3.2.53 \(\int x^2 (a+b \text {ArcCos}(c x))^3 \, dx\) [153]

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

[Out]

-4/3*a*b^2*x/c^2-2/27*b^3*(-c^2*x^2+1)^(3/2)/c^3-4/3*b^3*x*arccos(c*x)/c^2-2/9*b^2*x^3*(a+b*arccos(c*x))+1/3*x
^3*(a+b*arccos(c*x))^3+14/9*b^3*(-c^2*x^2+1)^(1/2)/c^3-2/3*b*(a+b*arccos(c*x))^2*(-c^2*x^2+1)^(1/2)/c^3-1/3*b*
x^2*(a+b*arccos(c*x))^2*(-c^2*x^2+1)^(1/2)/c

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Rubi [A]
time = 0.21, antiderivative size = 178, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4724, 4796, 4768, 4716, 267, 272, 45} \begin {gather*} -\frac {2}{9} b^2 x^3 (a+b \text {ArcCos}(c x))-\frac {b x^2 \sqrt {1-c^2 x^2} (a+b \text {ArcCos}(c x))^2}{3 c}-\frac {2 b \sqrt {1-c^2 x^2} (a+b \text {ArcCos}(c x))^2}{3 c^3}+\frac {1}{3} x^3 (a+b \text {ArcCos}(c x))^3-\frac {4 a b^2 x}{3 c^2}-\frac {4 b^3 x \text {ArcCos}(c x)}{3 c^2}-\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} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^2*(a + b*ArcCos[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
cCos[c*x])/(3*c^2) - (2*b^2*x^3*(a + b*ArcCos[c*x]))/9 - (2*b*Sqrt[1 - c^2*x^2]*(a + b*ArcCos[c*x])^2)/(3*c^3)
- (b*x^2*Sqrt[1 - c^2*x^2]*(a + b*ArcCos[c*x])^2)/(3*c) + (x^3*(a + b*ArcCos[c*x])^3)/3

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 4716

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*ArcCos[c*x])^n, x] + Dist[b*c*n, Int[
x*((a + b*ArcCos[c*x])^(n - 1)/Sqrt[1 - c^2*x^2]), x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]

Rule 4724

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*ArcCo
s[c*x])^n/(d*(m + 1))), x] + Dist[b*c*(n/(d*(m + 1))), Int[(d*x)^(m + 1)*((a + b*ArcCos[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 4768

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x^2)^(
p + 1)*((a + b*ArcCos[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*ArcCos[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 4796

Int[((a_.) + ArcCos[(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*ArcCos[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*ArcCos[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*ArcCos[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 x^2 \left (a+b \cos ^{-1}(c x)\right )^3 \, dx &=\frac {1}{3} x^3 \left (a+b \cos ^{-1}(c x)\right )^3+(b c) \int \frac {x^3 \left (a+b \cos ^{-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 \cos ^{-1}(c x)\right )^2}{3 c}+\frac {1}{3} x^3 \left (a+b \cos ^{-1}(c x)\right )^3-\frac {1}{3} \left (2 b^2\right ) \int x^2 \left (a+b \cos ^{-1}(c x)\right ) \, dx+\frac {(2 b) \int \frac {x \left (a+b \cos ^{-1}(c x)\right )^2}{\sqrt {1-c^2 x^2}} \, dx}{3 c}\\ &=-\frac {2}{9} b^2 x^3 \left (a+b \cos ^{-1}(c x)\right )-\frac {2 b \sqrt {1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{3 c^3}-\frac {b x^2 \sqrt {1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{3 c}+\frac {1}{3} x^3 \left (a+b \cos ^{-1}(c x)\right )^3-\frac {\left (4 b^2\right ) \int \left (a+b \cos ^{-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 \cos ^{-1}(c x)\right )-\frac {2 b \sqrt {1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{3 c^3}-\frac {b x^2 \sqrt {1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{3 c}+\frac {1}{3} x^3 \left (a+b \cos ^{-1}(c x)\right )^3-\frac {\left (4 b^3\right ) \int \cos ^{-1}(c x) \, dx}{3 c^2}-\frac {1}{9} \left (b^3 c\right ) \text {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 \cos ^{-1}(c x)}{3 c^2}-\frac {2}{9} b^2 x^3 \left (a+b \cos ^{-1}(c x)\right )-\frac {2 b \sqrt {1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{3 c^3}-\frac {b x^2 \sqrt {1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{3 c}+\frac {1}{3} x^3 \left (a+b \cos ^{-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 ) \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 )\\ &=-\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 \cos ^{-1}(c x)}{3 c^2}-\frac {2}{9} b^2 x^3 \left (a+b \cos ^{-1}(c x)\right )-\frac {2 b \sqrt {1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{3 c^3}-\frac {b x^2 \sqrt {1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{3 c}+\frac {1}{3} x^3 \left (a+b \cos ^{-1}(c x)\right )^3\\ \end {align*}

