3.91.87 \(\int \frac {1}{25} (25+200 x-60 x^2+4 x^3+500 x^4-60 x^5+200 x^7+(-200 x+20 x^2-100 x^4) \log (x)+(-200 x+30 x^2-250 x^4) \log ^2(x)+100 x \log ^3(x)+50 x \log ^4(x)) \, dx\)

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

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Rubi [B]  time = 0.19, antiderivative size = 73, normalized size of antiderivative = 3.04, number of steps used = 24, number of rules used = 5, integrand size = 84, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.060, Rules used = {12, 1594, 2356, 2304, 2305} \begin {gather*} x^8-\frac {2 x^6}{5}+4 x^5-2 x^5 \log ^2(x)+\frac {x^4}{25}-\frac {4 x^3}{5}+\frac {2}{5} x^3 \log ^2(x)+4 x^2+x^2 \log ^4(x)-4 x^2 \log ^2(x)+x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(25 + 200*x - 60*x^2 + 4*x^3 + 500*x^4 - 60*x^5 + 200*x^7 + (-200*x + 20*x^2 - 100*x^4)*Log[x] + (-200*x +
30*x^2 - 250*x^4)*Log[x]^2 + 100*x*Log[x]^3 + 50*x*Log[x]^4)/25,x]

[Out]

x + 4*x^2 - (4*x^3)/5 + x^4/25 + 4*x^5 - (2*x^6)/5 + x^8 - 4*x^2*Log[x]^2 + (2*x^3*Log[x]^2)/5 - 2*x^5*Log[x]^
2 + x^2*Log[x]^4

Rule 12

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

Rule 1594

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.) + (c_.)*(x_)^(r_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^
(q - p) + c*x^(r - p))^n, x] /; FreeQ[{a, b, c, p, q, r}, x] && IntegerQ[n] && PosQ[q - p] && PosQ[r - p]

Rule 2304

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Log[c*x^
n]))/(d*(m + 1)), x] - Simp[(b*n*(d*x)^(m + 1))/(d*(m + 1)^2), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1
]

Rule 2305

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Lo
g[c*x^n])^p)/(d*(m + 1)), x] - Dist[(b*n*p)/(m + 1), Int[(d*x)^m*(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{
a, b, c, d, m, n}, x] && NeQ[m, -1] && GtQ[p, 0]

