3.624 \(\int \frac {x^4 (a+b \sin ^{-1}(c x))}{d+e x^2} \, dx\)

Optimal. Leaf size=653 \[ \frac {(-d)^{3/2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac {\sqrt {e} e^{i \sin ^{-1}(c x)}}{-\sqrt {c^2 d+e}+i c \sqrt {-d}}\right )}{2 e^{5/2}}-\frac {(-d)^{3/2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1+\frac {\sqrt {e} e^{i \sin ^{-1}(c x)}}{-\sqrt {c^2 d+e}+i c \sqrt {-d}}\right )}{2 e^{5/2}}+\frac {(-d)^{3/2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac {\sqrt {e} e^{i \sin ^{-1}(c x)}}{\sqrt {c^2 d+e}+i c \sqrt {-d}}\right )}{2 e^{5/2}}-\frac {(-d)^{3/2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1+\frac {\sqrt {e} e^{i \sin ^{-1}(c x)}}{\sqrt {c^2 d+e}+i c \sqrt {-d}}\right )}{2 e^{5/2}}+\frac {x^3 \left (a+b \sin ^{-1}(c x)\right )}{3 e}-\frac {a d x}{e^2}+\frac {i b (-d)^{3/2} \text {Li}_2\left (-\frac {\sqrt {e} e^{i \sin ^{-1}(c x)}}{i c \sqrt {-d}-\sqrt {d c^2+e}}\right )}{2 e^{5/2}}-\frac {i b (-d)^{3/2} \text {Li}_2\left (\frac {\sqrt {e} e^{i \sin ^{-1}(c x)}}{i c \sqrt {-d}-\sqrt {d c^2+e}}\right )}{2 e^{5/2}}+\frac {i b (-d)^{3/2} \text {Li}_2\left (-\frac {\sqrt {e} e^{i \sin ^{-1}(c x)}}{i \sqrt {-d} c+\sqrt {d c^2+e}}\right )}{2 e^{5/2}}-\frac {i b (-d)^{3/2} \text {Li}_2\left (\frac {\sqrt {e} e^{i \sin ^{-1}(c x)}}{i \sqrt {-d} c+\sqrt {d c^2+e}}\right )}{2 e^{5/2}}-\frac {b d \sqrt {1-c^2 x^2}}{c e^2}-\frac {b \left (1-c^2 x^2\right )^{3/2}}{9 c^3 e}+\frac {b \sqrt {1-c^2 x^2}}{3 c^3 e}-\frac {b d x \sin ^{-1}(c x)}{e^2} \]

[Out]

-a*d*x/e^2-1/9*b*(-c^2*x^2+1)^(3/2)/c^3/e-b*d*x*arcsin(c*x)/e^2+1/3*x^3*(a+b*arcsin(c*x))/e+1/2*(-d)^(3/2)*(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^(5/2)-1/2*(-d)^(3/2
)*(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^(5/2)+1/2*(-d)
^(3/2)*(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^(5/2)-1/2
*(-d)^(3/2)*(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^(5/2
)+1/2*I*b*(-d)^(3/2)*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)-1
/2*I*b*(-d)^(3/2)*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)+1/2*I
*b*(-d)^(3/2)*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)-1/2*I*b*
(-d)^(3/2)*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)-b*d*(-c^2*x^
2+1)^(1/2)/c/e^2+1/3*b*(-c^2*x^2+1)^(1/2)/c^3/e

________________________________________________________________________________________

Rubi [A]  time = 1.05, antiderivative size = 653, normalized size of antiderivative = 1.00, number of steps used = 27, number of rules used = 12, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {4733, 4619, 261, 4627, 266, 43, 4667, 4741, 4521, 2190, 2279, 2391} \[ \frac {i b (-d)^{3/2} \text {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \sin ^{-1}(c x)}}{-\sqrt {c^2 d+e}+i c \sqrt {-d}}\right )}{2 e^{5/2}}-\frac {i b (-d)^{3/2} \text {PolyLog}\left (2,\frac {\sqrt {e} e^{i \sin ^{-1}(c x)}}{-\sqrt {c^2 d+e}+i c \sqrt {-d}}\right )}{2 e^{5/2}}+\frac {i b (-d)^{3/2} \text {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \sin ^{-1}(c x)}}{\sqrt {c^2 d+e}+i c \sqrt {-d}}\right )}{2 e^{5/2}}-\frac {i b (-d)^{3/2} \text {PolyLog}\left (2,\frac {\sqrt {e} e^{i \sin ^{-1}(c x)}}{\sqrt {c^2 d+e}+i c \sqrt {-d}}\right )}{2 e^{5/2}}+\frac {(-d)^{3/2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac {\sqrt {e} e^{i \sin ^{-1}(c x)}}{-\sqrt {c^2 d+e}+i c \sqrt {-d}}\right )}{2 e^{5/2}}-\frac {(-d)^{3/2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1+\frac {\sqrt {e} e^{i \sin ^{-1}(c x)}}{-\sqrt {c^2 d+e}+i c \sqrt {-d}}\right )}{2 e^{5/2}}+\frac {(-d)^{3/2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac {\sqrt {e} e^{i \sin ^{-1}(c x)}}{\sqrt {c^2 d+e}+i c \sqrt {-d}}\right )}{2 e^{5/2}}-\frac {(-d)^{3/2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1+\frac {\sqrt {e} e^{i \sin ^{-1}(c x)}}{\sqrt {c^2 d+e}+i c \sqrt {-d}}\right )}{2 e^{5/2}}+\frac {x^3 \left (a+b \sin ^{-1}(c x)\right )}{3 e}-\frac {a d x}{e^2}-\frac {b d \sqrt {1-c^2 x^2}}{c e^2}-\frac {b \left (1-c^2 x^2\right )^{3/2}}{9 c^3 e}+\frac {b \sqrt {1-c^2 x^2}}{3 c^3 e}-\frac {b d x \sin ^{-1}(c x)}{e^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((a*d*x)/e^2) - (b*d*Sqrt[1 - c^2*x^2])/(c*e^2) + (b*Sqrt[1 - c^2*x^2])/(3*c^3*e) - (b*(1 - c^2*x^2)^(3/2))/(
9*c^3*e) - (b*d*x*ArcSin[c*x])/e^2 + (x^3*(a + b*ArcSin[c*x]))/(3*e) + ((-d)^(3/2)*(a + b*ArcSin[c*x])*Log[1 -
(Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/(2*e^(5/2)) - ((-d)^(3/2)*(a + b*ArcSin[c*x])*
Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/(2*e^(5/2)) + ((-d)^(3/2)*(a + b*ArcSin
[c*x])*Log[1 - (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])])/(2*e^(5/2)) - ((-d)^(3/2)*(a + b
*ArcSin[c*x])*Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])])/(2*e^(5/2)) + ((I/2)*b*(-
d)^(3/2)*PolyLog[2, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e]))])/e^(5/2) - ((I/2)*b*(-d)^
(3/2)*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/e^(5/2) + ((I/2)*b*(-d)^(3/2)*
PolyLog[2, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e]))])/e^(5/2) - ((I/2)*b*(-d)^(3/2)*Pol
yLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])])/e^(5/2)

