### 3.36 $$\int \frac{d x^m+e \log ^{-1+q}(c x^n)}{x (a x^m+b \log ^q(c x^n))^2} \, dx$$

Optimal. Leaf size=74 $\left (d-\frac{a e m}{b n q}\right ) \text{CannotIntegrate}\left (\frac{x^{m-1}}{\left (a x^m+b \log ^q\left (c x^n\right )\right )^2},x\right )-\frac{e}{b n q \left (a x^m+b \log ^q\left (c x^n\right )\right )}$

[Out]

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

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Rubi [A]  time = 0.282174, 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 = {} $\int \frac{d x^m+e \log ^{-1+q}\left (c x^n\right )}{x \left (a x^m+b \log ^q\left (c x^n\right )\right )^2} \, dx$

Veriﬁcation is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 7.30777, size = 0, normalized size = 0. $\int \frac{d x^m+e \log ^{-1+q}\left (c x^n\right )}{x \left (a x^m+b \log ^q\left (c x^n\right )\right )^2} \, dx$

Veriﬁcation is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 7.905, size = 0, normalized size = 0. \begin{align*} \int{\frac{d{x}^{m}+e \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 ) ^{2}}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((d*x^m+e*ln(c*x^n)^(-1+q))/x/(a*x^m+b*ln(c*x^n)^q)^2,x)

[Out]

int((d*x^m+e*ln(c*x^n)^(-1+q))/x/(a*x^m+b*ln(c*x^n)^q)^2,x)

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^m+e*log(c*x^n)^(-1+q))/x/(a*x^m+b*log(c*x^n)^q)^2,x, algorithm="maxima")

[Out]

-(b*d*log(c) + b*d*log(x^n) - a*e)/(a^2*b*m*x^m*log(x^n) - (n*q - m*log(c))*a^2*b*x^m + (a*b^2*m*log(x^n) - (n
*q - m*log(c))*a*b^2)*(log(c) + log(x^n))^q) + integrate(-((e*m*n*(q - 1) - e*m^2*log(c))*a + (d*m*n*q*log(c)
- (q^2 - q)*d*n^2)*b + (b*d*m*n*q - a*e*m^2)*log(x^n))/(a^2*b*m^2*x*x^m*log(x^n)^2 - 2*(m*n*q - m^2*log(c))*a^
2*b*x*x^m*log(x^n) + (n^2*q^2 - 2*m*n*q*log(c) + m^2*log(c)^2)*a^2*b*x*x^m + (a*b^2*m^2*x*log(x^n)^2 - 2*(m*n*
q - m^2*log(c))*a*b^2*x*log(x^n) + (n^2*q^2 - 2*m*n*q*log(c) + m^2*log(c)^2)*a*b^2*x)*(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{d x^{m} + e \log \left (c x^{n}\right )^{q - 1}}{2 \, a b x x^{m} \log \left (c x^{n}\right )^{q} + a^{2} x x^{2 \, m} + b^{2} x \log \left (c x^{n}\right )^{2 \, q}}, x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^m+e*log(c*x^n)^(-1+q))/x/(a*x^m+b*log(c*x^n)^q)^2,x, algorithm="fricas")

[Out]

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

[Out]

Timed out

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^m+e*log(c*x^n)^(-1+q))/x/(a*x^m+b*log(c*x^n)^q)^2,x, algorithm="giac")

[Out]

integrate((d*x^m + e*log(c*x^n)^(q - 1))/((a*x^m + b*log(c*x^n)^q)^2*x), x)