3.1.75 \(\int \frac {(f+g x)^3 (a+b \text {ArcSin}(c x))^2}{(d-c^2 d x^2)^{3/2}} \, dx\) [75]

Optimal. Leaf size=738 \[ -\frac {2 a b g^3 x \sqrt {1-c^2 x^2}}{c^3 d \sqrt {d-c^2 d x^2}}-\frac {2 b^2 g^3 \left (1-c^2 x^2\right )}{c^4 d \sqrt {d-c^2 d x^2}}-\frac {2 b^2 g^3 x \sqrt {1-c^2 x^2} \text {ArcSin}(c x)}{c^3 d \sqrt {d-c^2 d x^2}}+\frac {g \left (3 c^2 f^2+g^2\right ) (a+b \text {ArcSin}(c x))^2}{c^4 d \sqrt {d-c^2 d x^2}}+\frac {f \left (f^2+\frac {3 g^2}{c^2}\right ) x (a+b \text {ArcSin}(c x))^2}{d \sqrt {d-c^2 d x^2}}-\frac {i f \left (c^2 f^2+3 g^2\right ) \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))^2}{c^3 d \sqrt {d-c^2 d x^2}}+\frac {g^3 \left (1-c^2 x^2\right ) (a+b \text {ArcSin}(c x))^2}{c^4 d \sqrt {d-c^2 d x^2}}-\frac {f g^2 \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))^3}{b c^3 d \sqrt {d-c^2 d x^2}}+\frac {4 i b g \left (3 c^2 f^2+g^2\right ) \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x)) \text {ArcTan}\left (e^{i \text {ArcSin}(c x)}\right )}{c^4 d \sqrt {d-c^2 d x^2}}+\frac {2 b f \left (c^2 f^2+3 g^2\right ) \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x)) \log \left (1+e^{2 i \text {ArcSin}(c x)}\right )}{c^3 d \sqrt {d-c^2 d x^2}}-\frac {2 i b^2 g \left (3 c^2 f^2+g^2\right ) \sqrt {1-c^2 x^2} \text {PolyLog}\left (2,-i e^{i \text {ArcSin}(c x)}\right )}{c^4 d \sqrt {d-c^2 d x^2}}+\frac {2 i b^2 g \left (3 c^2 f^2+g^2\right ) \sqrt {1-c^2 x^2} \text {PolyLog}\left (2,i e^{i \text {ArcSin}(c x)}\right )}{c^4 d \sqrt {d-c^2 d x^2}}-\frac {i b^2 f \left (c^2 f^2+3 g^2\right ) \sqrt {1-c^2 x^2} \text {PolyLog}\left (2,-e^{2 i \text {ArcSin}(c x)}\right )}{c^3 d \sqrt {d-c^2 d x^2}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.79, antiderivative size = 738, normalized size of antiderivative = 1.00, number of steps used = 23, number of rules used = 15, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.454, Rules used = {4861, 4859, 4847, 4745, 4765, 3800, 2221, 2317, 2438, 4767, 4749, 4266, 4737, 4715, 267} \begin {gather*} \frac {4 i b g \sqrt {1-c^2 x^2} \left (3 c^2 f^2+g^2\right ) \text {ArcTan}\left (e^{i \text {ArcSin}(c x)}\right ) (a+b \text {ArcSin}(c x))}{c^4 d \sqrt {d-c^2 d x^2}}+\frac {f x \left (\frac {3 g^2}{c^2}+f^2\right ) (a+b \text {ArcSin}(c x))^2}{d \sqrt {d-c^2 d x^2}}+\frac {g \left (3 c^2 f^2+g^2\right ) (a+b \text {ArcSin}(c x))^2}{c^4 d \sqrt {d-c^2 d x^2}}+\frac {g^3 \left (1-c^2 