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

Optimal. Leaf size=205 \[ \frac{\log ^2(f) (2 c d-b e)^2 \text{Unintegrable}\left (\frac{f^{a+b x+c x^2}}{d+e x},x\right )}{2 e^4}+\frac{c \log (f) \text{Unintegrable}\left (\frac{f^{a+b x+c x^2}}{d+e x},x\right )}{e^2}-\frac{\sqrt{\pi } \sqrt{c} \log ^{\frac{3}{2}}(f) f^{a-\frac{b^2}{4 c}} (2 c d-b e) \text{Erfi}\left (\frac{\sqrt{\log (f)} (b+2 c x)}{2 \sqrt{c}}\right )}{2 e^4}+\frac{\log (f) (2 c d-b e) f^{a+b x+c x^2}}{2 e^3 (d+e x)}-\frac{f^{a+b x+c x^2}}{2 e (d+e x)^2} \]

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

________________________________________________________________________________________

Rubi [A]  time = 0.208053, 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 = {} \[ \int \frac{f^{a+b x+c x^2}}{(d+e x)^3} \, dx \]

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[Int][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)

Rubi steps

\begin{align*} \int \frac{f^{a+b x+c x^2}}{(d+e x)^3} \, dx &=-\frac{f^{a+b x+c x^2}}{2 e (d+e x)^2}+\frac{(c \log (f)) \int \frac{f^{a+b x+c x^2}}{d+e x} \, dx}{e^2}-\frac{((2 c d-b e) \log (f)) \int \frac{f^{a+b x+c x^2}}{(d+e x)^2} \, dx}{2 e^2}\\ &=-\frac{f^{a+b x+c x^2}}{2 e (d+e x)^2}+\frac{(2 c d-b e) f^{a+b x+c x^2} \log (f)}{2 e^3 (d+e x)}+\frac{(c \log (f)) \int \frac{f^{a+b x+c x^2}}{d+e x} \, dx}{e^2}-\frac{\left (c (2 c d-b e) \log ^2(f)\right ) \int f^{a+b x+c x^2} \, dx}{e^4}+\frac{\left ((2 c d-b e)^2 \log ^2(f)\right ) \int \frac{f^{a+b x+c x^2}}{d+e x} \, dx}{2 e^4}\\ &=-\frac{f^{a+b x+c x^2}}{2 e (d+e x)^2}+\frac{(2 c d-b e) f^{a+b x+c x^2} \log (f)}{2 e^3 (d+e x)}+\frac{(c \log (f)) \int \frac{f^{a+b x+c x^2}}{d+e x} \, dx}{e^2}+\frac{\left ((2 c d-b e)^2 \log ^2(f)\right ) \int \frac{f^{a+b x+c x^2}}{d+e x} \, dx}{2 e^4}-\frac{\left (c (2 c d-b e) f^{a-\frac{b^2}{4 c}} \log ^2(f)\right ) \int f^{\frac{(b+2 c x)^2}{4 c}} \, dx}{e^4}\\ &=-\frac{f^{a+b x+c x^2}}{2 e (d+e x)^2}+\frac{(2 c d-b e) f^{a+b x+c x^2} \log (f)}{2 e^3 (d+e x)}-\frac{\sqrt{c} (2 c d-b e) f^{a-\frac{b^2}{4 c}} \sqrt{\pi } \text{erfi}\left (\frac{(b+2 c x) \sqrt{\log (f)}}{2 \sqrt{c}}\right ) \log ^{\frac{3}{2}}(f)}{2 e^4}+\frac{(c \log (f)) \int \frac{f^{a+b x+c x^2}}{d+e x} \, dx}{e^2}+\frac{\left ((2 c d-b e)^2 \log ^2(f)\right ) \int \frac{f^{a+b x+c x^2}}{d+e x} \, dx}{2 e^4}\\ \end{align*}

Mathematica [A]  time = 0.83303, 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]

________________________________________________________________________________________

Maple [A]  time = 0.049, size = 0, normalized size = 0. \begin{align*} \int{\frac{{f}^{c{x}^{2}+bx+a}}{ \left ( ex+d \right ) ^{3}}}\, 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)^3,x)

[Out]

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

________________________________________________________________________________________

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 )}^{3}}\,{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)^3,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\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 ) \end{align*}

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)

________________________________________________________________________________________

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 )^{3}}\, 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)**3,x)

[Out]

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

________________________________________________________________________________________

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 )}^{3}}\,{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)^3,x, algorithm="giac")

[Out]

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