Optimal. Leaf size=104 \[ \frac {3 \sqrt {\pi } f^a \text {erfi}\left (\sqrt {b} \sqrt {\log (f)} x^{n/2}\right )}{4 b^{5/2} n \log ^{\frac {5}{2}}(f)}-\frac {3 x^{n/2} f^{a+b x^n}}{2 b^2 n \log ^2(f)}+\frac {x^{3 n/2} f^{a+b x^n}}{b n \log (f)} \]
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Rubi [A] time = 0.11, antiderivative size = 104, 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 = {2213, 2211, 2204} \[ \frac {3 \sqrt {\pi } f^a \text {Erfi}\left (\sqrt {b} \sqrt {\log (f)} x^{n/2}\right )}{4 b^{5/2} n \log ^{\frac {5}{2}}(f)}-\frac {3 x^{n/2} f^{a+b x^n}}{2 b^2 n \log ^2(f)}+\frac {x^{3 n/2} f^{a+b x^n}}{b n \log (f)} \]
Antiderivative was successfully verified.
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Rule 2204
Rule 2211
Rule 2213
Rubi steps
\begin {align*} \int f^{a+b x^n} x^{-1+\frac {5 n}{2}} \, dx &=\frac {f^{a+b x^n} x^{3 n/2}}{b n \log (f)}-\frac {3 \int f^{a+b x^n} x^{-1+\frac {3 n}{2}} \, dx}{2 b \log (f)}\\ &=-\frac {3 f^{a+b x^n} x^{n/2}}{2 b^2 n \log ^2(f)}+\frac {f^{a+b x^n} x^{3 n/2}}{b n \log (f)}+\frac {3 \int f^{a+b x^n} x^{\frac {1}{2} (-2+n)} \, dx}{4 b^2 \log ^2(f)}\\ &=-\frac {3 f^{a+b x^n} x^{n/2}}{2 b^2 n \log ^2(f)}+\frac {f^{a+b x^n} x^{3 n/2}}{b n \log (f)}+\frac {3 \operatorname {Subst}\left (\int f^{a+b x^2} \, dx,x,x^{1+\frac {1}{2} (-2+n)}\right )}{2 b^2 n \log ^2(f)}\\ &=\frac {3 f^a \sqrt {\pi } \text {erfi}\left (\sqrt {b} x^{n/2} \sqrt {\log (f)}\right )}{4 b^{5/2} n \log ^{\frac {5}{2}}(f)}-\frac {3 f^{a+b x^n} x^{n/2}}{2 b^2 n \log ^2(f)}+\frac {f^{a+b x^n} x^{3 n/2}}{b n \log (f)}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 39, normalized size = 0.38 \[ -\frac {f^a x^{5 n/2} \Gamma \left (\frac {5}{2},-b x^n \log (f)\right )}{n \left (-b \log (f) x^n\right )^{5/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 82, normalized size = 0.79 \[ -\frac {3 \, \sqrt {\pi } \sqrt {-b \log \relax (f)} f^{a} \operatorname {erf}\left (\sqrt {-b \log \relax (f)} x^{\frac {1}{2} \, n}\right ) - 2 \, {\left (2 \, b^{2} x^{\frac {3}{2} \, n} \log \relax (f)^{2} - 3 \, b x^{\frac {1}{2} \, n} \log \relax (f)\right )} e^{\left (b x^{n} \log \relax (f) + a \log \relax (f)\right )}}{4 \, b^{3} n \log \relax (f)^{3}} \]
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 \[ \int f^{b x^{n} + a} x^{\frac {5}{2} \, n - 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.11, size = 96, normalized size = 0.92 \[ \frac {f^{a} f^{b \,x^{n}} x^{\frac {3 n}{2}}}{b n \ln \relax (f )}-\frac {3 f^{a} f^{b \,x^{n}} x^{\frac {n}{2}}}{2 b^{2} n \ln \relax (f )^{2}}+\frac {3 \sqrt {\pi }\, f^{a} \erf \left (\sqrt {-b \ln \relax (f )}\, x^{\frac {n}{2}}\right )}{4 \sqrt {-b \ln \relax (f )}\, b^{2} n \ln \relax (f )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.21, size = 33, normalized size = 0.32 \[ -\frac {f^{a} x^{\frac {5}{2} \, n} \Gamma \left (\frac {5}{2}, -b x^{n} \log \relax (f)\right )}{\left (-b x^{n} \log \relax (f)\right )^{\frac {5}{2}} n} \]
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 \[ \int f^{a+b\,x^n}\,x^{\frac {5\,n}{2}-1} \,d x \]
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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