Integrand size = 13, antiderivative size = 79 \[ \int f^{a+\frac {b}{x}} x^2 \, dx=\frac {1}{3} f^{a+\frac {b}{x}} x^3+\frac {1}{6} b f^{a+\frac {b}{x}} x^2 \log (f)+\frac {1}{6} b^2 f^{a+\frac {b}{x}} x \log ^2(f)-\frac {1}{6} b^3 f^a \operatorname {ExpIntegralEi}\left (\frac {b \log (f)}{x}\right ) \log ^3(f) \]
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Time = 0.04 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {2245, 2237, 2241} \[ \int f^{a+\frac {b}{x}} x^2 \, dx=-\frac {1}{6} b^3 f^a \log ^3(f) \operatorname {ExpIntegralEi}\left (\frac {b \log (f)}{x}\right )+\frac {1}{6} b^2 x \log ^2(f) f^{a+\frac {b}{x}}+\frac {1}{3} x^3 f^{a+\frac {b}{x}}+\frac {1}{6} b x^2 \log (f) f^{a+\frac {b}{x}} \]
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Rule 2237
Rule 2241
Rule 2245
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} f^{a+\frac {b}{x}} x^3+\frac {1}{3} (b \log (f)) \int f^{a+\frac {b}{x}} x \, dx \\ & = \frac {1}{3} f^{a+\frac {b}{x}} x^3+\frac {1}{6} b f^{a+\frac {b}{x}} x^2 \log (f)+\frac {1}{6} \left (b^2 \log ^2(f)\right ) \int f^{a+\frac {b}{x}} \, dx \\ & = \frac {1}{3} f^{a+\frac {b}{x}} x^3+\frac {1}{6} b f^{a+\frac {b}{x}} x^2 \log (f)+\frac {1}{6} b^2 f^{a+\frac {b}{x}} x \log ^2(f)+\frac {1}{6} \left (b^3 \log ^3(f)\right ) \int \frac {f^{a+\frac {b}{x}}}{x} \, dx \\ & = \frac {1}{3} f^{a+\frac {b}{x}} x^3+\frac {1}{6} b f^{a+\frac {b}{x}} x^2 \log (f)+\frac {1}{6} b^2 f^{a+\frac {b}{x}} x \log ^2(f)-\frac {1}{6} b^3 f^a \text {Ei}\left (\frac {b \log (f)}{x}\right ) \log ^3(f) \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.67 \[ \int f^{a+\frac {b}{x}} x^2 \, dx=\frac {1}{6} f^a \left (-b^3 \operatorname {ExpIntegralEi}\left (\frac {b \log (f)}{x}\right ) \log ^3(f)+f^{b/x} x \left (2 x^2+b x \log (f)+b^2 \log ^2(f)\right )\right ) \]
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Time = 0.08 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.97
method | result | size |
risch | \(\frac {f^{a} \operatorname {Ei}_{1}\left (-\frac {b \ln \left (f \right )}{x}\right ) b^{3} \ln \left (f \right )^{3}}{6}+\frac {f^{a} f^{\frac {b}{x}} x \,b^{2} \ln \left (f \right )^{2}}{6}+\frac {f^{a} f^{\frac {b}{x}} x^{2} b \ln \left (f \right )}{6}+\frac {f^{a} f^{\frac {b}{x}} x^{3}}{3}\) | \(77\) |
meijerg | \(f^{a} \ln \left (f \right )^{3} b^{3} \left (\frac {x^{3}}{3 b^{3} \ln \left (f \right )^{3}}+\frac {x^{2}}{2 b^{2} \ln \left (f \right )^{2}}+\frac {x}{2 b \ln \left (f \right )}+\frac {11}{36}+\frac {\ln \left (x \right )}{6}-\frac {\ln \left (-b \right )}{6}-\frac {\ln \left (\ln \left (f \right )\right )}{6}-\frac {x^{3} \left (\frac {22 b^{3} \ln \left (f \right )^{3}}{x^{3}}+\frac {36 b^{2} \ln \left (f \right )^{2}}{x^{2}}+\frac {36 b \ln \left (f \right )}{x}+24\right )}{72 b^{3} \ln \left (f \right )^{3}}+\frac {x^{3} \left (\frac {4 b^{2} \ln \left (f \right )^{2}}{x^{2}}+\frac {4 b \ln \left (f \right )}{x}+8\right ) {\mathrm e}^{\frac {b \ln \left (f \right )}{x}}}{24 b^{3} \ln \left (f \right )^{3}}+\frac {\ln \left (-\frac {b \ln \left (f \right )}{x}\right )}{6}+\frac {\operatorname {Ei}_{1}\left (-\frac {b \ln \left (f \right )}{x}\right )}{6}\right )\) | \(174\) |
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Time = 0.31 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.71 \[ \int f^{a+\frac {b}{x}} x^2 \, dx=-\frac {1}{6} \, b^{3} f^{a} {\rm Ei}\left (\frac {b \log \left (f\right )}{x}\right ) \log \left (f\right )^{3} + \frac {1}{6} \, {\left (b^{2} x \log \left (f\right )^{2} + b x^{2} \log \left (f\right ) + 2 \, x^{3}\right )} f^{\frac {a x + b}{x}} \]
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\[ \int f^{a+\frac {b}{x}} x^2 \, dx=\int f^{a + \frac {b}{x}} x^{2}\, dx \]
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Time = 0.23 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.28 \[ \int f^{a+\frac {b}{x}} x^2 \, dx=-b^{3} f^{a} \Gamma \left (-3, -\frac {b \log \left (f\right )}{x}\right ) \log \left (f\right )^{3} \]
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\[ \int f^{a+\frac {b}{x}} x^2 \, dx=\int { f^{a + \frac {b}{x}} x^{2} \,d x } \]
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Time = 0.22 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.84 \[ \int f^{a+\frac {b}{x}} x^2 \, dx=b^3\,f^a\,{\ln \left (f\right )}^3\,\left (f^{b/x}\,\left (\frac {x^2}{6\,b^2\,{\ln \left (f\right )}^2}+\frac {x^3}{3\,b^3\,{\ln \left (f\right )}^3}+\frac {x}{6\,b\,\ln \left (f\right )}\right )+\frac {\mathrm {expint}\left (-\frac {b\,\ln \left (f\right )}{x}\right )}{6}\right ) \]
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