\(\int \frac {1}{(f+g x)^2 (a+b \log (c (d+e x)^n))^3} \, dx\) [104]

Optimal result

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

[Out]

Unintegrable(1/(g*x+f)^2/(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 = {} \[ \int \frac {1}{(f+g x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3} \, dx=\int \frac {1}{(f+g x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3} \, dx \]

[In]

Int[1/((f + g*x)^2*(a + b*Log[c*(d + e*x)^n])^3),x]

[Out]

Defer[Int][1/((f + g*x)^2*(a + b*Log[c*(d + e*x)^n])^3), 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 )^3} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 2.72 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {1}{(f+g x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3} \, dx=\int \frac {1}{(f+g x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3} \, dx \]

[In]

Integrate[1/((f + g*x)^2*(a + b*Log[c*(d + e*x)^n])^3),x]

[Out]

Integrate[1/((f + g*x)^2*(a + b*Log[c*(d + e*x)^n])^3), x]

Maple [N/A]

Not integrable

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

\[\int \frac {1}{\left (g x +f \right )^{2} {\left (a +b \ln \left (c \left (e x +d \right )^{n}\right )\right )}^{3}}d x\]

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.31 (sec) , antiderivative size = 153, normalized size of antiderivative = 6.38 \[ \int \frac {1}{(f+g x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3} \, dx=\int { \frac {1}{{\left (g x + f\right )}^{2} {\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{3}} \,d x } \]

[In]

integrate(1/(g*x+f)^2/(a+b*log(c*(e*x+d)^n))^3,x, algorithm="fricas")

[Out]

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

Sympy [N/A]

Not integrable

Time = 35.27 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {1}{(f+g x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3} \, dx=\int \frac {1}{\left (a + b \log {\left (c \left (d + e x\right )^{n} \right )}\right )^{3} \left (f + g x\right )^{2}}\, dx \]

[In]

integrate(1/(g*x+f)**2/(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)**2), x)

Maxima [N/A]

Not integrable

Time = 0.30 (sec) , antiderivative size = 934, normalized size of antiderivative = 38.92 \[ \int \frac {1}{(f+g x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3} \, dx=\int { \frac {1}{{\left (g x + f\right )}^{2} {\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{3}} \,d x } \]

[In]

integrate(1/(g*x+f)^2/(a+b*log(c*(e*x+d)^n))^3,x, algorithm="maxima")

[Out]

1/2*((a*e^2*g - (e^2*g*n - e^2*g*log(c))*b)*x^2 - (d*e*f - 2*d^2*g)*a - (d*e*f*n + (d*e*f - 2*d^2*g)*log(c))*b
- ((e^2*f - 3*d*e*g)*a + (e^2*f*n + d*e*g*n + (e^2*f - 3*d*e*g)*log(c))*b)*x + (b*e^2*g*x^2 - (e^2*f - 3*d*e*
g)*b*x - (d*e*f - 2*d^2*g)*b)*log((e*x + d)^n))/(b^4*e^2*f^3*n^2*log(c)^2 + 2*a*b^3*e^2*f^3*n^2*log(c) + a^2*b
^2*e^2*f^3*n^2 + (b^4*e^2*g^3*n^2*log(c)^2 + 2*a*b^3*e^2*g^3*n^2*log(c) + a^2*b^2*e^2*g^3*n^2)*x^3 + 3*(b^4*e^
2*f*g^2*n^2*log(c)^2 + 2*a*b^3*e^2*f*g^2*n^2*log(c) + a^2*b^2*e^2*f*g^2*n^2)*x^2 + (b^4*e^2*g^3*n^2*x^3 + 3*b^
4*e^2*f*g^2*n^2*x^2 + 3*b^4*e^2*f^2*g*n^2*x + b^4*e^2*f^3*n^2)*log((e*x + d)^n)^2 + 3*(b^4*e^2*f^2*g*n^2*log(c
)^2 + 2*a*b^3*e^2*f^2*g*n^2*log(c) + a^2*b^2*e^2*f^2*g*n^2)*x + 2*(b^4*e^2*f^3*n^2*log(c) + a*b^3*e^2*f^3*n^2
+ (b^4*e^2*g^3*n^2*log(c) + a*b^3*e^2*g^3*n^2)*x^3 + 3*(b^4*e^2*f*g^2*n^2*log(c) + a*b^3*e^2*f*g^2*n^2)*x^2 +
3*(b^4*e^2*f^2*g*n^2*log(c) + a*b^3*e^2*f^2*g*n^2)*x)*log((e*x + d)^n)) + integrate(1/2*(e^2*g^2*x^2 + e^2*f^2
- 6*d*e*f*g + 6*d^2*g^2 - 2*(2*e^2*f*g - 3*d*e*g^2)*x)/(b^3*e^2*f^4*n^2*log(c) + a*b^2*e^2*f^4*n^2 + (b^3*e^2
*g^4*n^2*log(c) + a*b^2*e^2*g^4*n^2)*x^4 + 4*(b^3*e^2*f*g^3*n^2*log(c) + a*b^2*e^2*f*g^3*n^2)*x^3 + 6*(b^3*e^2
*f^2*g^2*n^2*log(c) + a*b^2*e^2*f^2*g^2*n^2)*x^2 + 4*(b^3*e^2*f^3*g*n^2*log(c) + a*b^2*e^2*f^3*g*n^2)*x + (b^3
*e^2*g^4*n^2*x^4 + 4*b^3*e^2*f*g^3*n^2*x^3 + 6*b^3*e^2*f^2*g^2*n^2*x^2 + 4*b^3*e^2*f^3*g*n^2*x + b^3*e^2*f^4*n
^2)*log((e*x + d)^n)), x)

Giac [N/A]

Not integrable

Time = 0.35 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {1}{(f+g x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3} \, dx=\int { \frac {1}{{\left (g x + f\right )}^{2} {\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{3}} \,d x } \]

[In]

integrate(1/(g*x+f)^2/(a+b*log(c*(e*x+d)^n))^3,x, algorithm="giac")

[Out]

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

Mupad [N/A]

Not integrable

Time = 1.29 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {1}{(f+g x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3} \, dx=\int \frac {1}{{\left (f+g\,x\right )}^2\,{\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^3} \,d x \]

[In]

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

[Out]

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