x^2\right ) (a+b \text {ArcSin}(c x))^2}{c^4 d \sqrt {d-c^2 d x^2}}-\frac {i f \sqrt {1-c^2 x^2} \left (c^2 f^2+3 g^2\right ) (a+b \text {ArcSin}(c x))^2}{c^3 d \sqrt {d-c^2 d x^2}}+\frac {2 b f \sqrt {1-c^2 x^2} \left (c^2 f^2+3 g^2\right ) \log \left (1+e^{2 i \text {ArcSin}(c x)}\right ) (a+b \text {ArcSin}(c x))}{c^3 d \sqrt {d-c^2 d x^2}}-\frac {f g^2 \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))^3}{b c^3 d \sqrt {d-c^2 d x^2}}-\frac {2 a b g^3 x \sqrt {1-c^2 x^2}}{c^3 d \sqrt {d-c^2 d x^2}}-\frac {2 i b^2 g \sqrt {1-c^2 x^2} \left (3 c^2 f^2+g^2\right ) \text {Li}_2\left (-i e^{i \text {ArcSin}(c x)}\right )}{c^4 d \sqrt {d-c^2 d x^2}}+\frac {2 i b^2 g \sqrt {1-c^2 x^2} \left (3 c^2 f^2+g^2\right ) \text {Li}_2\left (i e^{i \text {ArcSin}(c x)}\right )}{c^4 d \sqrt {d-c^2 d x^2}}-\frac {i b^2 f \sqrt {1-c^2 x^2} \left (c^2 f^2+3 g^2\right ) \text {Li}_2\left (-e^{2 i \text {ArcSin}(c x)}\right )}{c^3 d \sqrt {d-c^2 d x^2}}-\frac {2 b^2 g^3 x \sqrt {1-c^2 x^2} \text {ArcSin}(c x)}{c^3 d \sqrt {d-c^2 d x^2}}-\frac {2 b^2 g^3 \left (1-c^2 x^2\right )}{c^4 d \sqrt {d-c^2 d x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*a*b*g^3*x*Sqrt[1 - c^2*x^2])/(c^3*d*Sqrt[d - c^2*d*x^2]) - (2*b^2*g^3*(1 - c^2*x^2))/(c^4*d*Sqrt[d - c^2*d
*x^2]) - (2*b^2*g^3*x*Sqrt[1 - c^2*x^2]*ArcSin[c*x])/(c^3*d*Sqrt[d - c^2*d*x^2]) + (g*(3*c^2*f^2 + g^2)*(a + b
*ArcSin[c*x])^2)/(c^4*d*Sqrt[d - c^2*d*x^2]) + (f*(f^2 + (3*g^2)/c^2)*x*(a + b*ArcSin[c*x])^2)/(d*Sqrt[d - c^2
*d*x^2]) - (I*f*(c^2*f^2 + 3*g^2)*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/(c^3*d*Sqrt[d - c^2*d*x^2]) + (g^3*
(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/(c^4*d*Sqrt[d - c^2*d*x^2]) - (f*g^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]
)^3)/(b*c^3*d*Sqrt[d - c^2*d*x^2]) + ((4*I)*b*g*(3*c^2*f^2 + g^2)*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*ArcTan
[E^(I*ArcSin[c*x])])/(c^4*d*Sqrt[d - c^2*d*x^2]) + (2*b*f*(c^2*f^2 + 3*g^2)*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*
x])*Log[1 + E^((2*I)*ArcSin[c*x])])/(c^3*d*Sqrt[d - c^2*d*x^2]) - ((2*I)*b^2*g*(3*c^2*f^2 + g^2)*Sqrt[1 - c^2*
x^2]*PolyLog[2, (-I)*E^(I*ArcSin[c*x])])/(c^4*d*Sqrt[d - c^2*d*x^2]) + ((2*I)*b^2*g*(3*c^2*f^2 + g^2)*Sqrt[1 -
c^2*x^2]*PolyLog[2, I*E^(I*ArcSin[c*x])])/(c^4*d*Sqrt[d - c^2*d*x^2]) - (I*b^2*f*(c^2*f^2 + 3*g^2)*Sqrt[1 - c
^2*x^2]*PolyLog[2, -E^((2*I)*ArcSin[c*x])])/(c^3*d*Sqrt[d - c^2*d*x^2])

