frac {\left (b c^3 d\right ) \int \frac {1}{\left (\sqrt {-d} \sqrt {e}-e x\right ) \sqrt {1-c^2 x^2}} \, dx}{16 e^2 \left (c^2 d+e\right )}-\frac {\left (b c^3 d\right ) \int \frac {1}{\left (\sqrt {-d} \sqrt {e}+e x\right ) \sqrt {1-c^2 x^2}} \, dx}{16 e^2 \left (c^2 d+e\right )} \\ & = \frac {b c \sqrt {-d} \sqrt {1-c^2 x^2}}{16 e^2 \left (c^2 d+e\right ) \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {b c \sqrt {-d} \sqrt {1-c^2 x^2}}{16 e^2 \left (c^2 d+e\right ) \left (\sqrt {-d}+\sqrt {e} x\right )}-\frac {\sqrt {-d} (a+b \arcsin (c x))}{16 e^{5/2} \left (\sqrt {-d}-\sqrt {e} x\right )^2}+\frac {5 (a+b \arcsin (c x))}{16 e^{5/2} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {\sqrt {-d} (a+b \arcsin (c x))}{16 e^{5/2} \left (\sqrt {-d}+\sqrt {e} x\right )^2}-\frac {5 (a+b \arcsin (c x))}{16 e^{5/2} \left (\sqrt {-d}+\sqrt {e} x\right )}-\frac {5 b c \text {arctanh}\left (\frac {\sqrt {e}-c^2 \sqrt {-d} x}{\sqrt {c^2 d+e} \sqrt {1-c^2 x^2}}\right )}{16 e^{5/2} \sqrt {c^2 d+e}}-\frac {5 b c \text {arctanh}\left (\frac {\sqrt {e}+c^2 \sqrt {-d} x}{\sqrt {c^2 d+e} \sqrt {1-c^2 x^2}}\right )}{16 e^{5/2} \sqrt {c^2 d+e}}+\frac {(a+b \arcsin (c x)) \log \left (1-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{2 \sqrt {-d} e^{5/2}}-\frac {(a+b \arcsin (c x)) \log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{2 \sqrt {-d} e^{5/2}}+\frac {(a+b \arcsin (c x)) \log \left (1-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{2 \sqrt {-d} e^{5/2}}-\frac {(a+b \arcsin (c x)) \log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{2 \sqrt {-d} e^{5/2}}-\frac {b \text {Subst}\left (\int \log \left (1-\frac {\sqrt {e} e^{i x}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right ) \, dx,x,\arcsin (c x)\right )}{2 \sqrt {-d} e^{5/2}}+\frac {b \text {Subst}\left (\int \log \left (1+\frac {\sqrt {e} e^{i x}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right ) \, dx,x,\arcsin (c x)\right )}{2 \sqrt {-d} e^{5/2}}-\frac {b \text {Subst}\left (\int \log \left (1-\frac {\sqrt {e} e^{i x}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right ) \, dx,x,\arcsin (c x)\right )}{2 \sqrt {-d} e^{5/2}}+\frac {b \text {Subst}\left (\int \log \left (1+\frac {\sqrt {e} e^{i x}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right ) \, dx,x,\arcsin (c x)\right )}{2 \sqrt {-d} e^{5/2}}-\frac {3 \text {Subst}\left (\int \frac {(a+b x) \cos (x)}{c \sqrt {-d}-\sqrt {e} \sin (x)} \, dx,x,\arcsin (c x)\right )}{16 \sqrt {-d} e^2}-\frac {3 \text {Subst}\left (\int \frac {(a+b x) \cos (x)}{c \sqrt {-d}+\sqrt {e} \sin (x)} \, dx,x,\arcsin (c x)\right )}{16 \sqrt {-d} e^2}+\frac {\text {Subst}\left (\int \frac {(a+b x) \cos (x)}{c \sqrt {-d}-\sqrt {e} \sin (x)} \, dx,x,\arcsin (c x)\right )}{2 \sqrt {-d} e^2}+\frac {\text {Subst}\left (\int \frac {(a+b x) \cos (x)}{c \sqrt {-d}+\sqrt {e} \sin (x)} \, dx,x,\arcsin (c x)\right )}{2 \sqrt {-d} e^2}-\frac {\left (b c^3 d\right ) \text {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^2 \left (c^2 d+e\right )}+\frac {\left (b c^3 d\right ) \text {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^2 \left (c^2 d+e\right )} \\ & = \text {Too large to display} \\ \end{align*}

