\(\int \frac {a+b \arcsin (c x)}{x^3 (d+e x^2)^3} \, dx\) [645]

Optimal result

Integrand size = 21, antiderivative size = 783 \[ \int \frac {a+b \arcsin (c x)}{x^3 \left (d+e x^2\right )^3} \, dx=-\frac {b c \sqrt {1-c^2 x^2}}{2 d^3 x}+\frac {b c e^2 x \sqrt {1-c^2 x^2}}{8 d^3 \left (c^2 d+e\right ) \left (d+e x^2\right )}-\frac {a+b \arcsin (c x)}{2 d^3 x^2}-\frac {e (a+b \arcsin (c x))}{4 d^2 \left (d+e x^2\right )^2}-\frac {e (a+b \arcsin (c x))}{d^3 \left (d+e x^2\right )}+\frac {b c e \arctan \left (\frac {\sqrt {c^2 d+e} x}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{d^{7/2} \sqrt {c^2 d+e}}+\frac {b c e \left (2 c^2 d+e\right ) \arctan \left (\frac {\sqrt {c^2 d+e} x}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{8 d^{7/2} \left (c^2 d+e\right )^{3/2}}+\frac {3 e (a+b \arcsin (c x)) \log \left (1-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{2 d^4}+\frac {3 e (a+b \arcsin (c x)) \log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{2 d^4}+\frac {3 e (a+b \arcsin (c x)) \log \left (1-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{2 d^4}+\frac {3 e (a+b \arcsin (c x)) \log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{2 d^4}-\frac {3 e (a+b \arcsin (c x)) \log \left (1-e^{2 i \arcsin (c x)}\right )}{d^4}-\frac {3 i b e \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{2 d^4}-\frac {3 i b e \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{2 d^4}-\frac {3 i b e \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{2 d^4}-\frac {3 i b e \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{2 d^4}+\frac {3 i b e \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )}{2 d^4} \]

[Out]

1/2*(-a-b*arcsin(c*x))/d^3/x^2-1/4*e*(a+b*arcsin(c*x))/d^2/(e*x^2+d)^2-e*(a+b*arcsin(c*x))/d^3/(e*x^2+d)+1/8*b
*c*e*(2*c^2*d+e)*arctan(x*(c^2*d+e)^(1/2)/d^(1/2)/(-c^2*x^2+1)^(1/2))/d^(7/2)/(c^2*d+e)^(3/2)-3*e*(a+b*arcsin(
c*x))*ln(1-(I*c*x+(-c^2*x^2+1)^(1/2))^2)/d^4+3/2*e*(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^4+3/2*e*(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^4+3/2*e*(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^4+3/2*e*(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^4+3/2*I*b*e*polylog(2,(I*c*x+(-c^2*x^2+1)^(1/2))^2)/d^4-3/2*I*b*e*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^4-3/2*I*b*e*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^4-3/2*I*b*e*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^4-3/2*I*b*e*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^4
+b*c*e*arctan(x*(c^2*d+e)^(1/2)/d^(1/2)/(-c^2*x^2+1)^(1/2))/d^(7/2)/(c^2*d+e)^(1/2)-1/2*b*c*(-c^2*x^2+1)^(1/2)
/d^3/x+1/8*b*c*e^2*x*(-c^2*x^2+1)^(1/2)/d^3/(c^2*d+e)/(e*x^2+d)

Rubi [A] (verified)

