3.52.63 \(\int \frac {28800 x+97200 x^2+119700 x^3+63900 x^4+13500 x^5+900 x^6+(960+1200 x+360 x^2+120 x^3) \log ^2(2)-32 \log ^4(2)}{225+675 x+675 x^2+225 x^3} \, dx\)

Optimal. Leaf size=27 \[ \left (x^2+4 \left (2 x+\frac {\log ^2(2)}{5 (3+3 x)}\right )\right )^2 \]

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Rubi [A]  time = 0.10, antiderivative size = 49, normalized size of antiderivative = 1.81, number of steps used = 2, number of rules used = 1, integrand size = 73, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.014, Rules used = {2074} \begin {gather*} x^4+16 x^3+64 x^2+\frac {16 \log ^4(2)}{225 (x+1)^2}+\frac {8}{15} x \log ^2(2)-\frac {56 \log ^2(2)}{15 (x+1)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(28800*x + 97200*x^2 + 119700*x^3 + 63900*x^4 + 13500*x^5 + 900*x^6 + (960 + 1200*x + 360*x^2 + 120*x^3)*L
og[2]^2 - 32*Log[2]^4)/(225 + 675*x + 675*x^2 + 225*x^3),x]

[Out]

64*x^2 + 16*x^3 + x^4 + (8*x*Log[2]^2)/15 - (56*Log[2]^2)/(15*(1 + x)) + (16*Log[2]^4)/(225*(1 + x)^2)

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 (128 x+48 x^2+4 x^3+\frac {8 \log ^2(2)}{15}+\frac {56 \log ^2(2)}{15 (1+x)^2}-\frac {32 \log ^4(2)}{225 (1+x)^3}\right ) \, dx\\ &=64 x^2+16 x^3+x^4+\frac {8}{15} x \log ^2(2)-\frac {56 \log ^2(2)}{15 (1+x)}+\frac {16 \log ^4(2)}{225 (1+x)^2}\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.05, size = 57, normalized size = 2.11 \begin {gather*} \frac {4}{225} \left (-\frac {11025}{4}+3600 x^2+900 x^3+\frac {225 x^4}{4}+30 x \log ^2(2)+\frac {30 (-6+x) \log ^2(2)}{1+x}+\frac {4 \log ^4(2)}{(1+x)^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(28800*x + 97200*x^2 + 119700*x^3 + 63900*x^4 + 13500*x^5 + 900*x^6 + (960 + 1200*x + 360*x^2 + 120*
x^3)*Log[2]^2 - 32*Log[2]^4)/(225 + 675*x + 675*x^2 + 225*x^3),x]

[Out]

(4*(-11025/4 + 3600*x^2 + 900*x^3 + (225*x^4)/4 + 30*x*Log[2]^2 + (30*(-6 + x)*Log[2]^2)/(1 + x) + (4*Log[2]^4
)/(1 + x)^2))/225

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fricas [B]  time = 0.52, size = 63, normalized size = 2.33 \begin {gather*} \frac {225 \, x^{6} + 4050 \, x^{5} + 21825 \, x^{4} + 16 \, \log \relax (2)^{4} + 32400 \, x^{3} + 120 \, {\left (x^{3} + 2 \, x^{2} - 6 \, x - 7\right )} \log \relax (2)^{2} + 14400 \, x^{2}}{225 \, {\left (x^{2} + 2 \, x + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-32*log(2)^4+(120*x^3+360*x^2+1200*x+960)*log(2)^2+900*x^6+13500*x^5+63900*x^4+119700*x^3+97200*x^2
+28800*x)/(225*x^3+675*x^2+675*x+225),x, algorithm="fricas")

[Out]

1/225*(225*x^6 + 4050*x^5 + 21825*x^4 + 16*log(2)^4 + 32400*x^3 + 120*(x^3 + 2*x^2 - 6*x - 7)*log(2)^2 + 14400
*x^2)/(x^2 + 2*x + 1)

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giac [A]  time = 0.22, size = 48, normalized size = 1.78 \begin {gather*} x^{4} + 16 \, x^{3} + \frac {8}{15} \, x \log \relax (2)^{2} + 64 \, x^{2} + \frac {8 \, {\left (2 \, \log \relax (2)^{4} - 105 \, x \log \relax (2)^{2} - 105 \, \log \relax (2)^{2}\right )}}{225 \, {\left (x + 1\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-32*log(2)^4+(120*x^3+360*x^2+1200*x+960)*log(2)^2+900*x^6+13500*x^5+63900*x^4+119700*x^3+97200*x^2
+28800*x)/(225*x^3+675*x^2+675*x+225),x, algorithm="giac")

[Out]

x^4 + 16*x^3 + 8/15*x*log(2)^2 + 64*x^2 + 8/225*(2*log(2)^4 - 105*x*log(2)^2 - 105*log(2)^2)/(x + 1)^2

