3.77.33 \(\int \frac {436+784 x-13 x^2+4 x^3+2 x^4+(-16+8 x+7 x^2-2 x^3-x^4) \log (3)}{16-8 x-7 x^2+2 x^3+x^4} \, dx\)

Optimal. Leaf size=25 \[ 2 x+\frac {400+x}{4-x-x^2}-x \log (3) \]

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Rubi [A]  time = 0.10, antiderivative size = 27, normalized size of antiderivative = 1.08, number of steps used = 4, number of rules used = 4, integrand size = 64, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {1680, 1814, 21, 8} \begin {gather*} \frac {4 (x+400)}{17-4 \left (x+\frac {1}{2}\right )^2}+x (2-\log (3)) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(436 + 784*x - 13*x^2 + 4*x^3 + 2*x^4 + (-16 + 8*x + 7*x^2 - 2*x^3 - x^4)*Log[3])/(16 - 8*x - 7*x^2 + 2*x^
3 + x^4),x]

[Out]

(4*(400 + x))/(17 - 4*(1/2 + x)^2) + x*(2 - Log[3])

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

Rule 1680

Int[(Pq_)*(Q4_)^(p_), x_Symbol] :> With[{a = Coeff[Q4, x, 0], b = Coeff[Q4, x, 1], c = Coeff[Q4, x, 2], d = Co
eff[Q4, x, 3], e = Coeff[Q4, x, 4]}, Subst[Int[SimplifyIntegrand[(Pq /. x -> -(d/(4*e)) + x)*(a + d^4/(256*e^3
) - (b*d)/(8*e) + (c - (3*d^2)/(8*e))*x^2 + e*x^4)^p, x], x], x, d/(4*e) + x] /; EqQ[d^3 - 4*c*d*e + 8*b*e^2,
0] && NeQ[d, 0]] /; FreeQ[p, x] && PolyQ[Pq, x] && PolyQ[Q4, x, 4] &&  !IGtQ[p, 0]

Rule 1814

Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, a + b*x^2, x], f = Coeff[P
olynomialRemainder[Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 1]}, Simp[((a
*g - b*f*x)*(a + b*x^2)^(p + 1))/(2*a*b*(p + 1)), x] + Dist[1/(2*a*(p + 1)), Int[(a + b*x^2)^(p + 1)*ExpandToS
um[2*a*(p + 1)*Q + f*(2*p + 3), x], x], x]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && LtQ[p, -1]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\operatorname {Subst}\left (\int \frac {12784 x-8 x^2 (32-17 \log (3))+17 (38-17 \log (3))+16 x^4 (2-\log (3))}{\left (17-4 x^2\right )^2} \, dx,x,\frac {1}{2}+x\right )\\ &=\frac {4 (400+x)}{17-4 \left (\frac {1}{2}+x\right )^2}-\frac {1}{34} \operatorname {Subst}\left (\int \frac {-578 (2-\log (3))+136 x^2 (2-\log (3))}{17-4 x^2} \, dx,x,\frac {1}{2}+x\right )\\ &=\frac {4 (400+x)}{17-4 \left (\frac {1}{2}+x\right )^2}-(-2+\log (3)) \operatorname {Subst}\left (\int 1 \, dx,x,\frac {1}{2}+x\right )\\ &=\frac {4 (400+x)}{17-4 \left (\frac {1}{2}+x\right )^2}+x (2-\log (3))\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.04, size = 23, normalized size = 0.92 \begin {gather*} \frac {-400-x}{-4+x+x^2}+x (2-\log (3)) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(436 + 784*x - 13*x^2 + 4*x^3 + 2*x^4 + (-16 + 8*x + 7*x^2 - 2*x^3 - x^4)*Log[3])/(16 - 8*x - 7*x^2
+ 2*x^3 + x^4),x]

[Out]

(-400 - x)/(-4 + x + x^2) + x*(2 - Log[3])

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fricas [A]  time = 0.66, size = 38, normalized size = 1.52 \begin {gather*} \frac {2 \, x^{3} + 2 \, x^{2} - {\left (x^{3} + x^{2} - 4 \, x\right )} \log \relax (3) - 9 \, x - 400}{x^{2} + x - 4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^4-2*x^3+7*x^2+8*x-16)*log(3)+2*x^4+4*x^3-13*x^2+784*x+436)/(x^4+2*x^3-7*x^2-8*x+16),x, algorith
m="fricas")

[Out]

(2*x^3 + 2*x^2 - (x^3 + x^2 - 4*x)*log(3) - 9*x - 400)/(x^2 + x - 4)

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giac [A]  time = 0.20, size = 22, normalized size = 0.88 \begin {gather*} -x \log \relax (3) + 2 \, x - \frac {x + 400}{x^{2} + x - 4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^4-2*x^3+7*x^2+8*x-16)*log(3)+2*x^4+4*x^3-13*x^2+784*x+436)/(x^4+2*x^3-7*x^2-8*x+16),x, algorith
m="giac")

[Out]

-x*log(3) + 2*x - (x + 400)/(x^2 + x - 4)

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maple [A]  time = 0.05, size = 24, normalized size = 0.96

method	result	size
default	\(\frac {-x -400}{x^{2}+x -4}+2 x -x \ln \relax (3)\)	\(24\)
risch	\(\frac {-x -400}{x^{2}+x -4}+2 x -x \ln \relax (3)\)	\(24\)
norman	\(\frac {\left (2-\ln \relax (3)\right ) x^{3}+\left (5 \ln \relax (3)-11\right ) x -392-4 \ln \relax (3)}{x^{2}+x -4}\)	\(34\)
gosper	\(-\frac {x^{3} \ln \relax (3)-2 x^{3}-5 x \ln \relax (3)+4 \ln \relax (3)+11 x +392}{x^{2}+x -4}\)	\(36\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-x^4-2*x^3+7*x^2+8*x-16)*ln(3)+2*x^4+4*x^3-13*x^2+784*x+436)/(x^4+2*x^3-7*x^2-8*x+16),x,method=_RETURNVE
RBOSE)

[Out]

(-x-400)/(x^2+x-4)+2*x-x*ln(3)

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maxima [A]  time = 0.34, size = 21, normalized size = 0.84 \begin {gather*} -x {\left (\log \relax (3) - 2\right )} - \frac {x + 400}{x^{2} + x - 4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^4-2*x^3+7*x^2+8*x-16)*log(3)+2*x^4+4*x^3-13*x^2+784*x+436)/(x^4+2*x^3-7*x^2-8*x+16),x, algorith
m="maxima")

[Out]

-x*(log(3) - 2) - (x + 400)/(x^2 + x - 4)

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mupad [B]  time = 4.52, size = 21, normalized size = 0.84 \begin {gather*} -\frac {x+400}{x^2+x-4}-x\,\left (\ln \relax (3)-2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((784*x - log(3)*(2*x^3 - 7*x^2 - 8*x + x^4 + 16) - 13*x^2 + 4*x^3 + 2*x^4 + 436)/(2*x^3 - 7*x^2 - 8*x + x^
4 + 16),x)

[Out]

- (x + 400)/(x + x^2 - 4) - x*(log(3) - 2)

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sympy [A]  time = 0.16, size = 17, normalized size = 0.68 \begin {gather*} x \left (2 - \log {\relax (3 )}\right ) + \frac {- x - 400}{x^{2} + x - 4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x**4-2*x**3+7*x**2+8*x-16)*ln(3)+2*x**4+4*x**3-13*x**2+784*x+436)/(x**4+2*x**3-7*x**2-8*x+16),x)

[Out]

x*(2 - log(3)) + (-x - 400)/(x**2 + x - 4)

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