\(\int \frac {(f x)^m (a+b \log (c x^n))}{(d+e x^2)^2} \, dx\) [323]

Optimal result

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

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.04 (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 = {} \[ \int \frac {(f x)^m \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^2} \, dx=\int \frac {(f x)^m \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^2} \, dx \]

[In]

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

[Out]

Defer[Int][((f*x)^m*(a + b*Log[c*x^n]))/(d + e*x^2)^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^2\right )^2} \, dx \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(108\) vs. \(2(28)=56\).

Time = 0.14 (sec) , antiderivative size = 108, normalized size of antiderivative = 4.32 \[ \int \frac {(f x)^m \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^2} \, dx=\frac {x (f x)^m \left (-b n \, _3F_2\left (2,\frac {1}{2}+\frac {m}{2},\frac {1}{2}+\frac {m}{2};\frac {3}{2}+\frac {m}{2},\frac {3}{2}+\frac {m}{2};-\frac {e x^2}{d}\right )+(1+m) \operatorname {Hypergeometric2F1}\left (2,\frac {1+m}{2},\frac {3+m}{2},-\frac {e x^2}{d}\right ) \left (a+b \log \left (c x^n\right )\right )\right )}{d^2 (1+m)^2} \]

[In]

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

[Out]

(x*(f*x)^m*(-(b*n*HypergeometricPFQ[{2, 1/2 + m/2, 1/2 + m/2}, {3/2 + m/2, 3/2 + m/2}, -((e*x^2)/d)]) + (1 + m
)*Hypergeometric2F1[2, (1 + m)/2, (3 + m)/2, -((e*x^2)/d)]*(a + b*Log[c*x^n])))/(d^2*(1 + m)^2)

Maple [N/A]

Not integrable

Time = 0.04 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.76 \[ \int \frac {(f x)^m \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^2} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} \left (f x\right )^{m}}{{\left (e x^{2} + d\right )}^{2}} \,d x } \]

[In]

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

[Out]

integral(((f*x)^m*b*log(c*x^n) + (f*x)^m*a)/(e^2*x^4 + 2*d*e*x^2 + d^2), x)

Sympy [N/A]

Not integrable

Time = 162.22 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96 \[ \int \frac {(f x)^m \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^2} \, dx=\int \frac {\left (f x\right )^{m} \left (a + b \log {\left (c x^{n} \right )}\right )}{\left (d + e x^{2}\right )^{2}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.22 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {(f x)^m \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^2} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} \left (f x\right )^{m}}{{\left (e x^{2} + d\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.42 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {(f x)^m \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^2} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} \left (f x\right )^{m}}{{\left (e x^{2} + d\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 0.45 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {(f x)^m \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^2} \, dx=\int \frac {{\left (f\,x\right )}^m\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{{\left (e\,x^2+d\right )}^2} \,d x \]

[In]

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

[Out]

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