\(\int \frac {1}{(g+h x) (a+b \log (c (d (e+f x)^p)^q))^2} \, dx\) [453]

Optimal result

Integrand size = 28, antiderivative size = 28 \[ \int \frac {1}{(g+h x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2} \, dx=\text {Int}\left (\frac {1}{(g+h x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.05 (sec) , antiderivative size = 28, 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 = {} \[ \int \frac {1}{(g+h x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2} \, dx=\int \frac {1}{(g+h x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2} \, dx \]

[In]

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

[Out]

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

Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{(g+h x) \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 = 0.67 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {1}{(g+h x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2} \, dx=\int \frac {1}{(g+h x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A]

Not integrable

Time = 0.18 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.33 (sec) , antiderivative size = 71, normalized size of antiderivative = 2.54 \[ \int \frac {1}{(g+h x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2} \, dx=\int { \frac {1}{{\left (h x + g\right )} {\left (b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 5.01 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93 \[ \int \frac {1}{(g+h x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2} \, dx=\int \frac {1}{\left (a + b \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}\right )^{2} \left (g + h x\right )}\, dx \]

[In]

integrate(1/(h*x+g)/(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)), x)

Maxima [N/A]

Not integrable

Time = 1.15 (sec) , antiderivative size = 267, normalized size of antiderivative = 9.54 \[ \int \frac {1}{(g+h x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2} \, dx=\int { \frac {1}{{\left (h x + g\right )} {\left (b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.32 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {1}{(g+h x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2} \, dx=\int { \frac {1}{{\left (h x + g\right )} {\left (b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 1.24 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {1}{(g+h x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2} \, dx=\int \frac {1}{\left (g+h\,x\right )\,{\left (a+b\,\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^p\right )}^q\right )\right )}^2} \,d x \]

[In]

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

[Out]

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