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

Optimal. Leaf size=30 \[ \text{Unintegrable}\left (\frac{1}{(g+h x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.0621928, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., 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*}

Mathematica [A]  time = 18.2729, size = 0, normalized size = 0. \[ \int \frac{1}{(g+h x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2} \, dx \]

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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Maple [A]  time = 0.665, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( hx+g \right ) ^{2} \left ( a+b\ln \left ( c \left ( d \left ( fx+e \right ) ^{p} \right ) ^{q} \right ) \right ) ^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{f x + e}{a b f g^{2} p q +{\left (f g^{2} p q \log \left (c\right ) + f g^{2} p q \log \left (d^{q}\right )\right )} b^{2} +{\left (a b f h^{2} p q +{\left (f h^{2} p q \log \left (c\right ) + f h^{2} p q \log \left (d^{q}\right )\right )} b^{2}\right )} x^{2} + 2 \,{\left (a b f g h p q +{\left (f g h p q \log \left (c\right ) + f g h p q \log \left (d^{q}\right )\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 \log \left (c\right ) + f h^{3} p q \log \left (d^{q}\right )\right )} b^{2}\right )} x^{3} +{\left (f g^{3} p q \log \left (c\right ) + f g^{3} p q \log \left (d^{q}\right )\right )} b^{2} + 3 \,{\left (a b f g h^{2} p q +{\left (f g h^{2} p q \log \left (c\right ) + f g h^{2} p q \log \left (d^{q}\right )\right )} b^{2}\right )} x^{2} + 3 \,{\left (a b f g^{2} h p q +{\left (f g^{2} h p q \log \left (c\right ) + f g^{2} h p q \log \left (d^{q}\right )\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} \end{align*}

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*log(c) + f*g^2*p*q*log(d^q))*b^2 + (a*b*f*h^2*p*q + (f*h^2*p*q*log(c) +
f*h^2*p*q*log(d^q))*b^2)*x^2 + 2*(a*b*f*g*h*p*q + (f*g*h*p*q*log(c) + f*g*h*p*q*log(d^q))*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*log(c) + f*h^3*p*q*log(d^q))*b^2)*x^3 + (f*g^3*p*q*log(c) + f*g^3*p*q*log
(d^q))*b^2 + 3*(a*b*f*g*h^2*p*q + (f*g*h^2*p*q*log(c) + f*g*h^2*p*q*log(d^q))*b^2)*x^2 + 3*(a*b*f*g^2*h*p*q +
(f*g^2*h*p*q*log(c) + f*g^2*h*p*q*log(d^q))*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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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\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 ) \end{align*}

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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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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]

Timed out

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

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)