3.220 \(\int (2 x+x^3) (1+4 x^2+x^4) \, dx\)

Optimal. Leaf size=16 \[ \frac{1}{8} \left (x^4+4 x^2+1\right )^2 \]

[Out]

(1 + 4*x^2 + x^4)^2/8

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Rubi [A]  time = 0.0066653, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056, Rules used = {1588} \[ \frac{1}{8} \left (x^4+4 x^2+1\right )^2 \]

Antiderivative was successfully verified.

[In]

Int[(2*x + x^3)*(1 + 4*x^2 + x^4),x]

[Out]

(1 + 4*x^2 + x^4)^2/8

Rule 1588

Int[(Pp_)*(Qq_)^(m_.), x_Symbol] :> With[{p = Expon[Pp, x], q = Expon[Qq, x]}, Simp[(Coeff[Pp, x, p]*x^(p - q
+ 1)*Qq^(m + 1))/((p + m*q + 1)*Coeff[Qq, x, q]), x] /; NeQ[p + m*q + 1, 0] && EqQ[(p + m*q + 1)*Coeff[Qq, x,
q]*Pp, Coeff[Pp, x, p]*x^(p - q)*((p - q + 1)*Qq + (m + 1)*x*D[Qq, x])]] /; FreeQ[m, x] && PolyQ[Pp, x] && Pol
yQ[Qq, x] && NeQ[m, -1]

Rubi steps

\begin{align*} \int \left (2 x+x^3\right ) \left (1+4 x^2+x^4\right ) \, dx &=\frac{1}{8} \left (1+4 x^2+x^4\right )^2\\ \end{align*}

Mathematica [A]  time = 0.0014668, size = 21, normalized size = 1.31 \[ \frac{x^8}{8}+x^6+\frac{9 x^4}{4}+x^2 \]

Antiderivative was successfully verified.

[In]

Integrate[(2*x + x^3)*(1 + 4*x^2 + x^4),x]

[Out]

x^2 + (9*x^4)/4 + x^6 + x^8/8

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Maple [A]  time = 0.001, size = 18, normalized size = 1.1 \begin{align*}{\frac{{x}^{8}}{8}}+{x}^{6}+{\frac{9\,{x}^{4}}{4}}+{x}^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^3+2*x)*(x^4+4*x^2+1),x)

[Out]

1/8*x^8+x^6+9/4*x^4+x^2

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Maxima [A]  time = 0.985187, size = 19, normalized size = 1.19 \begin{align*} \frac{1}{8} \,{\left (x^{4} + 4 \, x^{2} + 1\right )}^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3+2*x)*(x^4+4*x^2+1),x, algorithm="maxima")

[Out]

1/8*(x^4 + 4*x^2 + 1)^2

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Fricas [A]  time = 1.10848, size = 42, normalized size = 2.62 \begin{align*} \frac{1}{8} x^{8} + x^{6} + \frac{9}{4} x^{4} + x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3+2*x)*(x^4+4*x^2+1),x, algorithm="fricas")

[Out]

1/8*x^8 + x^6 + 9/4*x^4 + x^2

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Sympy [A]  time = 0.053976, size = 17, normalized size = 1.06 \begin{align*} \frac{x^{8}}{8} + x^{6} + \frac{9 x^{4}}{4} + x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**3+2*x)*(x**4+4*x**2+1),x)

[Out]

x**8/8 + x**6 + 9*x**4/4 + x**2

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Giac [A]  time = 1.1962, size = 23, normalized size = 1.44 \begin{align*} \frac{1}{8} \, x^{8} + x^{6} + \frac{9}{4} \, x^{4} + x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3+2*x)*(x^4+4*x^2+1),x, algorithm="giac")

[Out]

1/8*x^8 + x^6 + 9/4*x^4 + x^2