\(\int \frac {f^{a+b x+c x^2}}{x^2} \, dx\) [431]

Optimal result

Integrand size = 16, antiderivative size = 16 \[ \int \frac {f^{a+b x+c x^2}}{x^2} \, dx=-\frac {f^{a+b x+c x^2}}{x}+\sqrt {c} f^{a-\frac {b^2}{4 c}} \sqrt {\pi } \text {erfi}\left (\frac {(b+2 c x) \sqrt {\log (f)}}{2 \sqrt {c}}\right ) \sqrt {\log (f)}+b \log (f) \text {Int}\left (\frac {f^{a+b x+c x^2}}{x},x\right ) \]

[Out]

-f^(c*x^2+b*x+a)/x+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)+b*ln(f
)*Unintegrable(f^(c*x^2+b*x+a)/x,x)

Rubi [N/A]

Not integrable

Time = 0.05 (sec) , antiderivative size = 16, 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 = {} \[ \int \frac {f^{a+b x+c x^2}}{x^2} \, dx=\int \frac {f^{a+b x+c x^2}}{x^2} \, dx \]

[In]

Int[f^(a + b*x + c*x^2)/x^2,x]

[Out]

-(f^(a + b*x + c*x^2)/x) + Sqrt[c]*f^(a - b^2/(4*c))*Sqrt[Pi]*Erfi[((b + 2*c*x)*Sqrt[Log[f]])/(2*Sqrt[c])]*Sqr
t[Log[f]] + b*Log[f]*Defer[Int][f^(a + b*x + c*x^2)/x, x]

Rubi steps \begin{align*} \text {integral}& = -\frac {f^{a+b x+c x^2}}{x}+(b \log (f)) \int \frac {f^{a+b x+c x^2}}{x} \, dx+(2 c \log (f)) \int f^{a+b x+c x^2} \, dx \\ & = -\frac {f^{a+b x+c x^2}}{x}+(b \log (f)) \int \frac {f^{a+b x+c x^2}}{x} \, dx+\left (2 c f^{a-\frac {b^2}{4 c}} \log (f)\right ) \int f^{\frac {(b+2 c x)^2}{4 c}} \, dx \\ & = -\frac {f^{a+b x+c x^2}}{x}+\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)}+(b \log (f)) \int \frac {f^{a+b x+c x^2}}{x} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 0.34 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int \frac {f^{a+b x+c x^2}}{x^2} \, dx=\int \frac {f^{a+b x+c x^2}}{x^2} \, dx \]

[In]

Integrate[f^(a + b*x + c*x^2)/x^2,x]

[Out]

Integrate[f^(a + b*x + c*x^2)/x^2, x]

Maple [N/A]

Not integrable

Time = 0.01 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00

\[\int \frac {f^{c \,x^{2}+b x +a}}{x^{2}}d x\]

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.35 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int \frac {f^{a+b x+c x^2}}{x^2} \, dx=\int { \frac {f^{c x^{2} + b x + a}}{x^{2}} \,d x } \]

[In]

integrate(f^(c*x^2+b*x+a)/x^2,x, algorithm="fricas")

[Out]

integral(f^(c*x^2 + b*x + a)/x^2, x)

Sympy [N/A]

Not integrable

Time = 0.29 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.94 \[ \int \frac {f^{a+b x+c x^2}}{x^2} \, dx=\int \frac {f^{a + b x + c x^{2}}}{x^{2}}\, dx \]

[In]

integrate(f**(c*x**2+b*x+a)/x**2,x)

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int \frac {f^{a+b x+c x^2}}{x^2} \, dx=\int { \frac {f^{c x^{2} + b x + a}}{x^{2}} \,d x } \]

[In]

integrate(f^(c*x^2+b*x+a)/x^2,x, algorithm="maxima")

[Out]

integrate(f^(c*x^2 + b*x + a)/x^2, x)

Giac [N/A]

Not integrable

Time = 0.38 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int \frac {f^{a+b x+c x^2}}{x^2} \, dx=\int { \frac {f^{c x^{2} + b x + a}}{x^{2}} \,d x } \]

[In]

integrate(f^(c*x^2+b*x+a)/x^2,x, algorithm="giac")

[Out]

integrate(f^(c*x^2 + b*x + a)/x^2, x)

Mupad [N/A]

Not integrable

Time = 0.31 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int \frac {f^{a+b x+c x^2}}{x^2} \, dx=\int \frac {f^{c\,x^2+b\,x+a}}{x^2} \,d x \]

[In]

int(f^(a + b*x + c*x^2)/x^2,x)

[Out]

int(f^(a + b*x + c*x^2)/x^2, x)