Optimal. Leaf size=96 \[ -\frac {f^2 F^{a+b c+b d x}}{b^2 d^2 \log ^2(F)}+e^2 F^{a+b c} \text {Ei}(b d x \log (F))+\frac {2 e f F^{a+b c+b d x}}{b d \log (F)}+\frac {f^2 x F^{a+b c+b d x}}{b d \log (F)} \]
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Rubi [A] time = 0.26, antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {2199, 2194, 2178, 2176} \[ -\frac {f^2 F^{a+b c+b d x}}{b^2 d^2 \log ^2(F)}+e^2 F^{a+b c} \text {Ei}(b d x \log (F))+\frac {2 e f F^{a+b c+b d x}}{b d \log (F)}+\frac {f^2 x F^{a+b c+b d x}}{b d \log (F)} \]
Antiderivative was successfully verified.
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Rule 2176
Rule 2178
Rule 2194
Rule 2199
Rubi steps
\begin {align*} \int \frac {F^{a+b (c+d x)} (e+f x)^2}{x} \, dx &=\int \left (2 e f F^{a+b c+b d x}+\frac {e^2 F^{a+b c+b d x}}{x}+f^2 F^{a+b c+b d x} x\right ) \, dx\\ &=e^2 \int \frac {F^{a+b c+b d x}}{x} \, dx+(2 e f) \int F^{a+b c+b d x} \, dx+f^2 \int F^{a+b c+b d x} x \, dx\\ &=e^2 F^{a+b c} \text {Ei}(b d x \log (F))+\frac {2 e f F^{a+b c+b d x}}{b d \log (F)}+\frac {f^2 F^{a+b c+b d x} x}{b d \log (F)}-\frac {f^2 \int F^{a+b c+b d x} \, dx}{b d \log (F)}\\ &=e^2 F^{a+b c} \text {Ei}(b d x \log (F))-\frac {f^2 F^{a+b c+b d x}}{b^2 d^2 \log ^2(F)}+\frac {2 e f F^{a+b c+b d x}}{b d \log (F)}+\frac {f^2 F^{a+b c+b d x} x}{b d \log (F)}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 54, normalized size = 0.56 \[ F^{a+b c} \left (\frac {f F^{b d x} (b d \log (F) (2 e+f x)-f)}{b^2 d^2 \log ^2(F)}+e^2 \text {Ei}(b d x \log (F))\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 75, normalized size = 0.78 \[ \frac {F^{b c + a} b^{2} d^{2} e^{2} {\rm Ei}\left (b d x \log \relax (F)\right ) \log \relax (F)^{2} - {\left (f^{2} - {\left (b d f^{2} x + 2 \, b d e f\right )} \log \relax (F)\right )} F^{b d x + b c + a}}{b^{2} d^{2} \log \relax (F)^{2}} \]
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 \[ \int \frac {{\left (f x + e\right )}^{2} F^{{\left (d x + c\right )} b + a}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 126, normalized size = 1.31 \[ -e^{2} F^{a} F^{b c} \Ei \left (1, -b d x \ln \relax (F )+b c \ln \relax (F )+a \ln \relax (F )-\left (b c +a \right ) \ln \relax (F )\right )+\frac {f^{2} x \,F^{b d x} F^{b c +a}}{b d \ln \relax (F )}+\frac {2 e f \,F^{b d x} F^{b c +a}}{b d \ln \relax (F )}-\frac {f^{2} F^{b d x} F^{b c +a}}{b^{2} d^{2} \ln \relax (F )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.05, size = 87, normalized size = 0.91 \[ F^{b c + a} e^{2} {\rm Ei}\left (b d x \log \relax (F)\right ) + \frac {2 \, F^{b d x + b c + a} e f}{b d \log \relax (F)} + \frac {{\left (F^{b c + a} b d x \log \relax (F) - F^{b c + a}\right )} F^{b d x} f^{2}}{b^{2} d^{2} \log \relax (F)^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.58, size = 80, normalized size = 0.83 \[ \frac {F^{a+b\,c}\,\left (b^2\,d^2\,e^2\,\mathrm {ei}\left (b\,d\,x\,\ln \relax (F)\right )\,{\ln \relax (F)}^2-F^{b\,d\,x}\,f^2+F^{b\,d\,x}\,b\,d\,f^2\,x\,\ln \relax (F)+2\,F^{b\,d\,x}\,b\,d\,e\,f\,\ln \relax (F)\right )}{b^2\,d^2\,{\ln \relax (F)}^2} \]
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 \[ \int \frac {F^{a + b \left (c + d x\right )} \left (e + f x\right )^{2}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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