3.97 \(\int \frac{(d-c^2 d x^2)^{5/2} (a+b \sin ^{-1}(c x))}{x^3} \, dx\)

Optimal. Leaf size=386 \[ -\frac{5 i b c^2 d^2 \sqrt{d-c^2 d x^2} \text{PolyLog}\left (2,-e^{i \sin ^{-1}(c x)}\right )}{2 \sqrt{1-c^2 x^2}}+\frac{5 i b c^2 d^2 \sqrt{d-c^2 d x^2} \text{PolyLog}\left (2,e^{i \sin ^{-1}(c x)}\right )}{2 \sqrt{1-c^2 x^2}}-\frac{5}{2} c^2 d^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )+\frac{5 c^2 d^2 \sqrt{d-c^2 d x^2} \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}}-\frac{5}{6} c^2 d \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )-\frac{\left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{2 x^2}-\frac{b c^5 d^2 x^3 \sqrt{d-c^2 d x^2}}{9 \sqrt{1-c^2 x^2}}+\frac{7 b c^3 d^2 x \sqrt{d-c^2 d x^2}}{3 \sqrt{1-c^2 x^2}}-\frac{b c d^2 \sqrt{d-c^2 d x^2}}{2 x \sqrt{1-c^2 x^2}} \]

[Out]

-(b*c*d^2*Sqrt[d - c^2*d*x^2])/(2*x*Sqrt[1 - c^2*x^2]) + (7*b*c^3*d^2*x*Sqrt[d - c^2*d*x^2])/(3*Sqrt[1 - c^2*x
^2]) - (b*c^5*d^2*x^3*Sqrt[d - c^2*d*x^2])/(9*Sqrt[1 - c^2*x^2]) - (5*c^2*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSi
n[c*x]))/2 - (5*c^2*d*(d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x]))/6 - ((d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x]
))/(2*x^2) + (5*c^2*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])*ArcTanh[E^(I*ArcSin[c*x])])/Sqrt[1 - c^2*x^2]
- (((5*I)/2)*b*c^2*d^2*Sqrt[d - c^2*d*x^2]*PolyLog[2, -E^(I*ArcSin[c*x])])/Sqrt[1 - c^2*x^2] + (((5*I)/2)*b*c^
2*d^2*Sqrt[d - c^2*d*x^2]*PolyLog[2, E^(I*ArcSin[c*x])])/Sqrt[1 - c^2*x^2]

________________________________________________________________________________________

Rubi [A]  time = 0.458406, antiderivative size = 386, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 9, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {4695, 4699, 4697, 4709, 4183, 2279, 2391, 8, 270} \[ -\frac{5 i b c^2 d^2 \sqrt{d-c^2 d x^2} \text{PolyLog}\left (2,-e^{i \sin ^{-1}(c x)}\right )}{2 \sqrt{1-c^2 x^2}}+\frac{5 i b c^2 d^2 \sqrt{d-c^2 d x^2} \text{PolyLog}\left (2,e^{i \sin ^{-1}(c x)}\right )}{2 \sqrt{1-c^2 x^2}}-\frac{5}{2} c^2 d^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )+\frac{5 c^2 d^2 \sqrt{d-c^2 d x^2} \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}}-\frac{5}{6} c^2 d \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )-\frac{\left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{2 x^2}-\frac{b c^5 d^2 x^3 \sqrt{d-c^2 d x^2}}{9 \sqrt{1-c^2 x^2}}+\frac{7 b c^3 d^2 x \sqrt{d-c^2 d x^2}}{3 \sqrt{1-c^2 x^2}}-\frac{b c d^2 \sqrt{d-c^2 d x^2}}{2 x \sqrt{1-c^2 x^2}} \]

Antiderivative was successfully verified.

