3.3.44 \(\int \frac {x^5 (a+b \text {ArcSin}(c x))^2}{(d-c^2 d x^2)^{3/2}} \, dx\) [244]

Optimal. Leaf size=549 \[ -\frac {16 a b x \sqrt {1-c^2 x^2}}{3 c^5 d \sqrt {d-c^2 d x^2}}-\frac {32 b^2 \left (1-c^2 x^2\right )}{9 c^6 d \sqrt {d-c^2 d x^2}}+\frac {2 b^2 \left (1-c^2 x^2\right )^2}{27 c^6 d \sqrt {d-c^2 d x^2}}-\frac {16 b^2 x \sqrt {1-c^2 x^2} \text {ArcSin}(c x)}{3 c^5 d \sqrt {d-c^2 d x^2}}+\frac {2 b x \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{c^5 d \sqrt {d-c^2 d x^2}}-\frac {2 b x^3 \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{9 c^3 d \sqrt {d-c^2 d x^2}}+\frac {x^4 (a+b \text {ArcSin}(c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {8 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^2}{3 c^6 d^2}+\frac {4 x^2 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^2}{3 c^4 d^2}+\frac {4 i b \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x)) \text {ArcTan}\left (e^{i \text {ArcSin}(c x)}\right )}{c^6 d \sqrt {d-c^2 d x^2}}-\frac {2 i b^2 \sqrt {1-c^2 x^2} \text {PolyLog}\left (2,-i e^{i \text {ArcSin}(c x)}\right )}{c^6 d \sqrt {d-c^2 d x^2}}+\frac {2 i b^2 \sqrt {1-c^2 x^2} \text {PolyLog}\left (2,i e^{i \text {ArcSin}(c x)}\right )}{c^6 d \sqrt {d-c^2 d x^2}} \]

[Out]

-32/9*b^2*(-c^2*x^2+1)/c^6/d/(-c^2*d*x^2+d)^(1/2)+2/27*b^2*(-c^2*x^2+1)^2/c^6/d/(-c^2*d*x^2+d)^(1/2)+x^4*(a+b*
arcsin(c*x))^2/c^2/d/(-c^2*d*x^2+d)^(1/2)-16/3*a*b*x*(-c^2*x^2+1)^(1/2)/c^5/d/(-c^2*d*x^2+d)^(1/2)-16/3*b^2*x*
arcsin(c*x)*(-c^2*x^2+1)^(1/2)/c^5/d/(-c^2*d*x^2+d)^(1/2)+2*b*x*(a+b*arcsin(c*x))*(-c^2*x^2+1)^(1/2)/c^5/d/(-c
^2*d*x^2+d)^(1/2)-2/9*b*x^3*(a+b*arcsin(c*x))*(-c^2*x^2+1)^(1/2)/c^3/d/(-c^2*d*x^2+d)^(1/2)+4*I*b*(a+b*arcsin(
c*x))*arctan(I*c*x+(-c^2*x^2+1)^(1/2))*(-c^2*x^2+1)^(1/2)/c^6/d/(-c^2*d*x^2+d)^(1/2)-2*I*b^2*polylog(2,-I*(I*c
*x+(-c^2*x^2+1)^(1/2)))*(-c^2*x^2+1)^(1/2)/c^6/d/(-c^2*d*x^2+d)^(1/2)+2*I*b^2*polylog(2,I*(I*c*x+(-c^2*x^2+1)^
(1/2)))*(-c^2*x^2+1)^(1/2)/c^6/d/(-c^2*d*x^2+d)^(1/2)+8/3*(a+b*arcsin(c*x))^2*(-c^2*d*x^2+d)^(1/2)/c^6/d^2+4/3
*x^2*(a+b*arcsin(c*x))^2*(-c^2*d*x^2+d)^(1/2)/c^4/d^2

