Integrand size = 19, antiderivative size = 96 \[ \int f^{a+b x^n} x^{-1-\frac {3 n}{2}} \, dx=-\frac {2 f^{a+b x^n} x^{-3 n/2}}{3 n}-\frac {4 b f^{a+b x^n} x^{-n/2} \log (f)}{3 n}+\frac {4 b^{3/2} f^a \sqrt {\pi } \text {erfi}\left (\sqrt {b} x^{n/2} \sqrt {\log (f)}\right ) \log ^{\frac {3}{2}}(f)}{3 n} \]
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Time = 0.06 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {2246, 2242, 2235} \[ \int f^{a+b x^n} x^{-1-\frac {3 n}{2}} \, dx=\frac {4 \sqrt {\pi } b^{3/2} f^a \log ^{\frac {3}{2}}(f) \text {erfi}\left (\sqrt {b} \sqrt {\log (f)} x^{n/2}\right )}{3 n}-\frac {2 x^{-3 n/2} f^{a+b x^n}}{3 n}-\frac {4 b \log (f) x^{-n/2} f^{a+b x^n}}{3 n} \]
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Rule 2235
Rule 2242
Rule 2246
Rubi steps \begin{align*} \text {integral}& = -\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*}
Time = 0.01 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.41 \[ \int f^{a+b x^n} x^{-1-\frac {3 n}{2}} \, dx=-\frac {f^a x^{-3 n/2} \Gamma \left (-\frac {3}{2},-b x^n \log (f)\right ) \left (-b x^n \log (f)\right )^{3/2}}{n} \]
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Time = 0.12 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.82
method | result | size |
meijerg | \(\frac {f^{a} \left (-b \right )^{\frac {3}{2}} \ln \left (f \right )^{\frac {3}{2}} \left (-\frac {2 x^{-\frac {3 n}{2}} \left (2 b \,x^{n} \ln \left (f \right )+1\right ) {\mathrm e}^{b \,x^{n} \ln \left (f \right )}}{3 \left (-b \right )^{\frac {3}{2}} \ln \left (f \right )^{\frac {3}{2}}}+\frac {4 b^{\frac {3}{2}} \sqrt {\pi }\, \operatorname {erfi}\left (x^{\frac {n}{2}} \sqrt {b}\, \sqrt {\ln \left (f \right )}\right )}{3 \left (-b \right )^{\frac {3}{2}}}\right )}{n}\) | \(79\) |
risch | \(-\frac {2 f^{a} x^{-\frac {3 n}{2}} f^{b \,x^{n}}}{3 n}-\frac {4 f^{a} \ln \left (f \right ) b \,x^{-\frac {n}{2}} f^{b \,x^{n}}}{3 n}+\frac {4 f^{a} \ln \left (f \right )^{2} b^{2} \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {-b \ln \left (f \right )}\, x^{\frac {n}{2}}\right )}{3 n \sqrt {-b \ln \left (f \right )}}\) | \(88\) |
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\[ \int f^{a+b x^n} x^{-1-\frac {3 n}{2}} \, dx=\int { f^{b x^{n} + a} x^{-\frac {3}{2} \, n - 1} \,d x } \]
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\[ \int f^{a+b x^n} x^{-1-\frac {3 n}{2}} \, dx=\int f^{a + b x^{n}} x^{- \frac {3 n}{2} - 1}\, dx \]
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Time = 0.23 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.36 \[ \int f^{a+b x^n} x^{-1-\frac {3 n}{2}} \, dx=-\frac {\left (-b x^{n} \log \left (f\right )\right )^{\frac {3}{2}} f^{a} \Gamma \left (-\frac {3}{2}, -b x^{n} \log \left (f\right )\right )}{n x^{\frac {3}{2} \, n}} \]
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\[ \int f^{a+b x^n} x^{-1-\frac {3 n}{2}} \, dx=\int { f^{b x^{n} + a} x^{-\frac {3}{2} \, n - 1} \,d x } \]
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Timed out. \[ \int f^{a+b x^n} x^{-1-\frac {3 n}{2}} \, dx=\int \frac {f^{a+b\,x^n}}{x^{\frac {3\,n}{2}+1}} \,d x \]
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