3.6.64 \(\int \frac {(d+c d x)^{5/2} (a+b \text {ArcSin}(c x))^2}{(e-c e x)^{3/2}} \, dx\) [564]

Optimal. Leaf size=918 \[ -\frac {8 a b d^4 x \left (1-c^2 x^2\right )^{3/2}}{(d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {8 b^2 d^4 \left (1-c^2 x^2\right )^2}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {b^2 d^4 x \left (1-c^2 x^2\right )^2}{4 (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {b^2 d^4 \left (1-c^2 x^2\right )^{3/2} \text {ArcSin}(c x)}{4 c (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {8 b^2 d^4 x \left (1-c^2 x^2\right )^{3/2} \text {ArcSin}(c x)}{(d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {b c d^4 x^2 \left (1-c^2 x^2\right )^{3/2} (a+b \text {ArcSin}(c x))}{2 (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {8 d^4 \left (1-c^2 x^2\right ) (a+b \text {ArcSin}(c x))^2}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {8 d^4 x \left (1-c^2 x^2\right ) (a+b \text {ArcSin}(c x))^2}{(d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {8 i d^4 \left (1-c^2 x^2\right )^{3/2} (a+b \text {ArcSin}(c x))^2}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {4 d^4 \left (1-c^2 x^2\right )^2 (a+b \text {ArcSin}(c x))^2}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {d^4 x \left (1-c^2 x^2\right )^2 (a+b \text {ArcSin}(c x))^2}{2 (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {5 d^4 \left (1-c^2 x^2\right )^{3/2} (a+b \text {ArcSin}(c x))^3}{2 b c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {32 i b d^4 \left (1-c^2 x^2\right )^{3/2} (a+b \text {ArcSin}(c x)) \text {ArcTan}\left (e^{i \text {ArcSin}(c x)}\right )}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {16 b d^4 \left (1-c^2 x^2\right )^{3/2} (a+b \text {ArcSin}(c x)) \log \left (1+e^{2 i \text {ArcSin}(c x)}\right )}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {16 i b^2 d^4 \left (1-c^2 x^2\right )^{3/2} \text {PolyLog}\left (2,-i e^{i \text {ArcSin}(c x)}\right )}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {16 i b^2 d^4 \left (1-c^2 x^2\right )^{3/2} \text {PolyLog}\left (2,i e^{i \text {ArcSin}(c x)}\right )}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {8 i b^2 d^4 \left (1-c^2 x^2\right )^{3/2} \text {PolyLog}\left (2,-e^{2 i \text {ArcSin}(c x)}\right )}{c (d+c d x)^{3/2} (e-c e x)^{3/2}} \]

[Out]