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 261

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 266

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

Rule 2190

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), x]
- Dist[(d*m)/(b*f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*Log[1 + (b*(F^(g*(e + f*x)))^n)/a], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2279

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 4521

Int[(Cos[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sin[(c_.) + (d_.)*(x_)]), x_Symbol] :>
-Simp[(I*(e + f*x)^(m + 1))/(b*f*(m + 1)), x] + (Dist[I, Int[((e + f*x)^m*E^(I*(c + d*x)))/(I*a - Rt[-a^2 + b^
2, 2] + b*E^(I*(c + d*x))), x], x] + Dist[I, Int[((e + f*x)^m*E^(I*(c + d*x)))/(I*a + Rt[-a^2 + b^2, 2] + b*E^
(I*(c + d*x))), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && NegQ[a^2 - b^2]

Rule 4619

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 4627

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 4667

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a
+ b*ArcSin[c*x])^n, (d + e*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, n}, x] && NeQ[c^2*d + e, 0] && IntegerQ[p]
&& (GtQ[p, 0] || IGtQ[n, 0])

Rule 4733

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Int[
ExpandIntegrand[(a + b*ArcSin[c*x])^n, (f*x)^m*(d + e*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[c^
2*d + e, 0] && IGtQ[n, 0] && IntegerQ[p] && IntegerQ[m]

Rule 4741

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)), x_Symbol] :> Subst[Int[((a + b*x)^n*Cos[x])/
(c*d + e*Sin[x]), x], x, ArcSin[c*x]] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[n, 0]

