Optimal. Leaf size=120 \[ -\frac{\log (f) (2 c d-b e) \text{Int}\left (\frac{f^{a+b x+c x^2}}{d+e x},x\right )}{e^2}+\frac{\sqrt{\pi } \sqrt{c} \sqrt{\log (f)} f^{a-\frac{b^2}{4 c}} \text{Erfi}\left (\frac{\sqrt{\log (f)} (b+2 c x)}{2 \sqrt{c}}\right )}{e^2}-\frac{f^{a+b x+c x^2}}{e (d+e x)} \]
[Out]
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Rubi [A] time = 0.186204, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0. \[ \text{Int}\left (\frac{f^{a+b x+c x^2}}{(d+e x)^2},x\right ) \]
Verification is Not applicable to the result.
[In] Int[f^(a + b*x + c*x^2)/(d + e*x)^2,x]
[Out]
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Rubi in Sympy [A] time = 0., size = 0, normalized size = 0. \[ \frac{2 c f^{a - \frac{b^{2}}{4 c}} \log{\left (f \right )} \int f^{\frac{b^{2}}{4 c} + b x + c x^{2}}\, dx}{e^{2}} - \frac{f^{a + b x + c x^{2}}}{e \left (d + e x\right )} + \frac{\left (b e - 2 c d\right ) \log{\left (f \right )} \int \frac{f^{a + b x + c x^{2}}}{d + e x}\, dx}{e^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(f**(c*x**2+b*x+a)/(e*x+d)**2,x)
[Out]
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Mathematica [A] time = 1.09558, size = 0, normalized size = 0. \[ \int \frac{f^{a+b x+c x^2}}{(d+e x)^2} \, dx \]
Verification is Not applicable to the result.
[In] Integrate[f^(a + b*x + c*x^2)/(d + e*x)^2,x]
[Out]
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Maple [A] time = 0.049, size = 0, normalized size = 0. \[ \int{\frac{{f}^{c{x}^{2}+bx+a}}{ \left ( ex+d \right ) ^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(f^(c*x^2+b*x+a)/(e*x+d)^2,x)
[Out]
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Maxima [A] time = 0., size = 0, normalized size = 0. \[ \int \frac{f^{c x^{2} + b x + a}}{{\left (e x + d\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(f^(c*x^2 + b*x + a)/(e*x + d)^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{f^{c x^{2} + b x + a}}{e^{2} x^{2} + 2 \, d e x + d^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(f^(c*x^2 + b*x + a)/(e*x + d)^2,x, algorithm="fricas")
[Out]
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Sympy [A] time = 0., size = 0, normalized size = 0. \[ \int \frac{f^{a + b x + c x^{2}}}{\left (d + e x\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(f**(c*x**2+b*x+a)/(e*x+d)**2,x)
[Out]
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GIAC/XCAS [A] time = 0., size = 0, normalized size = 0. \[ \int \frac{f^{c x^{2} + b x + a}}{{\left (e x + d\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(f^(c*x^2 + b*x + a)/(e*x + d)^2,x, algorithm="giac")
[Out]