\(\int (d+e x^n)^q (a+c x^{2 n})^p \, dx\) [59]

Optimal result

Integrand size = 21, antiderivative size = 21 \[ \int \left (d+e x^n\right )^q \left (a+c x^{2 n}\right )^p \, dx=\text {Int}\left (\left (d+e x^n\right )^q \left (a+c x^{2 n}\right )^p,x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.01 (sec) , antiderivative size = 21, 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 = {} \[ \int \left (d+e x^n\right )^q \left (a+c x^{2 n}\right )^p \, dx=\int \left (d+e x^n\right )^q \left (a+c x^{2 n}\right )^p \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 0.30 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int \left (d+e x^n\right )^q \left (a+c x^{2 n}\right )^p \, dx=\int \left (d+e x^n\right )^q \left (a+c x^{2 n}\right )^p \, dx \]

[In]

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

[Out]

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

Maple [N/A]

Not integrable

Time = 0.16 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.30 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int \left (d+e x^n\right )^q \left (a+c x^{2 n}\right )^p \, dx=\int { {\left (c x^{2 \, n} + a\right )}^{p} {\left (e x^{n} + d\right )}^{q} \,d x } \]

[In]

integrate((d+e*x^n)^q*(a+c*x^(2*n))^p,x, algorithm="fricas")

[Out]

integral((c*x^(2*n) + a)^p*(e*x^n + d)^q, x)

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int \left (d+e x^n\right )^q \left (a+c x^{2 n}\right )^p \, dx=\int { {\left (c x^{2 \, n} + a\right )}^{p} {\left (e x^{n} + d\right )}^{q} \,d x } \]

[In]

integrate((d+e*x^n)^q*(a+c*x^(2*n))^p,x, algorithm="maxima")

[Out]

integrate((c*x^(2*n) + a)^p*(e*x^n + d)^q, x)

Giac [N/A]

Not integrable

Time = 0.37 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int \left (d+e x^n\right )^q \left (a+c x^{2 n}\right )^p \, dx=\int { {\left (c x^{2 \, n} + a\right )}^{p} {\left (e x^{n} + d\right )}^{q} \,d x } \]

[In]

integrate((d+e*x^n)^q*(a+c*x^(2*n))^p,x, algorithm="giac")

[Out]

integrate((c*x^(2*n) + a)^p*(e*x^n + d)^q, x)

Mupad [N/A]

Not integrable

Time = 8.93 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int \left (d+e x^n\right )^q \left (a+c x^{2 n}\right )^p \, dx=\int {\left (a+c\,x^{2\,n}\right )}^p\,{\left (d+e\,x^n\right )}^q \,d x \]

[In]

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

[Out]

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