-8*a*b*d^4*x*(-c^2*x^2+1)^(3/2)/(c*d*x+d)^(3/2)/(-c*e*x+e)^(3/2)-8*b^2*d^4*(-c^2*x^2+1)^2/c/(c*d*x+d)^(3/2)/(-
c*e*x+e)^(3/2)-1/4*b^2*d^4*x*(-c^2*x^2+1)^2/(c*d*x+d)^(3/2)/(-c*e*x+e)^(3/2)+1/4*b^2*d^4*(-c^2*x^2+1)^(3/2)*ar
csin(c*x)/c/(c*d*x+d)^(3/2)/(-c*e*x+e)^(3/2)-8*b^2*d^4*x*(-c^2*x^2+1)^(3/2)*arcsin(c*x)/(c*d*x+d)^(3/2)/(-c*e*
x+e)^(3/2)-1/2*b*c*d^4*x^2*(-c^2*x^2+1)^(3/2)*(a+b*arcsin(c*x))/(c*d*x+d)^(3/2)/(-c*e*x+e)^(3/2)+8*d^4*(-c^2*x
^2+1)*(a+b*arcsin(c*x))^2/c/(c*d*x+d)^(3/2)/(-c*e*x+e)^(3/2)+8*d^4*x*(-c^2*x^2+1)*(a+b*arcsin(c*x))^2/(c*d*x+d
)^(3/2)/(-c*e*x+e)^(3/2)-8*I*b^2*d^4*(-c^2*x^2+1)^(3/2)*polylog(2,-(I*c*x+(-c^2*x^2+1)^(1/2))^2)/c/(c*d*x+d)^(
3/2)/(-c*e*x+e)^(3/2)+4*d^4*(-c^2*x^2+1)^2*(a+b*arcsin(c*x))^2/c/(c*d*x+d)^(3/2)/(-c*e*x+e)^(3/2)+1/2*d^4*x*(-
c^2*x^2+1)^2*(a+b*arcsin(c*x))^2/(c*d*x+d)^(3/2)/(-c*e*x+e)^(3/2)-5/2*d^4*(-c^2*x^2+1)^(3/2)*(a+b*arcsin(c*x))
^3/b/c/(c*d*x+d)^(3/2)/(-c*e*x+e)^(3/2)-16*I*b^2*d^4*(-c^2*x^2+1)^(3/2)*polylog(2,-I*(I*c*x+(-c^2*x^2+1)^(1/2)
))/c/(c*d*x+d)^(3/2)/(-c*e*x+e)^(3/2)+16*b*d^4*(-c^2*x^2+1)^(3/2)*(a+b*arcsin(c*x))*ln(1+(I*c*x+(-c^2*x^2+1)^(
1/2))^2)/c/(c*d*x+d)^(3/2)/(-c*e*x+e)^(3/2)+16*I*b^2*d^4*(-c^2*x^2+1)^(3/2)*polylog(2,I*(I*c*x+(-c^2*x^2+1)^(1
/2)))/c/(c*d*x+d)^(3/2)/(-c*e*x+e)^(3/2)-8*I*d^4*(-c^2*x^2+1)^(3/2)*(a+b*arcsin(c*x))^2/c/(c*d*x+d)^(3/2)/(-c*
e*x+e)^(3/2)+32*I*b*d^4*(-c^2*x^2+1)^(3/2)*(a+b*arcsin(c*x))*arctan(I*c*x+(-c^2*x^2+1)^(1/2))/c/(c*d*x+d)^(3/2
)/(-c*e*x+e)^(3/2)

