3.82 \(\int \frac{(d-c^2 d x^2)^{3/2} (a+b \sin ^{-1}(c x))}{x^3} \, dx\)

Optimal. Leaf size=297 \[ -\frac{3 i b c^2 d \sqrt{d-c^2 d x^2} \text{PolyLog}\left (2,-e^{i \sin ^{-1}(c x)}\right )}{2 \sqrt{1-c^2 x^2}}+\frac{3 i b c^2 d \sqrt{d-c^2 d x^2} \text{PolyLog}\left (2,e^{i \sin ^{-1}(c x)}\right )}{2 \sqrt{1-c^2 x^2}}-\frac{3}{2} c^2 d \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac{\left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{2 x^2}+\frac{3 c^2 d \sqrt{d-c^2 d x^2} \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}}+\frac{b c^3 d x \sqrt{d-c^2 d x^2}}{\sqrt{1-c^2 x^2}}-\frac{b c d \sqrt{d-c^2 d x^2}}{2 x \sqrt{1-c^2 x^2}} \]

[Out]

-(b*c*d*Sqrt[d - c^2*d*x^2])/(2*x*Sqrt[1 - c^2*x^2]) + (b*c^3*d*x*Sqrt[d - c^2*d*x^2])/Sqrt[1 - c^2*x^2] - (3*
c^2*d*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/2 - ((d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x]))/(2*x^2) + (3*c^
2*d*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])*ArcTanh[E^(I*ArcSin[c*x])])/Sqrt[1 - c^2*x^2] - (((3*I)/2)*b*c^2*d
*Sqrt[d - c^2*d*x^2]*PolyLog[2, -E^(I*ArcSin[c*x])])/Sqrt[1 - c^2*x^2] + (((3*I)/2)*b*c^2*d*Sqrt[d - c^2*d*x^2
]*PolyLog[2, E^(I*ArcSin[c*x])])/Sqrt[1 - c^2*x^2]

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Rubi [A]  time = 0.330104, antiderivative size = 297, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.296, Rules used = {4695, 4697, 4709, 4183, 2279, 2391, 8, 14} \[ -\frac{3 i b c^2 d \sqrt{d-c^2 d x^2} \text{PolyLog}\left (2,-e^{i \sin ^{-1}(c x)}\right )}{2 \sqrt{1-c^2 x^2}}+\frac{3 i b c^2 d \sqrt{d-c^2 d x^2} \text{PolyLog}\left (2,e^{i \sin ^{-1}(c x)}\right )}{2 \sqrt{1-c^2 x^2}}-\frac{3}{2} c^2 d \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac{\left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{2 x^2}+\frac{3 c^2 d \sqrt{d-c^2 d x^2} \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}}+\frac{b c^3 d x \sqrt{d-c^2 d x^2}}{\sqrt{1-c^2 x^2}}-\frac{b c d \sqrt{d-c^2 d x^2}}{2 x \sqrt{1-c^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(b*c*d*Sqrt[d - c^2*d*x^2])/(2*x*Sqrt[1 - c^2*x^2]) + (b*c^3*d*x*Sqrt[d - c^2*d*x^2])/Sqrt[1 - c^2*x^2] - (3*
c^2*d*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/2 - ((d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x]))/(2*x^2) + (3*c^
2*d*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])*ArcTanh[E^(I*ArcSin[c*x])])/Sqrt[1 - c^2*x^2] - (((3*I)/2)*b*c^2*d
*Sqrt[d - c^2*d*x^2]*PolyLog[2, -E^(I*ArcSin[c*x])])/Sqrt[1 - c^2*x^2] + (((3*I)/2)*b*c^2*d*Sqrt[d - c^2*d*x^2
]*PolyLog[2, E^(I*ArcSin[c*x])])/Sqrt[1 - c^2*x^2]

Rule 4695

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[
((f*x)^(m + 1)*(d + e*x^2)^p*(a + b*ArcSin[c*x])^n)/(f*(m + 1)), x] + (-Dist[(2*e*p)/(f^2*(m + 1)), Int[(f*x)^
(m + 2)*(d + e*x^2)^(p - 1)*(a + b*ArcSin[c*x])^n, x], x] - Dist[(b*c*n*d^IntPart[p]*(d + e*x^2)^FracPart[p])/
(f*(m + 1)*(1 - c^2*x^2)^FracPart[p]), Int[(f*x)^(m + 1)*(1 - c^2*x^2)^(p - 1/2)*(a + b*ArcSin[c*x])^(n - 1),
x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && GtQ[p, 0] && LtQ[m, -1]

