\(\int \frac {1}{(f+g x) (h+i x)^2 (a+b \log (c (d+e x)^n))^2} \, dx\) [241]

Optimal result

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

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.16 (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 = {} \[ \int \frac {1}{(f+g x) (h+i x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx=\int \frac {1}{(f+g x) (h+i x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx \]

[In]

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

[Out]

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

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {g^2}{(g h-f i)^2 (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)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}-\frac {g i}{(g h-f i)^2 (h+i x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}\right ) \, dx \\ & = \frac {g^2 \int \frac {1}{(f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx}{(g h-f i)^2}-\frac {(g 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)^2}-\frac {i \int \frac {1}{(h+i x)^2 \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 = 9.88 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.06 \[ \int \frac {1}{(f+g x) (h+i x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx=\int \frac {1}{(f+g x) (h+i x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A]

Not integrable

Time = 122.39 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.33 (sec) , antiderivative size = 203, normalized size of antiderivative = 6.55 \[ \int \frac {1}{(f+g x) (h+i x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx=\int { \frac {1}{{\left (g x + f\right )} {\left (i x + h\right )}^{2} {\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 42.43 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.94 \[ \int \frac {1}{(f+g x) (h+i x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx=\int \frac {1}{\left (a + b \log {\left (c \left (d + e x\right )^{n} \right )}\right )^{2} \left (f + g x\right ) \left (h + i x\right )^{2}}\, dx \]

[In]

integrate(1/(g*x+f)/(i*x+h)**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)*(h + i*x)**2), x)

Maxima [N/A]

Not integrable

Time = 0.31 (sec) , antiderivative size = 723, normalized size of antiderivative = 23.32 \[ \int \frac {1}{(f+g x) (h+i x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx=\int { \frac {1}{{\left (g x + f\right )} {\left (i x + h\right )}^{2} {\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

-(e*x + d)/(b^2*e*f*h^2*n*log(c) + a*b*e*f*h^2*n + (b^2*e*g*i^2*n*log(c) + a*b*e*g*i^2*n)*x^3 + ((2*g*h*i*n +
f*i^2*n)*b^2*e*log(c) + (2*g*h*i*n + f*i^2*n)*a*b*e)*x^2 + ((g*h^2*n + 2*f*h*i*n)*b^2*e*log(c) + (g*h^2*n + 2*
f*h*i*n)*a*b*e)*x + (b^2*e*g*i^2*n*x^3 + b^2*e*f*h^2*n + (2*g*h*i*n + f*i^2*n)*b^2*e*x^2 + (g*h^2*n + 2*f*h*i*
n)*b^2*e*x)*log((e*x + d)^n)) - integrate((2*e*g*i*x^2 - e*f*h + (g*h + 2*f*i)*d + (e*f*i + 3*d*g*i)*x)/(b^2*e
*f^2*h^3*n*log(c) + a*b*e*f^2*h^3*n + (b^2*e*g^2*i^3*n*log(c) + a*b*e*g^2*i^3*n)*x^5 + ((3*g^2*h*i^2*n + 2*f*g
*i^3*n)*b^2*e*log(c) + (3*g^2*h*i^2*n + 2*f*g*i^3*n)*a*b*e)*x^4 + ((3*g^2*h^2*i*n + 6*f*g*h*i^2*n + f^2*i^3*n)
*b^2*e*log(c) + (3*g^2*h^2*i*n + 6*f*g*h*i^2*n + f^2*i^3*n)*a*b*e)*x^3 + ((g^2*h^3*n + 6*f*g*h^2*i*n + 3*f^2*h
*i^2*n)*b^2*e*log(c) + (g^2*h^3*n + 6*f*g*h^2*i*n + 3*f^2*h*i^2*n)*a*b*e)*x^2 + ((2*f*g*h^3*n + 3*f^2*h^2*i*n)
*b^2*e*log(c) + (2*f*g*h^3*n + 3*f^2*h^2*i*n)*a*b*e)*x + (b^2*e*g^2*i^3*n*x^5 + b^2*e*f^2*h^3*n + (3*g^2*h*i^2
*n + 2*f*g*i^3*n)*b^2*e*x^4 + (3*g^2*h^2*i*n + 6*f*g*h*i^2*n + f^2*i^3*n)*b^2*e*x^3 + (g^2*h^3*n + 6*f*g*h^2*i
*n + 3*f^2*h*i^2*n)*b^2*e*x^2 + (2*f*g*h^3*n + 3*f^2*h^2*i*n)*b^2*e*x)*log((e*x + d)^n)), x)

Giac [N/A]

Not integrable

Time = 0.42 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.06 \[ \int \frac {1}{(f+g x) (h+i x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx=\int { \frac {1}{{\left (g x + f\right )} {\left (i x + h\right )}^{2} {\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 1.32 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.06 \[ \int \frac {1}{(f+g x) (h+i x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx=\int \frac {1}{\left (f+g\,x\right )\,{\left (h+i\,x\right )}^2\,{\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^2} \,d x \]

[In]

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

[Out]

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