Optimal. Leaf size=98 \[ -\frac{8 \text{Unintegrable}\left (\frac{1}{x^5 \left (a^2 c x^2+c\right )^{5/2} \sqrt{\tan ^{-1}(a x)}},x\right )}{a}-14 a \text{Unintegrable}\left (\frac{1}{x^3 \left (a^2 c x^2+c\right )^{5/2} \sqrt{\tan ^{-1}(a x)}},x\right )-\frac{2}{a c x^4 \left (a^2 c x^2+c\right )^{3/2} \sqrt{\tan ^{-1}(a x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.368755, 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 = {} \[ \int \frac{1}{x^4 \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^{3/2}} \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
Rubi steps
\begin{align*} \int \frac{1}{x^4 \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^{3/2}} \, dx &=-\frac{2}{a c x^4 \left (c+a^2 c x^2\right )^{3/2} \sqrt{\tan ^{-1}(a x)}}-\frac{8 \int \frac{1}{x^5 \left (c+a^2 c x^2\right )^{5/2} \sqrt{\tan ^{-1}(a x)}} \, dx}{a}-(14 a) \int \frac{1}{x^3 \left (c+a^2 c x^2\right )^{5/2} \sqrt{\tan ^{-1}(a x)}} \, dx\\ \end{align*}
Mathematica [A] time = 20.7907, size = 0, normalized size = 0. \[ \int \frac{1}{x^4 \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^{3/2}} \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 2.022, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{4}} \left ({a}^{2}c{x}^{2}+c \right ) ^{-{\frac{5}{2}}} \left ( \arctan \left ( ax \right ) \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
Giac [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (a^{2} c x^{2} + c\right )}^{\frac{5}{2}} x^{4} \arctan \left (a x\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]