Optimal. Leaf size=76 \[ \frac{5}{8 b^2 x \left (b+c x^2\right )}-\frac{15 \sqrt{c} \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{8 b^{7/2}}-\frac{15}{8 b^3 x}+\frac{1}{4 b x \left (b+c x^2\right )^2} \]
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Rubi [A] time = 0.0356755, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235, Rules used = {1584, 290, 325, 205} \[ \frac{5}{8 b^2 x \left (b+c x^2\right )}-\frac{15 \sqrt{c} \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{8 b^{7/2}}-\frac{15}{8 b^3 x}+\frac{1}{4 b x \left (b+c x^2\right )^2} \]
Antiderivative was successfully verified.
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Rule 1584
Rule 290
Rule 325
Rule 205
Rubi steps
\begin{align*} \int \frac{x^4}{\left (b x^2+c x^4\right )^3} \, dx &=\int \frac{1}{x^2 \left (b+c x^2\right )^3} \, dx\\ &=\frac{1}{4 b x \left (b+c x^2\right )^2}+\frac{5 \int \frac{1}{x^2 \left (b+c x^2\right )^2} \, dx}{4 b}\\ &=\frac{1}{4 b x \left (b+c x^2\right )^2}+\frac{5}{8 b^2 x \left (b+c x^2\right )}+\frac{15 \int \frac{1}{x^2 \left (b+c x^2\right )} \, dx}{8 b^2}\\ &=-\frac{15}{8 b^3 x}+\frac{1}{4 b x \left (b+c x^2\right )^2}+\frac{5}{8 b^2 x \left (b+c x^2\right )}-\frac{(15 c) \int \frac{1}{b+c x^2} \, dx}{8 b^3}\\ &=-\frac{15}{8 b^3 x}+\frac{1}{4 b x \left (b+c x^2\right )^2}+\frac{5}{8 b^2 x \left (b+c x^2\right )}-\frac{15 \sqrt{c} \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{8 b^{7/2}}\\ \end{align*}
Mathematica [A] time = 0.0413318, size = 68, normalized size = 0.89 \[ -\frac{8 b^2+25 b c x^2+15 c^2 x^4}{8 b^3 x \left (b+c x^2\right )^2}-\frac{15 \sqrt{c} \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{8 b^{7/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.055, size = 66, normalized size = 0.9 \begin{align*} -{\frac{1}{{b}^{3}x}}-{\frac{7\,{c}^{2}{x}^{3}}{8\,{b}^{3} \left ( c{x}^{2}+b \right ) ^{2}}}-{\frac{9\,cx}{8\,{b}^{2} \left ( c{x}^{2}+b \right ) ^{2}}}-{\frac{15\,c}{8\,{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 = 1.53436, size = 428, normalized size = 5.63 \begin{align*} \left [-\frac{30 \, c^{2} x^{4} + 50 \, b c x^{2} - 15 \,{\left (c^{2} x^{5} + 2 \, b c x^{3} + b^{2} x\right )} \sqrt{-\frac{c}{b}} \log \left (\frac{c x^{2} - 2 \, b x \sqrt{-\frac{c}{b}} - b}{c x^{2} + b}\right ) + 16 \, b^{2}}{16 \,{\left (b^{3} c^{2} x^{5} + 2 \, b^{4} c x^{3} + b^{5} x\right )}}, -\frac{15 \, c^{2} x^{4} + 25 \, b c x^{2} + 15 \,{\left (c^{2} x^{5} + 2 \, b c x^{3} + b^{2} x\right )} \sqrt{\frac{c}{b}} \arctan \left (x \sqrt{\frac{c}{b}}\right ) + 8 \, b^{2}}{8 \,{\left (b^{3} c^{2} x^{5} + 2 \, b^{4} c x^{3} + b^{5} x\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.782732, size = 114, normalized size = 1.5 \begin{align*} \frac{15 \sqrt{- \frac{c}{b^{7}}} \log{\left (- \frac{b^{4} \sqrt{- \frac{c}{b^{7}}}}{c} + x \right )}}{16} - \frac{15 \sqrt{- \frac{c}{b^{7}}} \log{\left (\frac{b^{4} \sqrt{- \frac{c}{b^{7}}}}{c} + x \right )}}{16} - \frac{8 b^{2} + 25 b c x^{2} + 15 c^{2} x^{4}}{8 b^{5} x + 16 b^{4} c x^{3} + 8 b^{3} c^{2} x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.2728, size = 77, normalized size = 1.01 \begin{align*} -\frac{15 \, c \arctan \left (\frac{c x}{\sqrt{b c}}\right )}{8 \, \sqrt{b c} b^{3}} - \frac{7 \, c^{2} x^{3} + 9 \, b c x}{8 \,{\left (c x^{2} + b\right )}^{2} b^{3}} - \frac{1}{b^{3} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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