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^(c*x^2+b*x+a)/e/(e*x+d)+f^(a-1/4/c*b^2)*erfi(1/2*(2*c*x+b)*ln(f)^(1/2)/c^(1/2))*c^(1/2)*Pi^(1/2)*ln(f)^(1/2
)/e^2-(-b*e+2*c*d)*ln(f)*Unintegrable(f^(c*x^2+b*x+a)/(e*x+d),x)/e^2

________________________________________________________________________________________

Rubi [A]  time = 0.10, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, 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 = 0.59, size = 0, normalized size = 0.00 \[ \int \frac {f^{a+b x+c x^2}}{(d+e x)^2} \, dx \]

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]

________________________________________________________________________________________

fricas [A]  time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {f^{c x^{2} + b x + a}}{e^{2} x^{2} + 2 \, d e x + d^{2}}, x\right ) \]

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)

________________________________________________________________________________________

giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {f^{c x^{2} + b x + a}}{{\left (e x + d\right )}^{2}}\,{d x} \]

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)

________________________________________________________________________________________

maple [A]  time = 0.10, size = 0, normalized size = 0.00 \[ \int \frac {f^{c \,x^{2}+b x +a}}{\left (e x +d \right )^{2}}\, dx \]

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)

________________________________________________________________________________________

maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {f^{c x^{2} + b x + a}}{{\left (e x + d\right )}^{2}}\,{d x} \]

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)

________________________________________________________________________________________

mupad [A]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {f^{c\,x^2+b\,x+a}}{{\left (d+e\,x\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {f^{a + b x + c x^{2}}}{\left (d + e x\right )^{2}}\, dx \]

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)

________________________________________________________________________________________