________________________________________________________________________________________

Rubi [A]
time = 0.52, antiderivative size = 549, normalized size of antiderivative = 1.00, number of steps used = 22, number of rules used = 12, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.414, Rules used = {4791, 4795, 4767, 4715, 267, 4723, 272, 45, 4749, 4266, 2317, 2438} \begin {gather*} \frac {4 i b \sqrt {1-c^2 x^2} \text {ArcTan}\left (e^{i \text {ArcSin}(c x)}\right ) (a+b \text {ArcSin}(c x))}{c^6 d \sqrt {d-c^2 d x^2}}+\frac {x^4 (a+b \text {ArcSin}(c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {8 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^2}{3 c^6 d^2}+\frac {2 b x \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{c^5 d \sqrt {d-c^2 d x^2}}+\frac {4 x^2 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^2}{3 c^4 d^2}-\frac {2 b x^3 \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{9 c^3 d \sqrt {d-c^2 d x^2}}-\frac {16 a b x \sqrt {1-c^2 x^2}}{3 c^5 d \sqrt {d-c^2 d x^2}}-\frac {2 i b^2 \sqrt {1-c^2 x^2} \text {Li}_2\left (-i e^{i \text {ArcSin}(c x)}\right )}{c^6 d \sqrt {d-c^2 d x^2}}+\frac {2 i b^2 \sqrt {1-c^2 x^2} \text {Li}_2\left (i e^{i \text {ArcSin}(c x)}\right )}{c^6 d \sqrt {d-c^2 d x^2}}-\frac {16 b^2 x \sqrt {1-c^2 x^2} \text {ArcSin}(c x)}{3 c^5 d \sqrt {d-c^2 d x^2}}+\frac {2 b^2 \left (1-c^2 x^2\right )^2}{27 c^6 d \sqrt {d-c^2 d x^2}}-\frac {32 b^2 \left (1-c^2 x^2\right )}{9 c^6 d \sqrt {d-c^2 d x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-16*a*b*x*Sqrt[1 - c^2*x^2])/(3*c^5*d*Sqrt[d - c^2*d*x^2]) - (32*b^2*(1 - c^2*x^2))/(9*c^6*d*Sqrt[d - c^2*d*x
^2]) + (2*b^2*(1 - c^2*x^2)^2)/(27*c^6*d*Sqrt[d - c^2*d*x^2]) - (16*b^2*x*Sqrt[1 - c^2*x^2]*ArcSin[c*x])/(3*c^
5*d*Sqrt[d - c^2*d*x^2]) + (2*b*x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(c^5*d*Sqrt[d - c^2*d*x^2]) - (2*b*x^
3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(9*c^3*d*Sqrt[d - c^2*d*x^2]) + (x^4*(a + b*ArcSin[c*x])^2)/(c^2*d*Sq
rt[d - c^2*d*x^2]) + (8*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(3*c^6*d^2) + (4*x^2*Sqrt[d - c^2*d*x^2]*(a
+ b*ArcSin[c*x])^2)/(3*c^4*d^2) + ((4*I)*b*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*ArcTan[E^(I*ArcSin[c*x])])/(
c^6*d*Sqrt[d - c^2*d*x^2]) - ((2*I)*b^2*Sqrt[1 - c^2*x^2]*PolyLog[2, (-I)*E^(I*ArcSin[c*x])])/(c^6*d*Sqrt[d -
c^2*d*x^2]) + ((2*I)*b^2*Sqrt[1 - c^2*x^2]*PolyLog[2, I*E^(I*ArcSin[c*x])])/(c^6*d*Sqrt[d - c^2*d*x^2])

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

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 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

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 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 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 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 4791

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/(2*e*(p + 1))), x] + (-Dist[f^2*((m - 1)/(2*e*(p + 1
))), Int[(f*x)^(m - 2)*(d + e*x^2)^(p + 1)*(a + b*ArcSin[c*x])^n, x], x] + Dist[b*f*(n/(2*c*(p + 1)))*Simp[(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}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && LtQ[p, -1] && IGtQ[m, 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]

