3.220 \(\int x (d-c^2 d x^2)^{3/2} (a+b \sin ^{-1}(c x))^2 \, dx\)

Optimal. Leaf size=279 \[ \frac{2 b c^3 d x^5 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 \sqrt{1-c^2 x^2}}-\frac{4 b c d x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{15 \sqrt{1-c^2 x^2}}+\frac{2 b d x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{5 c \sqrt{1-c^2 x^2}}-\frac{\left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2}{5 c^2 d}+\frac{2 b^2 d \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2}}{125 c^2}+\frac{16 b^2 d \sqrt{d-c^2 d x^2}}{75 c^2}+\frac{8 b^2 d \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}{225 c^2} \]

[Out]

(16*b^2*d*Sqrt[d - c^2*d*x^2])/(75*c^2) + (8*b^2*d*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2])/(225*c^2) + (2*b^2*d*(1
- c^2*x^2)^2*Sqrt[d - c^2*d*x^2])/(125*c^2) + (2*b*d*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(5*c*Sqrt[1 -
c^2*x^2]) - (4*b*c*d*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(15*Sqrt[1 - c^2*x^2]) + (2*b*c^3*d*x^5*Sqrt
[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(25*Sqrt[1 - c^2*x^2]) - ((d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x])^2)/(5
*c^2*d)

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Rubi [A]  time = 0.228064, antiderivative size = 279, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {4677, 194, 4645, 12, 1247, 698} \[ \frac{2 b c^3 d x^5 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 \sqrt{1-c^2 x^2}}-\frac{4 b c d x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{15 \sqrt{1-c^2 x^2}}+\frac{2 b d x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{5 c \sqrt{1-c^2 x^2}}-\frac{\left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2}{5 c^2 d}+\frac{2 b^2 d \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2}}{125 c^2}+\frac{16 b^2 d \sqrt{d-c^2 d x^2}}{75 c^2}+\frac{8 b^2 d \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}{225 c^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(16*b^2*d*Sqrt[d - c^2*d*x^2])/(75*c^2) + (8*b^2*d*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2])/(225*c^2) + (2*b^2*d*(1
- c^2*x^2)^2*Sqrt[d - c^2*d*x^2])/(125*c^2) + (2*b*d*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(5*c*Sqrt[1 -
c^2*x^2]) - (4*b*c*d*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(15*Sqrt[1 - c^2*x^2]) + (2*b*c^3*d*x^5*Sqrt
[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(25*Sqrt[1 - c^2*x^2]) - ((d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x])^2)/(5
*c^2*d)

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 194

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

Rule 4645

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> With[{u = IntHide[(d + e*x^2)
^p, x]}, Dist[a + b*ArcSin[c*x], u, x] - Dist[b*c, Int[SimplifyIntegrand[u/Sqrt[1 - c^2*x^2], x], x], x]] /; F
reeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 1247

Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[
Int[(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x]

Rule 698

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*
e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

