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 = 0.01, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 50, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.020, Rules used = {389}
\begin {gather*} \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 {gather*}
Antiderivative was successfully verified.
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Rule 389
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*}
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Mathematica [A]
time = 0.41, size = 52, normalized size = 0.98 \begin {gather*} \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 {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.13, size = 71, normalized size = 1.34
method | result | size |
gosper | \(\frac {\left (b \,x^{2}+a \right )^{1-\frac {2 a d -3 b c}{2 \left (a d -b c \right )}} \left (d \,x^{2}+c \right )^{1-\frac {3 a d -2 b c}{2 \left (a d -b c \right )}} x}{a c}\) | \(71\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.96, size = 91, normalized size = 1.72 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a + b x^{2}\right )^{- \frac {b c}{- 2 a d + 2 b c} - 1} \left (c + d x^{2}\right )^{\frac {a d}{- 2 a d + 2 b c} - 1}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.74, size = 131, normalized size = 2.47 \begin {gather*} \frac {x\,{\left (b\,x^2+a\right )}^{\frac {b\,c}{2\,a\,d-2\,b\,c}-1}+\frac {x^3\,{\left (b\,x^2+a\right )}^{\frac {b\,c}{2\,a\,d-2\,b\,c}-1}\,\left (a\,d+b\,c\right )}{a\,c}+\frac {b\,d\,x^5\,{\left (b\,x^2+a\right )}^{\frac {b\,c}{2\,a\,d-2\,b\,c}-1}}{a\,c}}{{\left (d\,x^2+c\right )}^{\frac {a\,d}{2\,a\,d-2\,b\,c}+1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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