\(\int \frac {a+b \log (c x^n)}{x^2 (d+e x^r)^2} \, dx\) [419]

Optimal result

Integrand size = 23, antiderivative size = 23 \[ \int \frac {a+b \log \left (c x^n\right )}{x^2 \left (d+e x^r\right )^2} \, dx=\text {Int}\left (\frac {a+b \log \left (c x^n\right )}{x^2 \left (d+e x^r\right )^2},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.04 (sec) , antiderivative size = 23, 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 = {} \[ \int \frac {a+b \log \left (c x^n\right )}{x^2 \left (d+e x^r\right )^2} \, dx=\int \frac {a+b \log \left (c x^n\right )}{x^2 \left (d+e x^r\right )^2} \, dx \]

[In]

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

[Out]

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

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(135\) vs. \(2(26)=52\).

Time = 0.14 (sec) , antiderivative size = 135, normalized size of antiderivative = 5.87 \[ \int \frac {a+b \log \left (c x^n\right )}{x^2 \left (d+e x^r\right )^2} \, dx=\frac {-b n (1+r) \left (d+e x^r\right ) \, _3F_2\left (1,-\frac {1}{r},-\frac {1}{r};1-\frac {1}{r},1-\frac {1}{r};-\frac {e x^r}{d}\right )+d \left (a+b \log \left (c x^n\right )\right )-\left (d+e x^r\right ) \operatorname {Hypergeometric2F1}\left (1,-\frac {1}{r},\frac {-1+r}{r},-\frac {e x^r}{d}\right ) \left (a-b n+a r+b (1+r) \log \left (c x^n\right )\right )}{d^2 r x \left (d+e x^r\right )} \]

[In]

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

[Out]

(-(b*n*(1 + r)*(d + e*x^r)*HypergeometricPFQ[{1, -r^(-1), -r^(-1)}, {1 - r^(-1), 1 - r^(-1)}, -((e*x^r)/d)]) +
d*(a + b*Log[c*x^n]) - (d + e*x^r)*Hypergeometric2F1[1, -r^(-1), (-1 + r)/r, -((e*x^r)/d)]*(a - b*n + a*r + b
*(1 + r)*Log[c*x^n]))/(d^2*r*x*(d + e*x^r))

Maple [N/A]

Not integrable

Time = 0.07 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.96 \[ \int \frac {a+b \log \left (c x^n\right )}{x^2 \left (d+e x^r\right )^2} \, dx=\int { \frac {b \log \left (c x^{n}\right ) + a}{{\left (e x^{r} + d\right )}^{2} x^{2}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 54.17 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96 \[ \int \frac {a+b \log \left (c x^n\right )}{x^2 \left (d+e x^r\right )^2} \, dx=\int \frac {a + b \log {\left (c x^{n} \right )}}{x^{2} \left (d + e x^{r}\right )^{2}}\, dx \]

[In]

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

[Out]

Integral((a + b*log(c*x**n))/(x**2*(d + e*x**r)**2), x)

Maxima [N/A]

Not integrable

Time = 0.24 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {a+b \log \left (c x^n\right )}{x^2 \left (d+e x^r\right )^2} \, dx=\int { \frac {b \log \left (c x^{n}\right ) + a}{{\left (e x^{r} + d\right )}^{2} x^{2}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.30 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {a+b \log \left (c x^n\right )}{x^2 \left (d+e x^r\right )^2} \, dx=\int { \frac {b \log \left (c x^{n}\right ) + a}{{\left (e x^{r} + d\right )}^{2} x^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 0.47 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {a+b \log \left (c x^n\right )}{x^2 \left (d+e x^r\right )^2} \, dx=\int \frac {a+b\,\ln \left (c\,x^n\right )}{x^2\,{\left (d+e\,x^r\right )}^2} \,d x \]

[In]

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

[Out]

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