Optimal. Leaf size=92 \[ \frac{(3 A c+b B) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{8 b^{5/2} c^{3/2}}+\frac{x (3 A c+b B)}{8 b^2 c \left (b+c x^2\right )}-\frac{x (b B-A c)}{4 b c \left (b+c x^2\right )^2} \]
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Rubi [A] time = 0.0452505, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {1584, 385, 199, 205} \[ \frac{(3 A c+b B) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{8 b^{5/2} c^{3/2}}+\frac{x (3 A c+b B)}{8 b^2 c \left (b+c x^2\right )}-\frac{x (b B-A c)}{4 b c \left (b+c x^2\right )^2} \]
Antiderivative was successfully verified.
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Rule 1584
Rule 385
Rule 199
Rule 205
Rubi steps
\begin{align*} \int \frac{x^6 \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^3} \, dx &=\int \frac{A+B x^2}{\left (b+c x^2\right )^3} \, dx\\ &=-\frac{(b B-A c) x}{4 b c \left (b+c x^2\right )^2}+\frac{(b B+3 A c) \int \frac{1}{\left (b+c x^2\right )^2} \, dx}{4 b c}\\ &=-\frac{(b B-A c) x}{4 b c \left (b+c x^2\right )^2}+\frac{(b B+3 A c) x}{8 b^2 c \left (b+c x^2\right )}+\frac{(b B+3 A c) \int \frac{1}{b+c x^2} \, dx}{8 b^2 c}\\ &=-\frac{(b B-A c) x}{4 b c \left (b+c x^2\right )^2}+\frac{(b B+3 A c) x}{8 b^2 c \left (b+c x^2\right )}+\frac{(b B+3 A c) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{8 b^{5/2} c^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0588319, size = 84, normalized size = 0.91 \[ \frac{x \left (b c \left (5 A+B x^2\right )+3 A c^2 x^2+b^2 (-B)\right )}{8 b^2 c \left (b+c x^2\right )^2}+\frac{(3 A c+b B) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{8 b^{5/2} c^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 90, normalized size = 1. \begin{align*}{\frac{1}{ \left ( c{x}^{2}+b \right ) ^{2}} \left ({\frac{ \left ( 3\,Ac+Bb \right ){x}^{3}}{8\,{b}^{2}}}+{\frac{ \left ( 5\,Ac-Bb \right ) x}{8\,bc}} \right ) }+{\frac{3\,A}{8\,{b}^{2}}\arctan \left ({cx{\frac{1}{\sqrt{bc}}}} \right ){\frac{1}{\sqrt{bc}}}}+{\frac{B}{8\,bc}\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 = 1.11134, size = 621, normalized size = 6.75 \begin{align*} \left [\frac{2 \,{\left (B b^{2} c^{2} + 3 \, A b c^{3}\right )} x^{3} -{\left ({\left (B b c^{2} + 3 \, A c^{3}\right )} x^{4} + B b^{3} + 3 \, A b^{2} c + 2 \,{\left (B b^{2} c + 3 \, A b c^{2}\right )} x^{2}\right )} \sqrt{-b c} \log \left (\frac{c x^{2} - 2 \, \sqrt{-b c} x - b}{c x^{2} + b}\right ) - 2 \,{\left (B b^{3} c - 5 \, A b^{2} c^{2}\right )} x}{16 \,{\left (b^{3} c^{4} x^{4} + 2 \, b^{4} c^{3} x^{2} + b^{5} c^{2}\right )}}, \frac{{\left (B b^{2} c^{2} + 3 \, A b c^{3}\right )} x^{3} +{\left ({\left (B b c^{2} + 3 \, A c^{3}\right )} x^{4} + B b^{3} + 3 \, A b^{2} c + 2 \,{\left (B b^{2} c + 3 \, A b c^{2}\right )} x^{2}\right )} \sqrt{b c} \arctan \left (\frac{\sqrt{b c} x}{b}\right ) -{\left (B b^{3} c - 5 \, A b^{2} c^{2}\right )} x}{8 \,{\left (b^{3} c^{4} x^{4} + 2 \, b^{4} c^{3} x^{2} + b^{5} c^{2}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.728523, size = 150, normalized size = 1.63 \begin{align*} - \frac{\sqrt{- \frac{1}{b^{5} c^{3}}} \left (3 A c + B b\right ) \log{\left (- b^{3} c \sqrt{- \frac{1}{b^{5} c^{3}}} + x \right )}}{16} + \frac{\sqrt{- \frac{1}{b^{5} c^{3}}} \left (3 A c + B b\right ) \log{\left (b^{3} c \sqrt{- \frac{1}{b^{5} c^{3}}} + x \right )}}{16} + \frac{x^{3} \left (3 A c^{2} + B b c\right ) + x \left (5 A b c - B b^{2}\right )}{8 b^{4} c + 16 b^{3} c^{2} x^{2} + 8 b^{2} c^{3} x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12735, size = 105, normalized size = 1.14 \begin{align*} \frac{{\left (B b + 3 \, A c\right )} \arctan \left (\frac{c x}{\sqrt{b c}}\right )}{8 \, \sqrt{b c} b^{2} c} + \frac{B b c x^{3} + 3 \, A c^{2} x^{3} - B b^{2} x + 5 \, A b c x}{8 \,{\left (c x^{2} + b\right )}^{2} b^{2} c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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