Optimal. Leaf size=51 \[ -\frac {a^x b^x}{2 x^2}-\frac {a^x b^x (\log (a)+\log (b))}{2 x}+\frac {1}{2} (\log (a)+\log (b))^2 \text {Ei}(x (\log (a)+\log (b))) \]
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Rubi [A] time = 0.09, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {2287, 2177, 2178} \[ -\frac {a^x b^x}{2 x^2}-\frac {a^x b^x (\log (a)+\log (b))}{2 x}+\frac {1}{2} (\log (a)+\log (b))^2 \text {Ei}(x (\log (a)+\log (b))) \]
Antiderivative was successfully verified.
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Rule 2177
Rule 2178
Rule 2287
Rubi steps
\begin {align*} \int \frac {a^x b^x}{x^3} \, dx &=\int \frac {e^{x (\log (a)+\log (b))}}{x^3} \, dx\\ &=-\frac {a^x b^x}{2 x^2}-\frac {1}{2} (-\log (a)-\log (b)) \int \frac {e^{x (\log (a)+\log (b))}}{x^2} \, dx\\ &=-\frac {a^x b^x}{2 x^2}-\frac {a^x b^x (\log (a)+\log (b))}{2 x}+\frac {1}{2} (\log (a)+\log (b))^2 \int \frac {e^{x (\log (a)+\log (b))}}{x} \, dx\\ &=-\frac {a^x b^x}{2 x^2}-\frac {a^x b^x (\log (a)+\log (b))}{2 x}+\frac {1}{2} \text {Ei}(x (\log (a)+\log (b))) (\log (a)+\log (b))^2\\ \end {align*}
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Mathematica [F] time = 0.06, size = 0, normalized size = 0.00 \[ \int \frac {a^x b^x}{x^3} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.42, size = 61, normalized size = 1.20 \[ -\frac {{\left (x \log \relax (a) + x \log \relax (b) + 1\right )} a^{x} b^{x} - {\left (x^{2} \log \relax (a)^{2} + 2 \, x^{2} \log \relax (a) \log \relax (b) + x^{2} \log \relax (b)^{2}\right )} {\rm Ei}\left (x \log \relax (a) + x \log \relax (b)\right )}{2 \, x^{2}} \]
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 \[ \int \frac {a^{x} b^{x}}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.07, size = 225, normalized size = 4.41 \[ \left (\frac {\ln \relax (a )}{\ln \relax (b )}+1\right )^{2} \left (-\frac {\Ei \left (1, -\left (\frac {\ln \relax (a )}{\ln \relax (b )}+1\right ) x \ln \relax (b )\right )}{2}+\frac {\ln \relax (x )}{2}-\frac {\ln \left (-\left (\frac {\ln \relax (a )}{\ln \relax (b )}+1\right ) x \ln \relax (b )\right )}{2}+\frac {\ln \left (\frac {\ln \relax (a )}{\ln \relax (b )}+1\right )}{2}+\frac {\ln \left (\ln \relax (b )\right )}{2}-\frac {1}{\left (\frac {\ln \relax (a )}{\ln \relax (b )}+1\right ) x \ln \relax (b )}-\frac {\left (3 \left (\frac {\ln \relax (a )}{\ln \relax (b )}+1\right ) x \ln \relax (b )+3\right ) {\mathrm e}^{\left (\frac {\ln \relax (a )}{\ln \relax (b )}+1\right ) x \ln \relax (b )}}{6 \left (\frac {\ln \relax (a )}{\ln \relax (b )}+1\right )^{2} x^{2} \ln \relax (b )^{2}}+\frac {9 \left (\frac {\ln \relax (a )}{\ln \relax (b )}+1\right )^{2} x^{2} \ln \relax (b )^{2}+12 \left (\frac {\ln \relax (a )}{\ln \relax (b )}+1\right ) x \ln \relax (b )+6}{12 \left (\frac {\ln \relax (a )}{\ln \relax (b )}+1\right )^{2} x^{2} \ln \relax (b )^{2}}-\frac {1}{2 \left (\frac {\ln \relax (a )}{\ln \relax (b )}+1\right )^{2} x^{2} \ln \relax (b )^{2}}-\frac {3}{4}+\frac {i \pi }{2}\right ) \ln \relax (b )^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.33, size = 19, normalized size = 0.37 \[ -{\left (\log \relax (a) + \log \relax (b)\right )}^{2} \Gamma \left (-2, -x {\left (\log \relax (a) + \log \relax (b)\right )}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 59, normalized size = 1.16 \[ -\frac {\mathrm {expint}\left (-x\,\left (\ln \relax (a)+\ln \relax (b)\right )\right )\,{\left (\ln \relax (a)+\ln \relax (b)\right )}^2}{2}-a^x\,b^x\,\left (\frac {1}{2\,x\,\left (\ln \relax (a)+\ln \relax (b)\right )}+\frac {1}{2\,x^2\,{\left (\ln \relax (a)+\ln \relax (b)\right )}^2}\right )\,{\left (\ln \relax (a)+\ln \relax (b)\right )}^2 \]
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 \[ \int \frac {a^{x} b^{x}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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