Rule 4697

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[((
f*x)^(m + 1)*Sqrt[d + e*x^2]*(a + b*ArcSin[c*x])^n)/(f*(m + 2)), x] + (Dist[Sqrt[d + e*x^2]/((m + 2)*Sqrt[1 -
c^2*x^2]), Int[((f*x)^m*(a + b*ArcSin[c*x])^n)/Sqrt[1 - c^2*x^2], x], x] - Dist[(b*c*n*Sqrt[d + e*x^2])/(f*(m
+ 2)*Sqrt[1 - c^2*x^2]), Int[(f*x)^(m + 1)*(a + b*ArcSin[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f, m}
, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] &&  !LtQ[m, -1] && (RationalQ[m] || EqQ[n, 1])

Rule 4709

Int[(((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Dist[1/(c^(m
+ 1)*Sqrt[d]), Subst[Int[(a + b*x)^n*Sin[x]^m, x], x, ArcSin[c*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2
*d + e, 0] && GtQ[d, 0] && IGtQ[n, 0] && IntegerQ[m]

Rule 4183

Int[csc[(e_.) + (f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(-2*(c + d*x)^m*ArcTanh[E^(I*(e + f*
x))])/f, x] + (-Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Log[1 - E^(I*(e + f*x))], x], x] + Dist[(d*m)/f, Int[(c +
d*x)^(m - 1)*Log[1 + E^(I*(e + f*x))], x], x]) /; FreeQ[{c, d, e, f}, 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 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

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

Mathematica [A]  time = 2.09342, size = 389, normalized size = 1.31 \[ \frac{b c^2 d^2 \sqrt{1-c^2 x^2} \left (-4 i \text{PolyLog}\left (2,-e^{i \sin ^{-1}(c x)}\right )+4 i \text{PolyLog}\left (2,e^{i \sin ^{-1}(c x)}\right )-4 \sin ^{-1}(c x) \log \left (1-e^{i \sin ^{-1}(c x)}\right )+4 \sin ^{-1}(c x) \log \left (1+e^{i \sin ^{-1}(c x)}\right )-2 \tan \left (\frac{1}{2} \sin ^{-1}(c x)\right )-2 \cot \left (\frac{1}{2} \sin ^{-1}(c x)\right )-\sin ^{-1}(c x) \csc ^2\left (\frac{1}{2} \sin ^{-1}(c x)\right )+\sin ^{-1}(c x) \sec ^2\left (\frac{1}{2} \sin ^{-1}(c x)\right )\right )}{8 \sqrt{d-c^2 d x^2}}+\frac{b c^2 d \sqrt{d-c^2 d x^2} \left (-i \text{PolyLog}\left (2,-e^{i \sin ^{-1}(c x)}\right )+i \text{PolyLog}\left (2,e^{i \sin ^{-1}(c x)}\right )-\sqrt{1-c^2 x^2} \sin ^{-1}(c x)+c x-\sin ^{-1}(c x) \log \left (1-e^{i \sin ^{-1}(c x)}\right )+\sin ^{-1}(c x) \log \left (1+e^{i \sin ^{-1}(c x)}\right )\right )}{\sqrt{1-c^2 x^2}}+\frac{3}{2} a c^2 d^{3/2} \log \left (\sqrt{d} \sqrt{d-c^2 d x^2}+d\right )-\frac{3}{2} a c^2 d^{3/2} \log (x)-\frac{a d \left (2 c^2 x^2+1\right ) \sqrt{d-c^2 d x^2}}{2 x^2} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-(a*d*(1 + 2*c^2*x^2)*Sqrt[d - c^2*d*x^2])/(2*x^2) - (3*a*c^2*d^(3/2)*Log[x])/2 + (3*a*c^2*d^(3/2)*Log[d + Sqr
t[d]*Sqrt[d - c^2*d*x^2]])/2 + (b*c^2*d*Sqrt[d - c^2*d*x^2]*(c*x - Sqrt[1 - c^2*x^2]*ArcSin[c*x] - ArcSin[c*x]
*Log[1 - E^(I*ArcSin[c*x])] + ArcSin[c*x]*Log[1 + E^(I*ArcSin[c*x])] - I*PolyLog[2, -E^(I*ArcSin[c*x])] + I*Po
lyLog[2, E^(I*ArcSin[c*x])]))/Sqrt[1 - c^2*x^2] + (b*c^2*d^2*Sqrt[1 - c^2*x^2]*(-2*Cot[ArcSin[c*x]/2] - ArcSin
[c*x]*Csc[ArcSin[c*x]/2]^2 - 4*ArcSin[c*x]*Log[1 - E^(I*ArcSin[c*x])] + 4*ArcSin[c*x]*Log[1 + E^(I*ArcSin[c*x]
)] - (4*I)*PolyLog[2, -E^(I*ArcSin[c*x])] + (4*I)*PolyLog[2, E^(I*ArcSin[c*x])] + ArcSin[c*x]*Sec[ArcSin[c*x]/
2]^2 - 2*Tan[ArcSin[c*x]/2]))/(8*Sqrt[d - c^2*d*x^2])

