\(\int \frac {x^4 (a+b \arcsin (c x))}{(d+e x^2)^2} \, dx\) [637]

Optimal result

Integrand size = 21, antiderivative size = 787 \[ \int \frac {x^4 (a+b \arcsin (c x))}{\left (d+e x^2\right )^2} \, dx=\frac {a x}{e^2}+\frac {b \sqrt {1-c^2 x^2}}{c e^2}+\frac {b x \arcsin (c x)}{e^2}-\frac {d (a+b \arcsin (c x))}{4 e^{5/2} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {d (a+b \arcsin (c x))}{4 e^{5/2} \left (\sqrt {-d}+\sqrt {e} x\right )}+\frac {b c d \text {arctanh}\left (\frac {\sqrt {e}-c^2 \sqrt {-d} x}{\sqrt {c^2 d+e} \sqrt {1-c^2 x^2}}\right )}{4 e^{5/2} \sqrt {c^2 d+e}}+\frac {b c d \text {arctanh}\left (\frac {\sqrt {e}+c^2 \sqrt {-d} x}{\sqrt {c^2 d+e} \sqrt {1-c^2 x^2}}\right )}{4 e^{5/2} \sqrt {c^2 d+e}}+\frac {3 \sqrt {-d} (a+b \arcsin (c x)) \log \left (1-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{4 e^{5/2}}-\frac {3 \sqrt {-d} (a+b \arcsin (c x)) \log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{4 e^{5/2}}+\frac {3 \sqrt {-d} (a+b \arcsin (c x)) \log \left (1-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{4 e^{5/2}}-\frac {3 \sqrt {-d} (a+b \arcsin (c x)) \log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{4 e^{5/2}}+\frac {3 i b \sqrt {-d} \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{4 e^{5/2}}-\frac {3 i b \sqrt {-d} \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{4 e^{5/2}}+\frac {3 i b \sqrt {-d} \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{4 e^{5/2}}-\frac {3 i b \sqrt {-d} \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{4 e^{5/2}} \]

[Out]

a*x/e^2+b*x*arcsin(c*x)/e^2+3/4*(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)^(1/2)/e^(5/2)-3/4*(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)^(1/2)/e^(5/2)+3/4*(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)^(1/2)/e^(5/2)-3/4*(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)^(1/2)/e^(5/2)+3/4*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)^(1/2)/e^(5/2)-3/4*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)^(1/2)/e^(5/2)+3/4*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)^(1/2)/e^(5/2)-3/4*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)^(1/2)/e^(5/2)-1/4*d*(a+b*arcsin(c*x))/e^(5/2)/((-d)^(1/2)-x*e^(1/2))+1/4*d*(a+b*arcsin(c*x
))/e^(5/2)/((-d)^(1/2)+x*e^(1/2))+1/4*b*c*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)^(1/2)+1/4*b*c*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)^(1/2)+b*(-c^2*x^2+1)^(1/2)/c/e^2

Rubi [A] (verified)

Time = 1.42 (sec) , antiderivative size = 787, normalized size of antiderivative = 1.00, number of steps used = 49, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {4817, 4715, 267, 4757, 4827, 739, 212, 4825, 4617, 2221, 2317, 2438} \[ \int \frac {x^4 (a+b \arcsin (c x))}{\left (d+e x^2\right )^2} \, dx=\frac {3 \sqrt {-d} (a+b \arcsin (c x)) \log \left (1-\frac {\sqrt {e} e^{i \arcsin (c x)}}{-\sqrt {c^2 d+e}+i c \sqrt {-d}}\right )}{4 e^{5/2}}-\frac {3 \sqrt {-d} (a+b \arcsin (c x)) \log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{-\sqrt {c^2 d+e}+i c \sqrt {-d}}\right )}{4 e^{5/2}}+\frac {3 \sqrt {-d} (a+b \arcsin (c x)) \log \left (1-\frac {\sqrt {e} e^{i \arcsin (c x)}}{\sqrt {c^2 d+e}+i c \sqrt {-d}}\right )}{4 e^{5/2}}-\frac {3 \sqrt {-d} (a+b \arcsin (c x)) \log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{\sqrt {c^2 d+e}+i c \sqrt {-d}}\right )}{4 e^{5/2}}-\frac {d (a+b \arcsin (c x))}{4 e^{5/2} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {d (a+b \arcsin (c x))}{4 e^{5/2} \left (\sqrt {-d}+\sqrt {e} x\right )}+\frac {a x}{e^2}+\frac {3 i b \sqrt {-d} \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {d c^2+e}}\right )}{4 e^{5/2}}-\frac {3 i b \sqrt {-d} \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {d c^2+e}}\right )}{4 e^{5/2}}+\frac {3 i b \sqrt {-d} \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i \sqrt {-d} c+\sqrt {d c^2+e}}\right )}{4 e^{5/2}}-\frac {3 i b \sqrt {-d} \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{i \sqrt {-d} c+\sqrt {d c^2+e}}\right )}{4 e^{5/2}}+\frac {b x \arcsin (c x)}{e^2}+\frac {b c d \text {arctanh}\left (\frac {\sqrt {e}-c^2 \sqrt {-d} x}{\sqrt {1-c^2 x^2} \sqrt {c^2 d+e}}\right )}{4 e^{5/2} \sqrt {c^2 d+e}}+\frac {b c d \text {arctanh}\left (\frac {c^2 \sqrt {-d} x+\sqrt {e}}{\sqrt {1-c^2 x^2} \sqrt {c^2 d+e}}\right )}{4 e^{5/2} \sqrt {c^2 d+e}}+\frac {b \sqrt {1-c^2 x^2}}{c e^2} \]

