\(\int \frac {(f x)^m (a+b \log (c x^n))^p}{(d+e x^r)^2} \, dx\) [453]

Optimal result

Integrand size = 27, antiderivative size = 27 \[ \int \frac {(f x)^m \left (a+b \log \left (c x^n\right )\right )^p}{\left (d+e x^r\right )^2} \, dx=\text {Int}\left (\frac {(f x)^m \left (a+b \log \left (c x^n\right )\right )^p}{\left (d+e x^r\right )^2},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.07 (sec) , antiderivative size = 27, 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 = {} \[ \int \frac {(f x)^m \left (a+b \log \left (c x^n\right )\right )^p}{\left (d+e x^r\right )^2} \, dx=\int \frac {(f x)^m \left (a+b \log \left (c x^n\right )\right )^p}{\left (d+e x^r\right )^2} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 1.34 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.07 \[ \int \frac {(f x)^m \left (a+b \log \left (c x^n\right )\right )^p}{\left (d+e x^r\right )^2} \, dx=\int \frac {(f x)^m \left (a+b \log \left (c x^n\right )\right )^p}{\left (d+e x^r\right )^2} \, dx \]

[In]

Integrate[((f*x)^m*(a + b*Log[c*x^n])^p)/(d + e*x^r)^2,x]

[Out]

Integrate[((f*x)^m*(a + b*Log[c*x^n])^p)/(d + e*x^r)^2, x]

Maple [N/A]

Not integrable

Time = 0.14 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00

\[\int \frac {\left (f x \right )^{m} {\left (a +b \ln \left (c \,x^{n}\right )\right )}^{p}}{\left (d +e \,x^{r}\right )^{2}}d x\]

[In]

int((f*x)^m*(a+b*ln(c*x^n))^p/(d+e*x^r)^2,x)

[Out]

int((f*x)^m*(a+b*ln(c*x^n))^p/(d+e*x^r)^2,x)

Fricas [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.56 \[ \int \frac {(f x)^m \left (a+b \log \left (c x^n\right )\right )^p}{\left (d+e x^r\right )^2} \, dx=\int { \frac {\left (f x\right )^{m} {\left (b \log \left (c x^{n}\right ) + a\right )}^{p}}{{\left (e x^{r} + d\right )}^{2}} \,d x } \]

[In]

integrate((f*x)^m*(a+b*log(c*x^n))^p/(d+e*x^r)^2,x, algorithm="fricas")

[Out]

integral((f*x)^m*(b*log(c*x^n) + a)^p/(e^2*x^(2*r) + 2*d*e*x^r + d^2), x)

Sympy [N/A]

Not integrable

Time = 0.00 (sec) , antiderivative size = 1, normalized size of antiderivative = 0.04 \[ \int \frac {(f x)^m \left (a+b \log \left (c x^n\right )\right )^p}{\left (d+e x^r\right )^2} \, dx=\int \frac {\left (f x\right )^{m} \left (a + b \log {\left (c x^{n} \right )}\right )^{p}}{\left (d + e x^{r}\right )^{2}} \, dx \]

[In]

integrate((f*x)**m*(a+b*ln(c*x**n))**p/(d+e*x**r)**2,x)

[Out]

integrate((f*x)**m*(a+b*ln(c*x**n))**p/(d+e*x**r)**2,x)

Maxima [F(-2)]

Exception generated. \[ \int \frac {(f x)^m \left (a+b \log \left (c x^n\right )\right )^p}{\left (d+e x^r\right )^2} \, dx=\text {Exception raised: RuntimeError} \]

[In]

integrate((f*x)^m*(a+b*log(c*x^n))^p/(d+e*x^r)^2,x, algorithm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: In function CAR, the value of the first argument is  0which is not
of the expected type LIST

Giac [N/A]

Not integrable

Time = 0.38 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.07 \[ \int \frac {(f x)^m \left (a+b \log \left (c x^n\right )\right )^p}{\left (d+e x^r\right )^2} \, dx=\int { \frac {\left (f x\right )^{m} {\left (b \log \left (c x^{n}\right ) + a\right )}^{p}}{{\left (e x^{r} + d\right )}^{2}} \,d x } \]

[In]

integrate((f*x)^m*(a+b*log(c*x^n))^p/(d+e*x^r)^2,x, algorithm="giac")

[Out]

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

Mupad [N/A]

Not integrable

Time = 0.43 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.07 \[ \int \frac {(f x)^m \left (a+b \log \left (c x^n\right )\right )^p}{\left (d+e x^r\right )^2} \, dx=\int \frac {{\left (f\,x\right )}^m\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^p}{{\left (d+e\,x^r\right )}^2} \,d x \]

[In]

int(((f*x)^m*(a + b*log(c*x^n))^p)/(d + e*x^r)^2,x)

[Out]

int(((f*x)^m*(a + b*log(c*x^n))^p)/(d + e*x^r)^2, x)