3.3.37 \(\int \frac {x^2 (3+2 x^2) (1+x^2+2 x^6)}{(1+x^2)^2 \sqrt {1+x^2+x^6}} \, dx\)

Optimal. Leaf size=23 \[ \frac {x^3 \sqrt {x^6+x^2+1}}{x^2+1} \]

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Rubi [A]  time = 0.03, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.025, Rules used = {1590} \begin {gather*} \frac {x^3 \sqrt {x^6+x^2+1}}{x^2+1} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x^2*(3 + 2*x^2)*(1 + x^2 + 2*x^6))/((1 + x^2)^2*Sqrt[1 + x^2 + x^6]),x]

[Out]

(x^3*Sqrt[1 + x^2 + x^6])/(1 + x^2)

Rule 1590

Int[(Pp_)*(Qq_)^(m_.)*(Rr_)^(n_.), x_Symbol] :> With[{p = Expon[Pp, x], q = Expon[Qq, x], r = Expon[Rr, x]}, S
imp[(Coeff[Pp, x, p]*x^(p - q - r + 1)*Qq^(m + 1)*Rr^(n + 1))/((p + m*q + n*r + 1)*Coeff[Qq, x, q]*Coeff[Rr, x
, r]), x] /; NeQ[p + m*q + n*r + 1, 0] && EqQ[(p + m*q + n*r + 1)*Coeff[Qq, x, q]*Coeff[Rr, x, r]*Pp, Coeff[Pp
, x, p]*x^(p - q - r)*((p - q - r + 1)*Qq*Rr + (m + 1)*x*Rr*D[Qq, x] + (n + 1)*x*Qq*D[Rr, x])]] /; FreeQ[{m, n
}, x] && PolyQ[Pp, x] && PolyQ[Qq, x] && PolyQ[Rr, x] && NeQ[m, -1] && NeQ[n, -1]

Rubi steps

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

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Mathematica [A]  time = 0.21, size = 25, normalized size = 1.09 \begin {gather*} \sqrt {x^6+x^2+1} \left (x-\frac {x}{x^2+1}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x^2*(3 + 2*x^2)*(1 + x^2 + 2*x^6))/((1 + x^2)^2*Sqrt[1 + x^2 + x^6]),x]

[Out]

Sqrt[1 + x^2 + x^6]*(x - x/(1 + x^2))

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IntegrateAlgebraic [A]  time = 4.89, size = 23, normalized size = 1.00 \begin {gather*} \frac {x^3 \sqrt {1+x^2+x^6}}{1+x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(x^2*(3 + 2*x^2)*(1 + x^2 + 2*x^6))/((1 + x^2)^2*Sqrt[1 + x^2 + x^6]),x]

[Out]

(x^3*Sqrt[1 + x^2 + x^6])/(1 + x^2)

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fricas [A]  time = 0.44, size = 21, normalized size = 0.91 \begin {gather*} \frac {\sqrt {x^{6} + x^{2} + 1} x^{3}}{x^{2} + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(2*x^2+3)*(2*x^6+x^2+1)/(x^2+1)^2/(x^6+x^2+1)^(1/2),x, algorithm="fricas")

[Out]

sqrt(x^6 + x^2 + 1)*x^3/(x^2 + 1)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (2 \, x^{6} + x^{2} + 1\right )} {\left (2 \, x^{2} + 3\right )} x^{2}}{\sqrt {x^{6} + x^{2} + 1} {\left (x^{2} + 1\right )}^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(2*x^2+3)*(2*x^6+x^2+1)/(x^2+1)^2/(x^6+x^2+1)^(1/2),x, algorithm="giac")

[Out]

integrate((2*x^6 + x^2 + 1)*(2*x^2 + 3)*x^2/(sqrt(x^6 + x^2 + 1)*(x^2 + 1)^2), x)

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maple [A]  time = 0.10, size = 22, normalized size = 0.96

method	result	size
gosper	\(\frac {x^{3} \sqrt {x^{6}+x^{2}+1}}{x^{2}+1}\)	\(22\)
trager	\(\frac {x^{3} \sqrt {x^{6}+x^{2}+1}}{x^{2}+1}\)	\(22\)
risch	\(\frac {x^{3} \sqrt {x^{6}+x^{2}+1}}{x^{2}+1}\)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(2*x^2+3)*(2*x^6+x^2+1)/(x^2+1)^2/(x^6+x^2+1)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (2 \, x^{6} + x^{2} + 1\right )} {\left (2 \, x^{2} + 3\right )} x^{2}}{\sqrt {x^{6} + x^{2} + 1} {\left (x^{2} + 1\right )}^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(2*x^2+3)*(2*x^6+x^2+1)/(x^2+1)^2/(x^6+x^2+1)^(1/2),x, algorithm="maxima")

[Out]

integrate((2*x^6 + x^2 + 1)*(2*x^2 + 3)*x^2/(sqrt(x^6 + x^2 + 1)*(x^2 + 1)^2), x)

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mupad [B]  time = 0.25, size = 21, normalized size = 0.91 \begin {gather*} \frac {x^3\,\sqrt {x^6+x^2+1}}{x^2+1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2*(2*x^2 + 3)*(x^2 + 2*x^6 + 1))/((x^2 + 1)^2*(x^2 + x^6 + 1)^(1/2)),x)

[Out]

(x^3*(x^2 + x^6 + 1)^(1/2))/(x^2 + 1)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2} \left (2 x^{2} + 3\right ) \left (2 x^{6} + x^{2} + 1\right )}{\left (x^{2} + 1\right )^{2} \sqrt {x^{6} + x^{2} + 1}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*(2*x**2+3)*(2*x**6+x**2+1)/(x**2+1)**2/(x**6+x**2+1)**(1/2),x)

[Out]

Integral(x**2*(2*x**2 + 3)*(2*x**6 + x**2 + 1)/((x**2 + 1)**2*sqrt(x**6 + x**2 + 1)), x)

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