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

Optimal. Leaf size=178 \[ -\frac{4 a b^2 x}{3 c^2}-\frac{2}{9} b^2 x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac{b x^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c}+\frac{2 b \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^3}+\frac{1}{3} x^3 \left (a+b \sin ^{-1}(c x)\right )^3+\frac{2 b^3 \left (1-c^2 x^2\right )^{3/2}}{27 c^3}-\frac{14 b^3 \sqrt{1-c^2 x^2}}{9 c^3}-\frac{4 b^3 x \sin ^{-1}(c x)}{3 c^2} \]

[Out]

(-4*a*b^2*x)/(3*c^2) - (14*b^3*Sqrt[1 - c^2*x^2])/(9*c^3) + (2*b^3*(1 - c^2*x^2)^(3/2))/(27*c^3) - (4*b^3*x*Ar
cSin[c*x])/(3*c^2) - (2*b^2*x^3*(a + b*ArcSin[c*x]))/9 + (2*b*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/(3*c^3)
+ (b*x^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/(3*c) + (x^3*(a + b*ArcSin[c*x])^3)/3

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Rubi [A]  time = 0.296877, antiderivative size = 178, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {4627, 4707, 4677, 4619, 261, 266, 43} \[ -\frac{4 a b^2 x}{3 c^2}-\frac{2}{9} b^2 x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac{b x^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c}+\frac{2 b \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^3}+\frac{1}{3} x^3 \left (a+b \sin ^{-1}(c x)\right )^3+\frac{2 b^3 \left (1-c^2 x^2\right )^{3/2}}{27 c^3}-\frac{14 b^3 \sqrt{1-c^2 x^2}}{9 c^3}-\frac{4 b^3 x \sin ^{-1}(c x)}{3 c^2} \]

Antiderivative was successfully verified.

[In]

Int[x^2*(a + b*ArcSin[c*x])^3,x]

[Out]

(-4*a*b^2*x)/(3*c^2) - (14*b^3*Sqrt[1 - c^2*x^2])/(9*c^3) + (2*b^3*(1 - c^2*x^2)^(3/2))/(27*c^3) - (4*b^3*x*Ar
cSin[c*x])/(3*c^2) - (2*b^2*x^3*(a + b*ArcSin[c*x]))/9 + (2*b*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/(3*c^3)
+ (b*x^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/(3*c) + (x^3*(a + b*ArcSin[c*x])^3)/3

Rule 4627

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 4707

Int[(((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[
(f*(f*x)^(m - 1)*Sqrt[d + e*x^2]*(a + b*ArcSin[c*x])^n)/(e*m), x] + (Dist[(f^2*(m - 1))/(c^2*m), Int[((f*x)^(m
- 2)*(a + b*ArcSin[c*x])^n)/Sqrt[d + e*x^2], x], x] + Dist[(b*f*n*Sqrt[1 - c^2*x^2])/(c*m*Sqrt[d + e*x^2]), I
nt[(f*x)^(m - 1)*(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] && GtQ[m, 1] && IntegerQ[m]

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 4619

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 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 266

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 43

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])

