Optimal. Leaf size=28 \[ f^{a+\frac {b}{x}} x-b f^a \text {Ei}\left (\frac {b \log (f)}{x}\right ) \log (f) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.02, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2237, 2241}
\begin {gather*} x f^{a+\frac {b}{x}}-b f^a \log (f) \text {Ei}\left (\frac {b \log (f)}{x}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2237
Rule 2241
Rubi steps
\begin {align*} \int f^{a+\frac {b}{x}} \, dx &=f^{a+\frac {b}{x}} x+(b \log (f)) \int \frac {f^{a+\frac {b}{x}}}{x} \, dx\\ &=f^{a+\frac {b}{x}} x-b f^a \text {Ei}\left (\frac {b \log (f)}{x}\right ) \log (f)\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.01, size = 28, normalized size = 1.00 \begin {gather*} f^{a+\frac {b}{x}} x-b f^a \text {Ei}\left (\frac {b \log (f)}{x}\right ) \log (f) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.06, size = 31, normalized size = 1.11
method | result | size |
risch | \(b \ln \left (f \right ) f^{a} \expIntegral \left (1, -\frac {b \ln \left (f \right )}{x}\right )+f^{a} f^{\frac {b}{x}} x\) | \(31\) |
meijerg | \(f^{a} b \ln \left (f \right ) \left (-\frac {x \left (2+\frac {2 b \ln \left (f \right )}{x}\right )}{2 b \ln \left (f \right )}+\frac {x \,{\mathrm e}^{\frac {b \ln \left (f \right )}{x}}}{b \ln \left (f \right )}+\ln \left (-\frac {b \ln \left (f \right )}{x}\right )+\expIntegral \left (1, -\frac {b \ln \left (f \right )}{x}\right )+1+\ln \left (x \right )-\ln \left (-b \right )-\ln \left (\ln \left (f \right )\right )+\frac {x}{\ln \left (f \right ) b}\right )\) | \(88\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.32, size = 18, normalized size = 0.64 \begin {gather*} -b f^{a} \Gamma \left (-1, -\frac {b \log \left (f\right )}{x}\right ) \log \left (f\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.39, size = 30, normalized size = 1.07 \begin {gather*} -b f^{a} {\rm Ei}\left (\frac {b \log \left (f\right )}{x}\right ) \log \left (f\right ) + f^{\frac {a x + b}{x}} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int f^{a + \frac {b}{x}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 3.60, size = 27, normalized size = 0.96 \begin {gather*} f^a\,\left (f^{b/x}\,x+b\,\ln \left (f\right )\,\mathrm {expint}\left (-\frac {b\,\ln \left (f\right )}{x}\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________