________________________________________________________________________________________

Rubi [A]
time = 0.87, antiderivative size = 918, normalized size of antiderivative = 1.00, number of steps used = 28, number of rules used = 19, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.594, Rules used = {4763, 4859, 4847, 4745, 4765, 3800, 2221, 2317, 2438, 4767, 4749, 4266, 4737, 4715, 267, 4795, 4723, 327, 222} \begin {gather*} -\frac {5 \left (1-c^2 x^2\right )^{3/2} (a+b \text {ArcSin}(c x))^3 d^4}{2 b c (c x d+d)^{3/2} (e-c e x)^{3/2}}-\frac {b^2 x \left (1-c^2 x^2\right )^2 d^4}{4 (c x d+d)^{3/2} (e-c e x)^{3/2}}-\frac {8 b^2 \left (1-c^2 x^2\right )^2 d^4}{c (c x d+d)^{3/2} (e-c e x)^{3/2}}+\frac {x \left (1-c^2 x^2\right )^2 (a+b \text {ArcSin}(c x))^2 d^4}{2 (c x d+d)^{3/2} (e-c e x)^{3/2}}+\frac {4 \left (1-c^2 x^2\right )^2 (a+b \text {ArcSin}(c x))^2 d^4}{c (c x d+d)^{3/2} (e-c e x)^{3/2}}-\frac {8 i \left (1-c^2 x^2\right )^{3/2} (a+b \text {ArcSin}(c x))^2 d^4}{c (c x d+d)^{3/2} (e-c e x)^{3/2}}+\frac {8 x \left (1-c^2 x^2\right ) (a+b \text {ArcSin}(c x))^2 d^4}{(c x d+d)^{3/2} (e-c e x)^{3/2}}+\frac {8 \left (1-c^2 x^2\right ) (a+b \text {ArcSin}(c x))^2 d^4}{c (c x d+d)^{3/2} (e-c e x)^{3/2}}-\frac {8 a b x \left (1-c^2 x^2\right )^{3/2} d^4}{(c x d+d)^{3/2} (e-c e x)^{3/2}}-\frac {8 b^2 x \left (1-c^2 x^2\right )^{3/2} \text {ArcSin}(c x) d^4}{(c x d+d)^{3/2} (e-c e x)^{3/2}}+\frac {b^2 \left (1-c^2 x^2\right )^{3/2} \text {ArcSin}(c x) d^4}{4 c (c x d+d)^{3/2} (e-c e x)^{3/2}}-\frac {b c x^2 \left (1-c^2 x^2\right )^{3/2} (a+b \text {ArcSin}(c x)) d^4}{2 (c x d+d)^{3/2} (e-c e x)^{3/2}}+\frac {32 i b \left (1-c^2 x^2\right )^{3/2} (a+b \text {ArcSin}(c x)) \text {ArcTan}\left (e^{i \text {ArcSin}(c x)}\right ) d^4}{c (c x d+d)^{3/2} (e-c e x)^{3/2}}+\frac {16 b \left (1-c^2 x^2\right )^{3/2} (a+b \text {ArcSin}(c x)) \log \left (1+e^{2 i \text {ArcSin}(c x)}\right ) d^4}{c (c x d+d)^{3/2} (e-c e x)^{3/2}}-\frac {16 i b^2 \left (1-c^2 x^2\right )^{3/2} \text {Li}_2\left (-i e^{i \text {ArcSin}(c x)}\right ) d^4}{c (c x d+d)^{3/2} (e-c e x)^{3/2}}+\frac {16 i b^2 \left (1-c^2 x^2\right )^{3/2} \text {Li}_2\left (i e^{i \text {ArcSin}(c x)}\right ) d^4}{c (c x d+d)^{3/2} (e-c e x)^{3/2}}-\frac {8 i b^2 \left (1-c^2 x^2\right )^{3/2} \text {Li}_2\left (-e^{2 i \text {ArcSin}(c x)}\right ) d^4}{c (c x d+d)^{3/2} (e-c e x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-8*a*b*d^4*x*(1 - c^2*x^2)^(3/2))/((d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)) - (8*b^2*d^4*(1 - c^2*x^2)^2)/(c*(d +
c*d*x)^(3/2)*(e - c*e*x)^(3/2)) - (b^2*d^4*x*(1 - c^2*x^2)^2)/(4*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)) + (b^2*
d^4*(1 - c^2*x^2)^(3/2)*ArcSin[c*x])/(4*c*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)) - (8*b^2*d^4*x*(1 - c^2*x^2)^(3
/2)*ArcSin[c*x])/((d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)) - (b*c*d^4*x^2*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x]))
/(2*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)) + (8*d^4*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/(c*(d + c*d*x)^(3/2)*(e
- c*e*x)^(3/2)) + (8*d^4*x*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/((d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)) - ((8*I
)*d^4*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x])^2)/(c*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)) + (4*d^4*(1 - c^2*x^2
)^2*(a + b*ArcSin[c*x])^2)/(c*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)) + (d^4*x*(1 - c^2*x^2)^2*(a + b*ArcSin[c*x]
)^2)/(2*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)) - (5*d^4*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x])^3)/(2*b*c*(d + c
*d*x)^(3/2)*(e - c*e*x)^(3/2)) + ((32*I)*b*d^4*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x])*ArcTan[E^(I*ArcSin[c*x]
)])/(c*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)) + (16*b*d^4*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x])*Log[1 + E^((2*
I)*ArcSin[c*x])])/(c*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)) - ((16*I)*b^2*d^4*(1 - c^2*x^2)^(3/2)*PolyLog[2, (-I
)*E^(I*ArcSin[c*x])])/(c*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)) + ((16*I)*b^2*d^4*(1 - c^2*x^2)^(3/2)*PolyLog[2,
I*E^(I*ArcSin[c*x])])/(c*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)) - ((8*I)*b^2*d^4*(1 - c^2*x^2)^(3/2)*PolyLog[2,
-E^((2*I)*ArcSin[c*x])])/(c*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2))

