Optimal. Leaf size=53 \[ \frac{x \left (a+b x^2\right )^{-\frac{b c}{2 b c-2 a d}} \left (c+d x^2\right )^{\frac{a d}{2 b c-2 a d}}}{a c} \]
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Rubi [A] time = 0.0200724, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 50, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.02, Rules used = {381} \[ \frac{x \left (a+b x^2\right )^{-\frac{b c}{2 b c-2 a d}} \left (c+d x^2\right )^{\frac{a d}{2 b c-2 a d}}}{a c} \]
Antiderivative was successfully verified.
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Rule 381
Rubi steps
\begin{align*} \int \left (a+b x^2\right )^{-1-\frac{b c}{2 b c-2 a d}} \left (c+d x^2\right )^{-1+\frac{a d}{2 b c-2 a d}} \, dx &=\frac{x \left (a+b x^2\right )^{-\frac{b c}{2 b c-2 a d}} \left (c+d x^2\right )^{\frac{a d}{2 b c-2 a d}}}{a c}\\ \end{align*}
Mathematica [A] time = 0.0311385, size = 52, normalized size = 0.98 \[ \frac{x \left (a+b x^2\right )^{\frac{b c}{2 a d-2 b c}} \left (c+d x^2\right )^{\frac{a d}{2 b c-2 a d}}}{a c} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 71, normalized size = 1.3 \begin{align*}{\frac{x}{ac} \left ( b{x}^{2}+a \right ) ^{1-{\frac{2\,ad-3\,bc}{2\,ad-2\,bc}}} \left ( d{x}^{2}+c \right ) ^{1-{\frac{3\,ad-2\,bc}{2\,ad-2\,bc}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b x^{2} + a\right )}^{-\frac{b c}{2 \,{\left (b c - a d\right )}} - 1}{\left (d x^{2} + c\right )}^{\frac{a d}{2 \,{\left (b c - a d\right )}} - 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.67663, size = 182, normalized size = 3.43 \begin{align*} \frac{b d x^{5} +{\left (b c + a d\right )} x^{3} + a c x}{{\left (b x^{2} + a\right )}^{\frac{3 \, b c - 2 \, a d}{2 \,{\left (b c - a d\right )}}}{\left (d x^{2} + c\right )}^{\frac{2 \, b c - 3 \, a d}{2 \,{\left (b c - a d\right )}}} a c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b x^{2} + a\right )}^{-\frac{b c}{2 \,{\left (b c - a d\right )}} - 1}{\left (d x^{2} + c\right )}^{\frac{a d}{2 \,{\left (b c - a d\right )}} - 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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