Rubi steps

\begin {align*} \int \frac {x^4 \left (a+b \sin ^{-1}(c x)\right )}{d+e x^2} \, dx &=\int \left (-\frac {d \left (a+b \sin ^{-1}(c x)\right )}{e^2}+\frac {x^2 \left (a+b \sin ^{-1}(c x)\right )}{e}+\frac {d^2 \left (a+b \sin ^{-1}(c x)\right )}{e^2 \left (d+e x^2\right )}\right ) \, dx\\ &=-\frac {d \int \left (a+b \sin ^{-1}(c x)\right ) \, dx}{e^2}+\frac {d^2 \int \frac {a+b \sin ^{-1}(c x)}{d+e x^2} \, dx}{e^2}+\frac {\int x^2 \left (a+b \sin ^{-1}(c x)\right ) \, dx}{e}\\ &=-\frac {a d x}{e^2}+\frac {x^3 \left (a+b \sin ^{-1}(c x)\right )}{3 e}-\frac {(b d) \int \sin ^{-1}(c x) \, dx}{e^2}+\frac {d^2 \int \left (\frac {\sqrt {-d} \left (a+b \sin ^{-1}(c x)\right )}{2 d \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {\sqrt {-d} \left (a+b \sin ^{-1}(c x)\right )}{2 d \left (\sqrt {-d}+\sqrt {e} x\right )}\right ) \, dx}{e^2}-\frac {(b c) \int \frac {x^3}{\sqrt {1-c^2 x^2}} \, dx}{3 e}\\ &=-\frac {a d x}{e^2}-\frac {b d x \sin ^{-1}(c x)}{e^2}+\frac {x^3 \left (a+b \sin ^{-1}(c x)\right )}{3 e}-\frac {(-d)^{3/2} \int \frac {a+b \sin ^{-1}(c x)}{\sqrt {-d}-\sqrt {e} x} \, dx}{2 e^2}-\frac {(-d)^{3/2} \int \frac {a+b \sin ^{-1}(c x)}{\sqrt {-d}+\sqrt {e} x} \, dx}{2 e^2}+\frac {(b c d) \int \frac {x}{\sqrt {1-c^2 x^2}} \, dx}{e^2}-\frac {(b c) \operatorname {Subst}\left (\int \frac {x}{\sqrt {1-c^2 x}} \, dx,x,x^2\right )}{6 e}\\ &=-\frac {a d x}{e^2}-\frac {b d \sqrt {1-c^2 x^2}}{c e^2}-\frac {b d x \sin ^{-1}(c x)}{e^2}+\frac {x^3 \left (a+b \sin ^{-1}(c x)\right )}{3 e}-\frac {(-d)^{3/2} \operatorname {Subst}\left (\int \frac {(a+b x) \cos (x)}{c \sqrt {-d}-\sqrt {e} \sin (x)} \, dx,x,\sin ^{-1}(c x)\right )}{2 e^2}-\frac {(-d)^{3/2} \operatorname {Subst}\left (\int \frac {(a+b x) \cos (x)}{c \sqrt {-d}+\sqrt {e} \sin (x)} \, dx,x,\sin ^{-1}(c x)\right )}{2 e^2}-\frac {(b c) \operatorname {Subst}\left (\int \left (\frac {1}{c^2 \sqrt {1-c^2 x}}-\frac {\sqrt {1-c^2 x}}{c^2}\right ) \, dx,x,x^2\right )}{6 e}\\ &=-\frac {a d x}{e^2}-\frac {b d \sqrt {1-c^2 x^2}}{c e^2}+\frac {b \sqrt {1-c^2 x^2}}{3 c^3 e}-\frac {b \left (1-c^2 x^2\right )^{3/2}}{9 c^3 e}-\frac {b d x \sin ^{-1}(c x)}{e^2}+\frac {x^3 \left (a+b \sin ^{-1}(c x)\right )}{3 e}-\frac {\left (i (-d)^{3/2}\right ) \operatorname {Subst}\left (\int \frac {e^{i x} (a+b x)}{i c \sqrt {-d}-\sqrt {c^2 d+e}-\sqrt {e} e^{i x}} \, dx,x,\sin ^{-1}(c x)\right )}{2 e^2}-\frac {\left (i (-d)^{3/2}\right ) \operatorname {Subst}\left (\int \frac {e^{i x} (a+b x)}{i c \sqrt {-d}+\sqrt {c^2 d+e}-\sqrt {e} e^{i x}} \, dx,x,\sin ^{-1}(c x)\right )}{2 e^2}-\frac {\left (i (-d)^{3/2}\right ) \operatorname {Subst}\left (\int \frac {e^{i x} (a+b x)}{i c \sqrt {-d}-\sqrt {c^2 d+e}+\sqrt {e} e^{i x}} \, dx,x,\sin ^{-1}(c x)\right )}{2 e^2}-\frac {\left (i (-d)^{3/2}\right ) \operatorname {Subst}\left (\int \frac {e^{i x} (a+b x)}{i c \sqrt {-d}+\sqrt {c^2 d+e}+\sqrt {e} e^{i x}} \, dx,x,\sin ^{-1}(c x)\right )}{2 e^2}\\ &=-\frac {a d x}{e^2}-\frac {b d \sqrt {1-c^2 x^2}}{c e^2}+\frac {b \sqrt {1-c^2 x^2}}{3 c^3 e}-\frac {b \left (1-c^2 x^2\right )^{3/2}}{9 c^3 e}-\frac {b d x \sin ^{-1}(c x)}{e^2}+\frac {x^3 \left (a+b \sin ^{-1}(c x)\right )}{3 e}+\frac {(-d)^{3/2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac {\sqrt {e} e^{i \sin ^{-1}(c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{2 e^{5/2}}-\frac {(-d)^{3/2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1+\frac {\sqrt {e} e^{i \sin ^{-1}(c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{2 e^{5/2}}+\frac {(-d)^{3/2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac {\sqrt {e} e^{i \sin ^{-1}(c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{2 e^{5/2}}-\frac {(-d)^{3/2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1+\frac {\sqrt {e} e^{i \sin ^{-1}(c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{2 e^{5/2}}-\frac {\left (b (-d)^{3/2}\right ) \operatorname {Subst}\left (\int \log \left (1-\frac {\sqrt {e} e^{i x}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{2 e^{5/2}}+\frac {\left (b (-d)^{3/2}\right ) \operatorname {Subst}\left (\int \log \left (1+\frac {\sqrt {e} e^{i x}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{2 e^{5/2}}-\frac {\left (b (-d)^{3/2}\right ) \operatorname {Subst}\left (\int \log \left (1-\frac {\sqrt {e} e^{i x}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{2 e^{5/2}}+\frac {\left (b (-d)^{3/2}\right ) \operatorname {Subst}\left (\int \log \left (1+\frac {\sqrt {e} e^{i x}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{2 e^{5/2}}\\ &=-\frac {a d x}{e^2}-\frac {b d \sqrt {1-c^2 x^2}}{c e^2}+\frac {b \sqrt {1-c^2 x^2}}{3 c^3 e}-\frac {b \left (1-c^2 x^2\right )^{3/2}}{9 c^3 e}-\frac {b d x \sin ^{-1}(c x)}{e^2}+\frac {x^3 \left (a+b \sin ^{-1}(c x)\right )}{3 e}+\frac {(-d)^{3/2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac {\sqrt {e} e^{i \sin ^{-1}(c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{2 e^{5/2}}-\frac {(-d)^{3/2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1+\frac {\sqrt {e} e^{i \sin ^{-1}(c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{2 e^{5/2}}+\frac {(-d)^{3/2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac {\sqrt {e} e^{i \sin ^{-1}(c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{2 e^{5/2}}-\frac {(-d)^{3/2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1+\frac {\sqrt {e} e^{i \sin ^{-1}(c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{2 e^{5/2}}+\frac {\left (i b (-d)^{3/2}\right ) \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt {e} x}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{2 e^{5/2}}-\frac {\left (i b (-d)^{3/2}\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt {e} x}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{2 e^{5/2}}+\frac {\left (i b (-d)^{3/2}\right ) \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt {e} x}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{2 e^{5/2}}-\frac {\left (i b (-d)^{3/2}\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt {e} x}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{2 e^{5/2}}\\ &=-\frac {a d x}{e^2}-\frac {b d \sqrt {1-c^2 x^2}}{c e^2}+\frac {b \sqrt {1-c^2 x^2}}{3 c^3 e}-\frac {b \left (1-c^2 x^2\right )^{3/2}}{9 c^3 e}-\frac {b d x \sin ^{-1}(c x)}{e^2}+\frac {x^3 \left (a+b \sin ^{-1}(c x)\right )}{3 e}+\frac {(-d)^{3/2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac {\sqrt {e} e^{i \sin ^{-1}(c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{2 e^{5/2}}-\frac {(-d)^{3/2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1+\frac {\sqrt {e} e^{i \sin ^{-1}(c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{2 e^{5/2}}+\frac {(-d)^{3/2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac {\sqrt {e} e^{i \sin ^{-1}(c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{2 e^{5/2}}-\frac {(-d)^{3/2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1+\frac {\sqrt {e} e^{i \sin ^{-1}(c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{2 e^{5/2}}+\frac {i b (-d)^{3/2} \text {Li}_2\left (-\frac {\sqrt {e} e^{i \sin ^{-1}(c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{2 e^{5/2}}-\frac {i b (-d)^{3/2} \text {Li}_2\left (\frac {\sqrt {e} e^{i \sin ^{-1}(c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{2 e^{5/2}}+\frac {i b (-d)^{3/2} \text {Li}_2\left (-\frac {\sqrt {e} e^{i \sin ^{-1}(c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{2 e^{5/2}}-\frac {i b (-d)^{3/2} \text {Li}_2\left (\frac {\sqrt {e} e^{i \sin ^{-1}(c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{2 e^{5/2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.97, size = 515, normalized size = 0.79 \[ \frac {a d^{3/2} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{e^{5/2}}-\frac {a d x}{e^2}+\frac {a x^3}{3 e}+\frac {b \left (d^{3/2} \left (-2 \text {Li}_2\left (\frac {\sqrt {e} e^{i \sin ^{-1}(c x)}}{\sqrt {d c^2+e}-c \sqrt {d}}\right )-2 \text {Li}_2\left (-\frac {\sqrt {e} e^{i \sin ^{-1}(c x)}}{\sqrt {d} c+\sqrt {d c^2+e}}\right )-\sin ^{-1}(c x) \left (\sin ^{-1}(c x)+2 i \left (\log \left (1+\frac {\sqrt {e} e^{i \sin ^{-1}(c x)}}{c \sqrt {d}-\sqrt {c^2 d+e}}\right )+\log \left (1+\frac {\sqrt {e} e^{i \sin ^{-1}(c x)}}{\sqrt {c^2 d+e}+c \sqrt {d}}\right )\right )\right )\right )+d^{3/2} \left (2 \text {Li}_2\left (\frac {\sqrt {e} e^{i \sin ^{-1}(c x)}}{c \sqrt {d}-\sqrt {d c^2+e}}\right )+2 \text {Li}_2\left (\frac {\sqrt {e} e^{i \sin ^{-1}(c x)}}{\sqrt {d} c+\sqrt {d c^2+e}}\right )+\sin ^{-1}(c x) \left (\sin ^{-1}(c x)+2 i \left (\log \left (1+\frac {\sqrt {e} e^{i \sin ^{-1}(c x)}}{\sqrt {c^2 d+e}-c \sqrt {d}}\right )+\log \left (1-\frac {\sqrt {e} e^{i \sin ^{-1}(c x)}}{\sqrt {c^2 d+e}+c \sqrt {d}}\right )\right )\right )\right )-\frac {4 d \sqrt {e} \left (\sqrt {1-c^2 x^2}+c x \sin ^{-1}(c x)\right )}{c}+\frac {4 e^{3/2} \left (3 c^3 x^3 \sin ^{-1}(c x)+\sqrt {1-c^2 x^2} \left (c^2 x^2+2\right )\right )}{9 c^3}\right )}{4 e^{5/2}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-((a*d*x)/e^2) + (a*x^3)/(3*e) + (a*d^(3/2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/e^(5/2) + (b*((-4*d*Sqrt[e]*(Sqrt[1 -
c^2*x^2] + c*x*ArcSin[c*x]))/c + (4*e^(3/2)*(Sqrt[1 - c^2*x^2]*(2 + c^2*x^2) + 3*c^3*x^3*ArcSin[c*x]))/(9*c^3
) + d^(3/2)*(-(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*Arc
Sin[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]))]) + d^(3/2)*(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] + S
qrt[c^2*d + e])])))/(4*e^(5/2))

