Optimal. Leaf size=100 \[ -\frac{63 c^2}{8 b^5 x}-\frac{63 c^{5/2} \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{8 b^{11/2}}+\frac{21 c}{8 b^4 x^3}+\frac{9}{8 b^2 x^5 \left (b+c x^2\right )}-\frac{63}{40 b^3 x^5}+\frac{1}{4 b x^5 \left (b+c x^2\right )^2} \]
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Rubi [A] time = 0.0491003, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {1593, 290, 325, 205} \[ -\frac{63 c^2}{8 b^5 x}-\frac{63 c^{5/2} \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{8 b^{11/2}}+\frac{21 c}{8 b^4 x^3}+\frac{9}{8 b^2 x^5 \left (b+c x^2\right )}-\frac{63}{40 b^3 x^5}+\frac{1}{4 b x^5 \left (b+c x^2\right )^2} \]
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 )^3} \, dx &=\int \frac{1}{x^6 \left (b+c x^2\right )^3} \, dx\\ &=\frac{1}{4 b x^5 \left (b+c x^2\right )^2}+\frac{9 \int \frac{1}{x^6 \left (b+c x^2\right )^2} \, dx}{4 b}\\ &=\frac{1}{4 b x^5 \left (b+c x^2\right )^2}+\frac{9}{8 b^2 x^5 \left (b+c x^2\right )}+\frac{63 \int \frac{1}{x^6 \left (b+c x^2\right )} \, dx}{8 b^2}\\ &=-\frac{63}{40 b^3 x^5}+\frac{1}{4 b x^5 \left (b+c x^2\right )^2}+\frac{9}{8 b^2 x^5 \left (b+c x^2\right )}-\frac{(63 c) \int \frac{1}{x^4 \left (b+c x^2\right )} \, dx}{8 b^3}\\ &=-\frac{63}{40 b^3 x^5}+\frac{21 c}{8 b^4 x^3}+\frac{1}{4 b x^5 \left (b+c x^2\right )^2}+\frac{9}{8 b^2 x^5 \left (b+c x^2\right )}+\frac{\left (63 c^2\right ) \int \frac{1}{x^2 \left (b+c x^2\right )} \, dx}{8 b^4}\\ &=-\frac{63}{40 b^3 x^5}+\frac{21 c}{8 b^4 x^3}-\frac{63 c^2}{8 b^5 x}+\frac{1}{4 b x^5 \left (b+c x^2\right )^2}+\frac{9}{8 b^2 x^5 \left (b+c x^2\right )}-\frac{\left (63 c^3\right ) \int \frac{1}{b+c x^2} \, dx}{8 b^5}\\ &=-\frac{63}{40 b^3 x^5}+\frac{21 c}{8 b^4 x^3}-\frac{63 c^2}{8 b^5 x}+\frac{1}{4 b x^5 \left (b+c x^2\right )^2}+\frac{9}{8 b^2 x^5 \left (b+c x^2\right )}-\frac{63 c^{5/2} \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{8 b^{11/2}}\\ \end{align*}
Mathematica [A] time = 0.053974, size = 90, normalized size = 0.9 \[ -\frac{168 b^2 c^2 x^4-24 b^3 c x^2+8 b^4+525 b c^3 x^6+315 c^4 x^8}{40 b^5 x^5 \left (b+c x^2\right )^2}-\frac{63 c^{5/2} \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{8 b^{11/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.056, size = 89, normalized size = 0.9 \begin{align*} -{\frac{1}{5\,{b}^{3}{x}^{5}}}-6\,{\frac{{c}^{2}}{{b}^{5}x}}+{\frac{c}{{b}^{4}{x}^{3}}}-{\frac{15\,{c}^{4}{x}^{3}}{8\,{b}^{5} \left ( c{x}^{2}+b \right ) ^{2}}}-{\frac{17\,{c}^{3}x}{8\,{b}^{4} \left ( c{x}^{2}+b \right ) ^{2}}}-{\frac{63\,{c}^{3}}{8\,{b}^{5}}\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 = 1.55846, size = 560, normalized size = 5.6 \begin{align*} \left [-\frac{630 \, c^{4} x^{8} + 1050 \, b c^{3} x^{6} + 336 \, b^{2} c^{2} x^{4} - 48 \, b^{3} c x^{2} + 16 \, b^{4} - 315 \,{\left (c^{4} x^{9} + 2 \, b c^{3} x^{7} + b^{2} c^{2} x^{5}\right )} \sqrt{-\frac{c}{b}} \log \left (\frac{c x^{2} - 2 \, b x \sqrt{-\frac{c}{b}} - b}{c x^{2} + b}\right )}{80 \,{\left (b^{5} c^{2} x^{9} + 2 \, b^{6} c x^{7} + b^{7} x^{5}\right )}}, -\frac{315 \, c^{4} x^{8} + 525 \, b c^{3} x^{6} + 168 \, b^{2} c^{2} x^{4} - 24 \, b^{3} c x^{2} + 8 \, b^{4} + 315 \,{\left (c^{4} x^{9} + 2 \, b c^{3} x^{7} + b^{2} c^{2} x^{5}\right )} \sqrt{\frac{c}{b}} \arctan \left (x \sqrt{\frac{c}{b}}\right )}{40 \,{\left (b^{5} c^{2} x^{9} + 2 \, b^{6} c x^{7} + b^{7} x^{5}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.75648, size = 150, normalized size = 1.5 \begin{align*} \frac{63 \sqrt{- \frac{c^{5}}{b^{11}}} \log{\left (- \frac{b^{6} \sqrt{- \frac{c^{5}}{b^{11}}}}{c^{3}} + x \right )}}{16} - \frac{63 \sqrt{- \frac{c^{5}}{b^{11}}} \log{\left (\frac{b^{6} \sqrt{- \frac{c^{5}}{b^{11}}}}{c^{3}} + x \right )}}{16} - \frac{8 b^{4} - 24 b^{3} c x^{2} + 168 b^{2} c^{2} x^{4} + 525 b c^{3} x^{6} + 315 c^{4} x^{8}}{40 b^{7} x^{5} + 80 b^{6} c x^{7} + 40 b^{5} c^{2} x^{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.30243, size = 108, normalized size = 1.08 \begin{align*} -\frac{63 \, c^{3} \arctan \left (\frac{c x}{\sqrt{b c}}\right )}{8 \, \sqrt{b c} b^{5}} - \frac{15 \, c^{4} x^{3} + 17 \, b c^{3} x}{8 \,{\left (c x^{2} + b\right )}^{2} b^{5}} - \frac{30 \, c^{2} x^{4} - 5 \, b c x^{2} + b^{2}}{5 \, b^{5} x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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