[In]

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

[Out]

(a*x)/e^2 + (b*Sqrt[1 - c^2*x^2])/(c*e^2) + (b*x*ArcSin[c*x])/e^2 - (d*(a + b*ArcSin[c*x]))/(4*e^(5/2)*(Sqrt[-
d] - Sqrt[e]*x)) + (d*(a + b*ArcSin[c*x]))/(4*e^(5/2)*(Sqrt[-d] + Sqrt[e]*x)) + (b*c*d*ArcTanh[(Sqrt[e] - c^2*
Sqrt[-d]*x)/(Sqrt[c^2*d + e]*Sqrt[1 - c^2*x^2])])/(4*e^(5/2)*Sqrt[c^2*d + e]) + (b*c*d*ArcTanh[(Sqrt[e] + c^2*
Sqrt[-d]*x)/(Sqrt[c^2*d + e]*Sqrt[1 - c^2*x^2])])/(4*e^(5/2)*Sqrt[c^2*d + e]) + (3*Sqrt[-d]*(a + b*ArcSin[c*x]
)*Log[1 - (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/(4*e^(5/2)) - (3*Sqrt[-d]*(a + b*ArcS
in[c*x])*Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/(4*e^(5/2)) + (3*Sqrt[-d]*(a +
b*ArcSin[c*x])*Log[1 - (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])])/(4*e^(5/2)) - (3*Sqrt[-
d]*(a + b*ArcSin[c*x])*Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])])/(4*e^(5/2)) + ((
(3*I)/4)*b*Sqrt[-d]*PolyLog[2, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e]))])/e^(5/2) - (((
3*I)/4)*b*Sqrt[-d]*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/e^(5/2) + (((3*I)
/4)*b*Sqrt[-d]*PolyLog[2, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e]))])/e^(5/2) - (((3*I)/
4)*b*Sqrt[-d]*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])])/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 267

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

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 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 4715