Mathematica [A] (verified)

Time = 4.47 (sec) , antiderivative size = 1014, normalized size of antiderivative = 0.94 \[ \int \frac {x^4 (a+b \arcsin (c x))}{\left (d+e x^2\right )^3} \, dx=\frac {-\frac {i b c \sqrt {d} \sqrt {e} \sqrt {1-c^2 x^2}}{\left (c^2 d+e\right ) \left (-i \sqrt {d}+\sqrt {e} x\right )}+\frac {i b c \sqrt {d} \sqrt {e} \sqrt {1-c^2 x^2}}{\left (c^2 d+e\right ) \left (i \sqrt {d}+\sqrt {e} x\right )}+\frac {4 a d \sqrt {e} x}{\left (d+e x^2\right )^2}-\frac {10 a \sqrt {e} x}{d+e x^2}+\frac {i b \sqrt {d} \arcsin (c x)}{\left (\sqrt {d}+i \sqrt {e} x\right )^2}+\frac {i b \sqrt {d} \arcsin (c x)}{\left (i \sqrt {d}+\sqrt {e} x\right )^2}-\frac {5 b \arcsin (c x)}{i \sqrt {d}+\sqrt {e} x}+\frac {6 a \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d}}-5 i b \left (\frac {\arcsin (c x)}{\sqrt {d}+i \sqrt {e} x}-\frac {c \arctan \left (\frac {i \sqrt {e}+c^2 \sqrt {d} x}{\sqrt {c^2 d+e} \sqrt {1-c^2 x^2}}\right )}{\sqrt {c^2 d+e}}\right )-\frac {5 b c \text {arctanh}\left (\frac {\sqrt {e}+i c^2 \sqrt {d} x}{\sqrt {c^2 d+e} \sqrt {1-c^2 x^2}}\right )}{\sqrt {c^2 d+e}}+\frac {3 i b \arcsin (c x) \left (\log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{-c \sqrt {d}+\sqrt {c^2 d+e}}\right )+\log \left (1-\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}+\sqrt {c^2 d+e}}\right )\right )}{\sqrt {d}}-\frac {3 i b \arcsin (c x) \left (\log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}-\sqrt {c^2 d+e}}\right )+\log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}+\sqrt {c^2 d+e}}\right )\right )}{\sqrt {d}}+\frac {b c^3 d \left (\log (4)+\log \left (\frac {e \sqrt {c^2 d+e} \left (\sqrt {e}-i c^2 \sqrt {d} x+\sqrt {c^2 d+e} \sqrt {1-c^2 x^2}\right )}{c^3 \left (d+i \sqrt {d} \sqrt {e} x\right )}\right )\right )}{\left (c^2 d+e\right )^{3/2}}+\frac {b c^3 d \left (\log (4)+\log \left (\frac {e \sqrt {c^2 d+e} \left (\sqrt {e}+i c^2 \sqrt {d} x+\sqrt {c^2 d+e} \sqrt {1-c^2 x^2}\right )}{c^3 \left (d-i \sqrt {d} \sqrt {e} x\right )}\right )\right )}{\left (c^2 d+e\right )^{3/2}}+\frac {3 b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}-\sqrt {c^2 d+e}}\right )}{\sqrt {d}}-\frac {3 b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{-c \sqrt {d}+\sqrt {c^2 d+e}}\right )}{\sqrt {d}}-\frac {3 b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}+\sqrt {c^2 d+e}}\right )}{\sqrt {d}}+\frac {3 b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}+\sqrt {c^2 d+e}}\right )}{\sqrt {d}}}{16 e^{5/2}} \]

[In]

Integrate[(x^4*(a + b*ArcSin[c*x]))/(d + e*x^2)^3,x]

[Out]

