Optimal. Leaf size=136 \[ \frac {1}{2} b^2 d^2 e^2 \log ^2(F) F^{a+b c} \text {Ei}(b d x \log (F))-\frac {e^2 F^{a+b c+b d x}}{2 x^2}-\frac {b d e^2 \log (F) F^{a+b c+b d x}}{2 x}+2 b d e f \log (F) F^{a+b c} \text {Ei}(b d x \log (F))-\frac {2 e f F^{a+b c+b d x}}{x}+f^2 F^{a+b c} \text {Ei}(b d x \log (F)) \]
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Rubi [A] time = 0.36, antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {2199, 2177, 2178} \[ \frac {1}{2} b^2 d^2 e^2 \log ^2(F) F^{a+b c} \text {Ei}(b d x \log (F))-\frac {e^2 F^{a+b c+b d x}}{2 x^2}-\frac {b d e^2 \log (F) F^{a+b c+b d x}}{2 x}+2 b d e f \log (F) F^{a+b c} \text {Ei}(b d x \log (F))-\frac {2 e f F^{a+b c+b d x}}{x}+f^2 F^{a+b c} \text {Ei}(b d x \log (F)) \]
Antiderivative was successfully verified.
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Rule 2177
Rule 2178
Rule 2199
Rubi steps
\begin {align*} \int \frac {F^{a+b (c+d x)} (e+f x)^2}{x^3} \, dx &=\int \left (\frac {e^2 F^{a+b c+b d x}}{x^3}+\frac {2 e f F^{a+b c+b d x}}{x^2}+\frac {f^2 F^{a+b c+b d x}}{x}\right ) \, dx\\ &=e^2 \int \frac {F^{a+b c+b d x}}{x^3} \, dx+(2 e f) \int \frac {F^{a+b c+b d x}}{x^2} \, dx+f^2 \int \frac {F^{a+b c+b d x}}{x} \, dx\\ &=-\frac {e^2 F^{a+b c+b d x}}{2 x^2}-\frac {2 e f F^{a+b c+b d x}}{x}+f^2 F^{a+b c} \text {Ei}(b d x \log (F))+\frac {1}{2} \left (b d e^2 \log (F)\right ) \int \frac {F^{a+b c+b d x}}{x^2} \, dx+(2 b d e f \log (F)) \int \frac {F^{a+b c+b d x}}{x} \, dx\\ &=-\frac {e^2 F^{a+b c+b d x}}{2 x^2}-\frac {2 e f F^{a+b c+b d x}}{x}+f^2 F^{a+b c} \text {Ei}(b d x \log (F))-\frac {b d e^2 F^{a+b c+b d x} \log (F)}{2 x}+2 b d e f F^{a+b c} \text {Ei}(b d x \log (F)) \log (F)+\frac {1}{2} \left (b^2 d^2 e^2 \log ^2(F)\right ) \int \frac {F^{a+b c+b d x}}{x} \, dx\\ &=-\frac {e^2 F^{a+b c+b d x}}{2 x^2}-\frac {2 e f F^{a+b c+b d x}}{x}+f^2 F^{a+b c} \text {Ei}(b d x \log (F))-\frac {b d e^2 F^{a+b c+b d x} \log (F)}{2 x}+2 b d e f F^{a+b c} \text {Ei}(b d x \log (F)) \log (F)+\frac {1}{2} b^2 d^2 e^2 F^{a+b c} \text {Ei}(b d x \log (F)) \log ^2(F)\\ \end {align*}
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Mathematica [A] time = 0.15, size = 76, normalized size = 0.56 \[ \frac {F^{a+b c} \left (x^2 \left (b^2 d^2 e^2 \log ^2(F)+4 b d e f \log (F)+2 f^2\right ) \text {Ei}(b d x \log (F))-e F^{b d x} (b d e x \log (F)+e+4 f x)\right )}{2 x^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 89, normalized size = 0.65 \[ \frac {{\left (b^{2} d^{2} e^{2} x^{2} \log \relax (F)^{2} + 4 \, b d e f x^{2} \log \relax (F) + 2 \, f^{2} x^{2}\right )} F^{b c + a} {\rm Ei}\left (b d x \log \relax (F)\right ) - {\left (b d e^{2} x \log \relax (F) + 4 \, e f x + e^{2}\right )} F^{b d x + b c + a}}{2 \, x^{2}} \]
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 \[ \int \frac {{\left (f x + e\right )}^{2} F^{{\left (d x + c\right )} b + a}}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 204, normalized size = 1.50 \[ -\frac {b^{2} d^{2} e^{2} F^{a} F^{b c} \Ei \left (1, -b d x \ln \relax (F )+b c \ln \relax (F )+a \ln \relax (F )-\left (b c +a \right ) \ln \relax (F )\right ) \ln \relax (F )^{2}}{2}-2 b d e f \,F^{a} F^{b c} \Ei \left (1, -b d x \ln \relax (F )+b c \ln \relax (F )+a \ln \relax (F )-\left (b c +a \right ) \ln \relax (F )\right ) \ln \relax (F )-\frac {b d \,e^{2} F^{b d x} F^{b c +a} \ln \relax (F )}{2 x}-f^{2} F^{a} F^{b c} \Ei \left (1, -b d x \ln \relax (F )+b c \ln \relax (F )+a \ln \relax (F )-\left (b c +a \right ) \ln \relax (F )\right )-\frac {2 e f \,F^{b d x} F^{b c +a}}{x}-\frac {e^{2} F^{b d x} F^{b c +a}}{2 x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.23, size = 74, normalized size = 0.54 \[ -F^{b c + a} b^{2} d^{2} e^{2} \Gamma \left (-2, -b d x \log \relax (F)\right ) \log \relax (F)^{2} + 2 \, F^{b c + a} b d e f \Gamma \left (-1, -b d x \log \relax (F)\right ) \log \relax (F) + F^{b c + a} f^{2} {\rm Ei}\left (b d x \log \relax (F)\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.60, size = 133, normalized size = 0.98 \[ F^{a+b\,c}\,f^2\,\mathrm {ei}\left (b\,d\,x\,\ln \relax (F)\right )-\frac {2\,F^{b\,d\,x}\,F^{a+b\,c}\,e\,f}{x}-F^{a+b\,c}\,b^2\,d^2\,e^2\,{\ln \relax (F)}^2\,\left (\frac {\mathrm {expint}\left (-b\,d\,x\,\ln \relax (F)\right )}{2}+F^{b\,d\,x}\,\left (\frac {1}{2\,b\,d\,x\,\ln \relax (F)}+\frac {1}{2\,b^2\,d^2\,x^2\,{\ln \relax (F)}^2}\right )\right )-2\,F^{a+b\,c}\,b\,d\,e\,f\,\ln \relax (F)\,\mathrm {expint}\left (-b\,d\,x\,\ln \relax (F)\right ) \]
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 \[ \int \frac {F^{a + b \left (c + d x\right )} \left (e + f x\right )^{2}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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