\(\int \frac {(f+g x)^m}{(a+b \log (c (d+e x)^n))^2} \, dx\) [164]

Optimal result

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

[Out]

Unintegrable((g*x+f)^m/(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 = {} \[ \int \frac {(f+g x)^m}{\left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx=\int \frac {(f+g x)^m}{\left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 5.81 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {(f+g x)^m}{\left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx=\int \frac {(f+g x)^m}{\left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A]

Not integrable

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

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.30 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.92 \[ \int \frac {(f+g x)^m}{\left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx=\int { \frac {{\left (g x + f\right )}^{m}}{{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Sympy [F(-2)]

Exception generated. \[ \int \frac {(f+g x)^m}{\left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx=\text {Exception raised: HeuristicGCDFailed} \]

[In]

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

[Out]

Exception raised: HeuristicGCDFailed >> no luck

Maxima [N/A]

Not integrable

Time = 0.31 (sec) , antiderivative size = 131, normalized size of antiderivative = 5.46 \[ \int \frac {(f+g x)^m}{\left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx=\int { \frac {{\left (g x + f\right )}^{m}}{{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

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

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 1.31 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {(f+g x)^m}{\left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx=\int \frac {{\left (f+g\,x\right )}^m}{{\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^2} \,d x \]

[In]

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

[Out]

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