\(\int \frac {(f x)^m (a+b \log (c x^n))}{d+e x^r} \, dx\) [444]

Optimal result

Integrand size = 25, antiderivative size = 25 \[ \int \frac {(f x)^m \left (a+b \log \left (c x^n\right )\right )}{d+e x^r} \, dx=\text {Int}\left (\frac {(f x)^m \left (a+b \log \left (c x^n\right )\right )}{d+e x^r},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.05 (sec) , antiderivative size = 25, 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 = {} \[ \int \frac {(f x)^m \left (a+b \log \left (c x^n\right )\right )}{d+e x^r} \, dx=\int \frac {(f x)^m \left (a+b \log \left (c x^n\right )\right )}{d+e x^r} \, dx \]

[In]

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

[Out]

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

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(111\) vs. \(2(28)=56\).

Time = 0.16 (sec) , antiderivative size = 111, normalized size of antiderivative = 4.44 \[ \int \frac {(f x)^m \left (a+b \log \left (c x^n\right )\right )}{d+e x^r} \, dx=\frac {x (f x)^m \left (-b n \, _3F_2\left (1,\frac {1}{r}+\frac {m}{r},\frac {1}{r}+\frac {m}{r};1+\frac {1}{r}+\frac {m}{r},1+\frac {1}{r}+\frac {m}{r};-\frac {e x^r}{d}\right )+(1+m) \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{r},\frac {1+m+r}{r},-\frac {e x^r}{d}\right ) \left (a+b \log \left (c x^n\right )\right )\right )}{d (1+m)^2} \]

[In]

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

[Out]

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

Maple [N/A]

Not integrable

Time = 0.08 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.32 \[ \int \frac {(f x)^m \left (a+b \log \left (c x^n\right )\right )}{d+e x^r} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} \left (f x\right )^{m}}{e x^{r} + d} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 6.05 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.88 \[ \int \frac {(f x)^m \left (a+b \log \left (c x^n\right )\right )}{d+e x^r} \, dx=\int \frac {\left (f x\right )^{m} \left (a + b \log {\left (c x^{n} \right )}\right )}{d + e x^{r}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {(f x)^m \left (a+b \log \left (c x^n\right )\right )}{d+e x^r} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} \left (f x\right )^{m}}{e x^{r} + d} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.33 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {(f x)^m \left (a+b \log \left (c x^n\right )\right )}{d+e x^r} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} \left (f x\right )^{m}}{e x^{r} + d} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 0.45 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {(f x)^m \left (a+b \log \left (c x^n\right )\right )}{d+e x^r} \, dx=\int \frac {{\left (f\,x\right )}^m\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{d+e\,x^r} \,d x \]

[In]

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

[Out]

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