Time = 0.81 (sec) , antiderivative size = 783, normalized size of antiderivative = 1.00, number of steps used = 34, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {4817, 4723, 270, 4721, 3798, 2221, 2317, 2438, 4813, 390, 385, 211, 4825, 4617} \[ \int \frac {a+b \arcsin (c x)}{x^3 \left (d+e x^2\right )^3} \, dx=\frac {3 e (a+b \arcsin (c x)) \log \left (1-\frac {\sqrt {e} e^{i \arcsin (c x)}}{-\sqrt {c^2 d+e}+i c \sqrt {-d}}\right )}{2 d^4}+\frac {3 e (a+b \arcsin (c x)) \log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{-\sqrt {c^2 d+e}+i c \sqrt {-d}}\right )}{2 d^4}+\frac {3 e (a+b \arcsin (c x)) \log \left (1-\frac {\sqrt {e} e^{i \arcsin (c x)}}{\sqrt {c^2 d+e}+i c \sqrt {-d}}\right )}{2 d^4}+\frac {3 e (a+b \arcsin (c x)) \log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{\sqrt {c^2 d+e}+i c \sqrt {-d}}\right )}{2 d^4}-\frac {3 e \log \left (1-e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))}{d^4}-\frac {e (a+b \arcsin (c x))}{d^3 \left (d+e x^2\right )}-\frac {a+b \arcsin (c x)}{2 d^3 x^2}-\frac {e (a+b \arcsin (c x))}{4 d^2 \left (d+e x^2\right )^2}-\frac {3 i b e \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {d c^2+e}}\right )}{2 d^4}-\frac {3 i b e \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {d c^2+e}}\right )}{2 d^4}-\frac {3 i b e \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i \sqrt {-d} c+\sqrt {d c^2+e}}\right )}{2 d^4}-\frac {3 i b e \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{i \sqrt {-d} c+\sqrt {d c^2+e}}\right )}{2 d^4}+\frac {3 i b e \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )}{2 d^4}+\frac {b c e \left (2 c^2 d+e\right ) \arctan \left (\frac {x \sqrt {c^2 d+e}}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{8 d^{7/2} \left (c^2 d+e\right )^{3/2}}+\frac {b c e \arctan \left (\frac {x \sqrt {c^2 d+e}}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{d^{7/2} \sqrt {c^2 d+e}}+\frac {b c e^2 x \sqrt {1-c^2 x^2}}{8 d^3 \left (c^2 d+e\right ) \left (d+e x^2\right )}-\frac {b c \sqrt {1-c^2 x^2}}{2 d^3 x} \]

[In]

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

[Out]

-1/2*(b*c*Sqrt[1 - c^2*x^2])/(d^3*x) + (b*c*e^2*x*Sqrt[1 - c^2*x^2])/(8*d^3*(c^2*d + e)*(d + e*x^2)) - (a + b*
ArcSin[c*x])/(2*d^3*x^2) - (e*(a + b*ArcSin[c*x]))/(4*d^2*(d + e*x^2)^2) - (e*(a + b*ArcSin[c*x]))/(d^3*(d + e
*x^2)) + (b*c*e*ArcTan[(Sqrt[c^2*d + e]*x)/(Sqrt[d]*Sqrt[1 - c^2*x^2])])/(d^(7/2)*Sqrt[c^2*d + e]) + (b*c*e*(2
*c^2*d + e)*ArcTan[(Sqrt[c^2*d + e]*x)/(Sqrt[d]*Sqrt[1 - c^2*x^2])])/(8*d^(7/2)*(c^2*d + e)^(3/2)) + (3*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*d^4) + (3*e*(a + b*A
rcSin[c*x])*Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/(2*d^4) + (3*e*(a + b*ArcSi
n[c*x])*Log[1 - (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])])/(2*d^4) + (3*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*d^4) - (3*e*(a + b*ArcSin[c*x])*
Log[1 - E^((2*I)*ArcSin[c*x])])/d^4 - (((3*I)/2)*b*e*PolyLog[2, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] -
Sqrt[c^2*d + e]))])/d^4 - (((3*I)/2)*b*e*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e
])])/d^4 - (((3*I)/2)*b*e*PolyLog[2, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e]))])/d^4 - (
((3*I)/2)*b*e*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])])/d^4 + (((3*I)/2)*b*e*P
olyLog[2, E^((2*I)*ArcSin[c*x])])/d^4

