Optimal. Leaf size=79 \[ \frac{\sqrt{c} \tan ^{-1}\left (\frac{\sqrt{d} x^2}{\sqrt{c}}\right )}{2 \sqrt{d} (b c-a d)}-\frac{\sqrt{a} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{2 \sqrt{b} (b c-a d)} \]
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Rubi [A] time = 0.0621678, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {465, 481, 205} \[ \frac{\sqrt{c} \tan ^{-1}\left (\frac{\sqrt{d} x^2}{\sqrt{c}}\right )}{2 \sqrt{d} (b c-a d)}-\frac{\sqrt{a} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{2 \sqrt{b} (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 465
Rule 481
Rule 205
Rubi steps
\begin{align*} \int \frac{x^5}{\left (a+b x^4\right ) \left (c+d x^4\right )} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^2}{\left (a+b x^2\right ) \left (c+d x^2\right )} \, dx,x,x^2\right )\\ &=-\frac{a \operatorname{Subst}\left (\int \frac{1}{a+b x^2} \, dx,x,x^2\right )}{2 (b c-a d)}+\frac{c \operatorname{Subst}\left (\int \frac{1}{c+d x^2} \, dx,x,x^2\right )}{2 (b c-a d)}\\ &=-\frac{\sqrt{a} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{2 \sqrt{b} (b c-a d)}+\frac{\sqrt{c} \tan ^{-1}\left (\frac{\sqrt{d} x^2}{\sqrt{c}}\right )}{2 \sqrt{d} (b c-a d)}\\ \end{align*}
Mathematica [A] time = 0.038372, size = 66, normalized size = 0.84 \[ \frac{\frac{\sqrt{c} \tan ^{-1}\left (\frac{\sqrt{d} x^2}{\sqrt{c}}\right )}{\sqrt{d}}-\frac{\sqrt{a} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{\sqrt{b}}}{2 b c-2 a d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 60, normalized size = 0.8 \begin{align*} -{\frac{c}{2\,ad-2\,bc}\arctan \left ({{x}^{2}d{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}+{\frac{a}{2\,ad-2\,bc}\arctan \left ({b{x}^{2}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \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 = 1.34681, size = 657, normalized size = 8.32 \begin{align*} \left [-\frac{\sqrt{-\frac{a}{b}} \log \left (\frac{b x^{4} + 2 \, b x^{2} \sqrt{-\frac{a}{b}} - a}{b x^{4} + a}\right ) + \sqrt{-\frac{c}{d}} \log \left (\frac{d x^{4} - 2 \, d x^{2} \sqrt{-\frac{c}{d}} - c}{d x^{4} + c}\right )}{4 \,{\left (b c - a d\right )}}, -\frac{2 \, \sqrt{\frac{a}{b}} \arctan \left (\frac{b x^{2} \sqrt{\frac{a}{b}}}{a}\right ) + \sqrt{-\frac{c}{d}} \log \left (\frac{d x^{4} - 2 \, d x^{2} \sqrt{-\frac{c}{d}} - c}{d x^{4} + c}\right )}{4 \,{\left (b c - a d\right )}}, \frac{2 \, \sqrt{\frac{c}{d}} \arctan \left (\frac{d x^{2} \sqrt{\frac{c}{d}}}{c}\right ) - \sqrt{-\frac{a}{b}} \log \left (\frac{b x^{4} + 2 \, b x^{2} \sqrt{-\frac{a}{b}} - a}{b x^{4} + a}\right )}{4 \,{\left (b c - a d\right )}}, -\frac{\sqrt{\frac{a}{b}} \arctan \left (\frac{b x^{2} \sqrt{\frac{a}{b}}}{a}\right ) - \sqrt{\frac{c}{d}} \arctan \left (\frac{d x^{2} \sqrt{\frac{c}{d}}}{c}\right )}{2 \,{\left (b c - a