\(\int \frac {1}{(g+h x) (i+j x)^2 (a+b \log (c (d (e+f x)^p)^q))^2} \, dx\) [547]

Optimal result

Integrand size = 35, antiderivative size = 35 \[ \int \frac {1}{(g+h x) (i+j x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2} \, dx=\text {Int}\left (\frac {1}{(g+h x) (i+j x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.22 (sec) , antiderivative size = 35, 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 = {} \[ \int \frac {1}{(g+h x) (i+j x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2} \, dx=\int \frac {1}{(g+h x) (i+j x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 19.11 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.06 \[ \int \frac {1}{(g+h x) (i+j x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2} \, dx=\int \frac {1}{(g+h x) (i+j x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A]

Not integrable

Time = 0.32 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.33 (sec) , antiderivative size = 211, normalized size of antiderivative = 6.03 \[ \int \frac {1}{(g+h x) (i+j x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2} \, dx=\int { \frac {1}{{\left (h x + g\right )} {\left (j x + i\right )}^{2} {\left (b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 112.69 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.91 \[ \int \frac {1}{(g+h x) (i+j x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2} \, dx=\int \frac {1}{\left (a + b \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}\right )^{2} \left (g + h x\right ) \left (i + j x\right )^{2}}\, dx \]

[In]

integrate(1/(h*x+g)/(j*x+i)**2/(a+b*ln(c*(d*(f*x+e)**p)**q))**2,x)

[Out]

Integral(1/((a + b*log(c*(d*(e + f*x)**p)**q))**2*(g + h*x)*(i + j*x)**2), x)

Maxima [N/A]

Not integrable

Time = 1.27 (sec) , antiderivative size = 1039, normalized size of antiderivative = 29.69 \[ \int \frac {1}{(g+h x) (i+j x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2} \, dx=\int { \frac {1}{{\left (h x + g\right )} {\left (j x + i\right )}^{2} {\left (b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.45 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.06 \[ \int \frac {1}{(g+h x) (i+j x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2} \, dx=\int { \frac {1}{{\left (h x + g\right )} {\left (j x + i\right )}^{2} {\left (b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 1.42 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.06 \[ \int \frac {1}{(g+h x) (i+j x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2} \, dx=\int \frac {1}{\left (g+h\,x\right )\,{\left (i+j\,x\right )}^2\,{\left (a+b\,\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^p\right )}^q\right )\right )}^2} \,d x \]

[In]

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

[Out]

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