3.40.51 \(\int (1-3 x+2 x^2+2 x^3-3 x^4+x^5+(8-16 x+16 x^3-8 x^4) \log (25)+(28-32 x-24 x^2+32 x^3-4 x^4) \log ^2(25)+(56-24 x-56 x^2+24 x^3) \log ^3(25)+(70+10 x-54 x^2+6 x^3) \log ^4(25)+(56+32 x-24 x^2) \log ^5(25)+(28+24 x-4 x^2) \log ^6(25)+(8+8 x) \log ^7(25)+(1+x) \log ^8(25)+(1-6 x+6 x^2+8 x^3-15 x^4+6 x^5+(8-32 x+64 x^3-40 x^4) \log (25)+(28-64 x-72 x^2+128 x^3-20 x^4) \log ^2(25)+(56-48 x-168 x^2+96 x^3) \log ^3(25)+(70+20 x-162 x^2+24 x^3) \log ^4(25)+(56+64 x-72 x^2) \log ^5(25)+(28+48 x-12 x^2) \log ^6(25)+(8+16 x) \log ^7(25)+(1+2 x) \log ^8(25)) \log (x)) \, dx\)

Optimal. Leaf size=20 \[ \left (x+x^2\right ) \left (x-(1+\log (25))^2\right )^4 \log (x) \]

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Rubi [B]  time = 0.30, antiderivative size = 464, normalized size of antiderivative = 23.20, number of steps used = 15, number of rules used = 3, integrand size = 315, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.010, Rules used = {2356, 2304, 2295} \begin {gather*} x^6 \log (x)-\frac {3 x^5}{5}-x^5 \left (3+4 \log ^2(25)+8 \log (25)\right ) \log (x)+\frac {1}{5} x^5 \left (3+4 \log ^2(25)+8 \log (25)\right )-\frac {4}{5} x^5 \log ^2(25)-\frac {8}{5} x^5 \log (25)+\frac {x^4}{2}+\frac {3}{2} x^4 \log ^4(25)+6 x^4 \log ^3(25)+2 x^4 (1+\log (25))^2 \left (1+3 \log ^2(25)+6 \log (25)\right ) \log (x)-\frac {1}{2} x^4 (1+\log (25))^2 \left (1+3 \log ^2(25)+6 \log (25)\right )+8 x^4 \log ^2(25)+4 x^4 \log (25)+\frac {2 x^3}{3}-\frac {4}{3} x^3 \log ^6(25)-8 x^3 \log ^5(25)-18 x^3 \log ^4(25)-\frac {56}{3} x^3 \log ^3(25)+2 x^3 (1+\log (25))^4 \left (1-2 \log ^2(25)-4 \log (25)\right ) \log (x)-\frac {2}{3} x^3 (1+\log (25))^4 \left (1-2 \log ^2(25)-4 \log (25)\right )-8 x^3 \log ^2(25)-\frac {3 x^2}{2}+12 x^2 \log ^6(25)+16 x^2 \log ^5(25)+5 x^2 \log ^4(25)-12 x^2 \log ^3(25)-x^2 (1+\log (25))^6 \left (3-\log ^2(25)-\log (625)\right ) \log (x)+\frac {1}{2} x^2 (1+\log (25))^6 \left (3-\log ^2(25)-\log (625)\right )-16 x^2 \log ^2(25)-8 x^2 \log (25)+x+\frac {1}{2} (x+1)^2 \log ^8(25)+4 (x+1)^2 \log ^7(25)+28 x \log ^6(25)+56 x \log ^5(25)+70 x \log ^4(25)+56 x \log ^3(25)+28 x \log ^2(25)+x (1+\log (25))^8 \log (x)-x (1+\log (25))^8+8 x \log (25) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1 - 3*x + 2*x^2 + 2*x^3 - 3*x^4 + x^5 + (8 - 16*x + 16*x^3 - 8*x^4)*Log[25] + (28 - 32*x - 24*x^2 + 32*x^3
- 4*x^4)*Log[25]^2 + (56 - 24*x - 56*x^2 + 24*x^3)*Log[25]^3 + (70 + 10*x - 54*x^2 + 6*x^3)*Log[25]^4 + (56 +
32*x - 24*x^2)*Log[25]^5 + (28 + 24*x - 4*x^2)*Log[25]^6 + (8 + 8*x)*Log[25]^7 + (1 + x)*Log[25]^8 + (1 - 6*x
+ 6*x^2 + 8*x^3 - 15*x^4 + 6*x^5 + (8 - 32*x + 64*x^3 - 40*x^4)*Log[25] + (28 - 64*x - 72*x^2 + 128*x^3 - 20*
x^4)*Log[25]^2 + (56 - 48*x - 168*x^2 + 96*x^3)*Log[25]^3 + (70 + 20*x - 162*x^2 + 24*x^3)*Log[25]^4 + (56 + 6
4*x - 72*x^2)*Log[25]^5 + (28 + 48*x - 12*x^2)*Log[25]^6 + (8 + 16*x)*Log[25]^7 + (1 + 2*x)*Log[25]^8)*Log[x],
x]

