\(\int \frac {x^3 (a+b \arcsin (c x))}{(d+e x^2)^2} \, dx\) [633]

Optimal result

Integrand size = 21, antiderivative size = 574 \[ \int \frac {x^3 (a+b \arcsin (c x))}{\left (d+e x^2\right )^2} \, dx=\frac {d (a+b \arcsin (c x))}{2 e^2 \left (d+e x^2\right )}-\frac {i (a+b \arcsin (c x))^2}{2 b e^2}-\frac {b c \sqrt {d} \arctan \left (\frac {\sqrt {c^2 d+e} x}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{2 e^2 \sqrt {c^2 d+e}}+\frac {(a+b \arcsin (c x)) \log \left (1-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{2 e^2}+\frac {(a+b \arcsin (c x)) \log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{2 e^2}+\frac {(a+b \arcsin (c x)) \log \left (1-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{2 e^2}+\frac {(a+b \arcsin (c x)) \log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{2 e^2}-\frac {i b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{2 e^2}-\frac {i b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{2 e^2}-\frac {i b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{2 e^2}-\frac {i b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{2 e^2} \]

[Out]

1/2*d*(a+b*arcsin(c*x))/e^2/(e*x^2+d)-1/2*I*(a+b*arcsin(c*x))^2/b/e^2+1/2*(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^2+1/2*(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^2+1/2*(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^2+1/2*(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^2-1/2*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^2-1/2*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^2-1/2*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^2-1/2*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^2-1/2*b*c*arctan(x*(c^2*d+e)^(1/2)/d^(1/2)/(-c
^2*x^2+1)^(1/2))*d^(1/2)/e^2/(c^2*d+e)^(1/2)

Rubi [A] (verified)

Time = 0.69 (sec) , antiderivative size = 574, normalized size of antiderivative = 1.00, number of steps used = 23, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {4817, 4813, 385, 211, 4825, 4617, 2221, 2317, 2438} \[ \int \frac {x^3 (a+b \arcsin (c x))}{\left (d+e x^2\right )^2} \, dx=\frac {(a+b \arcsin (c x)) \log \left (1-\frac {\sqrt {e} e^{i \arcsin (c x)}}{-\sqrt {c^2 d+e}+i c \sqrt {-d}}\right )}{2 e^2}+\frac {(a+b \arcsin (c x)) \log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{-\sqrt {c^2 d+e}+i c \sqrt {-d}}\right )}{2 e^2}+\frac {(a+b \arcsin (c x)) \log \left (1-\frac {\sqrt {e} e^{i \arcsin (c x)}}{\sqrt {c^2 d+e}+i c \sqrt {-d}}\right )}{2 e^2}+\frac {(a+b \arcsin (c x)) \log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{\sqrt {c^2 d+e}+i c \sqrt {-d}}\right )}{2 e^2}+\frac {d (a+b \arcsin (c x))}{2 e^2 \left (d+e x^2\right )}-\frac {i (a+b \arcsin (c x))^2}{2 b e^2}-\frac {i b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {d c^2+e}}\right )}{2 e^2}-\frac {i b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {d c^2+e}}\right )}{2 e^2}-\frac {i b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i \sqrt {-d} c+\sqrt {d c^2+e}}\right )}{2 e^2}-\frac {i b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{i \sqrt {-d} c+\sqrt {d c^2+e}}\right )}{2 e^2}-\frac {b c \sqrt {d} \arctan \left (\frac {x \sqrt {c^2 d+e}}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{2 e^2 \sqrt {c^2 d+e}} \]

[In]

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

[Out]

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

Rule 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 385

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

Rule 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 4813

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

Rule 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]

Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {d x (a+b \arcsin (c x))}{e \left (d+e x^2\right )^2}+\frac {x (a+b \arcsin (c x))}{e \left (d+e x^2\right )}\right ) \, dx \\ & = \frac {\int \frac {x (a+b \arcsin (c x))}{d+e x^2} \, dx}{e}-\frac {d \int \frac {x (a+b \arcsin (c x))}{\left (d+e x^2\right )^2} \, dx}{e} \\ & = \frac {d (a+b \arcsin (c x))}{2 e^2 \left (d+e x^2\right )}-\frac {(b c d) \int \frac {1}{\sqrt {1-c^2 x^2} \left (d+e x^2\right )} \, dx}{2 e^2}+\frac {\int \left (-\frac {a+b \arcsin (c x)}{2 \sqrt {e} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {a+b \arcsin (c x)}{2 \sqrt {e} \left (\sqrt {-d}+\sqrt {e} x\right )}\right ) \, dx}{e} \\ & = \frac {d (a+b \arcsin (c x))}{2 e^2 \left (d+e x^2\right )}-\frac {(b c d) \text {Subst}\left (\int \frac {1}{d-\left (-c^2 d-e\right ) x^2} \, dx,x,\frac {x}{\sqrt {1-c^2 x^2}}\right )}{2 e^2}-\frac {\int \frac {a+b \arcsin (c x)}{\sqrt {-d}-\sqrt {e} x} \, dx}{2 e^{3/2}}+\frac {\int \frac {a+b \arcsin (c x)}{\sqrt {-d}+\sqrt {e} x} \, dx}{2 e^{3/2}} \\ & = \frac {d (a+b \arcsin (c x))}{2 e^2 \left (d+e x^2\right )}-\frac {b c \sqrt {d} \arctan \left (\frac {\sqrt {c^2 d+e} x}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{2 e^2 \sqrt {c^2 d+e}}-\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 e^{3/2}}+\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 e^{3/2}} \\ & = \frac {d (a+b \arcsin (c x))}{2 e^2 \left (d+e x^2\right )}-\frac {i (a+b \arcsin (c x))^2}{2 b e^2}-\frac {b c \sqrt {d} \arctan \left (\frac {\sqrt {c^2 d+e} x}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{2 e^2 \sqrt {c^2 d+e}}-\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 e^{3/2}}-\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 e^{3/2}}+\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 e^{3/2}}+\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 e^{3/2}} \\ & = \frac {d (a+b \arcsin (c x))}{2 e^2 \left (d+e x^2\right )}-\frac {i (a+b \arcsin (c x))^2}{2 b e^2}-\frac {b c \sqrt {d} \arctan \left (\frac {\sqrt {c^2 d+e} x}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{2 e^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 e^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 e^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 e^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 e^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 e^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 e^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 e^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 e^2} \\ & = \frac {d (a+b \arcsin (c x))}{2 e^2 \left (d+e x^2\right )}-\frac {i (a+b \arcsin (c x))^2}{2 b e^2}-\frac {b c \sqrt {d} \arctan \left (\frac {\sqrt {c^2 d+e} x}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{2 e^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 e^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 e^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 e^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 e^2}+\frac {(i b) \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 )}{2 e^2}+\frac {(i b) \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 )}{2 e^2}+\frac {(i b) \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 )}{2 e^2}+\frac {(i b) \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 )}{2 e^2} \\ & = \frac {d (a+b \arcsin (c x))}{2 e^2 \left (d+e x^2\right )}-\frac {i (a+b \arcsin (c x))^2}{2 b e^2}-\frac {b c \sqrt {d} \arctan \left (\frac {\sqrt {c^2 d+e} x}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{2 e^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 e^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 e^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 e^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 e^2}-\frac {i b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{2 e^2}-\frac {i b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{2 e^2}-\frac {i b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{2 e^2}-\frac {i b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{2 e^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.87 (sec) , antiderivative size = 593, normalized size of antiderivative = 1.03 \[ \int \frac {x^3 (a+b \arcsin (c x))}{\left (d+e x^2\right )^2} \, dx=\frac {\frac {2 a d}{d+e x^2}+2 a \log \left (d+e x^2\right )+b \left (\sqrt {d} \left (\frac {\arcsin (c x)}{\sqrt {d}+i \sqrt {e} x}-\frac {c \arctan \left (\frac {i \sqrt {e}+c^2 \sqrt {d} x}{\sqrt {c^2 d+e} \sqrt {1-c^2 x^2}}\right )}{\sqrt {c^2 d+e}}\right )-i \sqrt {d} \left (-\frac {\arcsin (c x)}{i \sqrt {d}+\sqrt {e} x}-\frac {c \text {arctanh}\left (\frac {\sqrt {e}+i c^2 \sqrt {d} x}{\sqrt {c^2 d+e} \sqrt {1-c^2 x^2}}\right )}{\sqrt {c^2 d+e}}\right )-i \left (\arcsin (c x) \left (\arcsin (c x)+2 i \left (\log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}-\sqrt {c^2 d+e}}\right )+\log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}+\sqrt {c^2 d+e}}\right )\right )\right )+2 \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{-c \sqrt {d}+\sqrt {c^2 d+e}}\right )+2 \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}+\sqrt {c^2 d+e}}\right )\right )-i \left (\arcsin (c x) \left (\arcsin (c x)+2 i \left (\log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{-c \sqrt {d}+\sqrt {c^2 d+e}}\right )+\log \left (1-\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}+\sqrt {c^2 d+e}}\right )\right )\right )+2 \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}-\sqrt {c^2 d+e}}\right )+2 \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}+\sqrt {c^2 d+e}}\right )\right )\right )}{4 e^2} \]

