Optimal. Leaf size=81 \[ -\frac {f^{a+b x^2}}{6 x^6}-\frac {b f^{a+b x^2} \log (f)}{12 x^4}-\frac {b^2 f^{a+b x^2} \log ^2(f)}{12 x^2}+\frac {1}{12} b^3 f^a \text {Ei}\left (b x^2 \log (f)\right ) \log ^3(f) \]
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Rubi [A]
time = 0.07, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2245, 2241}
\begin {gather*} \frac {1}{12} b^3 f^a \log ^3(f) \text {Ei}\left (b x^2 \log (f)\right )-\frac {b^2 \log ^2(f) f^{a+b x^2}}{12 x^2}-\frac {f^{a+b x^2}}{6 x^6}-\frac {b \log (f) f^{a+b x^2}}{12 x^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 2241
Rule 2245
Rubi steps
\begin {align*} \int \frac {f^{a+b x^2}}{x^7} \, dx &=-\frac {f^{a+b x^2}}{6 x^6}+\frac {1}{3} (b \log (f)) \int \frac {f^{a+b x^2}}{x^5} \, dx\\ &=-\frac {f^{a+b x^2}}{6 x^6}-\frac {b f^{a+b x^2} \log (f)}{12 x^4}+\frac {1}{6} \left (b^2 \log ^2(f)\right ) \int \frac {f^{a+b x^2}}{x^3} \, dx\\ &=-\frac {f^{a+b x^2}}{6 x^6}-\frac {b f^{a+b x^2} \log (f)}{12 x^4}-\frac {b^2 f^{a+b x^2} \log ^2(f)}{12 x^2}+\frac {1}{6} \left (b^3 \log ^3(f)\right ) \int \frac {f^{a+b x^2}}{x} \, dx\\ &=-\frac {f^{a+b x^2}}{6 x^6}-\frac {b f^{a+b x^2} \log (f)}{12 x^4}-\frac {b^2 f^{a+b x^2} \log ^2(f)}{12 x^2}+\frac {1}{12} b^3 f^a \text {Ei}\left (b x^2 \log (f)\right ) \log ^3(f)\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 59, normalized size = 0.73 \begin {gather*} \frac {f^a \left (b^3 x^6 \text {Ei}\left (b x^2 \log (f)\right ) \log ^3(f)-f^{b x^2} \left (2+b x^2 \log (f)+b^2 x^4 \log ^2(f)\right )\right )}{12 x^6} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 79, normalized size = 0.98
method | result | size |
risch | \(-\frac {f^{a} f^{b \,x^{2}}}{6 x^{6}}-\frac {f^{a} \ln \left (f \right ) b \,f^{b \,x^{2}}}{12 x^{4}}-\frac {f^{a} \ln \left (f \right )^{2} b^{2} f^{b \,x^{2}}}{12 x^{2}}-\frac {f^{a} \ln \left (f \right )^{3} b^{3} \expIntegral \left (1, -b \,x^{2} \ln \left (f \right )\right )}{12}\) | \(79\) |
meijerg | \(-\frac {f^{a} b^{3} \ln \left (f \right )^{3} \left (-\frac {22 b^{3} x^{6} \ln \left (f \right )^{3}+36 b^{2} x^{4} \ln \left (f \right )^{2}+36 b \,x^{2} \ln \left (f \right )+24}{72 b^{3} x^{6} \ln \left (f \right )^{3}}+\frac {\left (4 b^{2} x^{4} \ln \left (f \right )^{2}+4 b \,x^{2} \ln \left (f \right )+8\right ) {\mathrm e}^{b \,x^{2} \ln \left (f \right )}}{24 b^{3} x^{6} \ln \left (f \right )^{3}}+\frac {\ln \left (-b \,x^{2} \ln \left (f \right )\right )}{6}+\frac {\expIntegral \left (1, -b \,x^{2} \ln \left (f \right )\right )}{6}+\frac {11}{36}-\frac {\ln \left (x \right )}{3}-\frac {\ln \left (-b \right )}{6}-\frac {\ln \left (\ln \left (f \right )\right )}{6}+\frac {1}{3 b^{3} x^{6} \ln \left (f \right )^{3}}+\frac {1}{2 b^{2} x^{4} \ln \left (f \right )^{2}}+\frac {1}{2 b \,x^{2} \ln \left (f \right )}\right )}{2}\) | \(177\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.32, size = 22, normalized size = 0.27 \begin {gather*} \frac {1}{2} \, b^{3} f^{a} \Gamma \left (-3, -b x^{2} \log \left (f\right )\right ) \log \left (f\right )^{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 59, normalized size = 0.73 \begin {gather*} \frac {b^{3} f^{a} x^{6} {\rm Ei}\left (b x^{2} \log \left (f\right )\right ) \log \left (f\right )^{3} - {\left (b^{2} x^{4} \log \left (f\right )^{2} + b x^{2} \log \left (f\right ) + 2\right )} f^{b x^{2} + a}}{12 \, x^{6}} \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 \frac {f^{a + b x^{2}}}{x^{7}}\, 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 = 3.54, size = 69, normalized size = 0.85 \begin {gather*} -\frac {b^3\,f^a\,{\ln \left (f\right )}^3\,\left (f^{b\,x^2}\,\left (\frac {1}{6\,b\,x^2\,\ln \left (f\right )}+\frac {1}{6\,b^2\,x^4\,{\ln \left (f\right )}^2}+\frac {1}{3\,b^3\,x^6\,{\ln \left (f\right )}^3}\right )+\frac {\mathrm {expint}\left (-b\,x^2\,\ln \left (f\right )\right )}{6}\right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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