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 270

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

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 390

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(-b)*x*(a + b*x^n)^(p + 1)*
((c + d*x^n)^(q + 1)/(a*n*(p + 1)*(b*c - a*d))), x] + Dist[(b*c + n*(p + 1)*(b*c - a*d))/(a*n*(p + 1)*(b*c - a
*d)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, n, q}, x] && NeQ[b*c - a*d, 0] && Eq
Q[n*(p + q + 2) + 1, 0] && (LtQ[p, -1] ||  !LtQ[q, -1]) && NeQ[p, -1]

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 3798

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (f_.)*(x_)], x_Symbol] :> Simp[I*((c + d*x)^(m + 1)/(d*(
m + 1))), x] - Dist[2*I, Int[(c + d*x)^m*E^(2*I*k*Pi)*(E^(2*I*(e + f*x))/(1 + E^(2*I*k*Pi)*E^(2*I*(e + f*x))))
, x], x] /; FreeQ[{c, d, e, f}, x] && IntegerQ[4*k] && IGtQ[m, 0]

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 4721

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

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 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 {a+b \arcsin (c x)}{d^3 x^3}-\frac {3 e (a+b \arcsin (c x))}{d^4 x}+\frac {e^2 x (a+b \arcsin (c x))}{d^2 \left (d+e x^2\right )^3}+\frac {2 e^2 x (a+b \arcsin (c x))}{d^3 \left (d+e x^2\right )^2}+\frac {3 e^2 x (a+b \arcsin (c x))}{d^4 \left (d+e x^2\right )}\right ) \, dx \\ & = \frac {\int \frac {a+b \arcsin (c x)}{x^3} \, dx}{d^3}-\frac {(3 e) \int \frac {a+b \arcsin (c x)}{x} \, dx}{d^4}+\frac {\left (3 e^2\right ) \int \frac {x (a+b \arcsin (c x))}{d+e x^2} \, dx}{d^4}+\frac {\left (2 e^2\right ) \int \frac {x (a+b \arcsin (c x))}{\left (d+e x^2\right )^2} \, dx}{d^3}+\frac {e^2 \int \frac {x (a+b \arcsin (c x))}{\left (d+e x^2\right )^3} \, dx}{d^2} \\ & = -\frac {a+b \arcsin (c x)}{2 d^3 x^2}-\frac {e (a+b \arcsin (c x))}{4 d^2 \left (d+e x^2\right )^2}-\frac {e (a+b \arcsin (c x))}{d^3 \left (d+e x^2\right )}+\frac {(b c) \int \frac {1}{x^2 \sqrt {1-c^2 x^2}} \, dx}{2 d^3}-\frac {(3 e) \text {Subst}(\int (a+b x) \cot (x) \, dx,x,\arcsin (c x))}{d^4}+\frac {(b c e) \int \frac {1}{\sqrt {1-c^2 x^2} \left (d+e x^2\right )} \, dx}{d^3}+\frac {(b c e) \int \frac {1}{\sqrt {1-c^2 x^2} \left (d+e x^2\right )^2} \, dx}{4 d^2}+\frac {\left (3 e^2\right ) \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}{d^4} \\ & = -\frac {b c \sqrt {1-c^2 x^2}}{2 d^3 x}+\frac {b c e^2 x \sqrt {1-c^2 x^2}}{8 d^3 \left (c^2 d+e\right ) \left (d+e x^2\right )}-\frac {a+b \arcsin (c x)}{2 d^3 x^2}-\frac {e (a+b \arcsin (c x))}{4 d^2 \left (d+e x^2\right )^2}-\frac {e (a+b \arcsin (c x))}{d^3 \left (d+e x^2\right )}+\frac {3 i e (a+b \arcsin (c x))^2}{2 b d^4}+\frac {(6 i e) \text {Subst}\left (\int \frac {e^{2 i x} (a+b x)}{1-e^{2 i x}} \, dx,x,\arcsin (c x)\right )}{d^4}+\frac {(b c e) \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 )}{d^3}-\frac {\left (3 e^{3/2}\right ) \int \frac {a+b \arcsin (c x)}{\sqrt {-d}-\sqrt {e} x} \, dx}{2 d^4}+\frac {\left (3 e^{3/2}\right ) \int \frac {a+b \arcsin (c x)}{\sqrt {-d}+\sqrt {e} x} \, dx}{2 d^4}+\frac {\left (b c e \left (2 c^2 d+e\right )\right ) \int \frac {1}{\sqrt {1-c^2 x^2} \left (d+e x^2\right )} \, dx}{8 d^3 \left (c^2 d+e\right )} \\ & = -\frac {b c \sqrt {1-c^2 x^2}}{2 d^3 x}+\frac {b c e^2 x \sqrt {1-c^2 x^2}}{8 d^3 \left (c^2 d+e\right ) \left (d+e x^2\right )}-\frac {a+b \arcsin (c x)}{2 d^3 x^2}-\frac {e (a+b \arcsin (c x))}{4 d^2 \left (d+e x^2\right )^2}-\frac {e (a+b \arcsin (c x))}{d^3 \left (d+e x^2\right )}+\frac {3 i e (a+b \arcsin (c x))^2}{2 b d^4}+\frac {b c e \arctan \left (\frac {\sqrt {c^2 d+e} x}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{d^{7/2} \sqrt {c^2 d+e}}-\frac {3 e (a+b \arcsin (c x)) \log \left (1-e^{2 i \arcsin (c x)}\right )}{d^4}+\frac {(3 b e) \text {Subst}\left (\int \log \left (1-e^{2 i x}\right ) \, dx,x,\arcsin (c x)\right )}{d^4}-\frac {\left (3 e^{3/2}\right ) \text {Subst}\left (\int \frac {(a+b x) \cos (x)}{c \sqrt {-d}-\sqrt {e} \sin (x)} \, dx,x,\arcsin (c x)\right )}{2 d^4}+\frac {\left (3 e^{3/2}\right ) \text {Subst}\left (\int \frac {(a+b x) \cos (x)}{c \sqrt {-d}+\sqrt {e} \sin (x)} \, dx,x,\arcsin (c x)\right )}{2 d^4}+\frac {\left (b c e \left (2 c^2 d+e\right )\right ) \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 )}{8 d^3 \left (c^2 d+e\right )} \\ & = -\frac {b c \sqrt {1-c^2 x^2}}{2 d^3 x}+\frac {b c e^2 x \sqrt {1-c^2 x^2}}{8 d^3 \left (c^2 d+e\right ) \left (d+e x^2\right )}-\frac {a+b \arcsin (c x)}{2 d^3 x^2}-\frac {e (a+b \arcsin (c x))}{4 d^2 \left (d+e x^2\right )^2}-\frac {e (a+b \arcsin (c x))}{d^3 \left (d+e x^2\right )}+\frac {b c e \arctan \left (\frac {\sqrt {c^2 d+e} x}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{d^{7/2} \sqrt {c^2 d+e}}+\frac {b c e \left (2 c^2 d+e\right ) \arctan \left (\frac {\sqrt {c^2 d+e} x}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{8 d^{7/2} \left (c^2 d+e\right )^{3/2}}-\frac {3 e (a+b \arcsin (c x)) \log \left (1-e^{2 i \arcsin (c x)}\right )}{d^4}-\frac {(3 i b e) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 i \arcsin (c x)}\right )}{2 d^4}-\frac {\left (3 i e^{3/2}\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 )}{2 d^4}-\frac {\left (3 i e^{3/2}\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 )}{2 d^4}+\frac {\left (3 i e^{3/2}\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 )}{2 d^4}+\frac {\left (3 i e^{3/2}\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 )}{2 d^4} \\ & = -\frac {b c \sqrt {1-c^2 x^2}}{2 d^3 x}+\frac {b c e^2 x \sqrt {1-c^2 x^2}}{8 d^3 \left (c^2 d+e\right ) \left (d+e x^2\right )}-\frac {a+b \arcsin (c x)}{2 d^3 x^2}-\frac {e (a+b \arcsin (c x))}{4 d^2 \left (d+e x^2\right )^2}-\frac {e (a+b \arcsin (c x))}{d^3 \left (d+e x^2\right )}+\frac {b c e \arctan \left (\frac {\sqrt {c^2 d+e} x}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{d^{7/2} \sqrt {c^2 d+e}}+\frac {b c e \left (2 c^2 d+e\right ) \arctan \left (\frac {\sqrt {c^2 d+e} x}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{8 d^{7/2} \left (c^2 d+e\right )^{3/2}}+\frac {3 e (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 d^4}+\frac {3 e (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 d^4}+\frac {3 e (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 d^4}+\frac {3 e (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 d^4}-\frac {3 e (a+b \arcsin (c x)) \log \left (1-e^{2 i \arcsin (c x)}\right )}{d^4}+\frac {3 i b e \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )}{2 d^4}-\frac {(3 b e) \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 d^4}-\frac {(3 b e) \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 d^4}-\frac {(3 b e) \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 d^4}-\frac {(3 b e) \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 d^4} \\ & = -\frac {b c \sqrt {1-c^2 x^2}}{2 d^3 x}+\frac {b c e^2 x \sqrt {1-c^2 x^2}}{8 d^3 \left (c^2 d+e\right ) \left (d+e x^2\right )}-\frac {a+b \arcsin (c x)}{2 d^3 x^2}-\frac {e (a+b \arcsin (c x))}{4 d^2 \left (d+e x^2\right )^2}-\frac {e (a+b \arcsin (c x))}{d^3 \left (d+e x^2\right )}+\frac {b c e \arctan \left (\frac {\sqrt {c^2 d+e} x}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{d^{7/2} \sqrt {c^2 d+e}}+\frac {b c e \left (2 c^2 d+e\right ) \arctan \left (\frac {\sqrt {c^2 d+e} x}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{8 d^{7/2} \left (c^2 d+e\right )^{3/2}}+\frac {3 e (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 d^4}+\frac {3 e (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 d^4}+\frac {3 e (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 d^4}+\frac {3 e (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 d^4}-\frac {3 e (a+b \arcsin (c x)) \log \left (1-e^{2 i \arcsin (c x)}\right )}{d^4}+\frac {3 i b e \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )}{2 d^4}+\frac {(3 i b e) \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 d^4}+\frac {(3 i b e) \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 d^4}+\frac {(3 i b e) \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 d^4}+\frac {(3 i b e) \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 d^4} \\ & = -\frac {b c \sqrt {1-c^2 x^2}}{2 d^3 x}+\frac {b c e^2 x \sqrt {1-c^2 x^2}}{8 d^3 \left (c^2 d+e\right ) \left (d+e x^2\right )}-\frac {a+b \arcsin (c x)}{2 d^3 x^2}-\frac {e (a+b \arcsin (c x))}{4 d^2 \left (d+e x^2\right )^2}-\frac {e (a+b \arcsin (c x))}{d^3 \left (d+e x^2\right )}+\frac {b c e \arctan \left (\frac {\sqrt {c^2 d+e} x}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{d^{7/2} \sqrt {c^2 d+e}}+\frac {b c e \left (2 c^2 d+e\right ) \arctan \left (\frac {\sqrt {c^2 d+e} x}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{8 d^{7/2} \left (c^2 d+e\right )^{3/2}}+\frac {3 e (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 d^4}+\frac {3 e (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 d^4}+\frac {3 e (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 d^4}+\frac {3 e (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 d^4}-\frac {3 e (a+b \arcsin (c x)) \log \left (1-e^{2 i \arcsin (c x)}\right )}{d^4}-\frac {3 i b e \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{2 d^4}-\frac {3 i b e \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{2 d^4}-\frac {3 i b e \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{2 d^4}-\frac {3 i b e \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{2 d^4}+\frac {3 i b e \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )}{2 d^4} \\ \end{align*}

