Optimal. Leaf size=68 \[ \frac{5 c^{3/2} \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{2 b^{7/2}}+\frac{5 c}{2 b^3 x}-\frac{5}{6 b^2 x^3}+\frac{1}{2 b x^3 \left (b+c x^2\right )} \]
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Rubi [A] time = 0.029305, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {1593, 290, 325, 205} \[ \frac{5 c^{3/2} \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{2 b^{7/2}}+\frac{5 c}{2 b^3 x}-\frac{5}{6 b^2 x^3}+\frac{1}{2 b x^3 \left (b+c x^2\right )} \]
Antiderivative was successfully verified.
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Rule 1593
Rule 290
Rule 325
Rule 205
Rubi steps
\begin{align*} \int \frac{1}{\left (b x^2+c x^4\right )^2} \, dx &=\int \frac{1}{x^4 \left (b+c x^2\right )^2} \, dx\\ &=\frac{1}{2 b x^3 \left (b+c x^2\right )}+\frac{5 \int \frac{1}{x^4 \left (b+c x^2\right )} \, dx}{2 b}\\ &=-\frac{5}{6 b^2 x^3}+\frac{1}{2 b x^3 \left (b+c x^2\right )}-\frac{(5 c) \int \frac{1}{x^2 \left (b+c x^2\right )} \, dx}{2 b^2}\\ &=-\frac{5}{6 b^2 x^3}+\frac{5 c}{2 b^3 x}+\frac{1}{2 b x^3 \left (b+c x^2\right )}+\frac{\left (5 c^2\right ) \int \frac{1}{b+c x^2} \, dx}{2 b^3}\\ &=-\frac{5}{6 b^2 x^3}+\frac{5 c}{2 b^3 x}+\frac{1}{2 b x^3 \left (b+c x^2\right )}+\frac{5 c^{3/2} \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{2 b^{7/2}}\\ \end{align*}
Mathematica [A] time = 0.0383423, size = 67, normalized size = 0.99 \[ \frac{c^2 x}{2 b^3 \left (b+c x^2\right )}+\frac{5 c^{3/2} \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{2 b^{7/2}}+\frac{2 c}{b^3 x}-\frac{1}{3 b^2 x^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.055, size = 59, normalized size = 0.9 \begin{align*} -{\frac{1}{3\,{b}^{2}{x}^{3}}}+2\,{\frac{c}{{b}^{3}x}}+{\frac{{c}^{2}x}{2\,{b}^{3} \left ( c{x}^{2}+b \right ) }}+{\frac{5\,{c}^{2}}{2\,{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 = 1.58413, size = 359, normalized size = 5.28 \begin{align*} \left [\frac{30 \, c^{2} x^{4} + 20 \, b c x^{2} + 15 \,{\left (c^{2} x^{5} + b c x^{3}\right )} \sqrt{-\frac{c}{b}} \log \left (\frac{c x^{2} + 2 \, b x \sqrt{-\frac{c}{b}} - b}{c x^{2} + b}\right ) - 4 \, b^{2}}{12 \,{\left (b^{3} c x^{5} + b^{4} x^{3}\right )}}, \frac{15 \, c^{2} x^{4} + 10 \, b c x^{2} + 15 \,{\left (c^{2} x^{5} + b c x^{3}\right )} \sqrt{\frac{c}{b}} \arctan \left (x \sqrt{\frac{c}{b}}\right ) - 2 \, b^{2}}{6 \,{\left (b^{3} c x^{5} + b^{4} x^{3}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.575817, size = 114, normalized size = 1.68 \begin{align*} - \frac{5 \sqrt{- \frac{c^{3}}{b^{7}}} \log{\left (- \frac{b^{4} \sqrt{- \frac{c^{3}}{b^{7}}}}{c^{2}} + x \right )}}{4} + \frac{5 \sqrt{- \frac{c^{3}}{b^{7}}} \log{\left (\frac{b^{4} \sqrt{- \frac{c^{3}}{b^{7}}}}{c^{2}} + x \right )}}{4} + \frac{- 2 b^{2} + 10 b c x^{2} + 15 c^{2} x^{4}}{6 b^{4} x^{3} + 6 b^{3} c x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.23951, size = 80, normalized size = 1.18 \begin{align*} \frac{5 \, c^{2} \arctan \left (\frac{c x}{\sqrt{b c}}\right )}{2 \, \sqrt{b c} b^{3}} + \frac{c^{2} x}{2 \,{\left (c x^{2} + b\right )} b^{3}} + \frac{6 \, c x^{2} - b}{3 \, b^{3} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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