\begin{align*} \int x \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx &=-\frac{\left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2}{5 c^2 d}+\frac{\left (2 b d \sqrt{d-c^2 d x^2}\right ) \int \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right ) \, dx}{5 c \sqrt{1-c^2 x^2}}\\ &=\frac{2 b d x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{5 c \sqrt{1-c^2 x^2}}-\frac{4 b c d x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{15 \sqrt{1-c^2 x^2}}+\frac{2 b c^3 d x^5 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 \sqrt{1-c^2 x^2}}-\frac{\left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2}{5 c^2 d}-\frac{\left (2 b^2 d \sqrt{d-c^2 d x^2}\right ) \int \frac{x \left (15-10 c^2 x^2+3 c^4 x^4\right )}{15 \sqrt{1-c^2 x^2}} \, dx}{5 \sqrt{1-c^2 x^2}}\\ &=\frac{2 b d x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{5 c \sqrt{1-c^2 x^2}}-\frac{4 b c d x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{15 \sqrt{1-c^2 x^2}}+\frac{2 b c^3 d x^5 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 \sqrt{1-c^2 x^2}}-\frac{\left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2}{5 c^2 d}-\frac{\left (2 b^2 d \sqrt{d-c^2 d x^2}\right ) \int \frac{x \left (15-10 c^2 x^2+3 c^4 x^4\right )}{\sqrt{1-c^2 x^2}} \, dx}{75 \sqrt{1-c^2 x^2}}\\ &=\frac{2 b d x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{5 c \sqrt{1-c^2 x^2}}-\frac{4 b c d x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{15 \sqrt{1-c^2 x^2}}+\frac{2 b c^3 d x^5 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 \sqrt{1-c^2 x^2}}-\frac{\left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2}{5 c^2 d}-\frac{\left (b^2 d \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{15-10 c^2 x+3 c^4 x^2}{\sqrt{1-c^2 x}} \, dx,x,x^2\right )}{75 \sqrt{1-c^2 x^2}}\\ &=\frac{2 b d x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{5 c \sqrt{1-c^2 x^2}}-\frac{4 b c d x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{15 \sqrt{1-c^2 x^2}}+\frac{2 b c^3 d x^5 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 \sqrt{1-c^2 x^2}}-\frac{\left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2}{5 c^2 d}-\frac{\left (b^2 d \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \left (\frac{8}{\sqrt{1-c^2 x}}+4 \sqrt{1-c^2 x}+3 \left (1-c^2 x\right )^{3/2}\right ) \, dx,x,x^2\right )}{75 \sqrt{1-c^2 x^2}}\\ &=\frac{16 b^2 d \sqrt{d-c^2 d x^2}}{75 c^2}+\frac{8 b^2 d \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}{225 c^2}+\frac{2 b^2 d \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2}}{125 c^2}+\frac{2 b d x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{5 c \sqrt{1-c^2 x^2}}-\frac{4 b c d x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{15 \sqrt{1-c^2 x^2}}+\frac{2 b c^3 d x^5 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 \sqrt{1-c^2 x^2}}-\frac{\left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2}{5 c^2 d}\\ \end{align*}

Mathematica [A]  time = 0.179759, size = 159, normalized size = 0.57 \[ \frac{2 b d \sqrt{d-c^2 d x^2} \left (15 a c x \left (3 c^4 x^4-10 c^2 x^2+15\right )+b \sqrt{1-c^2 x^2} \left (9 c^4 x^4-38 c^2 x^2+149\right )+15 b c x \left (3 c^4 x^4-10 c^2 x^2+15\right ) \sin ^{-1}(c x)\right )}{1125 c^2 \sqrt{1-c^2 x^2}}-\frac{\left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2}{5 c^2 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x])^2)/(5*c^2*d) + (2*b*d*Sqrt[d - c^2*d*x^2]*(15*a*c*x*(15 - 10*c^2*x
^2 + 3*c^4*x^4) + b*Sqrt[1 - c^2*x^2]*(149 - 38*c^2*x^2 + 9*c^4*x^4) + 15*b*c*x*(15 - 10*c^2*x^2 + 3*c^4*x^4)*
ArcSin[c*x]))/(1125*c^2*Sqrt[1 - c^2*x^2])