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 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 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 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 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]

Rule 4861

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

Rubi steps

\begin {align*} \int \frac {(f+g x)^3 \left (a+b \sin ^{-1}(c x)\right )^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx &=\frac {\sqrt {1-c^2 x^2} \int \frac {(f+g x)^3 \left (a+b \sin ^{-1}(c x)\right )^2}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{d \sqrt {d-c^2 d x^2}}\\ &=\frac {\sqrt {1-c^2 x^2} \int \left (\frac {\left (c^2 f^3+3 f g^2+g \left (3 c^2 f^2+g^2\right ) x\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 \left (1-c^2 x^2\right )^{3/2}}-\frac {3 f g^2 \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 \sqrt {1-c^2 x^2}}-\frac {g^3 x \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 \sqrt {1-c^2 x^2}}\right ) \, dx}{d \sqrt {d-c^2 d x^2}}\\ &=\frac {\sqrt {1-c^2 x^2} \int \frac {\left (c^2 f^3+3 f g^2+g \left (3 c^2 f^2+g^2\right ) x\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {\left (3 f g^2 \sqrt {1-c^2 x^2}\right ) \int \frac {\left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt {1-c^2 x^2}} \, dx}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {\left (g^3 \sqrt {1-c^2 x^2}\right ) \int \frac {x \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt {1-c^2 x^2}} \, dx}{c^2 d \sqrt {d-c^2 d x^2}}\\ &=\frac {g^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c^4 d \sqrt {d-c^2 d x^2}}-\frac {f g^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{b c^3 d \sqrt {d-c^2 d x^2}}+\frac {\sqrt {1-c^2 x^2} \int \left (\frac {c^2 f^3 \left (1+\frac {3 g^2}{c^2 f^2}\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{\left (1-c^2 x^2\right )^{3/2}}+\frac {g \left (3 c^2 f^2+g^2\right ) x \left (a+b \sin ^{-1}(c x)\right )^2}{\left (1-c^2 x^2\right )^{3/2}}\right ) \, dx}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {\left (2 b g^3 \sqrt {1-c^2 x^2}\right ) \int \left (a+b \sin ^{-1}(c x)\right ) \, dx}{c^3 d \sqrt {d-c^2 d x^2}}\\ &=-\frac {2 a b g^3 x \sqrt {1-c^2 x^2}}{c^3 d \sqrt {d-c^2 d x^2}}+\frac {g^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c^4 d \sqrt {d-c^2 d x^2}}-\frac {f g^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{b c^3 d \sqrt {d-c^2 d x^2}}-\frac {\left (2 b^2 g^3 \sqrt {1-c^2 x^2}\right ) \int \sin ^{-1}(c x) \, dx}{c^3 d \sqrt {d-c^2 d x^2}}+\frac {\left (g \left (3 c^2 f^2+g^2\right ) \sqrt {1-c^2 x^2}\right ) \int \frac {x \left (a+b \sin ^{-1}(c x)\right )^2}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {\left (f \left (c^2 f^2+3 g^2\right ) \sqrt {1-c^2 x^2}\right ) \int \frac {\left (a+b \sin ^{-1}(c x)\right )^2}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{c^2 d \sqrt {d-c^2 d x^2}}\\ &=-\frac {2 a b g^3 x \sqrt {1-c^2 x^2}}{c^3 d \sqrt {d-c^2 d x^2}}-\frac {2 b^2 g^3 x \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{c^3 d \sqrt {d-c^2 d x^2}}+\frac {g \left (3 c^2 f^2+g^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c^4 d \sqrt {d-c^2 d x^2}}+\frac {f \left (c^2 f^2+3 g^2\right ) x \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {g^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c^4 d \sqrt {d-c^2 d x^2}}-\frac {f g^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{b c^3 d \sqrt {d-c^2 d x^2}}+\frac {\left (2 b^2 g^3 \sqrt {1-c^2 x^2}\right ) \int \frac {x}{\sqrt {1-c^2 x^2}} \, dx}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {\left (2 b g \left (3 c^2 f^2+g^2\right ) \sqrt {1-c^2 x^2}\right ) \int \frac {a+b \sin ^{-1}(c x)}{1-c^2 x^2} \, dx}{c^3 d \sqrt {d-c^2 d x^2}}-\frac {\left (2 b f \left (c^2 f^2+3 g^2\right ) \sqrt {1-c^2 x^2}\right ) \int \frac {x \left (a+b \sin ^{-1}(c x)\right )}{1-c^2 x^2} \, dx}{c d \sqrt {d-c^2 d x^2}}\\ &=-\frac {2 a b g^3 x \sqrt {1-c^2 x^2}}{c^3 d \sqrt {d-c^2 d x^2}}-\frac {2 b^2 g^3 \left (1-c^2 x^2\right )}{c^4 d \sqrt {d-c^2 d x^2}}-\frac {2 b^2 g^3 x \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{c^3 d \sqrt {d-c^2 d x^2}}+\frac {g \left (3 c^2 f^2+g^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c^4 d \sqrt {d-c^2 d x^2}}+\frac {f \left (c^2 f^2+3 g^2\right ) x \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {g^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c^4 d \sqrt {d-c^2 d x^2}}-\frac {f g^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{b c^3 d \sqrt {d-c^2 d x^2}}-\frac {\left (2 b g \left (3 c^2 f^2+g^2\right ) \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int (a+b x) \sec (x) \, dx,x,\sin ^{-1}(c x)\right )}{c^4 d \sqrt {d-c^2 d x^2}}-\frac {\left (2 b f \left (c^2 f^2+3 g^2\right ) \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int (a+b x) \tan (x) \, dx,x,\sin ^{-1}(c x)\right )}{c^3 d \sqrt {d-c^2 d x^2}}\\ &=-\frac {2 a b g^3 x \sqrt {1-c^2 x^2}}{c^3 d \sqrt {d-c^2 d x^2}}-\frac {2 b^2 g^3 \left (1-c^2 x^2\right )}{c^4 d \sqrt {d-c^2 d x^2}}-\frac {2 b^2 g^3 x \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{c^3 d \sqrt {d-c^2 d x^2}}+\frac {g \left (3 c^2 f^2+g^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c^4 d \sqrt {d-c^2 d x^2}}+\frac {f \left (c^2 f^2+3 g^2\right ) x \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {i f \left (c^2 f^2+3 g^2\right ) \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{c^3 d \sqrt {d-c^2 d x^2}}+\frac {g^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c^4 d \sqrt {d-c^2 d x^2}}-\frac {f g^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{b c^3 d \sqrt {d-c^2 d x^2}}+\frac {4 i b g \left (3 c^2 f^2+g^2\right ) \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{c^4 d \sqrt {d-c^2 d x^2}}+\frac {\left (2 b^2 g \left (3 c^2 f^2+g^2\right ) \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \log \left (1-i e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{c^4 d \sqrt {d-c^2 d x^2}}-\frac {\left (2 b^2 g \left (3 c^2 f^2+g^2\right ) \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \log \left (1+i e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{c^4 d \sqrt {d-c^2 d x^2}}+\frac {\left (4 i b f \left (c^2 f^2+3 g^2\right ) \sqrt {1-c^2 x^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^3 d \sqrt {d-c^2 d x^2}}\\ &=-\frac {2 a b g^3 x \sqrt {1-c^2 x^2}}{c^3 d \sqrt {d-c^2 d x^2}}-\frac {2 b^2 g^3 \left (1-c^2 x^2\right )}{c^4 d \sqrt {d-c^2 d x^2}}-\frac {2 b^2 g^3 x \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{c^3 d \sqrt {d-c^2 d x^2}}+\frac {g \left (3 c^2 f^2+g^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c^4 d \sqrt {d-c^2 d x^2}}+\frac {f \left (c^2 f^2+3 g^2\right ) x \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {i f \left (c^2 f^2+3 g^2\right ) \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{c^3 d \sqrt {d-c^2 d x^2}}+\frac {g^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c^4 d \sqrt {d-c^2 d x^2}}-\frac {f g^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{b c^3 d \sqrt {d-c^2 d x^2}}+\frac {4 i b g \left (3 c^2 f^2+g^2\right ) \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{c^4 d \sqrt {d-c^2 d x^2}}+\frac {2 b f \left (c^2 f^2+3 g^2\right ) \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1+e^{2 i \sin ^{-1}(c x)}\right )}{c^3 d \sqrt {d-c^2 d x^2}}-\frac {\left (2 i b^2 g \left (3 c^2 f^2+g^2\right ) \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{c^4 d \sqrt {d-c^2 d x^2}}+\frac {\left (2 i b^2 g \left (3 c^2 f^2+g^2\right ) \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{c^4 d \sqrt {d-c^2 d x^2}}-\frac {\left (2 b^2 f \left (c^2 f^2+3 g^2\right ) \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \log \left (1+e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{c^3 d \sqrt {d-c^2 d x^2}}\\ &=-\frac {2 a b g^3 x \sqrt {1-c^2 x^2}}{c^3 d \sqrt {d-c^2 d x^2}}-\frac {2 b^2 g^3 \left (1-c^2 x^2\right )}{c^4 d \sqrt {d-c^2 d x^2}}-\frac {2 b^2 g^3 x \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{c^3 d \sqrt {d-c^2 d x^2}}+\frac {g \left (3 c^2 f^2+g^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c^4 d \sqrt {d-c^2 d x^2}}+\frac {f \left (c^2 f^2+3 g^2\right ) x \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {i f \left (c^2 f^2+3 g^2\right ) \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{c^3 d \sqrt {d-c^2 d x^2}}+\frac {g^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c^4 d \sqrt {d-c^2 d x^2}}-\frac {f g^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{b c^3 d \sqrt {d-c^2 d x^2}}+\frac {4 i b g \left (3 c^2 f^2+g^2\right ) \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{c^4 d \sqrt {d-c^2 d x^2}}+\frac {2 b f \left (c^2 f^2+3 g^2\right ) \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1+e^{2 i \sin ^{-1}(c x)}\right )}{c^3 d \sqrt {d-c^2 d x^2}}-\frac {2 i b^2 g \left (3 c^2 f^2+g^2\right ) \sqrt {1-c^2 x^2} \text {Li}_2\left (-i e^{i \sin ^{-1}(c x)}\right )}{c^4 d \sqrt {d-c^2 d x^2}}+\frac {2 i b^2 g \left (3 c^2 f^2+g^2\right ) \sqrt {1-c^2 x^2} \text {Li}_2\left (i e^{i \sin ^{-1}(c x)}\right )}{c^4 d \sqrt {d-c^2 d x^2}}+\frac {\left (i b^2 f \left (c^2 f^2+3 g^2\right ) \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 i \sin ^{-1}(c x)}\right )}{c^3 d \sqrt {d-c^2 d x^2}}\\ &=-\frac {2 a b g^3 x \sqrt {1-c^2 x^2}}{c^3 d \sqrt {d-c^2 d x^2}}-\frac {2 b^2 g^3 \left (1-c^2 x^2\right )}{c^4 d \sqrt {d-c^2 d x^2}}-\frac {2 b^2 g^3 x \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{c^3 d \sqrt {d-c^2 d x^2}}+\frac {g \left (3 c^2 f^2+g^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c^4 d \sqrt {d-c^2 d x^2}}+\frac {f \left (c^2 f^2+3 g^2\right ) x \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {i f \left (c^2 f^2+3 g^2\right ) \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{c^3 d \sqrt {d-c^2 d x^2}}+\frac {g^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c^4 d \sqrt {d-c^2 d x^2}}-\frac {f g^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{b c^3 d \sqrt {d-c^2 d x^2}}+\frac {4 i b g \left (3 c^2 f^2+g^2\right ) \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{c^4 d \sqrt {d-c^2 d x^2}}+\frac {2 b f \left (c^2 f^2+3 g^2\right ) \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1+e^{2 i \sin ^{-1}(c x)}\right )}{c^3 d \sqrt {d-c^2 d x^2}}-\frac {2 i b^2 g \left (3 c^2 f^2+g^2\right ) \sqrt {1-c^2 x^2} \text {Li}_2\left (-i e^{i \sin ^{-1}(c x)}\right )}{c^4 d \sqrt {d-c^2 d x^2}}+\frac {2 i b^2 g \left (3 c^2 f^2+g^2\right ) \sqrt {1-c^2 x^2} \text {Li}_2\left (i e^{i \sin ^{-1}(c x)}\right )}{c^4 d \sqrt {d-c^2 d x^2}}-\frac {i b^2 f \left (c^2 f^2+3 g^2\right ) \sqrt {1-c^2 x^2} \text {Li}_2\left (-e^{2 i \sin ^{-1}(c x)}\right )}{c^3 d \sqrt {d-c^2 d x^2}}\\ \end {align*}

