\(\int \frac {1}{(g+h x)^2 (a+b \log (c (d (e+f x)^p)^q))^3} \, dx\) [459]

Optimal result

Integrand size = 28, antiderivative size = 28 \[ \int \frac {1}{(g+h x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3} \, dx=\text {Int}\left (\frac {1}{(g+h x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3},x\right ) \]

[Out]

Unintegrable(1/(h*x+g)^2/(a+b*ln(c*(d*(f*x+e)^p)^q))^3,x)

Rubi [N/A]

Not integrable

Time = 0.04 (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 = {} \[ \int \frac {1}{(g+h x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3} \, dx=\int \frac {1}{(g+h x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3} \, dx \]

[In]

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

[Out]

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

Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{(g+h x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 23.06 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {1}{(g+h x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3} \, dx=\int \frac {1}{(g+h x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3} \, dx \]

[In]

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

[Out]

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

Maple [N/A]

Not integrable

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

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.31 (sec) , antiderivative size = 165, normalized size of antiderivative = 5.89 \[ \int \frac {1}{(g+h x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3} \, dx=\int { \frac {1}{{\left (h x + g\right )}^{2} {\left (b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a\right )}^{3}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 135.87 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.96 \[ \int \frac {1}{(g+h x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3} \, dx=\int \frac {1}{\left (a + b \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}\right )^{3} \left (g + h x\right )^{2}}\, dx \]

[In]

integrate(1/(h*x+g)**2/(a+b*ln(c*(d*(f*x+e)**p)**q))**3,x)

[Out]

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

Maxima [N/A]

Not integrable

Time = 1.58 (sec) , antiderivative size = 1486, normalized size of antiderivative = 53.07 \[ \int \frac {1}{(g+h x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3} \, dx=\int { \frac {1}{{\left (h x + g\right )}^{2} {\left (b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a\right )}^{3}} \,d x } \]

[In]

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

[Out]

1/2*((a*f^2*h - (f^2*h*p*q - f^2*h*q*log(d) - f^2*h*log(c))*b)*x^2 - (e*f*g - 2*e^2*h)*a - (e*f*g*p*q + (e*f*g
- 2*e^2*h)*log(c) + (e*f*g*q - 2*e^2*h*q)*log(d))*b - ((f^2*g - 3*e*f*h)*a + (f^2*g*p*q + e*f*h*p*q + (f^2*g
- 3*e*f*h)*log(c) + (f^2*g*q - 3*e*f*h*q)*log(d))*b)*x + (b*f^2*h*x^2 - (f^2*g - 3*e*f*h)*b*x - (e*f*g - 2*e^2
*h)*b)*log(((f*x + e)^p)^q))/(a^2*b^2*f^2*g^3*p^2*q^2 + 2*(f^2*g^3*p^2*q^3*log(d) + f^2*g^3*p^2*q^2*log(c))*a*
b^3 + (f^2*g^3*p^2*q^4*log(d)^2 + 2*f^2*g^3*p^2*q^3*log(c)*log(d) + f^2*g^3*p^2*q^2*log(c)^2)*b^4 + (a^2*b^2*f
^2*h^3*p^2*q^2 + 2*(f^2*h^3*p^2*q^3*log(d) + f^2*h^3*p^2*q^2*log(c))*a*b^3 + (f^2*h^3*p^2*q^4*log(d)^2 + 2*f^2
*h^3*p^2*q^3*log(c)*log(d) + f^2*h^3*p^2*q^2*log(c)^2)*b^4)*x^3 + 3*(a^2*b^2*f^2*g*h^2*p^2*q^2 + 2*(f^2*g*h^2*
p^2*q^3*log(d) + f^2*g*h^2*p^2*q^2*log(c))*a*b^3 + (f^2*g*h^2*p^2*q^4*log(d)^2 + 2*f^2*g*h^2*p^2*q^3*log(c)*lo
g(d) + f^2*g*h^2*p^2*q^2*log(c)^2)*b^4)*x^2 + (b^4*f^2*h^3*p^2*q^2*x^3 + 3*b^4*f^2*g*h^2*p^2*q^2*x^2 + 3*b^4*f
^2*g^2*h*p^2*q^2*x + b^4*f^2*g^3*p^2*q^2)*log(((f*x + e)^p)^q)^2 + 3*(a^2*b^2*f^2*g^2*h*p^2*q^2 + 2*(f^2*g^2*h
*p^2*q^3*log(d) + f^2*g^2*h*p^2*q^2*log(c))*a*b^3 + (f^2*g^2*h*p^2*q^4*log(d)^2 + 2*f^2*g^2*h*p^2*q^3*log(c)*l
og(d) + f^2*g^2*h*p^2*q^2*log(c)^2)*b^4)*x + 2*(a*b^3*f^2*g^3*p^2*q^2 + (f^2*g^3*p^2*q^3*log(d) + f^2*g^3*p^2*
q^2*log(c))*b^4 + (a*b^3*f^2*h^3*p^2*q^2 + (f^2*h^3*p^2*q^3*log(d) + f^2*h^3*p^2*q^2*log(c))*b^4)*x^3 + 3*(a*b
^3*f^2*g*h^2*p^2*q^2 + (f^2*g*h^2*p^2*q^3*log(d) + f^2*g*h^2*p^2*q^2*log(c))*b^4)*x^2 + 3*(a*b^3*f^2*g^2*h*p^2
*q^2 + (f^2*g^2*h*p^2*q^3*log(d) + f^2*g^2*h*p^2*q^2*log(c))*b^4)*x)*log(((f*x + e)^p)^q)) + integrate(1/2*(f^
2*h^2*x^2 + f^2*g^2 - 6*e*f*g*h + 6*e^2*h^2 - 2*(2*f^2*g*h - 3*e*f*h^2)*x)/(a*b^2*f^2*g^4*p^2*q^2 + (a*b^2*f^2
*h^4*p^2*q^2 + (f^2*h^4*p^2*q^3*log(d) + f^2*h^4*p^2*q^2*log(c))*b^3)*x^4 + (f^2*g^4*p^2*q^3*log(d) + f^2*g^4*
p^2*q^2*log(c))*b^3 + 4*(a*b^2*f^2*g*h^3*p^2*q^2 + (f^2*g*h^3*p^2*q^3*log(d) + f^2*g*h^3*p^2*q^2*log(c))*b^3)*
x^3 + 6*(a*b^2*f^2*g^2*h^2*p^2*q^2 + (f^2*g^2*h^2*p^2*q^3*log(d) + f^2*g^2*h^2*p^2*q^2*log(c))*b^3)*x^2 + 4*(a
*b^2*f^2*g^3*h*p^2*q^2 + (f^2*g^3*h*p^2*q^3*log(d) + f^2*g^3*h*p^2*q^2*log(c))*b^3)*x + (b^3*f^2*h^4*p^2*q^2*x
^4 + 4*b^3*f^2*g*h^3*p^2*q^2*x^3 + 6*b^3*f^2*g^2*h^2*p^2*q^2*x^2 + 4*b^3*f^2*g^3*h*p^2*q^2*x + b^3*f^2*g^4*p^2
*q^2)*log(((f*x + e)^p)^q)), x)

Giac [N/A]

Not integrable

Time = 0.40 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {1}{(g+h x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3} \, dx=\int { \frac {1}{{\left (h x + g\right )}^{2} {\left (b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a\right )}^{3}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 1.34 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {1}{(g+h x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3} \, dx=\int \frac {1}{{\left (g+h\,x\right )}^2\,{\left (a+b\,\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^p\right )}^q\right )\right )}^3} \,d x \]

[In]

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

[Out]

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