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Maple [C]  time = 0.306, size = 1224, normalized size = 4.4 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/5*a^2/c^2/d*(-c^2*d*x^2+d)^(5/2)+b^2*(-1/4000*(-d*(c^2*x^2-1))^(1/2)*(16*c^6*x^6-28*c^4*x^4-16*I*(-c^2*x^2+
1)^(1/2)*x^5*c^5+13*c^2*x^2+20*I*(-c^2*x^2+1)^(1/2)*x^3*c^3-5*I*(-c^2*x^2+1)^(1/2)*x*c-1)*(10*I*arcsin(c*x)+25
*arcsin(c*x)^2-2)*d/c^2/(c^2*x^2-1)+1/288*(-d*(c^2*x^2-1))^(1/2)*(4*c^4*x^4-5*c^2*x^2-4*I*(-c^2*x^2+1)^(1/2)*x
^3*c^3+3*I*(-c^2*x^2+1)^(1/2)*x*c+1)*(6*I*arcsin(c*x)+9*arcsin(c*x)^2-2)*d/c^2/(c^2*x^2-1)-1/16*(-d*(c^2*x^2-1
))^(1/2)*(c^2*x^2-I*(-c^2*x^2+1)^(1/2)*x*c-1)*(arcsin(c*x)^2-2+2*I*arcsin(c*x))*d/c^2/(c^2*x^2-1)-1/16*(-d*(c^
2*x^2-1))^(1/2)*(I*(-c^2*x^2+1)^(1/2)*x*c+c^2*x^2-1)*(arcsin(c*x)^2-2-2*I*arcsin(c*x))*d/c^2/(c^2*x^2-1)+1/288
*(-d*(c^2*x^2-1))^(1/2)*(4*I*(-c^2*x^2+1)^(1/2)*x^3*c^3+4*c^4*x^4-3*I*(-c^2*x^2+1)^(1/2)*x*c-5*c^2*x^2+1)*(-6*
I*arcsin(c*x)+9*arcsin(c*x)^2-2)*d/c^2/(c^2*x^2-1)-1/4000*(-d*(c^2*x^2-1))^(1/2)*(16*I*(-c^2*x^2+1)^(1/2)*x^5*
c^5+16*c^6*x^6-20*I*(-c^2*x^2+1)^(1/2)*x^3*c^3-28*c^4*x^4+5*I*(-c^2*x^2+1)^(1/2)*x*c+13*c^2*x^2-1)*(-10*I*arcs
in(c*x)+25*arcsin(c*x)^2-2)*d/c^2/(c^2*x^2-1))+2*a*b*(-1/800*(-d*(c^2*x^2-1))^(1/2)*(16*c^6*x^6-28*c^4*x^4-16*
I*(-c^2*x^2+1)^(1/2)*x^5*c^5+13*c^2*x^2+20*I*(-c^2*x^2+1)^(1/2)*x^3*c^3-5*I*(-c^2*x^2+1)^(1/2)*x*c-1)*(I+5*arc
sin(c*x))*d/c^2/(c^2*x^2-1)+1/96*(-d*(c^2*x^2-1))^(1/2)*(4*c^4*x^4-5*c^2*x^2-4*I*(-c^2*x^2+1)^(1/2)*x^3*c^3+3*
I*(-c^2*x^2+1)^(1/2)*x*c+1)*(I+3*arcsin(c*x))*d/c^2/(c^2*x^2-1)-1/16*(-d*(c^2*x^2-1))^(1/2)*(c^2*x^2-I*(-c^2*x
^2+1)^(1/2)*x*c-1)*(arcsin(c*x)+I)*d/c^2/(c^2*x^2-1)-1/16*(-d*(c^2*x^2-1))^(1/2)*(I*(-c^2*x^2+1)^(1/2)*x*c+c^2
*x^2-1)*(arcsin(c*x)-I)*d/c^2/(c^2*x^2-1)+1/96*(-d*(c^2*x^2-1))^(1/2)*(4*I*(-c^2*x^2+1)^(1/2)*x^3*c^3+4*c^4*x^
4-3*I*(-c^2*x^2+1)^(1/2)*x*c-5*c^2*x^2+1)*(-I+3*arcsin(c*x))*d/c^2/(c^2*x^2-1)-1/800*(-d*(c^2*x^2-1))^(1/2)*(1
6*I*(-c^2*x^2+1)^(1/2)*x^5*c^5+16*c^6*x^6-20*I*(-c^2*x^2+1)^(1/2)*x^3*c^3-28*c^4*x^4+5*I*(-c^2*x^2+1)^(1/2)*x*
c+13*c^2*x^2-1)*(-I+5*arcsin(c*x))*d/c^2/(c^2*x^2-1))

