\(\int \frac {f^{c (a+b x)^2}}{x^2} \, dx\) [200]

Optimal result

Integrand size = 15, antiderivative size = 15 \[ \int \frac {f^{c (a+b x)^2}}{x^2} \, dx=-\frac {f^{c (a+b x)^2}}{x}+b \sqrt {c} \sqrt {\pi } \text {erfi}\left (\sqrt {c} (a+b x) \sqrt {\log (f)}\right ) \sqrt {\log (f)}+2 a b c \log (f) \text {Int}\left (\frac {f^{c (a+b x)^2}}{x},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.03 (sec) , antiderivative size = 15, 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 = {} \[ \int \frac {f^{c (a+b x)^2}}{x^2} \, dx=\int \frac {f^{c (a+b x)^2}}{x^2} \, dx \]

[In]

Int[f^(c*(a + b*x)^2)/x^2,x]

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 0.50 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.13 \[ \int \frac {f^{c (a+b x)^2}}{x^2} \, dx=\int \frac {f^{c (a+b x)^2}}{x^2} \, dx \]

[In]

Integrate[f^(c*(a + b*x)^2)/x^2,x]

[Out]

Integrate[f^(c*(a + b*x)^2)/x^2, x]

Maple [N/A]

Not integrable

Time = 0.01 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.87 \[ \int \frac {f^{c (a+b x)^2}}{x^2} \, dx=\int { \frac {f^{{\left (b x + a\right )}^{2} c}}{x^{2}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 0.51 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.93 \[ \int \frac {f^{c (a+b x)^2}}{x^2} \, dx=\int \frac {f^{c \left (a + b x\right )^{2}}}{x^{2}}\, dx \]

[In]

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

[Out]

Integral(f**(c*(a + b*x)**2)/x**2, x)

Maxima [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.13 \[ \int \frac {f^{c (a+b x)^2}}{x^2} \, dx=\int { \frac {f^{{\left (b x + a\right )}^{2} c}}{x^{2}} \,d x } \]

[In]

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

[Out]

integrate(f^((b*x + a)^2*c)/x^2, x)

Giac [N/A]

Not integrable

Time = 0.33 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.13 \[ \int \frac {f^{c (a+b x)^2}}{x^2} \, dx=\int { \frac {f^{{\left (b x + a\right )}^{2} c}}{x^{2}} \,d x } \]

[In]

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

[Out]

integrate(f^((b*x + a)^2*c)/x^2, x)

Mupad [N/A]

Not integrable

Time = 0.15 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.13 \[ \int \frac {f^{c (a+b x)^2}}{x^2} \, dx=\int \frac {f^{c\,{\left (a+b\,x\right )}^2}}{x^2} \,d x \]

[In]

int(f^(c*(a + b*x)^2)/x^2,x)

[Out]

int(f^(c*(a + b*x)^2)/x^2, x)