Rule 222

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt[a])]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

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 327

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

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 3800

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (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*(e + f*x))/(1 + E^(2*I*(e + f*x)))), x], x] /; FreeQ[{c, d, e, f}, x] &&
IGtQ[m, 0]

Rule 4266

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

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 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 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 4745

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[x*((a + b*ArcSin[c
*x])^n/(d*Sqrt[d + e*x^2])), x] - Dist[b*c*(n/d)*Simp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]], Int[x*((a + b*ArcSin
[c*x])^(n - 1)/(1 - c^2*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0]

Rule 4749

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

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 4765

Int[(((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*(x_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Dist[-e^(-1), Subst[In
t[(a + b*x)^n*Tan[x], x], x, ArcSin[c*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 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 4795

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

Rule 4847

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

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)^{3/2}} \, dx &=\frac {\left (1-c^2 x^2\right )^{3/2} \int \frac {(d+c d x)^4 \left (a+b \sin ^{-1}(c x)\right )^2}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{(d+c d x)^{3/2} (e-c e x)^{3/2}}\\ &=\frac {\left (1-c^2 x^2\right )^{3/2} \int \left (\frac {8 \left (d^4+c d^4 x\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{\left (1-c^2 x^2\right )^{3/2}}-\frac {7 d^4 \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt {1-c^2 x^2}}-\frac {4 c d^4 x \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt {1-c^2 x^2}}-\frac {c^2 d^4 x^2 \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt {1-c^2 x^2}}\right ) \, dx}{(d+c d x)^{3/2} (e-c e x)^{3/2}}\\ &=\frac {\left (8 \left (1-c^2 x^2\right )^{3/2}\right ) \int \frac {\left (d^4+c d^4 x\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{(d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {\left (7 d^4 \left (1-c^2 x^2\right )^{3/2}\right ) \int \frac {\left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt {1-c^2 x^2}} \, dx}{(d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {\left (4 c d^4 \left (1-c^2 x^2\right )^{3/2}\right ) \int \frac {x \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt {1-c^2 x^2}} \, dx}{(d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {\left (c^2 d^4 \left (1-c^2 x^2\right )^{3/2}\right ) \int \frac {x^2 \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt {1-c^2 x^2}} \, dx}{(d+c d x)^{3/2} (e-c e x)^{3/2}}\\ &=\frac {4 d^4 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {d^4 x \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{2 (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {7 d^4 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {\left (8 \left (1-c^2 x^2\right )^{3/2}\right ) \int \left (\frac {d^4 \left (a+b \sin ^{-1}(c x)\right )^2}{\left (1-c^2 x^2\right )^{3/2}}+\frac {c d^4 x \left (a+b \sin ^{-1}(c x)\right )^2}{\left (1-c^2 x^2\right )^{3/2}}\right ) \, dx}{(d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {\left (d^4 \left (1-c^2 x^2\right )^{3/2}\right ) \int \frac {\left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt {1-c^2 x^2}} \, dx}{2 (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {\left (8 b d^4 \left (1-c^2 x^2\right )^{3/2}\right ) \int \left (a+b \sin ^{-1}(c x)\right ) \, dx}{(d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {\left (b c d^4 \left (1-c^2 x^2\right )^{3/2}\right ) \int x \left (a+b \sin ^{-1}(c x)\right ) \, dx}{(d+c d x)^{3/2} (e-c e x)^{3/2}}\\ &=-\frac {8 a b d^4 x \left (1-c^2 x^2\right )^{3/2}}{(d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {b c d^4 x^2 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{2 (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {4 d^4 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {d^4 x \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{2 (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {5 d^4 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^3}{2 b c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {\left (8 d^4 \left (1-c^2 x^2\right )^{3/2}\right ) \int \frac {\left (a+b \sin ^{-1}(c x)\right )^2}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{(d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {\left (8 b^2 d^4 \left (1-c^2 x^2\right )^{3/2}\right ) \int \sin ^{-1}(c x) \, dx}{(d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {\left (8 c d^4 \left (1-c^2 x^2\right )^{3/2}\right ) \int \frac {x \left (a+b \sin ^{-1}(c x)\right )^2}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{(d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {\left (b^2 c^2 d^4 \left (1-c^2 x^2\right )^{3/2}\right ) \int \frac {x^2}{\sqrt {1-c^2 x^2}} \, dx}{2 (d+c d x)^{3/2} (e-c e x)^{3/2}}\\ &=-\frac {8 a b d^4 x \left (1-c^2 x^2\right )^{3/2}}{(d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {b^2 d^4 x \left (1-c^2 x^2\right )^2}{4 (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {8 b^2 d^4 x \left (1-c^2 x^2\right )^{3/2} \sin ^{-1}(c x)}{(d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {b c d^4 x^2 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{2 (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {8 d^4 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {8 d^4 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{(d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {4 d^4 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {d^4 x \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{2 (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {5 d^4 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^3}{2 b c (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {\left (16 b d^4 \left (1-c^2 x^2\right )^{3/2}\right ) \int \frac {a+b \sin ^{-1}(c x)}{1-c^2 x^2} \, dx}{(d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {\left (b^2 d^4 \left (1-c^2 x^2\right )^{3/2}\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{4 (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {\left (16 b c d^4 \left (1-c^2 x^2\right )^{3/2}\right ) \int \frac {x \left (a+b \sin ^{-1}(c x)\right )}{1-c^2 x^2} \, dx}{(d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {\left (8 b^2 c d^4 \left (1-c^2 x^2\right )^{3/2}\right ) \int \frac {x}{\sqrt {1-c^2 x^2}} \, dx}{(d+c d x)^{3/2} (e-c e x)^{3/2}}\\ &=-\frac {8 a b d^4 x \left (1-c^2 x^2\right )^{3/2}}{(d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {8 b^2 d^4 \left (1-c^2 x^2\right )^2}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {b^2 d^4 x \left (1-c^2 x^2\right )^2}{4 (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {b^2 d^4 \left (1-c^2 x^2\right )^{3/2} \sin ^{-1}(c x)}{4 c (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {8 b^2 d^4 x \left (1-c^2 x^2\right )^{3/2} \sin ^{-1}(c x)}{(d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {b c d^4 x^2 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{2 (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {8 d^4 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {8 d^4 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{(d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {4 d^4 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {d^4 x \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{2 (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {5 d^4 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^3}{2 b c (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {\left (16 b d^4 \left (1-c^2 x^2\right )^{3/2}\right ) \text {Subst}\left (\int (a+b x) \sec (x) \, dx,x,\sin ^{-1}(c x)\right )}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {\left (16 b d^4 \left (1-c^2 x^2\right )^{3/2}\right ) \text {Subst}\left (\int (a+b x) \tan (x) \, dx,x,\sin ^{-1}(c x)\right )}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}\\ &=-\frac {8 a b d^4 x \left (1-c^2 x^2\right )^{3/2}}{(d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {8 b^2 d^4 \left (1-c^2 x^2\right )^2}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {b^2 d^4 x \left (1-c^2 x^2\right )^2}{4 (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {b^2 d^4 \left (1-c^2 x^2\right )^{3/2} \sin ^{-1}(c x)}{4 c (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {8 b^2 d^4 x \left (1-c^2 x^2\right )^{3/2} \sin ^{-1}(c x)}{(d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {b c d^4 x^2 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{2 (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {8 d^4 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {8 d^4 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{(d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {8 i d^4 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {4 d^4 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {d^4 x \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{2 (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {5 d^4 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^3}{2 b c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {32 i b d^4 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {\left (32 i