Optimal. Leaf size=73 \[ \frac {2}{3} \sqrt {\pi } b^{3/2} f^a \log ^{\frac {3}{2}}(f) \text {erfi}\left (\sqrt {b} x \sqrt {\log (f)}\right )-\frac {2 b \log (f) f^{a+b x^2}}{3 x}-\frac {f^{a+b x^2}}{3 x^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.05, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2214, 2204} \[ \frac {2}{3} \sqrt {\pi } b^{3/2} f^a \log ^{\frac {3}{2}}(f) \text {Erfi}\left (\sqrt {b} x \sqrt {\log (f)}\right )-\frac {f^{a+b x^2}}{3 x^3}-\frac {2 b \log (f) f^{a+b x^2}}{3 x} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2204
Rule 2214
Rubi steps
\begin {align*} \int \frac {f^{a+b x^2}}{x^4} \, dx &=-\frac {f^{a+b x^2}}{3 x^3}+\frac {1}{3} (2 b \log (f)) \int \frac {f^{a+b x^2}}{x^2} \, dx\\ &=-\frac {f^{a+b x^2}}{3 x^3}-\frac {2 b f^{a+b x^2} \log (f)}{3 x}+\frac {1}{3} \left (4 b^2 \log ^2(f)\right ) \int f^{a+b x^2} \, dx\\ &=-\frac {f^{a+b x^2}}{3 x^3}-\frac {2 b f^{a+b x^2} \log (f)}{3 x}+\frac {2}{3} b^{3/2} f^a \sqrt {\pi } \text {erfi}\left (\sqrt {b} x \sqrt {\log (f)}\right ) \log ^{\frac {3}{2}}(f)\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.04, size = 62, normalized size = 0.85 \[ \frac {1}{3} f^a \left (2 \sqrt {\pi } b^{3/2} \log ^{\frac {3}{2}}(f) \text {erfi}\left (\sqrt {b} x \sqrt {\log (f)}\right )-\frac {f^{b x^2} \left (2 b x^2 \log (f)+1\right )}{x^3}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.41, size = 57, normalized size = 0.78 \[ -\frac {2 \, \sqrt {\pi } \sqrt {-b \log \relax (f)} b f^{a} x^{3} \operatorname {erf}\left (\sqrt {-b \log \relax (f)} x\right ) \log \relax (f) + {\left (2 \, b x^{2} \log \relax (f) + 1\right )} f^{b x^{2} + a}}{3 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {f^{b x^{2} + a}}{x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.05, size = 67, normalized size = 0.92 \[ \frac {2 \sqrt {\pi }\, b^{2} f^{a} \erf \left (\sqrt {-b \ln \relax (f )}\, x \right ) \ln \relax (f )^{2}}{3 \sqrt {-b \ln \relax (f )}}-\frac {2 b \,f^{a} f^{b \,x^{2}} \ln \relax (f )}{3 x}-\frac {f^{a} f^{b \,x^{2}}}{3 x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.15, size = 28, normalized size = 0.38 \[ -\frac {\left (-b x^{2} \log \relax (f)\right )^{\frac {3}{2}} f^{a} \Gamma \left (-\frac {3}{2}, -b x^{2} \log \relax (f)\right )}{2 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 3.56, size = 70, normalized size = 0.96 \[ \frac {2\,b^2\,f^a\,\sqrt {\pi }\,\mathrm {erfi}\left (\frac {b\,x\,\ln \relax (f)}{\sqrt {b\,\ln \relax (f)}}\right )\,{\ln \relax (f)}^2}{3\,\sqrt {b\,\ln \relax (f)}}-\frac {\frac {f^a\,f^{b\,x^2}}{3}+\frac {2\,b\,f^a\,f^{b\,x^2}\,x^2\,\ln \relax (f)}{3}}{x^3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {f^{a + b x^{2}}}{x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________