3.28.34 \(\int \frac {76+106 x+102 x^2+81 x^3+36 x^4+6 x^5+(-8-12 x-6 x^2-x^3) \log (4)}{32+48 x+24 x^2+4 x^3} \, dx\)

Optimal. Leaf size=23 \[ x-\frac {1}{4} x \left (-5-2 x^2-\frac {2}{(2+x)^2}+\log (4)\right ) \]

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Rubi [A]  time = 0.08, antiderivative size = 35, normalized size of antiderivative = 1.52, number of steps used = 2, number of rules used = 1, integrand size = 61, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.016, Rules used = {2074} \begin {gather*} \frac {x^3}{2}+\frac {1}{2 (x+2)}-\frac {1}{(x+2)^2}+\frac {1}{4} x (9-\log (4)) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(76 + 106*x + 102*x^2 + 81*x^3 + 36*x^4 + 6*x^5 + (-8 - 12*x - 6*x^2 - x^3)*Log[4])/(32 + 48*x + 24*x^2 +
4*x^3),x]

[Out]

x^3/2 - (2 + x)^(-2) + 1/(2*(2 + x)) + (x*(9 - Log[4]))/4

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 (\frac {3 x^2}{2}+\frac {2}{(2+x)^3}-\frac {1}{2 (2+x)^2}+\frac {1}{4} (9-\log (4))\right ) \, dx\\ &=\frac {x^3}{2}-\frac {1}{(2+x)^2}+\frac {1}{2 (2+x)}+\frac {1}{4} x (9-\log (4))\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.05, size = 37, normalized size = 1.61 \begin {gather*} \frac {1}{4} \left (2 x^3+\frac {2 \left (x+17 (2+x)^2\right )}{(2+x)^2}-x (-9+\log (4))-\log (16)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(76 + 106*x + 102*x^2 + 81*x^3 + 36*x^4 + 6*x^5 + (-8 - 12*x - 6*x^2 - x^3)*Log[4])/(32 + 48*x + 24*
x^2 + 4*x^3),x]

[Out]

(2*x^3 + (2*(x + 17*(2 + x)^2))/(2 + x)^2 - x*(-9 + Log[4]) - Log[16])/4

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fricas [B]  time = 0.42, size = 52, normalized size = 2.26 \begin {gather*} \frac {2 \, x^{5} + 8 \, x^{4} + 17 \, x^{3} + 36 \, x^{2} - 2 \, {\left (x^{3} + 4 \, x^{2} + 4 \, x\right )} \log \relax (2) + 38 \, x}{4 \, {\left (x^{2} + 4 \, x + 4\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*(-x^3-6*x^2-12*x-8)*log(2)+6*x^5+36*x^4+81*x^3+102*x^2+106*x+76)/(4*x^3+24*x^2+48*x+32),x, algori
thm="fricas")

[Out]

1/4*(2*x^5 + 8*x^4 + 17*x^3 + 36*x^2 - 2*(x^3 + 4*x^2 + 4*x)*log(2) + 38*x)/(x^2 + 4*x + 4)

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giac [A]  time = 0.22, size = 22, normalized size = 0.96 \begin {gather*} \frac {1}{2} \, x^{3} - \frac {1}{2} \, x \log \relax (2) + \frac {9}{4} \, x + \frac {x}{2 \, {\left (x + 2\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*(-x^3-6*x^2-12*x-8)*log(2)+6*x^5+36*x^4+81*x^3+102*x^2+106*x+76)/(4*x^3+24*x^2+48*x+32),x, algori
thm="giac")

[Out]

1/2*x^3 - 1/2*x*log(2) + 9/4*x + 1/2*x/(x + 2)^2

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maple [A]  time = 0.04, size = 28, normalized size = 1.22

method	result	size
risch	\(\frac {x^{3}}{2}-\frac {x \ln \relax (2)}{2}+\frac {9 x}{4}+\frac {x}{2 x^{2}+8 x +8}\)	\(28\)
default	\(-\frac {1}{\left (2+x \right )^{2}}+\frac {1}{2 x +4}+\frac {x^{3}}{2}+\frac {9 x}{4}-\frac {x \ln \relax (2)}{2}\)	\(29\)
norman	\(\frac {\left (-\frac {\ln \relax (2)}{2}+\frac {17}{4}\right ) x^{3}+\left (-\frac {53}{2}+6 \ln \relax (2)\right ) x +2 x^{4}+\frac {x^{5}}{2}-36+8 \ln \relax (2)}{\left (2+x \right )^{2}}\)	\(41\)
gosper	\(-\frac {-2 x^{5}+2 x^{3} \ln \relax (2)-8 x^{4}-17 x^{3}-24 x \ln \relax (2)-32 \ln \relax (2)+106 x +144}{4 \left (x^{2}+4 x +4\right )}\)	\(49\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*(-x^3-6*x^2-12*x-8)*ln(2)+6*x^5+36*x^4+81*x^3+102*x^2+106*x+76)/(4*x^3+24*x^2+48*x+32),x,method=_RETURN
VERBOSE)

[Out]

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

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maxima [A]  time = 0.35, size = 28, normalized size = 1.22 \begin {gather*} \frac {1}{2} \, x^{3} - \frac {1}{4} \, x {\left (2 \, \log \relax (2) - 9\right )} + \frac {x}{2 \, {\left (x^{2} + 4 \, x + 4\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*(-x^3-6*x^2-12*x-8)*log(2)+6*x^5+36*x^4+81*x^3+102*x^2+106*x+76)/(4*x^3+24*x^2+48*x+32),x, algori
thm="maxima")

[Out]

1/2*x^3 - 1/4*x*(2*log(2) - 9) + 1/2*x/(x^2 + 4*x + 4)

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mupad [B]  time = 1.62, size = 29, normalized size = 1.26 \begin {gather*} \frac {x}{2\,\left (x^2+4\,x+4\right )}-x\,\left (\frac {\ln \relax (4)}{4}-\frac {9}{4}\right )+\frac {x^3}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((106*x - 2*log(2)*(12*x + 6*x^2 + x^3 + 8) + 102*x^2 + 81*x^3 + 36*x^4 + 6*x^5 + 76)/(48*x + 24*x^2 + 4*x^
3 + 32),x)

[Out]

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

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sympy [A]  time = 0.15, size = 26, normalized size = 1.13 \begin {gather*} \frac {x^{3}}{2} + x \left (\frac {9}{4} - \frac {\log {\relax (2 )}}{2}\right ) + \frac {x}{2 x^{2} + 8 x + 8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*(-x**3-6*x**2-12*x-8)*ln(2)+6*x**5+36*x**4+81*x**3+102*x**2+106*x+76)/(4*x**3+24*x**2+48*x+32),x)

[Out]

x**3/2 + x*(9/4 - log(2)/2) + x/(2*x**2 + 8*x + 8)

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