3.86.41 \(\int \frac {720+476 x-1680 x^2-1404 x^3-358 x^4-30 x^5+(40-202 x-180 x^2-47 x^3-4 x^4) \log (4)}{128+96 x+24 x^2+2 x^3} \, dx\)

Optimal. Leaf size=25 \[ \left (1-x \left (-\frac {1}{2}+x+\frac {3}{(4+x)^2}\right )\right ) (2+5 x+\log (4)) \]

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Rubi [B]  time = 0.11, antiderivative size = 51, normalized size of antiderivative = 2.04, number of steps used = 2, number of rules used = 1, integrand size = 66, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.015, Rules used = {2074} \begin {gather*} -5 x^3+\frac {1}{2} x^2 (1-2 \log (4))+x (6+\log (2))+\frac {3 (38-\log (4))}{x+4}-\frac {12 (18-\log (4))}{(x+4)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(720 + 476*x - 1680*x^2 - 1404*x^3 - 358*x^4 - 30*x^5 + (40 - 202*x - 180*x^2 - 47*x^3 - 4*x^4)*Log[4])/(1
28 + 96*x + 24*x^2 + 2*x^3),x]

[Out]

-5*x^3 + x*(6 + Log[2]) + (x^2*(1 - 2*Log[4]))/2 - (12*(18 - Log[4]))/(4 + x)^2 + (3*(38 - Log[4]))/(4 + x)

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 (-15 x^2+6 \left (1+\frac {\log (2)}{6}\right )+x (1-2 \log (4))+\frac {3 (-38+\log (4))}{(4+x)^2}-\frac {24 (-18+\log (4))}{(4+x)^3}\right ) \, dx\\ &=-5 x^3+x (6+\log (2))+\frac {1}{2} x^2 (1-2 \log (4))-\frac {12 (18-\log (4))}{(4+x)^2}+\frac {3 (38-\log (4))}{4+x}\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.09, size = 72, normalized size = 2.88 \begin {gather*} \frac {1}{2} \left (-10 x^3+x (12+\log (4))-x^2 (-1+\log (16))-\frac {2 \left (4624-2448 \log (4)+x^2 (304+94 \log (4)-56 \log (16))+720 \log (64)+x (2318-477 \log (4)-192 \log (16)+240 \log (64))\right )}{(4+x)^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(720 + 476*x - 1680*x^2 - 1404*x^3 - 358*x^4 - 30*x^5 + (40 - 202*x - 180*x^2 - 47*x^3 - 4*x^4)*Log[
4])/(128 + 96*x + 24*x^2 + 2*x^3),x]

[Out]

(-10*x^3 + x*(12 + Log[4]) - x^2*(-1 + Log[16]) - (2*(4624 - 2448*Log[4] + x^2*(304 + 94*Log[4] - 56*Log[16])
+ 720*Log[64] + x*(2318 - 477*Log[4] - 192*Log[16] + 240*Log[64])))/(4 + x)^2)/2

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fricas [B]  time = 0.75, size = 60, normalized size = 2.40 \begin {gather*} -\frac {10 \, x^{5} + 79 \, x^{4} + 140 \, x^{3} - 112 \, x^{2} + 2 \, {\left (2 \, x^{4} + 15 \, x^{3} + 24 \, x^{2} - 10 \, x\right )} \log \relax (2) - 420 \, x - 480}{2 \, {\left (x^{2} + 8 \, x + 16\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*(-4*x^4-47*x^3-180*x^2-202*x+40)*log(2)-30*x^5-358*x^4-1404*x^3-1680*x^2+476*x+720)/(2*x^3+24*x^2
+96*x+128),x, algorithm="fricas")

[Out]

-1/2*(10*x^5 + 79*x^4 + 140*x^3 - 112*x^2 + 2*(2*x^4 + 15*x^3 + 24*x^2 - 10*x)*log(2) - 420*x - 480)/(x^2 + 8*
x + 16)

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giac [A]  time = 0.13, size = 41, normalized size = 1.64 \begin {gather*} -5 \, x^{3} - 2 \, x^{2} \log \relax (2) + \frac {1}{2} \, x^{2} + x \log \relax (2) + 6 \, x - \frac {6 \, {\left (x \log \relax (2) - 19 \, x - 40\right )}}{{\left (x + 4\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*(-4*x^4-47*x^3-180*x^2-202*x+40)*log(2)-30*x^5-358*x^4-1404*x^3-1680*x^2+476*x+720)/(2*x^3+24*x^2
+96*x+128),x, algorithm="giac")

[Out]

-5*x^3 - 2*x^2*log(2) + 1/2*x^2 + x*log(2) + 6*x - 6*(x*log(2) - 19*x - 40)/(x + 4)^2