[Out]

x - (3*x^2)/2 + (2*x^3)/3 + x^4/2 - (3*x^5)/5 + 8*x*Log[25] - 8*x^2*Log[25] + 4*x^4*Log[25] - (8*x^5*Log[25])/
5 + 28*x*Log[25]^2 - 16*x^2*Log[25]^2 - 8*x^3*Log[25]^2 + 8*x^4*Log[25]^2 - (4*x^5*Log[25]^2)/5 + 56*x*Log[25]
^3 - 12*x^2*Log[25]^3 - (56*x^3*Log[25]^3)/3 + 6*x^4*Log[25]^3 + 70*x*Log[25]^4 + 5*x^2*Log[25]^4 - 18*x^3*Log
[25]^4 + (3*x^4*Log[25]^4)/2 + 56*x*Log[25]^5 + 16*x^2*Log[25]^5 - 8*x^3*Log[25]^5 + 28*x*Log[25]^6 + 12*x^2*L
og[25]^6 - (4*x^3*Log[25]^6)/3 + 4*(1 + x)^2*Log[25]^7 + ((1 + x)^2*Log[25]^8)/2 - x*(1 + Log[25])^8 - (2*x^3*
(1 + Log[25])^4*(1 - 4*Log[25] - 2*Log[25]^2))/3 - (x^4*(1 + Log[25])^2*(1 + 6*Log[25] + 3*Log[25]^2))/2 + (x^
5*(3 + 8*Log[25] + 4*Log[25]^2))/5 + (x^2*(1 + Log[25])^6*(3 - Log[25]^2 - Log[625]))/2 + x^6*Log[x] + x*(1 +
Log[25])^8*Log[x] + 2*x^3*(1 + Log[25])^4*(1 - 4*Log[25] - 2*Log[25]^2)*Log[x] + 2*x^4*(1 + Log[25])^2*(1 + 6*
Log[25] + 3*Log[25]^2)*Log[x] - x^5*(3 + 8*Log[25] + 4*Log[25]^2)*Log[x] - x^2*(1 + Log[25])^6*(3 - Log[25]^2
- Log[625])*Log[x]

