\(\int \frac {x (a+b \log (c x^n))}{(d+e x^r)^2} \, dx\) [414]

Optimal result

Integrand size = 21, antiderivative size = 21 \[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^r\right )^2} \, dx=\text {Int}\left (\frac {x \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^r\right )^2},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.03 (sec) , antiderivative size = 21, 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 = {} \[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^r\right )^2} \, dx=\int \frac {x \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^r\right )^2} \, dx \]

[In]

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

[Out]

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

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(140\) vs. \(2(24)=48\).

Time = 0.16 (sec) , antiderivative size = 140, normalized size of antiderivative = 6.67 \[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^r\right )^2} \, dx=\frac {x^2 \left (-b n (-2+r) \left (d+e x^r\right ) \, _3F_2\left (1,\frac {2}{r},\frac {2}{r};1+\frac {2}{r},1+\frac {2}{r};-\frac {e x^r}{d}\right )+4 d \left (a+b \log \left (c x^n\right )\right )+2 \left (d+e x^r\right ) \operatorname {Hypergeometric2F1}\left (1,\frac {2}{r},\frac {2+r}{r},-\frac {e x^r}{d}\right ) \left (-b n+a (-2+r)+b (-2+r) \log \left (c x^n\right )\right )\right )}{4 d^2 r \left (d+e x^r\right )} \]

[In]

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

[Out]

(x^2*(-(b*n*(-2 + r)*(d + e*x^r)*HypergeometricPFQ[{1, 2/r, 2/r}, {1 + 2/r, 1 + 2/r}, -((e*x^r)/d)]) + 4*d*(a
+ b*Log[c*x^n]) + 2*(d + e*x^r)*Hypergeometric2F1[1, 2/r, (2 + r)/r, -((e*x^r)/d)]*(-(b*n) + a*(-2 + r) + b*(-
2 + r)*Log[c*x^n])))/(4*d^2*r*(d + e*x^r))

Maple [N/A]

Not integrable

Time = 0.06 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.81 \[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^r\right )^2} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} x}{{\left (e x^{r} + d\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 10.05 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.95 \[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^r\right )^2} \, dx=\int \frac {x \left (a + b \log {\left (c x^{n} \right )}\right )}{\left (d + e x^{r}\right )^{2}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.23 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^r\right )^2} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} x}{{\left (e x^{r} + d\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.31 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^r\right )^2} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} x}{{\left (e x^{r} + d\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 0.55 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^r\right )^2} \, dx=\int \frac {x\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{{\left (d+e\,x^r\right )}^2} \,d x \]

[In]

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

[Out]

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