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

Optimal. Leaf size=30 \[ \text{Unintegrable}\left (\frac{1}{(g+h x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.062645, 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 = {} \[ \int \frac{1}{(g+h x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3} \, dx \]

Verification is Not applicable to the result.

[In]

Int[1/((g + h*x)^2*(a + b*Log[c*(d*(e + f*x)^p)^q])^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*} \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\\ \end{align*}

Mathematica [A]  time = 38.8578, size = 0, normalized size = 0. \[ \int \frac{1}{(g+h x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.648, 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 ) ^{3}}}\, 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))^3,x)

[Out]

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

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \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))^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{a^{3} h^{2} x^{2} + 2 \, a^{3} g h x + a^{3} g^{2} +{\left (b^{3} h^{2} x^{2} + 2 \, b^{3} g h x + b^{3} g^{2}\right )} \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right )^{3} + 3 \,{\left (a b^{2} h^{2} x^{2} + 2 \, a b^{2} g h x + a b^{2} g^{2}\right )} \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right )^{2} + 3 \,{\left (a^{2} b h^{2} x^{2} + 2 \, a^{2} b g h x + a^{2} 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))^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)

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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))**3,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 )}^{3}}\,{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))^3,x, algorithm="giac")

[Out]

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