Integrand size = 19, antiderivative size = 43 \[ \int f^{a+b x^n} x^{-1+\frac {n}{2}} \, dx=\frac {f^a \sqrt {\pi } \text {erfi}\left (\sqrt {b} x^{n/2} \sqrt {\log (f)}\right )}{\sqrt {b} n \sqrt {\log (f)}} \]
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Time = 0.02 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2242, 2235} \[ \int f^{a+b x^n} x^{-1+\frac {n}{2}} \, dx=\frac {\sqrt {\pi } f^a \text {erfi}\left (\sqrt {b} \sqrt {\log (f)} x^{n/2}\right )}{\sqrt {b} n \sqrt {\log (f)}} \]
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Rule 2235
Rule 2242
Rubi steps \begin{align*} \text {integral}& = \frac {2 \text {Subst}\left (\int f^{a+b x^2} \, dx,x,x^{n/2}\right )}{n} \\ & = \frac {f^a \sqrt {\pi } \text {erfi}\left (\sqrt {b} x^{n/2} \sqrt {\log (f)}\right )}{\sqrt {b} n \sqrt {\log (f)}} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.00 \[ \int f^{a+b x^n} x^{-1+\frac {n}{2}} \, dx=\frac {f^a \sqrt {\pi } \text {erfi}\left (\sqrt {b} x^{n/2} \sqrt {\log (f)}\right )}{\sqrt {b} n \sqrt {\log (f)}} \]
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Time = 0.10 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.74
method | result | size |
meijerg | \(\frac {f^{a} \operatorname {erfi}\left (x^{\frac {n}{2}} \sqrt {b}\, \sqrt {\ln \left (f \right )}\right ) \sqrt {\pi }}{n \sqrt {b}\, \sqrt {\ln \left (f \right )}}\) | \(32\) |
risch | \(\frac {f^{a} \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {-b \ln \left (f \right )}\, x^{\frac {n}{2}}\right )}{n \sqrt {-b \ln \left (f \right )}}\) | \(32\) |
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Time = 0.28 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.98 \[ \int f^{a+b x^n} x^{-1+\frac {n}{2}} \, dx=-\frac {\sqrt {\pi } \sqrt {-b \log \left (f\right )} f^{a} \operatorname {erf}\left (\sqrt {-b \log \left (f\right )} x x^{\frac {1}{2} \, n - 1}\right )}{b n \log \left (f\right )} \]
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\[ \int f^{a+b x^n} x^{-1+\frac {n}{2}} \, dx=\int f^{a + b x^{n}} x^{\frac {n}{2} - 1}\, dx \]
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Time = 0.23 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.88 \[ \int f^{a+b x^n} x^{-1+\frac {n}{2}} \, dx=\frac {\sqrt {\pi } f^{a} x^{\frac {1}{2} \, n} {\left (\operatorname {erf}\left (\sqrt {-b x^{n} \log \left (f\right )}\right ) - 1\right )}}{\sqrt {-b x^{n} \log \left (f\right )} n} \]
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Time = 0.32 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.77 \[ \int f^{a+b x^n} x^{-1+\frac {n}{2}} \, dx=-\frac {\sqrt {\pi } f^{a} \operatorname {erf}\left (-\sqrt {-b \log \left (f\right )} \sqrt {x^{n}}\right )}{\sqrt {-b \log \left (f\right )} n} \]
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Timed out. \[ \int f^{a+b x^n} x^{-1+\frac {n}{2}} \, dx=\int f^{a+b\,x^n}\,x^{\frac {n}{2}-1} \,d x \]
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