Optimal. Leaf size=57 \[ -\frac {F^a \left (c+d x^n\right )^{1+b \log (F)} \, _2F_1\left (1,1+b \log (F);2+b \log (F);1+\frac {d x^n}{c}\right )}{c n (1+b \log (F))} \]
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Rubi [A]
time = 0.04, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2306, 12, 272,
67} \begin {gather*} -\frac {F^a \left (c+d x^n\right )^{b \log (F)+1} \, _2F_1\left (1,b \log (F)+1;b \log (F)+2;\frac {d x^n}{c}+1\right )}{c n (b \log (F)+1)} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 67
Rule 272
Rule 2306
Rubi steps
\begin {align*} \int \frac {F^{a+b \log \left (c+d x^n\right )}}{x} \, dx &=\int \frac {F^a \left (c+d x^n\right )^{b \log (F)}}{x} \, dx\\ &=F^a \int \frac {\left (c+d x^n\right )^{b \log (F)}}{x} \, dx\\ &=\frac {F^a \text {Subst}\left (\int \frac {(c+d x)^{b \log (F)}}{x} \, dx,x,x^n\right )}{n}\\ &=-\frac {F^a \left (c+d x^n\right )^{1+b \log (F)} \, _2F_1\left (1,1+b \log (F);2+b \log (F);1+\frac {d x^n}{c}\right )}{c n (1+b \log (F))}\\ \end {align*}
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Mathematica [A]
time = 0.09, size = 50, normalized size = 0.88 \begin {gather*} -\frac {F^{a+b \log \left (c+d x^n\right )} \left (-1+\, _2F_1\left (1,b \log (F);1+b \log (F);1+\frac {d x^n}{c}\right )\right )}{b n \log (F)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {F^{a +b \ln \left (c +d \,x^{n}\right )}}{x}\, dx\]
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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {F^{a + b \log {\left (c + d x^{n} \right )}}}{x}\, dx \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 \frac {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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