3.55.6 \(\int \frac {29808+4317192 x+2861676 x^2-360810 x^3-637147 x^4-156751 x^5-12005 x^6}{-29808 x+1296 x^2+6480 x^3} \, dx\)

Optimal. Leaf size=35 \[ x-\frac {1}{4} \left (5+\frac {7 x}{6}\right )^4+\log \left (\frac {4+\frac {3-x}{5}}{x}-x\right ) \]

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Rubi [A]  time = 0.09, antiderivative size = 42, normalized size of antiderivative = 1.20, number of steps used = 4, number of rules used = 3, integrand size = 47, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.064, Rules used = {1594, 1628, 628} \begin {gather*} -\frac {2401 x^4}{5184}-\frac {1715 x^3}{216}-\frac {1225 x^2}{24}+\log \left (-5 x^2-x+23\right )-\frac {869 x}{6}-\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(29808 + 4317192*x + 2861676*x^2 - 360810*x^3 - 637147*x^4 - 156751*x^5 - 12005*x^6)/(-29808*x + 1296*x^2
+ 6480*x^3),x]

[Out]

(-869*x)/6 - (1225*x^2)/24 - (1715*x^3)/216 - (2401*x^4)/5184 - Log[x] + Log[23 - x - 5*x^2]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 1594

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.) + (c_.)*(x_)^(r_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^
(q - p) + c*x^(r - p))^n, x] /; FreeQ[{a, b, c, p, q, r}, x] && IntegerQ[n] && PosQ[q - p] && PosQ[r - p]

Rule 1628

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegra
nd[(d + e*x)^m*Pq*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {29808+4317192 x+2861676 x^2-360810 x^3-637147 x^4-156751 x^5-12005 x^6}{x \left (-29808+1296 x+6480 x^2\right )} \, dx\\ &=\int \left (-\frac {869}{6}-\frac {1}{x}-\frac {1225 x}{12}-\frac {1715 x^2}{72}-\frac {2401 x^3}{1296}+\frac {1+10 x}{-23+x+5 x^2}\right ) \, dx\\ &=-\frac {869 x}{6}-\frac {1225 x^2}{24}-\frac {1715 x^3}{216}-\frac {2401 x^4}{5184}-\log (x)+\int \frac {1+10 x}{-23+x+5 x^2} \, dx\\ &=-\frac {869 x}{6}-\frac {1225 x^2}{24}-\frac {1715 x^3}{216}-\frac {2401 x^4}{5184}-\log (x)+\log \left (23-x-5 x^2\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 42, normalized size = 1.20 \begin {gather*} \frac {-187704 x-66150 x^2-10290 x^3-\frac {2401 x^4}{4}-1296 \log (x)+1296 \log \left (23-x-5 x^2\right )}{1296} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(29808 + 4317192*x + 2861676*x^2 - 360810*x^3 - 637147*x^4 - 156751*x^5 - 12005*x^6)/(-29808*x + 129
6*x^2 + 6480*x^3),x]

[Out]

(-187704*x - 66150*x^2 - 10290*x^3 - (2401*x^4)/4 - 1296*Log[x] + 1296*Log[23 - x - 5*x^2])/1296

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fricas [A]  time = 0.50, size = 32, normalized size = 0.91 \begin {gather*} -\frac {2401}{5184} \, x^{4} - \frac {1715}{216} \, x^{3} - \frac {1225}{24} \, x^{2} - \frac {869}{6} \, x + \log \left (5 \, x^{2} + x - 23\right ) - \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-12005*x^6-156751*x^5-637147*x^4-360810*x^3+2861676*x^2+4317192*x+29808)/(6480*x^3+1296*x^2-29808*x
),x, algorithm="fricas")

[Out]

-2401/5184*x^4 - 1715/216*x^3 - 1225/24*x^2 - 869/6*x + log(5*x^2 + x - 23) - log(x)

