3.862 \(\int \frac{\tan ^{-1}(a x)^{5/2}}{x^3 (c+a^2 c x^2)} \, dx\)

Optimal. Leaf size=72 \[ -\frac{i a^2 \text{Unintegrable}\left (\frac{\tan ^{-1}(a x)^{5/2}}{x (a x+i)},x\right )}{c}+\frac{\text{Unintegrable}\left (\frac{\tan ^{-1}(a x)^{5/2}}{x^3},x\right )}{c}+\frac{2 i a^2 \tan ^{-1}(a x)^{7/2}}{7 c} \]

[Out]

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

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Rubi [A]  time = 0.186641, 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 = {} \[ \int \frac{\tan ^{-1}(a x)^{5/2}}{x^3 \left (c+a^2 c x^2\right )} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 1.98426, size = 0, normalized size = 0. \[ \int \frac{\tan ^{-1}(a x)^{5/2}}{x^3 \left (c+a^2 c x^2\right )} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[ArcTan[a*x]^(5/2)/(x^3*(c + a^2*c*x^2)),x]

[Out]

Integrate[ArcTan[a*x]^(5/2)/(x^3*(c + a^2*c*x^2)), x]

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Maple [A]  time = 0.604, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{3} \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^3/(a^2*c*x^2+c),x)

[Out]

int(arctan(a*x)^(5/2)/x^3/(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^3/(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^3/(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^{5} + x^{3}}\, dx}{c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(atan(a*x)**(5/2)/x**3/(a**2*c*x**2+c),x)

[Out]

Integral(atan(a*x)**(5/2)/(a**2*x**5 + x**3), 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^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctan(a*x)^(5/2)/x^3/(a^2*c*x^2+c),x, algorithm="giac")

[Out]

integrate(arctan(a*x)^(5/2)/((a^2*c*x^2 + c)*x^3), x)