Integrand size = 19, antiderivative size = 104 \[ \int f^{a+b x^n} x^{-1+\frac {5 n}{2}} \, dx=\frac {3 f^a \sqrt {\pi } \text {erfi}\left (\sqrt {b} x^{n/2} \sqrt {\log (f)}\right )}{4 b^{5/2} n \log ^{\frac {5}{2}}(f)}-\frac {3 f^{a+b x^n} x^{n/2}}{2 b^2 n \log ^2(f)}+\frac {f^{a+b x^n} x^{3 n/2}}{b n \log (f)} \]
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Time = 0.07 (sec) , antiderivative size = 104, 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 = {2244, 2242, 2235} \[ \int f^{a+b x^n} x^{-1+\frac {5 n}{2}} \, dx=\frac {3 \sqrt {\pi } f^a \text {erfi}\left (\sqrt {b} \sqrt {\log (f)} x^{n/2}\right )}{4 b^{5/2} n \log ^{\frac {5}{2}}(f)}-\frac {3 x^{n/2} f^{a+b x^n}}{2 b^2 n \log ^2(f)}+\frac {x^{3 n/2} f^{a+b x^n}}{b n \log (f)} \]
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Rule 2235
Rule 2242
Rule 2244
Rubi steps \begin{align*} \text {integral}& = \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 \text {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*}
Time = 0.01 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.38 \[ \int f^{a+b x^n} x^{-1+\frac {5 n}{2}} \, dx=-\frac {f^a x^{5 n/2} \Gamma \left (\frac {5}{2},-b x^n \log (f)\right )}{n \left (-b x^n \log (f)\right )^{5/2}} \]
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Time = 0.10 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.79
method | result | size |
meijerg | \(\frac {f^{a} \left (-\frac {x^{\frac {n}{2}} \left (-b \right )^{\frac {5}{2}} \sqrt {\ln \left (f \right )}\, \left (-10 b \,x^{n} \ln \left (f \right )+15\right ) {\mathrm e}^{b \,x^{n} \ln \left (f \right )}}{10 b^{2}}+\frac {3 \left (-b \right )^{\frac {5}{2}} \sqrt {\pi }\, \operatorname {erfi}\left (x^{\frac {n}{2}} \sqrt {b}\, \sqrt {\ln \left (f \right )}\right )}{4 b^{\frac {5}{2}}}\right )}{\left (-b \right )^{\frac {5}{2}} \ln \left (f \right )^{\frac {5}{2}} n}\) | \(82\) |
risch | \(\frac {f^{a} f^{b \,x^{n}} x^{\frac {3 n}{2}}}{n b \ln \left (f \right )}-\frac {3 f^{a} x^{\frac {n}{2}} f^{b \,x^{n}}}{2 n \ln \left (f \right )^{2} b^{2}}+\frac {3 f^{a} \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {-b \ln \left (f \right )}\, x^{\frac {n}{2}}\right )}{4 n \ln \left (f \right )^{2} b^{2} \sqrt {-b \ln \left (f \right )}}\) | \(96\) |
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Time = 0.27 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.79 \[ \int f^{a+b x^n} x^{-1+\frac {5 n}{2}} \, dx=-\frac {3 \, \sqrt {\pi } \sqrt {-b \log \left (f\right )} f^{a} \operatorname {erf}\left (\sqrt {-b \log \left (f\right )} x^{\frac {1}{2} \, n}\right ) - 2 \, {\left (2 \, b^{2} x^{\frac {3}{2} \, n} \log \left (f\right )^{2} - 3 \, b x^{\frac {1}{2} \, n} \log \left (f\right )\right )} e^{\left (b x^{n} \log \left (f\right ) + a \log \left (f\right )\right )}}{4 \, b^{3} n \log \left (f\right )^{3}} \]
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\[ \int f^{a+b x^n} x^{-1+\frac {5 n}{2}} \, dx=\int f^{a + b x^{n}} x^{\frac {5 n}{2} - 1}\, dx \]
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Time = 0.23 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.32 \[ \int f^{a+b x^n} x^{-1+\frac {5 n}{2}} \, dx=-\frac {f^{a} x^{\frac {5}{2} \, n} \Gamma \left (\frac {5}{2}, -b x^{n} \log \left (f\right )\right )}{\left (-b x^{n} \log \left (f\right )\right )^{\frac {5}{2}} n} \]
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\[ \int f^{a+b x^n} x^{-1+\frac {5 n}{2}} \, dx=\int { f^{b x^{n} + a} x^{\frac {5}{2} \, n - 1} \,d x } \]
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Timed out. \[ \int f^{a+b x^n} x^{-1+\frac {5 n}{2}} \, dx=\int f^{a+b\,x^n}\,x^{\frac {5\,n}{2}-1} \,d x \]
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