[In]

Int[((d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x]))/x^3,x]

[Out]

-(b*c*d^2*Sqrt[d - c^2*d*x^2])/(2*x*Sqrt[1 - c^2*x^2]) + (7*b*c^3*d^2*x*Sqrt[d - c^2*d*x^2])/(3*Sqrt[1 - c^2*x
^2]) - (b*c^5*d^2*x^3*Sqrt[d - c^2*d*x^2])/(9*Sqrt[1 - c^2*x^2]) - (5*c^2*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSi
n[c*x]))/2 - (5*c^2*d*(d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x]))/6 - ((d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x]
))/(2*x^2) + (5*c^2*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])*ArcTanh[E^(I*ArcSin[c*x])])/Sqrt[1 - c^2*x^2]
- (((5*I)/2)*b*c^2*d^2*Sqrt[d - c^2*d*x^2]*PolyLog[2, -E^(I*ArcSin[c*x])])/Sqrt[1 - c^2*x^2] + (((5*I)/2)*b*c^
2*d^2*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 4699

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 + 2*p + 1)), x] + (Dist[(2*d*p)/(m + 2*p + 1), Int[(
f*x)^m*(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 + 2*p + 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, m}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && GtQ[p, 0] &&  !LtQ[m, -1]
&& (RationalQ[m] || EqQ[n, 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 270

