3.2.79 \(\int x (1+2 x^2) \sqrt {-1+2 x^2+2 x^4} \, dx\)

Optimal. Leaf size=20 \[ \frac {1}{6} \left (2 x^4+2 x^2-1\right )^{3/2} \]

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Rubi [A]  time = 0.02, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {1247, 629} \begin {gather*} \frac {1}{6} \left (2 x^4+2 x^2-1\right )^{3/2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x*(1 + 2*x^2)*Sqrt[-1 + 2*x^2 + 2*x^4],x]

[Out]

(-1 + 2*x^2 + 2*x^4)^(3/2)/6

Rule 629

Int[((d_) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d*(a + b*x + c*x^2)^(p +
1))/(b*(p + 1)), x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[2*c*d - b*e, 0] && NeQ[p, -1]

Rule 1247

Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[
Int[(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x]

Rubi steps

\begin {align*} \int x \left (1+2 x^2\right ) \sqrt {-1+2 x^2+2 x^4} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int (1+2 x) \sqrt {-1+2 x+2 x^2} \, dx,x,x^2\right )\\ &=\frac {1}{6} \left (-1+2 x^2+2 x^4\right )^{3/2}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 20, normalized size = 1.00 \begin {gather*} \frac {1}{6} \left (2 x^4+2 x^2-1\right )^{3/2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-1 + 2*x^2 + 2*x^4)^(3/2)/6

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IntegrateAlgebraic [A]  time = 0.22, size = 20, normalized size = 1.00 \begin {gather*} \frac {1}{6} \left (-1+2 x^2+2 x^4\right )^{3/2} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[x*(1 + 2*x^2)*Sqrt[-1 + 2*x^2 + 2*x^4],x]

[Out]

(-1 + 2*x^2 + 2*x^4)^(3/2)/6

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fricas [A]  time = 0.45, size = 16, normalized size = 0.80 \begin {gather*} \frac {1}{6} \, {\left (2 \, x^{4} + 2 \, x^{2} - 1\right )}^{\frac {3}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(2*x^4 + 2*x^2 - 1)^(3/2)

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giac [A]  time = 0.26, size = 16, normalized size = 0.80 \begin {gather*} \frac {1}{6} \, {\left (2 \, x^{4} + 2 \, x^{2} - 1\right )}^{\frac {3}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(2*x^4 + 2*x^2 - 1)^(3/2)

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maple [A]  time = 0.05, size = 17, normalized size = 0.85

method	result	size
gosper	\(\frac {\left (2 x^{4}+2 x^{2}-1\right )^{\frac {3}{2}}}{6}\)	\(17\)
default	\(\frac {\left (2 x^{4}+2 x^{2}-1\right )^{\frac {3}{2}}}{6}\)	\(17\)
risch	\(\frac {\left (2 x^{4}+2 x^{2}-1\right )^{\frac {3}{2}}}{6}\)	\(17\)
trager	\(\left (\frac {1}{3} x^{4}+\frac {1}{3} x^{2}-\frac {1}{6}\right ) \sqrt {2 x^{4}+2 x^{2}-1}\)	\(28\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(2*x^2+1)*(2*x^4+2*x^2-1)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/6*(2*x^4+2*x^2-1)^(3/2)

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maxima [A]  time = 0.55, size = 16, normalized size = 0.80 \begin {gather*} \frac {1}{6} \, {\left (2 \, x^{4} + 2 \, x^{2} - 1\right )}^{\frac {3}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(2*x^4 + 2*x^2 - 1)^(3/2)

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mupad [B]  time = 0.17, size = 16, normalized size = 0.80 \begin {gather*} \frac {{\left (2\,x^4+2\,x^2-1\right )}^{3/2}}{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(2*x^2 + 1)*(2*x^2 + 2*x^4 - 1)^(1/2),x)

[Out]

(2*x^2 + 2*x^4 - 1)^(3/2)/6

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sympy [B]  time = 0.19, size = 56, normalized size = 2.80 \begin {gather*} \frac {x^{4} \sqrt {2 x^{4} + 2 x^{2} - 1}}{3} + \frac {x^{2} \sqrt {2 x^{4} + 2 x^{2} - 1}}{3} - \frac {\sqrt {2 x^{4} + 2 x^{2} - 1}}{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(2*x**2+1)*(2*x**4+2*x**2-1)**(1/2),x)

[Out]

x**4*sqrt(2*x**4 + 2*x**2 - 1)/3 + x**2*sqrt(2*x**4 + 2*x**2 - 1)/3 - sqrt(2*x**4 + 2*x**2 - 1)/6

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