Optimal. Leaf size=58 \[ -\frac{c^2}{b^3 x}-\frac{c^{5/2} \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{b^{7/2}}+\frac{c}{3 b^2 x^3}-\frac{1}{5 b x^5} \]
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Rubi [A] time = 0.0362493, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {1584, 325, 205} \[ -\frac{c^2}{b^3 x}-\frac{c^{5/2} \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{b^{7/2}}+\frac{c}{3 b^2 x^3}-\frac{1}{5 b x^5} \]
Antiderivative was successfully verified.
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Rule 1584
Rule 325
Rule 205
Rubi steps
\begin{align*} \int \frac{1}{x^4 \left (b x^2+c x^4\right )} \, dx &=\int \frac{1}{x^6 \left (b+c x^2\right )} \, dx\\ &=-\frac{1}{5 b x^5}-\frac{c \int \frac{1}{x^4 \left (b+c x^2\right )} \, dx}{b}\\ &=-\frac{1}{5 b x^5}+\frac{c}{3 b^2 x^3}+\frac{c^2 \int \frac{1}{x^2 \left (b+c x^2\right )} \, dx}{b^2}\\ &=-\frac{1}{5 b x^5}+\frac{c}{3 b^2 x^3}-\frac{c^2}{b^3 x}-\frac{c^3 \int \frac{1}{b+c x^2} \, dx}{b^3}\\ &=-\frac{1}{5 b x^5}+\frac{c}{3 b^2 x^3}-\frac{c^2}{b^3 x}-\frac{c^{5/2} \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{b^{7/2}}\\ \end{align*}
Mathematica [A] time = 0.0267664, size = 58, normalized size = 1. \[ -\frac{c^2}{b^3 x}-\frac{c^{5/2} \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{b^{7/2}}+\frac{c}{3 b^2 x^3}-\frac{1}{5 b x^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.05, size = 52, normalized size = 0.9 \begin{align*} -{\frac{1}{5\,b{x}^{5}}}-{\frac{{c}^{2}}{{b}^{3}x}}+{\frac{c}{3\,{b}^{2}{x}^{3}}}-{\frac{{c}^{3}}{{b}^{3}}\arctan \left ({cx{\frac{1}{\sqrt{bc}}}} \right ){\frac{1}{\sqrt{bc}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Fricas [A] time = 1.46194, size = 296, normalized size = 5.1 \begin{align*} \left [\frac{15 \, c^{2} x^{5} \sqrt{-\frac{c}{b}} \log \left (\frac{c x^{2} - 2 \, b x \sqrt{-\frac{c}{b}} - b}{c x^{2} + b}\right ) - 30 \, c^{2} x^{4} + 10 \, b c x^{2} - 6 \, b^{2}}{30 \, b^{3} x^{5}}, -\frac{15 \, c^{2} x^{5} \sqrt{\frac{c}{b}} \arctan \left (x \sqrt{\frac{c}{b}}\right ) + 15 \, c^{2} x^{4} - 5 \, b c x^{2} + 3 \, b^{2}}{15 \, b^{3} x^{5}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.549863, size = 100, normalized size = 1.72 \begin{align*} \frac{\sqrt{- \frac{c^{5}}{b^{7}}} \log{\left (- \frac{b^{4} \sqrt{- \frac{c^{5}}{b^{7}}}}{c^{3}} + x \right )}}{2} - \frac{\sqrt{- \frac{c^{5}}{b^{7}}} \log{\left (\frac{b^{4} \sqrt{- \frac{c^{5}}{b^{7}}}}{c^{3}} + x \right )}}{2} - \frac{3 b^{2} - 5 b c x^{2} + 15 c^{2} x^{4}}{15 b^{3} x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.23844, size = 70, normalized size = 1.21 \begin{align*} -\frac{c^{3} \arctan \left (\frac{c x}{\sqrt{b c}}\right )}{\sqrt{b c} b^{3}} - \frac{15 \, c^{2} x^{4} - 5 \, b c x^{2} + 3 \, b^{2}}{15 \, b^{3} x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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