3.81.70 \(\int \frac {-96 x+3021 x^2-15042 x^3-1911 x^4-60 x^5+(-1536 x+11328 x^2+1434 x^3+45 x^4) \log (4)}{256+32 x+x^2} \, dx\)

Optimal. Leaf size=27 \[ 3 \left (-x+5 x^2\right ) \left (\frac {x}{16+x}+x (-x+\log (4))\right ) \]

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Rubi [A]  time = 0.07, antiderivative size = 38, normalized size of antiderivative = 1.41, number of steps used = 3, number of rules used = 2, integrand size = 57, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.035, Rules used = {27, 1850} \begin {gather*} -15 x^4+3 x^3 (1+5 \log (4))+3 x^2 (5-\log (4))-243 x-\frac {62208}{x+16} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-96*x + 3021*x^2 - 15042*x^3 - 1911*x^4 - 60*x^5 + (-1536*x + 11328*x^2 + 1434*x^3 + 45*x^4)*Log[4])/(256
+ 32*x + x^2),x]

[Out]

-243*x - 15*x^4 - 62208/(16 + x) + 3*x^2*(5 - Log[4]) + 3*x^3*(1 + 5*Log[4])

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])

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-96 x+3021 x^2-15042 x^3-1911 x^4-60 x^5+\left (-1536 x+11328 x^2+1434 x^3+45 x^4\right ) \log (4)}{(16+x)^2} \, dx\\ &=\int \left (-243-60 x^3+\frac {62208}{(16+x)^2}-6 x (-5+\log (4))+9 x^2 (1+5 \log (4))\right ) \, dx\\ &=-243 x-15 x^4-\frac {62208}{16+x}+3 x^2 (5-\log (4))+3 x^3 (1+5 \log (4))\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.03, size = 42, normalized size = 1.56 \begin {gather*} -3 \left (-329200+81 x+5 x^4+\frac {20736}{16+x}+x^2 (-5+\log (4))-20736 \log (4)-x^3 (1+5 \log (4))\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-96*x + 3021*x^2 - 15042*x^3 - 1911*x^4 - 60*x^5 + (-1536*x + 11328*x^2 + 1434*x^3 + 45*x^4)*Log[4]
)/(256 + 32*x + x^2),x]

[Out]

-3*(-329200 + 81*x + 5*x^4 + 20736/(16 + x) + x^2*(-5 + Log[4]) - 20736*Log[4] - x^3*(1 + 5*Log[4]))

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fricas [A]  time = 0.73, size = 50, normalized size = 1.85 \begin {gather*} -\frac {3 \, {\left (5 \, x^{5} + 79 \, x^{4} - 21 \, x^{3} + x^{2} - 2 \, {\left (5 \, x^{4} + 79 \, x^{3} - 16 \, x^{2}\right )} \log \relax (2) + 1296 \, x + 20736\right )}}{x + 16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*(45*x^4+1434*x^3+11328*x^2-1536*x)*log(2)-60*x^5-1911*x^4-15042*x^3+3021*x^2-96*x)/(x^2+32*x+256)
,x, algorithm="fricas")

[Out]

-3*(5*x^5 + 79*x^4 - 21*x^3 + x^2 - 2*(5*x^4 + 79*x^3 - 16*x^2)*log(2) + 1296*x + 20736)/(x + 16)

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giac [A]  time = 0.22, size = 40, normalized size = 1.48 \begin {gather*} -15 \, x^{4} + 30 \, x^{3} \log \relax (2) + 3 \, x^{3} - 6 \, x^{2} \log \relax (2) + 15 \, x^{2} - 243 \, x - \frac {62208}{x + 16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*(45*x^4+1434*x^3+11328*x^2-1536*x)*log(2)-60*x^5-1911*x^4-15042*x^3+3021*x^2-96*x)/(x^2+32*x+256)
,x, algorithm="giac")

[Out]

-15*x^4 + 30*x^3*log(2) + 3*x^3 - 6*x^2*log(2) + 15*x^2 - 243*x - 62208/(x + 16)

