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

Optimal. Leaf size=486 \[ \frac{4 i b^2 e^3 \left (1-c^2 x^2\right )^{5/2} \text{PolyLog}\left (2,i e^{i \sin ^{-1}(c x)}\right )}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}+\frac{i e^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}-\frac{4 b e^3 \left (1-c^2 x^2\right )^{5/2} \log \left (1-i e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}+\frac{e^3 \left (1-c^2 x^2\right )^{5/2} \cot \left (\frac{1}{2} \sin ^{-1}(c x)+\frac{\pi }{4}\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}-\frac{2 b e^3 \left (1-c^2 x^2\right )^{5/2} \csc ^2\left (\frac{1}{2} \sin ^{-1}(c x)+\frac{\pi }{4}\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}-\frac{e^3 \left (1-c^2 x^2\right )^{5/2} \cot \left (\frac{1}{2} \sin ^{-1}(c x)+\frac{\pi }{4}\right ) \csc ^2\left (\frac{1}{2} \sin ^{-1}(c x)+\frac{\pi }{4}\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}-\frac{4 b^2 e^3 \left (1-c^2 x^2\right )^{5/2} \cot \left (\frac{1}{2} \sin ^{-1}(c x)+\frac{\pi }{4}\right )}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}} \]

[Out]

((I/3)*e^3*(1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x])^2)/(c*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) - (4*b^2*e^3*(1
- c^2*x^2)^(5/2)*Cot[Pi/4 + ArcSin[c*x]/2])/(3*c*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) + (e^3*(1 - c^2*x^2)^(5/
2)*(a + b*ArcSin[c*x])^2*Cot[Pi/4 + ArcSin[c*x]/2])/(3*c*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) - (2*b*e^3*(1 -
c^2*x^2)^(5/2)*(a + b*ArcSin[c*x])*Csc[Pi/4 + ArcSin[c*x]/2]^2)/(3*c*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) - (e
^3*(1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x])^2*Cot[Pi/4 + ArcSin[c*x]/2]*Csc[Pi/4 + ArcSin[c*x]/2]^2)/(3*c*(d +
c*d*x)^(5/2)*(e - c*e*x)^(5/2)) - (4*b*e^3*(1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x])*Log[1 - I*E^(I*ArcSin[c*x])
])/(3*c*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) + (((4*I)/3)*b^2*e^3*(1 - c^2*x^2)^(5/2)*PolyLog[2, I*E^(I*ArcSin
[c*x])])/(c*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2))

________________________________________________________________________________________

Rubi [A]  time = 1.11892, antiderivative size = 486, normalized size of antiderivative = 1., number of steps used = 20, number of rules used = 12, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {4673, 4775, 4773, 3318, 4186, 3767, 8, 4184, 3717, 2190, 2279, 2391} \[ \frac{4 i b^2 e^3 \left (1-c^2 x^2\right )^{5/2} \text{PolyLog}\left (2,i e^{i \sin ^{-1}(c x)}\right )}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}+\frac{i e^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}-\frac{4 b e^3 \left (1-c^2 x^2\right )^{5/2} \log \left (1-i e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}+\frac{e^3 \left (1-c^2 x^2\right )^{5/2} \cot \left (\frac{1}{2} \sin ^{-1}(c x)+\frac{\pi }{4}\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}-\frac{2 b e^3 \left (1-c^2 x^2\right )^{5/2} \csc ^2\left (\frac{1}{2} \sin ^{-1}(c x)+\frac{\pi }{4}\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}-\frac{e^3 \left (1-c^2 x^2\right )^{5/2} \cot \left (\frac{1}{2} \sin ^{-1}(c x)+\frac{\pi }{4}\right ) \csc ^2\left (\frac{1}{2} \sin ^{-1}(c x)+\frac{\pi }{4}\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}-\frac{4 b^2 e^3 \left (1-c^2 x^2\right )^{5/2} \cot \left (\frac{1}{2} \sin ^{-1}(c x)+\frac{\pi }{4}\right )}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((I/3)*e^3*(1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x])^2)/(c*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) - (4*b^2*e^3*(1
- c^2*x^2)^(5/2)*Cot[Pi/4 + ArcSin[c*x]/2])/(3*c*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) + (e^3*(1 - c^2*x^2)^(5/
2)*(a + b*ArcSin[c*x])^2*Cot[Pi/4 + ArcSin[c*x]/2])/(3*c*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) - (2*b*e^3*(1 -
c^2*x^2)^(5/2)*(a + b*ArcSin[c*x])*Csc[Pi/4 + ArcSin[c*x]/2]^2)/(3*c*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) - (e
^3*(1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x])^2*Cot[Pi/4 + ArcSin[c*x]/2]*Csc[Pi/4 + ArcSin[c*x]/2]^2)/(3*c*(d +
c*d*x)^(5/2)*(e - c*e*x)^(5/2)) - (4*b*e^3*(1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x])*Log[1 - I*E^(I*ArcSin[c*x])
])/(3*c*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) + (((4*I)/3)*b^2*e^3*(1 - c^2*x^2)^(5/2)*PolyLog[2, I*E^(I*ArcSin
[c*x])])/(c*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2))