________________________________________________________________________________________

fricas [F]  time = 0.53, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b x^{4} \arcsin \left (c x\right ) + a x^{4}}{e x^{2} + d}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Unab
le to divide, perhaps due to rounding error%%%{281474976710656,[0,8,0,0,0,256,8,72]%%%}+%%%{1125899906842624,[
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%%%{919123001344,[0,6,0,2,0,238,2,76]%%%}+%%%{8589934592,[0,6,0,2,0,236,1,77]%%%}+%%%{52776558133248,[0,4,4,0,
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%%%{2150644743929856,[0,4,4,0,4,250,11,65]%%%}+%%%{2625152730791936,[0,4,4,0,4,248,10,66]%%%}+%%%{221640928316
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}+%%%{8602578975719424,[0,4,3,1,3,250,11,65]%%%}+%%%{10500610923167744,[0,4,3,1,3,248,10,66]%%%}+%%%{886563713
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+%%%{19255479238656,[0,4,3,1,3,236,4,72]%%%}+%%%{1591419600896,[0,4,3,1,3,234,3,73]%%%}+%%%{66404220928,[0,4,3
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2462476640256,[0,2,5,1,5,252,15,59]%%%}+%%%{2057461133475840,[0,2,5,1,5,250,14,60]%%%}+%%%{3285340743794688,[0
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63]%%%}+%%%{2016200456404992,[0,2,5,1,5,242,10,64]%%%}+%%%{968709875171328,[0,2,5,1,5,240,9,65]%%%}+%%%{351898
146177024,[0,2,5,1,5,238,8,66]%%%}+%%%{95857982570496,[0,2,5,1,5,236,7,67]%%%}+%%%{19257115017216,[0,2,5,1,5,2
34,6,68]%%%}+%%%{2774390538240,[0,2,5,1,5,232,5,69]%%%}+%%%{273915838464,[0,2,5,1,5,230,4,70]%%%}+%%%{17176068
096,[0,2,5,1,5,228,3,71]%%%}+%%%{599851008,[0,2,5,1,5,226,2,72]%%%}+%%%{9928704,[0,2,5,1,5,224,1,73]%%%}+%%%{1
47456,[0,2,5,1,5,222,0,74]%%%}+%%%{65970697666560,[0,2,4,2,4,256,17,57]%%%}+%%%{560750930165760,[0,2,4,2,4,254
,16,58]%%%}+%%%{2181156191600640,[0,2,4,2,4,252,15,59]%%%}+%%%{5143652833689600,[0,2,4,2,4,250,14,60]%%%}+%%%{
8213351859486720,[0,2,4,2,4,248,13,61]%%%}+%%%{9393608872427520,[0,2,4,2,4,246,12,62]%%%}+%%%{7938210885795840
,[0,2,4,2,4,244,11,63]%%%}+%%%{5040501141012480,[0,2,4,2,4,242,10,64]%%%}+%%%{2421774687928320,[0,2,4,2,4,240,
9,65]%%%}+%%%{879745365442560,[0,2,4,2,4,238,8,66]%%%}+%%%{239644956426240,[0,2,4,2,4,236,7,67]%%%}+%%%{481427
87543040,[0,2,4,2,4,234,6,68]%%%}+%%%{6935976345600,[0,2,4,2,4,232,5,69]%%%}+%%%{684789596160,[0,2,4,2,4,230,4
,70]%%%}+%%%{42940170240,[0,2,4,2,4,228,3,71]%%%}+%%%{1499627520,[0,2,4,2,4,226,2,72]%%%}+%%%{24821760,[0,2,4,
2,4,224,1,73]%%%}+%%%{368640,[0,2,4,2,4,222,0,74]%%%}+%%%{87960930222080,[0,2,3,3,3,256,17,57]%%%}+%%%{7476679
06887680,[0,2,3,3,3,254,16,58]%%%}+%%%{2908208255467520,[0,2,3,3,3,252,15,59]%%%}+%%%{6858203778252800,[0,2,3,
3,3,250,14,60]%%%}+%%%{10951135812648960,[0,2,3,3,3,248,13,61]%%%}+%%%{12524811829903360,[0,2,3,3,3,246,12,62]
%%%}+%%%{10584281181061120,[0,2,3,3,3,244,11,63]%%%}+%%%{6720668188016640,[0,2,3,3,3,242,10,64]%%%}+%%%{322903
2917237760,[0,2,3,3,3,240,9,65]%%%}+%%%{1172993820590080,[0,2,3,3,3,238,8,66]%%%}+%%%{319526608568320,[0,2,3,3
,3,236,7,67]%%%}+%%%{64190383390720,[0,2,3,3,3,234,6,68]%%%}+%%%{9247968460800,[0,2,3,3,3,232,5,69]%%%}+%%%{91
3052794880,[0,2,3,3,3,230,4,70]%%%}+%%%{57253560320,[0,2,3,3,3,228,3,71]%%%}+%%%{1999503360,[0,2,3,3,3,226,2,7
2]%%%}+%%%{33095680,[0,2,3,3,3,224,1,73]%%%}+%%%{491520,[0,2,3,3,3,222,0,74]%%%}+%%%{65970697666560,[0,2,2,4,2