(((-I)*b*c*Sqrt[d]*Sqrt[e]*Sqrt[1 - c^2*x^2])/((c^2*d + e)*((-I)*Sqrt[d] + Sqrt[e]*x)) + (I*b*c*Sqrt[d]*Sqrt[e
]*Sqrt[1 - c^2*x^2])/((c^2*d + e)*(I*Sqrt[d] + Sqrt[e]*x)) + (4*a*d*Sqrt[e]*x)/(d + e*x^2)^2 - (10*a*Sqrt[e]*x
)/(d + e*x^2) + (I*b*Sqrt[d]*ArcSin[c*x])/(Sqrt[d] + I*Sqrt[e]*x)^2 + (I*b*Sqrt[d]*ArcSin[c*x])/(I*Sqrt[d] + S
qrt[e]*x)^2 - (5*b*ArcSin[c*x])/(I*Sqrt[d] + Sqrt[e]*x) + (6*a*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/Sqrt[d] - (5*I)*b*
(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]) - (5*b*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] + ((3*I)*b*ArcSin[c*x]*(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])]))/Sqrt[d] - ((3*I)*b*ArcSin[c*x]*(Log[1 + (Sqr
t[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])]))/Sqrt[d] + (b*c^3*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))]))/(c^2*d + e)^(3/2) + (b*c^3*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))]))/
(c^2*d + e)^(3/2) + (3*b*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] - Sqrt[c^2*d + e])])/Sqrt[d] - (3*b
*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(-(c*Sqrt[d]) + Sqrt[c^2*d + e])])/Sqrt[d] - (3*b*PolyLog[2, -((Sqrt[e
]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] + Sqrt[c^2*d + e]))])/Sqrt[d] + (3*b*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(c
*Sqrt[d] + Sqrt[c^2*d + e])])/Sqrt[d])/(16*e^(5/2))

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 9.73 (sec) , antiderivative size = 1748, normalized size of antiderivative = 1.62

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1748\)
default	\(\text {Expression too large to display}\)	\(1748\)
parts	\(\text {Expression too large to display}\)	\(1750\)

[In]

int(x^4*(a+b*arcsin(c*x))/(e*x^2+d)^3,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [F]

\[ \int \frac {x^4 (a+b \arcsin (c x))}{\left (d+e x^2\right )^3} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )} x^{4}}{{\left (e x^{2} + d\right )}^{3}} \,d x } \]

[In]

integrate(x^4*(a+b*arcsin(c*x))/(e*x^2+d)^3,x, algorithm="fricas")

[Out]

integral((b*x^4*arcsin(c*x) + a*x^4)/(e^3*x^6 + 3*d*e^2*x^4 + 3*d^2*e*x^2 + d^3), x)

Sympy [F]

\[ \int \frac {x^4 (a+b \arcsin (c x))}{\left (d+e x^2\right )^3} \, dx=\int \frac {x^{4} \left (a + b \operatorname {asin}{\left (c x \right )}\right )}{\left (d + e x^{2}\right )^{3}}\, dx \]

[In]

integrate(x**4*(a+b*asin(c*x))/(e*x**2+d)**3,x)

[Out]

Integral(x**4*(a + b*asin(c*x))/(d + e*x**2)**3, x)

Maxima [F(-2)]

Exception generated. \[ \int \frac {x^4 (a+b \arcsin (c x))}{\left (d+e x^2\right )^3} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate(x^4*(a+b*arcsin(c*x))/(e*x^2+d)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more
details)Is e

Giac [F]

\[ \int \frac {x^4 (a+b \arcsin (c x))}{\left (d+e x^2\right )^3} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )} x^{4}}{{\left (e x^{2} + d\right )}^{3}} \,d x } \]

[In]

integrate(x^4*(a+b*arcsin(c*x))/(e*x^2+d)^3,x, algorithm="giac")

[Out]

integrate((b*arcsin(c*x) + a)*x^4/(e*x^2 + d)^3, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {x^4 (a+b \arcsin (c x))}{\left (d+e x^2\right )^3} \, dx=\int \frac {x^4\,\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}{{\left (e\,x^2+d\right )}^3} \,d x \]

[In]

int((x^4*(a + b*asin(c*x)))/(d + e*x^2)^3,x)

[Out]

int((x^4*(a + b*asin(c*x)))/(d + e*x^2)^3, x)