Rule 4673

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_))^(p_)*((f_) + (g_.)*(x_))^(q_), x_Symbol] :> D
ist[((d + e*x)^q*(f + g*x)^q)/(1 - c^2*x^2)^q, Int[(d + e*x)^(p - q)*(1 - c^2*x^2)^q*(a + b*ArcSin[c*x])^n, x]
, x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && EqQ[e*f + d*g, 0] && EqQ[c^2*d^2 - e^2, 0] && HalfIntegerQ[p, q]
&& GeQ[p - q, 0]

Rule 4775

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :
> Int[ExpandIntegrand[(a + b*ArcSin[c*x])^n/Sqrt[d + e*x^2], (f + g*x)^m*(d + e*x^2)^(p + 1/2), x], x] /; Free
Q[{a, b, c, d, e, f, g}, x] && EqQ[c^2*d + e, 0] && IntegerQ[m] && ILtQ[p + 1/2, 0] && GtQ[d, 0] && IGtQ[n, 0]

Rule 4773

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

Rule 3318

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

Rule 4186

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((c_.) + (d_.)*(x_))^(m_), x_Symbol] :> -Simp[(b^2*(c + d*x)^m*Cot[e
+ f*x]*(b*Csc[e + f*x])^(n - 2))/(f*(n - 1)), x] + (Dist[(b^2*d^2*m*(m - 1))/(f^2*(n - 1)*(n - 2)), Int[(c + d
*x)^(m - 2)*(b*Csc[e + f*x])^(n - 2), x], x] + Dist[(b^2*(n - 2))/(n - 1), Int[(c + d*x)^m*(b*Csc[e + f*x])^(n
- 2), x], x] - Simp[(b^2*d*m*(c + d*x)^(m - 1)*(b*Csc[e + f*x])^(n - 2))/(f^2*(n - 1)*(n - 2)), x]) /; FreeQ[
{b, c, d, e, f}, x] && GtQ[n, 1] && NeQ[n, 2] && GtQ[m, 1]

Rule 3767

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rule 8

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

Rule 4184

Int[csc[(e_.) + (f_.)*(x_)]^2*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> -Simp[((c + d*x)^m*Cot[e + f*x])/f, x]
+ Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Cot[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 3717

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 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]