Rubi steps

\begin {align*} \int \frac {x^5 \left (a+b \sin ^{-1}(c x)\right )^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx &=\frac {x^4 \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {4 \int \frac {x^3 \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt {d-c^2 d x^2}} \, dx}{c^2 d}-\frac {\left (2 b \sqrt {1-c^2 x^2}\right ) \int \frac {x^4 \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 b x^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 c^3 d \sqrt {d-c^2 d x^2}}+\frac {x^4 \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {4 x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^4 d^2}-\frac {8 \int \frac {x \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt {d-c^2 d x^2}} \, dx}{3 c^4 d}-\frac {\left (2 b \sqrt {1-c^2 x^2}\right ) \int \frac {x^2 \left (a+b \sin ^{-1}(c x)\right )}{1-c^2 x^2} \, dx}{c^3 d \sqrt {d-c^2 d x^2}}-\frac {\left (8 b \sqrt {1-c^2 x^2}\right ) \int x^2 \left (a+b \sin ^{-1}(c x)\right ) \, dx}{3 c^3 d \sqrt {d-c^2 d x^2}}-\frac {\left (2 b^2 \sqrt {1-c^2 x^2}\right ) \int \frac {x^3}{\sqrt {1-c^2 x^2}} \, dx}{3 c^2 d \sqrt {d-c^2 d x^2}}\\ &=\frac {2 b x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c^5 d \sqrt {d-c^2 d x^2}}-\frac {2 b x^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 c^3 d \sqrt {d-c^2 d x^2}}+\frac {x^4 \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {8 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^6 d^2}+\frac {4 x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^4 d^2}-\frac {\left (2 b \sqrt {1-c^2 x^2}\right ) \int \frac {a+b \sin ^{-1}(c x)}{1-c^2 x^2} \, dx}{c^5 d \sqrt {d-c^2 d x^2}}-\frac {\left (16 b \sqrt {1-c^2 x^2}\right ) \int \left (a+b \sin ^{-1}(c x)\right ) \, dx}{3 c^5 d \sqrt {d-c^2 d x^2}}-\frac {\left (2 b^2 \sqrt {1-c^2 x^2}\right ) \int \frac {x}{\sqrt {1-c^2 x^2}} \, dx}{c^4 d \sqrt {d-c^2 d x^2}}-\frac {\left (b^2 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {x}{\sqrt {1-c^2 x}} \, dx,x,x^2\right )}{3 c^2 d \sqrt {d-c^2 d x^2}}+\frac {\left (8 b^2 \sqrt {1-c^2 x^2}\right ) \int \frac {x^3}{\sqrt {1-c^2 x^2}} \, dx}{9 c^2 d \sqrt {d-c^2 d x^2}}\\ &=-\frac {16 a b x \sqrt {1-c^2 x^2}}{3 c^5 d \sqrt {d-c^2 d x^2}}+\frac {2 b^2 \left (1-c^2 x^2\right )}{c^6 d \sqrt {d-c^2 d x^2}}+\frac {2 b x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c^5 d \sqrt {d-c^2 d x^2}}-\frac {2 b x^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 c^3 d \sqrt {d-c^2 d x^2}}+\frac {x^4 \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {8 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^6 d^2}+\frac {4 x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^4 d^2}-\frac {\left (2 b \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int (a+b x) \sec (x) \, dx,x,\sin ^{-1}(c x)\right )}{c^6 d \sqrt {d-c^2 d x^2}}-\frac {\left (16 b^2 \sqrt {1-c^2 x^2}\right ) \int \sin ^{-1}(c x) \, dx}{3 c^5 d \sqrt {d-c^2 d x^2}}-\frac {\left (b^2 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \left (\frac {1}{c^2 \sqrt {1-c^2 x}}-\frac {\sqrt {1-c^2 x}}{c^2}\right ) \, dx,x,x^2\right )}{3 c^2 d \sqrt {d-c^2 d x^2}}+\frac {\left (4 b^2 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {x}{\sqrt {1-c^2 x}} \, dx,x,x^2\right )}{9 c^2 d \sqrt {d-c^2 d x^2}}\\ &=-\frac {16 a b x \sqrt {1-c^2 