[In]

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

[Out]

((2*a*d)/(d + e*x^2) + 2*a*Log[d + e*x^2] + b*(Sqrt[d]*(ArcSin[c*x]/(Sqrt[d] + I*Sqrt[e]*x) - (c*ArcTan[(I*Sqr
t[e] + c^2*Sqrt[d]*x)/(Sqrt[c^2*d + e]*Sqrt[1 - c^2*x^2])])/Sqrt[c^2*d + e]) - I*Sqrt[d]*(-(ArcSin[c*x]/(I*Sqr
t[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]) - I*(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]))]) - I*(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*ArcS
in[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])])))/(4*e^2)

Maple [C] (warning: unable to verify)

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

Time = 1.85 (sec) , antiderivative size = 2101, normalized size of antiderivative = 3.66

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(2101\)
default	\(\text {Expression too large to display}\)	\(2101\)
parts	\(\text {Expression too large to display}\)	\(2113\)

[In]

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

[Out]

1/c^4*(1/2*a*c^6/e^2*d/(c^2*e*x^2+c^2*d)+1/2*a*c^4/e^2*ln(c^2*e*x^2+c^2*d)+b*c^4*(-I*(2*c^2*d+2*(d*c^2*(c^2*d+
e))^(1/2)+e)*arcsin(c*x)^2*d*c^2/e^4-1/2*I/e^2*sum((-_R1^2*e+4*c^2*d+2*e)/(-_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))-1/4*I*(d*c^2*(c^2*d+e))^(1/2)/c^2/d/e/(c^2*d+e)*arcsin(c*x)^2-1/8*I*(d*c^2*(c^2*d+e))^(1/2)/c^2/
d/e/(c^2*d+e)*polylog(2,e*(I*c*x+(-c^2*x^2+1)^(1/2))^2/(2*c^2*d+2*(d*c^2*(c^2*d+e))^(1/2)+e))+1/8*I*(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)*polylog(2,e*(I*c*x+(-c^2*x^2+1)^(1/2))^2
/(2*c^2*d-2*(d*c^2*(c^2*d+e))^(1/2)+e))/e^2/d/c^2/(c^2*d+e)+1/4*I*(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)*arcsin(c*x)^2/e^2/d/c^2/(c^2*d+e)+I*(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*arcsin(c*x)^2/e^4/(c^2*d+e)-(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*ln(1-e*(I*c*x+(-c^2*x^2+1)^(1/2))^2/(2*c^2*d-2*(d*c^2*(c^2
*d+e))^(1/2)+e))*arcsin(c*x)/e^4/(c^2*d+e)+1/2*I*(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*polylog(2,e*(I*c*x+(-c^2*x^2+1)^(1/2))^2/(2*c^2*d-2*(d*c^2*(c^2*d+e))^(1/2)+e))/e^4/(
