3.37 $$\int \frac{d x^m+e \log ^{-1+q}(c x^n)}{x (a x^m+b \log ^q(c x^n))^3} \, dx$$

Optimal. Leaf size=76 $\left (d-\frac{a e m}{b n q}\right ) \text{CannotIntegrate}\left (\frac{x^{m-1}}{\left (a x^m+b \log ^q\left (c x^n\right )\right )^3},x\right )-\frac{e}{2 b n q \left (a x^m+b \log ^q\left (c x^n\right )\right )^2}$

[Out]

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

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Rubi [A]  time = 0.249227, 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 = {} $\int \frac{d x^m+e \log ^{-1+q}\left (c x^n\right )}{x \left (a x^m+b \log ^q\left (c x^n\right )\right )^3} \, dx$

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)^3),x]

[Out]

-e/(2*b*n*q*(a*x^m + b*Log[c*x^n]^q)^2) + (d - (a*e*m)/(b*n*q))*Defer[Int][x^(-1 + m)/(a*x^m + b*Log[c*x^n]^q)
^3, 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 )^3} \, dx &=-\frac{e}{2 b n q \left (a x^m+b \log ^q\left (c x^n\right )\right )^2}-\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 )^3} \, dx\\ \end{align*}

Mathematica [A]  time = 66.9193, size = 0, normalized size = 0. $\int \frac{d x^m+e \log ^{-1+q}\left (c x^n\right )}{x \left (a x^m+b \log ^q\left (c x^n\right )\right )^3} \, dx$

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)^3),x]

[Out]

Integrate[(d*x^m + e*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 = 91.029, 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 ) ^{3}}}\, 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)^3,x)

[Out]

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

[Out]

-1/2*(a*b*d*m^2*x^m*log(x^n)^3 + (a^2*e*m^2 - (4*d*m*n*q - 3*d*m^2*log(c))*a*b)*x^m*log(x^n)^2 + ((2*e*m^2*log
(c) + e*m*n)*a^2 - (8*d*m*n*q*log(c) - 3*d*m^2*log(c)^2 - (3*q^2 - q)*d*n^2)*a*b)*x^m*log(x^n) - ((e*n^2*q^2 -
e*m^2*log(c)^2 - e*m*n*log(c))*a^2 + (4*d*m*n*q*log(c)^2 - d*m^2*log(c)^3 - (3*q^2 - q)*d*n^2*log(c))*a*b)*x^
m - ((e*m*n*(2*q - 1)*log(c) - 2*e*m^2*log(c)^2)*a*b + (2*d*m*n*q*log(c)^2 - (2*q^2 - q)*d*n^2*log(c))*b^2 + 2
*(b^2*d*m*n*q - a*b*e*m^2)*log(x^n)^2 + ((e*m*n*(2*q - 1) - 4*e*m^2*log(c))*a*b + (4*d*m*n*q*log(c) - (2*q^2 -
q)*d*n^2)*b^2)*log(x^n))*(log(c) + log(x^n))^q)/(a^4*b*m^3*x^(3*m)*log(x^n)^3 - 3*(m^2*n*q - m^3*log(c))*a^4*
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^4*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^4*b*x^(3*m) + (a^2*b^3*m^3*x^m*log(x^n)^3 - 3*(m^2*n*q
- m^3*log(c))*a^2*b^3*x^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^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^2*b^3*x^m)*(log(c) + log(x^n))^(2*q) + 2*
(a^3*b^2*m^3*x^(2*m)*log(x^n)^3 - 3*(m^2*n*q - m^3*log(c))*a^3*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^3*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^3*b^2*x^(2*m))*(log(c) + log(x^n))^q) - integrate(-1/2*(2*(b*d*m^3*n*q - a*e*m^4)*log(x^n)^3 + ((e
*m^3*n*(2*q - 3) - 6*e*m^4*log(c))*a + (6*d*m^3*n*q*log(c) - (2*q^2 - 3*q)*d*m^2*n^2)*b)*log(x^n)^2 + (e*m^3*n
*(2*q - 3)*log(c)^2 - 2*e*m^4*log(c)^3 + 2*(q^2 - 1)*e*m^2*n^2*log(c) - (2*q^3 - 3*q^2 + q)*e*m*n^3)*a + (2*d*
m^3*n*q*log(c)^3 - (2*q^2 - 3*q)*d*m^2*n^2*log(c)^2 - 2*(q^3 - q)*d*m*n^3*log(c) + (2*q^4 - 3*q^3 + q^2)*d*n^4
)*b + 2*((e*m^3*n*(2*q - 3)*log(c) - 3*e*m^4*log(c)^2 + (q^2 - 1)*e*m^2*n^2)*a + (3*d*m^3*n*q*log(c)^2 - (2*q^
2 - 3*q)*d*m^2*n^2*log(c) - (q^3 - q)*d*m*n^3)*b)*log(x^n))/(a^3*b*m^4*x*x^(2*m)*log(x^n)^4 - 4*(m^3*n*q - m^4
*log(c))*a^3*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^3*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^3*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^3*b*x*x^(2*m) + (a^
2*b^2*m^4*x*x^m*log(x^n)^4 - 4*(m^3*n*q - m^4*log(c))*a^2*b^2*x*x^m*log(x^n)^3 + 6*(m^2*n^2*q^2 - 2*m^3*n*q*lo
g(c) + m^4*log(c)^2)*a^2*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^2*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^2*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{d x^{m} + e \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((d*x^m+e*log(c*x^n)^(-1+q))/x/(a*x^m+b*log(c*x^n)^q)^3,x, algorithm="fricas")

[Out]

integral((d*x^m + e*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((d*x**m+e*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{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 )}^{3} 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)^3,x, algorithm="giac")

[Out]

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