Optimal. Leaf size=96 \[ -\frac {2 f^{a+b x^n} x^{-3 n/2}}{3 n}-\frac {4 b f^{a+b x^n} x^{-n/2} \log (f)}{3 n}+\frac {4 b^{3/2} f^a \sqrt {\pi } \text {erfi}\left (\sqrt {b} x^{n/2} \sqrt {\log (f)}\right ) \log ^{\frac {3}{2}}(f)}{3 n} \]
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Rubi [A]
time = 0.07, 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 = {2246, 2242,
2235} \begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 2235
Rule 2242
Rule 2246
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 ) \text {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 \begin {gather*} -\frac {f^a x^{-3 n/2} \Gamma \left (-\frac {3}{2},-b x^n \log (f)\right ) \left (-b x^n \log (f)\right )^{3/2}}{n} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 79, normalized size = 0.82
method | result | size |
meijerg | \(\frac {f^{a} \left (-b \right )^{\frac {3}{2}} \ln \left (f \right )^{\frac {3}{2}} \left (-\frac {2 x^{-\frac {3 n}{2}} \left (2 b \,x^{n} \ln \left (f \right )+1\right ) {\mathrm e}^{b \,x^{n} \ln \left (f \right )}}{3 \left (-b \right )^{\frac {3}{2}} \ln \left (f \right )^{\frac {3}{2}}}+\frac {4 b^{\frac {3}{2}} \sqrt {\pi }\, \erfi \left (x^{\frac {n}{2}} \sqrt {b}\, \sqrt {\ln \left (f \right )}\right )}{3 \left (-b \right )^{\frac {3}{2}}}\right )}{n}\) | \(79\) |
risch | \(-\frac {2 f^{a} x^{-\frac {3 n}{2}} f^{b \,x^{n}}}{3 n}-\frac {4 f^{a} \ln \left (f \right ) b \,x^{-\frac {n}{2}} f^{b \,x^{n}}}{3 n}+\frac {4 f^{a} \ln \left (f \right )^{2} b^{2} \sqrt {\pi }\, \erf \left (\sqrt {-b \ln \left (f \right )}\, x^{\frac {n}{2}}\right )}{3 n \sqrt {-b \ln \left (f \right )}}\) | \(88\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.33, size = 35, normalized size = 0.36 \begin {gather*} -\frac {\left (-b x^{n} \log \left (f\right )\right )^{\frac {3}{2}} f^{a} \Gamma \left (-\frac {3}{2}, -b x^{n} \log \left (f\right )\right )}{n x^{\frac {3}{2} \, n}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 66.73, size = 56, normalized size = 0.58 \begin {gather*} \begin {cases} - \frac {4 b f^{a} f^{b x^{n}} x^{- \frac {n}{2}} \log {\left (f \right )}}{3 n} - \frac {2 f^{a} f^{b x^{n}} x^{- \frac {3 n}{2}}}{3 n} & \text {for}\: n \neq 0 \\f^{a + b} \log {\left (x \right )} & \text {otherwise} \end {cases} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {f^{a+b\,x^n}}{x^{\frac {3\,n}{2}+1}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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