Optimal. Leaf size=61 \[ \frac {2 a^x b^{-x}}{(\log (a)-\log (b))^3}-\frac {2 a^x b^{-x} x}{(\log (a)-\log (b))^2}+\frac {a^x b^{-x} x^2}{\log (a)-\log (b)} \]
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Rubi [A]
time = 0.05, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2325, 2207,
2225} \begin {gather*} \frac {x^2 a^x b^{-x}}{\log (a)-\log (b)}-\frac {2 x a^x b^{-x}}{(\log (a)-\log (b))^2}+\frac {2 a^x b^{-x}}{(\log (a)-\log (b))^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 2207
Rule 2225
Rule 2325
Rubi steps
\begin {align*} \int a^x b^{-x} x^2 \, dx &=\int e^{x (\log (a)-\log (b))} x^2 \, dx\\ &=\frac {a^x b^{-x} x^2}{\log (a)-\log (b)}-\frac {2 \int e^{x (\log (a)-\log (b))} x \, dx}{\log (a)-\log (b)}\\ &=-\frac {2 a^x b^{-x} x}{(\log (a)-\log (b))^2}+\frac {a^x b^{-x} x^2}{\log (a)-\log (b)}+\frac {2 \int e^{x (\log (a)-\log (b))} \, dx}{(\log (a)-\log (b))^2}\\ &=\frac {2 a^x b^{-x}}{(\log (a)-\log (b))^3}-\frac {2 a^x b^{-x} x}{(\log (a)-\log (b))^2}+\frac {a^x b^{-x} x^2}{\log (a)-\log (b)}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 43, normalized size = 0.70 \begin {gather*} \frac {a^x b^{-x} \left (2-2 x (\log (a)-\log (b))+x^2 (\log (a)-\log (b))^2\right )}{(\log (a)-\log (b))^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 56, normalized size = 0.92
method | result | size |
risch | \(\frac {\left (\ln \left (a \right )^{2} x^{2}-2 \ln \left (b \right ) \ln \left (a \right ) x^{2}+\ln \left (b \right )^{2} x^{2}-2 \ln \left (a \right ) x +2 \ln \left (b \right ) x +2\right ) a^{x} b^{-x}}{\left (\ln \left (a \right )-\ln \left (b \right )\right )^{3}}\) | \(56\) |
gosper | \(\frac {\left (\ln \left (a \right )^{2} x^{2}-2 \ln \left (b \right ) \ln \left (a \right ) x^{2}+\ln \left (b \right )^{2} x^{2}-2 \ln \left (a \right ) x +2 \ln \left (b \right ) x +2\right ) a^{x} b^{-x}}{\left (\ln \left (a \right )-\ln \left (b \right )\right ) \left (\ln \left (a \right )^{2}-2 \ln \left (b \right ) \ln \left (a \right )+\ln \left (b \right )^{2}\right )}\) | \(73\) |
meijerg | \(-\frac {2-\frac {\left (3 x^{2} \ln \left (a \right )^{2} \left (1-\frac {\ln \left (b \right )}{\ln \left (a \right )}\right )^{2}-6 x \ln \left (a \right ) \left (1-\frac {\ln \left (b \right )}{\ln \left (a \right )}\right )+6\right ) {\mathrm e}^{x \ln \left (a \right ) \left (1-\frac {\ln \left (b \right )}{\ln \left (a \right )}\right )}}{3}}{\ln \left (a \right )^{3} \left (1-\frac {\ln \left (b \right )}{\ln \left (a \right )}\right )^{3}}\) | \(76\) |
norman | \(\left (\frac {x^{2} {\mathrm e}^{\ln \left (a \right ) x}}{\ln \left (a \right )-\ln \left (b \right )}-\frac {2 x \,{\mathrm e}^{\ln \left (a \right ) x}}{\ln \left (a \right )^{2}-2 \ln \left (b \right ) \ln \left (a \right )+\ln \left (b \right )^{2}}+\frac {2 \,{\mathrm e}^{\ln \left (a \right ) x}}{\left (\ln \left (a \right )^{2}-2 \ln \left (b \right ) \ln \left (a \right )+\ln \left (b \right )^{2}\right ) \left (\ln \left (a \right )-\ln \left (b \right )\right )}\right ) {\mathrm e}^{-\ln \left (b \right ) x}\) | \(86\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 72, normalized size = 1.18 \begin {gather*} \frac {{\left ({\left (\log \left (a\right )^{2} - 2 \, \log \left (a\right ) \log \left (b\right ) + \log \left (b\right )^{2}\right )} x^{2} - 2 \, x {\left (\log \left (a\right ) - \log \left (b\right )\right )} + 2\right )} e^{\left (x \log \left (a\right ) - x \log \left (b\right )\right )}}{\log \left (a\right )^{3} - 3 \, \log \left (a\right )^{2} \log \left (b\right ) + 3 \, \log \left (a\right ) \log \left (b\right )^{2} - \log \left (b\right )^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 75, normalized size = 1.23 \begin {gather*} \frac {{\left (x^{2} \log \left (a\right )^{2} + x^{2} \log \left (b\right )^{2} - 2 \, x \log \left (a\right ) - 2 \, {\left (x^{2} \log \left (a\right ) - x\right )} \log \left (b\right ) + 2\right )} a^{x}}{{\left (\log \left (a\right )^{3} - 3 \, \log \left (a\right )^{2} \log \left (b\right ) + 3 \, \log \left (a\right ) \log \left (b\right )^{2} - \log \left (b\right )^{3}\right )} b^{x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 333 vs.
\(2 (51) = 102\).
time = 0.43, size = 333, normalized size = 5.46 \begin {gather*} \begin {cases} \frac {a^{x} x^{2} \log {\left (a \right )}^{2}}{b^{x} \log {\left (a \right )}^{3} - 3 b^{x} \log {\left (a \right )}^{2} \log {\left (b \right )} + 3 b^{x} \log {\left (a \right )} \log {\left (b \right )}^{2} - b^{x} \log {\left (b \right )}^{3}} - \frac {2 a^{x} x^{2} \log {\left (a \right )} \log {\left (b \right )}}{b^{x} \log {\left (a \right )}^{3} - 3 b^{x} \log {\left (a \right )}^{2} \log {\left (b \right )} + 3 b^{x} \log {\left (a \right )} \log {\left (b \right )}^{2} - b^{x} \log {\left (b \right )}^{3}} + \frac {a^{x} x^{2} \log {\left (b \right )}^{2}}{b^{x} \log {\left (a \right )}^{3} - 3 b^{x} \log {\left (a \right )}^{2} \log {\left (b \right )} + 3 b^{x} \log {\left (a \right )} \log {\left (b \right )}^{2} - b^{x} \log {\left (b \right )}^{3}} - \frac {2 a^{x} x \log {\left (a \right )}}{b^{x} \log {\left (a \right )}^{3} - 3 b^{x} \log {\left (a \right )}^{2} \log {\left (b \right )} + 3 b^{x} \log {\left (a \right )} \log {\left (b \right )}^{2} - b^{x} \log {\left (b \right )}^{3}} + \frac {2 a^{x} x \log {\left (b \right )}}{b^{x} \log {\left (a \right )}^{3} - 3 b^{x} \log {\left (a \right )}^{2} \log {\left (b \right )} + 3 b^{x} \log {\left (a \right )} \log {\left (b \right )}^{2} - b^{x} \log {\left (b \right )}^{3}} + \frac {2 a^{x}}{b^{x} \log {\left (a \right )}^{3} - 3 b^{x} \log {\left (a \right )}^{2} \log {\left (b \right )} + 3 b^{x} \log {\left (a \right )} \log {\left (b \right )}^{2} - b^{x} \log {\left (b \right )}^{3}} & \text {for}\: a \neq b \\\frac {x^{3}}{3} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] Result contains complex when optimal does not.
time = 5.37, size = 1817, normalized size = 29.79 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.56, size = 43, normalized size = 0.70 \begin {gather*} \frac {a^x\,\left (x^2\,{\left (\ln \left (a\right )-\ln \left (b\right )\right )}^2-2\,x\,\left (\ln \left (a\right )-\ln \left (b\right )\right )+2\right )}{b^x\,{\left (\ln \left (a\right )-\ln \left (b\right )\right )}^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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