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

Optimal. Leaf size=12 \[ 2 \sqrt {x^3+x-1} \]

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Rubi [A]  time = 0.06, antiderivative size = 12, normalized size of antiderivative = 1.00, 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} \begin {gather*} 2 \sqrt {x^3+x-1} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 + 3*x^2)/Sqrt[-1 + x + x^3],x]

[Out]

2*Sqrt[-1 + x + x^3]

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 \frac {1+3 x^2}{\sqrt {-1+x+x^3}} \, dx &=2 \sqrt {-1+x+x^3}\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 12, normalized size = 1.00 \begin {gather*} 2 \sqrt {x^3+x-1} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

2*Sqrt[-1 + x + x^3]

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IntegrateAlgebraic [A]  time = 0.01, size = 12, normalized size = 1.00 \begin {gather*} 2 \sqrt {-1+x+x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

2*Sqrt[-1 + x + x^3]

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fricas [A]  time = 0.47, size = 10, normalized size = 0.83 \begin {gather*} 2 \, \sqrt {x^{3} + x - 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(x^3 + x - 1)

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giac [A]  time = 0.41, size = 10, normalized size = 0.83 \begin {gather*} 2 \, \sqrt {x^{3} + x - 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(x^3 + x - 1)

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maple [A]  time = 0.36, size = 11, normalized size = 0.92

method	result	size
gosper	\(2 \sqrt {x^{3}+x -1}\)	\(11\)
derivativedivides	\(2 \sqrt {x^{3}+x -1}\)	\(11\)
default	\(2 \sqrt {x^{3}+x -1}\)	\(11\)
trager	\(2 \sqrt {x^{3}+x -1}\)	\(11\)
risch	\(2 \sqrt {x^{3}+x -1}\)	\(11\)
elliptic	\(2 \sqrt {x^{3}+x -1}\)	\(11\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.61, size = 10, normalized size = 0.83 \begin {gather*} 2 \, \sqrt {x^{3} + x - 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(x^3 + x - 1)

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mupad [B]  time = 0.10, size = 10, normalized size = 0.83 \begin {gather*} 2\,\sqrt {x^3+x-1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(x + x^3 - 1)^(1/2)

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sympy [A]  time = 0.14, size = 10, normalized size = 0.83 \begin {gather*} 2 \sqrt {x^{3} + x - 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(x**3 + x - 1)

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