Optimal. Leaf size=96 \[ \frac {4 \sqrt {\pi } b^{3/2} f^a \log ^{\frac {3}{2}}(f) \text {erfi}\left (\sqrt {b} \sqrt {\log (f)} x^{n/2}\right )}{3 n}-\frac {2 x^{-3 n/2} f^{a+b x^n}}{3 n}-\frac {4 b \log (f) x^{-n/2} f^{a+b x^n}}{3 n} \]
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Rubi [A] time = 0.10, antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {2215, 2211, 2204} \[ \frac {4 \sqrt {\pi } b^{3/2} f^a \log ^{\frac {3}{2}}(f) \text {Erfi}\left (\sqrt {b} \sqrt {\log (f)} x^{n/2}\right )}{3 n}-\frac {2 x^{-3 n/2} f^{a+b x^n}}{3 n}-\frac {4 b \log (f) x^{-n/2} f^{a+b x^n}}{3 n} \]
Antiderivative was successfully verified.
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Rule 2204
Rule 2211
Rule 2215
Rubi steps
\begin {align*} \int f^{a+b x^n} x^{-1-\frac {3 n}{2}} \, dx &=-\frac {2 f^{a+b x^n} x^{-3 n/2}}{3 n}+\frac {1}{3} (2 b \log (f)) \int f^{a+b x^n} x^{-1-\frac {n}{2}} \, dx\\ &=-\frac {2 f^{a+b x^n} x^{-3 n/2}}{3 n}-\frac {4 b f^{a+b x^n} x^{-n/2} \log (f)}{3 n}+\frac {1}{3} \left (4 b^2 \log ^2(f)\right ) \int f^{a+b x^n} x^{\frac {1}{2} (-2+n)} \, dx\\ &=-\frac {2 f^{a+b x^n} x^{-3 n/2}}{3 n}-\frac {4 b f^{a+b x^n} x^{-n/2} \log (f)}{3 n}+\frac {\left (8 b^2 \log ^2(f)\right ) \operatorname {Subst}\left (\int f^{a+b x^2} \, dx,x,x^{1+\frac {1}{2} (-2+n)}\right )}{3 n}\\ &=-\frac {2 f^{a+b x^n} x^{-3 n/2}}{3 n}-\frac {4 b f^{a+b x^n} x^{-n/2} \log (f)}{3 n}+\frac {4 b^{3/2} f^a \sqrt {\pi } \text {erfi}\left (\sqrt {b} x^{n/2} \sqrt {\log (f)}\right ) \log ^{\frac {3}{2}}(f)}{3 n}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 39, normalized size = 0.41 \[ -\frac {f^a x^{-3 n/2} \left (-b \log (f) x^n\right )^{3/2} \Gamma \left (-\frac {3}{2},-b x^n \log (f)\right )}{n} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (f^{b x^{n} + a} x^{-\frac {3}{2} \, n - 1}, x\right ) \]
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 \[ \int f^{b x^{n} + a} x^{-\frac {3}{2} \, n - 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.10, size = 88, normalized size = 0.92 \[ \frac {4 \sqrt {\pi }\, b^{2} f^{a} \erf \left (\sqrt {-b \ln \relax (f )}\, x^{\frac {n}{2}}\right ) \ln \relax (f )^{2}}{3 \sqrt {-b \ln \relax (f )}\, n}-\frac {4 b \,f^{a} f^{b \,x^{n}} x^{-\frac {n}{2}} \ln \relax (f )}{3 n}-\frac {2 f^{a} f^{b \,x^{n}} x^{-\frac {3 n}{2}}}{3 n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.04, size = 35, normalized size = 0.36 \[ -\frac {\left (-b x^{n} \log \relax (f)\right )^{\frac {3}{2}} f^{a} \Gamma \left (-\frac {3}{2}, -b x^{n} \log \relax (f)\right )}{n x^{\frac {3}{2} \, n}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {f^{a+b\,x^n}}{x^{\frac {3\,n}{2}+1}} \,d x \]
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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