d\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 38.0516, size = 576, normalized size = 7.29 \begin{align*} \frac{\sqrt{- \frac{a}{b}} \log{\left (- \frac{2 a^{2} b d^{3} \left (- \frac{a}{b}\right )^{\frac{3}{2}}}{\left (a d - b c\right )^{3}} + \frac{4 a b^{2} c d^{2} \left (- \frac{a}{b}\right )^{\frac{3}{2}}}{\left (a d - b c\right )^{3}} - \frac{a d \sqrt{- \frac{a}{b}}}{a d - b c} - \frac{2 b^{3} c^{2} d \left (- \frac{a}{b}\right )^{\frac{3}{2}}}{\left (a d - b c\right )^{3}} - \frac{b c \sqrt{- \frac{a}{b}}}{a d - b c} + x^{2} \right )}}{4 \left (a d - b c\right )} - \frac{\sqrt{- \frac{a}{b}} \log{\left (\frac{2 a^{2} b d^{3} \left (- \frac{a}{b}\right )^{\frac{3}{2}}}{\left (a d - b c\right )^{3}} - \frac{4 a b^{2} c d^{2} \left (- \frac{a}{b}\right )^{\frac{3}{2}}}{\left (a d - b c\right )^{3}} + \frac{a d \sqrt{- \frac{a}{b}}}{a d - b c} + \frac{2 b^{3} c^{2} d \left (- \frac{a}{b}\right )^{\frac{3}{2}}}{\left (a d - b c\right )^{3}} + \frac{b c \sqrt{- \frac{a}{b}}}{a d - b c} + x^{2} \right )}}{4 \left (a d - b c\right )} + \frac{\sqrt{- \frac{c}{d}} \log{\left (- \frac{2 a^{2} b d^{3} \left (- \frac{c}{d}\right )^{\frac{3}{2}}}{\left (a d - b c\right )^{3}} + \frac{4 a b^{2} c d^{2} \left (- \frac{c}{d}\right )^{\frac{3}{2}}}{\left (a d - b c\right )^{3}} - \frac{a d \sqrt{- \frac{c}{d}}}{a d - b c} - \frac{2 b^{3} c^{2} d \left (- \frac{c}{d}\right )^{\frac{3}{2}}}{\left (a d - b c\right )^{3}} - \frac{b c \sqrt{- \frac{c}{d}}}{a d - b c} + x^{2} \right )}}{4 \left (a d - b c\right )} - \frac{\sqrt{- \frac{c}{d}} \log{\left (\frac{2 a^{2} b d^{3} \left (- \frac{c}{d}\right )^{\frac{3}{2}}}{\left (a d - b c\right )^{3}} - \frac{4 a b^{2} c d^{2} \left (- \frac{c}{d}\right )^{\frac{3}{2}}}{\left (a d - b c\right )^{3}} + \frac{a d \sqrt{- \frac{c}{d}}}{a d - b c} + \frac{2 b^{3} c^{2} d \left (- \frac{c}{d}\right )^{\frac{3}{2}}}{\left (a d - b c\right )^{3}} + \frac{b c \sqrt{- \frac{c}{d}}}{a d - b c} + x^{2} \right )}}{4 \left (a d - b c\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.14814, size = 271, normalized size = 3.43 \begin{align*} -\frac{\sqrt{c d} b x^{4}{\left | d \right |} \arctan \left (\frac{2 \, x^{2}}{\sqrt{\frac{2 \, b c + 2 \, a d + \sqrt{-16 \, a b c d + 4 \,{\left (b c + a d\right )}^{2}}}{b d}}}\right )}{b c d{\left | b c - a d \right |} + a d^{2}{\left | b c - a d \right |} +{\left (b c - a d\right )}^{2} d} + \frac{\sqrt{a b} d x^{4}{\left | b \right |} \arctan \left (\frac{2 \, x^{2}}{\sqrt{\frac{2 \, b c + 2 \, a d - \sqrt{-16 \, a b c d + 4 \,{\left (b c + a d\right )}^{2}}}{b d}}}\right )}{b^{2} c{\left | b c - a d \right |} + a b d{\left | b c - a d \right |} -{\left (b c - a d\right )}^{2} b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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