Optimal. Leaf size=39 \[ -\frac {f^a \left (-b \log (f) x^n\right )^{2/n} \Gamma \left (-\frac {2}{n},-b x^n \log (f)\right )}{n x^2} \]
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Rubi [A] time = 0.02, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {2218} \[ -\frac {f^a \left (-b \log (f) x^n\right )^{2/n} \text {Gamma}\left (-\frac {2}{n},-b \log (f) x^n\right )}{n x^2} \]
Antiderivative was successfully verified.
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Rule 2218
Rubi steps
\begin {align*} \int \frac {f^{a+b x^n}}{x^3} \, dx &=-\frac {f^a \Gamma \left (-\frac {2}{n},-b x^n \log (f)\right ) \left (-b x^n \log (f)\right )^{2/n}}{n x^2}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 39, normalized size = 1.00 \[ -\frac {f^a \left (-b \log (f) x^n\right )^{2/n} \Gamma \left (-\frac {2}{n},-b x^n \log (f)\right )}{n x^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.40, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {f^{b x^{n} + a}}{x^{3}}, x\right ) \]
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 \[ \int \frac {f^{b x^{n} + a}}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.04, size = 0, normalized size = 0.00 \[ \int \frac {f^{b \,x^{n}+a}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.16, size = 39, normalized size = 1.00 \[ -\frac {\left (-b x^{n} \log \relax (f)\right )^{\frac {2}{n}} f^{a} \Gamma \left (-\frac {2}{n}, -b x^{n} \log \relax (f)\right )}{n x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.52, size = 48, normalized size = 1.23 \[ -\frac {f^a\,{\mathrm {e}}^{\frac {b\,x^n\,\ln \relax (f)}{2}}\,{\mathrm {M}}_{\frac {1}{n}+\frac {1}{2},-\frac {1}{n}}\left (b\,x^n\,\ln \relax (f)\right )\,{\left (b\,x^n\,\ln \relax (f)\right )}^{\frac {1}{n}-\frac {1}{2}}}{2\,x^2} \]
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 \[ \begin {cases} \frac {b f^{a} f^{b x^{n}} n x^{n} \log {\relax (f )}}{2 n x^{2} - 4 x^{2}} - \frac {f^{a} f^{b x^{n}} n}{2 n x^{2} - 4 x^{2}} + \frac {2 f^{a} f^{b x^{n}}}{2 n x^{2} - 4 x^{2}} & \text {for}\: n \neq 2 \\\int \frac {f^{a + b x^{2}}}{x^{3}}\, dx & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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