Optimal. Leaf size=28 \[ x \left (a+b x^n\right )^{-1/n} \left (c+d x^n\right )^{-1/n} \]
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Rubi [A]
time = 0.07, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 48, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.021, Rules used = {1912}
\begin {gather*} x \left (a+b x^n\right )^{-1/n} \left (c+d x^n\right )^{-1/n} \end {gather*}
Antiderivative was successfully verified.
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Rule 1912
Rubi steps
\begin {align*} \int \left (a+b x^n\right )^{\frac {-1-n}{n}} \left (c+d x^n\right )^{\frac {-1-n}{n}} \left (a c-b d x^{2 n}\right ) \, dx &=x \left (a+b x^n\right )^{-1/n} \left (c+d x^n\right )^{-1/n}\\ \end {align*}
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Mathematica [A]
time = 0.28, size = 28, normalized size = 1.00 \begin {gather*} x \left (a+b x^n\right )^{-1/n} \left (c+d x^n\right )^{-1/n} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.15, size = 0, normalized size = 0.00 \[\int \left (a +b \,x^{n}\right )^{\frac {-1-n}{n}} \left (c +d \,x^{n}\right )^{\frac {-1-n}{n}} \left (a c -b d \,x^{2 n}\right )\, dx\]
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 [B] Leaf count of result is larger than twice the leaf count of optimal. 61 vs.
\(2 (28) = 56\).
time = 0.39, size = 61, normalized size = 2.18 \begin {gather*} \frac {b d x x^{2 \, n} + a c x + {\left (b c + a d\right )} x x^{n}}{{\left (b x^{n} + a\right )}^{\frac {n + 1}{n}} {\left (d x^{n} + c\right )}^{\frac {n + 1}{n}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 228 vs.
\(2 (28) = 56\).
time = 0.81, size = 228, normalized size = 8.14 \begin {gather*} b d x x^{2 \, n} e^{\left (-\frac {n \log \left (b x^{n} + a\right ) + \log \left (b x^{n} + a\right )}{n} - \frac {n \log \left (d x^{n} + c\right ) + \log \left (d x^{n} + c\right )}{n}\right )} + b c x x^{n} e^{\left (-\frac {n \log \left (b x^{n} + a\right ) + \log \left (b x^{n} + a\right )}{n} - \frac {n \log \left (d x^{n} + c\right ) + \log \left (d x^{n} + c\right )}{n}\right )} + a d x x^{n} e^{\left (-\frac {n \log \left (b x^{n} + a\right ) + \log \left (b x^{n} + a\right )}{n} - \frac {n \log \left (d x^{n} + c\right ) + \log \left (d x^{n} + c\right )}{n}\right )} + a c x e^{\left (-\frac {n \log \left (b x^{n} + a\right ) + \log \left (b x^{n} + a\right )}{n} - \frac {n \log \left (d x^{n} + c\right ) + \log \left (d x^{n} + c\right )}{n}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.20, size = 95, normalized size = 3.39 \begin {gather*} \frac {\frac {a\,c\,x}{{\left (a+b\,x^n\right )}^{\frac {n+1}{n}}}+\frac {x\,x^n\,\left (a\,d+b\,c\right )}{{\left (a+b\,x^n\right )}^{\frac {n+1}{n}}}+\frac {b\,d\,x\,x^{2\,n}}{{\left (a+b\,x^n\right )}^{\frac {n+1}{n}}}}{{\left (c+d\,x^n\right )}^{\frac {n+1}{n}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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