\(\int \frac {x^3 (a+b \arcsin (c x))}{d+e x^2} \, dx\) [625]

Optimal result

Integrand size = 21, antiderivative size = 559 \[ \int \frac {x^3 (a+b \arcsin (c x))}{d+e x^2} \, dx=\frac {b x \sqrt {1-c^2 x^2}}{4 c e}-\frac {b \arcsin (c x)}{4 c^2 e}+\frac {x^2 (a+b \arcsin (c x))}{2 e}+\frac {i d (a+b \arcsin (c x))^2}{2 b e^2}-\frac {d (a+b \arcsin (c x)) \log \left (1-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{2 e^2}-\frac {d (a+b \arcsin (c x)) \log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{2 e^2}-\frac {d (a+b \arcsin (c x)) \log \left (1-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{2 e^2}-\frac {d (a+b \arcsin (c x)) \log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{2 e^2}+\frac {i b d \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{2 e^2}+\frac {i b d \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{2 e^2}+\frac {i b d \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{2 e^2}+\frac {i b d \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{2 e^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.60 (sec) , antiderivative size = 559, 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, 4723, 327, 222, 4825, 4617, 2221, 2317, 2438} \[ \int \frac {x^3 (a+b \arcsin (c x))}{d+e x^2} \, dx=-\frac {d (a+b \arcsin (c x)) \log \left (1-\frac {\sqrt {e} e^{i \arcsin (c x)}}{-\sqrt {c^2 d+e}+i c \sqrt {-d}}\right )}{2 e^2}-\frac {d (a+b \arcsin (c x)) \log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{-\sqrt {c^2 d+e}+i c \sqrt {-d}}\right )}{2 e^2}-\frac {d (a+b \arcsin (c x)) \log \left (1-\frac {\sqrt {e} e^{i \arcsin (c x)}}{\sqrt {c^2 d+e}+i c \sqrt {-d}}\right )}{2 e^2}-\frac {d (a+b \arcsin (c x)) \log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{\sqrt {c^2 d+e}+i c \sqrt {-d}}\right )}{2 e^2}+\frac {i d (a+b \arcsin (c x))^2}{2 b e^2}+\frac {x^2 (a+b \arcsin (c x))}{2 e}+\frac {i b d \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {d c^2+e}}\right )}{2 e^2}+\frac {i b d \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {d c^2+e}}\right )}{2 e^2}+\frac {i b d \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i \sqrt {-d} c+\sqrt {d c^2+e}}\right )}{2 e^2}+\frac {i b d \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{i \sqrt {-d} c+\sqrt {d c^2+e}}\right )}{2 e^2}-\frac {b \arcsin (c x)}{4 c^2 e}+\frac {b x \sqrt {1-c^2 x^2}}{4 c e} \]

[In]

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

[Out]

(b*x*Sqrt[1 - c^2*x^2])/(4*c*e) - (b*ArcSin[c*x])/(4*c^2*e) + (x^2*(a + b*ArcSin[c*x]))/(2*e) + ((I/2)*d*(a +
b*ArcSin[c*x])^2)/(b*e^2) - (d*(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) - (d*(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) - (d*(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) - (d*(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*d*PolyLog[2, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e]))])/e^2 + ((I/2)*b*d*P
olyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/e^2 + ((I/2)*b*d*PolyLog[2, -((Sqrt[e
]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e]))])/e^2 + ((I/2)*b*d*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]
))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])])/e^2

Rule 222

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

Rule 327

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^n
)^(p + 1)/(b*(m + n*p + 1))), x] - Dist[a*c^n*((m - n + 1)/(b*(m + n*p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, 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 4723

