3.457 \(\int f^{b x+c x^2} (b+2 c x)^2 \, dx\)

Optimal. Leaf size=75 \[ \frac{(b+2 c x) f^{b x+c x^2}}{\log (f)}-\frac{\sqrt{\pi } \sqrt{c} f^{-\frac{b^2}{4 c}} \text{Erfi}\left (\frac{\sqrt{\log (f)} (b+2 c x)}{2 \sqrt{c}}\right )}{\log ^{\frac{3}{2}}(f)} \]

[Out]

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

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Rubi [A]  time = 0.101073, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ \frac{(b+2 c x) f^{b x+c x^2}}{\log (f)}-\frac{\sqrt{\pi } \sqrt{c} f^{-\frac{b^2}{4 c}} \text{Erfi}\left (\frac{\sqrt{\log (f)} (b+2 c x)}{2 \sqrt{c}}\right )}{\log ^{\frac{3}{2}}(f)} \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0375241, size = 84, normalized size = 1.12 \[ \frac{f^{-\frac{b^2}{4 c}} \left (\sqrt{\log (f)} (b+2 c x) f^{\frac{(b+2 c x)^2}{4 c}}-\sqrt{\pi } \sqrt{c} \text{Erfi}\left (\frac{\sqrt{\log (f)} (b+2 c x)}{2 \sqrt{c}}\right )\right )}{\log ^{\frac{3}{2}}(f)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.043, size = 84, normalized size = 1.1 \[ 2\,{\frac{cx{f}^{x \left ( cx+b \right ) }}{\ln \left ( f \right ) }}+{\frac{b{f}^{x \left ( cx+b \right ) }}{\ln \left ( f \right ) }}+{\frac{c\sqrt{\pi }}{\ln \left ( f \right ) }{f}^{-{\frac{{b}^{2}}{4\,c}}}{\it Erf} \left ( -\sqrt{-c\ln \left ( f \right ) }x+{\frac{b\ln \left ( f \right ) }{2}{\frac{1}{\sqrt{-c\ln \left ( f \right ) }}}} \right ){\frac{1}{\sqrt{-c\ln \left ( f \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.895085, size = 535, normalized size = 7.13 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*sqrt(pi)*b^2*erf(sqrt(-c*log(f))*x - 1/2*b*log(f)/sqrt(-c*log(f)))/(sqrt(-c*
log(f))*f^(1/4*b^2/c)) - (sqrt(pi)*(2*c*x*log(f) + b*log(f))*b*(erf(1/2*sqrt(-(2
*c*x*log(f) + b*log(f))^2/(c*log(f)))) - 1)*log(f)/((c*log(f))^(3/2)*sqrt(-(2*c*
x*log(f) + b*log(f))^2/(c*log(f)))) - 2*c*e^(1/4*(2*c*x*log(f) + b*log(f))^2/(c*
log(f)))*log(f)/(c*log(f))^(3/2))*b*c/(sqrt(c*log(f))*f^(1/4*b^2/c)) + 1/2*(sqrt
(pi)*(2*c*x*log(f) + b*log(f))*b^2*(erf(1/2*sqrt(-(2*c*x*log(f) + b*log(f))^2/(c
*log(f)))) - 1)*log(f)^2/((c*log(f))^(5/2)*sqrt(-(2*c*x*log(f) + b*log(f))^2/(c*
log(f)))) - 4*b*c*e^(1/4*(2*c*x*log(f) + b*log(f))^2/(c*log(f)))*log(f)^2/(c*log
(f))^(5/2) - 4*(2*c*x*log(f) + b*log(f))^3*gamma(3/2, -1/4*(2*c*x*log(f) + b*log
(f))^2/(c*log(f)))/((c*log(f))^(5/2)*(-(2*c*x*log(f) + b*log(f))^2/(c*log(f)))^(
3/2)))*c^2/(sqrt(c*log(f))*f^(1/4*b^2/c))

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Fricas [A]  time = 0.266306, size = 101, normalized size = 1.35 \[ \frac{{\left (2 \, c x + b\right )} \sqrt{-c \log \left (f\right )} f^{c x^{2} + b x} - \frac{\sqrt{\pi } c \operatorname{erf}\left (\frac{{\left (2 \, c x + b\right )} \sqrt{-c \log \left (f\right )}}{2 \, c}\right )}{f^{\frac{b^{2}}{4 \, c}}}}{\sqrt{-c \log \left (f\right )} \log \left (f\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.235435, size = 103, normalized size = 1.37 \[ \frac{c{\left (2 \, x + \frac{b}{c}\right )} e^{\left (c x^{2}{\rm ln}\left (f\right ) + b x{\rm ln}\left (f\right )\right )}}{{\rm ln}\left (f\right )} + \frac{\sqrt{\pi } c \operatorname{erf}\left (-\frac{1}{2} \, \sqrt{-c{\rm ln}\left (f\right )}{\left (2 \, x + \frac{b}{c}\right )}\right ) e^{\left (-\frac{b^{2}{\rm ln}\left (f\right )}{4 \, c}\right )}}{\sqrt{-c{\rm ln}\left (f\right )}{\rm ln}\left (f\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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