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maple [A]  time = 0.09, size = 44, normalized size = 1.63

method	result	size
default	\(x^{4}+16 x^{3}+64 x^{2}+\frac {8 x \ln \relax (2)^{2}}{15}+\frac {16 \ln \relax (2)^{4}}{225 \left (x +1\right )^{2}}-\frac {56 \ln \relax (2)^{2}}{15 \left (x +1\right )}\)	\(44\)
risch	\(x^{4}+\frac {8 x \ln \relax (2)^{2}}{15}+16 x^{3}+64 x^{2}+\frac {-\frac {56 x \ln \relax (2)^{2}}{15}+\frac {16 \ln \relax (2)^{4}}{225}-\frac {56 \ln \relax (2)^{2}}{15}}{x^{2}+2 x +1}\)	\(53\)
norman	\(\frac {x^{6}+\left (144+\frac {8 \ln \relax (2)^{2}}{15}\right ) x^{3}+\left (-128-\frac {16 \ln \relax (2)^{2}}{3}\right ) x +97 x^{4}+18 x^{5}-64-\frac {24 \ln \relax (2)^{2}}{5}+\frac {16 \ln \relax (2)^{4}}{225}}{\left (x +1\right )^{2}}\)	\(56\)
gosper	\(\frac {225 x^{6}+120 x^{3} \ln \relax (2)^{2}+4050 x^{5}+16 \ln \relax (2)^{4}+21825 x^{4}-1200 x \ln \relax (2)^{2}+32400 x^{3}-1080 \ln \relax (2)^{2}-28800 x -14400}{225 x^{2}+450 x +225}\)	\(66\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-32*ln(2)^4+(120*x^3+360*x^2+1200*x+960)*ln(2)^2+900*x^6+13500*x^5+63900*x^4+119700*x^3+97200*x^2+28800*x
)/(225*x^3+675*x^2+675*x+225),x,method=_RETURNVERBOSE)

[Out]

x^4+16*x^3+64*x^2+8/15*x*ln(2)^2+16/225*ln(2)^4/(x+1)^2-56/15*ln(2)^2/(x+1)

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maxima [B]  time = 0.35, size = 53, normalized size = 1.96 \begin {gather*} x^{4} + 16 \, x^{3} + \frac {8}{15} \, x \log \relax (2)^{2} + 64 \, x^{2} + \frac {8 \, {\left (2 \, \log \relax (2)^{4} - 105 \, x \log \relax (2)^{2} - 105 \, \log \relax (2)^{2}\right )}}{225 \, {\left (x^{2} + 2 \, x + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-32*log(2)^4+(120*x^3+360*x^2+1200*x+960)*log(2)^2+900*x^6+13500*x^5+63900*x^4+119700*x^3+97200*x^2
+28800*x)/(225*x^3+675*x^2+675*x+225),x, algorithm="maxima")

[Out]

x^4 + 16*x^3 + 8/15*x*log(2)^2 + 64*x^2 + 8/225*(2*log(2)^4 - 105*x*log(2)^2 - 105*log(2)^2)/(x^2 + 2*x + 1)

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mupad [B]  time = 3.24, size = 55, normalized size = 2.04 \begin {gather*} \frac {8\,x\,{\ln \relax (2)}^2}{15}-\frac {840\,x\,{\ln \relax (2)}^2+840\,{\ln \relax (2)}^2-16\,{\ln \relax (2)}^4}{225\,x^2+450\,x+225}+64\,x^2+16\,x^3+x^4 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((28800*x - 32*log(2)^4 + log(2)^2*(1200*x + 360*x^2 + 120*x^3 + 960) + 97200*x^2 + 119700*x^3 + 63900*x^4
+ 13500*x^5 + 900*x^6)/(675*x + 675*x^2 + 225*x^3 + 225),x)

[Out]

(8*x*log(2)^2)/15 - (840*x*log(2)^2 + 840*log(2)^2 - 16*log(2)^4)/(450*x + 225*x^2 + 225) + 64*x^2 + 16*x^3 +
x^4

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sympy [B]  time = 0.23, size = 54, normalized size = 2.00 \begin {gather*} x^{4} + 16 x^{3} + 64 x^{2} + \frac {8 x \log {\relax (2 )}^{2}}{15} + \frac {- 840 x \log {\relax (2 )}^{2} - 840 \log {\relax (2 )}^{2} + 16 \log {\relax (2 )}^{4}}{225 x^{2} + 450 x + 225} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-32*ln(2)**4+(120*x**3+360*x**2+1200*x+960)*ln(2)**2+900*x**6+13500*x**5+63900*x**4+119700*x**3+972
00*x**2+28800*x)/(225*x**3+675*x**2+675*x+225),x)

[Out]

x**4 + 16*x**3 + 64*x**2 + 8*x*log(2)**2/15 + (-840*x*log(2)**2 - 840*log(2)**2 + 16*log(2)**4)/(225*x**2 + 45
0*x + 225)

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