b d^4 \left (1-c^2 x^2\right )^{3/2}\right ) \text {Subst}\left (\int \frac {e^{2 i x} (a+b x)}{1+e^{2 i x}} \, dx,x,\sin ^{-1}(c x)\right )}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {\left (16 b^2 d^4 \left (1-c^2 x^2\right )^{3/2}\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)^{3/2} (e-c e x)^{3/2}}-\frac {\left (16 b^2 d^4 \left (1-c^2 x^2\right )^{3/2}\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)^{3/2} (e-c e x)^{3/2}}\\ &=-\frac {8 a b d^4 x \left (1-c^2 x^2\right )^{3/2}}{(d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {8 b^2 d^4 \left (1-c^2 x^2\right )^2}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {b^2 d^4 x \left (1-c^2 x^2\right )^2}{4 (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {b^2 d^4 \left (1-c^2 x^2\right )^{3/2} \sin ^{-1}(c x)}{4 c (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {8 b^2 d^4 x \left (1-c^2 x^2\right )^{3/2} \sin ^{-1}(c x)}{(d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {b c d^4 x^2 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{2 (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {8 d^4 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {8 d^4 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{(d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {8 i d^4 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {4 d^4 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {d^4 x \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{2 (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {5 d^4 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^3}{2 b c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {32 i b d^4 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {16 b d^4 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1+e^{2 i \sin ^{-1}(c x)}\right )}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {\left (16 i b^2 d^4 \left (1-c^2 x^2\right )^{3/2}\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)^{3/2} (e-c e x)^{3/2}}+\frac {\left (16 i b^2 d^4 \left (1-c^2 x^2\right )^{3/2}\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)^{3/2} (e-c e x)^{3/2}}-\frac {\left (16 b^2 d^4 \left (1-c^2 x^2\right )^{3/2}\right ) \text {Subst}\left (\int \log \left (1+e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}\\ &=-\frac {8 a b d^4 x \left (1-c^2 x^2\right )^{3/2}}{(d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {8 b^2 d^4 \left (1-c^2 x^2\right )^2}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {b^2 d^4 x \left (1-c^2 x^2\right )^2}{4 (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {b^2 d^4 \left (1-c^2 x^2\right )^{3/2} \sin ^{-1}(c x)}{4 c (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {8 b^2 d^4 x \left (1-c^2 x^2\right )^{3/2} \sin ^{-1}(c x)}{(d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {b c d^4 x^2 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{2 (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {8 d^4 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {8 d^4 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{(d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {8 i d^4 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {4 d^4 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {d^4 x \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{2 (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {5 d^4 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^3}{2 b c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {32 i b d^4 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {16 b d^4 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1+e^{2 i \sin ^{-1}(c x)}\right )}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {16 i b^2 d^4 \left (1-c^2 x^2\right )^{3/2} \text {Li}_2\left (-i e^{i \sin ^{-1}(c x)}\right )}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {16 i b^2 d^4 \left (1-c^2 x^2\right )^{3/2} \text {Li}_2\left (i e^{i \sin ^{-1}(c x)}\right )}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {\left (8 i b^2 d^4 \left (1-c^2 x^2\right )^{3/2}\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 i \sin ^{-1}(c x)}\right )}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}\\ &=-\frac {8 a b d^4 x \left (1-c^2 x^2\right )^{3/2}}{(d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {8 b^2 d^4 \left (1-c^2 x^2\right )^2}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {b^2 d^4 x \left (1-c^2 x^2\right )^2}{4 (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {b^2 d^4 \left (1-c^2 x^2\right )^{3/2} \sin ^{-1}(c x)}{4 c (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {8 b^2 d^4 x \left (1-c^2 x^2\right )^{3/2} \sin ^{-1}(c x)}{(d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {b c d^4 x^2 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{2 (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {8 