Optimal. Leaf size=217 \[ \frac {1}{6} b^3 d^3 e^2 \log ^3(F) F^{a+b c} \text {Ei}(b d x \log (F))-\frac {b^2 d^2 e^2 \log ^2(F) F^{a+b c+b d x}}{6 x}+b^2 d^2 e f \log ^2(F) F^{a+b c} \text {Ei}(b d x \log (F))-\frac {e^2 F^{a+b c+b d x}}{3 x^3}-\frac {b d e^2 \log (F) F^{a+b c+b d x}}{6 x^2}-\frac {e f F^{a+b c+b d x}}{x^2}-\frac {b d e f \log (F) F^{a+b c+b d x}}{x}+b d f^2 \log (F) F^{a+b c} \text {Ei}(b d x \log (F))-\frac {f^2 F^{a+b c+b d x}}{x} \]
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Rubi [A] time = 0.46, antiderivative size = 217, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {2199, 2177, 2178} \[ \frac {1}{6} b^3 d^3 e^2 \log ^3(F) F^{a+b c} \text {Ei}(b d x \log (F))-\frac {b^2 d^2 e^2 \log ^2(F) F^{a+b c+b d x}}{6 x}+b^2 d^2 e f \log ^2(F) F^{a+b c} \text {Ei}(b d x \log (F))-\frac {e^2 F^{a+b c+b d x}}{3 x^3}-\frac {b d e^2 \log (F) F^{a+b c+b d x}}{6 x^2}-\frac {e f F^{a+b c+b d x}}{x^2}-\frac {b d e f \log (F) F^{a+b c+b d x}}{x}+b d f^2 \log (F) F^{a+b c} \text {Ei}(b d x \log (F))-\frac {f^2 F^{a+b c+b d x}}{x} \]
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^4} \, dx &=\int \left (\frac {e^2 F^{a+b c+b d x}}{x^4}+\frac {2 e f F^{a+b c+b d x}}{x^3}+\frac {f^2 F^{a+b c+b d x}}{x^2}\right ) \, dx\\ &=e^2 \int \frac {F^{a+b c+b d x}}{x^4} \, dx+(2 e f) \int \frac {F^{a+b c+b d x}}{x^3} \, dx+f^2 \int \frac {F^{a+b c+b d x}}{x^2} \, dx\\ &=-\frac {e^2 F^{a+b c+b d x}}{3 x^3}-\frac {e f F^{a+b c+b d x}}{x^2}-\frac {f^2 F^{a+b c+b d x}}{x}+\frac {1}{3} \left (b d e^2 \log (F)\right ) \int \frac {F^{a+b c+b d x}}{x^3} \, dx+(b d e f \log (F)) \int \frac {F^{a+b c+b d x}}{x^2} \, dx+\left (b d f^2 \log (F)\right ) \int \frac {F^{a+b c+b d x}}{x} \, dx\\ &=-\frac {e^2 F^{a+b c+b d x}}{3 x^3}-\frac {e f F^{a+b c+b d x}}{x^2}-\frac {f^2 F^{a+b c+b d x}}{x}-\frac {b d e^2 F^{a+b c+b d x} \log (F)}{6 x^2}-\frac {b d e f F^{a+b c+b d x} \log (F)}{x}+b d f^2 F^{a+b c} \text {Ei}(b d x \log (F)) \log (F)+\frac {1}{6} \left (b^2 d^2 e^2 \log ^2(F)\right ) \int \frac {F^{a+b c+b d x}}{x^2} \, dx+\left (b^2 d^2 e f \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}}{3 x^3}-\frac {e f F^{a+b c+b d x}}{x^2}-\frac {f^2 F^{a+b c+b d x}}{x}-\frac {b d e^2 F^{a+b c+b d x} \log (F)}{6 x^2}-\frac {b d e f F^{a+b c+b d x} \log (F)}{x}+b d f^2 F^{a+b c} \text {Ei}(b d x \log (F)) \log (F)-\frac {b^2 d^2 e^2 F^{a+b c+b d x} \log ^2(F)}{6 x}+b^2 d^2 e f F^{a+b c} \text {Ei}(b d x \log (F)) \log ^2(F)+\frac {1}{6} \left (b^3 d^3 e^2 \log ^3(F)\right ) \int \frac {F^{a+b c+b d x}}{x} \, dx\\ &=-\frac {e^2 F^{a+b c+b d x}}{3 x^3}-\frac {e f F^{a+b c+b d x}}{x^2}-\frac {f^2 F^{a+b c+b d x}}{x}-\frac {b d e^2 F^{a+b c+b d x} \log (F)}{6 x^2}-\frac {b d e f F^{a+b c+b d x} \log (F)}{x}+b d f^2 F^{a+b c} \text {Ei}(b d x \log (F)) \log (F)-\frac {b^2 d^2 e^2 F^{a+b c+b d x} \log ^2(F)}{6 x}+b^2 d^2 e f F^{a+b c} \text {Ei}(b d x \log (F)) \log ^2(F)+\frac {1}{6} b^3 d^3 e^2 F^{a+b c} \text {Ei}(b d x \log (F)) \log ^3(F)\\ \end {align*}