Rule 2295

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

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 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} &=x-\frac {3 x^2}{2}+\frac {2 x^3}{3}+\frac {x^4}{2}-\frac {3 x^5}{5}+\frac {x^6}{6}+4 (1+x)^2 \log ^7(25)+\frac {1}{2} (1+x)^2 \log ^8(25)+\log (25) \int \left (8-16 x+16 x^3-8 x^4\right ) \, dx+\log ^2(25) \int \left (28-32 x-24 x^2+32 x^3-4 x^4\right ) \, dx+\log ^3(25) \int \left (56-24 x-56 x^2+24 x^3\right ) \, dx+\log ^4(25) \int \left (70+10 x-54 x^2+6 x^3\right ) \, dx+\log ^5(25) \int \left (56+32 x-24 x^2\right ) \, dx+\log ^6(25) \int \left (28+24 x-4 x^2\right ) \, dx+\int \left (1-6 x+6 x^2+8 x^3-15 x^4+6 x^5+\left (8-32 x+64 x^3-40 x^4\right ) \log (25)+\left (28-64 x-72 x^2+128 x^3-20 x^4\right ) \log ^2(25)+\left (56-48 x-168 x^2+96 x^3\right ) \log ^3(25)+\left (70+20 x-162 x^2+24 x^3\right ) \log ^4(25)+\left (56+64 x-72 x^2\right ) \log ^5(25)+\left (28+48 x-12 x^2\right ) \log ^6(25)+(8+16 x) \log ^7(25)+(1+2 x) \log ^8(25)\right ) \log (x) \, dx\\ &=x-\frac {3 x^2}{2}+\frac {2 x^3}{3}+\frac {x^4}{2}-\frac {3 x^5}{5}+\frac {x^6}{6}+8 x \log (25)-8 x^2 \log (25)+4 x^4 \log (25)-\frac {8}{5} x^5 \log (25)+28 x \log ^2(25)-16 x^2 \log ^2(25)-8 x^3 \log ^2(25)+8 x^4 \log ^2(25)-\frac {4}{5} x^5 \log ^2(25)+56 x \log ^3(25)-12 x^2 \log ^3(25)-\frac {56}{3} x^3 \log ^3(25)+6 x^4 \log ^3(25)+70 x \log ^4(25)+5 x^2 \log ^4(25)-18 x^3 \log ^4(25)+\frac {3}{2} x^4 \log ^4(25)+56 x \log ^5(25)+16 x^2 \log ^5(25)-8 x^3 \log ^5(25)+28 x \log ^6(25)+12 x^2 \log ^6(25)-\frac {4}{3} x^3 \log ^6(25)+4 (1+x)^2 \log ^7(25)+\frac {1}{2} (1+x)^2 \log ^8(25)+\int \left (6 x^5 \log (x)+(1+\log (25))^8 \log (x)-6 x^2 (1+\log (25))^4 \left (-1+4 \log (25)+2 \log ^2(25)\right ) \log (x)+8 x^3 (1+\log (25))^2 \left (1+6 \log (25)+3 \log ^2(25)\right ) \log (x)-5 x^4 \left (3+8 \log (25)+4 \log ^2(25)\right ) \log (x)+2 x (1+\log (25))^6 \left (-3+\log ^2(25)+\log (625)\right ) \log (x)\right ) \, dx\\ &=x-\frac {3 x^2}{2}+\frac {2 x^3}{3}+\frac {x^4}{2}-\frac {3 x^5}{5}+\frac {x^6}{6}+8 x \log (25)-8 x^2 \log (25)+4 x^4 \log (25)-\frac {8}{5} x^5 \log (25)+28 x \log ^2(25)-16 x^2 \log ^2(25)-8 x^3 \log ^2(25)+8 x^4 \log ^2(25)-\frac {4}{5} x^5 \log ^2(25)+56 x \log ^3(25)-12 x^2 \log ^3(25)-\frac {56}{3} x^3 \log ^3(25)+6 x^4 \log ^3(25)+70 x \log ^4(25)+5 x^2 \log ^4(25)-18 x^3 \log ^4(25)+\frac {3}{2} x^4 \log ^4(25)+56 x \log ^5(25)+16 x^2 \log ^5(25)-8 x^3 \log ^5(25)+28 x \log ^6(25)+12 x^2 \log ^6(25)-\frac {4}{3} x^3 \log ^6(25)+4 (1+x)^2 \log ^7(25)+\frac {1}{2} (1+x)^2 \log ^8(25)+6 \int x^5 \log (x) \, dx+(1+\log (25))^8 \int \log (x) \, dx+\left (6 (1+\log (25))^4 \left (1-4 \log (25)-2 \log ^2(25)\right )\right ) \int x^2 \log (x) \, dx+\left (8 (1+\log (25))^2 \left (1+6 \log (25)+3 \log ^2(25)\right )\right ) \int x^3 \log (x) \, dx-\left (5 \left (3+8 \log (25)+4 \log ^2(25)\right )\right ) \int x^4 \log (x) \, dx-\left (2 (1+\log (25))^6 \left (3-\log ^2(25)-\log (625)\right )\right ) \int x \log (x) \, dx\\ &=x-\frac {3 x^2}{2}+\frac {2 x^3}{3}+\frac {x^4}{2}-\frac {3 x^5}{5}+8 x \log (25)-8 x^2 \log (25)+4 x^4 \log (25)-\frac {8}{5} x^5 \log (25)+28 x \log ^2(25)-16 x^2 \log ^2(25)-8 x^3 \log ^2(25)+8 x^4 \log ^2(25)-\frac {4}{5} x^5 \log ^2(25)+56 x \log ^3(25)-12 x^2 \log ^3(25)-\frac {56}{3} x^3 \log ^3(25)+6 x^4 \log ^3(25)+70 x \log ^4(25)+5 x^2 \log ^4(25)-18 x^3 \log ^4(25)+\frac {3}{2} x^4 \log ^4(25)+56 x \log ^5(25)+16 x^2 \log ^5(25)-8 x^3 \log ^5(25)+28 x \log ^6(25)+12 x^2 \log ^6(25)-\frac {4}{3} x^3 \log ^6(25)+4 (1+x)^2 \log ^7(25)+\frac {1}{2} (1+x)^2 \log ^8(25)-x (1+\log (25))^8-\frac {2}{3} x^3 (1+\log (25))^4 \left (1-4 \log (25)-2 \log ^2(25)\right )-\frac {1}{2} x^4 (1+\log (25))^2 \left (1+6 \log (25)+3 \log ^2(25)\right )+\frac {1}{5} x^5 \left (3+8 \log (25)+4 \log ^2(25)\right )+\frac {1}{2} x^2 (1+\log (25))^6 \left (3-\log ^2(25)-\log (625)\right )+x^6 \log (x)+x (1+\log (25))^8 \log (x)+2 x^3 (1+\log (25))^4 \left (1-4 \log (25)-2 \log ^2(25)\right ) \log (x)+2 x^4 (1+\log (25))^2 \left (1+6 \log (25)+3 \log ^2(25)\right ) \log (x)-x^5 \left (3+8 \log (25)+4 \log ^2(25)\right ) \log (x)-x^2 (1+\log (25))^6 \left (3-\log ^2(25)-\log (625)\right ) \log (x)\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.23, size = 216, normalized size = 10.80 \begin {gather*} \frac {1}{30} x \left (-5 x^5+20 x^2 (1+\log (25))^4 \left (-1+4 \log (25)+2 \log ^2(25)\right )-15 x^3 (1+\log (25))^2 \left (1+6 \log (25)+3 \log ^2(25)\right )+6 x^4 \left (3+8 \log (25)+4 \log ^2(25)\right )-15 x (1+\log (25))^6 \left (-3+\log ^2(25)+\log (625)\right )+30 (1+\log (25))^8 \log (x)+15 x (1+\log (25))^6 \left (-3+\log ^2(25)+\log (625)\right ) (1+2 \log (x))-20 x^2 (1+\log (25))^4 \left (-1+4 \log (25)+2 \log ^2(25)\right ) (1+3 \log (x))+15 x^3 (1+\log (25))^2 \left (1+6 \log (25)+3 \log ^2(25)\right ) (1+4 \log (x))-6 x^4 \left (3+8 \log (25)+4 \log ^2(25)\right ) (1+5 \log (x))+5 x^5 (1+6 \log (x))\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1 - 3*x + 2*x^2 + 2*x^3 - 3*x^4 + x^5 + (8 - 16*x + 16*x^3 - 8*x^4)*Log[25] + (28 - 32*x - 24*x^2 +
32*x^3 - 4*x^4)*Log[25]^2 + (56 - 24*x - 56*x^2 + 24*x^3)*Log[25]^3 + (70 + 10*x - 54*x^2 + 6*x^3)*Log[25]^4 +
(56 + 32*x - 24*x^2)*Log[25]^5 + (28 + 24*x - 4*x^2)*Log[25]^6 + (8 + 8*x)*Log[25]^7 + (1 + x)*Log[25]^8 + (1
- 6*x + 6*x^2 + 8*x^3 - 15*x^4 + 6*x^5 + (8 - 32*x + 64*x^3 - 40*x^4)*Log[25] + (28 - 64*x - 72*x^2 + 128*x^3
- 20*x^4)*Log[25]^2 + (56 - 48*x - 168*x^2 + 96*x^3)*Log[25]^3 + (70 + 20*x - 162*x^2 + 24*x^3)*Log[25]^4 + (
56 + 64*x - 72*x^2)*Log[25]^5 + (28 + 48*x - 12*x^2)*Log[25]^6 + (8 + 16*x)*Log[25]^7 + (1 + 2*x)*Log[25]^8)*L
og[x],x]

[Out]

(x*(-5*x^5 + 20*x^2*(1 + Log[25])^4*(-1 + 4*Log[25] + 2*Log[25]^2) - 15*x^3*(1 + Log[25])^2*(1 + 6*Log[25] + 3
*Log[25]^2) + 6*x^4*(3 + 8*Log[25] + 4*Log[25]^2) - 15*x*(1 + Log[25])^6*(-3 + Log[25]^2 + Log[625]) + 30*(1 +
Log[25])^8*Log[x] + 15*x*(1 + Log[25])^6*(-3 + Log[25]^2 + Log[625])*(1 + 2*Log[x]) - 20*x^2*(1 + Log[25])^4*
(-1 + 4*Log[25] + 2*Log[25]^2)*(1 + 3*Log[x]) + 15*x^3*(1 + Log[25])^2*(1 + 6*Log[25] + 3*Log[25]^2)*(1 + 4*Lo
g[x]) - 6*x^4*(3 + 8*Log[25] + 4*Log[25]^2)*(1 + 5*Log[x]) + 5*x^5*(1 + 6*Log[x])))/30

