Optimal. Leaf size=58 \[ -\frac {1}{4} b^2 f^a \log ^2(f) \text {Ei}\left (\frac {b \log (f)}{x^2}\right )+\frac {1}{4} b x^2 \log (f) f^{a+\frac {b}{x^2}}+\frac {1}{4} x^4 f^{a+\frac {b}{x^2}} \]
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Rubi [A] time = 0.06, 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 = {2214, 2210} \[ -\frac {1}{4} b^2 f^a \log ^2(f) \text {Ei}\left (\frac {b \log (f)}{x^2}\right )+\frac {1}{4} x^4 f^{a+\frac {b}{x^2}}+\frac {1}{4} b x^2 \log (f) f^{a+\frac {b}{x^2}} \]
Antiderivative was successfully verified.
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Rule 2210
Rule 2214
Rubi steps
\begin {align*} \int f^{a+\frac {b}{x^2}} x^3 \, dx &=\frac {1}{4} f^{a+\frac {b}{x^2}} x^4+\frac {1}{2} (b \log (f)) \int f^{a+\frac {b}{x^2}} x \, dx\\ &=\frac {1}{4} f^{a+\frac {b}{x^2}} x^4+\frac {1}{4} b f^{a+\frac {b}{x^2}} x^2 \log (f)+\frac {1}{2} \left (b^2 \log ^2(f)\right ) \int \frac {f^{a+\frac {b}{x^2}}}{x} \, dx\\ &=\frac {1}{4} f^{a+\frac {b}{x^2}} x^4+\frac {1}{4} b f^{a+\frac {b}{x^2}} x^2 \log (f)-\frac {1}{4} b^2 f^a \text {Ei}\left (\frac {b \log (f)}{x^2}\right ) \log ^2(f)\\ \end {align*}
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Mathematica [A] time = 0.01, size = 44, normalized size = 0.76 \[ \frac {1}{4} f^a \left (x^2 f^{\frac {b}{x^2}} \left (b \log (f)+x^2\right )-b^2 \log ^2(f) \text {Ei}\left (\frac {b \log (f)}{x^2}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 47, normalized size = 0.81 \[ -\frac {1}{4} \, b^{2} f^{a} {\rm Ei}\left (\frac {b \log \relax (f)}{x^{2}}\right ) \log \relax (f)^{2} + \frac {1}{4} \, {\left (x^{4} + b x^{2} \log \relax (f)\right )} f^{\frac {a x^{2} + b}{x^{2}}} \]
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 \[ \int f^{a + \frac {b}{x^{2}}} x^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 57, normalized size = 0.98 \[ \frac {b^{2} f^{a} \Ei \left (1, -\frac {b \ln \relax (f )}{x^{2}}\right ) \ln \relax (f )^{2}}{4}+\frac {b \,x^{2} f^{a} f^{\frac {b}{x^{2}}} \ln \relax (f )}{4}+\frac {x^{4} f^{a} f^{\frac {b}{x^{2}}}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.21, size = 22, normalized size = 0.38 \[ \frac {1}{2} \, b^{2} f^{a} \Gamma \left (-2, -\frac {b \log \relax (f)}{x^{2}}\right ) \log \relax (f)^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.65, size = 57, normalized size = 0.98 \[ \frac {b^2\,f^a\,{\ln \relax (f)}^2\,\left (f^{\frac {b}{x^2}}\,\left (\frac {x^2}{2\,b\,\ln \relax (f)}+\frac {x^4}{2\,b^2\,{\ln \relax (f)}^2}\right )+\frac {\mathrm {expint}\left (-\frac {b\,\ln \relax (f)}{x^2}\right )}{2}\right )}{2} \]
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 \[ \int f^{a + \frac {b}{x^{2}}} x^{3}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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