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