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fricas [B]  time = 0.52, size = 187, normalized size = 9.35 \begin {gather*} {\left (256 \, {\left (x^{2} + x\right )} \log \relax (5)^{8} + 1024 \, {\left (x^{2} + x\right )} \log \relax (5)^{7} - 256 \, {\left (x^{3} - 6 \, x^{2} - 7 \, x\right )} \log \relax (5)^{6} + x^{6} - 256 \, {\left (3 \, x^{3} - 4 \, x^{2} - 7 \, x\right )} \log \relax (5)^{5} - 3 \, x^{5} + 32 \, {\left (3 \, x^{4} - 27 \, x^{3} + 5 \, x^{2} + 35 \, x\right )} \log \relax (5)^{4} + 2 \, x^{4} + 64 \, {\left (3 \, x^{4} - 7 \, x^{3} - 3 \, x^{2} + 7 \, x\right )} \log \relax (5)^{3} + 2 \, x^{3} - 16 \, {\left (x^{5} - 8 \, x^{4} + 6 \, x^{3} + 8 \, x^{2} - 7 \, x\right )} \log \relax (5)^{2} - 3 \, x^{2} - 16 \, {\left (x^{5} - 2 \, x^{4} + 2 \, x^{2} - x\right )} \log \relax (5) + x\right )} \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((256*(2*x+1)*log(5)^8+128*(16*x+8)*log(5)^7+64*(-12*x^2+48*x+28)*log(5)^6+32*(-72*x^2+64*x+56)*log(5
)^5+16*(24*x^3-162*x^2+20*x+70)*log(5)^4+8*(96*x^3-168*x^2-48*x+56)*log(5)^3+4*(-20*x^4+128*x^3-72*x^2-64*x+28
)*log(5)^2+2*(-40*x^4+64*x^3-32*x+8)*log(5)+6*x^5-15*x^4+8*x^3+6*x^2-6*x+1)*log(x)+256*(x+1)*log(5)^8+128*(8*x
+8)*log(5)^7+64*(-4*x^2+24*x+28)*log(5)^6+32*(-24*x^2+32*x+56)*log(5)^5+16*(6*x^3-54*x^2+10*x+70)*log(5)^4+8*(
24*x^3-56*x^2-24*x+56)*log(5)^3+4*(-4*x^4+32*x^3-24*x^2-32*x+28)*log(5)^2+2*(-8*x^4+16*x^3-16*x+8)*log(5)+x^5-
3*x^4+2*x^3+2*x^2-3*x+1,x, algorithm="fricas")

[Out]

(256*(x^2 + x)*log(5)^8 + 1024*(x^2 + x)*log(5)^7 - 256*(x^3 - 6*x^2 - 7*x)*log(5)^6 + x^6 - 256*(3*x^3 - 4*x^
2 - 7*x)*log(5)^5 - 3*x^5 + 32*(3*x^4 - 27*x^3 + 5*x^2 + 35*x)*log(5)^4 + 2*x^4 + 64*(3*x^4 - 7*x^3 - 3*x^2 +
7*x)*log(5)^3 + 2*x^3 - 16*(x^5 - 8*x^4 + 6*x^3 + 8*x^2 - 7*x)*log(5)^2 - 3*x^2 - 16*(x^5 - 2*x^4 + 2*x^2 - x)
*log(5) + x)*log(x)

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giac [B]  time = 0.22, size = 694, normalized size = 34.70 result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((256*(2*x+1)*log(5)^8+128*(16*x+8)*log(5)^7+64*(-12*x^2+48*x+28)*log(5)^6+32*(-72*x^2+64*x+56)*log(5
)^5+16*(24*x^3-162*x^2+20*x+70)*log(5)^4+8*(96*x^3-168*x^2-48*x+56)*log(5)^3+4*(-20*x^4+128*x^3-72*x^2-64*x+28
)*log(5)^2+2*(-40*x^4+64*x^3-32*x+8)*log(5)+6*x^5-15*x^4+8*x^3+6*x^2-6*x+1)*log(x)+256*(x+1)*log(5)^8+128*(8*x
+8)*log(5)^7+64*(-4*x^2+24*x+28)*log(5)^6+32*(-24*x^2+32*x+56)*log(5)^5+16*(6*x^3-54*x^2+10*x+70)*log(5)^4+8*(
24*x^3-56*x^2-24*x+56)*log(5)^3+4*(-4*x^4+32*x^3-24*x^2-32*x+28)*log(5)^2+2*(-8*x^4+16*x^3-16*x+8)*log(5)+x^5-
3*x^4+2*x^3+2*x^2-3*x+1,x, algorithm="giac")

[Out]