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Maxima [A]  time = 1.67837, size = 319, normalized size = 1.14 \begin{align*} -\frac{{\left (-c^{2} d x^{2} + d\right )}^{\frac{5}{2}} b^{2} \arcsin \left (c x\right )^{2}}{5 \, c^{2} d} + \frac{2}{1125} \, b^{2}{\left (\frac{9 \, \sqrt{-c^{2} x^{2} + 1} c^{2} d^{\frac{5}{2}} x^{4} - 38 \, \sqrt{-c^{2} x^{2} + 1} d^{\frac{5}{2}} x^{2} + \frac{149 \, \sqrt{-c^{2} x^{2} + 1} d^{\frac{5}{2}}}{c^{2}}}{d} + \frac{15 \,{\left (3 \, c^{4} d^{\frac{5}{2}} x^{5} - 10 \, c^{2} d^{\frac{5}{2}} x^{3} + 15 \, d^{\frac{5}{2}} x\right )} \arcsin \left (c x\right )}{c d}\right )} - \frac{2 \,{\left (-c^{2} d x^{2} + d\right )}^{\frac{5}{2}} a b \arcsin \left (c x\right )}{5 \, c^{2} d} - \frac{{\left (-c^{2} d x^{2} + d\right )}^{\frac{5}{2}} a^{2}}{5 \, c^{2} d} + \frac{2 \,{\left (3 \, c^{4} d^{\frac{5}{2}} x^{5} - 10 \, c^{2} d^{\frac{5}{2}} x^{3} + 15 \, d^{\frac{5}{2}} x\right )} a b}{75 \, c d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/5*(-c^2*d*x^2 + d)^(5/2)*b^2*arcsin(c*x)^2/(c^2*d) + 2/1125*b^2*((9*sqrt(-c^2*x^2 + 1)*c^2*d^(5/2)*x^4 - 38
*sqrt(-c^2*x^2 + 1)*d^(5/2)*x^2 + 149*sqrt(-c^2*x^2 + 1)*d^(5/2)/c^2)/d + 15*(3*c^4*d^(5/2)*x^5 - 10*c^2*d^(5/
2)*x^3 + 15*d^(5/2)*x)*arcsin(c*x)/(c*d)) - 2/5*(-c^2*d*x^2 + d)^(5/2)*a*b*arcsin(c*x)/(c^2*d) - 1/5*(-c^2*d*x
^2 + d)^(5/2)*a^2/(c^2*d) + 2/75*(3*c^4*d^(5/2)*x^5 - 10*c^2*d^(5/2)*x^3 + 15*d^(5/2)*x)*a*b/(c*d)

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Fricas [A]  time = 2.95549, size = 657, normalized size = 2.35 \begin{align*} -\frac{30 \,{\left (3 \, a b c^{5} d x^{5} - 10 \, a b c^{3} d x^{3} + 15 \, a b c d x +{\left (3 \, b^{2} c^{5} d x^{5} - 10 \, b^{2} c^{3} d x^{3} + 15 \, b^{2} c d x\right )} \arcsin \left (c x\right )\right )} \sqrt{-c^{2} d x^{2} + d} \sqrt{-c^{2} x^{2} + 1} +{\left (9 \,{\left (25 \, a^{2} - 2 \, b^{2}\right )} c^{6} d x^{6} -{\left (675 \, a^{2} - 94 \, b^{2}\right )} c^{4} d x^{4} +{\left (675 \, a^{2} - 374 \, b^{2}\right )} c^{2} d x^{2} + 225 \,{\left (b^{2} c^{6} d x^{6} - 3 \, b^{2} c^{4} d x^{4} + 3 \, b^{2} c^{2} d x^{2} - b^{2} d\right )} \arcsin \left (c x\right )^{2} -{\left (225 \, a^{2} - 298 \, b^{2}\right )} d + 450 \,{\left (a b c^{6} d x^{6} - 3 \, a b c^{4} d x^{4} + 3 \, a b c^{2} d x^{2} - a b d\right )} \arcsin \left (c x\right )\right )} \sqrt{-c^{2} d x^{2} + d}}{1125 \,{\left (c^{4} x^{2} - c^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/1125*(30*(3*a*b*c^5*d*x^5 - 10*a*b*c^3*d*x^3 + 15*a*b*c*d*x + (3*b^2*c^5*d*x^5 - 10*b^2*c^3*d*x^3 + 15*b^2*
c*d*x)*arcsin(c*x))*sqrt(-c^2*d*x^2 + d)*sqrt(-c^2*x^2 + 1) + (9*(25*a^2 - 2*b^2)*c^6*d*x^6 - (675*a^2 - 94*b^
2)*c^4*d*x^4 + (675*a^2 - 374*b^2)*c^2*d*x^2 + 225*(b^2*c^6*d*x^6 - 3*b^2*c^4*d*x^4 + 3*b^2*c^2*d*x^2 - b^2*d)
*arcsin(c*x)^2 - (225*a^2 - 298*b^2)*d + 450*(a*b*c^6*d*x^6 - 3*a*b*c^4*d*x^4 + 3*a*b*c^2*d*x^2 - a*b*d)*arcsi
n(c*x))*sqrt(-c^2*d*x^2 + d))/(c^4*x^2 - c^2)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (-c^{2} d x^{2} + d\right )}^{\frac{3}{2}}{\left (b \arcsin \left (c x\right ) + a\right )}^{2} x\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((-c^2*d*x^2 + d)^(3/2)*(b*arcsin(c*x) + a)^2*x, x)