Mathematica [A] (verified)

Time = 5.65 (sec) , antiderivative size = 1065, normalized size of antiderivative = 1.36 \[ \int \frac {a+b \arcsin (c x)}{x^3 \left (d+e x^2\right )^3} \, dx=\frac {-\frac {8 a d}{x^2}-\frac {4 a d^2 e}{\left (d+e x^2\right )^2}-\frac {16 a d e}{d+e x^2}-48 a e \log (x)+24 a e \log \left (d+e x^2\right )+b \left (-\frac {8 c d \sqrt {1-c^2 x^2}}{x}+\frac {c d e^{3/2} \sqrt {1-c^2 x^2}}{\left (c^2 d+e\right ) \left (-i \sqrt {d}+\sqrt {e} x\right )}+\frac {c d e^{3/2} \sqrt {1-c^2 x^2}}{\left (c^2 d+e\right ) \left (i \sqrt {d}+\sqrt {e} x\right )}-\frac {8 d \arcsin (c x)}{x^2}-\frac {9 \sqrt {d} e \arcsin (c x)}{\sqrt {d}-i \sqrt {e} x}-\frac {d e \arcsin (c x)}{\left (\sqrt {d}+i \sqrt {e} x\right )^2}-\frac {9 \sqrt {d} e \arcsin (c x)}{\sqrt {d}+i \sqrt {e} x}+\frac {d e \arcsin (c x)}{\left (i \sqrt {d}+\sqrt {e} x\right )^2}+\frac {9 c \sqrt {d} e \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}}-\frac {9 i c \sqrt {d} e \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}}+24 e \arcsin (c x) \log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}-\sqrt {c^2 d+e}}\right )+24 e \arcsin (c x) \log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{-c \sqrt {d}+\sqrt {c^2 d+e}}\right )+24 e \arcsin (c x) \log \left (1-\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}+\sqrt {c^2 d+e}}\right )+24 e \arcsin (c x) \log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}+\sqrt {c^2 d+e}}\right )-48 e \arcsin (c x) \log \left (1-e^{2 i \arcsin (c x)}\right )+\frac {i c^3 d^{3/2} e \log \left (\frac {e \sqrt {c^2 d+e} \left (\sqrt {e}-i c^2 \sqrt {d} x+\sqrt {c^2 d+e} \sqrt {1-c^2 x^2}\right )}{c^3 \left (d+i \sqrt {d} \sqrt {e} x\right )}\right )}{\left (c^2 d+e\right )^{3/2}}-\frac {i c^3 d^{3/2} e \log \left (\frac {e \sqrt {c^2 d+e} \left (\sqrt {e}+i c^2 \sqrt {d} x+\sqrt {c^2 d+e} \sqrt {1-c^2 x^2}\right )}{c^3 \left (d-i \sqrt {d} \sqrt {e} x\right )}\right )}{\left (c^2 d+e\right )^{3/2}}-24 i e \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}-\sqrt {c^2 d+e}}\right )-24 i e \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{-c \sqrt {d}+\sqrt {c^2 d+e}}\right )-24 i e \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}+\sqrt {c^2 d+e}}\right )-24 i e \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}+\sqrt {c^2 d+e}}\right )+24 i e \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )\right )}{16 d^4} \]

[In]

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

[Out]