Rubi steps

\begin{align*} \int \frac{\sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2}{(d+c d x)^{5/2}} \, dx &=\frac{\left (1-c^2 x^2\right )^{5/2} \int \frac{(e-c e x)^3 \left (a+b \sin ^{-1}(c x)\right )^2}{\left (1-c^2 x^2\right )^{5/2}} \, dx}{(d+c d x)^{5/2} (e-c e x)^{5/2}}\\ &=\frac{\left (1-c^2 x^2\right )^{5/2} \int \left (\frac{2 e^3 \left (a+b \sin ^{-1}(c x)\right )^2}{(1+c x)^2 \sqrt{1-c^2 x^2}}-\frac{e^3 \left (a+b \sin ^{-1}(c x)\right )^2}{(1+c x) \sqrt{1-c^2 x^2}}\right ) \, dx}{(d+c d x)^{5/2} (e-c e x)^{5/2}}\\ &=-\frac{\left (e^3 \left (1-c^2 x^2\right )^{5/2}\right ) \int \frac{\left (a+b \sin ^{-1}(c x)\right )^2}{(1+c x) \sqrt{1-c^2 x^2}} \, dx}{(d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac{\left (2 e^3 \left (1-c^2 x^2\right )^{5/2}\right ) \int \frac{\left (a+b \sin ^{-1}(c x)\right )^2}{(1+c x)^2 \sqrt{1-c^2 x^2}} \, dx}{(d+c d x)^{5/2} (e-c e x)^{5/2}}\\ &=-\frac{\left (e^3 \left (1-c^2 x^2\right )^{5/2}\right ) \operatorname{Subst}\left (\int \frac{(a+b x)^2}{c+c \sin (x)} \, dx,x,\sin ^{-1}(c x)\right )}{(d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac{\left (2 c e^3 \left (1-c^2 x^2\right )^{5/2}\right ) \operatorname{Subst}\left (\int \frac{(a+b x)^2}{(c+c \sin (x))^2} \, dx,x,\sin ^{-1}(c x)\right )}{(d+c d x)^{5/2} (e-c e x)^{5/2}}\\ &=-\frac{\left (e^3 \left (1-c^2 x^2\right )^{5/2}\right ) \operatorname{Subst}\left (\int (a+b x)^2 \csc ^2\left (\frac{\pi }{4}+\frac{x}{2}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{2 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac{\left (e^3 \left (1-c^2 x^2\right )^{5/2}\right ) \operatorname{Subst}\left (\int (a+b x)^2 \csc ^4\left (\frac{\pi }{4}+\frac{x}{2}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{2 c (d+c d x)^{5/2} (e-c e x)^{5/2}}\\ &=\frac{e^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2 \cot \left (\frac{\pi }{4}+\frac{1}{2} \sin ^{-1}(c x)\right )}{c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac{2 b e^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right ) \csc ^2\left (\frac{\pi }{4}+\frac{1}{2} \sin ^{-1}(c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac{e^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2 \cot \left (\frac{\pi }{4}+\frac{1}{2} \sin ^{-1}(c x)\right ) \csc ^2\left (\frac{\pi }{4}+\frac{1}{2} \sin ^{-1}(c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac{\left (e^3 \left (1-c^2 x^2\right )^{5/2}\right ) \operatorname{Subst}\left (\int (a+b x)^2 \csc ^2\left (\frac{\pi }{4}+\frac{x}{2}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac{\left (2 b e^3 \left (1-c^2 x^2\right )^{5/2}\right ) \operatorname{Subst}\left (\int (a+b x) \cot \left (\frac{\pi }{4}+\frac{x}{2}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac{\left (2 b^2 e^3 \left (1-c^2 x^2\right )^{5/2}\right ) \operatorname{Subst}\left (\int \csc ^2\left (\frac{\pi }{4}+\frac{x}{2}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}\\ &=\frac{i e^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2}{c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac{e^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2 \cot \left (\frac{\pi }{4}+\frac{1}{2} \sin ^{-1}(c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac{2 b e^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right ) \csc ^2\left (\frac{\pi }{4}+\frac{1}{2} \sin ^{-1}(c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac{e^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2 \cot \left (\frac{\pi }{4}+\frac{1}{2} \sin ^{-1}(c x)\right ) \csc ^2\left (\frac{\pi }{4}+\frac{1}{2} \sin ^{-1}(c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac{\left (4 b e^3 \left (1-c^2 x^2\right )^{5/2}\right ) \operatorname{Subst}\left (\int (a+b x) \cot \left (\frac{\pi }{4}+\frac{x}{2}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac{\left (4 b e^3 \left (1-c^2 x^2\right )^{5/2}\right ) \operatorname{Subst}\left (\int \frac{e^{i x} (a+b x)}{1-i e^{i x}} \, dx,x,\sin ^{-1}(c x)\right )}{c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac{\left (4 b^2 e^3 \left (1-c^2 x^2\right )^{5/2}\right ) \operatorname{Subst}\left (\int 1 \, dx,x,\cot \left (\frac{\pi }{4}+\frac{1}{2} \sin ^{-1}(c x)\right )\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}\\ &=\frac{i e^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac{4 b^2 e^3 \left (1-c^2 x^2\right )^{5/2} \cot \left (\frac{\pi }{4}+\frac{1}{2} \sin ^{-1}(c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac{e^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2 \cot \left (\frac{\pi }{4}+\frac{1}{2} \sin ^{-1}(c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac{2 b e^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right ) \csc ^2\left (\frac{\pi }{4}+\frac{1}{2} \sin ^{-1}(c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac{e^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2 \cot \left (\frac{\pi }{4}+\frac{1}{2} \sin ^{-1}(c x)\right ) \csc ^2\left (\frac{\pi }{4}+\frac{1}{2} \sin ^{-1}(c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac{4 b e^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-i e^{i \sin ^{-1}(c x)}\right )}{c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac{\left (8 b e^3 \left (1-c^2 x^2\right )^{5/2}\right ) \operatorname{Subst}\left (\int \frac{e^{i x} (a+b x)}{1-i e^{i x}} \, dx,x,\sin ^{-1}(c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac{\left (4 b^2 e^3 \left (1-c^2 x^2\right )^{5/2}\right ) \operatorname{Subst}\left (\int \log \left (1-i e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{c (d+c d x)^{5/2} (e-c e x)^{5/2}}\\ &=\frac{i e^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac{4 b^2 e^3 \left (1-c^2 x^2\right )^{5/2} \cot \left (\frac{\pi }{4}+\frac{1}{2} \sin ^{-1}(c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac{e^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2 \cot \left (\frac{\pi }{4}+\frac{1}{2} \sin ^{-1}(c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac{2 b e^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right ) \csc ^2\left (\frac{\pi }{4}+\frac{1}{2} \sin ^{-1}(c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac{e^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2 \cot \left (\frac{\pi }{4}+\frac{1}{2} \sin ^{-1}(c x)\right ) \csc ^2\left (\frac{\pi }{4}+\frac{1}{2} \sin ^{-1}(c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac{4 b e^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-i e^{i \sin ^{-1}(c x)}\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac{\left (4 i b^2 e^3 \left (1-c^2 x^2\right )^{5/2}\right ) \operatorname{Subst}\left (\int \frac{\log (1-i x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac{\left (8 b^2 e^3 \left (1-c^2 x^2\right )^{5/2}\right ) \operatorname{Subst}\left (\int \log \left (1-i e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}\\ &=\frac{i e^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac{4 b^2 e^3 \left (1-c^2 x^2\right )^{5/2} \cot \left (\frac{\pi }{4}+\frac{1}{2} \sin ^{-1}(c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac{e^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2 \cot \left (\frac{\pi }{4}+\frac{1}{2} \sin ^{-1}(c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac{2 b e^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right ) \csc ^2\left (\frac{\pi }{4}+\frac{1}{2} \sin ^{-1}(c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac{e^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2 \cot \left (\frac{\pi }{4}+\frac{1}{2} \sin ^{-1}(c x)\right ) \csc ^2\left (\frac{\pi }{4}+\frac{1}{2} \sin ^{-1}(c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac{4 b e^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-i e^{i \sin ^{-1}(c x)}\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac{4 i b^2 e^3 \left (1-c^2 x^2\right )^{5/2} \text{Li}_2\left (i e^{i \sin ^{-1}(c x)}\right )}{c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac{\left (8 i b^2 e^3 \left (1-c^2 x^2\right )^{5/2}\right ) \operatorname{Subst}\left (\int \frac{\log (1-i x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}\\ &=\frac{i e^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac{4 b^2 e^3 \left (1-c^2 x^2\right )^{5/2} \cot \left (\frac{\pi }{4}+\frac{1}{2} \sin ^{-1}(c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac{e^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2 \cot \left (\frac{\pi }{4}+\frac{1}{2} \sin ^{-1}(c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac{2 b e^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right ) \csc ^2\left (\frac{\pi }{4}+\frac{1}{2} \sin ^{-1}(c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac{e^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2 \cot \left (\frac{\pi }{4}+\frac{1}{2} \sin ^{-1}(c x)\right ) \csc ^2\left (\frac{\pi }{4}+\frac{1}{2} \sin ^{-1}(c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac{4 b e^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-i e^{i \sin ^{-1}(c x)}\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac{4 i b^2 e^3 \left (1-c^2 x^2\right )^{5/2} \text{Li}_2\left (i e^{i \sin ^{-1}(c x)}\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}\\ \end{align*}

