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

Optimal. Leaf size=641 \[ -\frac {b f \left (a+b \sin ^{-1}(c x)\right )}{3 c d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}+\frac {2 f x \left (a+b \sin ^{-1}(c x)\right )^2}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {2 i f \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c d^2 \sqrt {d-c^2 d x^2}}+\frac {f x \left (a+b \sin ^{-1}(c x)\right )^2}{3 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {4 b f \sqrt {1-c^2 x^2} \log \left (1+e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 c d^2 \sqrt {d-c^2 d x^2}}-\frac {b g x \left (a+b \sin ^{-1}(c x)\right )}{3 c d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}+\frac {g \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^2 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {2 i b g \sqrt {1-c^2 x^2} \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 c^2 d^2 \sqrt {d-c^2 d x^2}}-\frac {2 i b^2 f \sqrt {1-c^2 x^2} \text {Li}_2\left (-e^{2 i \sin ^{-1}(c x)}\right )}{3 c d^2 \sqrt {d-c^2 d x^2}}+\frac {b^2 f x}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {i b^2 g \sqrt {1-c^2 x^2} \text {Li}_2\left (-i e^{i \sin ^{-1}(c x)}\right )}{3 c^2 d^2 \sqrt {d-c^2 d x^2}}+\frac {i b^2 g \sqrt {1-c^2 x^2} \text {Li}_2\left (i e^{i \sin ^{-1}(c x)}\right )}{3 c^2 d^2 \sqrt {d-c^2 d x^2}}+\frac {b^2 g}{3 c^2 d^2 \sqrt {d-c^2 d x^2}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.78, antiderivative size = 641, normalized size of antiderivative = 1.00, number of steps used = 21, number of rules used = 14, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.452, Rules used = {4777, 4763, 4655, 4651, 4675, 3719, 2190, 2279, 2391, 4677, 191, 4657, 4181, 261} \[ -\frac {2 i b^2 f \sqrt {1-c^2 x^2} \text {PolyLog}\left (2,-e^{2 i \sin ^{-1}(c x)}\right )}{3 c d^2 \sqrt {d-c^2 d x^2}}-\frac {i b^2 g \sqrt {1-c^2 x^2} \text {PolyLog}\left (2,-i e^{i \sin ^{-1}(c x)}\right )}{3 c^2 d^2 \sqrt {d-c^2 d x^2}}+\frac {i b^2 g \sqrt {1-c^2 x^2} \text {PolyLog}\left (2,i e^{i \sin ^{-1}(c x)}\right )}{3 c^2 d^2 \sqrt {d-c^2 d x^2}}-\frac {b f \left (a+b \sin ^{-1}(c x)\right )}{3 c d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}+\frac {2 f x \left (a+b \sin ^{-1}(c x)\right )^2}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {2 i f \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c d^2 \sqrt {d-c^2 d x^2}}+\frac {f x \left (a+b \sin ^{-1}(c x)\right )^2}{3 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {4 b f \sqrt {1-c^2 x^2} \log \left (1+e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 c d^2 \sqrt {d-c^2 d x^2}}-\frac {b g x \left (a+b \sin ^{-1}(c x)\right )}{3 c d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}+\frac {g \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^2 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {2 i b g \sqrt {1-c^2 x^2} \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 c^2 d^2 \sqrt {d-c^2 d x^2}}+\frac {b^2 f x}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {b^2 g}{3 c^2 d^2 \sqrt {d-c^2 d x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 191

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^(p + 1))/a, x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rule 261

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 2190

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), x]
- Dist[(d*m)/(b*f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*Log[1 + (b*(F^(g*(e + f*x)))^n)/a], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2279

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

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 3719

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 4181

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 4651

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)/Sqrt[d], Int[(x*(a + b*ArcSin[c*x])^(n - 1))/(d + e*x^2), x], x
] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && GtQ[d, 0]

Rule 4655

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

Rule 4657

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 4675

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 4677

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*d^IntPart[p]*(d + e*x^2)^FracPart[p])/(2*c*(p + 1
)*(1 - c^2*x^2)^FracPart[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 4763

