3.449 \(\int \frac{f^{a+b x+c x^2}}{(d+e x)^3} \, dx\)

Optimal. Leaf size=207 \[ \frac{\log ^2(f) (2 c d-b e)^2 \text{Int}\left (\frac{f^{a+b x+c x^2}}{d+e x},x\right )}{2 e^4}+\frac{c \log (f) \text{Int}\left (\frac{f^{a+b x+c x^2}}{d+e x},x\right )}{e^2}-\frac{\sqrt{\pi } \sqrt{c} \log ^{\frac{3}{2}}(f) f^{a-\frac{b^2}{4 c}} (2 c d-b e) \text{Erfi}\left (\frac{\sqrt{\log (f)} (b+2 c x)}{2 \sqrt{c}}\right )}{2 e^4}+\frac{\log (f) (2 c d-b e) f^{a+b x+c x^2}}{2 e^3 (d+e x)}-\frac{f^{a+b x+c x^2}}{2 e (d+e x)^2} \]

[Out]

-f^(a + b*x + c*x^2)/(2*e*(d + e*x)^2) + ((2*c*d - b*e)*f^(a + b*x + c*x^2)*Log[
f])/(2*e^3*(d + e*x)) - (Sqrt[c]*(2*c*d - b*e)*f^(a - b^2/(4*c))*Sqrt[Pi]*Erfi[(
(b + 2*c*x)*Sqrt[Log[f]])/(2*Sqrt[c])]*Log[f]^(3/2))/(2*e^4) + (c*Log[f]*Uninteg
rable[f^(a + b*x + c*x^2)/(d + e*x), x])/e^2 + ((2*c*d - b*e)^2*Log[f]^2*Uninteg
rable[f^(a + b*x + c*x^2)/(d + e*x), x])/(2*e^4)

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Rubi [A]  time = 0.375792, 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. \[ \text{Int}\left (\frac{f^{a+b x+c x^2}}{(d+e x)^3},x\right ) \]

Verification is Not applicable to the result.

[In]  Int[f^(a + b*x + c*x^2)/(d + e*x)^3,x]

[Out]

-f^(a + b*x + c*x^2)/(2*e*(d + e*x)^2) + ((2*c*d - b*e)*f^(a + b*x + c*x^2)*Log[
f])/(2*e^3*(d + e*x)) - (Sqrt[c]*(2*c*d - b*e)*f^(a - b^2/(4*c))*Sqrt[Pi]*Erfi[(
(b + 2*c*x)*Sqrt[Log[f]])/(2*Sqrt[c])]*Log[f]^(3/2))/(2*e^4) + (c*Log[f]*Defer[I
nt][f^(a + b*x + c*x^2)/(d + e*x), x])/e^2 + ((2*c*d - b*e)^2*Log[f]^2*Defer[Int
][f^(a + b*x + c*x^2)/(d + e*x), x])/(2*e^4)

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Rubi in Sympy [A]  time = 0., size = 0, normalized size = 0. \[ \frac{c \log{\left (f \right )} \int \frac{f^{a + b x + c x^{2}}}{d + e x}\, dx}{e^{2}} + \frac{c f^{a - \frac{b^{2}}{4 c}} \left (b e - 2 c d\right ) \log{\left (f \right )}^{2} \int f^{\frac{b^{2}}{4 c} + b x + c x^{2}}\, dx}{e^{4}} - \frac{f^{a + b x + c x^{2}}}{2 e \left (d + e x\right )^{2}} - \frac{f^{a + b x + c x^{2}} \left (b e - 2 c d\right ) \log{\left (f \right )}}{2 e^{3} \left (d + e x\right )} + \frac{\left (b e - 2 c d\right )^{2} \log{\left (f \right )}^{2} \int \frac{f^{a + b x + c x^{2}}}{d + e x}\, dx}{2 e^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(f**(c*x**2+b*x+a)/(e*x+d)**3,x)

[Out]

c*log(f)*Integral(f**(a + b*x + c*x**2)/(d + e*x), x)/e**2 + c*f**(a - b**2/(4*c
))*(b*e - 2*c*d)*log(f)**2*Integral(f**(b**2/(4*c) + b*x + c*x**2), x)/e**4 - f*
*(a + b*x + c*x**2)/(2*e*(d + e*x)**2) - f**(a + b*x + c*x**2)*(b*e - 2*c*d)*log
(f)/(2*e**3*(d + e*x)) + (b*e - 2*c*d)**2*log(f)**2*Integral(f**(a + b*x + c*x**
2)/(d + e*x), x)/(2*e**4)

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Mathematica [A]  time = 1.00635, size = 0, normalized size = 0. \[ \int \frac{f^{a+b x+c x^2}}{(d+e x)^3} \, dx \]

Verification is Not applicable to the result.

[In]  Integrate[f^(a + b*x + c*x^2)/(d + e*x)^3,x]

[Out]

Integrate[f^(a + b*x + c*x^2)/(d + e*x)^3, x]

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Maple [A]  time = 0.073, size = 0, normalized size = 0. \[ \int{\frac{{f}^{c{x}^{2}+bx+a}}{ \left ( ex+d \right ) ^{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(f^(c*x^2+b*x+a)/(e*x+d)^3,x)

[Out]

int(f^(c*x^2+b*x+a)/(e*x+d)^3,x)

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Maxima [A]  time = 0., size = 0, normalized size = 0. \[ \int \frac{f^{c x^{2} + b x + a}}{{\left (e x + d\right )}^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f^(c*x^2 + b*x + a)/(e*x + d)^3,x, algorithm="maxima")

[Out]

integrate(f^(c*x^2 + b*x + a)/(e*x + d)^3, x)

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Fricas [A]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{f^{c x^{2} + b x + a}}{e^{3} x^{3} + 3 \, d e^{2} x^{2} + 3 \, d^{2} e x + d^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f^(c*x^2 + b*x + a)/(e*x + d)^3,x, algorithm="fricas")

[Out]

integral(f^(c*x^2 + b*x + a)/(e^3*x^3 + 3*d*e^2*x^2 + 3*d^2*e*x + d^3), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f**(c*x**2+b*x+a)/(e*x+d)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0., size = 0, normalized size = 0. \[ \int \frac{f^{c x^{2} + b x + a}}{{\left (e x + d\right )}^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f^(c*x^2 + b*x + a)/(e*x + d)^3,x, algorithm="giac")

[Out]

integrate(f^(c*x^2 + b*x + a)/(e*x + d)^3, x)