3.56.40 \(\int \frac {243+360 x+153 x^2-14 x^3-16 x^4+(-540-900 x-540 x^2-120 x^3) \log (2)+(675+1350 x+900 x^2+200 x^3) \log ^2(2)}{27+54 x+36 x^2+8 x^3} \, dx\)

Optimal. Leaf size=24 \[ x \left (5-x+\left (-2+\frac {x}{3+2 x}+5 \log (2)\right )^2\right ) \]

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Rubi [B]  time = 0.09, antiderivative size = 51, normalized size of antiderivative = 2.12, number of steps used = 2, number of rules used = 1, integrand size = 76, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.013, Rules used = {2074} \begin {gather*} -x^2-\frac {27}{8 (2 x+3)^2}+\frac {1}{4} x \left (29+100 \log ^2(2)-60 \log (2)\right )-\frac {45 (1-\log (16))}{8 (2 x+3)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(243 + 360*x + 153*x^2 - 14*x^3 - 16*x^4 + (-540 - 900*x - 540*x^2 - 120*x^3)*Log[2] + (675 + 1350*x + 900
*x^2 + 200*x^3)*Log[2]^2)/(27 + 54*x + 36*x^2 + 8*x^3),x]

[Out]

-x^2 - 27/(8*(3 + 2*x)^2) + (x*(29 - 60*Log[2] + 100*Log[2]^2))/4 - (45*(1 - Log[16]))/(8*(3 + 2*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 (-2 x+\frac {27}{2 (3+2 x)^3}+\frac {1}{4} \left (29-60 \log (2)+100 \log ^2(2)\right )-\frac {45 (-1+\log (16))}{4 (3+2 x)^2}\right ) \, dx\\ &=-x^2-\frac {27}{8 (3+2 x)^2}+\frac {1}{4} x \left (29-60 \log (2)+100 \log ^2(2)\right )-\frac {45 (1-\log (16))}{8 (3+2 x)}\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.04, size = 80, normalized size = 3.33 \begin {gather*} \frac {-32 x^4+8 x^3 \left (17-60 \log (2)+100 \log ^2(2)\right )+36 x^2 \left (29-60 \log (2)+100 \log ^2(2)\right )+27 \left (29-40 \log (2)+100 \log ^2(2)\right )+36 x \left (47-80 \log (2)+150 \log ^2(2)\right )}{8 (3+2 x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(243 + 360*x + 153*x^2 - 14*x^3 - 16*x^4 + (-540 - 900*x - 540*x^2 - 120*x^3)*Log[2] + (675 + 1350*x
+ 900*x^2 + 200*x^3)*Log[2]^2)/(27 + 54*x + 36*x^2 + 8*x^3),x]

[Out]

(-32*x^4 + 8*x^3*(17 - 60*Log[2] + 100*Log[2]^2) + 36*x^2*(29 - 60*Log[2] + 100*Log[2]^2) + 27*(29 - 40*Log[2]
+ 100*Log[2]^2) + 36*x*(47 - 80*Log[2] + 150*Log[2]^2))/(8*(3 + 2*x)^2)

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fricas [B]  time = 0.58, size = 73, normalized size = 3.04 \begin {gather*} -\frac {16 \, x^{4} - 68 \, x^{3} - 100 \, {\left (4 \, x^{3} + 12 \, x^{2} + 9 \, x\right )} \log \relax (2)^{2} - 312 \, x^{2} + 30 \, {\left (8 \, x^{3} + 24 \, x^{2} + 12 \, x - 9\right )} \log \relax (2) - 216 \, x + 81}{4 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((200*x^3+900*x^2+1350*x+675)*log(2)^2+(-120*x^3-540*x^2-900*x-540)*log(2)-16*x^4-14*x^3+153*x^2+360
*x+243)/(8*x^3+36*x^2+54*x+27),x, algorithm="fricas")

[Out]

-1/4*(16*x^4 - 68*x^3 - 100*(4*x^3 + 12*x^2 + 9*x)*log(2)^2 - 312*x^2 + 30*(8*x^3 + 24*x^2 + 12*x - 9)*log(2)
- 216*x + 81)/(4*x^2 + 12*x + 9)

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giac [A]  time = 0.17, size = 44, normalized size = 1.83 \begin {gather*} 25 \, x \log \relax (2)^{2} - x^{2} - 15 \, x \log \relax (2) + \frac {29}{4} \, x + \frac {9 \, {\left (20 \, x \log \relax (2) - 5 \, x + 30 \, \log \relax (2) - 9\right )}}{4 \, {\left (2 \, x + 3\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((200*x^3+900*x^2+1350*x+675)*log(2)^2+(-120*x^3-540*x^2-900*x-540)*log(2)-16*x^4-14*x^3+153*x^2+360
*x+243)/(8*x^3+36*x^2+54*x+27),x, algorithm="giac")

[Out]

