Optimal. Leaf size=43 \[ \frac{a^{3/2} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{b^{5/2}}+\frac{a}{b^2 x}-\frac{1}{3 b x^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0174545, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {263, 325, 205} \[ \frac{a^{3/2} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{b^{5/2}}+\frac{a}{b^2 x}-\frac{1}{3 b x^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 263
Rule 325
Rule 205
Rubi steps
\begin{align*} \int \frac{1}{\left (a+\frac{b}{x^2}\right ) x^6} \, dx &=\int \frac{1}{x^4 \left (b+a x^2\right )} \, dx\\ &=-\frac{1}{3 b x^3}-\frac{a \int \frac{1}{x^2 \left (b+a x^2\right )} \, dx}{b}\\ &=-\frac{1}{3 b x^3}+\frac{a}{b^2 x}+\frac{a^2 \int \frac{1}{b+a x^2} \, dx}{b^2}\\ &=-\frac{1}{3 b x^3}+\frac{a}{b^2 x}+\frac{a^{3/2} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{b^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.0205476, size = 43, normalized size = 1. \[ \frac{a^{3/2} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{b^{5/2}}+\frac{a}{b^2 x}-\frac{1}{3 b x^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.005, size = 39, normalized size = 0.9 \begin{align*}{\frac{{a}^{2}}{{b}^{2}}\arctan \left ({ax{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{1}{3\,b{x}^{3}}}+{\frac{a}{{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 = 1.4676, size = 234, normalized size = 5.44 \begin{align*} \left [\frac{3 \, a x^{3} \sqrt{-\frac{a}{b}} \log \left (\frac{a x^{2} + 2 \, b x \sqrt{-\frac{a}{b}} - b}{a x^{2} + b}\right ) + 6 \, a x^{2} - 2 \, b}{6 \, b^{2} x^{3}}, \frac{3 \, a x^{3} \sqrt{\frac{a}{b}} \arctan \left (x \sqrt{\frac{a}{b}}\right ) + 3 \, a x^{2} - b}{3 \, b^{2} x^{3}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] time = 0.493538, size = 87, normalized size = 2.02 \begin{align*} - \frac{\sqrt{- \frac{a^{3}}{b^{5}}} \log{\left (x - \frac{b^{3} \sqrt{- \frac{a^{3}}{b^{5}}}}{a^{2}} \right )}}{2} + \frac{\sqrt{- \frac{a^{3}}{b^{5}}} \log{\left (x + \frac{b^{3} \sqrt{- \frac{a^{3}}{b^{5}}}}{a^{2}} \right )}}{2} + \frac{3 a x^{2} - b}{3 b^{2} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.15485, size = 54, normalized size = 1.26 \begin{align*} \frac{a^{2} \arctan \left (\frac{a x}{\sqrt{a b}}\right )}{\sqrt{a b} b^{2}} + \frac{3 \, a x^{2} - b}{3 \, b^{2} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]