\(\int \frac {d x^m+e \log ^{-1+q}(c x^n)}{x (a x^m+b \log ^q(c x^n))^3} \, dx\) [37]

Optimal result

Integrand size = 40, antiderivative size = 40 \[ \int \frac {d x^m+e \log ^{-1+q}\left (c x^n\right )}{x \left (a x^m+b \log ^q\left (c x^n\right )\right )^3} \, dx=-\frac {e}{2 b n q \left (a x^m+b \log ^q\left (c x^n\right )\right )^2}+\left (d-\frac {a e m}{b n q}\right ) \text {Int}\left (\frac {x^{-1+m}}{\left (a x^m+b \log ^q\left (c x^n\right )\right )^3},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.17 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {d x^m+e \log ^{-1+q}\left (c x^n\right )}{x \left (a x^m+b \log ^q\left (c x^n\right )\right )^3} \, dx=\int \frac {d x^m+e \log ^{-1+q}\left (c x^n\right )}{x \left (a x^m+b \log ^q\left (c x^n\right )\right )^3} \, dx \]

[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]

-1/2*e/(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*} \text {integral}& = -\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 [N/A]

Not integrable

Time = 58.91 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.05 \[ \int \frac {d x^m+e \log ^{-1+q}\left (c x^n\right )}{x \left (a x^m+b \log ^q\left (c x^n\right )\right )^3} \, dx=\int \frac {d x^m+e \log ^{-1+q}\left (c x^n\right )}{x \left (a x^m+b \log ^q\left (c x^n\right )\right )^3} \, dx \]

[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]

Maple [N/A]

Not integrable

Time = 0.09 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.00

\[\int \frac {d \,x^{m}+e \ln \left (c \,x^{n}\right )^{-1+q}}{x \left (a \,x^{m}+b \ln \left (c \,x^{n}\right )^{q}\right )^{3}}d x\]

[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)

Fricas [N/A]

Not integrable

Time = 0.74 (sec) , antiderivative size = 89, normalized size of antiderivative = 2.22 \[ \int \frac {d x^m+e \log ^{-1+q}\left (c x^n\right )}{x \left (a x^m+b \log ^q\left (c x^n\right )\right )^3} \, dx=\int { \frac {d x^{m} + e \log \left (c x^{n}\right )^{q - 1}}{{\left (a x^{m} + b \log \left (c x^{n}\right )^{q}\right )}^{3} x} \,d x } \]

[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)

Sympy [F(-1)]

Timed out. \[ \int \frac {d x^m+e \log ^{-1+q}\left (c x^n\right )}{x \left (a x^m+b \log ^q\left (c x^n\right )\right )^3} \, dx=\text {Timed out} \]

[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

Maxima [N/A]

Not integrable

Time = 0.50 (sec) , antiderivative size = 1583, normalized size of antiderivative = 39.58 \[ \int \frac {d x^m+e \log ^{-1+q}\left (c x^n\right )}{x \left (a x^m+b \log ^q\left (c x^n\right )\right )^3} \, dx=\int { \frac {d x^{m} + e \log \left (c x^{n}\right )^{q - 1}}{{\left (a x^{m} + b \log \left (c x^{n}\right )^{q}\right )}^{3} x} \,d x } \]

[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)

Giac [N/A]

Not integrable

Time = 0.47 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.05 \[ \int \frac {d x^m+e \log ^{-1+q}\left (c x^n\right )}{x \left (a x^m+b \log ^q\left (c x^n\right )\right )^3} \, dx=\int { \frac {d x^{m} + e \log \left (c x^{n}\right )^{q - 1}}{{\left (a x^{m} + b \log \left (c x^{n}\right )^{q}\right )}^{3} x} \,d x } \]

[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)

Mupad [N/A]

Not integrable

Time = 1.95 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.05 \[ \int \frac {d x^m+e \log ^{-1+q}\left (c x^n\right )}{x \left (a x^m+b \log ^q\left (c x^n\right )\right )^3} \, dx=\int \frac {d\,x^m+e\,{\ln \left (c\,x^n\right )}^{q-1}}{x\,{\left (a\,x^m+b\,{\ln \left (c\,x^n\right )}^q\right )}^3} \,d x \]

[In]

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

[Out]

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