,256,17,57]%%%}+%%%{560750930165760,[0,2,2,4,2,254,16,58]%%%}+%%%{2181156191600640,[0,2,2,4,2,252,15,59]%%%}+%
%%{5143652833689600,[0,2,2,4,2,250,14,60]%%%}+%%%{8213351859486720,[0,2,2,4,2,248,13,61]%%%}+%%%{9393608872427
520,[0,2,2,4,2,246,12,62]%%%}+%%%{7938210885795840,[0,2,2,4,2,244,11,63]%%%}+%%%{5040501141012480,[0,2,2,4,2,2
42,10,64]%%%}+%%%{2421774687928320,[0,2,2,4,2,240,9,65]%%%}+%%%{879745365442560,[0,2,2,4,2,238,8,66]%%%}+%%%{2
39644956426240,[0,2,2,4,2,236,7,67]%%%}+%%%{48142787543040,[0,2,2,4,2,234,6,68]%%%}+%%%{6935976345600,[0,2,2,4
,2,232,5,69]%%%}+%%%{684789596160,[0,2,2,4,2,230,4,70]%%%}+%%%{42940170240,[0,2,2,4,2,228,3,71]%%%}+%%%{149962
7520,[0,2,2,4,2,226,2,72]%%%}+%%%{24821760,[0,2,2,4,2,224,1,73]%%%}+%%%{368640,[0,2,2,4,2,222,0,74]%%%}+%%%{26
388279066624,[0,2,1,5,1,256,17,57]%%%}+%%%{224300372066304,[0,2,1,5,1,254,16,58]%%%}+%%%{872462476640256,[0,2,
1,5,1,252,15,59]%%%}+%%%{2057461133475840,[0,2,1,5,1,250,14,60]%%%}+%%%{3285340743794688,[0,2,1,5,1,248,13,61]
%%%}+%%%{3757443548971008,[0,2,1,5,1,246,12,62]%%%}+%%%{3175284354318336,[0,2,1,5,1,244,11,63]%%%}+%%%{2016200
456404992,[0,2,1,5,1,242,10,64]%%%}+%%%{968709875171328,[0,2,1,5,1,240,9,65]%%%}+%%%{351898146177024,[0,2,1,5,
1,238,8,66]%%%}+%%%{95857982570496,[0,2,1,5,1,236,7,67]%%%}+%%%{19257115017216,[0,2,1,5,1,234,6,68]%%%}+%%%{27
74390538240,[0,2,1,5,1,232,5,69]%%%}+%%%{273915838464,[0,2,1,5,1,230,4,70]%%%}+%%%{17176068096,[0,2,1,5,1,228,
3,71]%%%}+%%%{599851008,[0,2,1,5,1,226,2,72]%%%}+%%%{9928704,[0,2,1,5,1,224,1,73]%%%}+%%%{147456,[0,2,1,5,1,22
2,0,74]%%%}+%%%{4398046511104,[0,2,0,6,0,256,17,57]%%%}+%%%{37383395344384,[0,2,0,6,0,254,16,58]%%%}+%%%{14541
0412773376,[0,2,0,6,0,252,15,59]%%%}+%%%{342910188912640,[0,2,0,6,0,250,14,60]%%%}+%%%{547556790632448,[0,2,0,
6,0,248,13,61]%%%}+%%%{626240591495168,[0,2,0,6,0,246,12,62]%%%}+%%%{529214059053056,[0,2,0,6,0,244,11,63]%%%}
+%%%{336033409400832,[0,2,0,6,0,242,10,64]%%%}+%%%{161451645861888,[0,2,0,6,0,240,9,65]%%%}+%%%{58649691029504
,[0,2,0,6,0,238,8,66]%%%}+%%%{15976330428416,[0,2,0,6,0,236,7,67]%%%}+%%%{3209519169536,[0,2,0,6,0,234,6,68]%%
%}+%%%{462398423040,[0,2,0,6,0,232,5,69]%%%}+%%%{45652639744,[0,2,0,6,0,230,4,70]%%%}+%%%{2862678016,[0,2,0,6,
0,228,3,71]%%%}+%%%{99975168,[0,2,0,6,0,226,2,72]%%%}+%%%{1654784,[0,2,0,6,0,224,1,73]%%%}+%%%{24576,[0,2,0,6,
0,222,0,74]%%%}+%%%{1048576,[0,0,8,0,8,236,10,62]%%%}+%%%{5242880,[0,0,8,0,8,234,9,63]%%%}+%%%{11403264,[0,0,8
,0,8,232,8,64]%%%}+%%%{14155776,[0,0,8,0,8,230,7,65]%%%}+%%%{11063296,[0,0,8,0,8,228,6,66]%%%}+%%%{5664768,[0,
0,8,0,8,226,5,67]%%%}+%%%{1916928,[0,0,8,0,8,224,4,68]%%%}+%%%{421888,[0,0,8,0,8,222,3,69]%%%}+%%%{57664,[0,0,
8,0,8,220,2,70]%%%}+%%%{4416,[0,0,8,0,8,218,1,71]%%%}+%%%{144,[0,0,8,0,8,216,0,72]%%%}+%%%{8388608,[0,0,7,1,7,
236,10,62]%%%}+%%%{41943040,[0,0,7,1,7,234,9,63]%%%}+%%%{91226112,[0,0,7,1,7,232,8,64]%%%}+%%%{113246208,[0,0,
7,1,7,230,7,65]%%%}+%%%{88506368,[0,0,7,1,7,228,6,66]%%%}+%%%{45318144,[0,0,7,1,7,226,5,67]%%%}+%%%{15335424,[
0,0,7,1,7,224,4,68]%%%}+%%%{3375104,[0,0,7,1,7,222,3,69]%%%}+%%%{461312,[0,0,7,1,7,220,2,70]%%%}+%%%{35328,[0,
0,7,1,7,218,1,71]%%%}+%%%{1152,[0,0,7,1,7,216,0,72]%%%}+%%%{29360128,[0,0,6,2,6,236,10,62]%%%}+%%%{146800640,[
0,0,6,2,6,234,9,63]%%%}+%%%{319291392,[0,0,6,2,6,232,8,64]%%%}+%%%{396361728,[0,0,6,2,6,230,7,65]%%%}+%%%{3097
72288,[0,0,6,2,6,228,6,66]%%%}+%%%{158613504,[0,0,6,2,6,226,5,67]%%%}+%%%{53673984,[0,0,6,2,6,224,4,68]%%%}+%%
