\(\int \frac {f^{a+b x+c x^2}}{x^2} \, dx\) [365]

Optimal result

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

-f^(c*x^2+b*x+a)/x+c^(1/2)*f^(a-1/4*b^2/c)*Pi^(1/2)*erfi(1/2*(2*c*x+b)*ln( 
f)^(1/2)/c^(1/2))*ln(f)^(1/2)+b*ln(f)*Defer(Int)(f^(c*x^2+b*x+a)/x,x)
 

Mathematica [N/A]

Not integrable

Time = 0.58 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int \frac {f^{a+b x+c x^2}}{x^2} \, dx=\int \frac {f^{a+b x+c x^2}}{x^2} \, dx \] Input:

Integrate[f^(a + b*x + c*x^2)/x^2,x]
 

Output:

Integrate[f^(a + b*x + c*x^2)/x^2, x]
 

Rubi [N/A]

Not integrable

Time = 0.57 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {2672, 2664, 2633, 2673}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 2672

\(\displaystyle 2 c \log (f) \int f^{c x^2+b x+a}dx+b \log (f) \int \frac {f^{c x^2+b x+a}}{x}dx-\frac {f^{a+b x+c x^2}}{x}\)

\(\Big \downarrow \) 2664

\(\displaystyle 2 c \log (f) f^{a-\frac {b^2}{4 c}} \int f^{\frac {(b+2 c x)^2}{4 c}}dx+b \log (f) \int \frac {f^{c x^2+b x+a}}{x}dx-\frac {f^{a+b x+c x^2}}{x}\)

\(\Big \downarrow \) 2633

\(\displaystyle b \log (f) \int \frac {f^{c x^2+b x+a}}{x}dx+\sqrt {\pi } \sqrt {c} \sqrt {\log (f)} f^{a-\frac {b^2}{4 c}} \text {erfi}\left (\frac {\sqrt {\log (f)} (b+2 c x)}{2 \sqrt {c}}\right )-\frac {f^{a+b x+c x^2}}{x}\)

\(\Big \downarrow \) 2673

\(\displaystyle b \log (f) \int \frac {f^{c x^2+b x+a}}{x}dx+\sqrt {\pi } \sqrt {c} \sqrt {\log (f)} f^{a-\frac {b^2}{4 c}} \text {erfi}\left (\frac {\sqrt {\log (f)} (b+2 c x)}{2 \sqrt {c}}\right )-\frac {f^{a+b x+c x^2}}{x}\)

Input:

Int[f^(a + b*x + c*x^2)/x^2,x]
 

Output:

$Aborted
 

Defintions of rubi rules used

rule 2633

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt 
[Pi]*(Erfi[(c + d*x)*Rt[b*Log[F], 2]]/(2*d*Rt[b*Log[F], 2])), x] /; FreeQ[{ 
F, a, b, c, d}, x] && PosQ[b]
 

rule 2664

Int[(F_)^((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[F^(a - b^2/ 
(4*c))   Int[F^((b + 2*c*x)^2/(4*c)), x], x] /; FreeQ[{F, a, b, c}, x]
 

rule 2672

Int[(F_)^((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)*((d_.) + (e_.)*(x_))^(m_), x_S 
ymbol] :> Simp[(d + e*x)^(m + 1)*(F^(a + b*x + c*x^2)/(e*(m + 1))), x] + (- 
Simp[2*c*(Log[F]/(e^2*(m + 1)))   Int[(d + e*x)^(m + 2)*F^(a + b*x + c*x^2) 
, x], x] - Simp[(b*e - 2*c*d)*(Log[F]/(e^2*(m + 1)))   Int[(d + e*x)^(m + 1 
)*F^(a + b*x + c*x^2), x], x]) /; FreeQ[{F, a, b, c, d, e}, x] && NeQ[b*e - 
 2*c*d, 0] && LtQ[m, -1]
 

rule 2673

Int[(F_)^((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)*((d_.) + (e_.)*(x_))^(m_.), x_ 
Symbol] :> Unintegrable[F^(a + b*x + c*x^2)*(d + e*x)^m, x] /; FreeQ[{F, a, 
 b, c, d, e, m}, x]
 

Maple [N/A]

Not integrable

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

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

Input:

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

Output:

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

Fricas [N/A]

Not integrable

Time = 0.08 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int \frac {f^{a+b x+c x^2}}{x^2} \, dx=\int { \frac {f^{c x^{2} + b x + a}}{x^{2}} \,d x } \] Input:

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

Output:

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

Sympy [N/A]

Not integrable

Time = 0.31 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.94 \[ \int \frac {f^{a+b x+c x^2}}{x^2} \, dx=\int \frac {f^{a + b x + c x^{2}}}{x^{2}}\, dx \] Input:

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

Output:

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

Maxima [N/A]

Not integrable

Time = 0.09 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int \frac {f^{a+b x+c x^2}}{x^2} \, dx=\int { \frac {f^{c x^{2} + b x + a}}{x^{2}} \,d x } \] Input:

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

Output:

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

Giac [N/A]

Not integrable

Time = 0.13 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int \frac {f^{a+b x+c x^2}}{x^2} \, dx=\int { \frac {f^{c x^{2} + b x + a}}{x^{2}} \,d x } \] Input:

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

Output:

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

Mupad [N/A]

Not integrable

Time = 0.16 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int \frac {f^{a+b x+c x^2}}{x^2} \, dx=\int \frac {f^{c\,x^2+b\,x+a}}{x^2} \,d x \] Input:

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

Output:

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

Reduce [N/A]

Not integrable

Time = 0.21 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.31 \[ \int \frac {f^{a+b x+c x^2}}{x^2} \, dx=f^{a} \left (\int \frac {f^{c \,x^{2}+b x}}{x^{2}}d x \right ) \] Input:

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

Output:

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