### 3.8 $$\int \frac{\log ^{-1+q}(c x^n)}{x (a x^m+b \log ^q(c x^n))^3} \, dx$$

Optimal. Leaf size=71 $-\frac{a m \text{CannotIntegrate}\left (\frac{x^{m-1}}{\left (a x^m+b \log ^q\left (c x^n\right )\right )^3},x\right )}{b n q}-\frac{1}{2 b n q \left (a x^m+b \log ^q\left (c x^n\right )\right )^2}$

[Out]

-((a*m*CannotIntegrate[x^(-1 + m)/(a*x^m + b*Log[c*x^n]^q)^3, x])/(b*n*q)) - 1/(2*b*n*q*(a*x^m + b*Log[c*x^n]^
q)^2)

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Rubi [A]  time = 0.245109, 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 = {} $\int \frac{\log ^{-1+q}\left (c x^n\right )}{x \left (a x^m+b \log ^q\left (c x^n\right )\right )^3} \, dx$

Veriﬁcation is Not applicable to the result.

[In]

Int[Log[c*x^n]^(-1 + q)/(x*(a*x^m + b*Log[c*x^n]^q)^3),x]

[Out]

-1/(2*b*n*q*(a*x^m + b*Log[c*x^n]^q)^2) - (a*m*Defer[Int][x^(-1 + m)/(a*x^m + b*Log[c*x^n]^q)^3, x])/(b*n*q)

Rubi steps

\begin{align*} \int \frac{\log ^{-1+q}\left (c x^n\right )}{x \left (a x^m+b \log ^q\left (c x^n\right )\right )^3} \, dx &=-\frac{1}{2 b n q \left (a x^m+b \log ^q\left (c x^n\right )\right )^2}-\frac{(a m) \int \frac{x^{-1+m}}{\left (a x^m+b \log ^q\left (c x^n\right )\right )^3} \, dx}{b n q}\\ \end{align*}

Mathematica [A]  time = 1.47245, size = 0, normalized size = 0. $\int \frac{\log ^{-1+q}\left (c x^n\right )}{x \left (a x^m+b \log ^q\left (c x^n\right )\right )^3} \, dx$

Veriﬁcation is Not applicable to the result.

[In]

Integrate[Log[c*x^n]^(-1 + q)/(x*(a*x^m + b*Log[c*x^n]^q)^3),x]

[Out]

Integrate[Log[c*x^n]^(-1 + q)/(x*(a*x^m + b*Log[c*x^n]^q)^3), x]

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Maple [A]  time = 49.67, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( \ln \left ( c{x}^{n} \right ) \right ) ^{-1+q}}{x \left ( a{x}^{m}+b \left ( \ln \left ( c{x}^{n} \right ) \right ) ^{q} \right ) ^{3}}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(ln(c*x^n)^(-1+q)/x/(a*x^m+b*ln(c*x^n)^q)^3,x)

[Out]

int(ln(c*x^n)^(-1+q)/x/(a*x^m+b*ln(c*x^n)^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(log(c*x^n)^(-1+q)/x/(a*x^m+b*log(c*x^n)^q)^3,x, algorithm="maxima")

[Out]