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Mathematica [A]
time = 2.36, size = 325, normalized size = 0.44 \begin {gather*} \frac {\sqrt {1-c^2 x^2} \left (2 g^3 \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))^2-\frac {2 c f g^2 (a+b \text {ArcSin}(c x))^3}{b}-4 b g^3 \left (a c x+b \sqrt {1-c^2 x^2}+b c x \text {ArcSin}(c x)\right )+(c f-g)^3 \left (-(a+b \text {ArcSin}(c x))^2 \cot \left (\frac {1}{4} (\pi +2 \text {ArcSin}(c x))\right )+i \left ((a+b \text {ArcSin}(c x)) \left (a+b \text {ArcSin}(c x)-4 i b \log \left (1+i e^{-i \text {ArcSin}(c x)}\right )\right )+4 b^2 \text {PolyLog}\left (2,-i e^{-i \text {ArcSin}(c x)}\right )\right )\right )-(c f+g)^3 \left (i \left ((a+b \text {ArcSin}(c x)) \left (a+b \text {ArcSin}(c x)+4 i b \log \left (1+i e^{i \text {ArcSin}(c x)}\right )\right )+4 b^2 \text {PolyLog}\left (2,-i e^{i \text {ArcSin}(c x)}\right )\right )-(a+b \text {ArcSin}(c x))^2 \tan \left (\frac {1}{4} (\pi +2 \text {ArcSin}(c x))\right )\right )\right )}{2 c^4 d \sqrt {d-c^2 d x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[1 - c^2*x^2]*(2*g^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2 - (2*c*f*g^2*(a + b*ArcSin[c*x])^3)/b - 4*b*
g^3*(a*c*x + b*Sqrt[1 - c^2*x^2] + b*c*x*ArcSin[c*x]) + (c*f - g)^3*(-((a + b*ArcSin[c*x])^2*Cot[(Pi + 2*ArcSi
n[c*x])/4]) + I*((a + b*ArcSin[c*x])*(a + b*ArcSin[c*x] - (4*I)*b*Log[1 + I/E^(I*ArcSin[c*x])]) + 4*b^2*PolyLo
g[2, (-I)/E^(I*ArcSin[c*x])])) - (c*f + g)^3*(I*((a + b*ArcSin[c*x])*(a + b*ArcSin[c*x] + (4*I)*b*Log[1 + I*E^
(I*ArcSin[c*x])]) + 4*b^2*PolyLog[2, (-I)*E^(I*ArcSin[c*x])]) - (a + b*ArcSin[c*x])^2*Tan[(Pi + 2*ArcSin[c*x])
/4])))/(2*c^4*d*Sqrt[d - c^2*d*x^2])