((-8*a*d)/x^2 - (4*a*d^2*e)/(d + e*x^2)^2 - (16*a*d*e)/(d + e*x^2) - 48*a*e*Log[x] + 24*a*e*Log[d + e*x^2] + b
*((-8*c*d*Sqrt[1 - c^2*x^2])/x + (c*d*e^(3/2)*Sqrt[1 - c^2*x^2])/((c^2*d + e)*((-I)*Sqrt[d] + Sqrt[e]*x)) + (c
*d*e^(3/2)*Sqrt[1 - c^2*x^2])/((c^2*d + e)*(I*Sqrt[d] + Sqrt[e]*x)) - (8*d*ArcSin[c*x])/x^2 - (9*Sqrt[d]*e*Arc
Sin[c*x])/(Sqrt[d] - I*Sqrt[e]*x) - (d*e*ArcSin[c*x])/(Sqrt[d] + I*Sqrt[e]*x)^2 - (9*Sqrt[d]*e*ArcSin[c*x])/(S
qrt[d] + I*Sqrt[e]*x) + (d*e*ArcSin[c*x])/(I*Sqrt[d] + Sqrt[e]*x)^2 + (9*c*Sqrt[d]*e*ArcTan[(I*Sqrt[e] + c^2*S
qrt[d]*x)/(Sqrt[c^2*d + e]*Sqrt[1 - c^2*x^2])])/Sqrt[c^2*d + e] - ((9*I)*c*Sqrt[d]*e*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] + 24*e*ArcSin[c*x]*Log[1 + (Sqrt[e]*E^(I*ArcS
in[c*x]))/(c*Sqrt[d] - Sqrt[c^2*d + e])] + 24*e*ArcSin[c*x]*Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(-(c*Sqrt[d])
+ Sqrt[c^2*d + e])] + 24*e*ArcSin[c*x]*Log[1 - (Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] + Sqrt[c^2*d + e])] + 24
*e*ArcSin[c*x]*Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] + Sqrt[c^2*d + e])] - 48*e*ArcSin[c*x]*Log[1 - E
^((2*I)*ArcSin[c*x])] + (I*c^3*d^(3/2)*e*Log[(e*Sqrt[c^2*d + e]*(Sqrt[e] - I*c^2*Sqrt[d]*x + Sqrt[c^2*d + e]*S
qrt[1 - c^2*x^2]))/(c^3*(d + I*Sqrt[d]*Sqrt[e]*x))])/(c^2*d + e)^(3/2) - (I*c^3*d^(3/2)*e*Log[(e*Sqrt[c^2*d +
e]*(Sqrt[e] + I*c^2*Sqrt[d]*x + Sqrt[c^2*d + e]*Sqrt[1 - c^2*x^2]))/(c^3*(d - I*Sqrt[d]*Sqrt[e]*x))])/(c^2*d +
e)^(3/2) - (24*I)*e*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] - Sqrt[c^2*d + e])] - (24*I)*e*PolyLog[
2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(-(c*Sqrt[d]) + Sqrt[c^2*d + e])] - (24*I)*e*PolyLog[2, -((Sqrt[e]*E^(I*ArcSin[
c*x]))/(c*Sqrt[d] + Sqrt[c^2*d + e]))] - (24*I)*e*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] + Sqrt[c^2
*d + e])] + (24*I)*e*PolyLog[2, E^((2*I)*ArcSin[c*x])]))/(16*d^4)

Maple [C] (warning: unable to verify)

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

Time = 2.52 (sec) , antiderivative size = 1344, normalized size of antiderivative = 1.72

method	result	size
parts	\(\text {Expression too large to display}\)	\(1344\)
derivativedivides	\(\text {Expression too large to display}\)	\(1395\)
default	\(\text {Expression too large to display}\)	\(1395\)

[In]

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

[Out]

