\(\int \frac {(f x)^m (a+b x^n+c x^{2 n})^p}{(d+e x^n)^2} \, dx\) [156]

Optimal result

Integrand size = 31, antiderivative size = 31 \[ \int \frac {(f x)^m \left (a+b x^n+c x^{2 n}\right )^p}{\left (d+e x^n\right )^2} \, dx=\text {Int}\left (\frac {(f x)^m \left (a+b x^n+c x^{2 n}\right )^p}{\left (d+e x^n\right )^2},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.02 (sec) , antiderivative size = 31, 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 = {} \[ \int \frac {(f x)^m \left (a+b x^n+c x^{2 n}\right )^p}{\left (d+e x^n\right )^2} \, dx=\int \frac {(f x)^m \left (a+b x^n+c x^{2 n}\right )^p}{\left (d+e x^n\right )^2} \, dx \]

[In]

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

[Out]

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

Rubi steps \begin{align*} \text {integral}& = \int \frac {(f x)^m \left (a+b x^n+c x^{2 n}\right )^p}{\left (d+e x^n\right )^2} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 0.93 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.06 \[ \int \frac {(f x)^m \left (a+b x^n+c x^{2 n}\right )^p}{\left (d+e x^n\right )^2} \, dx=\int \frac {(f x)^m \left (a+b x^n+c x^{2 n}\right )^p}{\left (d+e x^n\right )^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A]

Not integrable

Time = 0.14 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.31 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.48 \[ \int \frac {(f x)^m \left (a+b x^n+c x^{2 n}\right )^p}{\left (d+e x^n\right )^2} \, dx=\int { \frac {{\left (c x^{2 \, n} + b x^{n} + a\right )}^{p} \left (f x\right )^{m}}{{\left (e x^{n} + d\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Sympy [F(-2)]

Exception generated. \[ \int \frac {(f x)^m \left (a+b x^n+c x^{2 n}\right )^p}{\left (d+e x^n\right )^2} \, dx=\text {Exception raised: HeuristicGCDFailed} \]

[In]

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

[Out]

Exception raised: HeuristicGCDFailed >> no luck

Maxima [N/A]

Not integrable

Time = 0.24 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.06 \[ \int \frac {(f x)^m \left (a+b x^n+c x^{2 n}\right )^p}{\left (d+e x^n\right )^2} \, dx=\int { \frac {{\left (c x^{2 \, n} + b x^{n} + a\right )}^{p} \left (f x\right )^{m}}{{\left (e x^{n} + d\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.49 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.06 \[ \int \frac {(f x)^m \left (a+b x^n+c x^{2 n}\right )^p}{\left (d+e x^n\right )^2} \, dx=\int { \frac {{\left (c x^{2 \, n} + b x^{n} + a\right )}^{p} \left (f x\right )^{m}}{{\left (e x^{n} + d\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 9.02 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.06 \[ \int \frac {(f x)^m \left (a+b x^n+c x^{2 n}\right )^p}{\left (d+e x^n\right )^2} \, dx=\int \frac {{\left (f\,x\right )}^m\,{\left (a+b\,x^n+c\,x^{2\,n}\right )}^p}{{\left (d+e\,x^n\right )}^2} \,d x \]

[In]

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

[Out]

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