3.6.70 \(\int \frac {(d+c d x)^{5/2} (a+b \text {ArcSin}(c x))^2}{(e-c e x)^{5/2}} \, dx\) [570]

Optimal. Leaf size=730 \[ \frac {2 a b d^5 x \left (1-c^2 x^2\right )^{5/2}}{(d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {2 b^2 d^5 \left (1-c^2 x^2\right )^3}{c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {2 b^2 d^5 x \left (1-c^2 x^2\right )^{5/2} \text {ArcSin}(c x)}{(d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {28 i d^5 \left (1-c^2 x^2\right )^{5/2} (a+b \text {ArcSin}(c x))^2}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {d^5 \left (1-c^2 x^2\right )^3 (a+b \text {ArcSin}(c x))^2}{c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {5 d^5 \left (1-c^2 x^2\right )^{5/2} (a+b \text {ArcSin}(c x))^3}{3 b c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {112 b d^5 \left (1-c^2 x^2\right )^{5/2} (a+b \text {ArcSin}(c x)) \log \left (1-i e^{-i \text {ArcSin}(c x)}\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {112 i b^2 d^5 \left (1-c^2 x^2\right )^{5/2} \text {PolyLog}\left (2,i e^{-i \text {ArcSin}(c x)}\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {8 b d^5 \left (1-c^2 x^2\right )^{5/2} (a+b \text {ArcSin}(c x)) \sec ^2\left (\frac {\pi }{4}+\frac {1}{2} \text {ArcSin}(c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {16 b^2 d^5 \left (1-c^2 x^2\right )^{5/2} \tan \left (\frac {\pi }{4}+\frac {1}{2} \text {ArcSin}(c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {28 d^5 \left (1-c^2 x^2\right )^{5/2} (a+b \text {ArcSin}(c x))^2 \tan \left (\frac {\pi }{4}+\frac {1}{2} \text {ArcSin}(c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {4 d^5 \left (1-c^2 x^2\right )^{5/2} (a+b \text {ArcSin}(c x))^2 \sec ^2\left (\frac {\pi }{4}+\frac {1}{2} \text {ArcSin}(c x)\right ) \tan \left (\frac {\pi }{4}+\frac {1}{2} \text {ArcSin}(c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}} \]

[Out]

2*a*b*d^5*x*(-c^2*x^2+1)^(5/2)/(c*d*x+d)^(5/2)/(-c*e*x+e)^(5/2)+2*b^2*d^5*(-c^2*x^2+1)^3/c/(c*d*x+d)^(5/2)/(-c
*e*x+e)^(5/2)+2*b^2*d^5*x*(-c^2*x^2+1)^(5/2)*arcsin(c*x)/(c*d*x+d)^(5/2)/(-c*e*x+e)^(5/2)-28/3*I*d^5*(-c^2*x^2
+1)^(5/2)*(a+b*arcsin(c*x))^2/c/(c*d*x+d)^(5/2)/(-c*e*x+e)^(5/2)-d^5*(-c^2*x^2+1)^3*(a+b*arcsin(c*x))^2/c/(c*d
*x+d)^(5/2)/(-c*e*x+e)^(5/2)+5/3*d^5*(-c^2*x^2+1)^(5/2)*(a+b*arcsin(c*x))^3/b/c/(c*d*x+d)^(5/2)/(-c*e*x+e)^(5/
2)-112/3*b*d^5*(-c^2*x^2+1)^(5/2)*(a+b*arcsin(c*x))*ln(1-I/(I*c*x+(-c^2*x^2+1)^(1/2)))/c/(c*d*x+d)^(5/2)/(-c*e
*x+e)^(5/2)-112/3*I*b^2*d^5*(-c^2*x^2+1)^(5/2)*polylog(2,I/(I*c*x+(-c^2*x^2+1)^(1/2)))/c/(c*d*x+d)^(5/2)/(-c*e
*x+e)^(5/2)-8/3*b*d^5*(-c^2*x^2+1)^(5/2)*(a+b*arcsin(c*x))*sec(1/4*Pi+1/2*arcsin(c*x))^2/c/(c*d*x+d)^(5/2)/(-c
*e*x+e)^(5/2)+16/3*b^2*d^5*(-c^2*x^2+1)^(5/2)*tan(1/4*Pi+1/2*arcsin(c*x))/c/(c*d*x+d)^(5/2)/(-c*e*x+e)^(5/2)-2
8/3*d^5*(-c^2*x^2+1)^(5/2)*(a+b*arcsin(c*x))^2*tan(1/4*Pi+1/2*arcsin(c*x))/c/(c*d*x+d)^(5/2)/(-c*e*x+e)^(5/2)+
4/3*d^5*(-c^2*x^2+1)^(5/2)*(a+b*arcsin(c*x))^2*sec(1/4*Pi+1/2*arcsin(c*x))^2*tan(1/4*Pi+1/2*arcsin(c*x))/c/(c*
d*x+d)^(5/2)/(-c*e*x+e)^(5/2)

