Optimal. Leaf size=53 \[ \frac{c \log \left (c+d x^4\right )}{4 d (b c-a d)}-\frac{a \log \left (a+b x^4\right )}{4 b (b c-a d)} \]
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Rubi [A] time = 0.0499208, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {446, 72} \[ \frac{c \log \left (c+d x^4\right )}{4 d (b c-a d)}-\frac{a \log \left (a+b x^4\right )}{4 b (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 446
Rule 72
Rubi steps
\begin{align*} \int \frac{x^7}{\left (a+b x^4\right ) \left (c+d x^4\right )} \, dx &=\frac{1}{4} \operatorname{Subst}\left (\int \frac{x}{(a+b x) (c+d x)} \, dx,x,x^4\right )\\ &=\frac{1}{4} \operatorname{Subst}\left (\int \left (-\frac{a}{(b c-a d) (a+b x)}+\frac{c}{(b c-a d) (c+d x)}\right ) \, dx,x,x^4\right )\\ &=-\frac{a \log \left (a+b x^4\right )}{4 b (b c-a d)}+\frac{c \log \left (c+d x^4\right )}{4 d (b c-a d)}\\ \end{align*}
Mathematica [A] time = 0.0228855, size = 43, normalized size = 0.81 \[ -\frac{a d \log \left (a+b x^4\right )-b c \log \left (c+d x^4\right )}{4 b^2 c d-4 a b d^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 50, normalized size = 0.9 \begin{align*} -{\frac{c\ln \left ( d{x}^{4}+c \right ) }{ \left ( 4\,ad-4\,bc \right ) d}}+{\frac{a\ln \left ( b{x}^{4}+a \right ) }{ \left ( 4\,ad-4\,bc \right ) b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.932673, size = 66, normalized size = 1.25 \begin{align*} -\frac{a \log \left (b x^{4} + a\right )}{4 \,{\left (b^{2} c - a b d\right )}} + \frac{c \log \left (d x^{4} + c\right )}{4 \,{\left (b c d - a d^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.84223, size = 92, normalized size = 1.74 \begin{align*} -\frac{a d \log \left (b x^{4} + a\right ) - b c \log \left (d x^{4} + c\right )}{4 \,{\left (b^{2} c d - a b d^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 4.54636, size = 144, normalized size = 2.72 \begin{align*} \frac{a \log{\left (x^{4} + \frac{\frac{a^{3} d^{2}}{b \left (a d - b c\right )} - \frac{2 a^{2} c d}{a d - b c} + \frac{a b c^{2}}{a d - b c} + 2 a c}{a d + b c} \right )}}{4 b \left (a d - b c\right )} - \frac{c \log{\left (x^{4} + \frac{- \frac{a^{2} c d}{a d - b c} + \frac{2 a b c^{2}}{a d - b c} + 2 a c - \frac{b^{2} c^{3}}{d \left (a d - b c\right )}}{a d + b c} \right )}}{4 d \left (a d - b c\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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