3.33.36 \(\int \frac {-4050+(-180-90 x) \log ^2(2)+(48+88 x+60 x^2+18 x^3+2 x^4) \log ^4(2)}{(8+12 x+6 x^2+x^3) \log ^4(2)} \, dx\)

Optimal. Leaf size=18 \[ -3+\left (3+x+\frac {45}{(2+x) \log ^2(2)}\right )^2 \]

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Rubi [A]  time = 0.06, antiderivative size = 29, normalized size of antiderivative = 1.61, 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 = {12, 2074} \begin {gather*} x^2+6 x+\frac {2025}{(x+2)^2 \log ^4(2)}+\frac {90}{(x+2) \log ^2(2)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-4050 + (-180 - 90*x)*Log[2]^2 + (48 + 88*x + 60*x^2 + 18*x^3 + 2*x^4)*Log[2]^4)/((8 + 12*x + 6*x^2 + x^3
)*Log[2]^4),x]

[Out]

6*x + x^2 + 2025/((2 + x)^2*Log[2]^4) + 90/((2 + x)*Log[2]^2)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, 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} &=\frac {\int \frac {-4050+(-180-90 x) \log ^2(2)+\left (48+88 x+60 x^2+18 x^3+2 x^4\right ) \log ^4(2)}{8+12 x+6 x^2+x^3} \, dx}{\log ^4(2)}\\ &=\frac {\int \left (-\frac {4050}{(2+x)^3}-\frac {90 \log ^2(2)}{(2+x)^2}+6 \log ^4(2)+2 x \log ^4(2)\right ) \, dx}{\log ^4(2)}\\ &=6 x+x^2+\frac {2025}{(2+x)^2 \log ^4(2)}+\frac {90}{(2+x) \log ^2(2)}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.04, size = 29, normalized size = 1.61 \begin {gather*} 6 x+x^2+\frac {2025}{(2+x)^2 \log ^4(2)}+\frac {90}{(2+x) \log ^2(2)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-4050 + (-180 - 90*x)*Log[2]^2 + (48 + 88*x + 60*x^2 + 18*x^3 + 2*x^4)*Log[2]^4)/((8 + 12*x + 6*x^2
+ x^3)*Log[2]^4),x]

[Out]

6*x + x^2 + 2025/((2 + x)^2*Log[2]^4) + 90/((2 + x)*Log[2]^2)

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fricas [B]  time = 0.58, size = 48, normalized size = 2.67 \begin {gather*} \frac {{\left (x^{4} + 10 \, x^{3} + 28 \, x^{2} + 24 \, x\right )} \log \relax (2)^{4} + 90 \, {\left (x + 2\right )} \log \relax (2)^{2} + 2025}{{\left (x^{2} + 4 \, x + 4\right )} \log \relax (2)^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^4+18*x^3+60*x^2+88*x+48)*log(2)^4+(-90*x-180)*log(2)^2-4050)/(x^3+6*x^2+12*x+8)/log(2)^4,x, al
gorithm="fricas")

[Out]

((x^4 + 10*x^3 + 28*x^2 + 24*x)*log(2)^4 + 90*(x + 2)*log(2)^2 + 2025)/((x^2 + 4*x + 4)*log(2)^4)

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giac [B]  time = 0.22, size = 43, normalized size = 2.39 \begin {gather*} \frac {x^{2} \log \relax (2)^{4} + 6 \, x \log \relax (2)^{4} + \frac {45 \, {\left (2 \, x \log \relax (2)^{2} + 4 \, \log \relax (2)^{2} + 45\right )}}{{\left (x + 2\right )}^{2}}}{\log \relax (2)^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^4+18*x^3+60*x^2+88*x+48)*log(2)^4+(-90*x-180)*log(2)^2-4050)/(x^3+6*x^2+12*x+8)/log(2)^4,x, al
gorithm="giac")

[Out]

(x^2*log(2)^4 + 6*x*log(2)^4 + 45*(2*x*log(2)^2 + 4*log(2)^2 + 45)/(x + 2)^2)/log(2)^4