________________________________________________________________________________________

Rubi [A]
time = 0.91, antiderivative size = 730, normalized size of antiderivative = 1.00, number of steps used = 25, number of rules used = 16, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4763, 4859, 4737, 4767, 4715, 267, 4857, 3399, 4271, 3852, 8, 4269, 3798, 2221, 2317, 2438} \begin {gather*} \frac {5 d^5 \left (1-c^2 x^2\right )^{5/2} (a+b \text {ArcSin}(c x))^3}{3 b c (c d x+d)^{5/2} (e-c e x)^{5/2}}-\frac {d^5 \left (1-c^2 x^2\right )^3 (a+b \text {ArcSin}(c x))^2}{c (c d x+d)^{5/2} (e-c e x)^{5/2}}-\frac {28 i d^5 \left (1-c^2 x^2\right )^{5/2} (a+b \text {ArcSin}(c x))^2}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}-\frac {112 b d^5 \left (1-c^2 x^2\right )^{5/2} \log \left (1-i e^{-i \text {ArcSin}(c x)}\right ) (a+b \text {ArcSin}(c x))}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}-\frac {28 d^5 \left (1-c^2 x^2\right )^{5/2} \tan \left (\frac {1}{2} \text {ArcSin}(c x)+\frac {\pi }{4}\right ) (a+b \text {ArcSin}(c x))^2}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}-\frac {8 b d^5 \left (1-c^2 x^2\right )^{5/2} \sec ^2\left (\frac {1}{2} \text {ArcSin}(c x)+\frac {\pi }{4}\right ) (a+b \text {ArcSin}(c x))}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}+\frac {4 d^5 \left (1-c^2 x^2\right )^{5/2} \tan \left (\frac {1}{2} \text {ArcSin}(c x)+\frac {\pi }{4}\right ) \sec ^2\left (\frac {1}{2} \text {ArcSin}(c x)+\frac {\pi }{4}\right ) (a+b \text {ArcSin}(c x))^2}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}+\frac {2 a b d^5 x \left (1-c^2 x^2\right )^{5/2}}{(c d x+d)^{5/2} (e-c e x)^{5/2}}-\frac {112 i b^2 d^5 \left (1-c^2 x^2\right )^{5/2} \text {Li}_2\left (i e^{-i \text {ArcSin}(c x)}\right )}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}+\frac {2 b^2 d^5 x \left (1-c^2 x^2\right )^{5/2} \text {ArcSin}(c x)}{(c d x+d)^{5/2} (e-c e x)^{5/2}}+\frac {16 b^2 d^5 \left (1-c^2 x^2\right )^{5/2} \tan \left (\frac {1}{2} \text {ArcSin}(c x)+\frac {\pi }{4}\right )}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}+\frac {2 b^2 d^5 \left (1-c^2 x^2\right )^3}{c (c d x+d)^{5/2} (e-c e x)^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*a*b*d^5*x*(1 - c^2*x^2)^(5/2))/((d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) + (2*b^2*d^5*(1 - c^2*x^2)^3)/(c*(d +
c*d*x)^(5/2)*(e - c*e*x)^(5/2)) + (2*b^2*d^5*x*(1 - c^2*x^2)^(5/2)*ArcSin[c*x])/((d + c*d*x)^(5/2)*(e - c*e*x)
^(5/2)) - (((28*I)/3)*d^5*(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)) -
(d^5*(1 - c^2*x^2)^3*(a + b*ArcSin[c*x])^2)/(c*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) + (5*d^5*(1 - c^2*x^2)^(5
/2)*(a + b*ArcSin[c*x])^3)/(3*b*c*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) - (112*b*d^5*(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)) - (((112*I)/3)*b^2*d^5*(
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)) - (8*b*d^5*(1 - c^
2*x^2)^(5/2)*(a + b*ArcSin[c*x])*Sec[Pi/4 + ArcSin[c*x]/2]^2)/(3*c*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) + (16*
b^2*d^5*(1 - c^2*x^2)^(5/2)*Tan[Pi/4 + ArcSin[c*x]/2])/(3*c*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) - (28*d^5*(1
- c^2*x^2)^(5/2)*(a + b*ArcSin[c*x])^2*Tan[Pi/4 + ArcSin[c*x]/2])/(3*c*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) +
(4*d^5*(1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x])^2*Sec[Pi/4 + ArcSin[c*x]/2]^2*Tan[Pi/4 + ArcSin[c*x]/2])/(3*c*(
d + c*d*x)^(5/2)*(e - c*e*x)^(5/2))