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maple [A]  time = 0.07, size = 46, normalized size = 1.84

method	result	size
norman	\(\frac {\left (-15 \ln \relax (2)-70\right ) x^{3}+\left (-2 \ln \relax (2)-\frac {79}{2}\right ) x^{4}+\left (202 \ln \relax (2)-238\right ) x -5 x^{5}-656+384 \ln \relax (2)}{\left (4+x \right )^{2}}\)	\(46\)
risch	\(-2 x^{2} \ln \relax (2)-5 x^{3}+x \ln \relax (2)+\frac {x^{2}}{2}+6 x +\frac {\left (114-6 \ln \relax (2)\right ) x +240}{x^{2}+8 x +16}\)	\(47\)
default	\(-2 x^{2} \ln \relax (2)-5 x^{3}+x \ln \relax (2)+\frac {x^{2}}{2}+6 x -\frac {-114+6 \ln \relax (2)}{4+x}-\frac {-48 \ln \relax (2)+432}{2 \left (4+x \right )^{2}}\)	\(52\)
gosper	\(-\frac {4 x^{4} \ln \relax (2)+10 x^{5}+30 x^{3} \ln \relax (2)+79 x^{4}+140 x^{3}-404 x \ln \relax (2)-768 \ln \relax (2)+476 x +1312}{2 \left (x^{2}+8 x +16\right )}\)	\(56\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*(-4*x^4-47*x^3-180*x^2-202*x+40)*ln(2)-30*x^5-358*x^4-1404*x^3-1680*x^2+476*x+720)/(2*x^3+24*x^2+96*x+1
28),x,method=_RETURNVERBOSE)

[Out]

((-15*ln(2)-70)*x^3+(-2*ln(2)-79/2)*x^4+(202*ln(2)-238)*x-5*x^5-656+384*ln(2))/(4+x)^2

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maxima [A]  time = 0.36, size = 43, normalized size = 1.72 \begin {gather*} -5 \, x^{3} - \frac {1}{2} \, x^{2} {\left (4 \, \log \relax (2) - 1\right )} + x {\left (\log \relax (2) + 6\right )} - \frac {6 \, {\left (x {\left (\log \relax (2) - 19\right )} - 40\right )}}{x^{2} + 8 \, x + 16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*(-4*x^4-47*x^3-180*x^2-202*x+40)*log(2)-30*x^5-358*x^4-1404*x^3-1680*x^2+476*x+720)/(2*x^3+24*x^2
+96*x+128),x, algorithm="maxima")

[Out]

-5*x^3 - 1/2*x^2*(4*log(2) - 1) + x*(log(2) + 6) - 6*(x*(log(2) - 19) - 40)/(x^2 + 8*x + 16)

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mupad [B]  time = 5.24, size = 57, normalized size = 2.28 \begin {gather*} \frac {12\,\ln \relax (2)-2\,\ln \left (64\right )-x\,\left (\ln \left (64\right )-114\right )+240}{x^2+8\,x+16}-x^2\,\left (\frac {\ln \left (16\right )}{2}-\frac {1}{2}\right )+x\,\left (12\,\ln \left (16\right )-47\,\ln \relax (2)+6\right )-5\,x^3 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(2*log(2)*(202*x + 180*x^2 + 47*x^3 + 4*x^4 - 40) - 476*x + 1680*x^2 + 1404*x^3 + 358*x^4 + 30*x^5 - 720)
/(96*x + 24*x^2 + 2*x^3 + 128),x)

[Out]

(12*log(2) - 2*log(64) - x*(log(64) - 114) + 240)/(8*x + x^2 + 16) - x^2*(log(16)/2 - 1/2) + x*(12*log(16) - 4
7*log(2) + 6) - 5*x^3

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sympy [A]  time = 0.30, size = 44, normalized size = 1.76 \begin {gather*} - 5 x^{3} - x^{2} \left (- \frac {1}{2} + 2 \log {\relax (2 )}\right ) - x \left (-6 - \log {\relax (2 )}\right ) - \frac {x \left (-114 + 6 \log {\relax (2 )}\right ) - 240}{x^{2} + 8 x + 16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*(-4*x**4-47*x**3-180*x**2-202*x+40)*ln(2)-30*x**5-358*x**4-1404*x**3-1680*x**2+476*x+720)/(2*x**3
+24*x**2+96*x+128),x)

[Out]

-5*x**3 - x**2*(-1/2 + 2*log(2)) - x*(-6 - log(2)) - (x*(-114 + 6*log(2)) - 240)/(x**2 + 8*x + 16)

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