\(\int \frac {a+b \log (c (d+e x^m)^n)}{x \log ^3(f x^p)} \, dx\) [627]

Optimal result

Integrand size = 28, antiderivative size = 28 \[ \int \frac {a+b \log \left (c \left (d+e x^m\right )^n\right )}{x \log ^3\left (f x^p\right )} \, dx=-\frac {a+b \log \left (c \left (d+e x^m\right )^n\right )}{2 p \log ^2\left (f x^p\right )}+\frac {b e m n \text {Int}\left (\frac {x^{-1+m}}{\left (d+e x^m\right ) \log ^2\left (f x^p\right )},x\right )}{2 p} \]

[Out]

1/2*(-a-b*ln(c*(d+e*x^m)^n))/p/ln(f*x^p)^2+1/2*b*e*m*n*Unintegrable(x^(-1+m)/(d+e*x^m)/ln(f*x^p)^2,x)/p

Rubi [N/A]

Not integrable

Time = 0.08 (sec) , antiderivative size = 28, 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 = {} \[ \int \frac {a+b \log \left (c \left (d+e x^m\right )^n\right )}{x \log ^3\left (f x^p\right )} \, dx=\int \frac {a+b \log \left (c \left (d+e x^m\right )^n\right )}{x \log ^3\left (f x^p\right )} \, dx \]

[In]

Int[(a + b*Log[c*(d + e*x^m)^n])/(x*Log[f*x^p]^3),x]

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 8.83 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {a+b \log \left (c \left (d+e x^m\right )^n\right )}{x \log ^3\left (f x^p\right )} \, dx=\int \frac {a+b \log \left (c \left (d+e x^m\right )^n\right )}{x \log ^3\left (f x^p\right )} \, dx \]

[In]

Integrate[(a + b*Log[c*(d + e*x^m)^n])/(x*Log[f*x^p]^3),x]

[Out]

Integrate[(a + b*Log[c*(d + e*x^m)^n])/(x*Log[f*x^p]^3), x]

Maple [N/A]

Not integrable

Time = 0.06 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00

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

[In]

int((a+b*ln(c*(d+e*x^m)^n))/x/ln(f*x^p)^3,x)

[Out]

int((a+b*ln(c*(d+e*x^m)^n))/x/ln(f*x^p)^3,x)

Fricas [N/A]

Not integrable

Time = 0.31 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {a+b \log \left (c \left (d+e x^m\right )^n\right )}{x \log ^3\left (f x^p\right )} \, dx=\int { \frac {b \log \left ({\left (e x^{m} + d\right )}^{n} c\right ) + a}{x \log \left (f x^{p}\right )^{3}} \,d x } \]

[In]

integrate((a+b*log(c*(d+e*x^m)^n))/x/log(f*x^p)^3,x, algorithm="fricas")

[Out]

integral((b*log((e*x^m + d)^n*c) + a)/(x*log(f*x^p)^3), x)

Sympy [F(-1)]

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

[In]

integrate((a+b*ln(c*(d+e*x**m)**n))/x/ln(f*x**p)**3,x)

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 0.32 (sec) , antiderivative size = 238, normalized size of antiderivative = 8.50 \[ \int \frac {a+b \log \left (c \left (d+e x^m\right )^n\right )}{x \log ^3\left (f x^p\right )} \, dx=\int { \frac {b \log \left ({\left (e x^{m} + d\right )}^{n} c\right ) + a}{x \log \left (f x^{p}\right )^{3}} \,d x } \]

[In]

integrate((a+b*log(c*(d+e*x^m)^n))/x/log(f*x^p)^3,x, algorithm="maxima")

[Out]

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

Giac [N/A]

Not integrable

Time = 0.34 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {a+b \log \left (c \left (d+e x^m\right )^n\right )}{x \log ^3\left (f x^p\right )} \, dx=\int { \frac {b \log \left ({\left (e x^{m} + d\right )}^{n} c\right ) + a}{x \log \left (f x^{p}\right )^{3}} \,d x } \]

[In]

integrate((a+b*log(c*(d+e*x^m)^n))/x/log(f*x^p)^3,x, algorithm="giac")

[Out]

integrate((b*log((e*x^m + d)^n*c) + a)/(x*log(f*x^p)^3), x)

Mupad [N/A]

Not integrable

Time = 2.00 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {a+b \log \left (c \left (d+e x^m\right )^n\right )}{x \log ^3\left (f x^p\right )} \, dx=\int \frac {a+b\,\ln \left (c\,{\left (d+e\,x^m\right )}^n\right )}{x\,{\ln \left (f\,x^p\right )}^3} \,d x \]

[In]

int((a + b*log(c*(d + e*x^m)^n))/(x*log(f*x^p)^3),x)

[Out]

int((a + b*log(c*(d + e*x^m)^n))/(x*log(f*x^p)^3), x)