3.863 \(\int \frac{\tan ^{-1}(a x)^{5/2}}{x^4 (c+a^2 c x^2)} \, dx\)

Optimal. Leaf size=59 \[ -\frac{a^2 \text{Unintegrable}\left (\frac{\tan ^{-1}(a x)^{5/2}}{x^2},x\right )}{c}+\frac{\text{Unintegrable}\left (\frac{\tan ^{-1}(a x)^{5/2}}{x^4},x\right )}{c}+\frac{2 a^3 \tan ^{-1}(a x)^{7/2}}{7 c} \]

[Out]

(2*a^3*ArcTan[a*x]^(7/2))/(7*c) + Unintegrable[ArcTan[a*x]^(5/2)/x^4, x]/c - (a^2*Unintegrable[ArcTan[a*x]^(5/
2)/x^2, x])/c

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Rubi [A]  time = 0.17904, 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 = {} \[ \int \frac{\tan ^{-1}(a x)^{5/2}}{x^4 \left (c+a^2 c x^2\right )} \, dx \]

Verification is Not applicable to the result.

[In]

Int[ArcTan[a*x]^(5/2)/(x^4*(c + a^2*c*x^2)),x]

[Out]

(2*a^3*ArcTan[a*x]^(7/2))/(7*c) + Defer[Int][ArcTan[a*x]^(5/2)/x^4, x]/c - (a^2*Defer[Int][ArcTan[a*x]^(5/2)/x
^2, x])/c

Rubi steps

\begin{align*} \int \frac{\tan ^{-1}(a x)^{5/2}}{x^4 \left (c+a^2 c x^2\right )} \, dx &=-\left (a^2 \int \frac{\tan ^{-1}(a x)^{5/2}}{x^2 \left (c+a^2 c x^2\right )} \, dx\right )+\frac{\int \frac{\tan ^{-1}(a x)^{5/2}}{x^4} \, dx}{c}\\ &=a^4 \int \frac{\tan ^{-1}(a x)^{5/2}}{c+a^2 c x^2} \, dx+\frac{\int \frac{\tan ^{-1}(a x)^{5/2}}{x^4} \, dx}{c}-\frac{a^2 \int \frac{\tan ^{-1}(a x)^{5/2}}{x^2} \, dx}{c}\\ &=\frac{2 a^3 \tan ^{-1}(a x)^{7/2}}{7 c}+\frac{\int \frac{\tan ^{-1}(a x)^{5/2}}{x^4} \, dx}{c}-\frac{a^2 \int \frac{\tan ^{-1}(a x)^{5/2}}{x^2} \, dx}{c}\\ \end{align*}

Mathematica [A]  time = 3.45598, size = 0, normalized size = 0. \[ \int \frac{\tan ^{-1}(a x)^{5/2}}{x^4 \left (c+a^2 c x^2\right )} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[ArcTan[a*x]^(5/2)/(x^4*(c + a^2*c*x^2)),x]

[Out]

Integrate[ArcTan[a*x]^(5/2)/(x^4*(c + a^2*c*x^2)), x]

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Maple [A]  time = 0.487, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{4} \left ({a}^{2}c{x}^{2}+c \right ) } \left ( \arctan \left ( ax \right ) \right ) ^{{\frac{5}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arctan(a*x)^(5/2)/x^4/(a^2*c*x^2+c),x)

[Out]

int(arctan(a*x)^(5/2)/x^4/(a^2*c*x^2+c),x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctan(a*x)^(5/2)/x^4/(a^2*c*x^2+c),x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

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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(arctan(a*x)^(5/2)/x^4/(a^2*c*x^2+c),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{\operatorname{atan}^{\frac{5}{2}}{\left (a x \right )}}{a^{2} x^{6} + x^{4}}\, dx}{c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(atan(a*x)**(5/2)/x**4/(a**2*c*x**2+c),x)

[Out]

Integral(atan(a*x)**(5/2)/(a**2*x**6 + x**4), x)/c

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\arctan \left (a x\right )^{\frac{5}{2}}}{{\left (a^{2} c x^{2} + c\right )} x^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctan(a*x)^(5/2)/x^4/(a^2*c*x^2+c),x, algorithm="giac")

[Out]

integrate(arctan(a*x)^(5/2)/((a^2*c*x^2 + c)*x^4), x)