Rule 8

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

Rule 267

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 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 3399

Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[(2*a)^n, Int[(c
+ d*x)^m*Sin[(1/2)*(e + Pi*(a/(2*b))) + 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 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 3852

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 4269

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 4271

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]) /; Free
Q[{b, c, d, e, f}, x] && GtQ[n, 1] && NeQ[n, 2] && GtQ[m, 1]

Rule 4715

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 4737

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(1/(b*c*(n + 1)))*Si
mp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]]*(a + b*ArcSin[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c
^2*d + e, 0] && NeQ[n, -1]

Rule 4763

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 4767

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

Rule 4857

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 4859

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]

Rubi steps

\begin {align*} \int \frac {(d+c d x)^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2}{(e-c e x)^{5/2}} \, dx &=\frac {\left (1-c^2 x^2\right )^{5/2} \int \frac {(d+c d x)^5 \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 {5 d^5 \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt {1-c^2 x^2}}+\frac {c d^5 x \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt {1-c^2 x^2}}+\frac {8 d^5 \left (a+b \sin ^{-1}(c x)\right )^2}{(-1+c x)^2 \sqrt {1-c^2 x^2}}+\frac {12 d^5 \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 (5 d^5 \left (1-c^2 x^2\right )^{5/2}\right ) \int \frac {\left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt {1-c^2 x^2}} \, dx}{(d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {\left (8 d^5 \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 (12 d^5 \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 (c d^5 \left (1-c^2 x^2\right )^{5/2}\right ) \int \frac {x \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt {1-c^2 x^2}} \, dx}{(d+c d x)^{5/2} (e-c e x)^{5/2}}\\ &=-\frac {d^5 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2}{c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {5 d^5 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {\left (12 d^5 \left (1-c^2 x^2\right )^{5/2}\right ) \text {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 b d^5 \left (1-c^2 x^2\right )^{5/2}\right ) \int \left (a+b \sin ^{-1}(c x)\right ) \, dx}{(d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {\left (8 c d^5 \left (1-c^2 x^2\right )^{5/2}\right ) \text {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 {2 a b d^5 x \left (1-c^2 x^2\right )^{5/2}}{(d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {d^5 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2}{c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {5 d^5 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {\left (2 b^2 d^5 \left (1-c^2 x^2\right )^{5/2}\right ) \int \sin ^{-1}(c x) \, dx}{(d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {\left (2 d^5 \left (1-c^2 x^2\right )^{5/2}\right ) \text {Subst}\left (\int (a+b x)^2 \csc ^4\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 (6 d^5 \left (1-c^2 x^2\right )^{5/2}\right ) \text {Subst}\left (\int (a+b x)^2 \csc ^2\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 {2 a b d^5 x \left (1-c^2 x^2\right )^{5/2}}{(d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {2 b^2 d^5 x \left (1-c^2 x^2\right )^{5/2} \sin ^{-1}(c x)}{(d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {d^5 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2}{c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {5 d^5 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {8 b d^5 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right ) \sec ^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 {12 d^5 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2 \tan \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 {4 d^5 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2 \sec ^2\left (\frac {\pi }{4}+\frac {1}{2} \sin ^{-1}(c x)\right ) \tan \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 d^5 \left (1-c^2 x^2\right )^{5/2}\right ) \text {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 (24 b d^5 \left (1-c^2 x^2\right )^{5/2}\right ) \text {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 (8 b^2 d^5 \left (1-c^2 x^2\right )^{5/2}\right ) \text {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 {\left (2 b^2 c d^5 \left (1-c^2 x^2\right )^{5/2}\right ) \int \frac {x}{\sqrt {1-c^2 x^2}} \, dx}{(d+c d x)^{5/2} (e-c e x)^{5/2}}\\ &=\frac {2 a b d^5 x \left (1-c^2 x^2\right )^{5/2}}{(d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {2 b^2 d^5 \left (1-c^2 x^2\right )^3}{c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {2 b^2 d^5 x \left (1-c^2 x^2\right )^{5/2} \sin ^{-1}(c x)}{(d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {12 i d^5 \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 {d^5 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2}{c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {5 d^5 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {8 b d^5 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right ) \sec ^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 {28 d^5 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2 \tan \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 d^5 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2 \sec ^2\left (\frac {\pi }{4}+\frac {1}{2} \sin ^{-1}(c x)\right ) \tan \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 (16 b d^5 \left (1-c^2 x^2\right )^{5/2}\right ) \text {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 (48 b d^5 \left (1-c^2 x^2\right )^{5/2}\right ) \text {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 (16 b^2 d^5 \left (1-c^2 x^2\right )^{5/2}\right ) \text {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 {2 a b d^5 x \left (1-c^2 x^2\right )^{5/2}}{(d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {2 b^2 d^5 \left (1-c^2 x^2\right )^3}{c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {2 b^2 d^5 x \left (1-c^2 x^2\right )^{5/2} \sin ^{-1}(c x)}{(d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {28 i d^5 \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 {d^5 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2}{c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {5 d^5 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {16 b^2 d^5 \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 {48 b d^5 \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 {8 b d^5 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right ) \sec ^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 {28 d^5 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2 \tan \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 d^5 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2 \sec ^2\left (\frac {\pi }{4}+\frac {1}{2} \sin ^{-1}(c x)\right ) \tan \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 (32 b d^5 \left (1-c^2 x^2\right )^{5/2}\right ) \text {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 (48 b^2 d^5 \left (1-c^2 x^2\right )^{5/2}\right ) \text {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 {2 a b d^5 x \left (1-c^2 x^2\right )^{5/2}}{(d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {2 b^2 d^5 \left (1-c^2 x^2\right )^3}{c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {2 b^2 d^5 x \left (1-c^2 x^2\right )^{5/2} \sin ^{-1}(c x)}{(d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {28 i d^5 \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 {d^5 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2}{c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {5 d^5 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {16 b^2 d^5 \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 {112 b d^5 \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 {8 b d^5 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right ) \sec ^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 {28 d^5 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2 \tan \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 d^5 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2 \sec ^2\left (\frac {\pi }{4}+\frac {1}{2} \sin ^{-1}(c x)\right ) \tan \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 (48 i b^2 d^5 \left (1-c^2 x^2\right )^{5/2}\right ) \text {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 (32 b^2 d^5 \left (1-c^2 x^2\right )^{5/2}\right ) \text {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 {2 a b d^5 x \left (1-c^2 x^2\right )^{5/2}}{(d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {2 b^2 d^5 \left (1-c^2 x^2\right )^3}{c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {2 b^2 d^5 x \left (1-c^2 x^2\right )^{5/2} \sin ^{-1}(c x)}{(d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {28 i d^5 \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 {d^5 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2}{c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {5 d^5 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {16 b^2 d^5 \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 {112 b d^5 \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 {48 i b^2 d^5 \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 {8 b d^5 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right ) \sec ^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 {28 d^5 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2 \tan \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 d^5 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2 \sec ^2\left (\frac {\pi }{4}+\frac {1}{2} \sin ^{-1}(c x)\right ) \tan \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 (32 i b^2 d^5 \left (1-c^2 x^2\right )^{5/2}\right ) \text {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 {2 a b d^5 x \left (1-c^2 x^2\right )^{5/2}}{(d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {2 b^2 d^5 \left (1-c^2 x^2\right )^3}{c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {2 b^2 d^5 x \left (1-c^2 x^2\right )^{5/2} \sin ^{-1}(c x)}{(d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {28 i d^5 \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 {d^5 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2}{c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {5 d^5 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {16 b^2 d^5 \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 {112 b d^5 \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 {112 i b^2 d^5 \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}}-\frac {8 b d^5 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right ) \sec ^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 {28 d^5 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2 \tan \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 d^5 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2 \sec ^2\left (\frac {\pi }{4}+\frac {1}{2} \sin ^{-1}(c x)\right ) \tan \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}}\\ \end {align*}