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maple [A]  time = 0.34, size = 41, normalized size = 1.52

method	result	size
default	\(30 x^{3} \ln \relax (2)-15 x^{4}-6 x^{2} \ln \relax (2)+3 x^{3}+15 x^{2}-243 x -\frac {62208}{x +16}\)	\(41\)
risch	\(30 x^{3} \ln \relax (2)-15 x^{4}-6 x^{2} \ln \relax (2)+3 x^{3}+15 x^{2}-243 x -\frac {62208}{x +16}\)	\(41\)
gosper	\(\frac {3 x^{2} \left (10 x^{2} \ln \relax (2)-5 x^{3}+158 x \ln \relax (2)-79 x^{2}-32 \ln \relax (2)+21 x -1\right )}{x +16}\)	\(42\)
norman	\(\frac {\left (-3-96 \ln \relax (2)\right ) x^{2}+\left (63+474 \ln \relax (2)\right ) x^{3}+\left (30 \ln \relax (2)-237\right ) x^{4}-15 x^{5}}{x +16}\)	\(43\)
meijerg	\(16 \left (23040 \ln \relax (2)-489216\right ) \left (\frac {x \left (\frac {5}{4096} x^{3}-\frac {5}{128} x^{2}+\frac {15}{8} x +60\right )}{240+15 x}-4 \ln \left (1+\frac {x}{16}\right )\right )+16 \left (45888 \ln \relax (2)-240672\right ) \left (-\frac {x \left (-\frac {1}{128} x^{2}+\frac {3}{8} x +12\right )}{64 \left (1+\frac {x}{16}\right )}+3 \ln \left (1+\frac {x}{16}\right )\right )+16 \left (22656 \ln \relax (2)+3021\right ) \left (\frac {x \left (\frac {3 x}{16}+6\right )}{48+3 x}-2 \ln \left (1+\frac {x}{16}\right )\right )+16 \left (-192 \ln \relax (2)-6\right ) \left (-\frac {x}{16 \left (1+\frac {x}{16}\right )}+\ln \left (1+\frac {x}{16}\right )\right )+\frac {20480 x \left (-\frac {3}{65536} x^{4}+\frac {5}{4096} x^{3}-\frac {5}{128} x^{2}+\frac {15}{8} x +60\right )}{1+\frac {x}{16}}-19660800 \ln \left (1+\frac {x}{16}\right )\)	\(176\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*(45*x^4+1434*x^3+11328*x^2-1536*x)*ln(2)-60*x^5-1911*x^4-15042*x^3+3021*x^2-96*x)/(x^2+32*x+256),x,meth
od=_RETURNVERBOSE)

[Out]

30*x^3*ln(2)-15*x^4-6*x^2*ln(2)+3*x^3+15*x^2-243*x-62208/(x+16)

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maxima [A]  time = 0.35, size = 38, normalized size = 1.41 \begin {gather*} -15 \, x^{4} + 3 \, x^{3} {\left (10 \, \log \relax (2) + 1\right )} - 3 \, x^{2} {\left (2 \, \log \relax (2) - 5\right )} - 243 \, x - \frac {62208}{x + 16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*(45*x^4+1434*x^3+11328*x^2-1536*x)*log(2)-60*x^5-1911*x^4-15042*x^3+3021*x^2-96*x)/(x^2+32*x+256)
,x, algorithm="maxima")

[Out]

-15*x^4 + 3*x^3*(10*log(2) + 1) - 3*x^2*(2*log(2) - 5) - 243*x - 62208/(x + 16)

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mupad [B]  time = 0.11, size = 37, normalized size = 1.37 \begin {gather*} x^3\,\left (30\,\ln \relax (2)+3\right )-\frac {62208}{x+16}-x^2\,\left (6\,\ln \relax (2)-15\right )-243\,x-15\,x^4 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(96*x - 2*log(2)*(11328*x^2 - 1536*x + 1434*x^3 + 45*x^4) - 3021*x^2 + 15042*x^3 + 1911*x^4 + 60*x^5)/(32
*x + x^2 + 256),x)

[Out]

x^3*(30*log(2) + 3) - 62208/(x + 16) - x^2*(6*log(2) - 15) - 243*x - 15*x^4

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sympy [A]  time = 0.16, size = 36, normalized size = 1.33 \begin {gather*} - 15 x^{4} - x^{3} \left (- 30 \log {\relax (2 )} - 3\right ) - x^{2} \left (-15 + 6 \log {\relax (2 )}\right ) - 243 x - \frac {62208}{x + 16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*(45*x**4+1434*x**3+11328*x**2-1536*x)*ln(2)-60*x**5-1911*x**4-15042*x**3+3021*x**2-96*x)/(x**2+32
*x+256),x)

[Out]

-15*x**4 - x**3*(-30*log(2) - 3) - x**2*(-15 + 6*log(2)) - 243*x - 62208/(x + 16)

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