\(\int \frac {(f x)^m (a+b \log (c x^n))}{(d+e x^r)^2} \, dx\) [445]

Optimal result

Integrand size = 25, antiderivative size = 25 \[ \int \frac {(f x)^m \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^r\right )^2} \, dx=\text {Int}\left (\frac {(f x)^m \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^r\right )^2},x\right ) \]

[Out]

Unintegrable((f*x)^m*(a+b*ln(c*x^n))/(d+e*x^r)^2,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 = {} \[ \int \frac {(f x)^m \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^r\right )^2} \, dx=\int \frac {(f x)^m \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^r\right )^2} \, dx \]

[In]

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

[Out]

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

Rubi steps \begin{align*} \text {integral}& = \int \frac {(f x)^m \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. \(177\) vs. \(2(28)=56\).

Time = 0.30 (sec) , antiderivative size = 177, normalized size of antiderivative = 7.08 \[ \int \frac {(f x)^m \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^r\right )^2} \, dx=\frac {x (f x)^m \left (b n (1+m-r) \left (d+e x^r\right ) \, _3F_2\left (1,\frac {1}{r}+\frac {m}{r},\frac {1}{r}+\frac {m}{r};1+\frac {1}{r}+\frac {m}{r},1+\frac {1}{r}+\frac {m}{r};-\frac {e x^r}{d}\right )-(1+m) \left (-d (1+m) \left (a+b \log \left (c x^n\right )\right )+\left (d+e x^r\right ) \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{r},\frac {1+m+r}{r},-\frac {e x^r}{d}\right ) \left (b n+a (1+m-r)+b (1+m-r) \log \left (c x^n\right )\right )\right )\right )}{d^2 (1+m)^2 r \left (d+e x^r\right )} \]

[In]

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

[Out]

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

Maple [N/A]

Not integrable

Time = 0.06 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.84 \[ \int \frac {(f x)^m \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^r\right )^2} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} \left (f x\right )^{m}}{{\left (e x^{r} + d\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 23.53 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96 \[ \int \frac {(f x)^m \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^r\right )^2} \, dx=\int \frac {\left (f x\right )^{m} \left (a + b \log {\left (c x^{n} \right )}\right )}{\left (d + e x^{r}\right )^{2}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {(f x)^m \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^r\right )^2} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} \left (f x\right )^{m}}{{\left (e x^{r} + d\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.34 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {(f x)^m \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^r\right )^2} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} \left (f x\right )^{m}}{{\left (e x^{r} + d\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 0.52 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {(f x)^m \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^r\right )^2} \, dx=\int \frac {{\left (f\,x\right )}^m\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{{\left (d+e\,x^r\right )}^2} \,d x \]

[In]

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

[Out]

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