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Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(2312\) vs. \(2(730)=1460\).
time = 10.44, size = 2312, normalized size = 3.17 \begin {gather*} \text {Result too large to show} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(Sqrt[-(e*(-1 + c*x))]*Sqrt[d*(1 + c*x)]*(-((a^2*d^2)/e^3) + (8*a^2*d^2)/(3*e^3*(-1 + c*x)^2) + (28*a^2*d^2)/(
3*e^3*(-1 + c*x))))/c - (5*a^2*d^(5/2)*ArcTan[(c*x*Sqrt[-(e*(-1 + c*x))]*Sqrt[d*(1 + c*x)])/(Sqrt[d]*Sqrt[e]*(
-1 + c*x)*(1 + c*x))])/(c*e^(5/2)) + (a*b*d^2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*Sqrt[-(d*e*(1 - c^2*x^2))]*(Cos[
ArcSin[c*x]/2]*(-4 + 3*ArcSin[c*x] - 6*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]]) - Cos[(3*ArcSin[c*x])/2]*
(ArcSin[c*x] - 2*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]]) + 2*(2 + 2*ArcSin[c*x] + Sqrt[1 - c^2*x^2]*ArcS
in[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[Ar
cSin[c*x]/2]])*Sin[ArcSin[c*x]/2]))/(3*c*e^3*Sqrt[(-d - c*d*x)*(e - c*e*x)]*(Cos[ArcSin[c*x]/2] - Sin[ArcSin[c
*x]/2])^4*(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2])) + (a*b*d^2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*Sqrt[-(d*e*(1
- c^2*x^2))]*(Cos[ArcSin[c*x]/2]*(-8 - 6*ArcSin[c*x] + 9*ArcSin[c*x]^2 - 84*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSi
n[c*x]/2]]) + Cos[(3*ArcSin[c*x])/2]*(-(ArcSin[c*x]*(14 + 3*ArcSin[c*x])) + 28*Log[Cos[ArcSin[c*x]/2] - Sin[Ar
cSin[c*x]/2]]) + 2*(4 + 4*ArcSin[c*x] - 6*ArcSin[c*x]^2 + 56*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]] + Sq
rt[1 - c^2*x^2]*((14 - 3*ArcSin[c*x])*ArcSin[c*x] + 28*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]]))*Sin[ArcS
in[c*x]/2]))/(3*c*e^3*Sqrt[(-d - c*d*x)*(e - c*e*x)]*(Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2])^4*(Cos[ArcSin[c
*x]/2] + Sin[ArcSin[c*x]/2])) + (b^2*d^2*(1 + c*x)*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*Sqrt[-(d*e*(1 - c^2*x^2))]*
((-3*I)*Pi*ArcSin[c*x] + (4*ArcSin[c*x])/(-1 + c*x) - (1 - I)*ArcSin[c*x]^2 - (2*ArcSin[c*x]^2)/(-1 + c*x) - 4
*Pi*Log[1 + E^((-I)*ArcSin[c*x])] + 2*Pi*Log[1 + I*E^(I*ArcSin[c*x])] - 4*ArcSin[c*x]*Log[1 + I*E^(I*ArcSin[c*
x])] + 4*Pi*Log[Cos[ArcSin[c*x]/2]] - 2*Pi*Log[-Cos[(Pi + 2*ArcSin[c*x])/4]] + (4*I)*PolyLog[2, (-I)*E^(I*ArcS
in[c*x])] + (2*(4 + ArcSin[c*x]^2 + c*x*(-4 + ArcSin[c*x]^2))*Sin[ArcSin[c*x]/2])/(Cos[ArcSin[c*x]/2] - Sin[Ar
cSin[c*x]/2])^3))/(3*c*e^3*Sqrt[(-d - c*d*x)*(e - c*e*x)]*Sqrt[1 - c^2*x^2]*(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c