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

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 {a+b \arcsin (c x)}{e^2}+\frac {d^2 (a+b \arcsin (c x))}{e^2 \left (d+e x^2\right )^2}-\frac {2 d (a+b \arcsin (c x))}{e^2 \left (d+e x^2\right )}\right ) \, dx \\ & = \frac {\int (a+b \arcsin (c x)) \, dx}{e^2}-\frac {(2 d) \int \frac {a+b \arcsin (c x)}{d+e x^2} \, dx}{e^2}+\frac {d^2 \int \frac {a+b \arcsin (c x)}{\left (d+e x^2\right )^2} \, dx}{e^2} \\ & = \frac {a x}{e^2}+\frac {b \int \arcsin (c x) \, dx}{e^2}-\frac {(2 d) \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 {d^2 \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 {a x}{e^2}+\frac {b x \arcsin (c x)}{e^2}-\frac {(b c) \int \frac {x}{\sqrt {1-c^2 x^2}} \, dx}{e^2}-\frac {\sqrt {-d} \int \frac {a+b \arcsin (c x)}{\sqrt {-d}-\sqrt {e} x} \, dx}{e^2}-\frac {\sqrt {-d} \int \frac {a+b \arcsin (c x)}{\sqrt {-d}+\sqrt {e} x} \, dx}{e^2}-\frac {d \int \frac {a+b \arcsin (c x)}{\left (\sqrt {-d} \sqrt {e}-e x\right )^2} \, dx}{4 e}-\frac {d \int \frac {a+b \arcsin (c x)}{\left (\sqrt {-d} \sqrt {e}+e x\right )^2} \, dx}{4 e}-\frac {d \int \frac {a+b \arcsin (c x)}{-d e-e^2 x^2} \, dx}{2 e} \\ & = \frac {a x}{e^2}+\frac {b \sqrt {1-c^2 x^2}}{c e^2}+\frac {b x \arcsin (c x)}{e^2}-\frac {d (a+b \arcsin (c x))}{4 e^{5/2} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {d (a+b \arcsin (c x))}{4 e^{5/2} \left (\sqrt {-d}+\sqrt {e} x\right )}-\frac {\sqrt {-d} \text {Subst}\left (\int \frac {(a+b x) \cos (x)}{c \sqrt {-d}-\sqrt {e} \sin (x)} \, dx,x,\arcsin (c x)\right )}{e^2}-\frac {\sqrt {-d} \text {Subst}\left (\int \frac {(a+b x) \cos (x)}{c \sqrt {-d}+\sqrt {e} \sin (x)} \, dx,x,\arcsin (c x)\right )}{e^2}+\frac {(b c d) \int \frac {1}{\left (\sqrt {-d} \sqrt {e}-e x\right ) \sqrt {1-c^2 x^2}} \, dx}{4 e^2}-\frac {(b c d) \int \frac {1}{\left (\sqrt {-d} \sqrt {e}+e x\right ) \sqrt {1-c^2 x^2}} \, dx}{4 e^2}-\frac {d \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}{2 e} \\ & = \frac {a x}{e^2}+\frac {b \sqrt {1-c^2 x^2}}{c e^2}+\frac {b x \arcsin (c x)}{e^2}-\frac {d (a+b \arcsin (c x))}{4 e^{5/2} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {d (a+b \arcsin (c x))}{4 e^{5/2} \left (\sqrt {-d}+\sqrt {e} x\right )}-\frac {\left (i \sqrt {-d}\right ) \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 )}{e^2}-\frac {\left (i \sqrt {-d}\right ) \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 )}{e^2}-\frac {\left (i \sqrt {-d}\right ) \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 )}{e^2}-\frac {\left (i \sqrt {-d}\right ) \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 )}{e^2}+\frac {\sqrt {-d} \int \frac {a+b \arcsin (c x)}{\sqrt {-d}-\sqrt {e} x} \, dx}{4 e^2}+\frac {\sqrt {-d} \int \frac {a+b \arcsin (c x)}{\sqrt {-d}+\sqrt {e} x} \, dx}{4 e^2}-\frac {(b c d) \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 )}{4 e^2}+\frac {(b c d) \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 )}{4 e^2} \\ & = \frac {a x}{e^2}+\frac {b \sqrt {1-c^2 x^2}}{c e^2}+\frac {b x \arcsin (c x)}{e^2}-\frac {d (a+b \arcsin (c x))}{4 e^{5/2} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {d (a+b \arcsin (c x))}{4 e^{5/2} \left (\sqrt {-d}+\sqrt {e} x\right )}+\frac {b c d \text {arctanh}\left (\frac {\sqrt {e}-c^2 \sqrt {-d} x}{\sqrt {c^2 d+e} \sqrt {1-c^2 x^2}}\right )}{4 e^{5/2} \sqrt {c^2 d+e}}+\frac {b c d \text {arctanh}\left (\frac {\sqrt {e}+c^2 \sqrt {-d} x}{\sqrt {c^2 d+e} \sqrt {1-c^2 x^2}}\right )}{4 e^{5/2} \sqrt {c^2 d+e}}+\frac {\sqrt {-d} (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 )}{e^{5/2}}-\frac {\sqrt {-d} (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 )}{e^{5/2}}+\frac {\sqrt {-d} (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 )}{e^{5/2}}-\frac {\sqrt {-d} (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 )}{e^{5/2}}-\frac {\left (b \sqrt {-d}\right ) \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 )}{e^{5/2}}+\frac {\left (b \sqrt {-d}\right ) \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 )}{e^{5/2}}-\frac {\left (b \sqrt {-d}\right ) \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 )}{e^{5/2}}+\frac {\left (b \sqrt {-d}\right ) \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 )}{e^{5/2}}+\frac {\sqrt {-d} \text {Subst}\left (\int \frac {(a+b x) \cos (x)}{c \sqrt {-d}-\sqrt {e} \sin (x)} \, dx,x,\arcsin (c x)\right )}{4 e^2}+\frac {\sqrt {-d} \text {Subst}\left (\int \frac {(a+b x) \cos (x)}{c \sqrt {-d}+\sqrt {e} \sin (x)} \, dx,x,\arcsin (c x)\right )}{4 e^2} \\ & = \frac {a x}{e^2}+\frac {b \sqrt {1-c^2 x^2}}{c e^2}+\frac {b x \arcsin (c x)}{e^2}-\frac {d (a+b \arcsin (c x))}{4 e^{5/2} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {d (a+b \arcsin (c x))}{4 e^{5/2} \left (\sqrt {-d}+\sqrt {e} x\right )}+\frac {b c d \text {arctanh}\left (\frac {\sqrt {e}-c^2 \sqrt {-d} x}{\sqrt {c^2 d+e} \sqrt {1-c^2 x^2}}\right )}{4 e^{5/2} \sqrt {c^2 d+e}}+\frac {b c d \text {arctanh}\left (\frac {\sqrt {e}+c^2 \sqrt {-d} x}{\sqrt {c^2 d+e} \sqrt {1-c^2 x^2}}\right )}{4 e^{5/2} \sqrt {c^2 d+e}}+\frac {\sqrt {-d} (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 )}{e^{5/2}}-\frac {\sqrt {-d} (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 )}{e^{5/2}}+\frac {\sqrt {-d} (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 )}{e^{5/2}}-\frac {\sqrt {-d} (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 )}{e^{5/2}}+\frac {\left (i b \sqrt {-d}\right ) \text {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 \arcsin (c x)}\right )}{e^{5/2}}-\frac {\left (i b \sqrt {-d}\right ) \text {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 \arcsin (c x)}\right )}{e^{5/2}}+\frac {\left (i b \sqrt {-d}\right ) \text {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 \arcsin (c x)}\right )}{e^{5/2}}-\frac {\left (i b \sqrt {-d}\right ) \text {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 \arcsin (c x)}\right )}{e^{5/2}}+\frac {\left (i \sqrt {-d}\right ) \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 )}{4 e^2}+\frac {\left (i \sqrt {-d}\right ) \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 )}{4 e^2}+\frac {\left (i \sqrt {-d}\right ) \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 )}{4 e^2}+\frac {\left (i \sqrt {-d}\right ) \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 )}{4 e^2} \\ & = \frac {a x}{e^2}+\frac {b \sqrt {1-c^2 x^2}}{c e^2}+\frac {b x \arcsin (c x)}{e^2}-\frac {d (a+b \arcsin (c x))}{4 e^{5/2} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {d (a+b \arcsin (c x))}{4 e^{5/2} \left (\sqrt {-d}+\sqrt {e} x\right )}+\frac {b c d \text {arctanh}\left (\frac {\sqrt {e}-c^2 \sqrt {-d} x}{\sqrt {c^2 d+e} \sqrt {1-c^2 x^2}}\right )}{4 e^{5/2} \sqrt {c^2 d+e}}+\frac {b c d \text {arctanh}\left (\frac {\sqrt {e}+c^2 \sqrt {-d} x}{\sqrt {c^2 d+e} \sqrt {1-c^2 x^2}}\right )}{4 e^{5/2} \sqrt {c^2 d+e}}+\frac {3 \sqrt {-d} (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 )}{4 e^{5/2}}-\frac {3 \sqrt {-d} (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 )}{4 e^{5/2}}+\frac {3 \sqrt {-d} (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 )}{4 e^{5/2}}-\frac {3 \sqrt {-d} (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 )}{4 e^{5/2}}+\frac {i b \sqrt {-d} \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{e^{5/2}}-\frac {i b \sqrt {-d} \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{e^{5/2}}+\frac {i b \sqrt {-d} \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{e^{5/2}}-\frac {i b \sqrt {-d} \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{e^{5/2}}+\frac {\left (b \sqrt {-d}\right ) \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 )}{4 e^{5/2}}-\frac {\left (b \sqrt {-d}\right ) \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 )}{4 e^{5/2}}+\frac {\left (b \sqrt {-d}\right ) \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 )}{4 e^{5/2}}-\frac {\left (b \sqrt {-d}\right ) \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 )}{4 e^{5/2}} \\ & = \frac {a x}{e^2}+\frac {b \sqrt {1-c^2 x^2}}{c e^2}+\frac {b x \arcsin (c x)}{e^2}-\frac {d (a+b \arcsin (c x))}{4 e^{5/2} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {d (a+b \arcsin (c x))}{4 e^{5/2} \left (\sqrt {-d}+\sqrt {e} x\right )}+\frac {b c d \text {arctanh}\left (\frac {\sqrt {e}-c^2 \sqrt {-d} x}{\sqrt {c^2 d+e} \sqrt {1-c^2 x^2}}\right )}{4 e^{5/2} \sqrt {c^2 d+e}}+\frac {b c d \text {arctanh}\left (\frac {\sqrt {e}+c^2 \sqrt {-d} x}{\sqrt {c^2 d+e} \sqrt {1-c^2 x^2}}\right )}{4 e^{5/2} \sqrt {c^2 d+e}}+\frac {3 \sqrt {-d} (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 )}{4 e^{5/2}}-\frac {3 \sqrt {-d} (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 )}{4 e^{5/2}}+\frac {3 \sqrt {-d} (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 )}{4 e^{5/2}}-\frac {3 \sqrt {-d} (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 )}{4 e^{5/2}}+\frac {i b \sqrt {-d} \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{e^{5/2}}-\frac {i b \sqrt {-d} \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{e^{5/2}}+\frac {i b \sqrt {-d} \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{e^{5/2}}-\frac {i b \sqrt {-d} \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{e^{5/2}}-\frac {\left (i b \sqrt {-d}\right ) \text {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 \arcsin (c x)}\right )}{4 e^{5/2}}+\frac {\left (i b \sqrt {-d}\right ) \text {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 \arcsin (c x)}\right )}{4 e^{5/2}}-\frac {\left (i b \sqrt {-d}\right ) \text {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 \arcsin (c x)}\right )}{4 e^{5/2}}+\frac {\left (i b \sqrt {-d}\right ) \text {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 \arcsin (c x)}\right )}{4 e^{5/2}} \\ & = \frac {a x}{e^2}+\frac {b \sqrt {1-c^2 x^2}}{c e^2}+\frac {b x \arcsin (c x)}{e^2}-\frac {d (a+b \arcsin (c x))}{4 e^{5/2} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {d (a+b \arcsin (c x))}{4 e^{5/2} \left (\sqrt {-d}+\sqrt {e} x\right )}+\frac {b c d \text {arctanh}\left (\frac {\sqrt {e}-c^2 \sqrt {-d} x}{\sqrt {c^2 d+e} \sqrt {1-c^2 x^2}}\right )}{4 e^{5/2} \sqrt {c^2 d+e}}+\frac {b c d \text {arctanh}\left (\frac {\sqrt {e}+c^2 \sqrt {-d} x}{\sqrt {c^2 d+e} \sqrt {1-c^2 x^2}}\right )}{4 e^{5/2} \sqrt {c^2 d+e}}+\frac {3 \sqrt {-d} (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 )}{4 e^{5/2}}-\frac {3 \sqrt {-d} (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 )}{4 e^{5/2}}+\frac {3 \sqrt {-d} (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 )}{4 e^{5/2}}-\frac {3 \sqrt {-d} (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 )}{4 e^{5/2}}+\frac {3 i b \sqrt {-d} \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{4 e^{5/2}}-\frac {3 i b \sqrt {-d} \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{4 e^{5/2}}+\frac {3 i b \sqrt {-d} \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{4 e^{5/2}}-\frac {3 i b \sqrt {-d} \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{4 e^{5/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.12 (sec) , antiderivative size = 649, normalized size of antiderivative = 0.82 \[ \int \frac {x^4 (a+b \arcsin (c x))}{\left (d+e x^2\right )^2} \, dx=\frac {8 a \sqrt {e} x+\frac {4 a d \sqrt {e} x}{d+e x^2}-12 a \sqrt {d} \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )+b \left (\frac {8 \sqrt {e} \left (\sqrt {1-c^2 x^2}+c x \arcsin (c x)\right )}{c}+2 i d \left (\frac {\arcsin (c x)}{\sqrt {d}+i \sqrt {e} x}-\frac {c \arctan \left (\frac {i \sqrt {e}+c^2 \sqrt {d} x}{\sqrt {c^2 d+e} \sqrt {1-c^2 x^2}}\right )}{\sqrt {c^2 d+e}}\right )+2 d \left (\frac {\arcsin (c x)}{i \sqrt {d}+\sqrt {e} x}+\frac {c \text {arctanh}\left (\frac {\sqrt {e}+i c^2 \sqrt {d} x}{\sqrt {c^2 d+e} \sqrt {1-c^2 x^2}}\right )}{\sqrt {c^2 d+e}}\right )+3 \sqrt {d} \left (\arcsin (c x) \left (\arcsin (c x)+2 i \left (\log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}-\sqrt {c^2 d+e}}\right )+\log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}+\sqrt {c^2 d+e}}\right )\right )\right )+2 \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{-c \sqrt {d}+\sqrt {c^2 d+e}}\right )+2 \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}+\sqrt {c^2 d+e}}\right )\right )-3 \sqrt {d} \left (\arcsin (c x) \left (\arcsin (c x)+2 i \left (\log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{-c \sqrt {d}+\sqrt {c^2 d+e}}\right )+\log \left (1-\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}+\sqrt {c^2 d+e}}\right )\right )\right )+2 \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}-\sqrt {c^2 d+e}}\right )+2 \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}+\sqrt {c^2 d+e}}\right )\right )\right )}{8 e^{5/2}} \]

