\(\int \frac {a+b \log (c x^n)}{(d+e x^r)^2} \, dx\) [418]

Optimal result

Integrand size = 20, antiderivative size = 20 \[ \int \frac {a+b \log \left (c x^n\right )}{\left (d+e x^r\right )^2} \, dx=\text {Int}\left (\frac {a+b \log \left (c x^n\right )}{\left (d+e x^r\right )^2},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.01 (sec) , antiderivative size = 20, 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 = {} \[ \int \frac {a+b \log \left (c x^n\right )}{\left (d+e x^r\right )^2} \, dx=\int \frac {a+b \log \left (c x^n\right )}{\left (d+e x^r\right )^2} \, dx \]

[In]

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

[Out]

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

Rubi steps \begin{align*} \text {integral}& = \int \frac {a+b \log \left (c x^n\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. \(161\) vs. \(2(23)=46\).

Time = 1.71 (sec) , antiderivative size = 161, normalized size of antiderivative = 8.05 \[ \int \frac {a+b \log \left (c x^n\right )}{\left (d+e x^r\right )^2} \, dx=\frac {x \left (a d r \operatorname {Hypergeometric2F1}\left (2,\frac {1}{r},1+\frac {1}{r},-\frac {e x^r}{d}\right )+a e r x^r \operatorname {Hypergeometric2F1}\left (2,\frac {1}{r},1+\frac {1}{r},-\frac {e x^r}{d}\right )-b n (-1+r) \left (d+e x^r\right ) \, _3F_2\left (1,\frac {1}{r},\frac {1}{r};1+\frac {1}{r},1+\frac {1}{r};-\frac {e x^r}{d}\right )+b d \log \left (c x^n\right )-b \left (d+e x^r\right ) \operatorname {Hypergeometric2F1}\left (1,\frac {1}{r},1+\frac {1}{r},-\frac {e x^r}{d}\right ) \left (n-(-1+r) \log \left (c x^n\right )\right )\right )}{d^2 r \left (d+e x^r\right )} \]

[In]

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

[Out]

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

Maple [N/A]

Not integrable

Time = 0.05 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.75 \[ \int \frac {a+b \log \left (c x^n\right )}{\left (d+e x^r\right )^2} \, dx=\int { \frac {b \log \left (c x^{n}\right ) + a}{{\left (e x^{r} + d\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 9.13 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.95 \[ \int \frac {a+b \log \left (c x^n\right )}{\left (d+e x^r\right )^2} \, dx=\int \frac {a + b \log {\left (c x^{n} \right )}}{\left (d + e x^{r}\right )^{2}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.22 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {a+b \log \left (c x^n\right )}{\left (d+e x^r\right )^2} \, dx=\int { \frac {b \log \left (c x^{n}\right ) + a}{{\left (e x^{r} + d\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.40 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {a+b \log \left (c x^n\right )}{\left (d+e x^r\right )^2} \, dx=\int { \frac {b \log \left (c x^{n}\right ) + a}{{\left (e x^{r} + d\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 0.56 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {a+b \log \left (c x^n\right )}{\left (d+e x^r\right )^2} \, dx=\int \frac {a+b\,\ln \left (c\,x^n\right )}{{\left (d+e\,x^r\right )}^2} \,d x \]

[In]

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

[Out]

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