Mathematica [A]  time = 7.9466, size = 694, normalized size = 1.43 \[ -\frac{b^2 (c x-1) \sqrt{c d x+d} \sqrt{e-c e x} \sqrt{-d e \left (1-c^2 x^2\right )} \left (4 i \text{PolyLog}\left (2,i e^{i \sin ^{-1}(c x)}\right )+(1+i) \sin ^{-1}(c x)^2-i \pi \sin ^{-1}(c x)-4 \pi \log \left (1+e^{-i \sin ^{-1}(c x)}\right )-2 \left (2 \sin ^{-1}(c x)+\pi \right ) \log \left (1-i e^{i \sin ^{-1}(c x)}\right )+2 \pi \log \left (\sin \left (\frac{1}{4} \left (2 \sin ^{-1}(c x)+\pi \right )\right )\right )+\frac{4 \sin \left (\frac{1}{2} \sin ^{-1}(c x)\right ) \sin ^{-1}(c x)^2}{\left (\sin \left (\frac{1}{2} \sin ^{-1}(c x)\right )+\cos \left (\frac{1}{2} \sin ^{-1}(c x)\right )\right )^3}-\frac{2 \left (\sin ^{-1}(c x)+2\right ) \sin ^{-1}(c x)}{\left (\sin \left (\frac{1}{2} \sin ^{-1}(c x)\right )+\cos \left (\frac{1}{2} \sin ^{-1}(c x)\right )\right )^2}-\frac{2 \left (\sin ^{-1}(c x)^2-4\right ) \sin \left (\frac{1}{2} \sin ^{-1}(c x)\right )}{\sin \left (\frac{1}{2} \sin ^{-1}(c x)\right )+\cos \left (\frac{1}{2} \sin ^{-1}(c x)\right )}+4 \pi \log \left (\cos \left (\frac{1}{2} \sin ^{-1}(c x)\right )\right )\right )}{3 c d^3 \sqrt{1-c^2 x^2} \sqrt{-(c d x+d) (e-c e x)} \left (\cos \left (\frac{1}{2} \sin ^{-1}(c x)\right )-\sin \left (\frac{1}{2} \sin ^{-1}(c x)\right )\right )^2}+\frac{\sqrt{d (c x+1)} \sqrt{-e (c x-1)} \left (\frac{a^2}{3 d^3 (c x+1)}-\frac{2 a^2}{3 d^3 (c x+1)^2}\right )}{c}-\frac{a b \sqrt{c d x+d} \sqrt{e-c e x} \sqrt{-d e \left (1-c^2 x^2\right )} \left (\cos \left (\frac{1}{2} \sin ^{-1}(c x)\right )-\sin \left (\frac{1}{2} \sin ^{-1}(c x)\right )\right ) \left (2 \sin \left (\frac{1}{2} \sin ^{-1}(c x)\right ) \left (\sqrt{1-c^2 x^2} \sin ^{-1}(c x)-2 \sqrt{1-c^2 x^2} \log \left (\sin \left (\frac{1}{2} \sin ^{-1}(c x)\right )+\cos \left (\frac{1}{2} \sin ^{-1}(c x)\right )\right )+2 \sin ^{-1}(c x)-4 \log \left (\sin \left (\frac{1}{2} \sin ^{-1}(c x)\right )+\cos \left (\frac{1}{2} \sin ^{-1}(c x)\right )\right )-2\right )+\cos \left (\frac{3}{2} \sin ^{-1}(c x)\right ) \left (\sin ^{-1}(c x)+2 \log \left (\sin \left (\frac{1}{2} \sin ^{-1}(c x)\right )+\cos \left (\frac{1}{2} \sin ^{-1}(c x)\right )\right )\right )-\cos \left (\frac{1}{2} \sin ^{-1}(c x)\right ) \left (3 \sin ^{-1}(c x)+6 \log \left (\sin \left (\frac{1}{2} \sin ^{-1}(c x)\right )+\cos \left (\frac{1}{2} \sin ^{-1}(c x)\right )\right )+4\right )\right )}{3 c d^3 (c x-1) \sqrt{-(c d x+d) (e-c e x)} \left (\sin \left (\frac{1}{2} \sin ^{-1}(c x)\right )+\cos \left (\frac{1}{2} \sin ^{-1}(c x)\right )\right )^4} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(Sqrt[-(e*(-1 + c*x))]*Sqrt[d*(1 + c*x)]*((-2*a^2)/(3*d^3*(1 + c*x)^2) + a^2/(3*d^3*(1 + c*x))))/c - (a*b*Sqrt
[d + c*d*x]*Sqrt[e - c*e*x]*Sqrt[-(d*e*(1 - c^2*x^2))]*(Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2])*(Cos[(3*ArcSi
n[c*x])/2]*(ArcSin[c*x] + 2*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]]) - Cos[ArcSin[c*x]/2]*(4 + 3*ArcSin[c
*x] + 6*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]]) + 2*(-2 + 2*ArcSin[c*x] + Sqrt[1 - c^2*x^2]*ArcSin[c*x]
- 4*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]] - 2*Sqrt[1 - c^2*x^2]*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x
]/2]])*Sin[ArcSin[c*x]/2]))/(3*c*d^3*(-1 + c*x)*Sqrt[-((d + c*d*x)*(e - c*e*x))]*(Cos[ArcSin[c*x]/2] + Sin[Arc
Sin[c*x]/2])^4) - (b^2*(-1 + c*x)*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*Sqrt[-(d*e*(1 - c^2*x^2))]*((-I)*Pi*ArcSin[c
*x] + (1 + I)*ArcSin[c*x]^2 - 4*Pi*Log[1 + E^((-I)*ArcSin[c*x])] - 2*(Pi + 2*ArcSin[c*x])*Log[1 - I*E^(I*ArcSi
n[c*x])] + 4*Pi*Log[Cos[ArcSin[c*x]/2]] + 2*Pi*Log[Sin[(Pi + 2*ArcSin[c*x])/4]] + (4*I)*PolyLog[2, I*E^(I*ArcS
in[c*x])] + (4*ArcSin[c*x]^2*Sin[ArcSin[c*x]/2])/(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2])^3 - (2*ArcSin[c*x]*
(2 + ArcSin[c*x]))/(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2])^2 - (2*(-4 + ArcSin[c*x]^2)*Sin[ArcSin[c*x]/2])/(
Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2])))/(3*c*d^3*Sqrt[-((d + c*d*x)*(e - c*e*x))]*Sqrt[1 - c^2*x^2]*(Cos[Ar
cSin[c*x]/2] - Sin[ArcSin[c*x]/2])^2)

