\(\int (b x)^m \arccos (a x)^3 \, dx\) [120]

Optimal result

Integrand size = 12, antiderivative size = 12 \[ \int (b x)^m \arccos (a x)^3 \, dx=\frac {(b x)^{1+m} \arccos (a x)^3}{b (1+m)}+\frac {3 a \text {Int}\left (\frac {(b x)^{1+m} \arccos (a x)^2}{\sqrt {1-a^2 x^2}},x\right )}{b (1+m)} \]

[Out]

(b*x)^(1+m)*arccos(a*x)^3/b/(1+m)+3*a*Unintegrable((b*x)^(1+m)*arccos(a*x)^2/(-a^2*x^2+1)^(1/2),x)/b/(1+m)

Rubi [N/A]

Not integrable

Time = 0.08 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int (b x)^m \arccos (a x)^3 \, dx=\int (b x)^m \arccos (a x)^3 \, dx \]

[In]

Int[(b*x)^m*ArcCos[a*x]^3,x]

[Out]

((b*x)^(1 + m)*ArcCos[a*x]^3)/(b*(1 + m)) + (3*a*Defer[Int][((b*x)^(1 + m)*ArcCos[a*x]^2)/Sqrt[1 - a^2*x^2], x
])/(b*(1 + m))

Rubi steps \begin{align*} \text {integral}& = \frac {(b x)^{1+m} \arccos (a x)^3}{b (1+m)}+\frac {(3 a) \int \frac {(b x)^{1+m} \arccos (a x)^2}{\sqrt {1-a^2 x^2}} \, dx}{b (1+m)} \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 0.56 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17 \[ \int (b x)^m \arccos (a x)^3 \, dx=\int (b x)^m \arccos (a x)^3 \, dx \]

[In]

Integrate[(b*x)^m*ArcCos[a*x]^3,x]

[Out]

Integrate[(b*x)^m*ArcCos[a*x]^3, x]

Maple [N/A] (verified)

Not integrable

Time = 1.86 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00

\[\int \left (b x \right )^{m} \arccos \left (a x \right )^{3}d x\]

[In]

int((b*x)^m*arccos(a*x)^3,x)

[Out]

int((b*x)^m*arccos(a*x)^3,x)

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17 \[ \int (b x)^m \arccos (a x)^3 \, dx=\int { \left (b x\right )^{m} \arccos \left (a x\right )^{3} \,d x } \]

[In]

integrate((b*x)^m*arccos(a*x)^3,x, algorithm="fricas")

[Out]

integral((b*x)^m*arccos(a*x)^3, x)

Sympy [N/A]

Not integrable

Time = 2.90 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00 \[ \int (b x)^m \arccos (a x)^3 \, dx=\int \left (b x\right )^{m} \operatorname {acos}^{3}{\left (a x \right )}\, dx \]

[In]

integrate((b*x)**m*acos(a*x)**3,x)

[Out]

Integral((b*x)**m*acos(a*x)**3, x)

Maxima [N/A]

Not integrable

Time = 0.68 (sec) , antiderivative size = 115, normalized size of antiderivative = 9.58 \[ \int (b x)^m \arccos (a x)^3 \, dx=\int { \left (b x\right )^{m} \arccos \left (a x\right )^{3} \,d x } \]

[In]

integrate((b*x)^m*arccos(a*x)^3,x, algorithm="maxima")

[Out]

(b^m*x*x^m*arctan2(sqrt(a*x + 1)*sqrt(-a*x + 1), a*x)^3 - 3*(a*b^m*m + a*b^m)*integrate(sqrt(a*x + 1)*sqrt(-a*
x + 1)*x*x^m*arctan2(sqrt(a*x + 1)*sqrt(-a*x + 1), a*x)^2/((a^2*m + a^2)*x^2 - m - 1), x))/(m + 1)

Giac [N/A]

Not integrable

Time = 0.64 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17 \[ \int (b x)^m \arccos (a x)^3 \, dx=\int { \left (b x\right )^{m} \arccos \left (a x\right )^{3} \,d x } \]

[In]

integrate((b*x)^m*arccos(a*x)^3,x, algorithm="giac")

[Out]

integrate((b*x)^m*arccos(a*x)^3, x)

Mupad [N/A]

Not integrable

Time = 0.24 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17 \[ \int (b x)^m \arccos (a x)^3 \, dx=\int {\mathrm {acos}\left (a\,x\right )}^3\,{\left (b\,x\right )}^m \,d x \]

[In]

int(acos(a*x)^3*(b*x)^m,x)

[Out]

int(acos(a*x)^3*(b*x)^m, x)