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giac [A]  time = 0.21, size = 34, normalized size = 0.97 \begin {gather*} -\frac {2401}{5184} \, x^{4} - \frac {1715}{216} \, x^{3} - \frac {1225}{24} \, x^{2} - \frac {869}{6} \, x + \log \left ({\left | 5 \, x^{2} + x - 23 \right |}\right ) - \log \left ({\left | x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-12005*x^6-156751*x^5-637147*x^4-360810*x^3+2861676*x^2+4317192*x+29808)/(6480*x^3+1296*x^2-29808*x
),x, algorithm="giac")

[Out]

-2401/5184*x^4 - 1715/216*x^3 - 1225/24*x^2 - 869/6*x + log(abs(5*x^2 + x - 23)) - log(abs(x))

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maple [A]  time = 0.04, size = 33, normalized size = 0.94

method	result	size
default	\(-\frac {2401 x^{4}}{5184}-\frac {1715 x^{3}}{216}-\frac {1225 x^{2}}{24}-\frac {869 x}{6}-\ln \relax (x )+\ln \left (5 x^{2}+x -23\right )\)	\(33\)
norman	\(-\frac {2401 x^{4}}{5184}-\frac {1715 x^{3}}{216}-\frac {1225 x^{2}}{24}-\frac {869 x}{6}-\ln \relax (x )+\ln \left (5 x^{2}+x -23\right )\)	\(33\)
risch	\(-\frac {2401 x^{4}}{5184}-\frac {1715 x^{3}}{216}-\frac {1225 x^{2}}{24}-\frac {869 x}{6}-\ln \relax (x )+\ln \left (5 x^{2}+x -23\right )\)	\(33\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-12005*x^6-156751*x^5-637147*x^4-360810*x^3+2861676*x^2+4317192*x+29808)/(6480*x^3+1296*x^2-29808*x),x,me
thod=_RETURNVERBOSE)

[Out]

-2401/5184*x^4-1715/216*x^3-1225/24*x^2-869/6*x-ln(x)+ln(5*x^2+x-23)

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maxima [A]  time = 0.36, size = 32, normalized size = 0.91 \begin {gather*} -\frac {2401}{5184} \, x^{4} - \frac {1715}{216} \, x^{3} - \frac {1225}{24} \, x^{2} - \frac {869}{6} \, x + \log \left (5 \, x^{2} + x - 23\right ) - \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-12005*x^6-156751*x^5-637147*x^4-360810*x^3+2861676*x^2+4317192*x+29808)/(6480*x^3+1296*x^2-29808*x
),x, algorithm="maxima")

[Out]

-2401/5184*x^4 - 1715/216*x^3 - 1225/24*x^2 - 869/6*x + log(5*x^2 + x - 23) - log(x)

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mupad [B]  time = 0.08, size = 32, normalized size = 0.91 \begin {gather*} \ln \left (5\,x^2+x-23\right )-\frac {869\,x}{6}-\ln \relax (x)-\frac {1225\,x^2}{24}-\frac {1715\,x^3}{216}-\frac {2401\,x^4}{5184} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(360810*x^3 - 2861676*x^2 - 4317192*x + 637147*x^4 + 156751*x^5 + 12005*x^6 - 29808)/(1296*x^2 - 29808*x
+ 6480*x^3),x)

[Out]

log(x + 5*x^2 - 23) - (869*x)/6 - log(x) - (1225*x^2)/24 - (1715*x^3)/216 - (2401*x^4)/5184

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sympy [A]  time = 0.10, size = 37, normalized size = 1.06 \begin {gather*} - \frac {2401 x^{4}}{5184} - \frac {1715 x^{3}}{216} - \frac {1225 x^{2}}{24} - \frac {869 x}{6} - \log {\relax (x )} + \log {\left (5 x^{2} + x - 23 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-12005*x**6-156751*x**5-637147*x**4-360810*x**3+2861676*x**2+4317192*x+29808)/(6480*x**3+1296*x**2-
29808*x),x)

[Out]

-2401*x**4/5184 - 1715*x**3/216 - 1225*x**2/24 - 869*x/6 - log(x) + log(5*x**2 + x - 23)

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