3.78.52 \(\int \frac {2-156 x+122 x^2-8 x^3+1352 x^4-208 x^5+8 x^6}{1+52 x^2-4 x^3+676 x^4-104 x^5+4 x^6} \, dx\)

Optimal. Leaf size=24 \[ 2 x+\frac {3}{1+x^2+(25-2 x) x^2}+\log (2) \]

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Rubi [A]  time = 0.11, antiderivative size = 20, normalized size of antiderivative = 0.83, number of steps used = 3, number of rules used = 2, integrand size = 60, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.033, Rules used = {2074, 1588} \begin {gather*} \frac {3}{-2 x^3+26 x^2+1}+2 x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(2 - 156*x + 122*x^2 - 8*x^3 + 1352*x^4 - 208*x^5 + 8*x^6)/(1 + 52*x^2 - 4*x^3 + 676*x^4 - 104*x^5 + 4*x^6
),x]

[Out]

2*x + 3/(1 + 26*x^2 - 2*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]

Rule 2074

Int[(P_)^(p_)*(Q_)^(q_.), x_Symbol] :> With[{PP = Factor[P]}, Int[ExpandIntegrand[PP^p*Q^q, x], x] /;  !SumQ[N
onfreeFactors[PP, x]]] /; FreeQ[q, x] && PolyQ[P, x] && PolyQ[Q, x] && IntegerQ[p] && NeQ[P, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (2+\frac {6 x (-26+3 x)}{\left (-1-26 x^2+2 x^3\right )^2}\right ) \, dx\\ &=2 x+6 \int \frac {x (-26+3 x)}{\left (-1-26 x^2+2 x^3\right )^2} \, dx\\ &=2 x+\frac {3}{1+26 x^2-2 x^3}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 22, normalized size = 0.92 \begin {gather*} 2 \left (x-\frac {3}{2 \left (-1-26 x^2+2 x^3\right )}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2 - 156*x + 122*x^2 - 8*x^3 + 1352*x^4 - 208*x^5 + 8*x^6)/(1 + 52*x^2 - 4*x^3 + 676*x^4 - 104*x^5 +
4*x^6),x]

[Out]

2*(x - 3/(2*(-1 - 26*x^2 + 2*x^3)))

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fricas [A]  time = 0.74, size = 30, normalized size = 1.25 \begin {gather*} \frac {4 \, x^{4} - 52 \, x^{3} - 2 \, x - 3}{2 \, x^{3} - 26 \, x^{2} - 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((8*x^6-208*x^5+1352*x^4-8*x^3+122*x^2-156*x+2)/(4*x^6-104*x^5+676*x^4-4*x^3+52*x^2+1),x, algorithm="
fricas")

[Out]

(4*x^4 - 52*x^3 - 2*x - 3)/(2*x^3 - 26*x^2 - 1)

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giac [A]  time = 0.15, size = 20, normalized size = 0.83 \begin {gather*} 2 \, x - \frac {3}{2 \, x^{3} - 26 \, x^{2} - 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((8*x^6-208*x^5+1352*x^4-8*x^3+122*x^2-156*x+2)/(4*x^6-104*x^5+676*x^4-4*x^3+52*x^2+1),x, algorithm="
giac")

[Out]

2*x - 3/(2*x^3 - 26*x^2 - 1)

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maple [A]  time = 0.03, size = 19, normalized size = 0.79

method	result	size
default	\(2 x -\frac {3}{2 \left (x^{3}-13 x^{2}-\frac {1}{2}\right )}\)	\(19\)
risch	\(2 x -\frac {3}{2 \left (x^{3}-13 x^{2}-\frac {1}{2}\right )}\)	\(19\)
gosper	\(\frac {2 x \left (2 x^{3}-29 x^{2}+39 x -1\right )}{2 x^{3}-26 x^{2}-1}\)	\(33\)
norman	\(\frac {4 x^{4}-58 x^{3}+78 x^{2}-2 x}{2 x^{3}-26 x^{2}-1}\)	\(35\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((8*x^6-208*x^5+1352*x^4-8*x^3+122*x^2-156*x+2)/(4*x^6-104*x^5+676*x^4-4*x^3+52*x^2+1),x,method=_RETURNVERB
OSE)

[Out]

2*x-3/2/(x^3-13*x^2-1/2)

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maxima [A]  time = 0.36, size = 20, normalized size = 0.83 \begin {gather*} 2 \, x - \frac {3}{2 \, x^{3} - 26 \, x^{2} - 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((8*x^6-208*x^5+1352*x^4-8*x^3+122*x^2-156*x+2)/(4*x^6-104*x^5+676*x^4-4*x^3+52*x^2+1),x, algorithm="
maxima")

[Out]

2*x - 3/(2*x^3 - 26*x^2 - 1)

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mupad [B]  time = 0.08, size = 20, normalized size = 0.83 \begin {gather*} 2\,x+\frac {3}{2\,\left (-x^3+13\,x^2+\frac {1}{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((122*x^2 - 156*x - 8*x^3 + 1352*x^4 - 208*x^5 + 8*x^6 + 2)/(52*x^2 - 4*x^3 + 676*x^4 - 104*x^5 + 4*x^6 + 1
),x)

[Out]

2*x + 3/(2*(13*x^2 - x^3 + 1/2))

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sympy [A]  time = 0.10, size = 15, normalized size = 0.62 \begin {gather*} 2 x - \frac {3}{2 x^{3} - 26 x^{2} - 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((8*x**6-208*x**5+1352*x**4-8*x**3+122*x**2-156*x+2)/(4*x**6-104*x**5+676*x**4-4*x**3+52*x**2+1),x)

[Out]

2*x - 3/(2*x**3 - 26*x**2 - 1)

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