Rule 2356

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(Polyx_), x_Symbol] :> Int[ExpandIntegrand[Polyx*(a + b*Log[c*
x^n])^p, x], x] /; FreeQ[{a, b, c, n, p}, x] && PolynomialQ[Polyx, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{25} \int \left (25+200 x-60 x^2+4 x^3+500 x^4-60 x^5+200 x^7+\left (-200 x+20 x^2-100 x^4\right ) \log (x)+\left (-200 x+30 x^2-250 x^4\right ) \log ^2(x)+100 x \log ^3(x)+50 x \log ^4(x)\right ) \, dx\\ &=x+4 x^2-\frac {4 x^3}{5}+\frac {x^4}{25}+4 x^5-\frac {2 x^6}{5}+x^8+\frac {1}{25} \int \left (-200 x+20 x^2-100 x^4\right ) \log (x) \, dx+\frac {1}{25} \int \left (-200 x+30 x^2-250 x^4\right ) \log ^2(x) \, dx+2 \int x \log ^4(x) \, dx+4 \int x \log ^3(x) \, dx\\ &=x+4 x^2-\frac {4 x^3}{5}+\frac {x^4}{25}+4 x^5-\frac {2 x^6}{5}+x^8+2 x^2 \log ^3(x)+x^2 \log ^4(x)+\frac {1}{25} \int x \left (-200+20 x-100 x^3\right ) \log (x) \, dx+\frac {1}{25} \int x \left (-200+30 x-250 x^3\right ) \log ^2(x) \, dx-4 \int x \log ^3(x) \, dx-6 \int x \log ^2(x) \, dx\\ &=x+4 x^2-\frac {4 x^3}{5}+\frac {x^4}{25}+4 x^5-\frac {2 x^6}{5}+x^8-3 x^2 \log ^2(x)+x^2 \log ^4(x)+\frac {1}{25} \int \left (-200 x \log (x)+20 x^2 \log (x)-100 x^4 \log (x)\right ) \, dx+\frac {1}{25} \int \left (-200 x \log ^2(x)+30 x^2 \log ^2(x)-250 x^4 \log ^2(x)\right ) \, dx+6 \int x \log (x) \, dx+6 \int x \log ^2(x) \, dx\\ &=x+\frac {5 x^2}{2}-\frac {4 x^3}{5}+\frac {x^4}{25}+4 x^5-\frac {2 x^6}{5}+x^8+3 x^2 \log (x)+x^2 \log ^4(x)+\frac {4}{5} \int x^2 \log (x) \, dx+\frac {6}{5} \int x^2 \log ^2(x) \, dx-4 \int x^4 \log (x) \, dx-6 \int x \log (x) \, dx-8 \int x \log (x) \, dx-8 \int x \log ^2(x) \, dx-10 \int x^4 \log ^2(x) \, dx\\ &=x+6 x^2-\frac {8 x^3}{9}+\frac {x^4}{25}+\frac {104 x^5}{25}-\frac {2 x^6}{5}+x^8-4 x^2 \log (x)+\frac {4}{15} x^3 \log (x)-\frac {4}{5} x^5 \log (x)-4 x^2 \log ^2(x)+\frac {2}{5} x^3 \log ^2(x)-2 x^5 \log ^2(x)+x^2 \log ^4(x)-\frac {4}{5} \int x^2 \log (x) \, dx+4 \int x^4 \log (x) \, dx+8 \int x \log (x) \, dx\\ &=x+4 x^2-\frac {4 x^3}{5}+\frac {x^4}{25}+4 x^5-\frac {2 x^6}{5}+x^8-4 x^2 \log ^2(x)+\frac {2}{5} x^3 \log ^2(x)-2 x^5 \log ^2(x)+x^2 \log ^4(x)\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.02, size = 73, normalized size = 3.04 \begin {gather*} x+4 x^2-\frac {4 x^3}{5}+\frac {x^4}{25}+4 x^5-\frac {2 x^6}{5}+x^8-4 x^2 \log ^2(x)+\frac {2}{5} x^3 \log ^2(x)-2 x^5 \log ^2(x)+x^2 \log ^4(x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(25 + 200*x - 60*x^2 + 4*x^3 + 500*x^4 - 60*x^5 + 200*x^7 + (-200*x + 20*x^2 - 100*x^4)*Log[x] + (-2
00*x + 30*x^2 - 250*x^4)*Log[x]^2 + 100*x*Log[x]^3 + 50*x*Log[x]^4)/25,x]

[Out]

x + 4*x^2 - (4*x^3)/5 + x^4/25 + 4*x^5 - (2*x^6)/5 + x^8 - 4*x^2*Log[x]^2 + (2*x^3*Log[x]^2)/5 - 2*x^5*Log[x]^
2 + x^2*Log[x]^4

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fricas [B]  time = 0.86, size = 60, normalized size = 2.50 \begin {gather*} x^{8} - \frac {2}{5} \, x^{6} + x^{2} \log \relax (x)^{4} + 4 \, x^{5} + \frac {1}{25} \, x^{4} - \frac {4}{5} \, x^{3} - \frac {2}{5} \, {\left (5 \, x^{5} - x^{3} + 10 \, x^{2}\right )} \log \relax (x)^{2} + 4 \, x^{2} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2*x*log(x)^4+4*x*log(x)^3+1/25*(-250*x^4+30*x^2-200*x)*log(x)^2+1/25*(-100*x^4+20*x^2-200*x)*log(x)+
8*x^7-12/5*x^5+20*x^4+4/25*x^3-12/5*x^2+8*x+1,x, algorithm="fricas")

