Optimal. Leaf size=57 \[ -\frac{3 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{2 b^{5/2}}+\frac{1}{2 b x \left (a x^2+b\right )}-\frac{3}{2 b^2 x} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0177912, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {263, 290, 325, 205} \[ -\frac{3 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{2 b^{5/2}}+\frac{1}{2 b x \left (a x^2+b\right )}-\frac{3}{2 b^2 x} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 263
Rule 290
Rule 325
Rule 205
Rubi steps
\begin{align*} \int \frac{1}{\left (a+\frac{b}{x^2}\right )^2 x^6} \, dx &=\int \frac{1}{x^2 \left (b+a x^2\right )^2} \, dx\\ &=\frac{1}{2 b x \left (b+a x^2\right )}+\frac{3 \int \frac{1}{x^2 \left (b+a x^2\right )} \, dx}{2 b}\\ &=-\frac{3}{2 b^2 x}+\frac{1}{2 b x \left (b+a x^2\right )}-\frac{(3 a) \int \frac{1}{b+a x^2} \, dx}{2 b^2}\\ &=-\frac{3}{2 b^2 x}+\frac{1}{2 b x \left (b+a x^2\right )}-\frac{3 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{2 b^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.036331, size = 54, normalized size = 0.95 \[ -\frac{a x}{2 b^2 \left (a x^2+b\right )}-\frac{3 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{2 b^{5/2}}-\frac{1}{b^2 x} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.009, size = 46, normalized size = 0.8 \begin{align*} -{\frac{ax}{2\,{b}^{2} \left ( a{x}^{2}+b \right ) }}-{\frac{3\,a}{2\,{b}^{2}}\arctan \left ({ax{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{1}{{b}^{2}x}} \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: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.49645, size = 288, normalized size = 5.05 \begin{align*} \left [-\frac{6 \, a x^{2} - 3 \,{\left (a x^{3} + b x\right )} \sqrt{-\frac{a}{b}} \log \left (\frac{a x^{2} - 2 \, b x \sqrt{-\frac{a}{b}} - b}{a x^{2} + b}\right ) + 4 \, b}{4 \,{\left (a b^{2} x^{3} + b^{3} x\right )}}, -\frac{3 \, a x^{2} + 3 \,{\left (a x^{3} + b x\right )} \sqrt{\frac{a}{b}} \arctan \left (x \sqrt{\frac{a}{b}}\right ) + 2 \, b}{2 \,{\left (a b^{2} x^{3} + b^{3} x\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.602729, size = 90, normalized size = 1.58 \begin{align*} \frac{3 \sqrt{- \frac{a}{b^{5}}} \log{\left (x - \frac{b^{3} \sqrt{- \frac{a}{b^{5}}}}{a} \right )}}{4} - \frac{3 \sqrt{- \frac{a}{b^{5}}} \log{\left (x + \frac{b^{3} \sqrt{- \frac{a}{b^{5}}}}{a} \right )}}{4} - \frac{3 a x^{2} + 2 b}{2 a b^{2} x^{3} + 2 b^{3} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.16279, size = 63, normalized size = 1.11 \begin{align*} -\frac{3 \, a \arctan \left (\frac{a x}{\sqrt{a b}}\right )}{2 \, \sqrt{a b} b^{2}} - \frac{3 \, a x^{2} + 2 \, b}{2 \,{\left (a x^{3} + b x\right )} b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]