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 4777

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :
> Dist[(d^IntPart[p]*(d + e*x^2)^FracPart[p])/(1 - c^2*x^2)^FracPart[p], Int[(f + g*x)^m*(1 - c^2*x^2)^p*(a +
b*ArcSin[c*x])^n, x], x] /; FreeQ[{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) \left (a+b \sin ^{-1}(c x)\right )^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx &=\frac {\sqrt {1-c^2 x^2} \int \frac {(f+g x) \left (a+b \sin ^{-1}(c x)\right )^2}{\left (1-c^2 x^2\right )^{5/2}} \, dx}{d^2 \sqrt {d-c^2 d x^2}}\\ &=\frac {\sqrt {1-c^2 x^2} \int \left (\frac {f \left (a+b \sin ^{-1}(c x)\right )^2}{\left (1-c^2 x^2\right )^{5/2}}+\frac {g x \left (a+b \sin ^{-1}(c x)\right )^2}{\left (1-c^2 x^2\right )^{5/2}}\right ) \, dx}{d^2 \sqrt {d-c^2 d x^2}}\\ &=\frac {\left (f \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 )^{5/2}} \, dx}{d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (g \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 )^{5/2}} \, dx}{d^2 \sqrt {d-c^2 d x^2}}\\ &=\frac {g \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^2 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {f x \left (a+b \sin ^{-1}(c x)\right )^2}{3 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {\left (2 f \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}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (2 b c f \sqrt {1-c^2 x^2}\right ) \int \frac {x \left (a+b \sin ^{-1}(c x)\right )}{\left (1-c^2 x^2\right )^2} \, dx}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (2 b g \sqrt {1-c^2 x^2}\right ) \int \frac {a+b \sin ^{-1}(c x)}{\left (1-c^2 x^2\right )^2} \, dx}{3 c d^2 \sqrt {d-c^2 d x^2}}\\ &=-\frac {b f \left (a+b \sin ^{-1}(c x)\right )}{3 c d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}-\frac {b g x \left (a+b \sin ^{-1}(c x)\right )}{3 c d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}+\frac {2 f x \left (a+b \sin ^{-1}(c x)\right )^2}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {g \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^2 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {f x \left (a+b \sin ^{-1}(c x)\right )^2}{3 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {\left (b^2 f \sqrt {1-c^2 x^2}\right ) \int \frac {1}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (4 b c f \sqrt {1-c^2 x^2}\right ) \int \frac {x \left (a+b \sin ^{-1}(c x)\right )}{1-c^2 x^2} \, dx}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (b^2 g \sqrt {1-c^2 x^2}\right ) \int \frac {x}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (b g \sqrt {1-c^2 x^2}\right ) \int \frac {a+b \sin ^{-1}(c x)}{1-c^2 x^2} \, dx}{3 c d^2 \sqrt {d-c^2 d x^2}}\\ &=\frac {b^2 g}{3 c^2 d^2 \sqrt {d-c^2 d x^2}}+\frac {b^2 f x}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {b f \left (a+b \sin ^{-1}(c x)\right )}{3 c d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}-\frac {b g x \left (a+b \sin ^{-1}(c x)\right )}{3 c d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}+\frac {2 f x \left (a+b \sin ^{-1}(c x)\right )^2}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {g \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^2 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {f x \left (a+b \sin ^{-1}(c x)\right )^2}{3 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}-\frac {\left (4 b f \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int (a+b x) \tan (x) \, dx,x,\sin ^{-1}(c x)\right )}{3 c d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (b g \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int (a+b x) \sec (x) \, dx,x,\sin ^{-1}(c x)\right )}{3 c^2 d^2 \sqrt {d-c^2 d x^2}}\\ &=\frac {b^2 g}{3 c^2 d^2 \sqrt {d-c^2 d x^2}}+\frac {b^2 f x}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {b f \left (a+b \sin ^{-1}(c x)\right )}{3 c d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}-\frac {b g x \left (a+b \sin ^{-1}(c x)\right )}{3 c d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}+\frac {2 f x \left (a+b \sin ^{-1}(c x)\right )^2}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {g \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^2 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {f x \left (a+b \sin ^{-1}(c x)\right )^2}{3 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}-\frac {2 i f \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c d^2 \sqrt {d-c^2 d x^2}}+\frac {2 i b g \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{3 c^2 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (8 i b f \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {e^{2 i x} (a+b x)}{1+e^{2 i x}} \, dx,x,\sin ^{-1}(c x)\right )}{3 c d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (b^2 g \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \log \left (1-i e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{3 c^2 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (b^2 g \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \log \left (1+i e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{3 c^2 d^2 \sqrt {d-c^2 d x^2}}\\ &=\frac {b^2 g}{3 c^2 d^2 \sqrt {d-c^2 d x^2}}+\frac {b^2 f x}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {b f \left (a+b \sin ^{-1}(c x)\right )}{3 c d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}-\frac {b g x \left (a+b \sin ^{-1}(c x)\right )}{3 c d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}+\frac {2 f x \left (a+b \sin ^{-1}(c x)\right )^2}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {g \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^2 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {f x \left (a+b \sin ^{-1}(c x)\right )^2}{3 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}-\frac {2 i f \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c d^2 \sqrt {d-c^2 d x^2}}+\frac {2 i b g \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{3 c^2 d^2 \sqrt {d-c^2 d x^2}}+\frac {4 b f \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 )}{3 c d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (4 b^2 f \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \log \left (1+e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{3 c d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (i b^2 g \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{3 c^2 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (i b^2 g \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{3 c^2 d^2 \sqrt {d-c^2 d x^2}}\\ &=\frac {b^2 g}{3 c^2 d^2 \sqrt {d-c^2 d x^2}}+\frac {b^2 f x}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {b f \left (a+b \sin ^{-1}(c x)\right )}{3 c d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}-\frac {b g x \left (a+b \sin ^{-1}(c x)\right )}{3 c d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}+\frac {2 f x \left (a+b \sin ^{-1}(c x)\right )^2}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {g \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^2 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {f x \left (a+b \sin ^{-1}(c x)\right )^2}{3 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}-\frac {2 i f \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c d^2 \sqrt {d-c^2 d x^2}}+\frac {2 i b g \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{3 c^2 d^2 \sqrt {d-c^2 d x^2}}+\frac {4 b f \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 )}{3 c d^2 \sqrt {d-c^2 d x^2}}-\frac {i b^2 g \sqrt {1-c^2 x^2} \text {Li}_2\left (-i e^{i \sin ^{-1}(c x)}\right )}{3 c^2 d^2 \sqrt {d-c^2 d x^2}}+\frac {i b^2 g \sqrt {1-c^2 x^2} \text {Li}_2\left (i e^{i \sin ^{-1}(c x)}\right )}{3 c^2 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (2 i b^2 f \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 i \sin ^{-1}(c x)}\right )}{3 c d^2 \sqrt {d-c^2 d x^2}}\\ &=\frac {b^2 g}{3 c^2 d^2 \sqrt {d-c^2 d x^2}}+\frac {b^2 f x}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {b f \left (a+b \sin ^{-1}(c x)\right )}{3 c d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}-\frac {b g x \left (a+b \sin ^{-1}(c x)\right )}{3 c d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}+\frac {2 f x \left (a+b \sin ^{-1}(c x)\right )^2}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {g \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^2 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {f x \left (a+b \sin ^{-1}(c x)\right )^2}{3 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}-\frac {2 i f \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c d^2 \sqrt {d-c^2 d x^2}}+\frac {2 i b g \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{3 c^2 d^2 \sqrt {d-c^2 d x^2}}+\frac {4 b f \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 )}{3 c d^2 \sqrt {d-c^2 d x^2}}-\frac {i b^2 g \sqrt {1-c^2 x^2} \text {Li}_2\left (-i e^{i \sin ^{-1}(c x)}\right )}{3 c^2 d^2 \sqrt {d-c^2 d x^2}}+\frac {i b^2 g \sqrt {1-c^2 x^2} \text {Li}_2\left (i e^{i \sin ^{-1}(c x)}\right )}{3 c^2 d^2 \sqrt {d-c^2 d x^2}}-\frac {2 i b^2 f \sqrt {1-c^2 x^2} \text {Li}_2\left (-e^{2 i \sin ^{-1}(c x)}\right )}{3 c d^2 \sqrt {d-c^2 d x^2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 6.24, size = 683, normalized size = 1.07 \[ \frac {\sqrt {1-c^2 x^2} \left (-\frac {(c f-g) \left (-2 \left (-\tan \left (\frac {\pi }{4}-\frac {1}{2} \sin ^{-1}(c x)\right ) \left (a+b \sin ^{-1}(c x)\right )^2+i b \left (\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{b}-4 \left (i \log \left (1+e^{\frac {1}{2} i \left (\pi -2 \sin ^{-1}(c x)\right )}\right ) \left (a+b \sin ^{-1}(c x)\right )-b \text {Li}_2\left (-e^{\frac {1}{2} i \left (\pi -2 \sin ^{-1}(c x)\right )}\right )\right )\right )\right )+2 b \sec ^2\left (\frac {\pi }{4}-\frac {1}{2} \sin ^{-1}(c x)\right ) \left (a+b \sin ^{-1}(c x)\right )+\tan \left (\frac {\pi }{4}-\frac {1}{2} \sin ^{-1}(c x)\right ) \sec ^2\left (\frac {\pi }{4}-\frac {1}{2} \sin ^{-1}(c x)\right ) \left (a+b \sin ^{-1}(c x)\right )^2+4 b^2 \tan \left (\frac {\pi }{4}-\frac {1}{2} \sin ^{-1}(c x)\right )\right )}{24 c^2}-\frac {(c f+g) \left (2 \left (-\tan \left (\frac {1}{2} \sin ^{-1}(c x)+\frac {\pi }{4}\right ) \left (a+b \sin ^{-1}(c x)\right )^2+i b \left (\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{b}+4 \left (i \log \left (1+e^{\frac {1}{2} i \left (2 \sin ^{-1}(c x)+\pi \right )}\right ) \left (a+b \sin ^{-1}(c x)\right )+b \text {Li}_2\left (-e^{\frac {1}{2} i \left (2 \sin ^{-1}(c x)+\pi \right )}\right )\right )\right )\right )+2 b \sec ^2\left (\frac {1}{2} \sin ^{-1}(c x)+\frac {\pi }{4}\right ) \left (a+b \sin ^{-1}(c x)\right )-\tan \left (\frac {1}{2} \sin ^{-1}(c x)+\frac {\pi }{4}\right ) \sec ^2\left (\frac {1}{2} \sin ^{-1}(c x)+\frac {\pi }{4}\right ) \left (a+b \sin ^{-1}(c x)\right )^2-4 b^2 \tan \left (\frac {1}{2} \sin ^{-1}(c x)+\frac {\pi }{4}\right )\right )}{24 c^2}+\frac {f \left (-\tan \left (\frac {\pi }{4}-\frac {1}{2} \sin ^{-1}(c x)\right ) \left (a+b \sin ^{-1}(c x)\right )^2+i b \left (\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{b}-4 \left (i \log \left (1+e^{\frac {1}{2} i \left (\pi -2 \sin ^{-1}(c x)\right )}\right ) \left (a+b \sin ^{-1}(c x)\right )-b \text {Li}_2\left (-e^{\frac {1}{2} i \left (\pi -2 \sin ^{-1}(c x)\right )}\right )\right )\right )\right )}{4 c}-\frac {f \left (-\tan \left (\frac {1}{2} \sin ^{-1}(c x)+\frac {\pi }{4}\right ) \left (a+b \sin ^{-1}(c x)\right )^2+i b \left (\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{b}+4 \left (i \log \left (1+e^{\frac {1}{2} i \left (2 \sin ^{-1}(c x)+\pi \right )}\right ) \left (a+b \sin ^{-1}(c x)\right )+b \text {Li}_2\left (-e^{\frac {1}{2} i \left (2 \sin ^{-1}(c x)+\pi \right )}\right )\right )\right )\right )}{4 c}\right )}{d^2 \sqrt {d-c^2 d x^2}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(Sqrt[1 - c^2*x^2]*((f*(I*b*((a + b*ArcSin[c*x])^2/b - 4*(I*(a + b*ArcSin[c*x])*Log[1 + E^((I/2)*(Pi - 2*ArcSi
n[c*x]))] - b*PolyLog[2, -E^((I/2)*(Pi - 2*ArcSin[c*x]))])) - (a + b*ArcSin[c*x])^2*Tan[Pi/4 - ArcSin[c*x]/2])
)/(4*c) - ((c*f - g)*(2*b*(a + b*ArcSin[c*x])*Sec[Pi/4 - ArcSin[c*x]/2]^2 + 4*b^2*Tan[Pi/4 - ArcSin[c*x]/2] +
(a + b*ArcSin[c*x])^2*Sec[Pi/4 - ArcSin[c*x]/2]^2*Tan[Pi/4 - ArcSin[c*x]/2] - 2*(I*b*((a + b*ArcSin[c*x])^2/b
- 4*(I*(a + b*ArcSin[c*x])*Log[1 + E^((I/2)*(Pi - 2*ArcSin[c*x]))] - b*PolyLog[2, -E^((I/2)*(Pi - 2*ArcSin[c*x
]))])) - (a + b*ArcSin[c*x])^2*Tan[Pi/4 - ArcSin[c*x]/2])))/(24*c^2) - (f*(I*b*((a + b*ArcSin[c*x])^2/b + 4*(I
*(a + b*ArcSin[c*x])*Log[1 + E^((I/2)*(Pi + 2*ArcSin[c*x]))] + b*PolyLog[2, -E^((I/2)*(Pi + 2*ArcSin[c*x]))]))
- (a + b*ArcSin[c*x])^2*Tan[Pi/4 + ArcSin[c*x]/2]))/(4*c) - ((c*f + g)*(2*b*(a + b*ArcSin[c*x])*Sec[Pi/4 + Ar
cSin[c*x]/2]^2 - 4*b^2*Tan[Pi/4 + ArcSin[c*x]/2] - (a + b*ArcSin[c*x])^2*Sec[Pi/4 + ArcSin[c*x]/2]^2*Tan[Pi/4
+ ArcSin[c*x]/2] + 2*(I*b*((a + b*ArcSin[c*x])^2/b + 4*(I*(a + b*ArcSin[c*x])*Log[1 + E^((I/2)*(Pi + 2*ArcSin[
c*x]))] + b*PolyLog[2, -E^((I/2)*(Pi + 2*ArcSin[c*x]))])) - (a + b*ArcSin[c*x])^2*Tan[Pi/4 + ArcSin[c*x]/2])))
/(24*c^2)))/(d^2*Sqrt[d - c^2*d*x^2])

