Integrand size = 19, antiderivative size = 74 \[ \int f^{a+b x^n} x^{-1+\frac {3 n}{2}} \, dx=-\frac {f^a \sqrt {\pi } \text {erfi}\left (\sqrt {b} x^{n/2} \sqrt {\log (f)}\right )}{2 b^{3/2} n \log ^{\frac {3}{2}}(f)}+\frac {f^{a+b x^n} x^{n/2}}{b n \log (f)} \]
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Time = 0.04 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {2244, 2242, 2235} \[ \int f^{a+b x^n} x^{-1+\frac {3 n}{2}} \, dx=\frac {x^{n/2} f^{a+b x^n}}{b n \log (f)}-\frac {\sqrt {\pi } f^a \text {erfi}\left (\sqrt {b} \sqrt {\log (f)} x^{n/2}\right )}{2 b^{3/2} n \log ^{\frac {3}{2}}(f)} \]
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Rule 2235
Rule 2242
Rule 2244
Rubi steps \begin{align*} \text {integral}& = \frac {f^{a+b x^n} x^{n/2}}{b n \log (f)}-\frac {\int f^{a+b x^n} x^{\frac {1}{2} (-2+n)} \, dx}{2 b \log (f)} \\ & = \frac {f^{a+b x^n} x^{n/2}}{b n \log (f)}-\frac {\text {Subst}\left (\int f^{a+b x^2} \, dx,x,x^{1+\frac {1}{2} (-2+n)}\right )}{b n \log (f)} \\ & = -\frac {f^a \sqrt {\pi } \text {erfi}\left (\sqrt {b} x^{n/2} \sqrt {\log (f)}\right )}{2 b^{3/2} n \log ^{\frac {3}{2}}(f)}+\frac {f^{a+b x^n} x^{n/2}}{b n \log (f)} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.53 \[ \int f^{a+b x^n} x^{-1+\frac {3 n}{2}} \, dx=-\frac {f^a x^{3 n/2} \Gamma \left (\frac {3}{2},-b x^n \log (f)\right )}{n \left (-b x^n \log (f)\right )^{3/2}} \]
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Time = 0.11 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.91
method | result | size |
risch | \(\frac {f^{a} x^{\frac {n}{2}} f^{b \,x^{n}}}{n \ln \left (f \right ) b}-\frac {f^{a} \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {-b \ln \left (f \right )}\, x^{\frac {n}{2}}\right )}{2 n \ln \left (f \right ) b \sqrt {-b \ln \left (f \right )}}\) | \(67\) |
meijerg | \(\frac {f^{a} \left (\frac {x^{\frac {n}{2}} \left (-b \right )^{\frac {3}{2}} \sqrt {\ln \left (f \right )}\, {\mathrm e}^{b \,x^{n} \ln \left (f \right )}}{b}-\frac {\left (-b \right )^{\frac {3}{2}} \sqrt {\pi }\, \operatorname {erfi}\left (x^{\frac {n}{2}} \sqrt {b}\, \sqrt {\ln \left (f \right )}\right )}{2 b^{\frac {3}{2}}}\right )}{\left (-b \right )^{\frac {3}{2}} \ln \left (f \right )^{\frac {3}{2}} n}\) | \(71\) |
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Time = 0.30 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.86 \[ \int f^{a+b x^n} x^{-1+\frac {3 n}{2}} \, dx=\frac {2 \, b x^{\frac {1}{2} \, n} e^{\left (b x^{n} \log \left (f\right ) + a \log \left (f\right )\right )} \log \left (f\right ) + \sqrt {\pi } \sqrt {-b \log \left (f\right )} f^{a} \operatorname {erf}\left (\sqrt {-b \log \left (f\right )} x^{\frac {1}{2} \, n}\right )}{2 \, b^{2} n \log \left (f\right )^{2}} \]
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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 = 33, normalized size of antiderivative = 0.45 \[ \int f^{a+b x^n} x^{-1+\frac {3 n}{2}} \, dx=-\frac {f^{a} x^{\frac {3}{2} \, n} \Gamma \left (\frac {3}{2}, -b x^{n} \log \left (f\right )\right )}{\left (-b x^{n} \log \left (f\right )\right )^{\frac {3}{2}} n} \]
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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. \[ \int f^{a+b x^n} x^{-1+\frac {3 n}{2}} \, dx=\int f^{a+b\,x^n}\,x^{\frac {3\,n}{2}-1} \,d x \]
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