3.5.4 \(\int \frac {1}{16} (1383120-1839264 x+458643 x^2+(790272-1052352 x+262752 x^2) \log (5)+(169344-225696 x+56400 x^2) \log ^2(5)+(16128-21504 x+5376 x^2) \log ^3(5)+(576-768 x+192 x^2) \log ^4(5)) \, dx\)

Optimal. Leaf size=27 \[ 16+x \left (9+\left (\frac {x}{4}+2 (3-x) (7+\log (5))^2\right )^2\right ) \]

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Rubi [B]  time = 0.05, antiderivative size = 110, normalized size of antiderivative = 4.07, number of steps used = 6, number of rules used = 1, integrand size = 72, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.014, Rules used = {12} \begin {gather*} \frac {152881 x^3}{16}+4 x^3 \log ^4(5)+112 x^3 \log ^3(5)+1175 x^3 \log ^2(5)+5474 x^3 \log (5)-57477 x^2-24 x^2 \log ^4(5)-672 x^2 \log ^3(5)-7053 x^2 \log ^2(5)-32886 x^2 \log (5)+86445 x+36 x \log ^4(5)+1008 x \log ^3(5)+10584 x \log ^2(5)+49392 x \log (5) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1383120 - 1839264*x + 458643*x^2 + (790272 - 1052352*x + 262752*x^2)*Log[5] + (169344 - 225696*x + 56400*
x^2)*Log[5]^2 + (16128 - 21504*x + 5376*x^2)*Log[5]^3 + (576 - 768*x + 192*x^2)*Log[5]^4)/16,x]

[Out]

86445*x - 57477*x^2 + (152881*x^3)/16 + 49392*x*Log[5] - 32886*x^2*Log[5] + 5474*x^3*Log[5] + 10584*x*Log[5]^2
- 7053*x^2*Log[5]^2 + 1175*x^3*Log[5]^2 + 1008*x*Log[5]^3 - 672*x^2*Log[5]^3 + 112*x^3*Log[5]^3 + 36*x*Log[5]
^4 - 24*x^2*Log[5]^4 + 4*x^3*Log[5]^4

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} &=\frac {1}{16} \int \left (1383120-1839264 x+458643 x^2+\left (790272-1052352 x+262752 x^2\right ) \log (5)+\left (169344-225696 x+56400 x^2\right ) \log ^2(5)+\left (16128-21504 x+5376 x^2\right ) \log ^3(5)+\left (576-768 x+192 x^2\right ) \log ^4(5)\right ) \, dx\\ &=86445 x-57477 x^2+\frac {152881 x^3}{16}+\frac {1}{16} \log (5) \int \left (790272-1052352 x+262752 x^2\right ) \, dx+\frac {1}{16} \log ^2(5) \int \left (169344-225696 x+56400 x^2\right ) \, dx+\frac {1}{16} \log ^3(5) \int \left (16128-21504 x+5376 x^2\right ) \, dx+\frac {1}{16} \log ^4(5) \int \left (576-768 x+192 x^2\right ) \, dx\\ &=86445 x-57477 x^2+\frac {152881 x^3}{16}+49392 x \log (5)-32886 x^2 \log (5)+5474 x^3 \log (5)+10584 x \log ^2(5)-7053 x^2 \log ^2(5)+1175 x^3 \log ^2(5)+1008 x \log ^3(5)-672 x^2 \log ^3(5)+112 x^3 \log ^3(5)+36 x \log ^4(5)-24 x^2 \log ^4(5)+4 x^3 \log ^4(5)\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.03, size = 72, normalized size = 2.67 \begin {gather*} -3 x^2 (7+\log (5))^2 \left (391+112 \log (5)+8 \log ^2(5)\right )+\frac {1}{16} x^3 \left (391+112 \log (5)+8 \log ^2(5)\right )^2+9 x \left (9605+5488 \log (5)+1176 \log ^2(5)+112 \log ^3(5)+4 \log ^4(5)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1383120 - 1839264*x + 458643*x^2 + (790272 - 1052352*x + 262752*x^2)*Log[5] + (169344 - 225696*x +
56400*x^2)*Log[5]^2 + (16128 - 21504*x + 5376*x^2)*Log[5]^3 + (576 - 768*x + 192*x^2)*Log[5]^4)/16,x]

[Out]

-3*x^2*(7 + Log[5])^2*(391 + 112*Log[5] + 8*Log[5]^2) + (x^3*(391 + 112*Log[5] + 8*Log[5]^2)^2)/16 + 9*x*(9605
+ 5488*Log[5] + 1176*Log[5]^2 + 112*Log[5]^3 + 4*Log[5]^4)