256*x^2*log(5)^8*log(x) - 128*x^2*log(5)^8 - 256*x^3*log(5)^6*log(x) + 1024*x^2*log(5)^7*log(x) + 256*x*log(5)
^8*log(x) + 256/3*x^3*log(5)^6 - 512*x^2*log(5)^7 + 128*(x^2 + 2*x)*log(5)^8 - 256*x*log(5)^8 + 96*x^4*log(5)^
4*log(x) - 768*x^3*log(5)^5*log(x) + 1536*x^2*log(5)^6*log(x) + 1024*x*log(5)^7*log(x) - 24*x^4*log(5)^4 + 256
*x^3*log(5)^5 - 768*x^2*log(5)^6 + 512*(x^2 + 2*x)*log(5)^7 - 1024*x*log(5)^7 - 16*x^5*log(5)^2*log(x) + 192*x
^4*log(5)^3*log(x) - 864*x^3*log(5)^4*log(x) + 1024*x^2*log(5)^5*log(x) + 1792*x*log(5)^6*log(x) + 16/5*x^5*lo
g(5)^2 - 48*x^4*log(5)^3 + 288*x^3*log(5)^4 - 512*x^2*log(5)^5 - 256/3*(x^3 - 9*x^2 - 21*x)*log(5)^6 - 1792*x*
log(5)^6 + x^6*log(x) - 16*x^5*log(5)*log(x) + 128*x^4*log(5)^2*log(x) - 448*x^3*log(5)^3*log(x) + 160*x^2*log
(5)^4*log(x) + 1792*x*log(5)^5*log(x) + 16/5*x^5*log(5) - 32*x^4*log(5)^2 + 448/3*x^3*log(5)^3 - 80*x^2*log(5)
^4 - 256*(x^3 - 2*x^2 - 7*x)*log(5)^5 - 1792*x*log(5)^5 - 3*x^5*log(x) + 32*x^4*log(5)*log(x) - 96*x^3*log(5)^
2*log(x) - 192*x^2*log(5)^3*log(x) + 1120*x*log(5)^4*log(x) - 8*x^4*log(5) + 32*x^3*log(5)^2 + 96*x^2*log(5)^3
+ 8*(3*x^4 - 36*x^3 + 10*x^2 + 140*x)*log(5)^4 - 1120*x*log(5)^4 + 2*x^4*log(x) - 128*x^2*log(5)^2*log(x) + 4
48*x*log(5)^3*log(x) + 64*x^2*log(5)^2 + 16/3*(9*x^4 - 28*x^3 - 18*x^2 + 84*x)*log(5)^3 - 448*x*log(5)^3 + 2*x
^3*log(x) - 32*x^2*log(5)*log(x) + 112*x*log(5)^2*log(x) + 16*x^2*log(5) - 16/5*(x^5 - 10*x^4 + 10*x^3 + 20*x^
2 - 35*x)*log(5)^2 - 112*x*log(5)^2 - 3*x^2*log(x) + 16*x*log(5)*log(x) - 8/5*(2*x^5 - 5*x^4 + 10*x^2 - 10*x)*
log(5) - 16*x*log(5) + x*log(x)

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maple [B]  time = 0.06, size = 200, normalized size = 10.00

method	result	size
norman	\(x^{6} \ln \relax (x )+\left (-16 \ln \relax (5)-16 \ln \relax (5)^{2}-3\right ) x^{5} \ln \relax (x )+\left (96 \ln \relax (5)^{4}+192 \ln \relax (5)^{3}+128 \ln \relax (5)^{2}+32 \ln \relax (5)+2\right ) x^{4} \ln \relax (x )+\left (-256 \ln \relax (5)^{6}-768 \ln \relax (5)^{5}-864 \ln \relax (5)^{4}-448 \ln \relax (5)^{3}-96 \ln \relax (5)^{2}+2\right ) x^{3} \ln \relax (x )+\left (256 \ln \relax (5)^{8}+1024 \ln \relax (5)^{7}+1792 \ln \relax (5)^{6}+1792 \ln \relax (5)^{5}+1120 \ln \relax (5)^{4}+448 \ln \relax (5)^{3}+112 \ln \relax (5)^{2}+16 \ln \relax (5)+1\right ) x \ln \relax (x )+\left (-32 \ln \relax (5)+256 \ln \relax (5)^{8}+1024 \ln \relax (5)^{7}+160 \ln \relax (5)^{4}-192 \ln \relax (5)^{3}-128 \ln \relax (5)^{2}+1536 \ln \relax (5)^{6}+1024 \ln \relax (5)^{5}-3\right ) x^{2} \ln \relax (x )\)	\(200\)
risch	\(x \left (1-3 x +128 x^{3} \ln \relax (5)^{2}+1024 \ln \relax (5)^{7}+1792 \ln \relax (5)^{6}+1792 \ln \relax (5)^{5}+256 \ln \relax (5)^{8}-768 x^{2} \ln \relax (5)^{5}+1120 \ln \relax (5)^{4}+448 \ln \relax (5)^{3}+16 \ln \relax (5)+112 \ln \relax (5)^{2}+x^{5}-3 x^{4}+2 x^{3}+2 x^{2}+1536 x \ln \relax (5)^{6}+192 x^{3} \ln \relax (5)^{3}+160 x \ln \relax (5)^{4}-32 x \ln \relax (5)-96 x^{2} \ln \relax (5)^{2}+32 x^{3} \ln \relax (5)+96 x^{3} \ln \relax (5)^{4}+256 \ln \relax (5)^{8} x +1024 \ln \relax (5)^{7} x -256 \ln \relax (5)^{6} x^{2}+1024 \ln \relax (5)^{5} x -864 \ln \relax (5)^{4} x^{2}-192 \ln \relax (5)^{3} x -16 x^{4} \ln \relax (5)^{2}-128 x \ln \relax (5)^{2}-16 x^{4} \ln \relax (5)-448 x^{2} \ln \relax (5)^{3}\right ) \ln \relax (x )\)	\(223\)
default	\(-96 x^{3} \ln \relax (5)^{2} \ln \relax (x )-3 x^{2} \ln \relax (x )+2 x^{3} \ln \relax (x )+x^{6} \ln \relax (x )+112 \ln \relax (x ) \ln \relax (5)^{2} x +448 \ln \relax (x ) \ln \relax (5)^{3} x -128 \ln \relax (x ) \ln \relax (5)^{2} x^{2}+1120 \ln \relax (x ) \ln \relax (5)^{4} x -3 x^{5} \ln \relax (x )+1024 \ln \relax (x ) \ln \relax (5)^{7} x +1536 \ln \relax (x ) \ln \relax (5)^{6} x^{2}+1792 \ln \relax (x ) \ln \relax (5)^{6} x +1024 \ln \relax (x ) \ln \relax (5)^{5} x^{2}-864 \ln \relax (x ) \ln \relax (5)^{4} x^{3}+1792 \ln \relax (x ) \ln \relax (5)^{5} x +160 \ln \relax (x ) \ln \relax (5)^{4} x^{2}-448 \ln \relax (x ) \ln \relax (5)^{3} x^{3}+128 \ln \relax (x ) \ln \relax (5)^{2} x^{4}-192 \ln \relax (x ) \ln \relax (5)^{3} x^{2}+32 \ln \relax (x ) \ln \relax (5) x^{4}-16 \ln \relax (5) \ln \relax (x ) x^{5}+192 \ln \relax (5)^{3} x^{4} \ln \relax (x )-16 \ln \relax (5)^{2} \ln \relax (x ) x^{5}-768 \ln \relax (5)^{5} \ln \relax (x ) x^{3}+96 \ln \relax (5)^{4} x^{4} \ln \relax (x )+256 \ln \relax (5)^{8} \ln \relax (x ) x^{2}+1024 \ln \relax (5)^{7} \ln \relax (x ) x^{2}-256 \ln \relax (5)^{6} \ln \relax (x ) x^{3}+2 x^{4} \ln \relax (x )+x \ln \relax (x )+256 x \ln \relax (5)^{8} \ln \relax (x )+16 x \ln \relax (5) \ln \relax (x )-32 x^{2} \ln \relax (5) \ln \relax (x )\)	\(313\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((256*(2*x+1)*ln(5)^8+128*(16*x+8)*ln(5)^7+64*(-12*x^2+48*x+28)*ln(5)^6+32*(-72*x^2+64*x+56)*ln(5)^5+16*(24
*x^3-162*x^2+20*x+70)*ln(5)^4+8*(96*x^3-168*x^2-48*x+56)*ln(5)^3+4*(-20*x^4+128*x^3-72*x^2-64*x+28)*ln(5)^2+2*
(-40*x^4+64*x^3-32*x+8)*ln(5)+6*x^5-15*x^4+8*x^3+6*x^2-6*x+1)*ln(x)+256*(x+1)*ln(5)^8+128*(8*x+8)*ln(5)^7+64*(
-4*x^2+24*x+28)*ln(5)^6+32*(-24*x^2+32*x+56)*ln(5)^5+16*(6*x^3-54*x^2+10*x+70)*ln(5)^4+8*(24*x^3-56*x^2-24*x+5
6)*ln(5)^3+4*(-4*x^4+32*x^3-24*x^2-32*x+28)*ln(5)^2+2*(-8*x^4+16*x^3-16*x+8)*ln(5)+x^5-3*x^4+2*x^3+2*x^2-3*x+1
,x,method=_RETURNVERBOSE)

