3.71 \(\int \frac{F^{a+b (c+d x)} (e+f x)^2}{x^3} \, dx\)

Optimal. Leaf size=136 \[ \frac{1}{2} b^2 d^2 e^2 \log ^2(F) F^{a+b c} \text{ExpIntegralEi}(b d x \log (F))-\frac{e^2 F^{a+b c+b d x}}{2 x^2}-\frac{b d e^2 \log (F) F^{a+b c+b d x}}{2 x}+2 b d e f \log (F) F^{a+b c} \text{ExpIntegralEi}(b d x \log (F))-\frac{2 e f F^{a+b c+b d x}}{x}+f^2 F^{a+b c} \text{ExpIntegralEi}(b d x \log (F)) \]

[Out]

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

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Rubi [A]  time = 0.56159, antiderivative size = 136, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ \frac{1}{2} b^2 d^2 e^2 \log ^2(F) F^{a+b c} \text{ExpIntegralEi}(b d x \log (F))-\frac{e^2 F^{a+b c+b d x}}{2 x^2}-\frac{b d e^2 \log (F) F^{a+b c+b d x}}{2 x}+2 b d e f \log (F) F^{a+b c} \text{ExpIntegralEi}(b d x \log (F))-\frac{2 e f F^{a+b c+b d x}}{x}+f^2 F^{a+b c} \text{ExpIntegralEi}(b d x \log (F)) \]

Antiderivative was successfully verified.

[In]  Int[(F^(a + b*(c + d*x))*(e + f*x)^2)/x^3,x]

[Out]

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

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Rubi in Sympy [A]  time = 25.3953, size = 146, normalized size = 1.07 \[ \frac{F^{a + b c} b^{2} d^{2} e^{2} \log{\left (F \right )}^{2} \operatorname{Ei}{\left (b d x \log{\left (F \right )} \right )}}{2} + 2 F^{a + b c} b d e f \log{\left (F \right )} \operatorname{Ei}{\left (b d x \log{\left (F \right )} \right )} + F^{a + b c} f^{2} \operatorname{Ei}{\left (b d x \log{\left (F \right )} \right )} - \frac{F^{a + b c + b d x} b d e^{2} \log{\left (F \right )}}{2 x} - \frac{F^{a + b c + b d x} e^{2}}{2 x^{2}} - \frac{2 F^{a + b c + b d x} e f}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(F**(a+b*(d*x+c))*(f*x+e)**2/x**3,x)

[Out]

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

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Mathematica [A]  time = 0.0758798, size = 76, normalized size = 0.56 \[ \frac{F^{a+b c} \left (x^2 \left (b^2 d^2 e^2 \log ^2(F)+4 b d e f \log (F)+2 f^2\right ) \text{ExpIntegralEi}(b d x \log (F))-e F^{b d x} (b d e x \log (F)+e+4 f x)\right )}{2 x^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(F^(a + b*(c + d*x))*(e + f*x)^2)/x^3,x]

[Out]

(F^(a + b*c)*(-(e*F^(b*d*x)*(e + 4*f*x + b*d*e*x*Log[F])) + x^2*ExpIntegralEi[b*
d*x*Log[F]]*(2*f^2 + 4*b*d*e*f*Log[F] + b^2*d^2*e^2*Log[F]^2)))/(2*x^2)

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Maple [A]  time = 0.055, size = 195, normalized size = 1.4 \[ -{\frac{{e}^{2}{F}^{bdx+cb+a}}{2\,{x}^{2}}}-{\frac{bd{e}^{2}{F}^{bdx+cb+a}\ln \left ( F \right ) }{2\,x}}-{f}^{2}{F}^{cb+a}{\it Ei} \left ( 1,cb\ln \left ( F \right ) +\ln \left ( F \right ) a-bdx\ln \left ( F \right ) -\ln \left ( F \right ) \left ( cb+a \right ) \right ) -2\,{\frac{ef{F}^{bdx+cb+a}}{x}}-2\,\ln \left ( F \right ) bdfe{F}^{cb+a}{\it Ei} \left ( 1,cb\ln \left ( F \right ) +\ln \left ( F \right ) a-bdx\ln \left ( F \right ) -\ln \left ( F \right ) \left ( cb+a \right ) \right ) -{\frac{ \left ( \ln \left ( F \right ) \right ) ^{2}{b}^{2}{d}^{2}{e}^{2}{F}^{cb+a}{\it Ei} \left ( 1,cb\ln \left ( F \right ) +\ln \left ( F \right ) a-bdx\ln \left ( F \right ) -\ln \left ( F \right ) \left ( cb+a \right ) \right ) }{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(F^(a+b*(d*x+c))*(f*x+e)^2/x^3,x)

[Out]

-1/2*e^2*F^(b*d*x+b*c+a)/x^2-1/2*b*d*e^2*F^(b*d*x+b*c+a)*ln(F)/x-f^2*F^(b*c+a)*E
i(1,c*b*ln(F)+ln(F)*a-b*d*x*ln(F)-ln(F)*(b*c+a))-2*e*f*F^(b*d*x+b*c+a)/x-2*ln(F)
*b*d*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))-1/2*ln(F)^2
*b^2*d^2*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 = 0.851818, size = 100, normalized size = 0.74 \[ -F^{b c + a} b^{2} d^{2} e^{2} \Gamma \left (-2, -b d x \log \left (F\right )\right ) \log \left (F\right )^{2} + 2 \, F^{b c + a} b d e f \Gamma \left (-1, -b d x \log \left (F\right )\right ) \log \left (F\right ) + F^{b c + a} f^{2}{\rm Ei}\left (b d x \log \left (F\right )\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x + e)^2*F^((d*x + c)*b + a)/x^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.2842, size = 120, normalized size = 0.88 \[ \frac{{\left (b^{2} d^{2} e^{2} x^{2} \log \left (F\right )^{2} + 4 \, b d e f x^{2} \log \left (F\right ) + 2 \, f^{2} x^{2}\right )} F^{b c + a}{\rm Ei}\left (b d x \log \left (F\right )\right ) -{\left (b d e^{2} x \log \left (F\right ) + 4 \, e f x + e^{2}\right )} F^{b d x + b c + a}}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x + e)^2*F^((d*x + c)*b + a)/x^3,x, algorithm="fricas")

[Out]

1/2*((b^2*d^2*e^2*x^2*log(F)^2 + 4*b*d*e*f*x^2*log(F) + 2*f^2*x^2)*F^(b*c + a)*E
i(b*d*x*log(F)) - (b*d*e^2*x*log(F) + 4*e*f*x + e^2)*F^(b*d*x + b*c + a))/x^2

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{F^{a + b \left (c + d x\right )} \left (e + f x\right )^{2}}{x^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(F**(a+b*(d*x+c))*(f*x+e)**2/x**3,x)

[Out]

Integral(F**(a + b*(c + d*x))*(e + f*x)**2/x**3, x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (f x + e\right )}^{2} F^{{\left (d x + c\right )} b + a}}{x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x + e)^2*F^((d*x + c)*b + a)/x^3,x, algorithm="giac")

[Out]

integrate((f*x + e)^2*F^((d*x + c)*b + a)/x^3, x)