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

Rubi steps

\begin{align*} \int \frac{\left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{x^3} \, dx &=-\frac{\left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{2 x^2}-\frac{1}{2} \left (5 c^2 d\right ) \int \frac{\left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{x} \, dx+\frac{\left (b c d^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{\left (1-c^2 x^2\right )^2}{x^2} \, dx}{2 \sqrt{1-c^2 x^2}}\\ &=-\frac{5}{6} c^2 d \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )-\frac{\left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{2 x^2}-\frac{1}{2} \left (5 c^2 d^2\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^2 \sqrt{d-c^2 d x^2}\right ) \int \left (-2 c^2+\frac{1}{x^2}+c^4 x^2\right ) \, dx}{2 \sqrt{1-c^2 x^2}}+\frac{\left (5 b c^3 d^2 \sqrt{d-c^2 d x^2}\right ) \int \left (1-c^2 x^2\right ) \, dx}{6 \sqrt{1-c^2 x^2}}\\ &=-\frac{b c d^2 \sqrt{d-c^2 d x^2}}{2 x \sqrt{1-c^2 x^2}}-\frac{b c^3 d^2 x \sqrt{d-c^2 d x^2}}{6 \sqrt{1-c^2 x^2}}-\frac{b c^5 d^2 x^3 \sqrt{d-c^2 d x^2}}{9 \sqrt{1-c^2 x^2}}-\frac{5}{2} c^2 d^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac{5}{6} c^2 d \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )-\frac{\left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{2 x^2}-\frac{\left (5 c^2 d^2 \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 (5 b c^3 d^2 \sqrt{d-c^2 d x^2}\right ) \int 1 \, dx}{2 \sqrt{1-c^2 x^2}}\\ &=-\frac{b c d^2 \sqrt{d-c^2 d x^2}}{2 x \sqrt{1-c^2 x^2}}+\frac{7 b c^3 d^2 x \sqrt{d-c^2 d x^2}}{3 \sqrt{1-c^2 x^2}}-\frac{b c^5 d^2 x^3 \sqrt{d-c^2 d x^2}}{9 \sqrt{1-c^2 x^2}}-\frac{5}{2} c^2 d^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac{5}{6} c^2 d \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )-\frac{\left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{2 x^2}-\frac{\left (5 c^2 d^2 \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^2 \sqrt{d-c^2 d x^2}}{2 x \sqrt{1-c^2 x^2}}+\frac{7 b c^3 d^2 x \sqrt{d-c^2 d x^2}}{3 \sqrt{1-c^2 x^2}}-\frac{b c^5 d^2 x^3 \sqrt{d-c^2 d x^2}}{9 \sqrt{1-c^2 x^2}}-\frac{5}{2} c^2 d^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac{5}{6} c^2 d \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )-\frac{\left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{2 x^2}+\frac{5 c^2 d^2 \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 (5 b c^2 d^2 \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 (5 b c^2 d^2 \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^2 \sqrt{d-c^2 d x^2}}{2 x \sqrt{1-c^2 x^2}}+\frac{7 b c^3 d^2 x \sqrt{d-c^2 d x^2}}{3 \sqrt{1-c^2 x^2}}-\frac{b c^5 d^2 x^3 \sqrt{d-c^2 d x^2}}{9 \sqrt{1-c^2 x^2}}-\frac{5}{2} c^2 d^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac{5}{6} c^2 d \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )-\frac{\left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{2 x^2}+\frac{5 c^2 d^2 \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 (5 i b c^2 d^2 \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 (5 i b c^2 d^2 \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^2 \sqrt{d-c^2 d x^2}}{2 x \sqrt{1-c^2 x^2}}+\frac{7 b c^3 d^2 x \sqrt{d-c^2 d x^2}}{3 \sqrt{1-c^2 x^2}}-\frac{b c^5 d^2 x^3 \sqrt{d-c^2 d x^2}}{9 \sqrt{1-c^2 x^2}}-\frac{5}{2} c^2 d^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac{5}{6} c^2 d \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )-\frac{\left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{2 x^2}+\frac{5 c^2 d^2 \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{5 i b c^2 d^2 \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{5 i b c^2 d^2 \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 = 5.29996, size = 484, normalized size = 1.25 \[ \frac{144 b c^2 d^3 x^2 \sqrt{1-c^2 x^2} \left (-i \left (\text{PolyLog}\left (2,-e^{i \sin ^{-1}(c x)}\right )-\text{PolyLog}\left (2,e^{i \sin ^{-1}(c x)}\right )\right )-\sqrt{1-c^2 x^2} \sin ^{-1}(c x)+c x-\sin ^{-1}(c x) \left (\log \left (1-e^{i \sin ^{-1}(c x)}\right )-\log \left (1+e^{i \sin ^{-1}(c x)}\right )\right )\right )-9 b c^2 d^3 x^2 \sqrt{1-c^2 x^2} \left (4 i \text{PolyLog}\left (2,-e^{i \sin ^{-1}(c x)}\right )-4 i \text{PolyLog}\left (2,e^{i \sin ^{-1}(c x)}\right )+4 \sin ^{-1}(c x) \log \left (1-e^{i \sin ^{-1}(c x)}\right )-4 \sin ^{-1}(c x) \log \left (1+e^{i \sin ^{-1}(c x)}\right )+2 \tan \left (\frac{1}{2} \sin ^{-1}(c x)\right )+2 \cot \left (\frac{1}{2} \sin ^{-1}(c x)\right )+\sin ^{-1}(c x) \csc ^2\left (\frac{1}{2} \sin ^{-1}(c x)\right )-\sin ^{-1}(c x) \sec ^2\left (\frac{1}{2} \sin ^{-1}(c x)\right )\right )-12 a d^3 \left (c^2 x^2-1\right ) \left (2 c^4 x^4-14 c^2 x^2-3\right )-180 a c^2 d^{5/2} x^2 \log (x) \sqrt{d-c^2 d x^2}+180 a c^2 d^{5/2} x^2 \sqrt{d-c^2 d x^2} \log \left (\sqrt{d} \sqrt{d-c^2 d x^2}+d\right )+2 b c^2 d^3 x^2 \sqrt{1-c^2 x^2} \left (-3 \sin ^{-1}(c x) \left (3 \sqrt{1-c^2 x^2}+\cos \left (3 \sin ^{-1}(c x)\right )\right )+9 c x+\sin \left (3 \sin ^{-1}(c x)\right )\right )}{72 x^2 \sqrt{d-c^2 d x^2}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[((d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x]))/x^3,x]

[Out]

