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