d^4 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {8 d^4 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{(d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {8 i d^4 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {4 d^4 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {d^4 x \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{2 (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {5 d^4 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^3}{2 b c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {32 i b d^4 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {16 b d^4 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1+e^{2 i \sin ^{-1}(c x)}\right )}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {16 i b^2 d^4 \left (1-c^2 x^2\right )^{3/2} \text {Li}_2\left (-i e^{i \sin ^{-1}(c x)}\right )}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {16 i b^2 d^4 \left (1-c^2 x^2\right )^{3/2} \text {Li}_2\left (i e^{i \sin ^{-1}(c x)}\right )}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {8 i b^2 d^4 \left (1-c^2 x^2\right )^{3/2} \text {Li}_2\left (-e^{2 i \sin ^{-1}(c x)}\right )}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}\\ \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. \(2041\) vs. \(2(918)=1836\).
time = 11.01, size = 2041, normalized size = 2.22 \begin {gather*} \text {Result too large to show} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[-(e*(-1 + c*x))]*Sqrt[d*(1 + c*x)]*((4*a^2*d^2)/e^2 + (a^2*c*d^2*x)/(2*e^2) - (8*a^2*d^2)/(e^2*(-1 + c*x
))))/c + (15*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))])/(2*c*e^(3/2)) - (a*b*d^2*(1 + c*x)*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*Sqrt[-(d*e*(1 - c^2*x^2))]*(Cos[Ar
cSin[c*x]/2]*((-4 + ArcSin[c*x])*ArcSin[c*x] - 8*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]]) - (ArcSin[c*x]*
(4 + ArcSin[c*x]) - 8*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]])*Sin[ArcSin[c*x]/2]))/(c*e^2*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])*(Cos[ArcSin[c*x]/2] + Sin[ArcSin[
c*x]/2])^2) + (4*a*b*d^2*(1 + c*x)*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*Sqrt[-(d*e*(1 - c^2*x^2))]*(Cos[ArcSin[c*x]
/2]*(-(c*x) + 2*ArcSin[c*x] + Sqrt[1 - c^2*x^2]*ArcSin[c*x] - ArcSin[c*x]^2 + 4*Log[Cos[ArcSin[c*x]/2] - Sin[A
rcSin[c*x]/2]]) + (c*x + 2*ArcSin[c*x] - Sqrt[1 - c^2*x^2]*ArcSin[c*x] + ArcSin[c*x]^2 - 4*Log[Cos[ArcSin[c*x]
/2] - Sin[ArcSin[c*x]/2]])*Sin[ArcSin[c*x]/2]))/(c*e^2*Sqrt[(-d - c*d*x)*(e - c*e*x)]*Sqrt[1 - c^2*x^2]*(Cos[A
rcSin[c*x]/2] - Sin[ArcSin[c*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))]*((-18*I)*Pi*ArcSin[c*x] - (6 - 6*I)*ArcSin[c*x]^2 + ArcSin[
c*x]^3 - 24*Pi*Log[1 + E^((-I)*ArcSin[c*x])] + 12*(Pi - 2*ArcSin[c*x])*Log[1 + I*E^(I*ArcSin[c*x])] + 24*Pi*Lo
g[Cos[ArcSin[c*x]/2]] - 12*Pi*Log[-Cos[(Pi + 2*ArcSin[c*x])/4]] + (24*I)*PolyLog[2, (-I)*E^(I*ArcSin[c*x])] -
(12*ArcSin[c*x]^2*Sin[ArcSin[c*x]/2])/(Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2])))/(3*c*e^2*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))]*((96*c*x*ArcSin[c*x])/Sqrt[1 - c^2*x^2] - ((48 - 48*I)*ArcSin[c*x
]^2)/Sqrt[1 - c^2*x^2] + (20*ArcSin[c*x]^3)/Sqrt[1 - c^2*x^2] - 48*(-2 + ArcSin[c*x]^2) - 6*c*x*(-1 + 2*ArcSin
[c*x]^2) - (6*ArcSin[c*x]*Cos[2*ArcSin[c*x]])/Sqrt[1 - c^2*x^2] + (48*((-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*P
i*Log[-Cos[(Pi + 2*ArcSin[c*x])/4]] + (4*I)*PolyLog[2, (-I)*E^(I*ArcSin[c*x])]))/Sqrt[1 - c^2*x^2] - (96*ArcSi
n[c*x]^2*Sin[ArcSin[c*x]/2])/(Sqrt[1 - c^2*x^2]*(Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]))))/(24*c*e^2*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]*Sq
rt[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] - 3*ArcSin[c*x]^2 - ((6 -
6*I)*ArcSin[c*x]^2)/Sqrt[1 - c^2*x^2] + (2*ArcSin[c*x]^3)/Sqrt[1 - c^2*x^2] + (6*((-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]
- (12*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
^2*Sqrt[(-d - c*d*x)*(e - c*e*x)]*(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2])^2) + (a*b*d^2*(1 + c*x)*Sqrt[d + c
*d*x]*Sqrt[e - c*e*x]*Sqrt[-(d*e*(1 - c^2*x^2))]*((-15 + 14*ArcSin[c*x])*Cos[(3*ArcSin[c*x])/2] + Cos[(5*ArcSi
n[c*x])/2] + 2*ArcSin[c*x]*Cos[(5*ArcSin[c*x])/2] + Cos[ArcSin[c*x]/2]*(16 + 48*ArcSin[c*x] - 20*ArcSin[c*x]^2
+ 64*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]]) - 16*Sin[ArcSin[c*x]/2] + 48*ArcSin[c*x]*Sin[ArcSin[c*x]/2
] + 20*ArcSin[c*x]^2*Sin[ArcSin[c*x]/2] - 64*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]]*Sin[ArcSin[c*x]/2] -
15*Sin[(3*ArcSin[c*x])/2] - 14*ArcSin[c*x]*Sin[(3*ArcSin[c*x])/2] - Sin[(5*ArcSin[c*x])/2] + 2*ArcSin[c*x]*Si
n[(5*ArcSin[c*x])/2]))/(8*c*e^2*Sqrt[(-d - c*d*x)*(e - c*e*x)]*Sqrt[1 - c^2*x^2]*(Cos[ArcSin[c*x]/2] - Sin[Arc
Sin[c*x]/2])*(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2])^2)

