\(\int \frac {F^{c+d x}}{(a+b F^{c+d x})^3 x^2} \, dx\) [93]

Optimal result

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

[Out]

-1/2/b/d/(a+b*F^(d*x+c))^2/x^2/ln(F)-Unintegrable(1/(a+b*F^(d*x+c))^2/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 = {} \[ \int \frac {F^{c+d x}}{\left (a+b F^{c+d x}\right )^3 x^2} \, dx=\int \frac {F^{c+d x}}{\left (a+b F^{c+d x}\right )^3 x^2} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 0.90 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {F^{c+d x}}{\left (a+b F^{c+d x}\right )^3 x^2} \, dx=\int \frac {F^{c+d x}}{\left (a+b F^{c+d x}\right )^3 x^2} \, dx \]

[In]

Integrate[F^(c + d*x)/((a + b*F^(c + d*x))^3*x^2),x]

[Out]

Integrate[F^(c + d*x)/((a + b*F^(c + d*x))^3*x^2), x]

Maple [N/A]

Not integrable

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

\[\int \frac {F^{d x +c}}{\left (a +b \,F^{d x +c}\right )^{3} x^{2}}d x\]

[In]

int(F^(d*x+c)/(a+b*F^(d*x+c))^3/x^2,x)

[Out]

int(F^(d*x+c)/(a+b*F^(d*x+c))^3/x^2,x)

Fricas [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 72, normalized size of antiderivative = 3.00 \[ \int \frac {F^{c+d x}}{\left (a+b F^{c+d x}\right )^3 x^2} \, dx=\int { \frac {F^{d x + c}}{{\left (F^{d x + c} b + a\right )}^{3} x^{2}} \,d x } \]

[In]

integrate(F^(d*x+c)/(a+b*F^(d*x+c))^3/x^2,x, algorithm="fricas")

[Out]

integral(F^(d*x + c)/(3*F^(d*x + c)*a^2*b*x^2 + 3*F^(2*d*x + 2*c)*a*b^2*x^2 + F^(3*d*x + 3*c)*b^3*x^2 + a^3*x^
2), x)

Sympy [N/A]

Not integrable

Time = 2.84 (sec) , antiderivative size = 172, normalized size of antiderivative = 7.17 \[ \int \frac {F^{c+d x}}{\left (a+b F^{c+d x}\right )^3 x^2} \, dx=\frac {- 2 F^{c + d x} b - a d x \log {\left (F \right )} - 2 a}{4 F^{c + d x} a^{2} b^{2} d^{2} x^{3} \log {\left (F \right )}^{2} + 2 F^{2 c + 2 d x} a b^{3} d^{2} x^{3} \log {\left (F \right )}^{2} + 2 a^{3} b d^{2} x^{3} \log {\left (F \right )}^{2}} - \frac {\int \frac {d x \log {\left (F \right )}}{a x^{4} + b x^{4} e^{c \log {\left (F \right )}} e^{d x \log {\left (F \right )}}}\, dx + \int \frac {3}{a x^{4} + b x^{4} e^{c \log {\left (F \right )}} e^{d x \log {\left (F \right )}}}\, dx}{a b d^{2} \log {\left (F \right )}^{2}} \]

[In]

integrate(F**(d*x+c)/(a+b*F**(d*x+c))**3/x**2,x)

[Out]

(-2*F**(c + d*x)*b - a*d*x*log(F) - 2*a)/(4*F**(c + d*x)*a**2*b**2*d**2*x**3*log(F)**2 + 2*F**(2*c + 2*d*x)*a*
b**3*d**2*x**3*log(F)**2 + 2*a**3*b*d**2*x**3*log(F)**2) - (Integral(d*x*log(F)/(a*x**4 + b*x**4*exp(c*log(F))
*exp(d*x*log(F))), x) + Integral(3/(a*x**4 + b*x**4*exp(c*log(F))*exp(d*x*log(F))), x))/(a*b*d**2*log(F)**2)

Maxima [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 147, normalized size of antiderivative = 6.12 \[ \int \frac {F^{c+d x}}{\left (a+b F^{c+d x}\right )^3 x^2} \, dx=\int { \frac {F^{d x + c}}{{\left (F^{d x + c} b + a\right )}^{3} x^{2}} \,d x } \]

[In]

integrate(F^(d*x+c)/(a+b*F^(d*x+c))^3/x^2,x, algorithm="maxima")

[Out]

-1/2*(a*d*x*log(F) + 2*F^(d*x)*F^c*b + 2*a)/(2*F^(d*x)*F^c*a^2*b^2*d^2*x^3*log(F)^2 + F^(2*d*x)*F^(2*c)*a*b^3*
d^2*x^3*log(F)^2 + a^3*b*d^2*x^3*log(F)^2) - integrate((d*x*log(F) + 3)/(F^(d*x)*F^c*a*b^2*d^2*x^4*log(F)^2 +
a^2*b*d^2*x^4*log(F)^2), x)

Giac [N/A]

Not integrable

Time = 0.42 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {F^{c+d x}}{\left (a+b F^{c+d x}\right )^3 x^2} \, dx=\int { \frac {F^{d x + c}}{{\left (F^{d x + c} b + a\right )}^{3} x^{2}} \,d x } \]

[In]

integrate(F^(d*x+c)/(a+b*F^(d*x+c))^3/x^2,x, algorithm="giac")

[Out]

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

Mupad [N/A]

Not integrable

Time = 0.17 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {F^{c+d x}}{\left (a+b F^{c+d x}\right )^3 x^2} \, dx=\int \frac {F^{c+d\,x}}{x^2\,{\left (a+F^{c+d\,x}\,b\right )}^3} \,d x \]

[In]

int(F^(c + d*x)/(x^2*(a + F^(c + d*x)*b)^3),x)

[Out]

int(F^(c + d*x)/(x^2*(a + F^(c + d*x)*b)^3), x)