[Out]

x^8 - 2/5*x^6 + x^2*log(x)^4 + 4*x^5 + 1/25*x^4 - 4/5*x^3 - 2/5*(5*x^5 - x^3 + 10*x^2)*log(x)^2 + 4*x^2 + x

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giac [B]  time = 0.17, size = 65, normalized size = 2.71 \begin {gather*} x^{8} - 2 \, x^{5} \log \relax (x)^{2} - \frac {2}{5} \, x^{6} + x^{2} \log \relax (x)^{4} + 4 \, x^{5} + \frac {2}{5} \, x^{3} \log \relax (x)^{2} + \frac {1}{25} \, x^{4} - 4 \, x^{2} \log \relax (x)^{2} - \frac {4}{5} \, x^{3} + 4 \, x^{2} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2*x*log(x)^4+4*x*log(x)^3+1/25*(-250*x^4+30*x^2-200*x)*log(x)^2+1/25*(-100*x^4+20*x^2-200*x)*log(x)+
8*x^7-12/5*x^5+20*x^4+4/25*x^3-12/5*x^2+8*x+1,x, algorithm="giac")

[Out]

x^8 - 2*x^5*log(x)^2 - 2/5*x^6 + x^2*log(x)^4 + 4*x^5 + 2/5*x^3*log(x)^2 + 1/25*x^4 - 4*x^2*log(x)^2 - 4/5*x^3
+ 4*x^2 + x

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maple [B]  time = 0.03, size = 66, normalized size = 2.75

method	result	size
default	\(4 x^{2}+x -\frac {4 x^{3}}{5}+\frac {x^{4}}{25}+4 x^{5}-\frac {2 x^{6}}{5}+x^{8}+x^{2} \ln \relax (x )^{4}-2 x^{5} \ln \relax (x )^{2}+\frac {2 x^{3} \ln \relax (x )^{2}}{5}-4 x^{2} \ln \relax (x )^{2}\)	\(66\)
risch	\(4 x^{2}+x -\frac {4 x^{3}}{5}+\frac {x^{4}}{25}+4 x^{5}-\frac {2 x^{6}}{5}+x^{8}+x^{2} \ln \relax (x )^{4}-2 x^{5} \ln \relax (x )^{2}+\frac {2 x^{3} \ln \relax (x )^{2}}{5}-4 x^{2} \ln \relax (x )^{2}\)	\(66\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(2*x*ln(x)^4+4*x*ln(x)^3+1/25*(-250*x^4+30*x^2-200*x)*ln(x)^2+1/25*(-100*x^4+20*x^2-200*x)*ln(x)+8*x^7-12/5
*x^5+20*x^4+4/25*x^3-12/5*x^2+8*x+1,x,method=_RETURNVERBOSE)

[Out]

4*x^2+x-4/5*x^3+1/25*x^4+4*x^5-2/5*x^6+x^8+x^2*ln(x)^4-2*x^5*ln(x)^2+2/5*x^3*ln(x)^2-4*x^2*ln(x)^2

