3.70 \(\int \frac{F^{a+b (c+d x)} (e+f x)^2}{x^2} \, dx\)

Optimal. Leaf size=85 \[ b d e^2 \log (F) F^{a+b c} \text{ExpIntegralEi}(b d x \log (F))-\frac{e^2 F^{a+b c+b d x}}{x}+2 e f F^{a+b c} \text{ExpIntegralEi}(b d x \log (F))+\frac{f^2 F^{a+b c+b d x}}{b d \log (F)} \]

[Out]

-((e^2*F^(a + b*c + b*d*x))/x) + 2*e*f*F^(a + b*c)*ExpIntegralEi[b*d*x*Log[F]] +
(f^2*F^(a + b*c + b*d*x))/(b*d*Log[F]) + b*d*e^2*F^(a + b*c)*ExpIntegralEi[b*d*
x*Log[F]]*Log[F]

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Rubi [A]  time = 0.430445, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ b d e^2 \log (F) F^{a+b c} \text{ExpIntegralEi}(b d x \log (F))-\frac{e^2 F^{a+b c+b d x}}{x}+2 e f F^{a+b c} \text{ExpIntegralEi}(b d x \log (F))+\frac{f^2 F^{a+b c+b d x}}{b d \log (F)} \]

Antiderivative was successfully verified.

[In]  Int[(F^(a + b*(c + d*x))*(e + f*x)^2)/x^2,x]

[Out]

-((e^2*F^(a + b*c + b*d*x))/x) + 2*e*f*F^(a + b*c)*ExpIntegralEi[b*d*x*Log[F]] +
(f^2*F^(a + b*c + b*d*x))/(b*d*Log[F]) + b*d*e^2*F^(a + b*c)*ExpIntegralEi[b*d*
x*Log[F]]*Log[F]

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Rubi in Sympy [A]  time = 20.3791, size = 87, normalized size = 1.02 \[ F^{a + b c} b d e^{2} \log{\left (F \right )} \operatorname{Ei}{\left (b d x \log{\left (F \right )} \right )} + 2 F^{a + b c} e f \operatorname{Ei}{\left (b d x \log{\left (F \right )} \right )} - \frac{F^{a + b c + b d x} e^{2}}{x} + \frac{F^{a + b c + b d x} f^{2}}{b d \log{\left (F \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(F**(a+b*(d*x+c))*(f*x+e)**2/x**2,x)

[Out]

F**(a + b*c)*b*d*e**2*log(F)*Ei(b*d*x*log(F)) + 2*F**(a + b*c)*e*f*Ei(b*d*x*log(
F)) - F**(a + b*c + b*d*x)*e**2/x + F**(a + b*c + b*d*x)*f**2/(b*d*log(F))

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Mathematica [A]  time = 0.109195, size = 58, normalized size = 0.68 \[ F^{a+b c} \left (F^{b d x} \left (\frac{f^2}{b d \log (F)}-\frac{e^2}{x}\right )+e (b d e \log (F)+2 f) \text{ExpIntegralEi}(b d x \log (F))\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(F^(a + b*(c + d*x))*(e + f*x)^2)/x^2,x]

[Out]

F^(a + b*c)*(F^(b*d*x)*(-(e^2/x) + f^2/(b*d*Log[F])) + e*ExpIntegralEi[b*d*x*Log
[F]]*(2*f + b*d*e*Log[F]))

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Maple [A]  time = 0.047, size = 129, normalized size = 1.5 \[ -2\,fe{F}^{cb+a}{\it Ei} \left ( 1,cb\ln \left ( F \right ) +\ln \left ( F \right ) a-bdx\ln \left ( F \right ) -\ln \left ( F \right ) \left ( cb+a \right ) \right ) -{\frac{{e}^{2}{F}^{bdx+cb+a}}{x}}+{\frac{{f}^{2}{F}^{bdx+cb+a}}{bd\ln \left ( F \right ) }}-\ln \left ( F \right ) bd{e}^{2}{F}^{cb+a}{\it Ei} \left ( 1,cb\ln \left ( F \right ) +\ln \left ( F \right ) a-bdx\ln \left ( F \right ) -\ln \left ( F \right ) \left ( cb+a \right ) \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(F^(a+b*(d*x+c))*(f*x+e)^2/x^2,x)

[Out]

-2*f*e*F^(b*c+a)*Ei(1,c*b*ln(F)+ln(F)*a-b*d*x*ln(F)-ln(F)*(b*c+a))-e^2*F^(b*d*x+
b*c+a)/x+f^2*F^(b*d*x+b*c+a)/b/d/ln(F)-ln(F)*b*d*e^2*F^(b*c+a)*Ei(1,c*b*ln(F)+ln
(F)*a-b*d*x*ln(F)-ln(F)*(b*c+a))

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Maxima [A]  time = 0.824211, size = 92, normalized size = 1.08 \[ F^{b c + a} b d e^{2} \Gamma \left (-1, -b d x \log \left (F\right )\right ) \log \left (F\right ) + 2 \, F^{b c + a} e f{\rm Ei}\left (b d x \log \left (F\right )\right ) + \frac{F^{b d x + b c + a} f^{2}}{b d \log \left (F\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x + e)^2*F^((d*x + c)*b + a)/x^2,x, algorithm="maxima")

[Out]

F^(b*c + a)*b*d*e^2*gamma(-1, -b*d*x*log(F))*log(F) + 2*F^(b*c + a)*e*f*Ei(b*d*x
*log(F)) + F^(b*d*x + b*c + a)*f^2/(b*d*log(F))

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Fricas [A]  time = 0.270982, size = 112, normalized size = 1.32 \[ \frac{{\left (b^{2} d^{2} e^{2} x \log \left (F\right )^{2} + 2 \, b d e f x \log \left (F\right )\right )} F^{b c + a}{\rm Ei}\left (b d x \log \left (F\right )\right ) -{\left (b d e^{2} \log \left (F\right ) - f^{2} x\right )} F^{b d x + b c + a}}{b d x \log \left (F\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x + e)^2*F^((d*x + c)*b + a)/x^2,x, algorithm="fricas")

[Out]

((b^2*d^2*e^2*x*log(F)^2 + 2*b*d*e*f*x*log(F))*F^(b*c + a)*Ei(b*d*x*log(F)) - (b
*d*e^2*log(F) - f^2*x)*F^(b*d*x + b*c + a))/(b*d*x*log(F))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{F^{a + b \left (c + d x\right )} \left (e + f x\right )^{2}}{x^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(F**(a+b*(d*x+c))*(f*x+e)**2/x**2,x)

[Out]

Integral(F**(a + b*(c + d*x))*(e + f*x)**2/x**2, x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (f x + e\right )}^{2} F^{{\left (d x + c\right )} b + a}}{x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x + e)^2*F^((d*x + c)*b + a)/x^2,x, algorithm="giac")

[Out]

integrate((f*x + e)^2*F^((d*x + c)*b + a)/x^2, x)