3.32.60 \(\int \frac {-9-6 x-x^2-192 x^3+192 x^4-8 x^5-20 x^6+(48 x^3-48 x^4+2 x^5+5 x^6) \log (4)}{-36-24 x-4 x^2+(9+6 x+x^2) \log (4)} \, dx\)

Optimal. Leaf size=26 \[ -4+\frac {(-2+x)^2 x^4}{3+x}+\frac {x}{4-\log (4)} \]

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Rubi [A]  time = 0.17, antiderivative size = 42, normalized size of antiderivative = 1.62, number of steps used = 5, number of rules used = 4, integrand size = 78, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.051, Rules used = {1986, 27, 12, 1850} \begin {gather*} x^5-7 x^4+25 x^3-75 x^2+\frac {2025}{x+3}+\frac {x (901-450 \log (2))}{4-\log (4)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-9 - 6*x - x^2 - 192*x^3 + 192*x^4 - 8*x^5 - 20*x^6 + (48*x^3 - 48*x^4 + 2*x^5 + 5*x^6)*Log[4])/(-36 - 24
*x - 4*x^2 + (9 + 6*x + x^2)*Log[4]),x]

[Out]

-75*x^2 + 25*x^3 - 7*x^4 + x^5 + 2025/(3 + x) + (x*(901 - 450*Log[2]))/(4 - Log[4])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 1850

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x^n)^p, x], x] /; FreeQ[
{a, b, n}, x] && PolyQ[Pq, x] && (IGtQ[p, 0] || EqQ[n, 1])

Rule 1986

Int[(Pq_)*(u_)^(p_.), x_Symbol] :> Int[Pq*ExpandToSum[u, x]^p, x] /; FreeQ[p, x] && PolyQ[Pq, x] && QuadraticQ
[u, x] &&  !QuadraticMatchQ[u, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-9-6 x-x^2-192 x^3+192 x^4-8 x^5-20 x^6+\left (48 x^3-48 x^4+2 x^5+5 x^6\right ) \log (4)}{-9 (4-\log (4))-6 x (4-\log (4))+x^2 (-4+\log (4))} \, dx\\ &=\int \frac {-9-6 x-x^2-192 x^3+192 x^4-8 x^5-20 x^6+\left (48 x^3-48 x^4+2 x^5+5 x^6\right ) \log (4)}{(3+x)^2 (-4+\log (4))} \, dx\\ &=\frac {\int \frac {-9-6 x-x^2-192 x^3+192 x^4-8 x^5-20 x^6+\left (48 x^3-48 x^4+2 x^5+5 x^6\right ) \log (4)}{(3+x)^2} \, dx}{-4+\log (4)}\\ &=\frac {\int \left (-901 \left (1-\frac {450 \log (2)}{901}\right )-150 x (-4+\log (4))+75 x^2 (-4+\log (4))-28 x^3 (-4+\log (4))+5 x^4 (-4+\log (4))-\frac {2025 (-4+\log (4))}{(3+x)^2}\right ) \, dx}{-4+\log (4)}\\ &=-75 x^2+25 x^3-7 x^4+x^5+\frac {2025}{3+x}+\frac {x (901-450 \log (2))}{4-\log (4)}\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.03, size = 58, normalized size = 2.23 \begin {gather*} -\frac {42129+x^2-4 x^4 (-4+\log (4))+4 x^5 (-4+\log (4))-x^6 (-4+\log (4))-10530 \log (4)-6 x (-2341+585 \log (4))}{(3+x) (-4+\log (4))} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-9 - 6*x - x^2 - 192*x^3 + 192*x^4 - 8*x^5 - 20*x^6 + (48*x^3 - 48*x^4 + 2*x^5 + 5*x^6)*Log[4])/(-3
6 - 24*x - 4*x^2 + (9 + 6*x + x^2)*Log[4]),x]

[Out]

-((42129 + x^2 - 4*x^4*(-4 + Log[4]) + 4*x^5*(-4 + Log[4]) - x^6*(-4 + Log[4]) - 10530*Log[4] - 6*x*(-2341 + 5
85*Log[4]))/((3 + x)*(-4 + Log[4])))