x^2}}{3 c^5 d \sqrt {d-c^2 d x^2}}+\frac {8 b^2 \left (1-c^2 x^2\right )}{3 c^6 d \sqrt {d-c^2 d x^2}}-\frac {2 b^2 \left (1-c^2 x^2\right )^2}{9 c^6 d \sqrt {d-c^2 d x^2}}-\frac {16 b^2 x \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{3 c^5 d \sqrt {d-c^2 d x^2}}+\frac {2 b x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c^5 d \sqrt {d-c^2 d x^2}}-\frac {2 b x^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 c^3 d \sqrt {d-c^2 d x^2}}+\frac {x^4 \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {8 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^6 d^2}+\frac {4 x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^4 d^2}+\frac {4 i b \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^6 d \sqrt {d-c^2 d x^2}}+\frac {\left (2 b^2 \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^6 d \sqrt {d-c^2 d x^2}}-\frac {\left (2 b^2 \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^6 d \sqrt {d-c^2 d x^2}}+\frac {\left (16 b^2 \sqrt {1-c^2 x^2}\right ) \int \frac {x}{\sqrt {1-c^2 x^2}} \, dx}{3 c^4 d \sqrt {d-c^2 d x^2}}+\frac {\left (4 b^2 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \left (\frac {1}{c^2 \sqrt {1-c^2 x}}-\frac {\sqrt {1-c^2 x}}{c^2}\right ) \, dx,x,x^2\right )}{9 c^2 d \sqrt {d-c^2 d x^2}}\\ &=-\frac {16 a b x \sqrt {1-c^2 x^2}}{3 c^5 d \sqrt {d-c^2 d x^2}}-\frac {32 b^2 \left (1-c^2 x^2\right )}{9 c^6 d \sqrt {d-c^2 d x^2}}+\frac {2 b^2 \left (1-c^2 x^2\right )^2}{27 c^6 d \sqrt {d-c^2 d x^2}}-\frac {16 b^2 x \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{3 c^5 d \sqrt {d-c^2 d x^2}}+\frac {2 b x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c^5 d \sqrt {d-c^2 d x^2}}-\frac {2 b x^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 c^3 d \sqrt {d-c^2 d x^2}}+\frac {x^4 \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {8 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^6 d^2}+\frac {4 x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^4 d^2}+\frac {4 i b \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^6 d \sqrt {d-c^2 d x^2}}-\frac {\left (2 i b^2 \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^6 d \sqrt {d-c^2 d x^2}}+\frac {\left (2 i b^2 \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^6 d \sqrt {d-c^2 d x^2}}\\ &=-\frac {16 a b x \sqrt {1-c^2 x^2}}{3 c^5 d \sqrt {d-c^2 d x^2}}-\frac {32 b^2 \left (1-c^2 x^2\right )}{9 c^6 d \sqrt {d-c^2 d x^2}}+\frac {2 b^2 \left (1-c^2 x^2\right )^2}{27 c^6 d \sqrt {d-c^2 d x^2}}-\frac {16 b^2 x \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{3 c^5 d \sqrt {d-c^2 d x^2}}+\frac {2 b x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c^5 d \sqrt {d-c^2 d x^2}}-\frac {2 b x^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 c^3 d \sqrt {d-c^2 d x^2}}+\frac {x^4 \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {8 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^6 d^2}+\frac {4 x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^4 d^2}+\frac {4 i b \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^6 d \sqrt {d-c^2 d x^2}}-\frac {2 i b^2 \sqrt {1-c^2 x^2} \text {Li}_2\left (-i e^{i \sin ^{-1}(c x)}\right )}{c^6 d \sqrt {d-c^2 d x^2}}+\frac {2 i b^2 \sqrt {1-c^2 x^2} \text {Li}_2\left (i e^{i \sin ^{-1}(c x)}\right )}{c^6 d \sqrt {d-c^2 d x^2}}\\ \end {align*}

