Optimal. Leaf size=43 \[ \frac {\sqrt {\pi } f^a \text {erfi}\left (\sqrt {b} \sqrt {\log (f)} x^{n/2}\right )}{\sqrt {b} n \sqrt {\log (f)}} \]
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Rubi [A] time = 0.04, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2211, 2204} \[ \frac {\sqrt {\pi } f^a \text {Erfi}\left (\sqrt {b} \sqrt {\log (f)} x^{n/2}\right )}{\sqrt {b} n \sqrt {\log (f)}} \]
Antiderivative was successfully verified.
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Rule 2204
Rule 2211
Rubi steps
\begin {align*} \int f^{a+b x^n} x^{-1+\frac {n}{2}} \, dx &=\frac {2 \operatorname {Subst}\left (\int f^{a+b x^2} \, dx,x,x^{n/2}\right )}{n}\\ &=\frac {f^a \sqrt {\pi } \text {erfi}\left (\sqrt {b} x^{n/2} \sqrt {\log (f)}\right )}{\sqrt {b} n \sqrt {\log (f)}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 43, normalized size = 1.00 \[ \frac {\sqrt {\pi } f^a \text {erfi}\left (\sqrt {b} \sqrt {\log (f)} x^{n/2}\right )}{\sqrt {b} n \sqrt {\log (f)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 42, normalized size = 0.98 \[ -\frac {\sqrt {\pi } \sqrt {-b \log \relax (f)} f^{a} \operatorname {erf}\left (\sqrt {-b \log \relax (f)} x x^{\frac {1}{2} \, n - 1}\right )}{b n \log \relax (f)} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.28, size = 33, normalized size = 0.77 \[ -\frac {\sqrt {\pi } f^{a} \operatorname {erf}\left (-\sqrt {-b \log \relax (f)} \sqrt {x^{n}}\right )}{\sqrt {-b \log \relax (f)} n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.10, size = 32, normalized size = 0.74 \[ \frac {\sqrt {\pi }\, f^{a} \erf \left (\sqrt {-b \ln \relax (f )}\, x^{\frac {n}{2}}\right )}{\sqrt {-b \ln \relax (f )}\, n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.94, size = 38, normalized size = 0.88 \[ \frac {\sqrt {\pi } f^{a} x^{\frac {1}{2} \, n} {\left (\operatorname {erf}\left (\sqrt {-b x^{n} \log \relax (f)}\right ) - 1\right )}}{\sqrt {-b x^{n} \log \relax (f)} n} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int f^{a+b\,x^n}\,x^{\frac {n}{2}-1} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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