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

Optimal. Leaf size=96 \[ -\frac{f^2 F^{a+b c+b d x}}{b^2 d^2 \log ^2(F)}+e^2 F^{a+b c} \text{ExpIntegralEi}(b d x \log (F))+\frac{2 e f F^{a+b c+b d x}}{b d \log (F)}+\frac{f^2 x F^{a+b c+b d x}}{b d \log (F)} \]

[Out]

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

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Rubi [A]  time = 0.406891, antiderivative size = 96, 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 \[ -\frac{f^2 F^{a+b c+b d x}}{b^2 d^2 \log ^2(F)}+e^2 F^{a+b c} \text{ExpIntegralEi}(b d x \log (F))+\frac{2 e f F^{a+b c+b d x}}{b d \log (F)}+\frac{f^2 x F^{a+b c+b d x}}{b d \log (F)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 24.0848, size = 94, normalized size = 0.98 \[ F^{a + b c} e^{2} \operatorname{Ei}{\left (b d x \log{\left (F \right )} \right )} + \frac{2 F^{a + b c + b d x} e f}{b d \log{\left (F \right )}} + \frac{F^{a + b c + b d x} f^{2} x}{b d \log{\left (F \right )}} - \frac{F^{a + b c + b d x} f^{2}}{b^{2} d^{2} \log{\left (F \right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0982223, size = 54, normalized size = 0.56 \[ F^{a+b c} \left (\frac{f F^{b d x} (b d \log (F) (2 e+f x)-f)}{b^2 d^2 \log ^2(F)}+e^2 \text{ExpIntegralEi}(b d x \log (F))\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.033, size = 119, normalized size = 1.2 \[ -{e}^{2}{F}^{cb+a}{\it Ei} \left ( 1,cb\ln \left ( F \right ) +\ln \left ( F \right ) a-bdx\ln \left ( F \right ) -\ln \left ( F \right ) \left ( cb+a \right ) \right ) +{\frac{{f}^{2}{F}^{bdx+cb+a}x}{bd\ln \left ( F \right ) }}-{\frac{{f}^{2}{F}^{bdx+cb+a}}{ \left ( \ln \left ( F \right ) \right ) ^{2}{b}^{2}{d}^{2}}}+2\,{\frac{ef{F}^{bdx+cb+a}}{bd\ln \left ( F \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-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))+f^2*F^(b*d*x+b*
c+a)*x/b/d/ln(F)-f^2*F^(b*d*x+b*c+a)/b^2/d^2/ln(F)^2+2*e*f*F^(b*d*x+b*c+a)/b/d/l
n(F)

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Maxima [A]  time = 0.80998, size = 117, normalized size = 1.22 \[ F^{b c + a} e^{2}{\rm Ei}\left (b d x \log \left (F\right )\right ) + \frac{2 \, F^{b d x + b c + a} e f}{b d \log \left (F\right )} + \frac{{\left (F^{b c + a} b d x \log \left (F\right ) - F^{b c + a}\right )} F^{b d x} f^{2}}{b^{2} d^{2} \log \left (F\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.263736, size = 101, normalized size = 1.05 \[ \frac{F^{b c + a} b^{2} d^{2} e^{2}{\rm Ei}\left (b d x \log \left (F\right )\right ) \log \left (F\right )^{2} -{\left (f^{2} -{\left (b d f^{2} x + 2 \, b d e f\right )} \log \left (F\right )\right )} F^{b d x + b c + a}}{b^{2} d^{2} \log \left (F\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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