3.202 \(\int \frac{1}{x^2 (a+b \cos ^{-1}(c x))^{5/2}} \, dx\)

Optimal. Leaf size=18 \[ \text{Unintegrable}\left (\frac{1}{x^2 \left (a+b \cos ^{-1}(c x)\right )^{5/2}},x\right ) \]

[Out]

Unintegrable[1/(x^2*(a + b*ArcCos[c*x])^(5/2)), x]

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Rubi [A]  time = 0.0405848, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{1}{x^2 \left (a+b \cos ^{-1}(c x)\right )^{5/2}} \, dx \]

Verification is Not applicable to the result.

[In]

Int[1/(x^2*(a + b*ArcCos[c*x])^(5/2)),x]

[Out]

Defer[Int][1/(x^2*(a + b*ArcCos[c*x])^(5/2)), x]

Rubi steps

\begin{align*} \int \frac{1}{x^2 \left (a+b \cos ^{-1}(c x)\right )^{5/2}} \, dx &=\int \frac{1}{x^2 \left (a+b \cos ^{-1}(c x)\right )^{5/2}} \, dx\\ \end{align*}

Mathematica [A]  time = 12.702, size = 0, normalized size = 0. \[ \int \frac{1}{x^2 \left (a+b \cos ^{-1}(c x)\right )^{5/2}} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[1/(x^2*(a + b*ArcCos[c*x])^(5/2)),x]

[Out]

Integrate[1/(x^2*(a + b*ArcCos[c*x])^(5/2)), x]

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Maple [A]  time = 0.272, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2}} \left ( a+b\arccos \left ( cx \right ) \right ) ^{-{\frac{5}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^2/(a+b*arccos(c*x))^(5/2),x)

[Out]

int(1/x^2/(a+b*arccos(c*x))^(5/2),x)

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b \arccos \left (c x\right ) + a\right )}^{\frac{5}{2}} x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(a+b*arccos(c*x))^(5/2),x, algorithm="maxima")

[Out]

integrate(1/((b*arccos(c*x) + a)^(5/2)*x^2), x)

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(a+b*arccos(c*x))^(5/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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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(1/x**2/(a+b*acos(c*x))**(5/2),x)

[Out]

Timed out

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b \arccos \left (c x\right ) + a\right )}^{\frac{5}{2}} x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(a+b*arccos(c*x))^(5/2),x, algorithm="giac")

[Out]

integrate(1/((b*arccos(c*x) + a)^(5/2)*x^2), x)