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*ArcSi
n[c*x])^n/(d*(m + 1))), x] - Dist[b*c*(n/(d*(m + 1))), Int[(d*x)^(m + 1)*((a + b*ArcSin[c*x])^(n - 1)/Sqrt[1 -
c^2*x^2]), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -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 {x (a+b \arcsin (c x))}{e}-\frac {d x (a+b \arcsin (c x))}{e \left (d+e x^2\right )}\right ) \, dx \\ & = \frac {\int x (a+b \arcsin (c x)) \, dx}{e}-\frac {d \int \frac {x (a+b \arcsin (c x))}{d+e x^2} \, dx}{e} \\ & = \frac {x^2 (a+b \arcsin (c x))}{2 e}-\frac {(b c) \int \frac {x^2}{\sqrt {1-c^2 x^2}} \, dx}{2 e}-\frac {d \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 {b x \sqrt {1-c^2 x^2}}{4 c e}+\frac {x^2 (a+b \arcsin (c x))}{2 e}+\frac {d \int \frac {a+b \arcsin (c x)}{\sqrt {-d}-\sqrt {e} x} \, dx}{2 e^{3/2}}-\frac {d \int \frac {a+b \arcsin (c x)}{\sqrt {-d}+\sqrt {e} x} \, dx}{2 e^{3/2}}-\frac {b \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{4 c e} \\ & = \frac {b x \sqrt {1-c^2 x^2}}{4 c e}-\frac {b \arcsin (c x)}{4 c^2 e}+\frac {x^2 (a+b \arcsin (c x))}{2 e}+\frac {d \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 \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 {b x \sqrt {1-c^2 x^2}}{4 c e}-\frac {b \arcsin (c x)}{4 c^2 e}+\frac {x^2 (a+b \arcsin (c x))}{2 e}+\frac {i d (a+b \arcsin (c x))^2}{2 b e^2}+\frac {(i d) \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 d) \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 d) \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 d) \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 {b x \sqrt {1-c^2 x^2}}{4 c e}-\frac {b \arcsin (c x)}{4 c^2 e}+\frac {x^2 (a+b \arcsin (c x))}{2 e}+\frac {i d (a+b \arcsin (c x))^2}{2 b e^2}-\frac {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 )}{2 e^2}-\frac {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 )}{2 e^2}-\frac {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 )}{2 e^2}-\frac {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 )}{2 e^2}+\frac {(b d) \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 d) \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 d) \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 d) \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 x \sqrt {1-c^2 x^2}}{4 c e}-\frac {b \arcsin (c x)}{4 c^2 e}+\frac {x^2 (a+b \arcsin (c x))}{2 e}+\frac {i d (a+b \arcsin (c x))^2}{2 b e^2}-\frac {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 )}{2 e^2}-\frac {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 )}{2 e^2}-\frac {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 )}{2 e^2}-\frac {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 )}{2 e^2}-\frac {(i b d) \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 d) \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 d) \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 d) \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 {b x \sqrt {1-c^2 x^2}}{4 c e}-\frac {b \arcsin (c x)}{4 c^2 e}+\frac {x^2 (a+b \arcsin (c x))}{2 e}+\frac {i d (a+b \arcsin (c x))^2}{2 b e^2}-\frac {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 )}{2 e^2}-\frac {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 )}{2 e^2}-\frac {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 )}{2 e^2}-\frac {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 )}{2 e^2}+\frac {i b d \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 d \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 d \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 d \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 = 0.30 (sec) , antiderivative size = 475, normalized size of antiderivative = 0.85 \[ \int \frac {x^3 (a+b \arcsin (c x))}{d+e x^2} \, dx=\frac {2 a c^2 e x^2-2 a c^2 d \log \left (d+e x^2\right )+b \left (e \left (c x \sqrt {1-c^2 x^2}+2 c^2 x^2 \arcsin (c x)-2 \arctan \left (\frac {c x}{-1+\sqrt {1-c^2 x^2}}\right )\right )+i c^2 d \left (\arcsin (c x) \left (\arcsin (c x)+2 i \left (\log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}-\sqrt {c^2 d+e}}\right )+\log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}+\sqrt {c^2 d+e}}\right )\right )\right )+2 \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{-c \sqrt {d}+\sqrt {c^2 d+e}}\right )+2 \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}+\sqrt {c^2 d+e}}\right )\right )+i c^2 d \left (\arcsin (c x) \left (\arcsin (c x)+2 i \left (\log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{-c \sqrt {d}+\sqrt {c^2 d+e}}\right )+\log \left (1-\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}+\sqrt {c^2 d+e}}\right )\right )\right )+2 \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}-\sqrt {c^2 d+e}}\right )+2 \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}+\sqrt {c^2 d+e}}\right )\right )\right )}{4 c^2 e^2} \]

[In]

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

[Out]

(2*a*c^2*e*x^2 - 2*a*c^2*d*Log[d + e*x^2] + b*(e*(c*x*Sqrt[1 - c^2*x^2] + 2*c^2*x^2*ArcSin[c*x] - 2*ArcTan[(c*
x)/(-1 + Sqrt[1 - c^2*x^2])]) + I*c^2*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*Pol
yLog[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*c^2*d*(ArcSin[c*x]*(ArcSin[c*x] + (2*I)*(Log[1 + (Sqrt[e]*E^(I*ArcSi
n[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*Arc
Sin[c*x]))/(c*Sqrt[d] + Sqrt[c^2*d + e])])))/(4*c^2*e^2)

Maple [C] (warning: unable to verify)

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

Time = 2.30 (sec) , antiderivative size = 2088, normalized size of antiderivative = 3.74

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(2088\)
default	\(\text {Expression too large to display}\)	\(2088\)
parts	\(\text {Expression too large to display}\)	\(2095\)

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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