3.453 \(\int \frac{f^{a+b x+c x^2}}{b+2 c x} \, dx\)

Optimal. Leaf size=39 \[ \frac{f^{a-\frac{b^2}{4 c}} \text{ExpIntegralEi}\left (\frac{\log (f) (b+2 c x)^2}{4 c}\right )}{4 c} \]

[Out]

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

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Rubi [A]  time = 0.0604502, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048 \[ \frac{f^{a-\frac{b^2}{4 c}} \text{ExpIntegralEi}\left (\frac{\log (f) (b+2 c x)^2}{4 c}\right )}{4 c} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 5.74068, size = 29, normalized size = 0.74 \[ \frac{f^{a - \frac{b^{2}}{4 c}} \operatorname{Ei}{\left (\frac{\left (b + 2 c x\right )^{2} \log{\left (f \right )}}{4 c} \right )}}{4 c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

f**(a - b**2/(4*c))*Ei((b + 2*c*x)**2*log(f)/(4*c))/(4*c)

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Mathematica [A]  time = 0.020148, size = 39, normalized size = 1. \[ \frac{f^{a-\frac{b^2}{4 c}} \text{ExpIntegralEi}\left (\frac{\log (f) (b+2 c x)^2}{4 c}\right )}{4 c} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.026, size = 40, normalized size = 1. \[ -{\frac{1}{4\,c}{f}^{{\frac{4\,ac-{b}^{2}}{4\,c}}}{\it Ei} \left ( 1,-{\frac{ \left ( 2\,cx+b \right ) ^{2}\ln \left ( f \right ) }{4\,c}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4/c*f^(1/4*(4*a*c-b^2)/c)*Ei(1,-1/4*(2*c*x+b)^2*ln(f)/c)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{f^{c x^{2} + b x + a}}{2 \, c x + b}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.264996, size = 63, normalized size = 1.62 \[ \frac{{\rm Ei}\left (\frac{{\left (4 \, c^{2} x^{2} + 4 \, b c x + b^{2}\right )} \log \left (f\right )}{4 \, c}\right )}{4 \, c f^{\frac{b^{2} - 4 \, a c}{4 \, c}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*Ei(1/4*(4*c^2*x^2 + 4*b*c*x + b^2)*log(f)/c)/(c*f^(1/4*(b^2 - 4*a*c)/c))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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