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Mathematica [A]
time = 0.13, size = 218, normalized size = 1.22 \begin {gather*} \frac {9 a^3 c^3 x^3-9 a^2 b \sqrt {1-c^2 x^2} \left (2+c^2 x^2\right )-6 a b^2 c x \left (6+c^2 x^2\right )+2 b^3 \sqrt {1-c^2 x^2} \left (20+c^2 x^2\right )-3 b \left (-9 a^2 c^3 x^3+6 a b \sqrt {1-c^2 x^2} \left (2+c^2 x^2\right )+2 b^2 c x \left (6+c^2 x^2\right )\right ) \text {ArcCos}(c x)-9 b^2 \left (-3 a c^3 x^3+b \sqrt {1-c^2 x^2} \left (2+c^2 x^2\right )\right ) \text {ArcCos}(c x)^2+9 b^3 c^3 x^3 \text {ArcCos}(c x)^3}{27 c^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(9*a^3*c^3*x^3 - 9*a^2*b*Sqrt[1 - c^2*x^2]*(2 + c^2*x^2) - 6*a*b^2*c*x*(6 + c^2*x^2) + 2*b^3*Sqrt[1 - c^2*x^2]
*(20 + c^2*x^2) - 3*b*(-9*a^2*c^3*x^3 + 6*a*b*Sqrt[1 - c^2*x^2]*(2 + c^2*x^2) + 2*b^2*c*x*(6 + c^2*x^2))*ArcCo
s[c*x] - 9*b^2*(-3*a*c^3*x^3 + b*Sqrt[1 - c^2*x^2]*(2 + c^2*x^2))*ArcCos[c*x]^2 + 9*b^3*c^3*x^3*ArcCos[c*x]^3)
/(27*c^3)

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Maple [A]
time = 0.05, size = 235, normalized size = 1.32

method	result	size
derivativedivides	\(\frac {\frac {c^{3} x^{3} a^{3}}{3}+b^{3} \left (\frac {c^{3} x^{3} \arccos \left (c x \right )^{3}}{3}-\frac {\arccos \left (c x \right )^{2} \left (c^{2} x^{2}+2\right ) \sqrt {-c^{2} x^{2}+1}}{3}+\frac {4 \sqrt {-c^{2} x^{2}+1}}{3}-\frac {4 c x \arccos \left (c x \right )}{3}-\frac {2 c^{3} x^{3} \arccos \left (c x \right )}{9}+\frac {2 \left (c^{2} x^{2}+2\right ) \sqrt {-c^{2} x^{2}+1}}{27}\right )+3 a \,b^{2} \left (\frac {c^{3} x^{3} \arccos \left (c x \right )^{2}}{3}-\frac {2 \arccos \left (c x \right ) \left (c^{2} x^{2}+2\right ) \sqrt {-c^{2} x^{2}+1}}{9}-\frac {2 c^{3} x^{3}}{27}-\frac {4 c x}{9}\right )+3 a^{2} b \left (\frac {c^{3} x^{3} \arccos \left (c x \right )}{3}-\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{9}-\frac {2 \sqrt {-c^{2} x^{2}+1}}{9}\right )}{c^{3}}\)	\(235\)
default	\(\frac {\frac {c^{3} x^{3} a^{3}}{3}+b^{3} \left (\frac {c^{3} x^{3} \arccos \left (c x \right )^{3}}{3}-\frac {\arccos \left (c x \right )^{2} \left (c^{2} x^{2}+2\right ) \sqrt {-c^{2} x^{2}+1}}{3}+\frac {4 \sqrt {-c^{2} x^{2}+1}}{3}-\frac {4 c x \arccos \left (c x \right )}{3}-\frac {2 c^{3} x^{3} \arccos \left (c x \right )}{9}+\frac {2 \left (c^{2} x^{2}+2\right ) \sqrt {-c^{2} x^{2}+1}}{27}\right )+3 a \,b^{2} \left (\frac {c^{3} x^{3} \arccos \left (c x \right )^{2}}{3}-\frac {2 \arccos \left (c x \right ) \left (c^{2} x^{2}+2\right ) \sqrt {-c^{2} x^{2}+1}}{9}-\frac {2 c^{3} x^{3}}{27}-\frac {4 c x}{9}\right )+3 a^{2} b \left (\frac {c^{3} x^{3} \arccos \left (c x \right )}{3}-\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{9}-\frac {2 \sqrt {-c^{2} x^{2}+1}}{9}\right )}{c^{3}}\)	\(235\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(a+b*arccos(c*x))^3,x,method=_RETURNVERBOSE)

[Out]

1/c^3*(1/3*c^3*x^3*a^3+b^3*(1/3*c^3*x^3*arccos(c*x)^3-1/3*arccos(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*arccos(c*x)-2/9*c^3*x^3*arccos(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*arccos(c*x)^2-2/9*arccos(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*arccos(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 = 0.48, size = 273, normalized size = 1.53 \begin {gather*} \frac {1}{3} \, b^{3} x^{3} \arccos \left (c x\right )^{3} + a b^{2} x^{3} \arccos \left (c x\right )^{2} + \frac {1}{3} \, a^{3} x^{3} + \frac {1}{3} \, {\left (3 \, x^{3} \arccos \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 )} \arccos \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 )} \arccos \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 )} \arccos \left (c x\right )}{c^{3}}\right )}\right )} b^{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*b^3*x^3*arccos(c*x)^3 + a*b^2*x^3*arccos(c*x)^2 + 1/3*a^3*x^3 + 1/3*(3*x^3*arccos(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)*arccos(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)*arccos(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)*arccos(c
*x)/c^3))*b^3