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Mathematica [A]
time = 0.47, size = 453, normalized size = 0.83 \begin {gather*} \frac {576 a^2-378 b^2-288 a^2 c^2 x^2-72 a^2 c^4 x^4+810 a b \text {ArcSin}(c x)+405 b^2 \text {ArcSin}(c x)^2-376 b^2 \cos (2 \text {ArcSin}(c x))+360 a b \text {ArcSin}(c x) \cos (2 \text {ArcSin}(c x))+180 b^2 \text {ArcSin}(c x)^2 \cos (2 \text {ArcSin}(c x))+2 b^2 \cos (4 \text {ArcSin}(c x))-18 a b \text {ArcSin}(c x) \cos (4 \text {ArcSin}(c x))-9 b^2 \text {ArcSin}(c x)^2 \cos (4 \text {ArcSin}(c x))-432 b^2 \sqrt {1-c^2 x^2} \text {ArcSin}(c x) \log \left (1-i e^{i \text {ArcSin}(c x)}\right )+432 b^2 \sqrt {1-c^2 x^2} \text {ArcSin}(c x) \log \left (1+i e^{i \text {ArcSin}(c x)}\right )+432 a b \sqrt {1-c^2 x^2} \log \left (\cos \left (\frac {1}{2} \text {ArcSin}(c x)\right )-\sin \left (\frac {1}{2} \text {ArcSin}(c x)\right )\right )-432 a b \sqrt {1-c^2 x^2} \log \left (\cos \left (\frac {1}{2} \text {ArcSin}(c x)\right )+\sin \left (\frac {1}{2} \text {ArcSin}(c x)\right )\right )-432 i b^2 \sqrt {1-c^2 x^2} \text {PolyLog}\left (2,-i e^{i \text {ArcSin}(c x)}\right )+432 i b^2 \sqrt {1-c^2 x^2} \text {PolyLog}\left (2,i e^{i \text {ArcSin}(c x)}\right )-372 a b \sin (2 \text {ArcSin}(c x))-372 b^2 \text {ArcSin}(c x) \sin (2 \text {ArcSin}(c x))+6 a b \sin (4 \text {ArcSin}(c x))+6 b^2 \text {ArcSin}(c x) \sin (4 \text {ArcSin}(c x))}{216 c^6 d \sqrt {d-c^2 d x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(576*a^2 - 378*b^2 - 288*a^2*c^2*x^2 - 72*a^2*c^4*x^4 + 810*a*b*ArcSin[c*x] + 405*b^2*ArcSin[c*x]^2 - 376*b^2*
Cos[2*ArcSin[c*x]] + 360*a*b*ArcSin[c*x]*Cos[2*ArcSin[c*x]] + 180*b^2*ArcSin[c*x]^2*Cos[2*ArcSin[c*x]] + 2*b^2
*Cos[4*ArcSin[c*x]] - 18*a*b*ArcSin[c*x]*Cos[4*ArcSin[c*x]] - 9*b^2*ArcSin[c*x]^2*Cos[4*ArcSin[c*x]] - 432*b^2
*Sqrt[1 - c^2*x^2]*ArcSin[c*x]*Log[1 - I*E^(I*ArcSin[c*x])] + 432*b^2*Sqrt[1 - c^2*x^2]*ArcSin[c*x]*Log[1 + I*
E^(I*ArcSin[c*x])] + 432*a*b*Sqrt[1 - c^2*x^2]*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]] - 432*a*b*Sqrt[1 -
c^2*x^2]*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]] - (432*I)*b^2*Sqrt[1 - c^2*x^2]*PolyLog[2, (-I)*E^(I*Ar
cSin[c*x])] + (432*I)*b^2*Sqrt[1 - c^2*x^2]*PolyLog[2, I*E^(I*ArcSin[c*x])] - 372*a*b*Sin[2*ArcSin[c*x]] - 372
*b^2*ArcSin[c*x]*Sin[2*ArcSin[c*x]] + 6*a*b*Sin[4*ArcSin[c*x]] + 6*b^2*ArcSin[c*x]*Sin[4*ArcSin[c*x]])/(216*c^
6*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. 1086 vs. \(2 (516 ) = 1032\).
time = 0.62, size = 1087, normalized size = 1.98