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fricas [B]  time = 0.76, size = 60, normalized size = 2.31 \begin {gather*} -\frac {4 \, x^{6} - 16 \, x^{5} + 16 \, x^{4} + x^{2} - 2 \, {\left (x^{6} - 4 \, x^{5} + 4 \, x^{4} + 675 \, x + 2025\right )} \log \relax (2) + 2703 \, x + 8100}{2 \, {\left ({\left (x + 3\right )} \log \relax (2) - 2 \, x - 6\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*(5*x^6+2*x^5-48*x^4+48*x^3)*log(2)-20*x^6-8*x^5+192*x^4-192*x^3-x^2-6*x-9)/(2*(x^2+6*x+9)*log(2)-
4*x^2-24*x-36),x, algorithm="fricas")

[Out]

-1/2*(4*x^6 - 16*x^5 + 16*x^4 + x^2 - 2*(x^6 - 4*x^5 + 4*x^4 + 675*x + 2025)*log(2) + 2703*x + 8100)/((x + 3)*
log(2) - 2*x - 6)

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giac [B]  time = 0.39, size = 269, normalized size = 10.35 \begin {gather*} \frac {2 \, x^{5} \log \relax (2)^{5} - 20 \, x^{5} \log \relax (2)^{4} - 14 \, x^{4} \log \relax (2)^{5} + 80 \, x^{5} \log \relax (2)^{3} + 140 \, x^{4} \log \relax (2)^{4} + 50 \, x^{3} \log \relax (2)^{5} - 160 \, x^{5} \log \relax (2)^{2} - 560 \, x^{4} \log \relax (2)^{3} - 500 \, x^{3} \log \relax (2)^{4} - 150 \, x^{2} \log \relax (2)^{5} + 160 \, x^{5} \log \relax (2) + 1120 \, x^{4} \log \relax (2)^{2} + 2000 \, x^{3} \log \relax (2)^{3} + 1500 \, x^{2} \log \relax (2)^{4} + 450 \, x \log \relax (2)^{5} - 64 \, x^{5} - 1120 \, x^{4} \log \relax (2) - 4000 \, x^{3} \log \relax (2)^{2} - 6000 \, x^{2} \log \relax (2)^{3} - 4501 \, x \log \relax (2)^{4} + 448 \, x^{4} + 4000 \, x^{3} \log \relax (2) + 12000 \, x^{2} \log \relax (2)^{2} + 18008 \, x \log \relax (2)^{3} - 1600 \, x^{3} - 12000 \, x^{2} \log \relax (2) - 36024 \, x \log \relax (2)^{2} + 4800 \, x^{2} + 36032 \, x \log \relax (2) - 14416 \, x}{2 \, {\left (\log \relax (2)^{5} - 10 \, \log \relax (2)^{4} + 40 \, \log \relax (2)^{3} - 80 \, \log \relax (2)^{2} + 80 \, \log \relax (2) - 32\right )}} + \frac {2025}{x + 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*(5*x^6+2*x^5-48*x^4+48*x^3)*log(2)-20*x^6-8*x^5+192*x^4-192*x^3-x^2-6*x-9)/(2*(x^2+6*x+9)*log(2)-
4*x^2-24*x-36),x, algorithm="giac")

[Out]

1/2*(2*x^5*log(2)^5 - 20*x^5*log(2)^4 - 14*x^4*log(2)^5 + 80*x^5*log(2)^3 + 140*x^4*log(2)^4 + 50*x^3*log(2)^5
- 160*x^5*log(2)^2 - 560*x^4*log(2)^3 - 500*x^3*log(2)^4 - 150*x^2*log(2)^5 + 160*x^5*log(2) + 1120*x^4*log(2
)^2 + 2000*x^3*log(2)^3 + 1500*x^2*log(2)^4 + 450*x*log(2)^5 - 64*x^5 - 1120*x^4*log(2) - 4000*x^3*log(2)^2 -
6000*x^2*log(2)^3 - 4501*x*log(2)^4 + 448*x^4 + 4000*x^3*log(2) + 12000*x^2*log(2)^2 + 18008*x*log(2)^3 - 1600
*x^3 - 12000*x^2*log(2) - 36024*x*log(2)^2 + 4800*x^2 + 36032*x*log(2) - 14416*x)/(log(2)^5 - 10*log(2)^4 + 40
*log(2)^3 - 80*log(2)^2 + 80*log(2) - 32) + 2025/(x + 3)

