Optimal. Leaf size=58 \[ -\frac {f^{a+b x^3}}{6 x^6}-\frac {b f^{a+b x^3} \log (f)}{6 x^3}+\frac {1}{6} b^2 f^a \text {Ei}\left (b x^3 \log (f)\right ) \log ^2(f) \]
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Rubi [A]
time = 0.04, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps
used = 3, 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}{6} b^2 f^a \log ^2(f) \text {Ei}\left (b x^3 \log (f)\right )-\frac {b \log (f) f^{a+b x^3}}{6 x^3}-\frac {f^{a+b x^3}}{6 x^6} \end {gather*}
Antiderivative was successfully verified.
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Rule 2241
Rule 2245
Rubi steps
\begin {align*} \int \frac {f^{a+b x^3}}{x^7} \, dx &=-\frac {f^{a+b x^3}}{6 x^6}+\frac {1}{2} (b \log (f)) \int \frac {f^{a+b x^3}}{x^4} \, dx\\ &=-\frac {f^{a+b x^3}}{6 x^6}-\frac {b f^{a+b x^3} \log (f)}{6 x^3}+\frac {1}{2} \left (b^2 \log ^2(f)\right ) \int \frac {f^{a+b x^3}}{x} \, dx\\ &=-\frac {f^{a+b x^3}}{6 x^6}-\frac {b f^{a+b x^3} \log (f)}{6 x^3}+\frac {1}{6} b^2 f^a \text {Ei}\left (b x^3 \log (f)\right ) \log ^2(f)\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 48, normalized size = 0.83 \begin {gather*} \frac {f^a \left (b^2 x^6 \text {Ei}\left (b x^3 \log (f)\right ) \log ^2(f)-f^{b x^3} \left (1+b x^3 \log (f)\right )\right )}{6 x^6} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(140\) vs.
\(2(52)=104\).
time = 0.03, size = 141, normalized size = 2.43
method | result | size |
meijerg | \(\frac {f^{a} b^{2} \ln \left (f \right )^{2} \left (\frac {9 b^{2} x^{6} \ln \left (f \right )^{2}+12 b \,x^{3} \ln \left (f \right )+6}{12 b^{2} x^{6} \ln \left (f \right )^{2}}-\frac {\left (3+3 b \,x^{3} \ln \left (f \right )\right ) {\mathrm e}^{b \,x^{3} \ln \left (f \right )}}{6 b^{2} x^{6} \ln \left (f \right )^{2}}-\frac {\ln \left (-b \,x^{3} \ln \left (f \right )\right )}{2}-\frac {\expIntegral \left (1, -b \,x^{3} \ln \left (f \right )\right )}{2}-\frac {3}{4}+\frac {3 \ln \left (x \right )}{2}+\frac {\ln \left (-b \right )}{2}+\frac {\ln \left (\ln \left (f \right )\right )}{2}-\frac {1}{2 b^{2} x^{6} \ln \left (f \right )^{2}}-\frac {1}{x^{3} \ln \left (f \right ) b}\right )}{3}\) | \(141\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.32, size = 22, normalized size = 0.38 \begin {gather*} -\frac {1}{3} \, b^{2} f^{a} \Gamma \left (-2, -b x^{3} \log \left (f\right )\right ) \log \left (f\right )^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 48, normalized size = 0.83 \begin {gather*} \frac {b^{2} f^{a} x^{6} {\rm Ei}\left (b x^{3} \log \left (f\right )\right ) \log \left (f\right )^{2} - {\left (b x^{3} \log \left (f\right ) + 1\right )} f^{b x^{3} + a}}{6 \, 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^{3}}}{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 = 3.32, size = 57, normalized size = 0.98 \begin {gather*} -\frac {b^2\,f^a\,{\ln \left (f\right )}^2\,\left (f^{b\,x^3}\,\left (\frac {1}{2\,b\,x^3\,\ln \left (f\right )}+\frac {1}{2\,b^2\,x^6\,{\ln \left (f\right )}^2}\right )+\frac {\mathrm {expint}\left (-b\,x^3\,\ln \left (f\right )\right )}{2}\right )}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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