Optimal. Leaf size=77 \[ \frac {F^a (d x)^{m+1} \left (c+d x^n\right )^{b \log (F)} \left (\frac {d x^n}{c}+1\right )^{-b \log (F)} \, _2F_1\left (\frac {m+1}{n},-b \log (F);\frac {m+n+1}{n};-\frac {d x^n}{c}\right )}{d (m+1)} \]
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Rubi [A] time = 0.05, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2274, 12, 365, 364} \[ \frac {F^a (d x)^{m+1} \left (c+d x^n\right )^{b \log (F)} \left (\frac {d x^n}{c}+1\right )^{-b \log (F)} \, _2F_1\left (\frac {m+1}{n},-b \log (F);\frac {m+n+1}{n};-\frac {d x^n}{c}\right )}{d (m+1)} \]
Antiderivative was successfully verified.
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Rule 12
Rule 364
Rule 365
Rule 2274
Rubi steps
\begin {align*} \int F^{a+b \log \left (c+d x^n\right )} (d x)^m \, dx &=\int F^a (d x)^m \left (c+d x^n\right )^{b \log (F)} \, dx\\ &=F^a \int (d x)^m \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 (d x)^m \left (1+\frac {d x^n}{c}\right )^{b \log (F)} \, dx\\ &=\frac {F^a (d x)^{1+m} \left (c+d x^n\right )^{b \log (F)} \left (1+\frac {d x^n}{c}\right )^{-b \log (F)} \, _2F_1\left (\frac {1+m}{n},-b \log (F);\frac {1+m+n}{n};-\frac {d x^n}{c}\right )}{d (1+m)}\\ \end {align*}
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Mathematica [A] time = 0.17, size = 94, normalized size = 1.22 \[ -\frac {x (d x)^m \left (c+d x^n\right ) \left (-\frac {d x^n}{c}\right )^{-\frac {m+1}{n}} F^{a+b \log \left (c+d x^n\right )} \, _2F_1\left (1-\frac {m+1}{n},b \log (F)+1;b \log (F)+2;\frac {d x^n}{c}+1\right )}{c n (b \log (F)+1)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (d x\right )^{m} F^{b \log \left (d x^{n} + c\right ) + a}, x\right ) \]
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 \[ \int \left (d x\right )^{m} F^{b \log \left (d x^{n} + c\right ) + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.08, size = 0, normalized size = 0.00 \[ \int F^{b \ln \left (d \,x^{n}+c \right )+a} \left (d x \right )^{m}\, dx \]
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 \[ \int \left (d x\right )^{m} F^{b \log \left (d x^{n} + c\right ) + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int F^{a+b\,\ln \left (c+d\,x^n\right )}\,{\left (d\,x\right )}^m \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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