Optimal. Leaf size=86 \[ -\frac {3 f^{a+b x^2}}{b^4 \log ^4(f)}+\frac {3 x^2 f^{a+b x^2}}{b^3 \log ^3(f)}-\frac {3 x^4 f^{a+b x^2}}{2 b^2 \log ^2(f)}+\frac {x^6 f^{a+b x^2}}{2 b \log (f)} \]
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Rubi [A] time = 0.09, antiderivative size = 86, 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 = {2212, 2209} \[ -\frac {3 x^4 f^{a+b x^2}}{2 b^2 \log ^2(f)}+\frac {3 x^2 f^{a+b x^2}}{b^3 \log ^3(f)}-\frac {3 f^{a+b x^2}}{b^4 \log ^4(f)}+\frac {x^6 f^{a+b x^2}}{2 b \log (f)} \]
Antiderivative was successfully verified.
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Rule 2209
Rule 2212
Rubi steps
\begin {align*} \int f^{a+b x^2} x^7 \, dx &=\frac {f^{a+b x^2} x^6}{2 b \log (f)}-\frac {3 \int f^{a+b x^2} x^5 \, dx}{b \log (f)}\\ &=-\frac {3 f^{a+b x^2} x^4}{2 b^2 \log ^2(f)}+\frac {f^{a+b x^2} x^6}{2 b \log (f)}+\frac {6 \int f^{a+b x^2} x^3 \, dx}{b^2 \log ^2(f)}\\ &=\frac {3 f^{a+b x^2} x^2}{b^3 \log ^3(f)}-\frac {3 f^{a+b x^2} x^4}{2 b^2 \log ^2(f)}+\frac {f^{a+b x^2} x^6}{2 b \log (f)}-\frac {6 \int f^{a+b x^2} x \, dx}{b^3 \log ^3(f)}\\ &=-\frac {3 f^{a+b x^2}}{b^4 \log ^4(f)}+\frac {3 f^{a+b x^2} x^2}{b^3 \log ^3(f)}-\frac {3 f^{a+b x^2} x^4}{2 b^2 \log ^2(f)}+\frac {f^{a+b x^2} x^6}{2 b \log (f)}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 53, normalized size = 0.62 \[ \frac {f^{a+b x^2} \left (b^3 x^6 \log ^3(f)-3 b^2 x^4 \log ^2(f)+6 b x^2 \log (f)-6\right )}{2 b^4 \log ^4(f)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 51, normalized size = 0.59 \[ \frac {{\left (b^{3} x^{6} \log \relax (f)^{3} - 3 \, b^{2} x^{4} \log \relax (f)^{2} + 6 \, b x^{2} \log \relax (f) - 6\right )} f^{b x^{2} + a}}{2 \, b^{4} \log \relax (f)^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.39, size = 55, normalized size = 0.64 \[ \frac {{\left (b^{3} x^{6} \log \relax (f)^{3} - 3 \, b^{2} x^{4} \log \relax (f)^{2} + 6 \, b x^{2} \log \relax (f) - 6\right )} e^{\left (b x^{2} \log \relax (f) + a \log \relax (f)\right )}}{2 \, b^{4} \log \relax (f)^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 52, normalized size = 0.60 \[ \frac {\left (b^{3} x^{6} \ln \relax (f )^{3}-3 b^{2} x^{4} \ln \relax (f )^{2}+6 b \,x^{2} \ln \relax (f )-6\right ) f^{b \,x^{2}+a}}{2 b^{4} \ln \relax (f )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.45, size = 62, normalized size = 0.72 \[ \frac {{\left (b^{3} f^{a} x^{6} \log \relax (f)^{3} - 3 \, b^{2} f^{a} x^{4} \log \relax (f)^{2} + 6 \, b f^{a} x^{2} \log \relax (f) - 6 \, f^{a}\right )} f^{b x^{2}}}{2 \, b^{4} \log \relax (f)^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.50, size = 52, normalized size = 0.60 \[ -\frac {f^{b\,x^2+a}\,\left (-\frac {b^3\,x^6\,{\ln \relax (f)}^3}{2}+\frac {3\,b^2\,x^4\,{\ln \relax (f)}^2}{2}-3\,b\,x^2\,\ln \relax (f)+3\right )}{b^4\,{\ln \relax (f)}^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.14, size = 68, normalized size = 0.79 \[ \begin {cases} \frac {f^{a + b x^{2}} \left (b^{3} x^{6} \log {\relax (f )}^{3} - 3 b^{2} x^{4} \log {\relax (f )}^{2} + 6 b x^{2} \log {\relax (f )} - 6\right )}{2 b^{4} \log {\relax (f )}^{4}} & \text {for}\: 2 b^{4} \log {\relax (f )}^{4} \neq 0 \\\frac {x^{8}}{8} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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