Optimal. Leaf size=64 \[ -\frac{3 x}{8 c^2 \left (b+c x^2\right )}+\frac{3 \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{8 \sqrt{b} c^{5/2}}-\frac{x^3}{4 c \left (b+c x^2\right )^2} \]
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Rubi [A] time = 0.0255314, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {1584, 288, 205} \[ -\frac{3 x}{8 c^2 \left (b+c x^2\right )}+\frac{3 \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{8 \sqrt{b} c^{5/2}}-\frac{x^3}{4 c \left (b+c x^2\right )^2} \]
Antiderivative was successfully verified.
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Rule 1584
Rule 288
Rule 205
Rubi steps
\begin{align*} \int \frac{x^{10}}{\left (b x^2+c x^4\right )^3} \, dx &=\int \frac{x^4}{\left (b+c x^2\right )^3} \, dx\\ &=-\frac{x^3}{4 c \left (b+c x^2\right )^2}+\frac{3 \int \frac{x^2}{\left (b+c x^2\right )^2} \, dx}{4 c}\\ &=-\frac{x^3}{4 c \left (b+c x^2\right )^2}-\frac{3 x}{8 c^2 \left (b+c x^2\right )}+\frac{3 \int \frac{1}{b+c x^2} \, dx}{8 c^2}\\ &=-\frac{x^3}{4 c \left (b+c x^2\right )^2}-\frac{3 x}{8 c^2 \left (b+c x^2\right )}+\frac{3 \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{8 \sqrt{b} c^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.0426801, size = 55, normalized size = 0.86 \[ \frac{3 \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{8 \sqrt{b} c^{5/2}}-\frac{3 b x+5 c x^3}{8 c^2 \left (b+c x^2\right )^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.048, size = 47, normalized size = 0.7 \begin{align*}{\frac{1}{ \left ( c{x}^{2}+b \right ) ^{2}} \left ( -{\frac{5\,{x}^{3}}{8\,c}}-{\frac{3\,bx}{8\,{c}^{2}}} \right ) }+{\frac{3}{8\,{c}^{2}}\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 = 1.59582, size = 404, normalized size = 6.31 \begin{align*} \left [-\frac{10 \, b c^{2} x^{3} + 6 \, b^{2} c x + 3 \,{\left (c^{2} x^{4} + 2 \, b c x^{2} + b^{2}\right )} \sqrt{-b c} \log \left (\frac{c x^{2} - 2 \, \sqrt{-b c} x - b}{c x^{2} + b}\right )}{16 \,{\left (b c^{5} x^{4} + 2 \, b^{2} c^{4} x^{2} + b^{3} c^{3}\right )}}, -\frac{5 \, b c^{2} x^{3} + 3 \, b^{2} c x - 3 \,{\left (c^{2} x^{4} + 2 \, b c x^{2} + b^{2}\right )} \sqrt{b c} \arctan \left (\frac{\sqrt{b c} x}{b}\right )}{8 \,{\left (b c^{5} x^{4} + 2 \, b^{2} c^{4} x^{2} + b^{3} c^{3}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.532172, size = 109, normalized size = 1.7 \begin{align*} - \frac{3 \sqrt{- \frac{1}{b c^{5}}} \log{\left (- b c^{2} \sqrt{- \frac{1}{b c^{5}}} + x \right )}}{16} + \frac{3 \sqrt{- \frac{1}{b c^{5}}} \log{\left (b c^{2} \sqrt{- \frac{1}{b c^{5}}} + x \right )}}{16} - \frac{3 b x + 5 c x^{3}}{8 b^{2} c^{2} + 16 b c^{3} x^{2} + 8 c^{4} x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.28926, size = 61, normalized size = 0.95 \begin{align*} \frac{3 \, \arctan \left (\frac{c x}{\sqrt{b c}}\right )}{8 \, \sqrt{b c} c^{2}} - \frac{5 \, c x^{3} + 3 \, b x}{8 \,{\left (c x^{2} + b\right )}^{2} c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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