Optimal. Leaf size=37 \[ \frac {\sqrt {\pi } f^a \text {erfi}\left (\sqrt {b} x \sqrt {\log (f)}\right )}{2 \sqrt {b} \sqrt {\log (f)}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2204} \[ \frac {\sqrt {\pi } f^a \text {Erfi}\left (\sqrt {b} x \sqrt {\log (f)}\right )}{2 \sqrt {b} \sqrt {\log (f)}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2204
Rubi steps
\begin {align*} \int f^{a+b x^2} \, dx &=\frac {f^a \sqrt {\pi } \text {erfi}\left (\sqrt {b} x \sqrt {\log (f)}\right )}{2 \sqrt {b} \sqrt {\log (f)}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.00, size = 37, normalized size = 1.00 \[ \frac {\sqrt {\pi } f^a \text {erfi}\left (\sqrt {b} x \sqrt {\log (f)}\right )}{2 \sqrt {b} \sqrt {\log (f)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.42, size = 32, normalized size = 0.86 \[ -\frac {\sqrt {\pi } \sqrt {-b \log \relax (f)} f^{a} \operatorname {erf}\left (\sqrt {-b \log \relax (f)} x\right )}{2 \, b \log \relax (f)} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.28, size = 26, normalized size = 0.70 \[ -\frac {\sqrt {\pi } f^{a} \operatorname {erf}\left (-\sqrt {-b \log \relax (f)} x\right )}{2 \, \sqrt {-b \log \relax (f)}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.04, size = 26, normalized size = 0.70 \[ \frac {\sqrt {\pi }\, f^{a} \erf \left (\sqrt {-b \ln \relax (f )}\, x \right )}{2 \sqrt {-b \ln \relax (f )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.92, size = 25, normalized size = 0.68 \[ \frac {\sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-b \log \relax (f)} x\right )}{2 \, \sqrt {-b \log \relax (f)}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 3.55, size = 26, normalized size = 0.70 \[ \frac {f^a\,\sqrt {\pi }\,\mathrm {erfi}\left (\frac {b\,x\,\ln \relax (f)}{\sqrt {b\,\ln \relax (f)}}\right )}{2\,\sqrt {b\,\ln \relax (f)}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int f^{a + b x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________