3.15.17 \(\int \frac {1}{9} (288 x+2688 x^3+4704 x^5+(-1800 x^2-7000 x^4) \log (2)+2500 x^3 \log ^2(2)) \, dx\)

Optimal. Leaf size=22 \[ \left (4 x+x^3-\frac {25}{3} x^2 (-x+\log (2))\right )^2 \]

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Rubi [B]  time = 0.02, antiderivative size = 46, normalized size of antiderivative = 2.09, number of steps used = 4, number of rules used = 2, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.049, Rules used = {6, 12} \begin {gather*} \frac {784 x^6}{9}-\frac {1400}{9} x^5 \log (2)+\frac {1}{9} x^4 \left (672+625 \log ^2(2)\right )-\frac {200}{3} x^3 \log (2)+16 x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(288*x + 2688*x^3 + 4704*x^5 + (-1800*x^2 - 7000*x^4)*Log[2] + 2500*x^3*Log[2]^2)/9,x]

[Out]

16*x^2 + (784*x^6)/9 - (200*x^3*Log[2])/3 - (1400*x^5*Log[2])/9 + (x^4*(672 + 625*Log[2]^2))/9

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, x]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {1}{9} \left (288 x+4704 x^5+\left (-1800 x^2-7000 x^4\right ) \log (2)+x^3 \left (2688+2500 \log ^2(2)\right )\right ) \, dx\\ &=\frac {1}{9} \int \left (288 x+4704 x^5+\left (-1800 x^2-7000 x^4\right ) \log (2)+x^3 \left (2688+2500 \log ^2(2)\right )\right ) \, dx\\ &=16 x^2+\frac {784 x^6}{9}+\frac {1}{9} x^4 \left (672+625 \log ^2(2)\right )+\frac {1}{9} \log (2) \int \left (-1800 x^2-7000 x^4\right ) \, dx\\ &=16 x^2+\frac {784 x^6}{9}-\frac {200}{3} x^3 \log (2)-\frac {1400}{9} x^5 \log (2)+\frac {1}{9} x^4 \left (672+625 \log ^2(2)\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 21, normalized size = 0.95 \begin {gather*} \frac {1}{9} x^2 \left (12+28 x^2-25 x \log (2)\right )^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(288*x + 2688*x^3 + 4704*x^5 + (-1800*x^2 - 7000*x^4)*Log[2] + 2500*x^3*Log[2]^2)/9,x]

[Out]

(x^2*(12 + 28*x^2 - 25*x*Log[2])^2)/9

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fricas [A]  time = 0.76, size = 40, normalized size = 1.82 \begin {gather*} \frac {784}{9} \, x^{6} + \frac {625}{9} \, x^{4} \log \relax (2)^{2} + \frac {224}{3} \, x^{4} + 16 \, x^{2} - \frac {200}{9} \, {\left (7 \, x^{5} + 3 \, x^{3}\right )} \log \relax (2) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2500/9*x^3*log(2)^2+1/9*(-7000*x^4-1800*x^2)*log(2)+1568/3*x^5+896/3*x^3+32*x,x, algorithm="fricas")

[Out]

784/9*x^6 + 625/9*x^4*log(2)^2 + 224/3*x^4 + 16*x^2 - 200/9*(7*x^5 + 3*x^3)*log(2)

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giac [A]  time = 0.31, size = 40, normalized size = 1.82 \begin {gather*} \frac {784}{9} \, x^{6} + \frac {625}{9} \, x^{4} \log \relax (2)^{2} + \frac {224}{3} \, x^{4} + 16 \, x^{2} - \frac {200}{9} \, {\left (7 \, x^{5} + 3 \, x^{3}\right )} \log \relax (2) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2500/9*x^3*log(2)^2+1/9*(-7000*x^4-1800*x^2)*log(2)+1568/3*x^5+896/3*x^3+32*x,x, algorithm="giac")

[Out]

