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

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]

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

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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]

-(f^(a + b*x + c*x^2)/(e*(d + e*x))) + (Sqrt[c]*f^(a - b^2/(4*c))*Sqrt[Pi]*Erfi[
((b + 2*c*x)*Sqrt[Log[f]])/(2*Sqrt[c])]*Sqrt[Log[f]])/e^2 - ((2*c*d - b*e)*Log[f
]*Defer[Int][f^(a + b*x + c*x^2)/(d + e*x), x])/e^2

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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]

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

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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]

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

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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]

int(f^(c*x^2+b*x+a)/(e*x+d)^2,x)

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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]

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

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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]

integral(f^(c*x^2 + b*x + a)/(e^2*x^2 + 2*d*e*x + d^2), x)

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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]

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

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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]

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