Optimal. Leaf size=65 \[ \frac {1}{2} F^a x^2 \left (c+d x^n\right )^{b \log (F)} \left (1+\frac {d x^n}{c}\right )^{-b \log (F)} \, _2F_1\left (\frac {2}{n},-b \log (F);\frac {2+n}{n};-\frac {d x^n}{c}\right ) \]
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Rubi [A]
time = 0.02, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2306, 12, 372,
371} \begin {gather*} \frac {1}{2} x^2 F^a \left (c+d x^n\right )^{b \log (F)} \left (\frac {d x^n}{c}+1\right )^{-b \log (F)} \, _2F_1\left (\frac {2}{n},-b \log (F);\frac {n+2}{n};-\frac {d x^n}{c}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 371
Rule 372
Rule 2306
Rubi steps
\begin {align*} \int F^{a+b \log \left (c+d x^n\right )} x \, dx &=\int F^a x \left (c+d x^n\right )^{b \log (F)} \, dx\\ &=F^a \int x \left (c+d x^n\right )^{b \log (F)} \, dx\\ &=\left (F^a \left (c+d x^n\right )^{b \log (F)} \left (1+\frac {d x^n}{c}\right )^{-b \log (F)}\right ) \int x \left (1+\frac {d x^n}{c}\right )^{b \log (F)} \, dx\\ &=\frac {1}{2} F^a x^2 \left (c+d x^n\right )^{b \log (F)} \left (1+\frac {d x^n}{c}\right )^{-b \log (F)} \, _2F_1\left (\frac {2}{n},-b \log (F);\frac {2+n}{n};-\frac {d x^n}{c}\right )\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 85, normalized size = 1.31 \begin {gather*} -\frac {F^{a+b \log \left (c+d x^n\right )} x^2 \left (-\frac {d x^n}{c}\right )^{-2/n} \left (c+d x^n\right ) \, _2F_1\left (\frac {-2+n}{n},1+b \log (F);2+b \log (F);1+\frac {d x^n}{c}\right )}{c n (1+b \log (F))} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int F^{a +b \ln \left (c +d \,x^{n}\right )} x\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [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.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int F^{a+b\,\ln \left (c+d\,x^n\right )}\,x \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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