\(\int \frac {-1+4 x^3}{\sqrt {-1-2 x+2 x^4}} \, dx\) [55]

Optimal result

Integrand size = 22, antiderivative size = 14 \[ \int \frac {-1+4 x^3}{\sqrt {-1-2 x+2 x^4}} \, dx=\sqrt {-1-2 x+2 x^4} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {1602} \[ \int \frac {-1+4 x^3}{\sqrt {-1-2 x+2 x^4}} \, dx=\sqrt {2 x^4-2 x-1} \]

[In]

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

[Out]

Sqrt[-1 - 2*x + 2*x^4]

Rule 1602

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*} \text {integral}& = \sqrt {-1-2 x+2 x^4} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \frac {-1+4 x^3}{\sqrt {-1-2 x+2 x^4}} \, dx=\sqrt {-1-2 x+2 x^4} \]

[In]

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

[Out]

Sqrt[-1 - 2*x + 2*x^4]

Maple [A] (verified)

Time = 0.88 (sec) , antiderivative size = 13, normalized size of antiderivative = 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	\(\sqrt {2 x^{4}-2 x -1}\)	\(13\)
pseudoelliptic	\(\sqrt {2 x^{4}-2 x -1}\)	\(13\)

[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)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \frac {-1+4 x^3}{\sqrt {-1-2 x+2 x^4}} \, dx=\sqrt {2 \, x^{4} - 2 \, x - 1} \]

[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)

Sympy [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \frac {-1+4 x^3}{\sqrt {-1-2 x+2 x^4}} \, dx=\sqrt {2 x^{4} - 2 x - 1} \]

[In]

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

[Out]

sqrt(2*x**4 - 2*x - 1)

Maxima [A] (verification not implemented)

none

Time = 0.17 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \frac {-1+4 x^3}{\sqrt {-1-2 x+2 x^4}} \, dx=\sqrt {2 \, x^{4} - 2 \, x - 1} \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \frac {-1+4 x^3}{\sqrt {-1-2 x+2 x^4}} \, dx=\sqrt {2 \, x^{4} - 2 \, x - 1} \]

[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)

Mupad [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \frac {-1+4 x^3}{\sqrt {-1-2 x+2 x^4}} \, dx=\sqrt {2\,x^4-2\,x-1} \]

[In]

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

[Out]

(2*x^4 - 2*x - 1)^(1/2)