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Mathematica [A] time = 0.23, size = 116, normalized size = 0.53 \[ \frac {F^{a+b c} \left (b d x^3 \log (F) \left (b^2 d^2 e^2 \log ^2(F)+6 b d e f \log (F)+6 f^2\right ) \text {Ei}(b d x \log (F))-F^{b d x} \left (b^2 d^2 e^2 x^2 \log ^2(F)+b d e x \log (F) (e+6 f x)+2 \left (e^2+3 e f x+3 f^2 x^2\right )\right )\right )}{6 x^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 137, normalized size = 0.63 \[ \frac {{\left (b^{3} d^{3} e^{2} x^{3} \log \relax (F)^{3} + 6 \, b^{2} d^{2} e f x^{3} \log \relax (F)^{2} + 6 \, b d f^{2} x^{3} \log \relax (F)\right )} F^{b c + a} {\rm Ei}\left (b d x \log \relax (F)\right ) - {\left (b^{2} d^{2} e^{2} x^{2} \log \relax (F)^{2} + 6 \, f^{2} x^{2} + 6 \, e f x + 2 \, e^{2} + {\left (6 \, b d e f x^{2} + b d e^{2} x\right )} \log \relax (F)\right )} F^{b d x + b c + a}}{6 \, x^{3}} \]
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 \[ \int \frac {{\left (f x + e\right )}^{2} F^{{\left (d x + c\right )} b + a}}{x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 290, normalized size = 1.34 \[ -\frac {b^{3} d^{3} e^{2} F^{a} F^{b c} \Ei \left (1, -b d x \ln \relax (F )+b c \ln \relax (F )+a \ln \relax (F )-\left (b c +a \right ) \ln \relax (F )\right ) \ln \relax (F )^{3}}{6}-b^{2} d^{2} e f \,F^{a} F^{b c} \Ei \left (1, -b d x \ln \relax (F )+b c \ln \relax (F )+a \ln \relax (F )-\left (b c +a \right ) \ln \relax (F )\right ) \ln \relax (F )^{2}-\frac {b^{2} d^{2} e^{2} F^{b d x} F^{b c +a} \ln \relax (F )^{2}}{6 x}-b d \,f^{2} F^{a} F^{b c} \Ei \left (1, -b d x \ln \relax (F )+b c \ln \relax (F )+a \ln \relax (F )-\left (b c +a \right ) \ln \relax (F )\right ) \ln \relax (F )-\frac {b d e f \,F^{b d x} F^{b c +a} \ln \relax (F )}{x}-\frac {b d \,e^{2} F^{b d x} F^{b c +a} \ln \relax (F )}{6 x^{2}}-\frac {f^{2} F^{b d x} F^{b c +a}}{x}-\frac {e f \,F^{b d x} F^{b c +a}}{x^{2}}-\frac {e^{2} F^{b d x} F^{b c +a}}{3 x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.05, size = 85, normalized size = 0.39 \[ F^{b c + a} b^{3} d^{3} e^{2} \Gamma \left (-3, -b d x \log \relax (F)\right ) \log \relax (F)^{3} - 2 \, F^{b c + a} b^{2} d^{2} e f \Gamma \left (-2, -b d x \log \relax (F)\right ) \log \relax (F)^{2} + F^{b c + a} b d f^{2} \Gamma \left (-1, -b d x \log \relax (F)\right ) \log \relax (F) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.66, size = 202, normalized size = 0.93 \[ -\frac {F^{b\,d\,x}\,F^{a+b\,c}\,f^2}{x}-F^{a+b\,c}\,b^3\,d^3\,e^2\,{\ln \relax (F)}^3\,\left (F^{b\,d\,x}\,\left (\frac {1}{6\,b\,d\,x\,\ln \relax (F)}+\frac {1}{6\,b^2\,d^2\,x^2\,{\ln \relax (F)}^2}+\frac {1}{3\,b^3\,d^3\,x^3\,{\ln \relax (F)}^3}\right )+\frac {\mathrm {expint}\left (-b\,d\,x\,\ln \relax (F)\right )}{6}\right )-F^{a+b\,c}\,b\,d\,f^2\,\ln \relax (F)\,\mathrm {expint}\left (-b\,d\,x\,\ln \relax (F)\right )-2\,F^{a+b\,c}\,b^2\,d^2\,e\,f\,{\ln \relax (F)}^2\,\left (\frac {\mathrm {expint}\left (-b\,d\,x\,\ln \relax (F)\right )}{2}+F^{b\,d\,x}\,\left (\frac {1}{2\,b\,d\,x\,\ln \relax (F)}+\frac {1}{2\,b^2\,d^2\,x^2\,{\ln \relax (F)}^2}\right )\right ) \]
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 \[ \int \frac {F^{a + b \left (c + d x\right )} \left (e + f x\right )^{2}}{x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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