\(\int \frac {(a+b x^n+c x^{2 n})^p}{(d+e x^n)^2} \, dx\) [95]

Optimal result

Integrand size = 26, antiderivative size = 26 \[ \int \frac {\left (a+b x^n+c x^{2 n}\right )^p}{\left (d+e x^n\right )^2} \, dx=\text {Int}\left (\frac {\left (a+b x^n+c x^{2 n}\right )^p}{\left (d+e x^n\right )^2},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.01 (sec) , antiderivative size = 26, 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 = {} \[ \int \frac {\left (a+b x^n+c x^{2 n}\right )^p}{\left (d+e x^n\right )^2} \, dx=\int \frac {\left (a+b x^n+c x^{2 n}\right )^p}{\left (d+e x^n\right )^2} \, dx \]

[In]

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

[Out]

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

Rubi steps \begin{align*} \text {integral}& = \int \frac {\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 = 0.62 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.08 \[ \int \frac {\left (a+b x^n+c x^{2 n}\right )^p}{\left (d+e x^n\right )^2} \, dx=\int \frac {\left (a+b x^n+c x^{2 n}\right )^p}{\left (d+e x^n\right )^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A]

Not integrable

Time = 0.08 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.58 \[ \int \frac {\left (a+b x^n+c x^{2 n}\right )^p}{\left (d+e x^n\right )^2} \, dx=\int { \frac {{\left (c x^{2 \, n} + b x^{n} + a\right )}^{p}}{{\left (e x^{n} + d\right )}^{2}} \,d x } \]

[In]

integrate((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/(e^2*x^(2*n) + 2*d*e*x^n + d^2), x)

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a+b x^n+c x^{2 n}\right )^p}{\left (d+e x^n\right )^2} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.08 \[ \int \frac {\left (a+b x^n+c x^{2 n}\right )^p}{\left (d+e x^n\right )^2} \, dx=\int { \frac {{\left (c x^{2 \, n} + b x^{n} + a\right )}^{p}}{{\left (e x^{n} + d\right )}^{2}} \,d x } \]

[In]

integrate((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/(e*x^n + d)^2, x)

Giac [N/A]

Not integrable

Time = 0.37 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.08 \[ \int \frac {\left (a+b x^n+c x^{2 n}\right )^p}{\left (d+e x^n\right )^2} \, dx=\int { \frac {{\left (c x^{2 \, n} + b x^{n} + a\right )}^{p}}{{\left (e x^{n} + d\right )}^{2}} \,d x } \]

[In]

integrate((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/(e*x^n + d)^2, x)

Mupad [N/A]

Not integrable

Time = 11.18 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.08 \[ \int \frac {\left (a+b x^n+c x^{2 n}\right )^p}{\left (d+e x^n\right )^2} \, dx=\int \frac {{\left (a+b\,x^n+c\,x^{2\,n}\right )}^p}{{\left (d+e\,x^n\right )}^2} \,d x \]

[In]

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

[Out]

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