\(\int \frac {1}{(f+g x) (h+i x) (a+b \log (c (d+e x)^n))^2} \, dx\) [240]

Optimal result

Integrand size = 31, antiderivative size = 31 \[ \int \frac {1}{(f+g x) (h+i x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx=\frac {g \text {Int}\left (\frac {1}{(f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2},x\right )}{g h-f i}-\frac {i \text {Int}\left (\frac {1}{(h+i x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2},x\right )}{g h-f i} \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.12 (sec) , antiderivative size = 31, 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 = {} \[ \int \frac {1}{(f+g x) (h+i x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx=\int \frac {1}{(f+g x) (h+i x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 4.73 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.06 \[ \int \frac {1}{(f+g x) (h+i x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx=\int \frac {1}{(f+g x) (h+i x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A]

Not integrable

Time = 0.03 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.33 (sec) , antiderivative size = 118, normalized size of antiderivative = 3.81 \[ \int \frac {1}{(f+g x) (h+i x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx=\int { \frac {1}{{\left (g x + f\right )} {\left (i x + h\right )} {\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 8.40 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.87 \[ \int \frac {1}{(f+g x) (h+i x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx=\int \frac {1}{\left (a + b \log {\left (c \left (d + e x\right )^{n} \right )}\right )^{2} \left (f + g x\right ) \left (h + i x\right )}\, dx \]

[In]

integrate(1/(g*x+f)/(i*x+h)/(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)*(h + i*x)), x)

Maxima [N/A]

Not integrable

Time = 0.29 (sec) , antiderivative size = 488, normalized size of antiderivative = 15.74 \[ \int \frac {1}{(f+g x) (h+i x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx=\int { \frac {1}{{\left (g x + f\right )} {\left (i x + h\right )} {\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.38 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.06 \[ \int \frac {1}{(f+g x) (h+i x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx=\int { \frac {1}{{\left (g x + f\right )} {\left (i x + h\right )} {\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 1.33 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.06 \[ \int \frac {1}{(f+g x) (h+i x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx=\int \frac {1}{\left (f+g\,x\right )\,\left (h+i\,x\right )\,{\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^2} \,d x \]

[In]

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

[Out]

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