method	result	size
default	\(a^{2} \left (-\frac {x^{4}}{3 c^{2} d \sqrt {-c^{2} d \,x^{2}+d}}+\frac {-\frac {4 x^{2}}{3 c^{2} d \sqrt {-c^{2} d \,x^{2}+d}}+\frac {8}{3 d \,c^{4} \sqrt {-c^{2} d \,x^{2}+d}}}{c^{2}}\right )-\frac {b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right ) \sin \left (4 \arcsin \left (c x \right )\right )}{36 d^{2} c^{6} \left (c^{2} x^{2}-1\right )}-\frac {94 b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, x^{2}}{27 d^{2} c^{4} \left (c^{2} x^{2}-1\right )}-\frac {2 b^{2} \sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{2} c^{6} \left (c^{2} x^{2}-1\right )}+\frac {2 b^{2} \sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right ) \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{2} c^{6} \left (c^{2} x^{2}-1\right )}+\frac {2 i b^{2} \sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \dilog \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{2} c^{6} \left (c^{2} x^{2}-1\right )}-\frac {2 i b^{2} \sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \dilog \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{2} c^{6} \left (c^{2} x^{2}-1\right )}+\frac {31 b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, x}{9 d^{2} c^{5} \left (c^{2} x^{2}-1\right )}-\frac {65 b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right )^{2}}{24 d^{2} c^{6} \left (c^{2} x^{2}-1\right )}+\frac {377 b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}}{108 d^{2} c^{6} \left (c^{2} x^{2}-1\right )}+\frac {5 b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right )^{2} x^{2}}{3 d^{2} c^{4} \left (c^{2} x^{2}-1\right )}-\frac {b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \cos \left (4 \arcsin \left (c x \right )\right )}{108 d^{2} c^{6} \left (c^{2} x^{2}-1\right )}+\frac {b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \cos \left (4 \arcsin \left (c x \right )\right ) \arcsin \left (c x \right )^{2}}{24 d^{2} c^{6} \left (c^{2} x^{2}-1\right )}+\frac {10 a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right ) x^{2}}{3 d^{2} c^{4} \left (c^{2} x^{2}-1\right )}-\frac {65 a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right )}{12 d^{2} c^{6} \left (c^{2} x^{2}-1\right )}+\frac {2 a b \sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}+i\right )}{d^{2} c^{6} \left (c^{2} x^{2}-1\right )}-\frac {a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sin \left (4 \arcsin \left (c x \right )\right )}{36 d^{2} c^{6} \left (c^{2} x^{2}-1\right )}+\frac {a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right ) \cos \left (4 \arcsin \left (c x \right )\right )}{12 d^{2} c^{6} \left (c^{2} x^{2}-1\right )}-\frac {2 a b \sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-i\right )}{d^{2} c^{6} \left (c^{2} x^{2}-1\right )}+\frac {31 a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, x}{9 d^{2} c^{5} \left (c^{2} x^{2}-1\right )}\)	\(1087\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(a+b*arcsin(c*x))^2/(-c^2*d*x^2+d)^(3/2),x,method=_RETURNVERBOSE)

