Optimal. Leaf size=79 \[ \frac{7 b^2 x}{2 c^4}-\frac{7 b^{5/2} \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{2 c^{9/2}}-\frac{7 b x^3}{6 c^3}-\frac{x^7}{2 c \left (b+c x^2\right )}+\frac{7 x^5}{10 c^2} \]
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Rubi [A] time = 0.0387618, antiderivative size = 79, 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, 288, 302, 205} \[ \frac{7 b^2 x}{2 c^4}-\frac{7 b^{5/2} \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{2 c^{9/2}}-\frac{7 b x^3}{6 c^3}-\frac{x^7}{2 c \left (b+c x^2\right )}+\frac{7 x^5}{10 c^2} \]
Antiderivative was successfully verified.
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Rule 1584
Rule 288
Rule 302
Rule 205
Rubi steps
\begin{align*} \int \frac{x^{12}}{\left (b x^2+c x^4\right )^2} \, dx &=\int \frac{x^8}{\left (b+c x^2\right )^2} \, dx\\ &=-\frac{x^7}{2 c \left (b+c x^2\right )}+\frac{7 \int \frac{x^6}{b+c x^2} \, dx}{2 c}\\ &=-\frac{x^7}{2 c \left (b+c x^2\right )}+\frac{7 \int \left (\frac{b^2}{c^3}-\frac{b x^2}{c^2}+\frac{x^4}{c}-\frac{b^3}{c^3 \left (b+c x^2\right )}\right ) \, dx}{2 c}\\ &=\frac{7 b^2 x}{2 c^4}-\frac{7 b x^3}{6 c^3}+\frac{7 x^5}{10 c^2}-\frac{x^7}{2 c \left (b+c x^2\right )}-\frac{\left (7 b^3\right ) \int \frac{1}{b+c x^2} \, dx}{2 c^4}\\ &=\frac{7 b^2 x}{2 c^4}-\frac{7 b x^3}{6 c^3}+\frac{7 x^5}{10 c^2}-\frac{x^7}{2 c \left (b+c x^2\right )}-\frac{7 b^{5/2} \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{2 c^{9/2}}\\ \end{align*}
Mathematica [A] time = 0.0524963, size = 71, normalized size = 0.9 \[ \frac{x \left (\frac{15 b^3}{b+c x^2}+90 b^2-20 b c x^2+6 c^2 x^4\right )}{30 c^4}-\frac{7 b^{5/2} \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{2 c^{9/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.052, size = 68, normalized size = 0.9 \begin{align*}{\frac{{x}^{5}}{5\,{c}^{2}}}-{\frac{2\,b{x}^{3}}{3\,{c}^{3}}}+3\,{\frac{{b}^{2}x}{{c}^{4}}}+{\frac{{b}^{3}x}{2\,{c}^{4} \left ( c{x}^{2}+b \right ) }}-{\frac{7\,{b}^{3}}{2\,{c}^{4}}\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 = 1.47675, size = 409, normalized size = 5.18 \begin{align*} \left [\frac{12 \, c^{3} x^{7} - 28 \, b c^{2} x^{5} + 140 \, b^{2} c x^{3} + 210 \, b^{3} x + 105 \,{\left (b^{2} c x^{2} + b^{3}\right )} \sqrt{-\frac{b}{c}} \log \left (\frac{c x^{2} - 2 \, c x \sqrt{-\frac{b}{c}} - b}{c x^{2} + b}\right )}{60 \,{\left (c^{5} x^{2} + b c^{4}\right )}}, \frac{6 \, c^{3} x^{7} - 14 \, b c^{2} x^{5} + 70 \, b^{2} c x^{3} + 105 \, b^{3} x - 105 \,{\left (b^{2} c x^{2} + b^{3}\right )} \sqrt{\frac{b}{c}} \arctan \left (\frac{c x \sqrt{\frac{b}{c}}}{b}\right )}{30 \,{\left (c^{5} x^{2} + b c^{4}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.482897, size = 124, normalized size = 1.57 \begin{align*} \frac{b^{3} x}{2 b c^{4} + 2 c^{5} x^{2}} + \frac{3 b^{2} x}{c^{4}} - \frac{2 b x^{3}}{3 c^{3}} + \frac{7 \sqrt{- \frac{b^{5}}{c^{9}}} \log{\left (x - \frac{c^{4} \sqrt{- \frac{b^{5}}{c^{9}}}}{b^{2}} \right )}}{4} - \frac{7 \sqrt{- \frac{b^{5}}{c^{9}}} \log{\left (x + \frac{c^{4} \sqrt{- \frac{b^{5}}{c^{9}}}}{b^{2}} \right )}}{4} + \frac{x^{5}}{5 c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.21212, size = 99, normalized size = 1.25 \begin{align*} -\frac{7 \, b^{3} \arctan \left (\frac{c x}{\sqrt{b c}}\right )}{2 \, \sqrt{b c} c^{4}} + \frac{b^{3} x}{2 \,{\left (c x^{2} + b\right )} c^{4}} + \frac{3 \, c^{8} x^{5} - 10 \, b c^{7} x^{3} + 45 \, b^{2} c^{6} x}{15 \, c^{10}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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