\(\int \sqrt {b x} \arccos (a x)^n \, dx\) [137]

Optimal result

Integrand size = 14, antiderivative size = 14 \[ \int \sqrt {b x} \arccos (a x)^n \, dx=\text {Int}\left (\sqrt {b x} \arccos (a x)^n,x\right ) \]

[Out]

Unintegrable(arccos(a*x)^n*(b*x)^(1/2),x)

Rubi [N/A]

Not integrable

Time = 0.01 (sec) , antiderivative size = 14, 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 = {} \[ \int \sqrt {b x} \arccos (a x)^n \, dx=\int \sqrt {b x} \arccos (a x)^n \, dx \]

[In]

Int[Sqrt[b*x]*ArcCos[a*x]^n,x]

[Out]

Defer[Int][Sqrt[b*x]*ArcCos[a*x]^n, x]

Rubi steps \begin{align*} \text {integral}& = \int \sqrt {b x} \arccos (a x)^n \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 3.15 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.14 \[ \int \sqrt {b x} \arccos (a x)^n \, dx=\int \sqrt {b x} \arccos (a x)^n \, dx \]

[In]

Integrate[Sqrt[b*x]*ArcCos[a*x]^n,x]

[Out]

Integrate[Sqrt[b*x]*ArcCos[a*x]^n, x]

Maple [N/A] (verified)

Not integrable

Time = 0.56 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86

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

[In]

int(arccos(a*x)^n*(b*x)^(1/2),x)

[Out]

int(arccos(a*x)^n*(b*x)^(1/2),x)

Fricas [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \sqrt {b x} \arccos (a x)^n \, dx=\int { \sqrt {b x} \arccos \left (a x\right )^{n} \,d x } \]

[In]

integrate(arccos(a*x)^n*(b*x)^(1/2),x, algorithm="fricas")

[Out]

integral(sqrt(b*x)*arccos(a*x)^n, x)

Sympy [N/A]

Not integrable

Time = 3.96 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \sqrt {b x} \arccos (a x)^n \, dx=\int \sqrt {b x} \operatorname {acos}^{n}{\left (a x \right )}\, dx \]

[In]

integrate(acos(a*x)**n*(b*x)**(1/2),x)

[Out]

Integral(sqrt(b*x)*acos(a*x)**n, x)

Maxima [F(-2)]

Exception generated. \[ \int \sqrt {b x} \arccos (a x)^n \, dx=\text {Exception raised: RuntimeError} \]

[In]

integrate(arccos(a*x)^n*(b*x)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negative exponent.

Giac [N/A]

Not integrable

Time = 0.60 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \sqrt {b x} \arccos (a x)^n \, dx=\int { \sqrt {b x} \arccos \left (a x\right )^{n} \,d x } \]

[In]

integrate(arccos(a*x)^n*(b*x)^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(b*x)*arccos(a*x)^n, x)

Mupad [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \sqrt {b x} \arccos (a x)^n \, dx=\int {\mathrm {acos}\left (a\,x\right )}^n\,\sqrt {b\,x} \,d x \]

[In]

int(acos(a*x)^n*(b*x)^(1/2),x)

[Out]

int(acos(a*x)^n*(b*x)^(1/2), x)