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fricas [B]  time = 0.62, size = 87, normalized size = 3.22 \begin {gather*} 4 \, {\left (x^{3} - 6 \, x^{2} + 9 \, x\right )} \log \relax (5)^{4} + 112 \, {\left (x^{3} - 6 \, x^{2} + 9 \, x\right )} \log \relax (5)^{3} + \frac {152881}{16} \, x^{3} + {\left (1175 \, x^{3} - 7053 \, x^{2} + 10584 \, x\right )} \log \relax (5)^{2} - 57477 \, x^{2} + 14 \, {\left (391 \, x^{3} - 2349 \, x^{2} + 3528 \, x\right )} \log \relax (5) + 86445 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/16*(192*x^2-768*x+576)*log(5)^4+1/16*(5376*x^2-21504*x+16128)*log(5)^3+1/16*(56400*x^2-225696*x+16
9344)*log(5)^2+1/16*(262752*x^2-1052352*x+790272)*log(5)+458643/16*x^2-114954*x+86445,x, algorithm="fricas")

[Out]

4*(x^3 - 6*x^2 + 9*x)*log(5)^4 + 112*(x^3 - 6*x^2 + 9*x)*log(5)^3 + 152881/16*x^3 + (1175*x^3 - 7053*x^2 + 105
84*x)*log(5)^2 - 57477*x^2 + 14*(391*x^3 - 2349*x^2 + 3528*x)*log(5) + 86445*x

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giac [B]  time = 0.28, size = 87, normalized size = 3.22 \begin {gather*} 4 \, {\left (x^{3} - 6 \, x^{2} + 9 \, x\right )} \log \relax (5)^{4} + 112 \, {\left (x^{3} - 6 \, x^{2} + 9 \, x\right )} \log \relax (5)^{3} + \frac {152881}{16} \, x^{3} + {\left (1175 \, x^{3} - 7053 \, x^{2} + 10584 \, x\right )} \log \relax (5)^{2} - 57477 \, x^{2} + 14 \, {\left (391 \, x^{3} - 2349 \, x^{2} + 3528 \, x\right )} \log \relax (5) + 86445 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/16*(192*x^2-768*x+576)*log(5)^4+1/16*(5376*x^2-21504*x+16128)*log(5)^3+1/16*(56400*x^2-225696*x+16
9344)*log(5)^2+1/16*(262752*x^2-1052352*x+790272)*log(5)+458643/16*x^2-114954*x+86445,x, algorithm="giac")

[Out]

4*(x^3 - 6*x^2 + 9*x)*log(5)^4 + 112*(x^3 - 6*x^2 + 9*x)*log(5)^3 + 152881/16*x^3 + (1175*x^3 - 7053*x^2 + 105
84*x)*log(5)^2 - 57477*x^2 + 14*(391*x^3 - 2349*x^2 + 3528*x)*log(5) + 86445*x

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maple [B]  time = 0.03, size = 84, normalized size = 3.11

method	result	size
norman	\(\left (-24 \ln \relax (5)^{4}-672 \ln \relax (5)^{3}-7053 \ln \relax (5)^{2}-32886 \ln \relax (5)-57477\right ) x^{2}+\left (4 \ln \relax (5)^{4}+112 \ln \relax (5)^{3}+1175 \ln \relax (5)^{2}+5474 \ln \relax (5)+\frac {152881}{16}\right ) x^{3}+\left (36 \ln \relax (5)^{4}+1008 \ln \relax (5)^{3}+10584 \ln \relax (5)^{2}+49392 \ln \relax (5)+86445\right ) x\)	\(84\)
default	\(\frac {\ln \relax (5)^{4} \left (64 x^{3}-384 x^{2}+576 x \right )}{16}+\frac {\ln \relax (5)^{3} \left (1792 x^{3}-10752 x^{2}+16128 x \right )}{16}+\frac {\ln \relax (5)^{2} \left (18800 x^{3}-112848 x^{2}+169344 x \right )}{16}+\frac {\ln \relax (5) \left (87584 x^{3}-526176 x^{2}+790272 x \right )}{16}+\frac {152881 x^{3}}{16}-57477 x^{2}+86445 x\)	\(93\)
gosper	\(\frac {x \left (64 \ln \relax (5)^{4} x^{2}-384 x \ln \relax (5)^{4}+1792 x^{2} \ln \relax (5)^{3}+576 \ln \relax (5)^{4}-10752 \ln \relax (5)^{3} x +18800 x^{2} \ln \relax (5)^{2}+16128 \ln \relax (5)^{3}-112848 x \ln \relax (5)^{2}+87584 x^{2} \ln \relax (5)+169344 \ln \relax (5)^{2}-526176 x \ln \relax (5)+152881 x^{2}+790272 \ln \relax (5)-919632 x +1383120\right )}{16}\)	\(96\)
risch	\(4 x^{3} \ln \relax (5)^{4}-24 \ln \relax (5)^{4} x^{2}+36 x \ln \relax (5)^{4}+112 x^{3} \ln \relax (5)^{3}-672 x^{2} \ln \relax (5)^{3}+1008 \ln \relax (5)^{3} x +1175 x^{3} \ln \relax (5)^{2}-7053 x^{2} \ln \relax (5)^{2}+10584 x \ln \relax (5)^{2}+5474 x^{3} \ln \relax (5)-32886 x^{2} \ln \relax (5)+49392 x \ln \relax (5)+\frac {152881 x^{3}}{16}-57477 x^{2}+86445 x\)	\(109\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/16*(192*x^2-768*x+576)*ln(5)^4+1/16*(5376*x^2-21504*x+16128)*ln(5)^3+1/16*(56400*x^2-225696*x+169344)*ln
(5)^2+1/16*(262752*x^2-1052352*x+790272)*ln(5)+458643/16*x^2-114954*x+86445,x,method=_RETURNVERBOSE)