[Out]

x^6*ln(x)+(-16*ln(5)-16*ln(5)^2-3)*x^5*ln(x)+(96*ln(5)^4+192*ln(5)^3+128*ln(5)^2+32*ln(5)+2)*x^4*ln(x)+(-256*l
n(5)^6-768*ln(5)^5-864*ln(5)^4-448*ln(5)^3-96*ln(5)^2+2)*x^3*ln(x)+(256*ln(5)^8+1024*ln(5)^7+1792*ln(5)^6+1792
*ln(5)^5+1120*ln(5)^4+448*ln(5)^3+112*ln(5)^2+16*ln(5)+1)*x*ln(x)+(-32*ln(5)+256*ln(5)^8+1024*ln(5)^7+160*ln(5
)^4-192*ln(5)^3-128*ln(5)^2+1536*ln(5)^6+1024*ln(5)^5-3)*x^2*ln(x)

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maxima [B]  time = 0.45, size = 559, normalized size = 27.95 \begin {gather*} 128 \, {\left (x^{2} + 2 \, x\right )} \log \relax (5)^{8} + 512 \, {\left (x^{2} + 2 \, x\right )} \log \relax (5)^{7} - \frac {256}{3} \, {\left (x^{3} - 9 \, x^{2} - 21 \, x\right )} \log \relax (5)^{6} + \frac {1}{5} \, {\left (16 \, \log \relax (5)^{2} + 16 \, \log \relax (5) + 3\right )} x^{5} - 256 \, {\left (x^{3} - 2 \, x^{2} - 7 \, x\right )} \log \relax (5)^{5} - \frac {1}{2} \, {\left (48 \, \log \relax (5)^{4} + 96 \, \log \relax (5)^{3} + 64 \, \log \relax (5)^{2} + 16 \, \log \relax (5) + 1\right )} x^{4} - \frac {3}{5} \, x^{5} + 8 \, {\left (3 \, x^{4} - 36 \, x^{3} + 10 \, x^{2} + 140 \, x\right )} \log \relax (5)^{4} + \frac {2}{3} \, {\left (128 \, \log \relax (5)^{6} + 384 \, \log \relax (5)^{5} + 432 \, \log \relax (5)^{4} + 224 \, \log \relax (5)^{3} + 48 \, \log \relax (5)^{2} - 1\right )} x^{3} + \frac {1}{2} \, x^{4} + \frac {16}{3} \, {\left (9 \, x^{4} - 28 \, x^{3} - 18 \, x^{2} + 84 \, x\right )} \log \relax (5)^{3} - \frac {1}{2} \, {\left (256 \, \log \relax (5)^{8} + 1024 \, \log \relax (5)^{7} + 1536 \, \log \relax (5)^{6} + 1024 \, \log \relax (5)^{5} + 160 \, \log \relax (5)^{4} - 192 \, \log \relax (5)^{3} - 128 \, \log \relax (5)^{2} - 32 \, \log \relax (5) - 3\right )} x^{2} + \frac {2}{3} \, x^{3} - \frac {16}{5} \, {\left (x^{5} - 10 \, x^{4} + 10 \, x^{3} + 20 \, x^{2} - 35 \, x\right )} \log \relax (5)^{2} - {\left (256 \, \log \relax (5)^{8} + 1024 \, \log \relax (5)^{7} + 1792 \, \log \relax (5)^{6} + 1792 \, \log \relax (5)^{5} + 1120 \, \log \relax (5)^{4} + 448 \, \log \relax (5)^{3} + 112 \, \log \relax (5)^{2} + 16 \, \log \relax (5) + 1\right )} x - \frac {3}{2} \, x^{2} - \frac {8}{5} \, {\left (2 \, x^{5} - 5 \, x^{4} + 10 \, x^{2} - 10 \, x\right )} \log \relax (5) + {\left (256 \, {\left (x^{2} + x\right )} \log \relax (5)^{8} + 1024 \, {\left (x^{2} + x\right )} \log \relax (5)^{7} - 256 \, {\left (x^{3} - 6 \, x^{2} - 7 \, x\right )} \log \relax (5)^{6} + x^{6} - 256 \, {\left (3 \, x^{3} - 4 \, x^{2} - 7 \, x\right )} \log \relax (5)^{5} - 3 \, x^{5} + 32 \, {\left (3 \, x^{4} - 27 \, x^{3} + 5 \, x^{2} + 35 \, x\right )} \log \relax (5)^{4} + 2 \, x^{4} + 64 \, {\left (3 \, x^{4} - 7 \, x^{3} - 3 \, x^{2} + 7 \, x\right )} \log \relax (5)^{3} + 2 \, x^{3} - 16 \, {\left (x^{5} - 8 \, x^{4} + 6 \, x^{3} + 8 \, x^{2} - 7 \, x\right )} \log \relax (5)^{2} - 3 \, x^{2} - 16 \, {\left (x^{5} - 2 \, x^{4} + 2 \, x^{2} - x\right )} \log \relax (5) + x\right )} \log \relax (x) + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((256*(2*x+1)*log(5)^8+128*(16*x+8)*log(5)^7+64*(-12*x^2+48*x+28)*log(5)^6+32*(-72*x^2+64*x+56)*log(5
)^5+16*(24*x^3-162*x^2+20*x+70)*log(5)^4+8*(96*x^3-168*x^2-48*x+56)*log(5)^3+4*(-20*x^4+128*x^3-72*x^2-64*x+28
)*log(5)^2+2*(-40*x^4+64*x^3-32*x+8)*log(5)+6*x^5-15*x^4+8*x^3+6*x^2-6*x+1)*log(x)+256*(x+1)*log(5)^8+128*(8*x
+8)*log(5)^7+64*(-4*x^2+24*x+28)*log(5)^6+32*(-24*x^2+32*x+56)*log(5)^5+16*(6*x^3-54*x^2+10*x+70)*log(5)^4+8*(
24*x^3-56*x^2-24*x+56)*log(5)^3+4*(-4*x^4+32*x^3-24*x^2-32*x+28)*log(5)^2+2*(-8*x^4+16*x^3-16*x+8)*log(5)+x^5-
3*x^4+2*x^3+2*x^2-3*x+1,x, algorithm="maxima")

[Out]

128*(x^2 + 2*x)*log(5)^8 + 512*(x^2 + 2*x)*log(5)^7 - 256/3*(x^3 - 9*x^2 - 21*x)*log(5)^6 + 1/5*(16*log(5)^2 +
16*log(5) + 3)*x^5 - 256*(x^3 - 2*x^2 - 7*x)*log(5)^5 - 1/2*(48*log(5)^4 + 96*log(5)^3 + 64*log(5)^2 + 16*log
(5) + 1)*x^4 - 3/5*x^5 + 8*(3*x^4 - 36*x^3 + 10*x^2 + 140*x)*log(5)^4 + 2/3*(128*log(5)^6 + 384*log(5)^5 + 432
*log(5)^4 + 224*log(5)^3 + 48*log(5)^2 - 1)*x^3 + 1/2*x^4 + 16/3*(9*x^4 - 28*x^3 - 18*x^2 + 84*x)*log(5)^3 - 1
/2*(256*log(5)^8 + 1024*log(5)^7 + 1536*log(5)^6 + 1024*log(5)^5 + 160*log(5)^4 - 192*log(5)^3 - 128*log(5)^2
- 32*log(5) - 3)*x^2 + 2/3*x^3 - 16/5*(x^5 - 10*x^4 + 10*x^3 + 20*x^2 - 35*x)*log(5)^2 - (256*log(5)^8 + 1024*
log(5)^7 + 1792*log(5)^6 + 1792*log(5)^5 + 1120*log(5)^4 + 448*log(5)^3 + 112*log(5)^2 + 16*log(5) + 1)*x - 3/
2*x^2 - 8/5*(2*x^5 - 5*x^4 + 10*x^2 - 10*x)*log(5) + (256*(x^2 + x)*log(5)^8 + 1024*(x^2 + x)*log(5)^7 - 256*(
x^3 - 6*x^2 - 7*x)*log(5)^6 + x^6 - 256*(3*x^3 - 4*x^2 - 7*x)*log(5)^5 - 3*x^5 + 32*(3*x^4 - 27*x^3 + 5*x^2 +
35*x)*log(5)^4 + 2*x^4 + 64*(3*x^4 - 7*x^3 - 3*x^2 + 7*x)*log(5)^3 + 2*x^3 - 16*(x^5 - 8*x^4 + 6*x^3 + 8*x^2 -
7*x)*log(5)^2 - 3*x^2 - 16*(x^5 - 2*x^4 + 2*x^2 - x)*log(5) + x)*log(x) + x

