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

Optimal. Leaf size=31 \[ \text {Int}\left (\frac {1}{(g+h x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.06, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {1}{(g+h x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 18.60, size = 0, normalized size = 0.00 \[ \int \frac {1}{(g+h x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{a^{2} h^{2} x^{2} + 2 \, a^{2} g h x + a^{2} g^{2} + {\left (b^{2} h^{2} x^{2} + 2 \, b^{2} g h x + b^{2} g^{2}\right )} \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right )^{2} + 2 \, {\left (a b h^{2} x^{2} + 2 \, a b g h x + a b g^{2}\right )} \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right )}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (h x + g\right )}^{2} {\left (b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.35, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (h x +g \right )^{2} \left (b \ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right )+a \right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {f x + e}{a b f g^{2} p q + {\left (f g^{2} p q^{2} \log \relax (d) + f g^{2} p q \log \relax (c)\right )} b^{2} + {\left (a b f h^{2} p q + {\left (f h^{2} p q^{2} \log \relax (d) + f h^{2} p q \log \relax (c)\right )} b^{2}\right )} x^{2} + 2 \, {\left (a b f g h p q + {\left (f g h p q^{2} \log \relax (d) + f g h p q \log \relax (c)\right )} b^{2}\right )} x + {\left (b^{2} f h^{2} p q x^{2} + 2 \, b^{2} f g h p q x + b^{2} f g^{2} p q\right )} \log \left ({\left ({\left (f x + e\right )}^{p}\right )}^{q}\right )} - \int \frac {f h x - f g + 2 \, e h}{a b f g^{3} p q + {\left (a b f h^{3} p q + {\left (f h^{3} p q^{2} \log \relax (d) + f h^{3} p q \log \relax (c)\right )} b^{2}\right )} x^{3} + {\left (f g^{3} p q^{2} \log \relax (d) + f g^{3} p q \log \relax (c)\right )} b^{2} + 3 \, {\left (a b f g h^{2} p q + {\left (f g h^{2} p q^{2} \log \relax (d) + f g h^{2} p q \log \relax (c)\right )} b^{2}\right )} x^{2} + 3 \, {\left (a b f g^{2} h p q + {\left (f g^{2} h p q^{2} \log \relax (d) + f g^{2} h p q \log \relax (c)\right )} b^{2}\right )} x + {\left (b^{2} f h^{3} p q x^{3} + 3 \, b^{2} f g h^{2} p q x^{2} + 3 \, b^{2} f g^{2} h p q x + b^{2} f g^{3} p q\right )} \log \left ({\left ({\left (f x + e\right )}^{p}\right )}^{q}\right )}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [A]  time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {1}{{\left (g+h\,x\right )}^2\,{\left (a+b\,\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^p\right )}^q\right )\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (a + b \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}\right )^{2} \left (g + h x\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(h*x+g)**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)**2), x)

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