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maxima [B]  time = 0.35, size = 153, normalized size = 6.38 \begin {gather*} x^{8} - \frac {2}{25} \, {\left (25 \, \log \relax (x)^{2} - 10 \, \log \relax (x) + 2\right )} x^{5} - \frac {2}{5} \, x^{6} + \frac {104}{25} \, x^{5} + \frac {2}{45} \, {\left (9 \, \log \relax (x)^{2} - 6 \, \log \relax (x) + 2\right )} x^{3} + \frac {1}{25} \, x^{4} + \frac {1}{2} \, {\left (2 \, \log \relax (x)^{4} - 4 \, \log \relax (x)^{3} + 6 \, \log \relax (x)^{2} - 6 \, \log \relax (x) + 3\right )} x^{2} + \frac {1}{2} \, {\left (4 \, \log \relax (x)^{3} - 6 \, \log \relax (x)^{2} + 6 \, \log \relax (x) - 3\right )} x^{2} - 2 \, {\left (2 \, \log \relax (x)^{2} - 2 \, \log \relax (x) + 1\right )} x^{2} - \frac {8}{9} \, x^{3} + 6 \, x^{2} - \frac {4}{15} \, {\left (3 \, x^{5} - x^{3} + 15 \, x^{2}\right )} \log \relax (x) + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2*x*log(x)^4+4*x*log(x)^3+1/25*(-250*x^4+30*x^2-200*x)*log(x)^2+1/25*(-100*x^4+20*x^2-200*x)*log(x)+
8*x^7-12/5*x^5+20*x^4+4/25*x^3-12/5*x^2+8*x+1,x, algorithm="maxima")

[Out]

x^8 - 2/25*(25*log(x)^2 - 10*log(x) + 2)*x^5 - 2/5*x^6 + 104/25*x^5 + 2/45*(9*log(x)^2 - 6*log(x) + 2)*x^3 + 1
/25*x^4 + 1/2*(2*log(x)^4 - 4*log(x)^3 + 6*log(x)^2 - 6*log(x) + 3)*x^2 + 1/2*(4*log(x)^3 - 6*log(x)^2 + 6*log
(x) - 3)*x^2 - 2*(2*log(x)^2 - 2*log(x) + 1)*x^2 - 8/9*x^3 + 6*x^2 - 4/15*(3*x^5 - x^3 + 15*x^2)*log(x) + x

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mupad [B]  time = 7.31, size = 65, normalized size = 2.71 \begin {gather*} x^8-\frac {2\,x^6}{5}-2\,x^5\,{\ln \relax (x)}^2+4\,x^5+\frac {x^4}{25}+\frac {2\,x^3\,{\ln \relax (x)}^2}{5}-\frac {4\,x^3}{5}+x^2\,{\ln \relax (x)}^4-4\,x^2\,{\ln \relax (x)}^2+4\,x^2+x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(8*x + 4*x*log(x)^3 + 2*x*log(x)^4 - (log(x)^2*(200*x - 30*x^2 + 250*x^4))/25 - (12*x^2)/5 + (4*x^3)/25 + 2
0*x^4 - (12*x^5)/5 + 8*x^7 - (log(x)*(200*x - 20*x^2 + 100*x^4))/25 + 1,x)

[Out]

x - 4*x^2*log(x)^2 + (2*x^3*log(x)^2)/5 + x^2*log(x)^4 - 2*x^5*log(x)^2 + 4*x^2 - (4*x^3)/5 + x^4/25 + 4*x^5 -
(2*x^6)/5 + x^8

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sympy [B]  time = 0.18, size = 63, normalized size = 2.62 \begin {gather*} x^{8} - \frac {2 x^{6}}{5} + 4 x^{5} + \frac {x^{4}}{25} - \frac {4 x^{3}}{5} + x^{2} \log {\relax (x )}^{4} + 4 x^{2} + x + \left (- 2 x^{5} + \frac {2 x^{3}}{5} - 4 x^{2}\right ) \log {\relax (x )}^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2*x*ln(x)**4+4*x*ln(x)**3+1/25*(-250*x**4+30*x**2-200*x)*ln(x)**2+1/25*(-100*x**4+20*x**2-200*x)*ln(
x)+8*x**7-12/5*x**5+20*x**4+4/25*x**3-12/5*x**2+8*x+1,x)

[Out]

x**8 - 2*x**6/5 + 4*x**5 + x**4/25 - 4*x**3/5 + x**2*log(x)**4 + 4*x**2 + x + (-2*x**5 + 2*x**3/5 - 4*x**2)*lo
g(x)**2

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