[In]

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

[Out]

(8*a*Sqrt[e]*x + (4*a*d*Sqrt[e]*x)/(d + e*x^2) - 12*a*Sqrt[d]*ArcTan[(Sqrt[e]*x)/Sqrt[d]] + b*((8*Sqrt[e]*(Sqr
t[1 - c^2*x^2] + c*x*ArcSin[c*x]))/c + (2*I)*d*(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]) + 2*d*(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]) + 3*Sqrt[d]*
(ArcSin[c*x]*(ArcSin[c*x] + (2*I)*(Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] - Sqrt[c^2*d + e])] + Log[1
+ (Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] + Sqrt[c^2*d + e])])) + 2*PolyLog[2, (Sqrt[e]*E^(I*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]))]) -
3*Sqrt[d]*(ArcSin[c*x]*(ArcSin[c*x] + (2*I)*(Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(-(c*Sqrt[d]) + Sqrt[c^2*d +
e])] + Log[1 - (Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] + Sqrt[c^2*d + e])])) + 2*PolyLog[2, (Sqrt[e]*E^(I*ArcSi
n[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
])])))/(8*e^(5/2))

Maple [C] (warning: unable to verify)

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

Time = 4.87 (sec) , antiderivative size = 898, normalized size of antiderivative = 1.14

method	result	size
parts	\(a \left (\frac {x}{e^{2}}-\frac {d \left (-\frac {x}{2 \left (e \,x^{2}+d \right )}+\frac {3 \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{2 \sqrt {d e}}\right )}{e^{2}}\right )+\frac {b \left (\frac {\left (-i \sqrt {-c^{2} x^{2}+1}+c x \right ) c^{4} \left (\arcsin \left (c x \right )+i\right )}{2 e^{2}}+\frac {\left (c x +i \sqrt {-c^{2} x^{2}+1}\right ) c^{4} \left (\arcsin \left (c x \right )-i\right )}{2 e^{2}}+\frac {c^{7} \arcsin \left (c x \right ) d x}{2 e^{2} \left (c^{2} e \,x^{2}+c^{2} d \right )}-\frac {\sqrt {-e \left (2 c^{2} d -2 \sqrt {d \,c^{2} \left (c^{2} d +e \right )}+e \right )}\, \left (2 d^{2} c^{4}+2 \sqrt {d \,c^{2} \left (c^{2} d +e \right )}\, d \,c^{2}+2 c^{2} e d +\sqrt {d \,c^{2} \left (c^{2} d +e \right )}\, e \right ) d \,c^{6} \arctan \left (\frac {e \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )}{\sqrt {\left (-2 c^{2} d +2 \sqrt {d \,c^{2} \left (c^{2} d +e \right )}-e \right ) e}}\right )}{2 e^{5} \left (c^{2} d +e \right )}+\frac {\sqrt {-e \left (2 c^{2} d -2 \sqrt {d \,c^{2} \left (c^{2} d +e \right )}+e \right )}\, \left (2 c^{2} d +2 \sqrt {d \,c^{2} \left (c^{2} d +e \right )}+e \right ) \arctan \left (\frac {e \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )}{\sqrt {\left (-2 c^{2} d +2 \sqrt {d \,c^{2} \left (c^{2} d +e \right )}-e \right ) e}}\right ) d \,c^{6}}{2 e^{5}}-\frac {\sqrt {\left (2 c^{2} d +2 \sqrt {d \,c^{2} \left (c^{2} d +e \right )}+e \right ) e}\, \left (-2 \sqrt {d \,c^{2} \left (c^{2} d +e \right )}\, d \,c^{2}+2 d^{2} c^{4}+2 c^{2} e d -\sqrt {d \,c^{2} \left (c^{2} d +e \right )}\, e \right ) d \,c^{6} \operatorname {arctanh}\left (\frac {e \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )}{\sqrt {\left (2 c^{2} d +2 \sqrt {d \,c^{2} \left (c^{2} d +e \right )}+e \right ) e}}\right )}{2 e^{5} \left (c^{2} d +e \right )}+\frac {\sqrt {\left (2 c^{2} d +2 \sqrt {d \,c^{2} \left (c^{2} d +e \right )}+e \right ) e}\, \left (2 c^{2} d -2 \sqrt {d \,c^{2} \left (c^{2} d +e \right )}+e \right ) \operatorname {arctanh}\left (\frac {e \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )}{\sqrt {\left (2 c^{2} d +2 \sqrt {d \,c^{2} \left (c^{2} d +e \right )}+e \right ) e}}\right ) d \,c^{6}}{2 e^{5}}-\frac {3 d \,c^{6} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (-4 c^{2} d -2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {i \arcsin \left (c x \right ) \ln \left (\frac {\textit {\_R1} -i c x -\sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -i c x -\sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )}{\textit {\_R1} \left (\textit {\_R1}^{2} e -2 c^{2} d -e \right )}\right )}{4 e^{2}}-\frac {3 d \,c^{6} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (-4 c^{2} d -2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\textit {\_R1} \left (i \arcsin \left (c x \right ) \ln \left (\frac {\textit {\_R1} -i c x -\sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -i c x -\sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )\right )}{\textit {\_R1}^{2} e -2 c^{2} d -e}\right )}{4 e^{2}}\right )}{c^{5}}\)	\(898\)
