\(\int \frac {f^{\frac {c}{(a+b x)^2}}}{x^2} \, dx\) [230]

Optimal result

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

[Out]

CannotIntegrate(f^(c/(b*x+a)^2)/x^2,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 = {} \[ \int \frac {f^{\frac {c}{(a+b x)^2}}}{x^2} \, dx=\int \frac {f^{\frac {c}{(a+b x)^2}}}{x^2} \, dx \]

[In]

Int[f^(c/(a + b*x)^2)/x^2,x]

[Out]

Defer[Int][f^(c/(a + b*x)^2)/x^2, x]

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

Mathematica [N/A]

Not integrable

Time = 0.29 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.13 \[ \int \frac {f^{\frac {c}{(a+b x)^2}}}{x^2} \, dx=\int \frac {f^{\frac {c}{(a+b x)^2}}}{x^2} \, dx \]

[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

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

[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 = 0.25 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.87 \[ \int \frac {f^{\frac {c}{(a+b x)^2}}}{x^2} \, dx=\int { \frac {f^{\frac {c}{{\left (b x + a\right )}^{2}}}}{x^{2}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 1.63 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.93 \[ \int \frac {f^{\frac {c}{(a+b x)^2}}}{x^2} \, dx=\int \frac {f^{\frac {c}{\left (a + b x\right )^{2}}}}{x^{2}}\, dx \]

[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 = 0.25 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.13 \[ \int \frac {f^{\frac {c}{(a+b x)^2}}}{x^2} \, dx=\int { \frac {f^{\frac {c}{{\left (b x + a\right )}^{2}}}}{x^{2}} \,d x } \]

[In]

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

[Out]

integrate(f^(c/(b*x + a)^2)/x^2, x)

Giac [N/A]

Not integrable

Time = 0.36 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.13 \[ \int \frac {f^{\frac {c}{(a+b x)^2}}}{x^2} \, dx=\int { \frac {f^{\frac {c}{{\left (b x + a\right )}^{2}}}}{x^{2}} \,d x } \]

[In]

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

[Out]

integrate(f^(c/(b*x + a)^2)/x^2, x)

Mupad [N/A]

Not integrable

Time = 0.24 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.13 \[ \int \frac {f^{\frac {c}{(a+b x)^2}}}{x^2} \, dx=\int \frac {f^{\frac {c}{{\left (a+b\,x\right )}^2}}}{x^2} \,d x \]

[In]

int(f^(c/(a + b*x)^2)/x^2,x)

[Out]

int(f^(c/(a + b*x)^2)/x^2, x)