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Maple [F]
time = 0.24, size = 0, normalized size = 0.00 \[\int \frac {\left (c d x +d \right )^{\frac {5}{2}} \left (a +b \arcsin \left (c x \right )\right )^{2}}{\left (-c e x +e \right )^{\frac {3}{2}}}\, dx\]

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)^(3/2),x)

[Out]

int((c*d*x+d)^(5/2)*(a+b*arcsin(c*x))^2/(-c*e*x+e)^(3/2),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)^(3/2),x, algorithm="maxima")

[Out]

-1/2*(c^2*d^3*x^3*e^(-1)/sqrt(-c^2*d*x^2*e + d*e) + 8*c*d^3*x^2*e^(-1)/sqrt(-c^2*d*x^2*e + d*e) - 17*d^3*x*e^(
-1)/sqrt(-c^2*d*x^2*e + d*e) + 15*d^(5/2)*arcsin(c*x)*e^(-3/2)/c - 24*d^3*e^(-1)/(sqrt(-c^2*d*x^2*e + d*e)*c))
*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, sqrt(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^2*x^2*e^2 - 2*c*x*e^2 + e^2), 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)^(3/2),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^(-2)/(c^2*
x^2 - 2*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)**(3/2),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)^(3/2),x, algorithm="giac")

[Out]

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

[Out]

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

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