derivativedivides	\(\frac {a \,c^{4} \left (\frac {c x}{e^{2}}-\frac {d \,c^{2} \left (-\frac {c x}{2 \left (c^{2} e \,x^{2}+c^{2} d \right )}+\frac {3 \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{2 c \sqrt {d e}}\right )}{e^{2}}\right )+b \,c^{4} \left (\frac {\left (-i \sqrt {-c^{2} x^{2}+1}+c x \right ) \left (\arcsin \left (c x \right )+i\right )}{2 e^{2}}+\frac {\left (c x +i \sqrt {-c^{2} x^{2}+1}\right ) \left (\arcsin \left (c x \right )-i\right )}{2 e^{2}}+\frac {c^{3} d \arcsin \left (c x \right ) x}{2 e^{2} \left (c^{2} e \,x^{2}+c^{2} d \right )}-\frac {\sqrt {\left (2 c^{2} d +2 \sqrt {d \,c^{2} \left (c^{2} d +e \right )}+e \right ) e}\, \left (-2 \sqrt {d \,c^{2} \left (c^{2} d +e \right )}\, d \,c^{2}+2 d^{2} c^{4}+2 c^{2} e d -\sqrt {d \,c^{2} \left (c^{2} d +e \right )}\, e \right ) d \,c^{2} \operatorname {arctanh}\left (\frac {e \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )}{\sqrt {\left (2 c^{2} d +2 \sqrt {d \,c^{2} \left (c^{2} d +e \right )}+e \right ) e}}\right )}{2 e^{5} \left (c^{2} d +e \right )}+\frac {\sqrt {\left (2 c^{2} d +2 \sqrt {d \,c^{2} \left (c^{2} d +e \right )}+e \right ) e}\, \left (2 c^{2} d -2 \sqrt {d \,c^{2} \left (c^{2} d +e \right )}+e \right ) \operatorname {arctanh}\left (\frac {e \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )}{\sqrt {\left (2 c^{2} d +2 \sqrt {d \,c^{2} \left (c^{2} d +e \right )}+e \right ) e}}\right ) d \,c^{2}}{2 e^{5}}-\frac {\sqrt {-e \left (2 c^{2} d -2 \sqrt {d \,c^{2} \left (c^{2} d +e \right )}+e \right )}\, \left (2 d^{2} c^{4}+2 \sqrt {d \,c^{2} \left (c^{2} d +e \right )}\, d \,c^{2}+2 c^{2} e d +\sqrt {d \,c^{2} \left (c^{2} d +e \right )}\, e \right ) d \,c^{2} \arctan \left (\frac {e \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )}{\sqrt {\left (-2 c^{2} d +2 \sqrt {d \,c^{2} \left (c^{2} d +e \right )}-e \right ) e}}\right )}{2 e^{5} \left (c^{2} d +e \right )}+\frac {\sqrt {-e \left (2 c^{2} d -2 \sqrt {d \,c^{2} \left (c^{2} d +e \right )}+e \right )}\, \left (2 c^{2} d +2 \sqrt {d \,c^{2} \left (c^{2} d +e \right )}+e \right ) \arctan \left (\frac {e \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )}{\sqrt {\left (-2 c^{2} d +2 \sqrt {d \,c^{2} \left (c^{2} d +e \right )}-e \right ) e}}\right ) d \,c^{2}}{2 e^{5}}+\frac {3 c^{2} d \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (-4 c^{2} d -2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {i \arcsin \left (c x \right ) \ln \left (\frac {\textit {\_R1} -i c x -\sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -i c x -\sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )}{\textit {\_R1} \left (-\textit {\_R1}^{2} e +2 c^{2} d +e \right )}\right )}{4 e^{2}}+\frac {3 c^{2} d \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (-4 c^{2} d -2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\textit {\_R1} \left (i \arcsin \left (c x \right ) \ln \left (\frac {\textit {\_R1} -i c x -\sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -i c x -\sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )\right )}{-\textit {\_R1}^{2} e +2 c^{2} d +e}\right )}{4 e^{2}}\right )}{c^{5}}\)	\(912\)
