\(\int (c+e x^2)^q (a+c x^2+b x^4)^p \, dx\) [399]

Optimal result

Integrand size = 24, antiderivative size = 24 \[ \int \left (c+e x^2\right )^q \left (a+c x^2+b x^4\right )^p \, dx=\text {Int}\left (\left (c+e x^2\right )^q \left (a+c x^2+b x^4\right )^p,x\right ) \]

[Out]

Unintegrable((e*x^2+c)^q*(b*x^4+c*x^2+a)^p,x)

Rubi [N/A]

Not integrable

Time = 0.01 (sec) , antiderivative size = 24, 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 = {} \[ \int \left (c+e x^2\right )^q \left (a+c x^2+b x^4\right )^p \, dx=\int \left (c+e x^2\right )^q \left (a+c x^2+b x^4\right )^p \, dx \]

[In]

Int[(c + e*x^2)^q*(a + c*x^2 + b*x^4)^p,x]

[Out]

Defer[Int][(c + e*x^2)^q*(a + c*x^2 + b*x^4)^p, x]

Rubi steps \begin{align*} \text {integral}& = \int \left (c+e x^2\right )^q \left (a+c x^2+b x^4\right )^p \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 0.88 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \left (c+e x^2\right )^q \left (a+c x^2+b x^4\right )^p \, dx=\int \left (c+e x^2\right )^q \left (a+c x^2+b x^4\right )^p \, dx \]

[In]

Integrate[(c + e*x^2)^q*(a + c*x^2 + b*x^4)^p,x]

[Out]

Integrate[(c + e*x^2)^q*(a + c*x^2 + b*x^4)^p, x]

Maple [N/A]

Not integrable

Time = 0.08 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00

\[\int \left (e \,x^{2}+c \right )^{q} \left (b \,x^{4}+c \,x^{2}+a \right )^{p}d x\]

[In]

int((e*x^2+c)^q*(b*x^4+c*x^2+a)^p,x)

[Out]

int((e*x^2+c)^q*(b*x^4+c*x^2+a)^p,x)

Fricas [N/A]

Not integrable

Time = 0.55 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \left (c+e x^2\right )^q \left (a+c x^2+b x^4\right )^p \, dx=\int { {\left (b x^{4} + c x^{2} + a\right )}^{p} {\left (e x^{2} + c\right )}^{q} \,d x } \]

[In]

integrate((e*x^2+c)^q*(b*x^4+c*x^2+a)^p,x, algorithm="fricas")

[Out]

integral((b*x^4 + c*x^2 + a)^p*(e*x^2 + c)^q, x)

Sympy [F(-1)]

Timed out. \[ \int \left (c+e x^2\right )^q \left (a+c x^2+b x^4\right )^p \, dx=\text {Timed out} \]

[In]

integrate((e*x**2+c)**q*(b*x**4+c*x**2+a)**p,x)

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 0.24 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \left (c+e x^2\right )^q \left (a+c x^2+b x^4\right )^p \, dx=\int { {\left (b x^{4} + c x^{2} + a\right )}^{p} {\left (e x^{2} + c\right )}^{q} \,d x } \]

[In]

integrate((e*x^2+c)^q*(b*x^4+c*x^2+a)^p,x, algorithm="maxima")

[Out]

integrate((b*x^4 + c*x^2 + a)^p*(e*x^2 + c)^q, x)

Giac [N/A]

Not integrable

Time = 1.44 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \left (c+e x^2\right )^q \left (a+c x^2+b x^4\right )^p \, dx=\int { {\left (b x^{4} + c x^{2} + a\right )}^{p} {\left (e x^{2} + c\right )}^{q} \,d x } \]

[In]

integrate((e*x^2+c)^q*(b*x^4+c*x^2+a)^p,x, algorithm="giac")

[Out]

integrate((b*x^4 + c*x^2 + a)^p*(e*x^2 + c)^q, x)

Mupad [N/A]

Not integrable

Time = 8.07 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \left (c+e x^2\right )^q \left (a+c x^2+b x^4\right )^p \, dx=\int {\left (e\,x^2+c\right )}^q\,{\left (b\,x^4+c\,x^2+a\right )}^p \,d x \]

[In]

int((c + e*x^2)^q*(a + b*x^4 + c*x^2)^p,x)

[Out]

int((c + e*x^2)^q*(a + b*x^4 + c*x^2)^p, x)