-1/2*(a*m^2*x^m*log(x^n)^2 + (2*m^2*log(c) + m*n)*a*x^m*log(x^n) - (n^2*q^2 - m^2*log(c)^2 - m*n*log(c))*a*x^m
+ (2*b*m^2*log(x^n)^2 - (m*n*(2*q - 1) - 4*m^2*log(c))*b*log(x^n) - (m*n*(2*q - 1)*log(c) - 2*m^2*log(c)^2)*b
)*(log(c) + log(x^n))^q)/(a^3*b*m^3*x^(3*m)*log(x^n)^3 - 3*(m^2*n*q - m^3*log(c))*a^3*b*x^(3*m)*log(x^n)^2 + 3
*(m*n^2*q^2 - 2*m^2*n*q*log(c) + m^3*log(c)^2)*a^3*b*x^(3*m)*log(x^n) - (n^3*q^3 - 3*m*n^2*q^2*log(c) + 3*m^2*
n*q*log(c)^2 - m^3*log(c)^3)*a^3*b*x^(3*m) + (a*b^3*m^3*x^m*log(x^n)^3 - 3*(m^2*n*q - m^3*log(c))*a*b^3*x^m*lo
g(x^n)^2 + 3*(m*n^2*q^2 - 2*m^2*n*q*log(c) + m^3*log(c)^2)*a*b^3*x^m*log(x^n) - (n^3*q^3 - 3*m*n^2*q^2*log(c)
+ 3*m^2*n*q*log(c)^2 - m^3*log(c)^3)*a*b^3*x^m)*(log(c) + log(x^n))^(2*q) + 2*(a^2*b^2*m^3*x^(2*m)*log(x^n)^3
- 3*(m^2*n*q - m^3*log(c))*a^2*b^2*x^(2*m)*log(x^n)^2 + 3*(m*n^2*q^2 - 2*m^2*n*q*log(c) + m^3*log(c)^2)*a^2*b^
2*x^(2*m)*log(x^n) - (n^3*q^3 - 3*m*n^2*q^2*log(c) + 3*m^2*n*q*log(c)^2 - m^3*log(c)^3)*a^2*b^2*x^(2*m))*(log(
c) + log(x^n))^q) - integrate(-1/2*(m^3*n*(2*q - 3)*log(c)^2 - 2*m^4*log(c)^3 - 2*m^4*log(x^n)^3 + 2*(q^2 - 1)
*m^2*n^2*log(c) - (2*q^3 - 3*q^2 + q)*m*n^3 + (m^3*n*(2*q - 3) - 6*m^4*log(c))*log(x^n)^2 + 2*(m^3*n*(2*q - 3)
*log(c) - 3*m^4*log(c)^2 + (q^2 - 1)*m^2*n^2)*log(x^n))/(a^2*b*m^4*x*x^(2*m)*log(x^n)^4 - 4*(m^3*n*q - m^4*log
(c))*a^2*b*x*x^(2*m)*log(x^n)^3 + 6*(m^2*n^2*q^2 - 2*m^3*n*q*log(c) + m^4*log(c)^2)*a^2*b*x*x^(2*m)*log(x^n)^2
- 4*(m*n^3*q^3 - 3*m^2*n^2*q^2*log(c) + 3*m^3*n*q*log(c)^2 - m^4*log(c)^3)*a^2*b*x*x^(2*m)*log(x^n) + (n^4*q^
4 - 4*m*n^3*q^3*log(c) + 6*m^2*n^2*q^2*log(c)^2 - 4*m^3*n*q*log(c)^3 + m^4*log(c)^4)*a^2*b*x*x^(2*m) + (a*b^2*
m^4*x*x^m*log(x^n)^4 - 4*(m^3*n*q - m^4*log(c))*a*b^2*x*x^m*log(x^n)^3 + 6*(m^2*n^2*q^2 - 2*m^3*n*q*log(c) + m
^4*log(c)^2)*a*b^2*x*x^m*log(x^n)^2 - 4*(m*n^3*q^3 - 3*m^2*n^2*q^2*log(c) + 3*m^3*n*q*log(c)^2 - m^4*log(c)^3)
*a*b^2*x*x^m*log(x^n) + (n^4*q^4 - 4*m*n^3*q^3*log(c) + 6*m^2*n^2*q^2*log(c)^2 - 4*m^3*n*q*log(c)^3 + m^4*log(
c)^4)*a*b^2*x*x^m)*(log(c) + log(x^n))^q), x)

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\log \left (c x^{n}\right )^{q - 1}}{3 \, a b^{2} x x^{m} \log \left (c x^{n}\right )^{2 \, q} + 3 \, a^{2} b x x^{2 \, m} \log \left (c x^{n}\right )^{q} + a^{3} x x^{3 \, m} + b^{3} x \log \left (c x^{n}\right )^{3 \, q}}, x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(c*x^n)^(-1+q)/x/(a*x^m+b*log(c*x^n)^q)^3,x, algorithm="fricas")

[Out]

integral(log(c*x^n)^(q - 1)/(3*a*b^2*x*x^m*log(c*x^n)^(2*q) + 3*a^2*b*x*x^(2*m)*log(c*x^n)^q + a^3*x*x^(3*m) +
b^3*x*log(c*x^n)^(3*q)), 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(ln(c*x**n)**(-1+q)/x/(a*x**m+b*ln(c*x**n)**q)**3,x)

[Out]

Timed out

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\log \left (c x^{n}\right )^{q - 1}}{{\left (a x^{m} + b \log \left (c x^{n}\right )^{q}\right )}^{3} x}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(c*x^n)^(-1+q)/x/(a*x^m+b*log(c*x^n)^q)^3,x, algorithm="giac")

[Out]

integrate(log(c*x^n)^(q - 1)/((a*x^m + b*log(c*x^n)^q)^3*x), x)