Rubi steps

\begin{align*} \int x^2 \left (a+b \sin ^{-1}(c x)\right )^3 \, dx &=\frac{1}{3} x^3 \left (a+b \sin ^{-1}(c x)\right )^3-(b c) \int \frac{x^3 \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt{1-c^2 x^2}} \, dx\\ &=\frac{b x^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c}+\frac{1}{3} x^3 \left (a+b \sin ^{-1}(c x)\right )^3-\frac{1}{3} \left (2 b^2\right ) \int x^2 \left (a+b \sin ^{-1}(c x)\right ) \, dx-\frac{(2 b) \int \frac{x \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt{1-c^2 x^2}} \, dx}{3 c}\\ &=-\frac{2}{9} b^2 x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac{2 b \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^3}+\frac{b x^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c}+\frac{1}{3} x^3 \left (a+b \sin ^{-1}(c x)\right )^3-\frac{\left (4 b^2\right ) \int \left (a+b \sin ^{-1}(c x)\right ) \, dx}{3 c^2}+\frac{1}{9} \left (2 b^3 c\right ) \int \frac{x^3}{\sqrt{1-c^2 x^2}} \, dx\\ &=-\frac{4 a b^2 x}{3 c^2}-\frac{2}{9} b^2 x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac{2 b \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^3}+\frac{b x^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c}+\frac{1}{3} x^3 \left (a+b \sin ^{-1}(c x)\right )^3-\frac{\left (4 b^3\right ) \int \sin ^{-1}(c x) \, dx}{3 c^2}+\frac{1}{9} \left (b^3 c\right ) \operatorname{Subst}\left (\int \frac{x}{\sqrt{1-c^2 x}} \, dx,x,x^2\right )\\ &=-\frac{4 a b^2 x}{3 c^2}-\frac{4 b^3 x \sin ^{-1}(c x)}{3 c^2}-\frac{2}{9} b^2 x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac{2 b \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^3}+\frac{b x^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c}+\frac{1}{3} x^3 \left (a+b \sin ^{-1}(c x)\right )^3+\frac{\left (4 b^3\right ) \int \frac{x}{\sqrt{1-c^2 x^2}} \, dx}{3 c}+\frac{1}{9} \left (b^3 c\right ) \operatorname{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 )\\ &=-\frac{4 a b^2 x}{3 c^2}-\frac{14 b^3 \sqrt{1-c^2 x^2}}{9 c^3}+\frac{2 b^3 \left (1-c^2 x^2\right )^{3/2}}{27 c^3}-\frac{4 b^3 x \sin ^{-1}(c x)}{3 c^2}-\frac{2}{9} b^2 x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac{2 b \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^3}+\frac{b x^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c}+\frac{1}{3} x^3 \left (a+b \sin ^{-1}(c x)\right )^3\\ \end{align*}

Mathematica [A]  time = 0.40945, size = 163, normalized size = 0.92 \[ \frac{1}{27} \left (\frac{b \left (9 c^2 x^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2-2 b \left (3 c^3 x^3 \left (a+b \sin ^{-1}(c x)\right )+b \sqrt{1-c^2 x^2} \left (c^2 x^2+2\right )\right )+18 \left (\sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2-2 b \left (a c x+b \sqrt{1-c^2 x^2}+b c x \sin ^{-1}(c x)\right )\right )\right )}{c^3}+9 x^3 \left (a+b \sin ^{-1}(c x)\right )^3\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(9*x^3*(a + b*ArcSin[c*x])^3 + (b*(9*c^2*x^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2 - 2*b*(b*Sqrt[1 - c^2*x^2
]*(2 + c^2*x^2) + 3*c^3*x^3*(a + b*ArcSin[c*x])) + 18*(Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2 - 2*b*(a*c*x +
b*Sqrt[1 - c^2*x^2] + b*c*x*ArcSin[c*x]))))/c^3)/27

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Maple [A]  time = 0.029, size = 235, normalized size = 1.3 \begin{align*}{\frac{1}{{c}^{3}} \left ({\frac{{a}^{3}{c}^{3}{x}^{3}}{3}}+{b}^{3} \left ({\frac{{c}^{3}{x}^{3} \left ( \arcsin \left ( cx \right ) \right ) ^{3}}{3}}+{\frac{ \left ( \arcsin \left ( cx \right ) \right ) ^{2} \left ({c}^{2}{x}^{2}+2 \right ) }{3}\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{4}{3}\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{4\,cx\arcsin \left ( cx \right ) }{3}}-{\frac{2\,{c}^{3}{x}^{3}\arcsin \left ( cx \right ) }{9}}-{\frac{2\,{c}^{2}{x}^{2}+4}{27}\sqrt{-{c}^{2}{x}^{2}+1}} \right ) +3\,a{b}^{2} \left ( 1/3\,{c}^{3}{x}^{3} \left ( \arcsin \left ( cx \right ) \right ) ^{2}+2/9\,\arcsin \left ( cx \right ) \left ({c}^{2}{x}^{2}+2 \right ) \sqrt{-{c}^{2}{x}^{2}+1}-{\frac{2\,{c}^{3}{x}^{3}}{27}}-4/9\,cx \right ) +3\,{a}^{2}b \left ( 1/3\,{c}^{3}{x}^{3}\arcsin \left ( cx \right ) +1/9\,{c}^{2}{x}^{2}\sqrt{-{c}^{2}{x}^{2}+1}+2/9\,\sqrt{-{c}^{2}{x}^{2}+1} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/c^3*(1/3*a^3*c^3*x^3+b^3*(1/3*c^3*x^3*arcsin(c*x)^3+1/3*arcsin(c*x)^2*(c^2*x^2+2)*(-c^2*x^2+1)^(1/2)-4/3*(-c
^2*x^2+1)^(1/2)-4/3*c*x*arcsin(c*x)-2/9*c^3*x^3*arcsin(c*x)-2/27*(c^2*x^2+2)*(-c^2*x^2+1)^(1/2))+3*a*b^2*(1/3*
c^3*x^3*arcsin(c*x)^2+2/9*arcsin(c*x)*(c^2*x^2+2)*(-c^2*x^2+1)^(1/2)-2/27*c^3*x^3-4/9*c*x)+3*a^2*b*(1/3*c^3*x^
3*arcsin(c*x)+1/9*c^2*x^2*(-c^2*x^2+1)^(1/2)+2/9*(-c^2*x^2+1)^(1/2)))

