3.72.5 \(\int \frac {1}{5} (109-400 x+390 x^2-160 x^3+25 x^4+(600 x^2-640 x^3+200 x^4) \log (5)+400 x^4 \log ^2(5)) \, dx\)

Optimal. Leaf size=24 \[ x \left (-\frac {16}{5}+\left (1+(2-x)^2+4 x^2 \log (5)\right )^2\right ) \]

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Rubi [B]  time = 0.02, antiderivative size = 54, normalized size of antiderivative = 2.25, number of steps used = 4, number of rules used = 2, integrand size = 52, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {6, 12} \begin {gather*} x^5 \left (1+16 \log ^2(5)\right )+8 x^5 \log (5)-8 x^4-32 x^4 \log (5)+26 x^3+40 x^3 \log (5)-40 x^2+\frac {109 x}{5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(109 - 400*x + 390*x^2 - 160*x^3 + 25*x^4 + (600*x^2 - 640*x^3 + 200*x^4)*Log[5] + 400*x^4*Log[5]^2)/5,x]

[Out]

(109*x)/5 - 40*x^2 + 26*x^3 - 8*x^4 + 40*x^3*Log[5] - 32*x^4*Log[5] + 8*x^5*Log[5] + x^5*(1 + 16*Log[5]^2)

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}{5} \left (109-400 x+390 x^2-160 x^3+\left (600 x^2-640 x^3+200 x^4\right ) \log (5)+x^4 \left (25+400 \log ^2(5)\right )\right ) \, dx\\ &=\frac {1}{5} \int \left (109-400 x+390 x^2-160 x^3+\left (600 x^2-640 x^3+200 x^4\right ) \log (5)+x^4 \left (25+400 \log ^2(5)\right )\right ) \, dx\\ &=\frac {109 x}{5}-40 x^2+26 x^3-8 x^4+x^5 \left (1+16 \log ^2(5)\right )+\frac {1}{5} \log (5) \int \left (600 x^2-640 x^3+200 x^4\right ) \, dx\\ &=\frac {109 x}{5}-40 x^2+26 x^3-8 x^4+40 x^3 \log (5)-32 x^4 \log (5)+8 x^5 \log (5)+x^5 \left (1+16 \log ^2(5)\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 41, normalized size = 1.71 \begin {gather*} \frac {109 x}{5}-40 x^2+2 x^3 (13+20 \log (5))-8 x^4 (1+\log (625))+x^5 (1+\log (625))^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(109 - 400*x + 390*x^2 - 160*x^3 + 25*x^4 + (600*x^2 - 640*x^3 + 200*x^4)*Log[5] + 400*x^4*Log[5]^2)
/5,x]

[Out]

(109*x)/5 - 40*x^2 + 2*x^3*(13 + 20*Log[5]) - 8*x^4*(1 + Log[625]) + x^5*(1 + Log[625])^2

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fricas [B]  time = 1.00, size = 49, normalized size = 2.04 \begin {gather*} 16 \, x^{5} \log \relax (5)^{2} + x^{5} - 8 \, x^{4} + 26 \, x^{3} - 40 \, x^{2} + 8 \, {\left (x^{5} - 4 \, x^{4} + 5 \, x^{3}\right )} \log \relax (5) + \frac {109}{5} \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(80*x^4*log(5)^2+1/5*(200*x^4-640*x^3+600*x^2)*log(5)+5*x^4-32*x^3+78*x^2-80*x+109/5,x, algorithm="fr
icas")

[Out]

16*x^5*log(5)^2 + x^5 - 8*x^4 + 26*x^3 - 40*x^2 + 8*(x^5 - 4*x^4 + 5*x^3)*log(5) + 109/5*x

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giac [B]  time = 0.21, size = 49, normalized size = 2.04 \begin {gather*} 16 \, x^{5} \log \relax (5)^{2} + x^{5} - 8 \, x^{4} + 26 \, x^{3} - 40 \, x^{2} + 8 \, {\left (x^{5} - 4 \, x^{4} + 5 \, x^{3}\right )} \log \relax (5) + \frac {109}{5} \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(80*x^4*log(5)^2+1/5*(200*x^4-640*x^3+600*x^2)*log(5)+5*x^4-32*x^3+78*x^2-80*x+109/5,x, algorithm="gi
ac")

[Out]

