\(\int \frac {F^{c+d x}}{(a+b F^{c+d x})^2 x^2} \, dx\) [87]

Optimal result

Integrand size = 24, antiderivative size = 24 \[ \int \frac {F^{c+d x}}{\left (a+b F^{c+d x}\right )^2 x^2} \, dx=-\frac {1}{b d \left (a+b F^{c+d x}\right ) x^2 \log (F)}-\frac {2 \text {Int}\left (\frac {1}{\left (a+b F^{c+d x}\right ) x^3},x\right )}{b d \log (F)} \]

[Out]

-1/b/d/(a+b*F^(d*x+c))/x^2/ln(F)-2*Unintegrable(1/(a+b*F^(d*x+c))/x^3,x)/b/d/ln(F)

Rubi [N/A]

Not integrable

Time = 0.07 (sec) , antiderivative size = 24, 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 = {} \[ \int \frac {F^{c+d x}}{\left (a+b F^{c+d x}\right )^2 x^2} \, dx=\int \frac {F^{c+d x}}{\left (a+b F^{c+d x}\right )^2 x^2} \, dx \]

[In]

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

[Out]

-(1/(b*d*(a + b*F^(c + d*x))*x^2*Log[F])) - (2*Defer[Int][1/((a + b*F^(c + d*x))*x^3), x])/(b*d*Log[F])

Rubi steps \begin{align*} \text {integral}& = -\frac {1}{b d \left (a+b F^{c+d x}\right ) x^2 \log (F)}-\frac {2 \int \frac {1}{\left (a+b F^{c+d x}\right ) x^3} \, dx}{b d \log (F)} \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 0.31 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {F^{c+d x}}{\left (a+b F^{c+d x}\right )^2 x^2} \, dx=\int \frac {F^{c+d x}}{\left (a+b F^{c+d x}\right )^2 x^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A]

Not integrable

Time = 0.01 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 51, normalized size of antiderivative = 2.12 \[ \int \frac {F^{c+d x}}{\left (a+b F^{c+d x}\right )^2 x^2} \, dx=\int { \frac {F^{d x + c}}{{\left (F^{d x + c} b + a\right )}^{2} x^{2}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 0.81 (sec) , antiderivative size = 70, normalized size of antiderivative = 2.92 \[ \int \frac {F^{c+d x}}{\left (a+b F^{c+d x}\right )^2 x^2} \, dx=- \frac {1}{F^{c + d x} b^{2} d x^{2} \log {\left (F \right )} + a b d x^{2} \log {\left (F \right )}} - \frac {2 \int \frac {1}{a x^{3} + b x^{3} e^{c \log {\left (F \right )}} e^{d x \log {\left (F \right )}}}\, dx}{b d \log {\left (F \right )}} \]

[In]

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

[Out]

-1/(F**(c + d*x)*b**2*d*x**2*log(F) + a*b*d*x**2*log(F)) - 2*Integral(1/(a*x**3 + b*x**3*exp(c*log(F))*exp(d*x
*log(F))), x)/(b*d*log(F))

Maxima [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 67, normalized size of antiderivative = 2.79 \[ \int \frac {F^{c+d x}}{\left (a+b F^{c+d x}\right )^2 x^2} \, dx=\int { \frac {F^{d x + c}}{{\left (F^{d x + c} b + a\right )}^{2} x^{2}} \,d x } \]

[In]

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

[Out]

-1/(F^(d*x)*F^c*b^2*d*x^2*log(F) + a*b*d*x^2*log(F)) - 2*integrate(1/(F^(d*x)*F^c*b^2*d*x^3*log(F) + a*b*d*x^3
*log(F)), x)

Giac [N/A]

Not integrable

Time = 0.42 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {F^{c+d x}}{\left (a+b F^{c+d x}\right )^2 x^2} \, dx=\int { \frac {F^{d x + c}}{{\left (F^{d x + c} b + a\right )}^{2} x^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 0.15 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {F^{c+d x}}{\left (a+b F^{c+d x}\right )^2 x^2} \, dx=\int \frac {F^{c+d\,x}}{x^2\,{\left (a+F^{c+d\,x}\,b\right )}^2} \,d x \]

[In]

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

[Out]

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