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Maxima [A]  time = 1.88801, size = 369, normalized size = 2.07 \begin{align*} \frac{1}{3} \, b^{3} x^{3} \arcsin \left (c x\right )^{3} + a b^{2} x^{3} \arcsin \left (c x\right )^{2} + \frac{1}{3} \, a^{3} x^{3} + \frac{1}{3} \,{\left (3 \, x^{3} \arcsin \left (c x\right ) + c{\left (\frac{\sqrt{-c^{2} x^{2} + 1} x^{2}}{c^{2}} + \frac{2 \, \sqrt{-c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} a^{2} b + \frac{2}{9} \,{\left (3 \, c{\left (\frac{\sqrt{-c^{2} x^{2} + 1} x^{2}}{c^{2}} + \frac{2 \, \sqrt{-c^{2} x^{2} + 1}}{c^{4}}\right )} \arcsin \left (c x\right ) - \frac{c^{2} x^{3} + 6 \, x}{c^{2}}\right )} a b^{2} + \frac{1}{27} \,{\left (9 \, c{\left (\frac{\sqrt{-c^{2} x^{2} + 1} x^{2}}{c^{2}} + \frac{2 \, \sqrt{-c^{2} x^{2} + 1}}{c^{4}}\right )} \arcsin \left (c x\right )^{2} - 2 \, c{\left (\frac{\sqrt{-c^{2} x^{2} + 1} x^{2} + \frac{20 \, \sqrt{-c^{2} x^{2} + 1}}{c^{2}}}{c^{2}} + \frac{3 \,{\left (c^{2} x^{3} + 6 \, x\right )} \arcsin \left (c x\right )}{c^{3}}\right )}\right )} b^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*b^3*x^3*arcsin(c*x)^3 + a*b^2*x^3*arcsin(c*x)^2 + 1/3*a^3*x^3 + 1/3*(3*x^3*arcsin(c*x) + c*(sqrt(-c^2*x^2
+ 1)*x^2/c^2 + 2*sqrt(-c^2*x^2 + 1)/c^4))*a^2*b + 2/9*(3*c*(sqrt(-c^2*x^2 + 1)*x^2/c^2 + 2*sqrt(-c^2*x^2 + 1)/
c^4)*arcsin(c*x) - (c^2*x^3 + 6*x)/c^2)*a*b^2 + 1/27*(9*c*(sqrt(-c^2*x^2 + 1)*x^2/c^2 + 2*sqrt(-c^2*x^2 + 1)/c
^4)*arcsin(c*x)^2 - 2*c*((sqrt(-c^2*x^2 + 1)*x^2 + 20*sqrt(-c^2*x^2 + 1)/c^2)/c^2 + 3*(c^2*x^3 + 6*x)*arcsin(c
*x)/c^3))*b^3