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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 2662 vs. \(2 (731 ) = 1462\).
time = 1.00, size = 2663, normalized size = 3.61

method	result	size
default	\(\text {Expression too large to display}\)	\(2663\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((g*x+f)^3*(a+b*arcsin(c*x))^2/(-c^2*d*x^2+d)^(3/2),x,method=_RETURNVERBOSE)

[Out]

6*I*b^2*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/c^2/d^2/(c^2*x^2-1)*f^2*g*dilog(1+I*(I*c*x+(-c^2*x^2+1)^(1/2
)))+3*I*b^2*(-d*(c^2*x^2-1))^(1/2)/c^3/d^2/(c^2*x^2-1)*arcsin(c*x)^2*(-c^2*x^2+1)^(1/2)*f*g^2+2*b^2*(-d*(c^2*x
^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/c^4/d^2/(c^2*x^2-1)*g^3*arcsin(c*x)*ln(1-I*(I*c*x+(-c^2*x^2+1)^(1/2)))-3*b^2*(
-d*(c^2*x^2-1))^(1/2)/c^2/d^2/(c^2*x^2-1)*arcsin(c*x)^2*x*f*g^2+I*b^2*(-d*(c^2*x^2-1))^(1/2)/c/d^2/(c^2*x^2-1)
*arcsin(c*x)^2*(-c^2*x^2+1)^(1/2)*f^3-2*b^2*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/c/d^2/(c^2*x^2-1)*f^3*ar
csin(c*x)*ln(1+(I*c*x+(-c^2*x^2+1)^(1/2))^2)-2*I*b^2*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/c^4/d^2/(c^2*x^
2-1)*g^3*dilog(1-I*(I*c*x+(-c^2*x^2+1)^(1/2)))-2*b^2*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/c^4/d^2/(c^2*x^
2-1)*g^3*arcsin(c*x)*ln(1+I*(I*c*x+(-c^2*x^2+1)^(1/2)))+b^2*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/c^3/d^2/
(c^2*x^2-1)*arcsin(c*x)^3*f*g^2+2*a^2*g^3/d/c^4/(-c^2*d*x^2+d)^(1/2)-4*a*b*(-d*(c^2*x^2-1))^(1/2)*g^3/c^4/d^2/
(c^2*x^2-1)*arcsin(c*x)-2*a*b*(-d*(c^2*x^2-1))^(1/2)/d^2/(c^2*x^2-1)*arcsin(c*x)*x*f^3+a^2*f^3*x/d/(-c^2*d*x^2
+d)^(1/2)-6*I*b^2*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/c^2/d^2/(c^2*x^2-1)*f^2*g*dilog(1-I*(I*c*x+(-c^2*x
^2+1)^(1/2)))-6*b^2*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/c^2/d^2/(c^2*x^2-1)*f^2*g*arcsin(c*x)*ln(1+I*(I*
c*x+(-c^2*x^2+1)^(1/2)))+3*I*b^2*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/c^3/d^2/(c^2*x^2-1)*f*g^2*polylog(2
,-(I*c*x+(-c^2*x^2+1)^(1/2))^2)+6*b^2*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/c^2/d^2/(c^2*x^2-1)*f^2*g*arcs
in(c*x)*ln(1-I*(I*c*x+(-c^2*x^2+1)^(1/2)))+3*a^2*f^2*g/c^2/d/(-c^2*d*x^2+d)^(1/2)-a^2*g^3*x^2/c^2/d/(-c^2*d*x^
2+d)^(1/2)+2*b^2*(-d*(c^2*x^2-1))^(1/2)*g^3/c^4/d^2/(c^2*x^2-1)+6*I*a*b*(-c^2*x^2+1)^(1/2)*(-d*(c^2*x^2-1))^(1
/2)/c^3/d^2/(c^2*x^2-1)*f*arcsin(c*x)*g^2+b^2*(-d*(c^2*x^2-1))^(1/2)*g^3/c^2/d^2/(c^2*x^2-1)*arcsin(c*x)^2*x^2
-3*b^2*(-d*(c^2*x^2-1))^(1/2)/c^2/d^2/(c^2*x^2-1)*arcsin(c*x)^2*f^2*g-3*a^2*f*g^2/c^2/d/(c^2*d)^(1/2)*arctan((
c^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2))+3*a^2*f*g^2*x/c^2/d/(-c^2*d*x^2+d)^(1/2)-6*b^2*(-d*(c^2*x^2-1))^(1/2)*(-c
^2*x^2+1)^(1/2)/c^3/d^2/(c^2*x^2-1)*f*g^2*arcsin(c*x)*ln(1+(I*c*x+(-c^2*x^2+1)^(1/2))^2)+2*a*b*(-d*(c^2*x^2-1)
)^(1/2)*g^3/c^2/d^2/(c^2*x^2-1)*arcsin(c*x)*x^2-6*a*b*(-d*(c^2*x^2-1))^(1/2)/c^2/d^2/(c^2*x^2-1)*arcsin(c*x)*f
^2*g-2*a*b*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)*ln(I*c*x+(-c^2*x^2+1)^(1/2)+I)/c/d^2/(c^2*x^2-1)*f^3+2*a*
b*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)*ln(I*c*x+(-c^2*x^2+1)^(1/2)+I)/c^4/d^2/(c^2*x^2-1)*g^3-2*a*b*(-d*(
c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/c/d^2/(c^2*x^2-1)*ln(I*c*x+(-c^2*x^2+1)^(1/2)-I)*f^3-2*a*b*(-d*(c^2*x^2-1
))^(1/2)*(-c^2*x^2+1)^(1/2)/c^4/d^2/(c^2*x^2-1)*ln(I*c*x+(-c^2*x^2+1)^(1/2)-I)*g^3-6*a*b*(-d*(c^2*x^2-1))^(1/2
)/c^2/d^2/(c^2*x^2-1)*arcsin(c*x)*x*f*g^2+6*a*b*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)*ln(I*c*x+(-c^2*x^2+1
)^(1/2)+I)/c^2/d^2/(c^2*x^2-1)*f^2*g-6*a*b*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)*ln(I*c*x+(-c^2*x^2+1)^(1/
2)+I)/c^3/d^2/(c^2*x^2-1)*f*g^2-6*a*b*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/c^2/d^2/(c^2*x^2-1)*ln(I*c*x+(
-c^2*x^2+1)^(1/2)-I)*f^2*g-6*a*b*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/c^3/d^2/(c^2*x^2-1)*ln(I*c*x+(-c^2*
x^2+1)^(1/2)-I)*f*g^2+3*a*b*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/c^3/d^2/(c^2*x^2-1)*arcsin(c*x)^2*f*g^2+
2*I*a*b*(-c^2*x^2+1)^(1/2)*(-d*(c^2*x^2-1))^(1/2)/c/d^2/(c^2*x^2-1)*f^3*arcsin(c*x)+2*b^2*(-d*(c^2*x^2-1))^(1/
2)*g^3/c^3/d^2/(c^2*x^2-1)*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*x+I*b^2*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/c/
d^2/(c^2*x^2-1)*f^3*polylog(2,-(I*c*x+(-c^2*x^2+1)^(1/2))^2)+2*I*b^2*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)
/c^4/d^2/(c^2*x^2-1)*g^3*dilog(1+I*(I*c*x+(-c^2*x^2+1)^(1/2)))+2*a*b*(-d*(c^2*x^2-1))^(1/2)*g^3/c^3/d^2/(c^2*x
^2-1)*(-c^2*x^2+1)^(1/2)*x-2*b^2*(-d*(c^2*x^2-1))^(1/2)*g^3/c^2/d^2/(c^2*x^2-1)*x^2-2*b^2*(-d*(c^2*x^2-1))^(1/
2)*g^3/c^4/d^2/(c^2*x^2-1)*arcsin(c*x)^2-b^2*(-d*(c^2*x^2-1))^(1/2)/d^2/(c^2*x^2-1)*arcsin(c*x)^2*x*f^3