(-12*a*d^3*(-1 + c^2*x^2)*(-3 - 14*c^2*x^2 + 2*c^4*x^4) - 180*a*c^2*d^(5/2)*x^2*Sqrt[d - c^2*d*x^2]*Log[x] + 1
80*a*c^2*d^(5/2)*x^2*Sqrt[d - c^2*d*x^2]*Log[d + Sqrt[d]*Sqrt[d - c^2*d*x^2]] + 144*b*c^2*d^3*x^2*Sqrt[1 - c^2
*x^2]*(c*x - Sqrt[1 - c^2*x^2]*ArcSin[c*x] - ArcSin[c*x]*(Log[1 - E^(I*ArcSin[c*x])] - Log[1 + E^(I*ArcSin[c*x
])]) - I*(PolyLog[2, -E^(I*ArcSin[c*x])] - PolyLog[2, E^(I*ArcSin[c*x])])) + 2*b*c^2*d^3*x^2*Sqrt[1 - c^2*x^2]
*(9*c*x - 3*ArcSin[c*x]*(3*Sqrt[1 - c^2*x^2] + Cos[3*ArcSin[c*x]]) + Sin[3*ArcSin[c*x]]) - 9*b*c^2*d^3*x^2*Sqr
t[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]))/(72*x^2*Sqrt[d - c^2*d*x^2])

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Maple [A]  time = 0.273, size = 704, normalized size = 1.8 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-c^2*d*x^2+d)^(5/2)*(a+b*arcsin(c*x))/x^3,x)

[Out]

-1/2*a/d/x^2*(-c^2*d*x^2+d)^(7/2)-1/2*a*c^2*(-c^2*d*x^2+d)^(5/2)-5/6*a*c^2*d*(-c^2*d*x^2+d)^(3/2)+5/2*a*c^2*d^
(5/2)*ln((2*d+2*d^(1/2)*(-c^2*d*x^2+d)^(1/2))/x)-5/2*a*c^2*(-c^2*d*x^2+d)^(1/2)*d^2-5*b*(-d*(c^2*x^2-1))^(1/2)
*(-c^2*x^2+1)^(1/2)*c^2*d^2/(2*c^2*x^2-2)*arcsin(c*x)*ln(1+I*c*x+(-c^2*x^2+1)^(1/2))+5*b*(-d*(c^2*x^2-1))^(1/2
)*(-c^2*x^2+1)^(1/2)*c^2*d^2/(2*c^2*x^2-2)*arcsin(c*x)*ln(1-I*c*x-(-c^2*x^2+1)^(1/2))+1/9*b*(-d*(c^2*x^2-1))^(
1/2)*c^5*d^2/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x^3-7/3*b*(-d*(c^2*x^2-1))^(1/2)*c^3*d^2/(c^2*x^2-1)*(-c^2*x^2+1)^
(1/2)*x+11/6*b*(-d*(c^2*x^2-1))^(1/2)*c^2*d^2/(c^2*x^2-1)*arcsin(c*x)+1/2*b*d^2*(-d*(c^2*x^2-1))^(1/2)/x^2/(c^
2*x^2-1)*arcsin(c*x)+1/3*b*(-d*(c^2*x^2-1))^(1/2)*c^6*d^2/(c^2*x^2-1)*arcsin(c*x)*x^4-8/3*b*(-d*(c^2*x^2-1))^(
1/2)*c^4*d^2/(c^2*x^2-1)*arcsin(c*x)*x^2+1/2*b*d^2*(-d*(c^2*x^2-1))^(1/2)/x/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*c+5
*I*b*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)*c^2*d^2/(2*c^2*x^2-2)*polylog(2,-I*c*x-(-c^2*x^2+1)^(1/2))-5*I*
b*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)*c^2*d^2/(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)^(5/2)*(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^{4} d^{2} x^{4} - 2 \, a c^{2} d^{2} x^{2} + a d^{2} +{\left (b c^{4} d^{2} x^{4} - 2 \, b c^{2} d^{2} x^{2} + b d^{2}\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)^(5/2)*(a+b*arcsin(c*x))/x^3,x, algorithm="fricas")

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c**2*d*x**2+d)**(5/2)*(a+b*asin(c*x))/x**3,x)

[Out]

Timed out

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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{5}{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)^(5/2)*(a+b*arcsin(c*x))/x^3,x, algorithm="giac")

[Out]

integrate((-c^2*d*x^2 + d)^(5/2)*(b*arcsin(c*x) + a)/x^3, x)