%{11812864,[0,0,6,2,6,222,3,69]%%%}+%%%{1614592,[0,0,6,2,6,220,2,70]%%%}+%%%{123648,[0,0,6,2,6,218,1,71]%%%}+%
%%{4032,[0,0,6,2,6,216,0,72]%%%}+%%%{58720256,[0,0,5,3,5,236,10,62]%%%}+%%%{293601280,[0,0,5,3,5,234,9,63]%%%}
+%%%{638582784,[0,0,5,3,5,232,8,64]%%%}+%%%{792723456,[0,0,5,3,5,230,7,65]%%%}+%%%{619544576,[0,0,5,3,5,228,6,
66]%%%}+%%%{317227008,[0,0,5,3,5,226,5,67]%%%}+%%%{107347968,[0,0,5,3,5,224,4,68]%%%}+%%%{23625728,[0,0,5,3,5,
222,3,69]%%%}+%%%{3229184,[0,0,5,3,5,220,2,70]%%%}+%%%{247296,[0,0,5,3,5,218,1,71]%%%}+%%%{8064,[0,0,5,3,5,216
,0,72]%%%}+%%%{73400320,[0,0,4,4,4,236,10,62]%%%}+%%%{367001600,[0,0,4,4,4,234,9,63]%%%}+%%%{798228480,[0,0,4,
4,4,232,8,64]%%%}+%%%{990904320,[0,0,4,4,4,230,7,65]%%%}+%%%{774430720,[0,0,4,4,4,228,6,66]%%%}+%%%{396533760,
[0,0,4,4,4,226,5,67]%%%}+%%%{134184960,[0,0,4,4,4,224,4,68]%%%}+%%%{29532160,[0,0,4,4,4,222,3,69]%%%}+%%%{4036
480,[0,0,4,4,4,220,2,70]%%%}+%%%{309120,[0,0,4,4,4,218,1,71]%%%}+%%%{10080,[0,0,4,4,4,216,0,72]%%%}+%%%{587202
56,[0,0,3,5,3,236,10,62]%%%}+%%%{293601280,[0,0,3,5,3,234,9,63]%%%}+%%%{638582784,[0,0,3,5,3,232,8,64]%%%}+%%%
{792723456,[0,0,3,5,3,230,7,65]%%%}+%%%{619544576,[0,0,3,5,3,228,6,66]%%%}+%%%{317227008,[0,0,3,5,3,226,5,67]%
%%}+%%%{107347968,[0,0,3,5,3,224,4,68]%%%}+%%%{23625728,[0,0,3,5,3,222,3,69]%%%}+%%%{3229184,[0,0,3,5,3,220,2,
70]%%%}+%%%{247296,[0,0,3,5,3,218,1,71]%%%}+%%%{8064,[0,0,3,5,3,216,0,72]%%%}+%%%{29360128,[0,0,2,6,2,236,10,6
2]%%%}+%%%{146800640,[0,0,2,6,2,234,9,63]%%%}+%%%{319291392,[0,0,2,6,2,232,8,64]%%%}+%%%{396361728,[0,0,2,6,2,
230,7,65]%%%}+%%%{309772288,[0,0,2,6,2,228,6,66]%%%}+%%%{158613504,[0,0,2,6,2,226,5,67]%%%}+%%%{53673984,[0,0,
2,6,2,224,4,68]%%%}+%%%{11812864,[0,0,2,6,2,222,3,69]%%%}+%%%{1614592,[0,0,2,6,2,220,2,70]%%%}+%%%{123648,[0,0
,2,6,2,218,1,71]%%%}+%%%{4032,[0,0,2,6,2,216,0,72]%%%}+%%%{8388608,[0,0,1,7,1,236,10,62]%%%}+%%%{41943040,[0,0
,1,7,1,234,9,63]%%%}+%%%{91226112,[0,0,1,7,1,232,8,64]%%%}+%%%{113246208,[0,0,1,7,1,230,7,65]%%%}+%%%{88506368
,[0,0,1,7,1,228,6,66]%%%}+%%%{45318144,[0,0,1,7,1,226,5,67]%%%}+%%%{15335424,[0,0,1,7,1,224,4,68]%%%}+%%%{3375
104,[0,0,1,7,1,222,3,69]%%%}+%%%{461312,[0,0,1,7,1,220,2,70]%%%}+%%%{35328,[0,0,1,7,1,218,1,71]%%%}+%%%{1152,[
0,0,1,7,1,216,0,72]%%%}+%%%{1048576,[0,0,0,8,0,236,10,62]%%%}+%%%{5242880,[0,0,0,8,0,234,9,63]%%%}+%%%{1140326
4,[0,0,0,8,0,232,8,64]%%%}+%%%{14155776,[0,0,0,8,0,230,7,65]%%%}+%%%{11063296,[0,0,0,8,0,228,6,66]%%%}+%%%{566
4768,[0,0,0,8,0,226,5,67]%%%}+%%%{1916928,[0,0,0,8,0,224,4,68]%%%}+%%%{421888,[0,0,0,8,0,222,3,69]%%%}+%%%{576
64,[0,0,0,8,0,220,2,70]%%%}+%%%{4416,[0,0,0,8,0,218,1,71]%%%}+%%%{144,[0,0,0,8,0,216,0,72]%%%} / %%%{-4096,[0,
2,0,0,0,64,2,18]%%%}+%%%{-4096,[0,2,0,0,0,62,1,19]%%%}+%%%{-1024,[0,2,0,0,0,60,0,20]%%%}+%%%{-1024,[0,0,2,0,2,
64,5,13]%%%}+%%%{-2560,[0,0,2,0,2,62,4,14]%%%}+%%%{-2368,[0,0,2,0,2,60,3,15]%%%}+%%%{-992,[0,0,2,0,2,58,2,16]%
%%}+%%%{-184,[0,0,2,0,2,56,1,17]%%%}+%%%{-12,[0,0,2,0,2,54,0,18]%%%}+%%%{-2048,[0,0,1,1,1,64,5,13]%%%}+%%%{-51
20,[0,0,1,1,1,62,4,14]%%%}+%%%{-4736,[0,0,1,1,1,60,3,15]%%%}+%%%{-1984,[0,0,1,1,1,58,2,16]%%%}+%%%{-368,[0,0,1
,1,1,56,1,17]%%%}+%%%{-24,[0,0,1,1,1,54,0,18]%%%}+%%%{-1024,[0,0,0,2,0,64,5,13]%%%}+%%%{-2560,[0,0,0,2,0,62,4,
14]%%%}+%%%{-2368,[0,0,0,2,0,60,3,15]%%%}+%%%{-992,[0,0,0,2,0,58,2,16]%%%}+%%%{-184,[0,0,0,2,0,56,1,17]%%%}+%%
%{-12,[0,0,0,2,0,54,0,18]%%%} Error: Bad Argument Value