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Fricas [A]
time = 1.38, size = 195, normalized size = 1.10 \begin {gather*} \frac {9 \, b^{3} c^{3} x^{3} \arccos \left (c x\right )^{3} + 27 \, a b^{2} c^{3} x^{3} \arccos \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 )} \arccos \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 )} \arccos \left (c x\right )^{2} + 18 \, {\left (a b^{2} c^{2} x^{2} + 2 \, a b^{2}\right )} \arccos \left (c x\right )\right )} \sqrt {-c^{2} x^{2} + 1}}{27 \, c^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/27*(9*b^3*c^3*x^3*arccos(c*x)^3 + 27*a*b^2*c^3*x^3*arccos(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)*arccos(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)*arccos(c*x)^2 + 18*(a*b^2*c^2*x^2 + 2*a*b^2)*arccos(c*x))*sqrt(-c^2*x^2 + 1))/c^3

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Sympy [A]
time = 0.41, size = 333, normalized size = 1.87 \begin {gather*} \begin {cases} \frac {a^{3} x^{3}}{3} + a^{2} b x^{3} \operatorname {acos}{\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 {acos}^{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 {acos}{\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 {acos}{\left (c x \right )}}{3 c^{3}} + \frac {b^{3} x^{3} \operatorname {acos}^{3}{\left (c x \right )}}{3} - \frac {2 b^{3} x^{3} \operatorname {acos}{\left (c x \right )}}{9} - \frac {b^{3} x^{2} \sqrt {- c^{2} x^{2} + 1} \operatorname {acos}^{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 {acos}{\left (c x \right )}}{3 c^{2}} - \frac {2 b^{3} \sqrt {- c^{2} x^{2} + 1} \operatorname {acos}^{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 {x^{3} \left (a + \frac {\pi b}{2}\right )^{3}}{3} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 0.44, size = 289, normalized size = 1.62 \begin {gather*} \frac {1}{3} \, b^{3} x^{3} \arccos \left (c x\right )^{3} + a b^{2} x^{3} \arccos \left (c x\right )^{2} + a^{2} b x^{3} \arccos \left (c x\right ) - \frac {2}{9} \, b^{3} x^{3} \arccos \left (c x\right ) - \frac {\sqrt {-c^{2} x^{2} + 1} b^{3} x^{2} \arccos \left (c x\right )^{2}}{3 \, c} + \frac {1}{3} \, a^{3} x^{3} - \frac {2}{9} \, a b^{2} x^{3} - \frac {2 \, \sqrt {-c^{2} x^{2} + 1} a b^{2} x^{2} \arccos \left (c x\right )}{3 \, c} - \frac {\sqrt {-c^{2} x^{2} + 1} a^{2} b x^{2}}{3 \, c} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1} b^{3} x^{2}}{27 \, c} - \frac {4 \, b^{3} x \arccos \left (c x\right )}{3 \, c^{2}} - \frac {2 \, \sqrt {-c^{2} x^{2} + 1} b^{3} \arccos \left (c x\right )^{2}}{3 \, c^{3}} - \frac {4 \, a b^{2} x}{3 \, c^{2}} - \frac {4 \, \sqrt {-c^{2} x^{2} + 1} a b^{2} \arccos \left (c x\right )}{3 \, c^{3}} - \frac {2 \, \sqrt {-c^{2} x^{2} + 1} a^{2} b}{3 \, c^{3}} + \frac {40 \, \sqrt {-c^{2} x^{2} + 1} b^{3}}{27 \, c^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*b^3*x^3*arccos(c*x)^3 + a*b^2*x^3*arccos(c*x)^2 + a^2*b*x^3*arccos(c*x) - 2/9*b^3*x^3*arccos(c*x) - 1/3*sq
rt(-c^2*x^2 + 1)*b^3*x^2*arccos(c*x)^2/c + 1/3*a^3*x^3 - 2/9*a*b^2*x^3 - 2/3*sqrt(-c^2*x^2 + 1)*a*b^2*x^2*arcc
os(c*x)/c - 1/3*sqrt(-c^2*x^2 + 1)*a^2*b*x^2/c + 2/27*sqrt(-c^2*x^2 + 1)*b^3*x^2/c - 4/3*b^3*x*arccos(c*x)/c^2
- 2/3*sqrt(-c^2*x^2 + 1)*b^3*arccos(c*x)^2/c^3 - 4/3*a*b^2*x/c^2 - 4/3*sqrt(-c^2*x^2 + 1)*a*b^2*arccos(c*x)/c
^3 - 2/3*sqrt(-c^2*x^2 + 1)*a^2*b/c^3 + 40/27*sqrt(-c^2*x^2 + 1)*b^3/c^3

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int x^2\,{\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )}^3 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(a + b*acos(c*x))^3,x)

[Out]

int(x^2*(a + b*acos(c*x))^3, x)

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