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maple [A]  time = 0.61, size = 40, normalized size = 1.54

method	result	size
norman	\(\frac {x^{6}+4 x^{4}-4 x^{5}-\frac {x^{2}}{2 \left (\ln \relax (2)-2\right )}+\frac {9}{2 \left (\ln \relax (2)-2\right )}}{3+x}\)	\(40\)
gosper	\(\frac {2 x^{6} \ln \relax (2)-8 x^{5} \ln \relax (2)-4 x^{6}+8 x^{4} \ln \relax (2)+16 x^{5}-16 x^{4}-x^{2}+9}{2 x \ln \relax (2)+6 \ln \relax (2)-4 x -12}\)	\(61\)
default	\(\frac {2 x^{5} \ln \relax (2)-14 x^{4} \ln \relax (2)-4 x^{5}+50 x^{3} \ln \relax (2)+28 x^{4}-150 x^{2} \ln \relax (2)-100 x^{3}+450 x \ln \relax (2)+300 x^{2}-901 x -\frac {-4050 \ln \relax (2)+8100}{3+x}}{2 \ln \relax (2)-4}\)	\(79\)
risch	\(\frac {2 \ln \relax (2) x^{5}}{2 \ln \relax (2)-4}-\frac {14 \ln \relax (2) x^{4}}{2 \ln \relax (2)-4}-\frac {4 x^{5}}{2 \ln \relax (2)-4}+\frac {50 \ln \relax (2) x^{3}}{2 \ln \relax (2)-4}+\frac {28 x^{4}}{2 \ln \relax (2)-4}-\frac {150 \ln \relax (2) x^{2}}{2 \ln \relax (2)-4}-\frac {100 x^{3}}{2 \ln \relax (2)-4}+\frac {450 x \ln \relax (2)}{2 \ln \relax (2)-4}+\frac {300 x^{2}}{2 \ln \relax (2)-4}-\frac {901 x}{2 \ln \relax (2)-4}+\frac {4050 \ln \relax (2)^{2}}{\left (2 \ln \relax (2)-4\right ) \left (x \ln \relax (2)+3 \ln \relax (2)-2 x -6\right )}-\frac {16200 \ln \relax (2)}{\left (2 \ln \relax (2)-4\right ) \left (x \ln \relax (2)+3 \ln \relax (2)-2 x -6\right )}+\frac {16200}{\left (2 \ln \relax (2)-4\right ) \left (x \ln \relax (2)+3 \ln \relax (2)-2 x -6\right )}\)	\(219\)
meijerg	\(-\frac {x}{\left (2 \ln \relax (2)-4\right ) \left (1+\frac {x}{3}\right )}-\frac {3 \left (\frac {\left (x +6\right ) x}{3 x +9}-2 \ln \left (1+\frac {x}{3}\right )\right )}{2 \ln \relax (2)-4}-\frac {6 \left (-\frac {x}{3 \left (1+\frac {x}{3}\right )}+\ln \left (1+\frac {x}{3}\right )\right )}{2 \ln \relax (2)-4}+\frac {243 \left (10 \ln \relax (2)-20\right ) \left (\frac {x \left (\frac {14}{243} x^{5}-\frac {7}{27} x^{4}+\frac {35}{27} x^{3}-\frac {70}{9} x^{2}+70 x +420\right )}{210+70 x}-6 \ln \left (1+\frac {x}{3}\right )\right )}{2 \ln \relax (2)-4}+\frac {81 \left (4 \ln \relax (2)-8\right ) \left (-\frac {x \left (-\frac {1}{27} x^{4}+\frac {5}{27} x^{3}-\frac {10}{9} x^{2}+10 x +60\right )}{36 \left (1+\frac {x}{3}\right )}+5 \ln \left (1+\frac {x}{3}\right )\right )}{2 \ln \relax (2)-4}+\frac {27 \left (-96 \ln \relax (2)+192\right ) \left (\frac {x \left (\frac {5}{27} x^{3}-\frac {10}{9} x^{2}+10 x +60\right )}{15 x +45}-4 \ln \left (1+\frac {x}{3}\right )\right )}{2 \ln \relax (2)-4}+\frac {9 \left (96 \ln \relax (2)-192\right ) \left (-\frac {x \left (-\frac {2}{9} x^{2}+2 x +12\right )}{12 \left (1+\frac {x}{3}\right )}+3 \ln \left (1+\frac {x}{3}\right )\right )}{2 \ln \relax (2)-4}\)	\(289\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*(5*x^6+2*x^5-48*x^4+48*x^3)*ln(2)-20*x^6-8*x^5+192*x^4-192*x^3-x^2-6*x-9)/(2*(x^2+6*x+9)*ln(2)-4*x^2-24
*x-36),x,method=_RETURNVERBOSE)