784/9*x^6 + 625/9*x^4*log(2)^2 + 224/3*x^4 + 16*x^2 - 200/9*(7*x^5 + 3*x^3)*log(2)

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maple [A]  time = 0.03, size = 20, normalized size = 0.91

method	result	size
gosper	\(\frac {x^{2} \left (25 x \ln \relax (2)-28 x^{2}-12\right )^{2}}{9}\)	\(20\)
norman	\(\left (\frac {625 \ln \relax (2)^{2}}{9}+\frac {224}{3}\right ) x^{4}+16 x^{2}+\frac {784 x^{6}}{9}-\frac {200 x^{3} \ln \relax (2)}{3}-\frac {1400 x^{5} \ln \relax (2)}{9}\)	\(38\)
risch	\(\frac {625 x^{4} \ln \relax (2)^{2}}{9}-\frac {1400 x^{5} \ln \relax (2)}{9}-\frac {200 x^{3} \ln \relax (2)}{3}+\frac {784 x^{6}}{9}+\frac {224 x^{4}}{3}+16 x^{2}\)	\(40\)
default	\(\frac {625 x^{4} \ln \relax (2)^{2}}{9}+\frac {\ln \relax (2) \left (-1400 x^{5}-600 x^{3}\right )}{9}+\frac {784 x^{6}}{9}+\frac {224 x^{4}}{3}+16 x^{2}\)	\(41\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(2500/9*x^3*ln(2)^2+1/9*(-7000*x^4-1800*x^2)*ln(2)+1568/3*x^5+896/3*x^3+32*x,x,method=_RETURNVERBOSE)

[Out]

1/9*x^2*(25*x*ln(2)-28*x^2-12)^2

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maxima [A]  time = 0.36, size = 40, normalized size = 1.82 \begin {gather*} \frac {784}{9} \, x^{6} + \frac {625}{9} \, x^{4} \log \relax (2)^{2} + \frac {224}{3} \, x^{4} + 16 \, x^{2} - \frac {200}{9} \, {\left (7 \, x^{5} + 3 \, x^{3}\right )} \log \relax (2) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2500/9*x^3*log(2)^2+1/9*(-7000*x^4-1800*x^2)*log(2)+1568/3*x^5+896/3*x^3+32*x,x, algorithm="maxima")

[Out]

784/9*x^6 + 625/9*x^4*log(2)^2 + 224/3*x^4 + 16*x^2 - 200/9*(7*x^5 + 3*x^3)*log(2)

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mupad [B]  time = 0.04, size = 37, normalized size = 1.68 \begin {gather*} \frac {784\,x^6}{9}-\frac {1400\,\ln \relax (2)\,x^5}{9}+\left (\frac {625\,{\ln \relax (2)}^2}{9}+\frac {224}{3}\right )\,x^4-\frac {200\,\ln \relax (2)\,x^3}{3}+16\,x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(32*x + (2500*x^3*log(2)^2)/9 - (log(2)*(1800*x^2 + 7000*x^4))/9 + (896*x^3)/3 + (1568*x^5)/3,x)

[Out]

x^4*((625*log(2)^2)/9 + 224/3) - (1400*x^5*log(2))/9 - (200*x^3*log(2))/3 + 16*x^2 + (784*x^6)/9

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sympy [B]  time = 0.06, size = 46, normalized size = 2.09 \begin {gather*} \frac {784 x^{6}}{9} - \frac {1400 x^{5} \log {\relax (2 )}}{9} + x^{4} \left (\frac {625 \log {\relax (2 )}^{2}}{9} + \frac {224}{3}\right ) - \frac {200 x^{3} \log {\relax (2 )}}{3} + 16 x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2500/9*x**3*ln(2)**2+1/9*(-7000*x**4-1800*x**2)*ln(2)+1568/3*x**5+896/3*x**3+32*x,x)

[Out]

784*x**6/9 - 1400*x**5*log(2)/9 + x**4*(625*log(2)**2/9 + 224/3) - 200*x**3*log(2)/3 + 16*x**2

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