*x]/2])^2) + (b^2*d^2*(1 + c*x)*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*Sqrt[-(d*e*(1 - c^2*x^2))]*(6 + (6*c*x*ArcSin[
c*x])/Sqrt[1 - c^2*x^2] - (2*(-2 + ArcSin[c*x])*ArcSin[c*x])/((-1 + c*x)*Sqrt[1 - c^2*x^2]) - 3*ArcSin[c*x]^2
- ((13 - 13*I)*ArcSin[c*x]^2)/Sqrt[1 - c^2*x^2] + (3*ArcSin[c*x]^3)/Sqrt[1 - c^2*x^2] + (13*((-3*I)*Pi*ArcSin[
c*x] - 4*Pi*Log[1 + E^((-I)*ArcSin[c*x])] + 2*(Pi - 2*ArcSin[c*x])*Log[1 + I*E^(I*ArcSin[c*x])] + 4*Pi*Log[Cos
[ArcSin[c*x]/2]] - 2*Pi*Log[-Cos[(Pi + 2*ArcSin[c*x])/4]] + (4*I)*PolyLog[2, (-I)*E^(I*ArcSin[c*x])]))/Sqrt[1
- c^2*x^2] + (4*ArcSin[c*x]^2*Sin[ArcSin[c*x]/2])/(Sqrt[1 - c^2*x^2]*(Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2])
^3) + (2*(4 - 13*ArcSin[c*x]^2)*Sin[ArcSin[c*x]/2])/(Sqrt[1 - c^2*x^2]*(Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2
]))))/(3*c*e^3*Sqrt[(-d - c*d*x)*(e - c*e*x)]*(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2])^2) + (2*b^2*d^2*(1 + c
*x)*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*Sqrt[-(d*e*(1 - c^2*x^2))]*((-21*I)*Pi*ArcSin[c*x] - (2*(-2 + ArcSin[c*x])
*ArcSin[c*x])/(-1 + c*x) - (7 - 7*I)*ArcSin[c*x]^2 + ArcSin[c*x]^3 - 28*Pi*Log[1 + E^((-I)*ArcSin[c*x])] + 14*
(Pi - 2*ArcSin[c*x])*Log[1 + I*E^(I*ArcSin[c*x])] + 28*Pi*Log[Cos[ArcSin[c*x]/2]] - 14*Pi*Log[-Cos[(Pi + 2*Arc
Sin[c*x])/4]] + (28*I)*PolyLog[2, (-I)*E^(I*ArcSin[c*x])] + (4*ArcSin[c*x]^2*Sin[ArcSin[c*x]/2])/(Cos[ArcSin[c
*x]/2] - Sin[ArcSin[c*x]/2])^3 + (2*(4 - 7*ArcSin[c*x]^2)*Sin[ArcSin[c*x]/2])/(Cos[ArcSin[c*x]/2] - Sin[ArcSin
[c*x]/2])))/(3*c*e^3*Sqrt[(-d - c*d*x)*(e - c*e*x)]*Sqrt[1 - c^2*x^2]*(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]
)^2) + (a*b*d^2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*Sqrt[-(d*e*(1 - c^2*x^2))]*(3*Cos[(5*ArcSin[c*x])/2] + 3*ArcSi
n[c*x]*Cos[(5*ArcSin[c*x])/2] + Cos[ArcSin[c*x]/2]*(-20 - 24*ArcSin[c*x] + 27*ArcSin[c*x]^2 - 156*Log[Cos[ArcS
in[c*x]/2] - Sin[ArcSin[c*x]/2]]) + Cos[(3*ArcSin[c*x])/2]*(9 - 35*ArcSin[c*x] - 9*ArcSin[c*x]^2 + 52*Log[Cos[
ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]]) + 20*Sin[ArcSin[c*x]/2] - 24*ArcSin[c*x]*Sin[ArcSin[c*x]/2] - 27*ArcSin[
c*x]^2*Sin[ArcSin[c*x]/2] + 156*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]]*Sin[ArcSin[c*x]/2] + 9*Sin[(3*Arc
Sin[c*x])/2] + 35*ArcSin[c*x]*Sin[(3*ArcSin[c*x])/2] - 9*ArcSin[c*x]^2*Sin[(3*ArcSin[c*x])/2] + 52*Log[Cos[Arc
Sin[c*x]/2] - Sin[ArcSin[c*x]/2]]*Sin[(3*ArcSin[c*x])/2] - 3*Sin[(5*ArcSin[c*x])/2] + 3*ArcSin[c*x]*Sin[(5*Arc
Sin[c*x])/2]))/(6*c*e^3*Sqrt[(-d - c*d*x)*(e - c*e*x)]*(Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2])^4*(Cos[ArcSin
[c*x]/2] + Sin[ArcSin[c*x]/2]))

