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

Optimal. Leaf size=84 \[ \frac{\sqrt{\pi } \sqrt{\log (f)} f^{a-\frac{b^2}{4 c}} \text{Erfi}\left (\frac{\sqrt{\log (f)} (b+2 c x)}{2 \sqrt{c}}\right )}{4 c^{3/2}}-\frac{f^{a+b x+c x^2}}{2 c (b+2 c x)} \]

[Out]

-f^(a + b*x + c*x^2)/(2*c*(b + 2*c*x)) + (f^(a - b^2/(4*c))*Sqrt[Pi]*Erfi[((b +
2*c*x)*Sqrt[Log[f]])/(2*Sqrt[c])]*Sqrt[Log[f]])/(4*c^(3/2))

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Rubi [A]  time = 0.0949937, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ \frac{\sqrt{\pi } \sqrt{\log (f)} f^{a-\frac{b^2}{4 c}} \text{Erfi}\left (\frac{\sqrt{\log (f)} (b+2 c x)}{2 \sqrt{c}}\right )}{4 c^{3/2}}-\frac{f^{a+b x+c x^2}}{2 c (b+2 c x)} \]

Antiderivative was successfully verified.

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

[Out]

-f^(a + b*x + c*x^2)/(2*c*(b + 2*c*x)) + (f^(a - b^2/(4*c))*Sqrt[Pi]*Erfi[((b +
2*c*x)*Sqrt[Log[f]])/(2*Sqrt[c])]*Sqrt[Log[f]])/(4*c^(3/2))

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{f^{a - \frac{b^{2}}{4 c}} \log{\left (f \right )} \int f^{\frac{b^{2}}{4 c} + b x + c x^{2}}\, dx}{2 c} - \frac{f^{a + b x + c x^{2}}}{2 c \left (b + 2 c x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0646938, size = 96, normalized size = 1.14 \[ \frac{f^{a-\frac{b^2}{4 c}} \left (\sqrt{\pi } \sqrt{\log (f)} (b+2 c x) \text{Erfi}\left (\frac{\sqrt{\log (f)} (b+2 c x)}{2 \sqrt{c}}\right )-2 \sqrt{c} f^{\frac{(b+2 c x)^2}{4 c}}\right )}{4 c^{3/2} (b+2 c x)} \]

Antiderivative was successfully verified.

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

[Out]

(f^(a - b^2/(4*c))*(-2*Sqrt[c]*f^((b + 2*c*x)^2/(4*c)) + Sqrt[Pi]*(b + 2*c*x)*Er
fi[((b + 2*c*x)*Sqrt[Log[f]])/(2*Sqrt[c])]*Sqrt[Log[f]]))/(4*c^(3/2)*(b + 2*c*x)
)

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Maple [A]  time = 0.075, size = 81, normalized size = 1. \[ -{\frac{{f}^{c{x}^{2}+bx+a}}{2\,c \left ( 2\,cx+b \right ) }}+{\frac{\ln \left ( f \right ) \sqrt{\pi }}{4\,{c}^{2}}{f}^{{\frac{4\,ac-{b}^{2}}{4\,c}}}{\it Erf} \left ({\frac{2\,cx+b}{2}\sqrt{-{\frac{\ln \left ( f \right ) }{c}}}} \right ){\frac{1}{\sqrt{-{\frac{\ln \left ( f \right ) }{c}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.272434, size = 123, normalized size = 1.46 \[ \frac{\frac{\sqrt{\pi }{\left (2 \, c x + b\right )} \operatorname{erf}\left (\frac{{\left (2 \, c x + b\right )} \sqrt{-c \log \left (f\right )}}{2 \, c}\right ) \log \left (f\right )}{f^{\frac{b^{2} - 4 \, a c}{4 \, c}}} - 2 \, \sqrt{-c \log \left (f\right )} f^{c x^{2} + b x + a}}{4 \,{\left (2 \, c^{2} x + b c\right )} \sqrt{-c \log \left (f\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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