3.46.26 \(\int \frac {-25920-60480 x-52560 x^2-20160 x^3-2880 x^4+(-72-168 x-72 x^2) \log (2)}{180 x^2+420 x^3+365 x^4+140 x^5+20 x^6} \, dx\)

Optimal. Leaf size=25 \[ \frac {12 \left (12+\frac {\log (2)}{5 (2+x) (3+2 x)}\right )}{x} \]

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Rubi [A]  time = 0.09, antiderivative size = 36, normalized size of antiderivative = 1.44, number of steps used = 2, number of rules used = 1, integrand size = 62, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.016, Rules used = {2074} \begin {gather*} \frac {6 \log (2)}{5 (x+2)}-\frac {16 \log (2)}{5 (2 x+3)}+\frac {720+\log (4)}{5 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-25920 - 60480*x - 52560*x^2 - 20160*x^3 - 2880*x^4 + (-72 - 168*x - 72*x^2)*Log[2])/(180*x^2 + 420*x^3 +
365*x^4 + 140*x^5 + 20*x^6),x]

[Out]

(6*Log[2])/(5*(2 + x)) - (16*Log[2])/(5*(3 + 2*x)) + (720 + Log[4])/(5*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 (-\frac {6 \log (2)}{5 (2+x)^2}+\frac {32 \log (2)}{5 (3+2 x)^2}+\frac {-720-\log (4)}{5 x^2}\right ) \, dx\\ &=\frac {6 \log (2)}{5 (2+x)}-\frac {16 \log (2)}{5 (3+2 x)}+\frac {720+\log (4)}{5 x}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.04, size = 31, normalized size = 1.24 \begin {gather*} \frac {4 \left (3240+3780 x+1080 x^2+\log (512)\right )}{15 x \left (6+7 x+2 x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-25920 - 60480*x - 52560*x^2 - 20160*x^3 - 2880*x^4 + (-72 - 168*x - 72*x^2)*Log[2])/(180*x^2 + 420
*x^3 + 365*x^4 + 140*x^5 + 20*x^6),x]

[Out]

(4*(3240 + 3780*x + 1080*x^2 + Log[512]))/(15*x*(6 + 7*x + 2*x^2))

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fricas [A]  time = 0.65, size = 30, normalized size = 1.20 \begin {gather*} \frac {12 \, {\left (120 \, x^{2} + 420 \, x + \log \relax (2) + 360\right )}}{5 \, {\left (2 \, x^{3} + 7 \, x^{2} + 6 \, x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-72*x^2-168*x-72)*log(2)-2880*x^4-20160*x^3-52560*x^2-60480*x-25920)/(20*x^6+140*x^5+365*x^4+420*x
^3+180*x^2),x, algorithm="fricas")

[Out]

12/5*(120*x^2 + 420*x + log(2) + 360)/(2*x^3 + 7*x^2 + 6*x)

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giac [A]  time = 0.19, size = 30, normalized size = 1.20 \begin {gather*} \frac {12 \, {\left (120 \, x^{2} + 420 \, x + \log \relax (2) + 360\right )}}{5 \, {\left (2 \, x^{3} + 7 \, x^{2} + 6 \, x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-72*x^2-168*x-72)*log(2)-2880*x^4-20160*x^3-52560*x^2-60480*x-25920)/(20*x^6+140*x^5+365*x^4+420*x
^3+180*x^2),x, algorithm="giac")

[Out]

12/5*(120*x^2 + 420*x + log(2) + 360)/(2*x^3 + 7*x^2 + 6*x)

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maple [A]  time = 0.06, size = 30, normalized size = 1.20

method	result	size
gosper	\(\frac {288 x^{2}+1008 x +864+\frac {12 \ln \relax (2)}{5}}{x \left (2 x^{2}+7 x +6\right )}\)	\(30\)
norman	\(\frac {288 x^{2}+1008 x +864+\frac {12 \ln \relax (2)}{5}}{x \left (2 x^{2}+7 x +6\right )}\)	\(31\)
risch	\(\frac {288 x^{2}+1008 x +864+\frac {12 \ln \relax (2)}{5}}{x \left (2 x^{2}+7 x +6\right )}\)	\(32\)
default	\(-\frac {24 \left (-\frac {\ln \relax (2)}{12}-30\right )}{5 x}-\frac {16 \ln \relax (2)}{5 \left (2 x +3\right )}+\frac {6 \ln \relax (2)}{5 \left (2+x \right )}\)	\(33\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-72*x^2-168*x-72)*ln(2)-2880*x^4-20160*x^3-52560*x^2-60480*x-25920)/(20*x^6+140*x^5+365*x^4+420*x^3+180*
x^2),x,method=_RETURNVERBOSE)

[Out]

12/5/x*(120*x^2+ln(2)+420*x+360)/(2*x^2+7*x+6)

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maxima [A]  time = 0.46, size = 30, normalized size = 1.20 \begin {gather*} \frac {12 \, {\left (120 \, x^{2} + 420 \, x + \log \relax (2) + 360\right )}}{5 \, {\left (2 \, x^{3} + 7 \, x^{2} + 6 \, x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-72*x^2-168*x-72)*log(2)-2880*x^4-20160*x^3-52560*x^2-60480*x-25920)/(20*x^6+140*x^5+365*x^4+420*x
^3+180*x^2),x, algorithm="maxima")

[Out]

12/5*(120*x^2 + 420*x + log(2) + 360)/(2*x^3 + 7*x^2 + 6*x)

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mupad [B]  time = 0.12, size = 29, normalized size = 1.16 \begin {gather*} \frac {1440\,x^2+5040\,x+\ln \left (4096\right )+4320}{5\,\left (2\,x^3+7\,x^2+6\,x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(60480*x + log(2)*(168*x + 72*x^2 + 72) + 52560*x^2 + 20160*x^3 + 2880*x^4 + 25920)/(180*x^2 + 420*x^3 +
365*x^4 + 140*x^5 + 20*x^6),x)

[Out]

(5040*x + log(4096) + 1440*x^2 + 4320)/(5*(6*x + 7*x^2 + 2*x^3))

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sympy [A]  time = 0.55, size = 31, normalized size = 1.24 \begin {gather*} - \frac {- 1440 x^{2} - 5040 x - 4320 - 12 \log {\relax (2 )}}{10 x^{3} + 35 x^{2} + 30 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-72*x**2-168*x-72)*ln(2)-2880*x**4-20160*x**3-52560*x**2-60480*x-25920)/(20*x**6+140*x**5+365*x**4
+420*x**3+180*x**2),x)

[Out]

-(-1440*x**2 - 5040*x - 4320 - 12*log(2))/(10*x**3 + 35*x**2 + 30*x)

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