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maple [B]  time = 0.07, size = 38, normalized size = 2.11

method	result	size
risch	\(x^{2}+6 x +\frac {90 x \ln \relax (2)^{2}+180 \ln \relax (2)^{2}+2025}{\ln \relax (2)^{4} \left (x^{2}+4 x +4\right )}\)	\(38\)
default	\(\frac {x^{2} \ln \relax (2)^{4}+6 x \ln \relax (2)^{4}+\frac {90 \ln \relax (2)^{2}}{2+x}+\frac {2025}{\left (2+x \right )^{2}}}{\ln \relax (2)^{4}}\)	\(40\)
gosper	\(\frac {\ln \relax (2)^{4} x^{4}+10 x^{3} \ln \relax (2)^{4}-88 x \ln \relax (2)^{4}-112 \ln \relax (2)^{4}+90 x \ln \relax (2)^{2}+180 \ln \relax (2)^{2}+2025}{\ln \relax (2)^{4} \left (x^{2}+4 x +4\right )}\)	\(61\)
norman	\(\frac {x^{4} \ln \relax (2)^{3}-2 \ln \relax (2) \left (44 \ln \relax (2)^{2}-45\right ) x +10 x^{3} \ln \relax (2)^{3}-\frac {112 \ln \relax (2)^{4}-180 \ln \relax (2)^{2}-2025}{\ln \relax (2)}}{\left (2+x \right )^{2} \ln \relax (2)^{3}}\)	\(62\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*x^4+18*x^3+60*x^2+88*x+48)*ln(2)^4+(-90*x-180)*ln(2)^2-4050)/(x^3+6*x^2+12*x+8)/ln(2)^4,x,method=_RETU
RNVERBOSE)

[Out]

x^2+6*x+1/ln(2)^4*(90*x*ln(2)^2+180*ln(2)^2+2025)/(x^2+4*x+4)

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maxima [B]  time = 0.36, size = 48, normalized size = 2.67 \begin {gather*} \frac {x^{2} \log \relax (2)^{4} + 6 \, x \log \relax (2)^{4} + \frac {45 \, {\left (2 \, x \log \relax (2)^{2} + 4 \, \log \relax (2)^{2} + 45\right )}}{x^{2} + 4 \, x + 4}}{\log \relax (2)^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^4+18*x^3+60*x^2+88*x+48)*log(2)^4+(-90*x-180)*log(2)^2-4050)/(x^3+6*x^2+12*x+8)/log(2)^4,x, al
gorithm="maxima")

[Out]

(x^2*log(2)^4 + 6*x*log(2)^4 + 45*(2*x*log(2)^2 + 4*log(2)^2 + 45)/(x^2 + 4*x + 4))/log(2)^4

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mupad [B]  time = 0.12, size = 58, normalized size = 3.22 \begin {gather*} \frac {28\,x^2\,{\ln \relax (2)}^4+10\,x^3\,{\ln \relax (2)}^4+x^4\,{\ln \relax (2)}^4+90\,x\,{\ln \relax (2)}^2+24\,x\,{\ln \relax (2)}^4+180\,{\ln \relax (2)}^2+2025}{{\ln \relax (2)}^4\,{\left (x+2\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(2)^2*(90*x + 180) - log(2)^4*(88*x + 60*x^2 + 18*x^3 + 2*x^4 + 48) + 4050)/(log(2)^4*(12*x + 6*x^2 +
x^3 + 8)),x)

[Out]

(28*x^2*log(2)^4 + 10*x^3*log(2)^4 + x^4*log(2)^4 + 90*x*log(2)^2 + 24*x*log(2)^4 + 180*log(2)^2 + 2025)/(log(
2)^4*(x + 2)^2)

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sympy [B]  time = 0.30, size = 46, normalized size = 2.56 \begin {gather*} x^{2} + 6 x + \frac {90 x \log {\relax (2 )}^{2} + 180 \log {\relax (2 )}^{2} + 2025}{x^{2} \log {\relax (2 )}^{4} + 4 x \log {\relax (2 )}^{4} + 4 \log {\relax (2 )}^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x**4+18*x**3+60*x**2+88*x+48)*ln(2)**4+(-90*x-180)*ln(2)**2-4050)/(x**3+6*x**2+12*x+8)/ln(2)**4,
x)

[Out]

x**2 + 6*x + (90*x*log(2)**2 + 180*log(2)**2 + 2025)/(x**2*log(2)**4 + 4*x*log(2)**4 + 4*log(2)**4)

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