Optimal. Leaf size=46 \[ \frac {(c+d x) F^{a+\frac {b}{c+d x}}}{d}-\frac {b F^a \log (F) \text {Ei}\left (\frac {b \log (F)}{c+d x}\right )}{d} \]
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Rubi [A] time = 0.05, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2206, 2210} \[ \frac {(c+d x) F^{a+\frac {b}{c+d x}}}{d}-\frac {b F^a \log (F) \text {Ei}\left (\frac {b \log (F)}{c+d x}\right )}{d} \]
Antiderivative was successfully verified.
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Rule 2206
Rule 2210
Rubi steps
\begin {align*} \int F^{a+\frac {b}{c+d x}} \, dx &=\frac {F^{a+\frac {b}{c+d x}} (c+d x)}{d}+(b \log (F)) \int \frac {F^{a+\frac {b}{c+d x}}}{c+d x} \, dx\\ &=\frac {F^{a+\frac {b}{c+d x}} (c+d x)}{d}-\frac {b F^a \text {Ei}\left (\frac {b \log (F)}{c+d x}\right ) \log (F)}{d}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 42, normalized size = 0.91 \[ \frac {F^a \left ((c+d x) F^{\frac {b}{c+d x}}-b \log (F) \text {Ei}\left (\frac {b \log (F)}{c+d x}\right )\right )}{d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 51, normalized size = 1.11 \[ -\frac {F^{a} b {\rm Ei}\left (\frac {b \log \relax (F)}{d x + c}\right ) \log \relax (F) - {\left (d x + c\right )} F^{\frac {a d x + a c + b}{d x + c}}}{d} \]
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 \[ \int F^{a + \frac {b}{d x + c}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.12, size = 61, normalized size = 1.33 \[ \frac {b \,F^{a} \Ei \left (1, -\frac {b \ln \relax (F )}{d x +c}\right ) \ln \relax (F )}{d}+x \,F^{a} F^{\frac {b}{d x +c}}+\frac {c \,F^{a} F^{\frac {b}{d x +c}}}{d} \]
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 \[ F^{a} b d \int \frac {F^{\frac {b}{d x + c}} x}{d^{2} x^{2} + 2 \, c d x + c^{2}}\,{d x} \log \relax (F) + F^{a} F^{\frac {b}{d x + c}} x \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.40, size = 47, normalized size = 1.02 \[ \frac {F^a\,F^{\frac {b}{c+d\,x}}\,\left (c+d\,x\right )}{d}+\frac {F^a\,b\,\ln \relax (F)\,\mathrm {expint}\left (-\frac {b\,\ln \relax (F)}{c+d\,x}\right )}{d} \]
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 \[ \int F^{a + \frac {b}{c + d x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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