\(\int \frac {1}{(f+g x) (a+b \log (c (d+e x)^n))^3} \, dx\) [103]

Optimal result

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

[Out]

Unintegrable(1/(g*x+f)/(a+b*ln(c*(e*x+d)^n))^3,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 = {} \[ \int \frac {1}{(f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3} \, dx=\int \frac {1}{(f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3} \, dx \]

[In]

Int[1/((f + g*x)*(a + b*Log[c*(d + e*x)^n])^3),x]

[Out]

Defer[Int][1/((f + g*x)*(a + b*Log[c*(d + e*x)^n])^3), x]

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

Mathematica [N/A]

Not integrable

Time = 0.34 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {1}{(f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3} \, dx=\int \frac {1}{(f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3} \, dx \]

[In]

Integrate[1/((f + g*x)*(a + b*Log[c*(d + e*x)^n])^3),x]

[Out]

Integrate[1/((f + g*x)*(a + b*Log[c*(d + e*x)^n])^3), x]

Maple [N/A]

Not integrable

Time = 0.02 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00

\[\int \frac {1}{\left (g x +f \right ) {\left (a +b \ln \left (c \left (e x +d \right )^{n}\right )\right )}^{3}}d x\]

[In]

int(1/(g*x+f)/(a+b*ln(c*(e*x+d)^n))^3,x)

[Out]

int(1/(g*x+f)/(a+b*ln(c*(e*x+d)^n))^3,x)

Fricas [N/A]

Not integrable

Time = 0.30 (sec) , antiderivative size = 95, normalized size of antiderivative = 3.96 \[ \int \frac {1}{(f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3} \, dx=\int { \frac {1}{{\left (g x + f\right )} {\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{3}} \,d x } \]

[In]

integrate(1/(g*x+f)/(a+b*log(c*(e*x+d)^n))^3,x, algorithm="fricas")

[Out]

integral(1/(a^3*g*x + a^3*f + (b^3*g*x + b^3*f)*log((e*x + d)^n*c)^3 + 3*(a*b^2*g*x + a*b^2*f)*log((e*x + d)^n
*c)^2 + 3*(a^2*b*g*x + a^2*b*f)*log((e*x + d)^n*c)), x)

Sympy [N/A]

Not integrable

Time = 6.42 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \frac {1}{(f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3} \, dx=\int \frac {1}{\left (a + b \log {\left (c \left (d + e x\right )^{n} \right )}\right )^{3} \left (f + g x\right )}\, dx \]

[In]

integrate(1/(g*x+f)/(a+b*ln(c*(e*x+d)**n))**3,x)

[Out]

Integral(1/((a + b*log(c*(d + e*x)**n))**3*(f + g*x)), x)

Maxima [N/A]

Not integrable

Time = 0.30 (sec) , antiderivative size = 703, normalized size of antiderivative = 29.29 \[ \int \frac {1}{(f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3} \, dx=\int { \frac {1}{{\left (g x + f\right )} {\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{3}} \,d x } \]

[In]

integrate(1/(g*x+f)/(a+b*log(c*(e*x+d)^n))^3,x, algorithm="maxima")

[Out]

-1/2*(b*e^2*g*n*x^2 + (d*e*f - d^2*g)*a + (d*e*f*n + (d*e*f - d^2*g)*log(c))*b + ((e^2*f - d*e*g)*a + (e^2*f*n
+ d*e*g*n + (e^2*f - d*e*g)*log(c))*b)*x + ((e^2*f - d*e*g)*b*x + (d*e*f - d^2*g)*b)*log((e*x + d)^n))/(b^4*e
^2*f^2*n^2*log(c)^2 + 2*a*b^3*e^2*f^2*n^2*log(c) + a^2*b^2*e^2*f^2*n^2 + (b^4*e^2*g^2*n^2*log(c)^2 + 2*a*b^3*e
^2*g^2*n^2*log(c) + a^2*b^2*e^2*g^2*n^2)*x^2 + (b^4*e^2*g^2*n^2*x^2 + 2*b^4*e^2*f*g*n^2*x + b^4*e^2*f^2*n^2)*l
og((e*x + d)^n)^2 + 2*(b^4*e^2*f*g*n^2*log(c)^2 + 2*a*b^3*e^2*f*g*n^2*log(c) + a^2*b^2*e^2*f*g*n^2)*x + 2*(b^4
*e^2*f^2*n^2*log(c) + a*b^3*e^2*f^2*n^2 + (b^4*e^2*g^2*n^2*log(c) + a*b^3*e^2*g^2*n^2)*x^2 + 2*(b^4*e^2*f*g*n^
2*log(c) + a*b^3*e^2*f*g*n^2)*x)*log((e*x + d)^n)) + integrate(1/2*(e^2*f^2 - 3*d*e*f*g + 2*d^2*g^2 - (e^2*f*g
- d*e*g^2)*x)/(b^3*e^2*f^3*n^2*log(c) + a*b^2*e^2*f^3*n^2 + (b^3*e^2*g^3*n^2*log(c) + a*b^2*e^2*g^3*n^2)*x^3
+ 3*(b^3*e^2*f*g^2*n^2*log(c) + a*b^2*e^2*f*g^2*n^2)*x^2 + 3*(b^3*e^2*f^2*g*n^2*log(c) + a*b^2*e^2*f^2*g*n^2)*
x + (b^3*e^2*g^3*n^2*x^3 + 3*b^3*e^2*f*g^2*n^2*x^2 + 3*b^3*e^2*f^2*g*n^2*x + b^3*e^2*f^3*n^2)*log((e*x + d)^n)
), x)

Giac [N/A]

Not integrable

Time = 0.33 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {1}{(f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3} \, dx=\int { \frac {1}{{\left (g x + f\right )} {\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{3}} \,d x } \]

[In]

integrate(1/(g*x+f)/(a+b*log(c*(e*x+d)^n))^3,x, algorithm="giac")

[Out]

integrate(1/((g*x + f)*(b*log((e*x + d)^n*c) + a)^3), x)

Mupad [N/A]

Not integrable

Time = 1.28 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {1}{(f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3} \, dx=\int \frac {1}{\left (f+g\,x\right )\,{\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^3} \,d x \]

[In]

int(1/((f + g*x)*(a + b*log(c*(d + e*x)^n))^3),x)

[Out]

int(1/((f + g*x)*(a + b*log(c*(d + e*x)^n))^3), x)