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Maple [F]
time = 0.25, size = 0, normalized size = 0.00 \[\int \frac {\left (c d x +d \right )^{\frac {5}{2}} \left (a +b \arcsin \left (c x \right )\right )^{2}}{\left (-c e x +e \right )^{\frac {5}{2}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(15*d^(5/2)*arcsin(c*x)*e^(-5/2)/c - 3*(-c^2*d*x^2*e + d*e)^(5/2)/(c^5*x^4*e^5 - 4*c^4*x^3*e^5 + 6*c^3*x^2
*e^5 - 4*c^2*x*e^5 + c*e^5) - 5*(-c^2*d*x^2*e + d*e)^(3/2)*d/(c^4*x^3*e^4 - 3*c^3*x^2*e^4 + 3*c^2*x*e^4 - c*e^
4) + 10*sqrt(-c^2*d*x^2*e + d*e)*d^2/(c^3*x^2*e^3 - 2*c^2*x*e^3 + c*e^3) + 35*sqrt(-c^2*d*x^2*e + d*e)*d^2/(c^
2*x*e^3 - c*e^3))*a^2 - sqrt(d)*e^(1/2)*integrate(((b^2*c^2*d^2*x^2 + 2*b^2*c*d^2*x + b^2*d^2)*arctan2(c*x, sq
rt(c*x + 1)*sqrt(-c*x + 1))^2 + 2*(a*b*c^2*d^2*x^2 + 2*a*b*c*d^2*x + a*b*d^2)*arctan2(c*x, sqrt(c*x + 1)*sqrt(
-c*x + 1)))*sqrt(c*x + 1)*sqrt(-c*x + 1)/(c^3*x^3*e^3 - 3*c^2*x^2*e^3 + 3*c*x*e^3 - e^3), x)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: SystemError >> excessive stack use: stack is 6189 deep

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,{\left (d+c\,d\,x\right )}^{5/2}}{{\left (e-c\,e\,x\right )}^{5/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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