[Out]

(-24*ln(5)^4-672*ln(5)^3-7053*ln(5)^2-32886*ln(5)-57477)*x^2+(4*ln(5)^4+112*ln(5)^3+1175*ln(5)^2+5474*ln(5)+15
2881/16)*x^3+(36*ln(5)^4+1008*ln(5)^3+10584*ln(5)^2+49392*ln(5)+86445)*x

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maxima [B]  time = 0.36, size = 87, normalized size = 3.22 \begin {gather*} 4 \, {\left (x^{3} - 6 \, x^{2} + 9 \, x\right )} \log \relax (5)^{4} + 112 \, {\left (x^{3} - 6 \, x^{2} + 9 \, x\right )} \log \relax (5)^{3} + \frac {152881}{16} \, x^{3} + {\left (1175 \, x^{3} - 7053 \, x^{2} + 10584 \, x\right )} \log \relax (5)^{2} - 57477 \, x^{2} + 14 \, {\left (391 \, x^{3} - 2349 \, x^{2} + 3528 \, x\right )} \log \relax (5) + 86445 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/16*(192*x^2-768*x+576)*log(5)^4+1/16*(5376*x^2-21504*x+16128)*log(5)^3+1/16*(56400*x^2-225696*x+16
9344)*log(5)^2+1/16*(262752*x^2-1052352*x+790272)*log(5)+458643/16*x^2-114954*x+86445,x, algorithm="maxima")

[Out]

4*(x^3 - 6*x^2 + 9*x)*log(5)^4 + 112*(x^3 - 6*x^2 + 9*x)*log(5)^3 + 152881/16*x^3 + (1175*x^3 - 7053*x^2 + 105
84*x)*log(5)^2 - 57477*x^2 + 14*(391*x^3 - 2349*x^2 + 3528*x)*log(5) + 86445*x

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mupad [B]  time = 0.45, size = 84, normalized size = 3.11 \begin {gather*} \left (5474\,\ln \relax (5)+1175\,{\ln \relax (5)}^2+112\,{\ln \relax (5)}^3+4\,{\ln \relax (5)}^4+\frac {152881}{16}\right )\,x^3+\left (-32886\,\ln \relax (5)-7053\,{\ln \relax (5)}^2-672\,{\ln \relax (5)}^3-24\,{\ln \relax (5)}^4-57477\right )\,x^2+\left (49392\,\ln \relax (5)+10584\,{\ln \relax (5)}^2+1008\,{\ln \relax (5)}^3+36\,{\ln \relax (5)}^4+86445\right )\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(5)*(262752*x^2 - 1052352*x + 790272))/16 - 114954*x + (log(5)^4*(192*x^2 - 768*x + 576))/16 + (log(5)
^3*(5376*x^2 - 21504*x + 16128))/16 + (log(5)^2*(56400*x^2 - 225696*x + 169344))/16 + (458643*x^2)/16 + 86445,
x)

[Out]

x*(49392*log(5) + 10584*log(5)^2 + 1008*log(5)^3 + 36*log(5)^4 + 86445) - x^2*(32886*log(5) + 7053*log(5)^2 +
672*log(5)^3 + 24*log(5)^4 + 57477) + x^3*(5474*log(5) + 1175*log(5)^2 + 112*log(5)^3 + 4*log(5)^4 + 152881/16
)

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sympy [B]  time = 0.09, size = 92, normalized size = 3.41 \begin {gather*} x^{3} \left (4 \log {\relax (5 )}^{4} + 112 \log {\relax (5 )}^{3} + 1175 \log {\relax (5 )}^{2} + 5474 \log {\relax (5 )} + \frac {152881}{16}\right ) + x^{2} \left (-57477 - 32886 \log {\relax (5 )} - 7053 \log {\relax (5 )}^{2} - 672 \log {\relax (5 )}^{3} - 24 \log {\relax (5 )}^{4}\right ) + x \left (36 \log {\relax (5 )}^{4} + 1008 \log {\relax (5 )}^{3} + 10584 \log {\relax (5 )}^{2} + 49392 \log {\relax (5 )} + 86445\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/16*(192*x**2-768*x+576)*ln(5)**4+1/16*(5376*x**2-21504*x+16128)*ln(5)**3+1/16*(56400*x**2-225696*x
+169344)*ln(5)**2+1/16*(262752*x**2-1052352*x+790272)*ln(5)+458643/16*x**2-114954*x+86445,x)

[Out]

x**3*(4*log(5)**4 + 112*log(5)**3 + 1175*log(5)**2 + 5474*log(5) + 152881/16) + x**2*(-57477 - 32886*log(5) -
7053*log(5)**2 - 672*log(5)**3 - 24*log(5)**4) + x*(36*log(5)**4 + 1008*log(5)**3 + 10584*log(5)**2 + 49392*lo
g(5) + 86445)

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