c^2*d+e)-1/4*(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)*ln(1-e*(I*c*x+(-c
^2*x^2+1)^(1/2))^2/(2*c^2*d-2*(d*c^2*(c^2*d+e))^(1/2)+e))*arcsin(c*x)/e^2/d/c^2/(c^2*d+e)+1/4*(d*c^2*(c^2*d+e)
)^(1/2)/c^2/d/e/(c^2*d+e)*arcsin(c*x)*ln(1-e*(I*c*x+(-c^2*x^2+1)^(1/2))^2/(2*c^2*d+2*(d*c^2*(c^2*d+e))^(1/2)+e
))+I*(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)*arcsin(c*x)^2/e^3/(c^2*d+
e)+1/2*(d*c^2*(c^2*d+e))^(1/2)/e^2/(c^2*d+e)*arcsin(c*x)*ln(1-e*(I*c*x+(-c^2*x^2+1)^(1/2))^2/(2*c^2*d+2*(d*c^2
*(c^2*d+e))^(1/2)+e))-(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)*ln(1-e*(
I*c*x+(-c^2*x^2+1)^(1/2))^2/(2*c^2*d-2*(d*c^2*(c^2*d+e))^(1/2)+e))*arcsin(c*x)/e^3/(c^2*d+e)-1/2*I*(d*c^2*(c^2
*d+e))^(1/2)/e^2/(c^2*d+e)*arcsin(c*x)^2+1/2*arcsin(c*x)/e^2*d*c^2/(c^2*e*x^2+c^2*d)+1/2*I*(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)*polylog(2,e*(I*c*x+(-c^2*x^2+1)^(1/2))^2/(2*c^2*d
-2*(d*c^2*(c^2*d+e))^(1/2)+e))/e^3/(c^2*d+e)-1/2*I*(d*c^2*(c^2*d+e))^(1/2)/e^2/(c^2*d+e)*arctanh(1/4*(4*c^2*d-
2*e*(I*c*x+(-c^2*x^2+1)^(1/2))^2+2*e)/(c^4*d^2+c^2*d*e)^(1/2))-1/4*I*(d*c^2*(c^2*d+e))^(1/2)/e^2/(c^2*d+e)*pol
ylog(2,e*(I*c*x+(-c^2*x^2+1)^(1/2))^2/(2*c^2*d+2*(d*c^2*(c^2*d+e))^(1/2)+e))-1/2*I*(2*c^2*d+2*(d*c^2*(c^2*d+e)
)^(1/2)+e)*arcsin(c*x)^2/e^3+1/2*(2*c^2*d+2*(d*c^2*(c^2*d+e))^(1/2)+e)*ln(1-e*(I*c*x+(-c^2*x^2+1)^(1/2))^2/(2*
c^2*d-2*(d*c^2*(c^2*d+e))^(1/2)+e))*arcsin(c*x)/e^3-1/4*I*(2*c^2*d+2*(d*c^2*(c^2*d+e))^(1/2)+e)*polylog(2,e*(I
*c*x+(-c^2*x^2+1)^(1/2))^2/(2*c^2*d-2*(d*c^2*(c^2*d+e))^(1/2)+e))/e^3-1/2*I*arcsin(c*x)^2/e^2-1/2*I*(2*c^2*d+2
*(d*c^2*(c^2*d+e))^(1/2)+e)*polylog(2,e*(I*c*x+(-c^2*x^2+1)^(1/2))^2/(2*c^2*d-2*(d*c^2*(c^2*d+e))^(1/2)+e))*d*
c^2/e^4+(2*c^2*d+2*(d*c^2*(c^2*d+e))^(1/2)+e)*ln(1-e*(I*c*x+(-c^2*x^2+1)^(1/2))^2/(2*c^2*d-2*(d*c^2*(c^2*d+e))
^(1/2)+e))*c^2*d*arcsin(c*x)/e^4))

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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