[Out]

(x^6+4*x^4-4*x^5-1/2/(ln(2)-2)*x^2+9/2/(ln(2)-2))/(3+x)

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maxima [B]  time = 0.80, size = 61, normalized size = 2.35 \begin {gather*} \frac {2 \, x^{5} {\left (\log \relax (2) - 2\right )} - 14 \, x^{4} {\left (\log \relax (2) - 2\right )} + 50 \, x^{3} {\left (\log \relax (2) - 2\right )} - 150 \, x^{2} {\left (\log \relax (2) - 2\right )} + x {\left (450 \, \log \relax (2) - 901\right )}}{2 \, {\left (\log \relax (2) - 2\right )}} + \frac {2025}{x + 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*(5*x^6+2*x^5-48*x^4+48*x^3)*log(2)-20*x^6-8*x^5+192*x^4-192*x^3-x^2-6*x-9)/(2*(x^2+6*x+9)*log(2)-
4*x^2-24*x-36),x, algorithm="maxima")

[Out]

1/2*(2*x^5*(log(2) - 2) - 14*x^4*(log(2) - 2) + 50*x^3*(log(2) - 2) - 150*x^2*(log(2) - 2) + x*(450*log(2) - 9
01))/(log(2) - 2) + 2025/(x + 3)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {6\,x-2\,\ln \relax (2)\,\left (5\,x^6+2\,x^5-48\,x^4+48\,x^3\right )+x^2+192\,x^3-192\,x^4+8\,x^5+20\,x^6+9}{24\,x+4\,x^2-2\,\ln \relax (2)\,\left (x^2+6\,x+9\right )+36} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((6*x - 2*log(2)*(48*x^3 - 48*x^4 + 2*x^5 + 5*x^6) + x^2 + 192*x^3 - 192*x^4 + 8*x^5 + 20*x^6 + 9)/(24*x +
4*x^2 - 2*log(2)*(6*x + x^2 + 9) + 36),x)

[Out]

int((6*x - 2*log(2)*(48*x^3 - 48*x^4 + 2*x^5 + 5*x^6) + x^2 + 192*x^3 - 192*x^4 + 8*x^5 + 20*x^6 + 9)/(24*x +
4*x^2 - 2*log(2)*(6*x + x^2 + 9) + 36), x)

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sympy [B]  time = 0.54, size = 44, normalized size = 1.69 \begin {gather*} x^{5} - 7 x^{4} + 25 x^{3} - 75 x^{2} + x \left (\frac {450 \log {\relax (2 )}}{-4 + 2 \log {\relax (2 )}} - \frac {901}{-4 + 2 \log {\relax (2 )}}\right ) + \frac {2025}{x + 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*(5*x**6+2*x**5-48*x**4+48*x**3)*ln(2)-20*x**6-8*x**5+192*x**4-192*x**3-x**2-6*x-9)/(2*(x**2+6*x+9
)*ln(2)-4*x**2-24*x-36),x)

[Out]

x**5 - 7*x**4 + 25*x**3 - 75*x**2 + x*(450*log(2)/(-4 + 2*log(2)) - 901/(-4 + 2*log(2))) + 2025/(x + 3)

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