25*x*log(2)^2 - x^2 - 15*x*log(2) + 29/4*x + 9/4*(20*x*log(2) - 5*x + 30*log(2) - 9)/(2*x + 3)^2

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maple [A]  time = 0.08, size = 46, normalized size = 1.92

method	result	size
default	\(-x^{2}+\frac {29 x}{4}+25 x \ln \relax (2)^{2}-15 x \ln \relax (2)-\frac {27}{8 \left (2 x +3\right )^{2}}-\frac {-\frac {45 \ln \relax (2)}{2}+\frac {45}{8}}{2 x +3}\)	\(46\)
risch	\(25 x \ln \relax (2)^{2}-15 x \ln \relax (2)-x^{2}+\frac {29 x}{4}+\frac {\frac {\left (45 \ln \relax (2)-\frac {45}{4}\right ) x}{4}-\frac {81}{16}+\frac {135 \ln \relax (2)}{8}}{x^{2}+3 x +\frac {9}{4}}\)	\(48\)
norman	\(\frac {-4 x^{4}+\left (100 \ln \relax (2)^{2}-60 \ln \relax (2)+17\right ) x^{3}-\frac {783}{4}+\left (-675 \ln \relax (2)^{2}+450 \ln \relax (2)-180\right ) x -675 \ln \relax (2)^{2}+\frac {945 \ln \relax (2)}{2}}{\left (2 x +3\right )^{2}}\)	\(56\)
gosper	\(\frac {400 x^{3} \ln \relax (2)^{2}-240 x^{3} \ln \relax (2)-16 x^{4}-2700 x \ln \relax (2)^{2}+68 x^{3}-2700 \ln \relax (2)^{2}+1800 x \ln \relax (2)+1890 \ln \relax (2)-720 x -783}{16 x^{2}+48 x +36}\)	\(68\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((200*x^3+900*x^2+1350*x+675)*ln(2)^2+(-120*x^3-540*x^2-900*x-540)*ln(2)-16*x^4-14*x^3+153*x^2+360*x+243)/
(8*x^3+36*x^2+54*x+27),x,method=_RETURNVERBOSE)

[Out]

-x^2+29/4*x+25*x*ln(2)^2-15*x*ln(2)-27/8/(2*x+3)^2-(-45/2*ln(2)+45/8)/(2*x+3)

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maxima [B]  time = 0.38, size = 50, normalized size = 2.08 \begin {gather*} \frac {1}{4} \, {\left (100 \, \log \relax (2)^{2} - 60 \, \log \relax (2) + 29\right )} x - x^{2} + \frac {9 \, {\left (5 \, x {\left (4 \, \log \relax (2) - 1\right )} + 30 \, \log \relax (2) - 9\right )}}{4 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((200*x^3+900*x^2+1350*x+675)*log(2)^2+(-120*x^3-540*x^2-900*x-540)*log(2)-16*x^4-14*x^3+153*x^2+360
*x+243)/(8*x^3+36*x^2+54*x+27),x, algorithm="maxima")

[Out]

1/4*(100*log(2)^2 - 60*log(2) + 29)*x - x^2 + 9/4*(5*x*(4*log(2) - 1) + 30*log(2) - 9)/(4*x^2 + 12*x + 9)

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mupad [B]  time = 3.49, size = 47, normalized size = 1.96 \begin {gather*} x\,\left (25\,{\ln \relax (2)}^2-15\,\ln \relax (2)+\frac {29}{4}\right )+\frac {270\,\ln \relax (2)+x\,\left (180\,\ln \relax (2)-45\right )-81}{16\,x^2+48\,x+36}-x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((360*x - log(2)*(900*x + 540*x^2 + 120*x^3 + 540) + log(2)^2*(1350*x + 900*x^2 + 200*x^3 + 675) + 153*x^2
- 14*x^3 - 16*x^4 + 243)/(54*x + 36*x^2 + 8*x^3 + 27),x)

[Out]

x*(25*log(2)^2 - 15*log(2) + 29/4) + (270*log(2) + x*(180*log(2) - 45) - 81)/(48*x + 16*x^2 + 36) - x^2

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sympy [B]  time = 0.28, size = 46, normalized size = 1.92 \begin {gather*} - x^{2} - x \left (- 25 \log {\relax (2 )}^{2} - \frac {29}{4} + 15 \log {\relax (2 )}\right ) - \frac {x \left (45 - 180 \log {\relax (2 )}\right ) - 270 \log {\relax (2 )} + 81}{16 x^{2} + 48 x + 36} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((200*x**3+900*x**2+1350*x+675)*ln(2)**2+(-120*x**3-540*x**2-900*x-540)*ln(2)-16*x**4-14*x**3+153*x*
*2+360*x+243)/(8*x**3+36*x**2+54*x+27),x)

[Out]

-x**2 - x*(-25*log(2)**2 - 29/4 + 15*log(2)) - (x*(45 - 180*log(2)) - 270*log(2) + 81)/(16*x**2 + 48*x + 36)

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