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Fricas [A]  time = 1.80444, size = 441, normalized size = 2.48 \begin{align*} \frac{9 \, b^{3} c^{3} x^{3} \arcsin \left (c x\right )^{3} + 27 \, a b^{2} c^{3} x^{3} \arcsin \left (c x\right )^{2} + 3 \,{\left (3 \, a^{3} - 2 \, a b^{2}\right )} c^{3} x^{3} - 36 \, a b^{2} c x + 3 \,{\left ({\left (9 \, a^{2} b - 2 \, b^{3}\right )} c^{3} x^{3} - 12 \, b^{3} c x\right )} \arcsin \left (c x\right ) +{\left ({\left (9 \, a^{2} b - 2 \, b^{3}\right )} c^{2} x^{2} + 18 \, a^{2} b - 40 \, b^{3} + 9 \,{\left (b^{3} c^{2} x^{2} + 2 \, b^{3}\right )} \arcsin \left (c x\right )^{2} + 18 \,{\left (a b^{2} c^{2} x^{2} + 2 \, a b^{2}\right )} \arcsin \left (c x\right )\right )} \sqrt{-c^{2} x^{2} + 1}}{27 \, c^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/27*(9*b^3*c^3*x^3*arcsin(c*x)^3 + 27*a*b^2*c^3*x^3*arcsin(c*x)^2 + 3*(3*a^3 - 2*a*b^2)*c^3*x^3 - 36*a*b^2*c*
x + 3*((9*a^2*b - 2*b^3)*c^3*x^3 - 12*b^3*c*x)*arcsin(c*x) + ((9*a^2*b - 2*b^3)*c^2*x^2 + 18*a^2*b - 40*b^3 +
9*(b^3*c^2*x^2 + 2*b^3)*arcsin(c*x)^2 + 18*(a*b^2*c^2*x^2 + 2*a*b^2)*arcsin(c*x))*sqrt(-c^2*x^2 + 1))/c^3

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Sympy [A]  time = 3.74468, size = 328, normalized size = 1.84 \begin{align*} \begin{cases} \frac{a^{3} x^{3}}{3} + a^{2} b x^{3} \operatorname{asin}{\left (c x \right )} + \frac{a^{2} b x^{2} \sqrt{- c^{2} x^{2} + 1}}{3 c} + \frac{2 a^{2} b \sqrt{- c^{2} x^{2} + 1}}{3 c^{3}} + a b^{2} x^{3} \operatorname{asin}^{2}{\left (c x \right )} - \frac{2 a b^{2} x^{3}}{9} + \frac{2 a b^{2} x^{2} \sqrt{- c^{2} x^{2} + 1} \operatorname{asin}{\left (c x \right )}}{3 c} - \frac{4 a b^{2} x}{3 c^{2}} + \frac{4 a b^{2} \sqrt{- c^{2} x^{2} + 1} \operatorname{asin}{\left (c x \right )}}{3 c^{3}} + \frac{b^{3} x^{3} \operatorname{asin}^{3}{\left (c x \right )}}{3} - \frac{2 b^{3} x^{3} \operatorname{asin}{\left (c x \right )}}{9} + \frac{b^{3} x^{2} \sqrt{- c^{2} x^{2} + 1} \operatorname{asin}^{2}{\left (c x \right )}}{3 c} - \frac{2 b^{3} x^{2} \sqrt{- c^{2} x^{2} + 1}}{27 c} - \frac{4 b^{3} x \operatorname{asin}{\left (c x \right )}}{3 c^{2}} + \frac{2 b^{3} \sqrt{- c^{2} x^{2} + 1} \operatorname{asin}^{2}{\left (c x \right )}}{3 c^{3}} - \frac{40 b^{3} \sqrt{- c^{2} x^{2} + 1}}{27 c^{3}} & \text{for}\: c \neq 0 \\\frac{a^{3} x^{3}}{3} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a**3*x**3/3 + a**2*b*x**3*asin(c*x) + a**2*b*x**2*sqrt(-c**2*x**2 + 1)/(3*c) + 2*a**2*b*sqrt(-c**2*
x**2 + 1)/(3*c**3) + a*b**2*x**3*asin(c*x)**2 - 2*a*b**2*x**3/9 + 2*a*b**2*x**2*sqrt(-c**2*x**2 + 1)*asin(c*x)
/(3*c) - 4*a*b**2*x/(3*c**2) + 4*a*b**2*sqrt(-c**2*x**2 + 1)*asin(c*x)/(3*c**3) + b**3*x**3*asin(c*x)**3/3 - 2
*b**3*x**3*asin(c*x)/9 + b**3*x**2*sqrt(-c**2*x**2 + 1)*asin(c*x)**2/(3*c) - 2*b**3*x**2*sqrt(-c**2*x**2 + 1)/
(27*c) - 4*b**3*x*asin(c*x)/(3*c**2) + 2*b**3*sqrt(-c**2*x**2 + 1)*asin(c*x)**2/(3*c**3) - 40*b**3*sqrt(-c**2*
x**2 + 1)/(27*c**3), Ne(c, 0)), (a**3*x**3/3, True))

