\(\int \frac {1}{(f+g x)^2 (a+b \log (c (d+e x)^n))^2} \, dx\) [99]

Optimal result

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

[Out]

Unintegrable(1/(g*x+f)^2/(a+b*ln(c*(e*x+d)^n))^2,x)

Rubi [N/A]

Not integrable

Time = 0.02 (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 = {} \[ \int \frac {1}{(f+g x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx=\int \frac {1}{(f+g x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx \]

[In]

Int[1/((f + g*x)^2*(a + b*Log[c*(d + e*x)^n])^2),x]

[Out]

Defer[Int][1/((f + g*x)^2*(a + b*Log[c*(d + e*x)^n])^2), x]

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

Mathematica [N/A]

Not integrable

Time = 2.11 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {1}{(f+g x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx=\int \frac {1}{(f+g x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx \]

[In]

Integrate[1/((f + g*x)^2*(a + b*Log[c*(d + e*x)^n])^2),x]

[Out]

Integrate[1/((f + g*x)^2*(a + b*Log[c*(d + e*x)^n])^2), x]

Maple [N/A]

Not integrable

Time = 0.03 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00

\[\int \frac {1}{\left (g x +f \right )^{2} {\left (a +b \ln \left (c \left (e x +d \right )^{n}\right )\right )}^{2}}d x\]

[In]

int(1/(g*x+f)^2/(a+b*ln(c*(e*x+d)^n))^2,x)

[Out]

int(1/(g*x+f)^2/(a+b*ln(c*(e*x+d)^n))^2,x)

Fricas [N/A]

Not integrable

Time = 0.29 (sec) , antiderivative size = 104, normalized size of antiderivative = 4.33 \[ \int \frac {1}{(f+g x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx=\int { \frac {1}{{\left (g x + f\right )}^{2} {\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2}} \,d x } \]

[In]

integrate(1/(g*x+f)^2/(a+b*log(c*(e*x+d)^n))^2,x, algorithm="fricas")

[Out]

integral(1/(a^2*g^2*x^2 + 2*a^2*f*g*x + a^2*f^2 + (b^2*g^2*x^2 + 2*b^2*f*g*x + b^2*f^2)*log((e*x + d)^n*c)^2 +
2*(a*b*g^2*x^2 + 2*a*b*f*g*x + a*b*f^2)*log((e*x + d)^n*c)), x)

Sympy [N/A]

Not integrable

Time = 11.14 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {1}{(f+g x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx=\int \frac {1}{\left (a + b \log {\left (c \left (d + e x\right )^{n} \right )}\right )^{2} \left (f + g x\right )^{2}}\, dx \]

[In]

integrate(1/(g*x+f)**2/(a+b*ln(c*(e*x+d)**n))**2,x)

[Out]

Integral(1/((a + b*log(c*(d + e*x)**n))**2*(f + g*x)**2), x)

Maxima [N/A]

Not integrable

Time = 0.29 (sec) , antiderivative size = 289, normalized size of antiderivative = 12.04 \[ \int \frac {1}{(f+g x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx=\int { \frac {1}{{\left (g x + f\right )}^{2} {\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2}} \,d x } \]

[In]

integrate(1/(g*x+f)^2/(a+b*log(c*(e*x+d)^n))^2,x, algorithm="maxima")

[Out]

-(e*x + d)/(b^2*e*f^2*n*log(c) + a*b*e*f^2*n + (b^2*e*g^2*n*log(c) + a*b*e*g^2*n)*x^2 + 2*(b^2*e*f*g*n*log(c)
+ a*b*e*f*g*n)*x + (b^2*e*g^2*n*x^2 + 2*b^2*e*f*g*n*x + b^2*e*f^2*n)*log((e*x + d)^n)) - integrate((e*g*x - e*
f + 2*d*g)/(b^2*e*f^3*n*log(c) + a*b*e*f^3*n + (b^2*e*g^3*n*log(c) + a*b*e*g^3*n)*x^3 + 3*(b^2*e*f*g^2*n*log(c
) + a*b*e*f*g^2*n)*x^2 + 3*(b^2*e*f^2*g*n*log(c) + a*b*e*f^2*g*n)*x + (b^2*e*g^3*n*x^3 + 3*b^2*e*f*g^2*n*x^2 +
3*b^2*e*f^2*g*n*x + b^2*e*f^3*n)*log((e*x + d)^n)), x)

Giac [N/A]

Not integrable

Time = 0.34 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {1}{(f+g x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx=\int { \frac {1}{{\left (g x + f\right )}^{2} {\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2}} \,d x } \]

[In]

integrate(1/(g*x+f)^2/(a+b*log(c*(e*x+d)^n))^2,x, algorithm="giac")

[Out]

integrate(1/((g*x + f)^2*(b*log((e*x + d)^n*c) + a)^2), x)

Mupad [N/A]

Not integrable

Time = 1.30 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {1}{(f+g x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx=\int \frac {1}{{\left (f+g\,x\right )}^2\,{\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^2} \,d x \]

[In]

int(1/((f + g*x)^2*(a + b*log(c*(d + e*x)^n))^2),x)

[Out]

int(1/((f + g*x)^2*(a + b*log(c*(d + e*x)^n))^2), x)