________________________________________________________________________________________

maple [C]  time = 3.85, size = 376, normalized size = 0.58 \[ \frac {a \,x^{3}}{3 e}-\frac {a d x}{e^{2}}+\frac {a \,d^{2} \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{e^{2} \sqrt {d e}}-\frac {b d \sqrt {-c^{2} x^{2}+1}}{c \,e^{2}}-\frac {b d x \arcsin \left (c x \right )}{e^{2}}+\frac {b \sqrt {-c^{2} x^{2}+1}}{4 c^{3} e}+\frac {b \arcsin \left (c x \right ) x}{4 c^{2} e}+\frac {c b \,d^{2} \left (\munderset {\textit {\_R1} =\RootOf \left (e \,\textit {\_Z}^{4}+\left (-4 c^{2} d -2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {i \arcsin \left (c x \right ) \ln \left (\frac {\textit {\_R1} -i c x -\sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )+\dilog \left (\frac {\textit {\_R1} -i c x -\sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )}{\textit {\_R1} \left (\textit {\_R1}^{2} e -2 c^{2} d -e \right )}\right )}{2 e^{2}}+\frac {c b \,d^{2} \left (\munderset {\textit {\_R1} =\RootOf \left (e \,\textit {\_Z}^{4}+\left (-4 c^{2} d -2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\textit {\_R1} \left (i \arcsin \left (c x \right ) \ln \left (\frac {\textit {\_R1} -i c x -\sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )+\dilog \left (\frac {\textit {\_R1} -i c x -\sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )\right )}{\textit {\_R1}^{2} e -2 c^{2} d -e}\right )}{2 e^{2}}-\frac {b \cos \left (3 \arcsin \left (c x \right )\right )}{36 c^{3} e}-\frac {b \arcsin \left (c x \right ) \sin \left (3 \arcsin \left (c x \right )\right )}{12 c^{3} e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*a/e*x^3-a*d*x/e^2+a*d^2/e^2/(d*e)^(1/2)*arctan(e*x/(d*e)^(1/2))-b*d*(-c^2*x^2+1)^(1/2)/c/e^2-b*d*x*arcsin(
c*x)/e^2+1/4*b*(-c^2*x^2+1)^(1/2)/c^3/e+1/4/c^2*b/e*arcsin(c*x)*x+1/2*c*b*d^2/e^2*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*c*b*d^2/e^2*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))-1/36/c
^3*b/e*cos(3*arcsin(c*x))-1/12/c^3*b*arcsin(c*x)/e*sin(3*arcsin(c*x))

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{4} \left (a + b \operatorname {asin}{\left (c x \right )}\right )}{d + e x^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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