Optimal. Leaf size=105 \[ -\frac {15 f^a \sqrt {\pi } \text {erfi}\left (\sqrt {b} x \sqrt {\log (f)}\right )}{16 b^{7/2} \log ^{\frac {7}{2}}(f)}+\frac {15 f^{a+b x^2} x}{8 b^3 \log ^3(f)}-\frac {5 f^{a+b x^2} x^3}{4 b^2 \log ^2(f)}+\frac {f^{a+b x^2} x^5}{2 b \log (f)} \]
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Rubi [A]
time = 0.08, antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2243, 2235}
\begin {gather*} -\frac {15 \sqrt {\pi } f^a \text {Erfi}\left (\sqrt {b} x \sqrt {\log (f)}\right )}{16 b^{7/2} \log ^{\frac {7}{2}}(f)}+\frac {15 x f^{a+b x^2}}{8 b^3 \log ^3(f)}-\frac {5 x^3 f^{a+b x^2}}{4 b^2 \log ^2(f)}+\frac {x^5 f^{a+b x^2}}{2 b \log (f)} \end {gather*}
Antiderivative was successfully verified.
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Rule 2235
Rule 2243
Rubi steps
\begin {align*} \int f^{a+b x^2} x^6 \, dx &=\frac {f^{a+b x^2} x^5}{2 b \log (f)}-\frac {5 \int f^{a+b x^2} x^4 \, dx}{2 b \log (f)}\\ &=-\frac {5 f^{a+b x^2} x^3}{4 b^2 \log ^2(f)}+\frac {f^{a+b x^2} x^5}{2 b \log (f)}+\frac {15 \int f^{a+b x^2} x^2 \, dx}{4 b^2 \log ^2(f)}\\ &=\frac {15 f^{a+b x^2} x}{8 b^3 \log ^3(f)}-\frac {5 f^{a+b x^2} x^3}{4 b^2 \log ^2(f)}+\frac {f^{a+b x^2} x^5}{2 b \log (f)}-\frac {15 \int f^{a+b x^2} \, dx}{8 b^3 \log ^3(f)}\\ &=-\frac {15 f^a \sqrt {\pi } \text {erfi}\left (\sqrt {b} x \sqrt {\log (f)}\right )}{16 b^{7/2} \log ^{\frac {7}{2}}(f)}+\frac {15 f^{a+b x^2} x}{8 b^3 \log ^3(f)}-\frac {5 f^{a+b x^2} x^3}{4 b^2 \log ^2(f)}+\frac {f^{a+b x^2} x^5}{2 b \log (f)}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 83, normalized size = 0.79 \begin {gather*} \frac {f^a \left (-15 \sqrt {\pi } \text {erfi}\left (\sqrt {b} x \sqrt {\log (f)}\right )+2 \sqrt {b} f^{b x^2} x \sqrt {\log (f)} \left (15-10 b x^2 \log (f)+4 b^2 x^4 \log ^2(f)\right )\right )}{16 b^{7/2} \log ^{\frac {7}{2}}(f)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 87, normalized size = 0.83
method | result | size |
meijerg | \(-\frac {f^{a} \left (\frac {x \left (-b \right )^{\frac {7}{2}} \sqrt {\ln \left (f \right )}\, \left (28 b^{2} x^{4} \ln \left (f \right )^{2}-70 b \,x^{2} \ln \left (f \right )+105\right ) {\mathrm e}^{b \,x^{2} \ln \left (f \right )}}{28 b^{3}}-\frac {15 \left (-b \right )^{\frac {7}{2}} \sqrt {\pi }\, \erfi \left (x \sqrt {b}\, \sqrt {\ln \left (f \right )}\right )}{8 b^{\frac {7}{2}}}\right )}{2 \ln \left (f \right )^{\frac {7}{2}} b^{3} \sqrt {-b}}\) | \(87\) |
risch | \(\frac {f^{a} x^{5} f^{b \,x^{2}}}{2 \ln \left (f \right ) b}-\frac {5 f^{a} x^{3} f^{b \,x^{2}}}{4 \ln \left (f \right )^{2} b^{2}}+\frac {15 f^{a} x \,f^{b \,x^{2}}}{8 \ln \left (f \right )^{3} b^{3}}-\frac {15 f^{a} \sqrt {\pi }\, \erf \left (\sqrt {-b \ln \left (f \right )}\, x \right )}{16 \ln \left (f \right )^{3} b^{3} \sqrt {-b \ln \left (f \right )}}\) | \(98\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 82, normalized size = 0.78 \begin {gather*} \frac {{\left (4 \, b^{2} f^{a} x^{5} \log \left (f\right )^{2} - 10 \, b f^{a} x^{3} \log \left (f\right ) + 15 \, f^{a} x\right )} f^{b x^{2}}}{8 \, b^{3} \log \left (f\right )^{3}} - \frac {15 \, \sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-b \log \left (f\right )} x\right )}{16 \, \sqrt {-b \log \left (f\right )} b^{3} \log \left (f\right )^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 77, normalized size = 0.73 \begin {gather*} \frac {15 \, \sqrt {\pi } \sqrt {-b \log \left (f\right )} f^{a} \operatorname {erf}\left (\sqrt {-b \log \left (f\right )} x\right ) + 2 \, {\left (4 \, b^{3} x^{5} \log \left (f\right )^{3} - 10 \, b^{2} x^{3} \log \left (f\right )^{2} + 15 \, b x \log \left (f\right )\right )} f^{b x^{2} + a}}{16 \, b^{4} \log \left (f\right )^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int f^{a + b x^{2}} x^{6}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 2.56, size = 80, normalized size = 0.76 \begin {gather*} \frac {15 \, \sqrt {\pi } f^{a} \operatorname {erf}\left (-\sqrt {-b \log \left (f\right )} x\right )}{16 \, \sqrt {-b \log \left (f\right )} b^{3} \log \left (f\right )^{3}} + \frac {{\left (4 \, b^{2} x^{5} \log \left (f\right )^{2} - 10 \, b x^{3} \log \left (f\right ) + 15 \, x\right )} e^{\left (b x^{2} \log \left (f\right ) + a \log \left (f\right )\right )}}{8 \, b^{3} \log \left (f\right )^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.52, size = 98, normalized size = 0.93 \begin {gather*} \frac {15\,f^a\,f^{b\,x^2}\,x}{8\,b^3\,{\ln \left (f\right )}^3}+\frac {f^a\,f^{b\,x^2}\,x^5}{2\,b\,\ln \left (f\right )}-\frac {5\,f^a\,f^{b\,x^2}\,x^3}{4\,b^2\,{\ln \left (f\right )}^2}-\frac {15\,f^a\,\sqrt {\pi }\,\mathrm {erfi}\left (\frac {b\,x\,\ln \left (f\right )}{\sqrt {b\,\ln \left (f\right )}}\right )}{16\,b^3\,{\ln \left (f\right )}^3\,\sqrt {b\,\ln \left (f\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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