________________________________________________________________________________________

fricas [F]  time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-c^{2} d x^{2} + d} {\left (a^{2} g x + a^{2} f + {\left (b^{2} g x + b^{2} f\right )} \arcsin \left (c x\right )^{2} + 2 \, {\left (a b g x + a b f\right )} \arcsin \left (c x\right )\right )}}{c^{6} d^{3} x^{6} - 3 \, c^{4} d^{3} x^{4} + 3 \, c^{2} d^{3} x^{2} - d^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-sqrt(-c^2*d*x^2 + d)*(a^2*g*x + a^2*f + (b^2*g*x + b^2*f)*arcsin(c*x)^2 + 2*(a*b*g*x + a*b*f)*arcsin
(c*x))/(c^6*d^3*x^6 - 3*c^4*d^3*x^4 + 3*c^2*d^3*x^2 - d^3), x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [B]  time = 0.89, size = 5897, normalized size = 9.20 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{3} \, a b c f {\left (\frac {1}{c^{4} d^{\frac {5}{2}} x^{2} - c^{2} d^{\frac {5}{2}}} + \frac {2 \, \log \left (c x + 1\right )}{c^{2} d^{\frac {5}{2}}} + \frac {2 \, \log \left (c x - 1\right )}{c^{2} d^{\frac {5}{2}}}\right )} + \frac {2}{3} \, a b f {\left (\frac {2 \, x}{\sqrt {-c^{2} d x^{2} + d} d^{2}} + \frac {x}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} d}\right )} \arcsin \left (c x\right ) + \frac {1}{3} \, a^{2} f {\left (\frac {2 \, x}{\sqrt {-c^{2} d x^{2} + d} d^{2}} + \frac {x}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} d}\right )} + \sqrt {d} \int \frac {2 \, a b g x \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right ) + {\left (b^{2} g x + b^{2} f\right )} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )^{2}}{{\left (c^{4} d^{3} x^{4} - 2 \, c^{2} d^{3} x^{2} + d^{3}\right )} \sqrt {c x + 1} \sqrt {-c x + 1}}\,{d x} + \frac {a^{2} g}{3 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} c^{2} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*a*b*c*f*(1/(c^4*d^(5/2)*x^2 - c^2*d^(5/2)) + 2*log(c*x + 1)/(c^2*d^(5/2)) + 2*log(c*x - 1)/(c^2*d^(5/2)))
+ 2/3*a*b*f*(2*x/(sqrt(-c^2*d*x^2 + d)*d^2) + x/((-c^2*d*x^2 + d)^(3/2)*d))*arcsin(c*x) + 1/3*a^2*f*(2*x/(sqrt
(-c^2*d*x^2 + d)*d^2) + x/((-c^2*d*x^2 + d)^(3/2)*d)) + sqrt(d)*integrate((2*a*b*g*x*arctan2(c*x, sqrt(c*x + 1
)*sqrt(-c*x + 1)) + (b^2*g*x + b^2*f)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2)/((c^4*d^3*x^4 - 2*c^2*d^3*
x^2 + d^3)*sqrt(c*x + 1)*sqrt(-c*x + 1)), x) + 1/3*a^2*g/((-c^2*d*x^2 + d)^(3/2)*c^2*d)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\left (f+g\,x\right )\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2}{{\left (d-c^2\,d\,x^2\right )}^{5/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2} \left (f + g x\right )}{\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a + b*asin(c*x))**2*(f + g*x)/(-d*(c*x - 1)*(c*x + 1))**(5/2), x)

________________________________________________________________________________________