default	\(\frac {a \,c^{4} \left (\frac {c x}{e^{2}}-\frac {d \,c^{2} \left (-\frac {c x}{2 \left (c^{2} e \,x^{2}+c^{2} d \right )}+\frac {3 \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{2 c \sqrt {d e}}\right )}{e^{2}}\right )+b \,c^{4} \left (\frac {\left (-i \sqrt {-c^{2} x^{2}+1}+c x \right ) \left (\arcsin \left (c x \right )+i\right )}{2 e^{2}}+\frac {\left (c x +i \sqrt {-c^{2} x^{2}+1}\right ) \left (\arcsin \left (c x \right )-i\right )}{2 e^{2}}+\frac {c^{3} d \arcsin \left (c x \right ) x}{2 e^{2} \left (c^{2} e \,x^{2}+c^{2} d \right )}-\frac {\sqrt {\left (2 c^{2} d +2 \sqrt {d \,c^{2} \left (c^{2} d +e \right )}+e \right ) e}\, \left (-2 \sqrt {d \,c^{2} \left (c^{2} d +e \right )}\, d \,c^{2}+2 d^{2} c^{4}+2 c^{2} e d -\sqrt {d \,c^{2} \left (c^{2} d +e \right )}\, e \right ) d \,c^{2} \operatorname {arctanh}\left (\frac {e \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )}{\sqrt {\left (2 c^{2} d +2 \sqrt {d \,c^{2} \left (c^{2} d +e \right )}+e \right ) e}}\right )}{2 e^{5} \left (c^{2} d +e \right )}+\frac {\sqrt {\left (2 c^{2} d +2 \sqrt {d \,c^{2} \left (c^{2} d +e \right )}+e \right ) e}\, \left (2 c^{2} d -2 \sqrt {d \,c^{2} \left (c^{2} d +e \right )}+e \right ) \operatorname {arctanh}\left (\frac {e \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )}{\sqrt {\left (2 c^{2} d +2 \sqrt {d \,c^{2} \left (c^{2} d +e \right )}+e \right ) e}}\right ) d \,c^{2}}{2 e^{5}}-\frac {\sqrt {-e \left (2 c^{2} d -2 \sqrt {d \,c^{2} \left (c^{2} d +e \right )}+e \right )}\, \left (2 d^{2} c^{4}+2 \sqrt {d \,c^{2} \left (c^{2} d +e \right )}\, d \,c^{2}+2 c^{2} e d +\sqrt {d \,c^{2} \left (c^{2} d +e \right )}\, e \right ) d \,c^{2} \arctan \left (\frac {e \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )}{\sqrt {\left (-2 c^{2} d +2 \sqrt {d \,c^{2} \left (c^{2} d +e \right )}-e \right ) e}}\right )}{2 e^{5} \left (c^{2} d +e \right )}+\frac {\sqrt {-e \left (2 c^{2} d -2 \sqrt {d \,c^{2} \left (c^{2} d +e \right )}+e \right )}\, \left (2 c^{2} d +2 \sqrt {d \,c^{2} \left (c^{2} d +e \right )}+e \right ) \arctan \left (\frac {e \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )}{\sqrt {\left (-2 c^{2} d +2 \sqrt {d \,c^{2} \left (c^{2} d +e \right )}-e \right ) e}}\right ) d \,c^{2}}{2 e^{5}}+\frac {3 c^{2} d \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (-4 c^{2} d -2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {i \arcsin \left (c x \right ) \ln \left (\frac {\textit {\_R1} -i c x -\sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -i c x -\sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )}{\textit {\_R1} \left (-\textit {\_R1}^{2} e +2 c^{2} d +e \right )}\right )}{4 e^{2}}+\frac {3 c^{2} d \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (-4 c^{2} d -2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\textit {\_R1} \left (i \arcsin \left (c x \right ) \ln \left (\frac {\textit {\_R1} -i c x -\sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -i c x -\sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )\right )}{-\textit {\_R1}^{2} e +2 c^{2} d +e}\right )}{4 e^{2}}\right )}{c^{5}}\)	\(912\)

[In]

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

[Out]

a*(1/e^2*x-1/e^2*d*(-1/2*x/(e*x^2+d)+3/2/(d*e)^(1/2)*arctan(e*x/(d*e)^(1/2))))+b/c^5*(1/2*(-I*(-c^2*x^2+1)^(1/
2)+c*x)*c^4*(arcsin(c*x)+I)/e^2+1/2*(c*x+I*(-c^2*x^2+1)^(1/2))*c^4*(arcsin(c*x)-I)/e^2+1/2*c^7*arcsin(c*x)/e^2
*d*x/(c^2*e*x^2+c^2*d)-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^6*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))/e^5/(c^2*d+e)+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)*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))*d
*c^6/e^5-1/2*((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^6*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))/e^5/(c^2*d+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^6/e^5-3/4*d/
e^2*c^6*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))-3/4*d/e^2*c^6*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)))

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F(-2)]

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

[In]

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

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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