3.1.55 \(\int \frac {-1+4 x^3}{\sqrt {-1-2 x+2 x^4}} \, dx\)

Optimal. Leaf size=14 \[ \sqrt {2 x^4-2 x-1} \]

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Rubi [A]  time = 0.01, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {1588} \begin {gather*} \sqrt {2 x^4-2 x-1} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-1 + 4*x^3)/Sqrt[-1 - 2*x + 2*x^4],x]

[Out]

Sqrt[-1 - 2*x + 2*x^4]

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+4 x^3}{\sqrt {-1-2 x+2 x^4}} \, dx &=\sqrt {-1-2 x+2 x^4}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 14, normalized size = 1.00 \begin {gather*} \sqrt {2 x^4-2 x-1} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Sqrt[-1 - 2*x + 2*x^4]

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IntegrateAlgebraic [A]  time = 0.01, size = 14, normalized size = 1.00 \begin {gather*} \sqrt {-1-2 x+2 x^4} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(-1 + 4*x^3)/Sqrt[-1 - 2*x + 2*x^4],x]

[Out]

Sqrt[-1 - 2*x + 2*x^4]

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fricas [A]  time = 0.45, size = 12, normalized size = 0.86 \begin {gather*} \sqrt {2 \, x^{4} - 2 \, x - 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(2*x^4 - 2*x - 1)

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giac [A]  time = 0.44, size = 12, normalized size = 0.86 \begin {gather*} \sqrt {2 \, x^{4} - 2 \, x - 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(2*x^4 - 2*x - 1)

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maple [A]  time = 0.90, size = 13, normalized size = 0.93

method	result	size
gosper	\(\sqrt {2 x^{4}-2 x -1}\)	\(13\)
default	\(\sqrt {2 x^{4}-2 x -1}\)	\(13\)
trager	\(\sqrt {2 x^{4}-2 x -1}\)	\(13\)
risch	\(\sqrt {2 x^{4}-2 x -1}\)	\(13\)
elliptic	\(\sqrt {2 x^{4}-2 x -1}\)	\(13\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*x^3-1)/(2*x^4-2*x-1)^(1/2),x,method=_RETURNVERBOSE)

[Out]

(2*x^4-2*x-1)^(1/2)

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maxima [A]  time = 0.44, size = 12, normalized size = 0.86 \begin {gather*} \sqrt {2 \, x^{4} - 2 \, x - 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(2*x^4 - 2*x - 1)

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mupad [B]  time = 0.20, size = 12, normalized size = 0.86 \begin {gather*} \sqrt {2\,x^4-2\,x-1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*x^3 - 1)/(2*x^4 - 2*x - 1)^(1/2),x)

[Out]

(2*x^4 - 2*x - 1)^(1/2)

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sympy [A]  time = 0.16, size = 12, normalized size = 0.86 \begin {gather*} \sqrt {2 x^{4} - 2 x - 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x**3-1)/(2*x**4-2*x-1)**(1/2),x)

[Out]

sqrt(2*x**4 - 2*x - 1)

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