-1/2*a/d^3/x^2-3*a/d^4*e*ln(x)-1/4*a*e/d^2/(e*x^2+d)^2+3/2*a*e/d^4*ln(e*x^2+d)-a*e/d^3/(e*x^2+d)+b*c^2*(-1/8*(
-4*I*c^8*d*e^2*x^6-8*I*c^8*d^2*e*x^4-3*I*c^6*d^2*e*x^2+4*(-c^2*x^2+1)^(1/2)*c^7*d^3*x+8*(-c^2*x^2+1)^(1/2)*c^7
*d^2*e*x^3+4*(-c^2*x^2+1)^(1/2)*c^7*d*e^2*x^5+4*c^6*d^3*arcsin(c*x)+18*arcsin(c*x)*c^6*d^2*e*x^2+12*arcsin(c*x
)*c^6*d*e^2*x^4-6*I*c^6*d*e^2*x^4-3*I*e^3*c^6*x^6-4*I*c^8*d^3*x^2+4*(-c^2*x^2+1)^(1/2)*c^5*d^2*e*x+7*(-c^2*x^2
+1)^(1/2)*c^5*d*e^2*x^3+3*(-c^2*x^2+1)^(1/2)*e^3*c^5*x^5+4*c^4*d^2*e*arcsin(c*x)+18*arcsin(c*x)*c^4*d*e^2*x^2+
12*arcsin(c*x)*e^3*c^4*x^4)/c^2/x^2/d^3/(c^2*e*x^2+c^2*d)^2/(c^2*d+e)-9/8*I*(d*c^2*(c^2*d+e))^(1/2)/(c^2*d+e)^
2/d^4/c^2*arctanh(1/4*(2*e*(I*c*x+(-c^2*x^2+1)^(1/2))^2-4*c^2*d-2*e)/(c^4*d^2+c^2*d*e)^(1/2))*e^2-3*I/(c^2*d+e
)/d^3*e*dilog(I*c*x+(-c^2*x^2+1)^(1/2))+3*I/(c^2*d+e)*e/d^3*dilog(1+I*c*x+(-c^2*x^2+1)^(1/2))-3/4*I/(c^2*d+e)*
e/d^3*sum((_R1^2*e-4*c^2*d-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))-3/4*I/(c^2*d+e)*e^2/d^3*sum((_R
1^2-1)/(_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/(c^2*d+e)/d^3*e*arcsin(c*x)*ln(1+I*c*x+(-c^2*x^2+1)^
(1/2))-3/4*I/(c^2*d+e)*e^2/d^4/c^2*sum((_R1^2*e-4*c^2*d-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))-3/
4*I/(c^2*d+e)*e^3/d^4/c^2*sum((_R1^2-1)/(_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/(c^2*d+e)*e^2/d^4/c
^2*arcsin(c*x)*ln(1+I*c*x+(-c^2*x^2+1)^(1/2))-5/4*I*(d*c^2*(c^2*d+e))^(1/2)/(c^2*d+e)^2/d^3*arctanh(1/4*(2*e*(
I*c*x+(-c^2*x^2+1)^(1/2))^2-4*c^2*d-2*e)/(c^4*d^2+c^2*d*e)^(1/2))*e-3*I/(c^2*d+e)*e^2/d^4/c^2*dilog(I*c*x+(-c^
2*x^2+1)^(1/2))+3*I/(c^2*d+e)*e^2/d^4/c^2*dilog(1+I*c*x+(-c^2*x^2+1)^(1/2)))

Fricas [F]

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

[In]

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

[Out]

integral((b*arcsin(c*x) + a)/(e^3*x^9 + 3*d*e^2*x^7 + 3*d^2*e*x^5 + d^3*x^3), x)

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

-1/4*a*((6*e^2*x^4 + 9*d*e*x^2 + 2*d^2)/(d^3*e^2*x^6 + 2*d^4*e*x^4 + d^5*x^2) - 6*e*log(e*x^2 + d)/d^4 + 12*e*
log(x)/d^4) + b*integrate(arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))/(e^3*x^9 + 3*d*e^2*x^7 + 3*d^2*e*x^5 + d^
3*x^3), x)

Giac [F(-1)]

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

[In]

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

[Out]

Timed out

Mupad [F(-1)]

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

[In]

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

[Out]

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