16*x^5*log(5)^2 + x^5 - 8*x^4 + 26*x^3 - 40*x^2 + 8*(x^5 - 4*x^4 + 5*x^3)*log(5) + 109/5*x

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maple [B]  time = 0.02, size = 46, normalized size = 1.92

method	result	size
norman	\(\left (-32 \ln \relax (5)-8\right ) x^{4}+\left (40 \ln \relax (5)+26\right ) x^{3}+\left (16 \ln \relax (5)^{2}+8 \ln \relax (5)+1\right ) x^{5}+\frac {109 x}{5}-40 x^{2}\)	\(46\)
default	\(16 x^{5} \ln \relax (5)^{2}+\frac {\ln \relax (5) \left (40 x^{5}-160 x^{4}+200 x^{3}\right )}{5}+x^{5}-8 x^{4}+26 x^{3}-40 x^{2}+\frac {109 x}{5}\)	\(52\)
risch	\(16 x^{5} \ln \relax (5)^{2}+8 x^{5} \ln \relax (5)-32 x^{4} \ln \relax (5)+40 x^{3} \ln \relax (5)+x^{5}-8 x^{4}+26 x^{3}-40 x^{2}+\frac {109 x}{5}\)	\(53\)
gosper	\(\frac {x \left (80 x^{4} \ln \relax (5)^{2}+40 x^{4} \ln \relax (5)-160 x^{3} \ln \relax (5)+5 x^{4}+200 x^{2} \ln \relax (5)-40 x^{3}+130 x^{2}-200 x +109\right )}{5}\)	\(54\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(80*x^4*ln(5)^2+1/5*(200*x^4-640*x^3+600*x^2)*ln(5)+5*x^4-32*x^3+78*x^2-80*x+109/5,x,method=_RETURNVERBOSE)

[Out]

(-32*ln(5)-8)*x^4+(40*ln(5)+26)*x^3+(16*ln(5)^2+8*ln(5)+1)*x^5+109/5*x-40*x^2

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maxima [B]  time = 0.37, size = 49, normalized size = 2.04 \begin {gather*} 16 \, x^{5} \log \relax (5)^{2} + x^{5} - 8 \, x^{4} + 26 \, x^{3} - 40 \, x^{2} + 8 \, {\left (x^{5} - 4 \, x^{4} + 5 \, x^{3}\right )} \log \relax (5) + \frac {109}{5} \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(80*x^4*log(5)^2+1/5*(200*x^4-640*x^3+600*x^2)*log(5)+5*x^4-32*x^3+78*x^2-80*x+109/5,x, algorithm="ma
xima")

[Out]

16*x^5*log(5)^2 + x^5 - 8*x^4 + 26*x^3 - 40*x^2 + 8*(x^5 - 4*x^4 + 5*x^3)*log(5) + 109/5*x

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mupad [B]  time = 4.09, size = 46, normalized size = 1.92 \begin {gather*} \left (8\,\ln \relax (5)+16\,{\ln \relax (5)}^2+1\right )\,x^5+\left (-32\,\ln \relax (5)-8\right )\,x^4+\left (40\,\ln \relax (5)+26\right )\,x^3-40\,x^2+\frac {109\,x}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(80*x^4*log(5)^2 - 80*x + (log(5)*(600*x^2 - 640*x^3 + 200*x^4))/5 + 78*x^2 - 32*x^3 + 5*x^4 + 109/5,x)

[Out]

(109*x)/5 - x^4*(32*log(5) + 8) + x^3*(40*log(5) + 26) + x^5*(8*log(5) + 16*log(5)^2 + 1) - 40*x^2

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sympy [B]  time = 0.08, size = 48, normalized size = 2.00 \begin {gather*} x^{5} \left (1 + 8 \log {\relax (5 )} + 16 \log {\relax (5 )}^{2}\right ) + x^{4} \left (- 32 \log {\relax (5 )} - 8\right ) + x^{3} \left (26 + 40 \log {\relax (5 )}\right ) - 40 x^{2} + \frac {109 x}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(80*x**4*ln(5)**2+1/5*(200*x**4-640*x**3+600*x**2)*ln(5)+5*x**4-32*x**3+78*x**2-80*x+109/5,x)

[Out]

x**5*(1 + 8*log(5) + 16*log(5)**2) + x**4*(-32*log(5) - 8) + x**3*(26 + 40*log(5)) - 40*x**2 + 109*x/5

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