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Maple [F]  time = 0.267, size = 0, normalized size = 0. \begin{align*} \int{ \left ( a+b\arcsin \left ( cx \right ) \right ) ^{2}\sqrt{-cex+e} \left ( cdx+d \right ) ^{-{\frac{5}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arcsin(c*x))^2*(-c*e*x+e)^(1/2)/(c*d*x+d)^(5/2),x)

[Out]

int((a+b*arcsin(c*x))^2*(-c*e*x+e)^(1/2)/(c*d*x+d)^(5/2),x)

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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((a+b*arcsin(c*x))^2*(-c*e*x+e)^(1/2)/(c*d*x+d)^(5/2),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 (b^{2} \arcsin \left (c x\right )^{2} + 2 \, a b \arcsin \left (c x\right ) + a^{2}\right )} \sqrt{c d x + d} \sqrt{-c e x + e}}{c^{3} d^{3} x^{3} + 3 \, c^{2} d^{3} x^{2} + 3 \, c d^{3} x + d^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsin(c*x))^2*(-c*e*x+e)^(1/2)/(c*d*x+d)^(5/2),x, algorithm="fricas")

[Out]

integral((b^2*arcsin(c*x)^2 + 2*a*b*arcsin(c*x) + a^2)*sqrt(c*d*x + d)*sqrt(-c*e*x + e)/(c^3*d^3*x^3 + 3*c^2*d
^3*x^2 + 3*c*d^3*x + d^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((a+b*asin(c*x))**2*(-c*e*x+e)**(1/2)/(c*d*x+d)**(5/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{-c e x + e}{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{{\left (c d x + d\right )}^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsin(c*x))^2*(-c*e*x+e)^(1/2)/(c*d*x+d)^(5/2),x, algorithm="giac")

[Out]

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