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Giac [B]  time = 1.45838, size = 497, normalized size = 2.79 \begin{align*} \frac{1}{3} \, a^{3} x^{3} + \frac{{\left (c^{2} x^{2} - 1\right )} b^{3} x \arcsin \left (c x\right )^{3}}{3 \, c^{2}} + \frac{{\left (c^{2} x^{2} - 1\right )} a b^{2} x \arcsin \left (c x\right )^{2}}{c^{2}} + \frac{b^{3} x \arcsin \left (c x\right )^{3}}{3 \, c^{2}} + \frac{{\left (c^{2} x^{2} - 1\right )} a^{2} b x \arcsin \left (c x\right )}{c^{2}} - \frac{2 \,{\left (c^{2} x^{2} - 1\right )} b^{3} x \arcsin \left (c x\right )}{9 \, c^{2}} + \frac{a b^{2} x \arcsin \left (c x\right )^{2}}{c^{2}} - \frac{{\left (-c^{2} x^{2} + 1\right )}^{\frac{3}{2}} b^{3} \arcsin \left (c x\right )^{2}}{3 \, c^{3}} - \frac{2 \,{\left (c^{2} x^{2} - 1\right )} a b^{2} x}{9 \, c^{2}} + \frac{a^{2} b x \arcsin \left (c x\right )}{c^{2}} - \frac{14 \, b^{3} x \arcsin \left (c x\right )}{9 \, c^{2}} - \frac{2 \,{\left (-c^{2} x^{2} + 1\right )}^{\frac{3}{2}} a b^{2} \arcsin \left (c x\right )}{3 \, c^{3}} + \frac{\sqrt{-c^{2} x^{2} + 1} b^{3} \arcsin \left (c x\right )^{2}}{c^{3}} - \frac{14 \, a b^{2} x}{9 \, c^{2}} - \frac{{\left (-c^{2} x^{2} + 1\right )}^{\frac{3}{2}} a^{2} b}{3 \, c^{3}} + \frac{2 \,{\left (-c^{2} x^{2} + 1\right )}^{\frac{3}{2}} b^{3}}{27 \, c^{3}} + \frac{2 \, \sqrt{-c^{2} x^{2} + 1} a b^{2} \arcsin \left (c x\right )}{c^{3}} + \frac{\sqrt{-c^{2} x^{2} + 1} a^{2} b}{c^{3}} - \frac{14 \, \sqrt{-c^{2} x^{2} + 1} b^{3}}{9 \, c^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*a^3*x^3 + 1/3*(c^2*x^2 - 1)*b^3*x*arcsin(c*x)^3/c^2 + (c^2*x^2 - 1)*a*b^2*x*arcsin(c*x)^2/c^2 + 1/3*b^3*x*
arcsin(c*x)^3/c^2 + (c^2*x^2 - 1)*a^2*b*x*arcsin(c*x)/c^2 - 2/9*(c^2*x^2 - 1)*b^3*x*arcsin(c*x)/c^2 + a*b^2*x*
arcsin(c*x)^2/c^2 - 1/3*(-c^2*x^2 + 1)^(3/2)*b^3*arcsin(c*x)^2/c^3 - 2/9*(c^2*x^2 - 1)*a*b^2*x/c^2 + a^2*b*x*a
rcsin(c*x)/c^2 - 14/9*b^3*x*arcsin(c*x)/c^2 - 2/3*(-c^2*x^2 + 1)^(3/2)*a*b^2*arcsin(c*x)/c^3 + sqrt(-c^2*x^2 +
1)*b^3*arcsin(c*x)^2/c^3 - 14/9*a*b^2*x/c^2 - 1/3*(-c^2*x^2 + 1)^(3/2)*a^2*b/c^3 + 2/27*(-c^2*x^2 + 1)^(3/2)*
b^3/c^3 + 2*sqrt(-c^2*x^2 + 1)*a*b^2*arcsin(c*x)/c^3 + sqrt(-c^2*x^2 + 1)*a^2*b/c^3 - 14/9*sqrt(-c^2*x^2 + 1)*
b^3/c^3