\(\int \frac {a+b \log (c x^n)}{d+e x^r} \, dx\) [411]

Optimal result

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

[Out]

Unintegrable((a+b*ln(c*x^n))/(d+e*x^r),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 = {} \[ \int \frac {a+b \log \left (c x^n\right )}{d+e x^r} \, dx=\int \frac {a+b \log \left (c x^n\right )}{d+e x^r} \, dx \]

[In]

Int[(a + b*Log[c*x^n])/(d + e*x^r),x]

[Out]

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

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(69\) vs. \(2(23)=46\).

Time = 0.07 (sec) , antiderivative size = 69, normalized size of antiderivative = 3.45 \[ \int \frac {a+b \log \left (c x^n\right )}{d+e x^r} \, dx=\frac {x \left (-b n \, _3F_2\left (1,\frac {1}{r},\frac {1}{r};1+\frac {1}{r},1+\frac {1}{r};-\frac {e x^r}{d}\right )+\operatorname {Hypergeometric2F1}\left (1,\frac {1}{r},1+\frac {1}{r},-\frac {e x^r}{d}\right ) \left (a+b \log \left (c x^n\right )\right )\right )}{d} \]

[In]

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

[Out]

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

Maple [N/A]

Not integrable

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

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {a+b \log \left (c x^n\right )}{d+e x^r} \, dx=\int { \frac {b \log \left (c x^{n}\right ) + a}{e x^{r} + d} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 1.59 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.85 \[ \int \frac {a+b \log \left (c x^n\right )}{d+e x^r} \, dx=\int \frac {a + b \log {\left (c x^{n} \right )}}{d + e x^{r}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.23 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {a+b \log \left (c x^n\right )}{d+e x^r} \, dx=\int { \frac {b \log \left (c x^{n}\right ) + a}{e x^{r} + d} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.33 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {a+b \log \left (c x^n\right )}{d+e x^r} \, dx=\int { \frac {b \log \left (c x^{n}\right ) + a}{e x^{r} + d} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 0.65 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {a+b \log \left (c x^n\right )}{d+e x^r} \, dx=\int \frac {a+b\,\ln \left (c\,x^n\right )}{d+e\,x^r} \,d x \]

[In]

int((a + b*log(c*x^n))/(d + e*x^r),x)

[Out]

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