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Maple [B]  time = 0.227, size = 574, normalized size = 1.9 \begin{align*} -{\frac{a}{2\,d{x}^{2}} \left ( -{c}^{2}d{x}^{2}+d \right ) ^{{\frac{5}{2}}}}-{\frac{a{c}^{2}}{2} \left ( -{c}^{2}d{x}^{2}+d \right ) ^{{\frac{3}{2}}}}+{\frac{3\,a{c}^{2}}{2}{d}^{{\frac{3}{2}}}\ln \left ({\frac{1}{x} \left ( 2\,d+2\,\sqrt{d}\sqrt{-{c}^{2}d{x}^{2}+d} \right ) } \right ) }-{\frac{3\,a{c}^{2}d}{2}\sqrt{-{c}^{2}d{x}^{2}+d}}-{\frac{b{c}^{4}d\arcsin \left ( cx \right ){x}^{2}}{{c}^{2}{x}^{2}-1}\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }}-{\frac{b{c}^{3}dx}{{c}^{2}{x}^{2}-1}\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-{c}^{2}{x}^{2}+1}}+{\frac{b{c}^{2}d\arcsin \left ( cx \right ) }{2\,{c}^{2}{x}^{2}-2}\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }}+{\frac{bdc}{2\,x \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-{c}^{2}{x}^{2}+1}}+{\frac{bd\arcsin \left ( cx \right ) }{2\,{x}^{2} \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }}-3\,{\frac{b\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-{c}^{2}{x}^{2}+1}{c}^{2}d\arcsin \left ( cx \right ) \ln \left ( 1+icx+\sqrt{-{c}^{2}{x}^{2}+1} \right ) }{2\,{c}^{2}{x}^{2}-2}}+3\,{\frac{b\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-{c}^{2}{x}^{2}+1}{c}^{2}d\arcsin \left ( cx \right ) \ln \left ( 1-icx-\sqrt{-{c}^{2}{x}^{2}+1} \right ) }{2\,{c}^{2}{x}^{2}-2}}+{\frac{3\,ib{c}^{2}d}{2\,{c}^{2}{x}^{2}-2}\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-{c}^{2}{x}^{2}+1}{\it polylog} \left ( 2,-icx-\sqrt{-{c}^{2}{x}^{2}+1} \right ) }-{\frac{3\,ib{c}^{2}d}{2\,{c}^{2}{x}^{2}-2}\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-{c}^{2}{x}^{2}+1}{\it polylog} \left ( 2,icx+\sqrt{-{c}^{2}{x}^{2}+1} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-c^2*d*x^2+d)^(3/2)*(a+b*arcsin(c*x))/x^3,x)

[Out]

-1/2*a/d/x^2*(-c^2*d*x^2+d)^(5/2)-1/2*a*c^2*(-c^2*d*x^2+d)^(3/2)+3/2*a*c^2*d^(3/2)*ln((2*d+2*d^(1/2)*(-c^2*d*x
^2+d)^(1/2))/x)-3/2*a*c^2*(-c^2*d*x^2+d)^(1/2)*d-b*(-d*(c^2*x^2-1))^(1/2)*c^4*d/(c^2*x^2-1)*arcsin(c*x)*x^2-b*
(-d*(c^2*x^2-1))^(1/2)*c^3*d/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x+1/2*b*(-d*(c^2*x^2-1))^(1/2)*c^2*d/(c^2*x^2-1)*a
rcsin(c*x)+1/2*b*d*(-d*(c^2*x^2-1))^(1/2)/x/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*c+1/2*b*d*(-d*(c^2*x^2-1))^(1/2)/x^
2/(c^2*x^2-1)*arcsin(c*x)-3*b*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)*c^2*d/(2*c^2*x^2-2)*arcsin(c*x)*ln(1+I
*c*x+(-c^2*x^2+1)^(1/2))+3*b*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)*c^2*d/(2*c^2*x^2-2)*arcsin(c*x)*ln(1-I*
c*x-(-c^2*x^2+1)^(1/2))+3*I*b*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)*c^2*d/(2*c^2*x^2-2)*polylog(2,-I*c*x-(
-c^2*x^2+1)^(1/2))-3*I*b*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)*c^2*d/(2*c^2*x^2-2)*polylog(2,I*c*x+(-c^2*x
^2+1)^(1/2))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{{\left (a c^{2} d x^{2} - a d +{\left (b c^{2} d x^{2} - b d\right )} \arcsin \left (c x\right )\right )} \sqrt{-c^{2} d x^{2} + d}}{x^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac{3}{2}} \left (a + b \operatorname{asin}{\left (c x \right )}\right )}{x^{3}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((-d*(c*x - 1)*(c*x + 1))**(3/2)*(a + b*asin(c*x))/x**3, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (-c^{2} d x^{2} + d\right )}^{\frac{3}{2}}{\left (b \arcsin \left (c x\right ) + a\right )}}{x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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