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

[Out]

-a^2*g^3*(x^2/(sqrt(-c^2*d*x^2 + d)*c^2*d) - 2/(sqrt(-c^2*d*x^2 + d)*c^4*d)) + 3*a^2*f*g^2*(x/(sqrt(-c^2*d*x^2
+ d)*c^2*d) - arcsin(c*x)/(c^3*d^(3/2))) + 2*a*b*f^3*x*arcsin(c*x)/(sqrt(-c^2*d*x^2 + d)*d) + a^2*f^3*x/(sqrt
(-c^2*d*x^2 + d)*d) - a*b*f^3*log(x^2 - 1/c^2)/(c*d^(3/2)) + 3*a^2*f^2*g/(sqrt(-c^2*d*x^2 + d)*c^2*d) - sqrt(d
)*integrate(((b^2*g^3*x^3 + 3*b^2*f*g^2*x^2 + 3*b^2*f^2*g*x + b^2*f^3)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x +
1))^2 + 2*(a*b*g^3*x^3 + 3*a*b*f*g^2*x^2 + 3*a*b*f^2*g*x)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)))/((c^2*d^
2*x^2 - d^2)*sqrt(c*x + 1)*sqrt(-c*x + 1)), 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((g*x+f)^3*(a+b*arcsin(c*x))^2/(-c^2*d*x^2+d)^(3/2),x, algorithm="fricas")

[Out]

integral((a^2*g^3*x^3 + 3*a^2*f*g^2*x^2 + 3*a^2*f^2*g*x + a^2*f^3 + (b^2*g^3*x^3 + 3*b^2*f*g^2*x^2 + 3*b^2*f^2
*g*x + b^2*f^3)*arcsin(c*x)^2 + 2*(a*b*g^3*x^3 + 3*a*b*f*g^2*x^2 + 3*a*b*f^2*g*x + a*b*f^3)*arcsin(c*x))*sqrt(
-c^2*d*x^2 + d)/(c^4*d^2*x^4 - 2*c^2*d^2*x^2 + d^2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> Invalid comparison of non-real zoo

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Warning, integration of abs or sign assumes constant sign by intervals (correct if the argument is real):Ch
eck [abs(t_

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (f+g\,x\right )}^3\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2}{{\left (d-c^2\,d\,x^2\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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