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mupad [B]  time = 2.69, size = 22, normalized size = 1.10 \begin {gather*} x\,\ln \relax (x)\,\left (x+1\right )\,{\left (\ln \left (625\right )-x+4\,{\ln \relax (5)}^2+1\right )}^4 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(256*log(5)^8*(x + 1) - 3*x - 4*log(5)^2*(32*x + 24*x^2 - 32*x^3 + 4*x^4 - 28) + 128*log(5)^7*(8*x + 8) + l
og(x)*(256*log(5)^8*(2*x + 1) - 4*log(5)^2*(64*x + 72*x^2 - 128*x^3 + 20*x^4 - 28) - 6*x + 128*log(5)^7*(16*x
+ 8) - 2*log(5)*(32*x - 64*x^3 + 40*x^4 - 8) + 64*log(5)^6*(48*x - 12*x^2 + 28) + 32*log(5)^5*(64*x - 72*x^2 +
56) + 16*log(5)^4*(20*x - 162*x^2 + 24*x^3 + 70) - 8*log(5)^3*(48*x + 168*x^2 - 96*x^3 - 56) + 6*x^2 + 8*x^3
- 15*x^4 + 6*x^5 + 1) - 2*log(5)*(16*x - 16*x^3 + 8*x^4 - 8) + 64*log(5)^6*(24*x - 4*x^2 + 28) + 32*log(5)^5*(
32*x - 24*x^2 + 56) + 16*log(5)^4*(10*x - 54*x^2 + 6*x^3 + 70) - 8*log(5)^3*(24*x + 56*x^2 - 24*x^3 - 56) + 2*
x^2 + 2*x^3 - 3*x^4 + x^5 + 1,x)

[Out]

x*log(x)*(x + 1)*(log(625) - x + 4*log(5)^2 + 1)^4

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sympy [B]  time = 0.44, size = 282, normalized size = 14.10 \begin {gather*} \left (x^{6} - 16 x^{5} \log {\relax (5 )}^{2} - 16 x^{5} \log {\relax (5 )} - 3 x^{5} + 2 x^{4} + 32 x^{4} \log {\relax (5 )} + 128 x^{4} \log {\relax (5 )}^{2} + 96 x^{4} \log {\relax (5 )}^{4} + 192 x^{4} \log {\relax (5 )}^{3} - 768 x^{3} \log {\relax (5 )}^{5} - 864 x^{3} \log {\relax (5 )}^{4} - 256 x^{3} \log {\relax (5 )}^{6} - 448 x^{3} \log {\relax (5 )}^{3} - 96 x^{3} \log {\relax (5 )}^{2} + 2 x^{3} - 192 x^{2} \log {\relax (5 )}^{3} - 128 x^{2} \log {\relax (5 )}^{2} - 32 x^{2} \log {\relax (5 )} - 3 x^{2} + 160 x^{2} \log {\relax (5 )}^{4} + 1024 x^{2} \log {\relax (5 )}^{5} + 256 x^{2} \log {\relax (5 )}^{8} + 1536 x^{2} \log {\relax (5 )}^{6} + 1024 x^{2} \log {\relax (5 )}^{7} + x + 16 x \log {\relax (5 )} + 112 x \log {\relax (5 )}^{2} + 448 x \log {\relax (5 )}^{3} + 1120 x \log {\relax (5 )}^{4} + 256 x \log {\relax (5 )}^{8} + 1792 x \log {\relax (5 )}^{5} + 1024 x \log {\relax (5 )}^{7} + 1792 x \log {\relax (5 )}^{6}\right ) \log {\relax (x )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((256*(2*x+1)*ln(5)**8+128*(16*x+8)*ln(5)**7+64*(-12*x**2+48*x+28)*ln(5)**6+32*(-72*x**2+64*x+56)*ln(
5)**5+16*(24*x**3-162*x**2+20*x+70)*ln(5)**4+8*(96*x**3-168*x**2-48*x+56)*ln(5)**3+4*(-20*x**4+128*x**3-72*x**
2-64*x+28)*ln(5)**2+2*(-40*x**4+64*x**3-32*x+8)*ln(5)+6*x**5-15*x**4+8*x**3+6*x**2-6*x+1)*ln(x)+256*(x+1)*ln(5
)**8+128*(8*x+8)*ln(5)**7+64*(-4*x**2+24*x+28)*ln(5)**6+32*(-24*x**2+32*x+56)*ln(5)**5+16*(6*x**3-54*x**2+10*x
+70)*ln(5)**4+8*(24*x**3-56*x**2-24*x+56)*ln(5)**3+4*(-4*x**4+32*x**3-24*x**2-32*x+28)*ln(5)**2+2*(-8*x**4+16*
x**3-16*x+8)*ln(5)+x**5-3*x**4+2*x**3+2*x**2-3*x+1,x)

[Out]

(x**6 - 16*x**5*log(5)**2 - 16*x**5*log(5) - 3*x**5 + 2*x**4 + 32*x**4*log(5) + 128*x**4*log(5)**2 + 96*x**4*l
og(5)**4 + 192*x**4*log(5)**3 - 768*x**3*log(5)**5 - 864*x**3*log(5)**4 - 256*x**3*log(5)**6 - 448*x**3*log(5)
**3 - 96*x**3*log(5)**2 + 2*x**3 - 192*x**2*log(5)**3 - 128*x**2*log(5)**2 - 32*x**2*log(5) - 3*x**2 + 160*x**
2*log(5)**4 + 1024*x**2*log(5)**5 + 256*x**2*log(5)**8 + 1536*x**2*log(5)**6 + 1024*x**2*log(5)**7 + x + 16*x*
log(5) + 112*x*log(5)**2 + 448*x*log(5)**3 + 1120*x*log(5)**4 + 256*x*log(5)**8 + 1792*x*log(5)**5 + 1024*x*lo
g(5)**7 + 1792*x*log(5)**6)*log(x)

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