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

Optimal. Leaf size=119 \[ -\frac{\log (f) (2 c d-b e) \text{Unintegrable}\left (\frac{f^{a+b x+c x^2}}{d+e x},x\right )}{e^2}+\frac{\sqrt{\pi } \sqrt{c} \sqrt{\log (f)} f^{a-\frac{b^2}{4 c}} \text{Erfi}\left (\frac{\sqrt{\log (f)} (b+2 c x)}{2 \sqrt{c}}\right )}{e^2}-\frac{f^{a+b x+c x^2}}{e (d+e x)} \]

[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 = 0.103133, 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., Rules used = {} \[ \int \frac{f^{a+b x+c x^2}}{(d+e x)^2} \, dx \]

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

Rubi steps

\begin{align*} \int \frac{f^{a+b x+c x^2}}{(d+e x)^2} \, dx &=-\frac{f^{a+b x+c x^2}}{e (d+e x)}+\frac{(2 c \log (f)) \int f^{a+b x+c x^2} \, dx}{e^2}-\frac{((2 c d-b e) \log (f)) \int \frac{f^{a+b x+c x^2}}{d+e x} \, dx}{e^2}\\ &=-\frac{f^{a+b x+c x^2}}{e (d+e x)}-\frac{((2 c d-b e) \log (f)) \int \frac{f^{a+b x+c x^2}}{d+e x} \, dx}{e^2}+\frac{\left (2 c f^{a-\frac{b^2}{4 c}} \log (f)\right ) \int f^{\frac{(b+2 c x)^2}{4 c}} \, dx}{e^2}\\ &=-\frac{f^{a+b x+c x^2}}{e (d+e x)}+\frac{\sqrt{c} f^{a-\frac{b^2}{4 c}} \sqrt{\pi } \text{erfi}\left (\frac{(b+2 c x) \sqrt{\log (f)}}{2 \sqrt{c}}\right ) \sqrt{\log (f)}}{e^2}-\frac{((2 c d-b e) \log (f)) \int \frac{f^{a+b x+c x^2}}{d+e x} \, dx}{e^2}\\ \end{align*}

Mathematica [A]  time = 0.583072, 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 = 0.04, size = 0, normalized size = 0. \begin{align*} \int{\frac{{f}^{c{x}^{2}+bx+a}}{ \left ( ex+d \right ) ^{2}}}\, dx \end{align*}

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. \begin{align*} \int \frac{f^{c x^{2} + b x + a}}{{\left (e x + d\right )}^{2}}\,{d x} \end{align*}

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. \begin{align*}{\rm integral}\left (\frac{f^{c x^{2} + b x + a}}{e^{2} x^{2} + 2 \, d e x + d^{2}}, x\right ) \end{align*}

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. \begin{align*} \int \frac{f^{a + b x + c x^{2}}}{\left (d + e x\right )^{2}}\, dx \end{align*}

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

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)