[Out]

a^2*(-1/3*x^4/c^2/d/(-c^2*d*x^2+d)^(1/2)+4/3/c^2*(-x^2/c^2/d/(-c^2*d*x^2+d)^(1/2)+2/d/c^4/(-c^2*d*x^2+d)^(1/2)
))-1/36*b^2*(-d*(c^2*x^2-1))^(1/2)/d^2/c^6/(c^2*x^2-1)*arcsin(c*x)*sin(4*arcsin(c*x))-94/27*b^2*(-d*(c^2*x^2-1
))^(1/2)/d^2/c^4/(c^2*x^2-1)*x^2-2*b^2*(-c^2*x^2+1)^(1/2)*(-d*(c^2*x^2-1))^(1/2)/d^2/c^6/(c^2*x^2-1)*arcsin(c*
x)*ln(1+I*(I*c*x+(-c^2*x^2+1)^(1/2)))+2*b^2*(-c^2*x^2+1)^(1/2)*(-d*(c^2*x^2-1))^(1/2)/d^2/c^6/(c^2*x^2-1)*arcs
in(c*x)*ln(1-I*(I*c*x+(-c^2*x^2+1)^(1/2)))-2*I*b^2*(-c^2*x^2+1)^(1/2)*(-d*(c^2*x^2-1))^(1/2)/d^2/c^6/(c^2*x^2-
1)*dilog(1-I*(I*c*x+(-c^2*x^2+1)^(1/2)))+2*I*b^2*(-c^2*x^2+1)^(1/2)*(-d*(c^2*x^2-1))^(1/2)/d^2/c^6/(c^2*x^2-1)
*dilog(1+I*(I*c*x+(-c^2*x^2+1)^(1/2)))+31/9*b^2*(-d*(c^2*x^2-1))^(1/2)/d^2/c^5/(c^2*x^2-1)*arcsin(c*x)*(-c^2*x
^2+1)^(1/2)*x-65/24*b^2*(-d*(c^2*x^2-1))^(1/2)/d^2/c^6/(c^2*x^2-1)*arcsin(c*x)^2+377/108*b^2*(-d*(c^2*x^2-1))^
(1/2)/d^2/c^6/(c^2*x^2-1)+5/3*b^2*(-d*(c^2*x^2-1))^(1/2)/d^2/c^4/(c^2*x^2-1)*arcsin(c*x)^2*x^2-1/108*b^2*(-d*(
c^2*x^2-1))^(1/2)/d^2/c^6/(c^2*x^2-1)*cos(4*arcsin(c*x))+1/24*b^2*(-d*(c^2*x^2-1))^(1/2)/d^2/c^6/(c^2*x^2-1)*c
os(4*arcsin(c*x))*arcsin(c*x)^2+10/3*a*b*(-d*(c^2*x^2-1))^(1/2)/d^2/c^4/(c^2*x^2-1)*arcsin(c*x)*x^2-65/12*a*b*
(-d*(c^2*x^2-1))^(1/2)/d^2/c^6/(c^2*x^2-1)*arcsin(c*x)+2*a*b*(-c^2*x^2+1)^(1/2)*(-d*(c^2*x^2-1))^(1/2)/d^2/c^6
/(c^2*x^2-1)*ln(I*c*x+(-c^2*x^2+1)^(1/2)+I)-1/36*a*b*(-d*(c^2*x^2-1))^(1/2)/d^2/c^6/(c^2*x^2-1)*sin(4*arcsin(c
*x))+1/12*a*b*(-d*(c^2*x^2-1))^(1/2)/d^2/c^6/(c^2*x^2-1)*arcsin(c*x)*cos(4*arcsin(c*x))-2*a*b*(-c^2*x^2+1)^(1/
2)*(-d*(c^2*x^2-1))^(1/2)/d^2/c^6/(c^2*x^2-1)*ln(I*c*x+(-c^2*x^2+1)^(1/2)-I)+31/9*a*b*(-d*(c^2*x^2-1))^(1/2)/d
^2/c^5/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*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(x^5*(a+b*arcsin(c*x))^2/(-c^2*d*x^2+d)^(3/2),x, algorithm="maxima")

[Out]

-1/3*a^2*(x^4/(sqrt(-c^2*d*x^2 + d)*c^2*d) + 4*x^2/(sqrt(-c^2*d*x^2 + d)*c^4*d) - 8/(sqrt(-c^2*d*x^2 + d)*c^6*
d)) + 1/3*((b^2*c^4*x^4 + 4*b^2*c^2*x^2 - 8*b^2)*sqrt(c*x + 1)*sqrt(-c*x + 1)*sqrt(d)*arctan2(c*x, sqrt(c*x +
1)*sqrt(-c*x + 1))^2 + 3*(c^8*d^2*x^2 - c^6*d^2)*integrate(2/3*(3*sqrt(c*x + 1)*sqrt(-c*x + 1)*a*b*c^5*sqrt(d)
*x^5*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)) - (b^2*c^6*x^6 + 3*b^2*c^4*x^4 - 12*b^2*c^2*x^2 + 8*b^2)*sqrt(
d)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)))/(c^9*d^2*x^4 - 2*c^7*d^2*x^2 + c^5*d^2), x))/(c^8*d^2*x^2 - c^6
*d^2)

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

[Out]

integral((b^2*x^5*arcsin(c*x)^2 + 2*a*b*x^5*arcsin(c*x) + a^2*x^5)*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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{5} \left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}}{\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**5*(a + b*asin(c*x))**2/(-d*(c*x - 1)*(c*x + 1))**(3/2), x)

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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(x^5*(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:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^5\,{\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((x^5*(a + b*asin(c*x))^2)/(d - c^2*d*x^2)^(3/2),x)

[Out]

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

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