Optimal. Leaf size=79 \[ -\frac {1}{6} b^3 f^a \log ^3(f) \text {Ei}\left (\frac {b \log (f)}{x}\right )+\frac {1}{6} b^2 x \log ^2(f) f^{a+\frac {b}{x}}+\frac {1}{3} x^3 f^{a+\frac {b}{x}}+\frac {1}{6} b x^2 \log (f) f^{a+\frac {b}{x}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.06, antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {2214, 2206, 2210} \[ -\frac {1}{6} b^3 f^a \log ^3(f) \text {Ei}\left (\frac {b \log (f)}{x}\right )+\frac {1}{6} b^2 x \log ^2(f) f^{a+\frac {b}{x}}+\frac {1}{3} x^3 f^{a+\frac {b}{x}}+\frac {1}{6} b x^2 \log (f) f^{a+\frac {b}{x}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2206
Rule 2210
Rule 2214
Rubi steps
\begin {align*} \int f^{a+\frac {b}{x}} x^2 \, dx &=\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*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 53, normalized size = 0.67 \[ \frac {1}{6} f^a \left (x f^{b/x} \left (b^2 \log ^2(f)+b x \log (f)+2 x^2\right )-b^3 \log ^3(f) \text {Ei}\left (\frac {b \log (f)}{x}\right )\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.41, size = 56, normalized size = 0.71 \[ -\frac {1}{6} \, b^{3} f^{a} {\rm Ei}\left (\frac {b \log \relax (f)}{x}\right ) \log \relax (f)^{3} + \frac {1}{6} \, {\left (b^{2} x \log \relax (f)^{2} + b x^{2} \log \relax (f) + 2 \, x^{3}\right )} f^{\frac {a x + b}{x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int f^{a + \frac {b}{x}} x^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.09, size = 77, normalized size = 0.97 \[ \frac {b^{3} f^{a} \Ei \left (1, -\frac {b \ln \relax (f )}{x}\right ) \ln \relax (f )^{3}}{6}+\frac {b^{2} x \,f^{a} f^{\frac {b}{x}} \ln \relax (f )^{2}}{6}+\frac {b \,x^{2} f^{a} f^{\frac {b}{x}} \ln \relax (f )}{6}+\frac {x^{3} f^{a} f^{\frac {b}{x}}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.97, size = 22, normalized size = 0.28 \[ -b^{3} f^{a} \Gamma \left (-3, -\frac {b \log \relax (f)}{x}\right ) \log \relax (f)^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 3.60, size = 66, normalized size = 0.84 \[ b^3\,f^a\,{\ln \relax (f)}^3\,\left (f^{b/x}\,\left (\frac {x^2}{6\,b^2\,{\ln \relax (f)}^2}+\frac {x^3}{3\,b^3\,{\ln \relax (f)}^3}+\frac {x}{6\,b\,\ln \relax (f)}\right )+\frac {\mathrm {expint}\left (-\frac {b\,\ln \relax (f)}{x}\right )}{6}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int f^{a + \frac {b}{x}} x^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________