\(\int \frac {f^{a+b x+c x^2}}{(d+e x)^2} \, dx\) [448]

Optimal result

Integrand size = 20, antiderivative size = 20 \[ \int \frac {f^{a+b x+c x^2}}{(d+e x)^2} \, dx=-\frac {f^{a+b x+c x^2}}{e (d+e x)}+\frac {\sqrt {c} f^{a-\frac {b^2}{4 c}} \sqrt {\pi } \text {erfi}\left (\frac {(b+2 c x) \sqrt {\log (f)}}{2 \sqrt {c}}\right ) \sqrt {\log (f)}}{e^2}-\frac {(2 c d-b e) \log (f) \text {Int}\left (\frac {f^{a+b x+c x^2}}{d+e x},x\right )}{e^2} \]

[Out]

-f^(c*x^2+b*x+a)/e/(e*x+d)+f^(a-1/4*b^2/c)*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 [N/A]

Not integrable

Time = 0.08 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {f^{a+b x+c x^2}}{(d+e x)^2} \, dx=\int \frac {f^{a+b x+c x^2}}{(d+e x)^2} \, dx \]

[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*} \text {integral}& = -\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 [N/A]

Not integrable

Time = 0.70 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {f^{a+b x+c x^2}}{(d+e x)^2} \, dx=\int \frac {f^{a+b x+c x^2}}{(d+e x)^2} \, dx \]

[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]

Maple [N/A]

Not integrable

Time = 0.03 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00

\[\int \frac {f^{c \,x^{2}+b x +a}}{\left (e x +d \right )^{2}}d x\]

[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)

Fricas [N/A]

Not integrable

Time = 0.29 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.65 \[ \int \frac {f^{a+b x+c x^2}}{(d+e x)^2} \, dx=\int { \frac {f^{c x^{2} + b x + a}}{{\left (e x + d\right )}^{2}} \,d x } \]

[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)

Sympy [N/A]

Not integrable

Time = 0.54 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.95 \[ \int \frac {f^{a+b x+c x^2}}{(d+e x)^2} \, dx=\int \frac {f^{a + b x + c x^{2}}}{\left (d + e x\right )^{2}}\, dx \]

[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)

Maxima [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {f^{a+b x+c x^2}}{(d+e x)^2} \, dx=\int { \frac {f^{c x^{2} + b x + a}}{{\left (e x + d\right )}^{2}} \,d x } \]

[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)

Giac [N/A]

Not integrable

Time = 0.44 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {f^{a+b x+c x^2}}{(d+e x)^2} \, dx=\int { \frac {f^{c x^{2} + b x + a}}{{\left (e x + d\right )}^{2}} \,d x } \]

[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)

Mupad [N/A]

Not integrable

Time = 1.38 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {f^{a+b x+c x^2}}{(d+e x)^2} \, dx=\int \frac {f^{c\,x^2+b\,x+a}}{{\left (d+e\,x\right )}^2} \,d x \]

[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)