3.96.38 \(\int \frac {953694 x+1262196 x^2+694960 x^3+196000 x^4+28000 x^5+1600 x^6+(-33614 x-48020 x^2-27440 x^3-7840 x^4-1120 x^5-64 x^6) \log (x)+(-33614 x-48020 x^2-27440 x^3-7840 x^4-1120 x^5-64 x^6) \log ^2(x)}{16807+24010 x+13720 x^2+3920 x^3+560 x^4+32 x^5} \, dx\)

Optimal. Leaf size=28 \[ e^4+x^2 \left (\left (5+\frac {16}{(7+2 x)^2}\right )^2-\log ^2(x)\right ) \]

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Rubi [B]  time = 0.56, antiderivative size = 67, normalized size of antiderivative = 2.39, number of steps used = 18, number of rules used = 7, integrand size = 123, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {6688, 12, 6742, 43, 37, 2304, 2305} \begin {gather*} \frac {24820 x^4}{(2 x+7)^4}+25 x^2-x^2 \log ^2(x)+\frac {42875}{2 x+7}-\frac {908087}{2 (2 x+7)^2}+\frac {2127419}{(2 x+7)^3}-\frac {14885661}{4 (2 x+7)^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(953694*x + 1262196*x^2 + 694960*x^3 + 196000*x^4 + 28000*x^5 + 1600*x^6 + (-33614*x - 48020*x^2 - 27440*x
^3 - 7840*x^4 - 1120*x^5 - 64*x^6)*Log[x] + (-33614*x - 48020*x^2 - 27440*x^3 - 7840*x^4 - 1120*x^5 - 64*x^6)*
Log[x]^2)/(16807 + 24010*x + 13720*x^2 + 3920*x^3 + 560*x^4 + 32*x^5),x]

[Out]

25*x^2 - 14885661/(4*(7 + 2*x)^4) + (24820*x^4)/(7 + 2*x)^4 + 2127419/(7 + 2*x)^3 - 908087/(2*(7 + 2*x)^2) + 4
2875/(7 + 2*x) - x^2*Log[x]^2

Rule 12

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

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

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 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 x \left (476847+631098 x+347480 x^2+98000 x^3+14000 x^4+800 x^5-(7+2 x)^5 \log (x)-(7+2 x)^5 \log ^2(x)\right )}{(7+2 x)^5} \, dx\\ &=2 \int \frac {x \left (476847+631098 x+347480 x^2+98000 x^3+14000 x^4+800 x^5-(7+2 x)^5 \log (x)-(7+2 x)^5 \log ^2(x)\right )}{(7+2 x)^5} \, dx\\ &=2 \int \left (\frac {476847 x}{(7+2 x)^5}+\frac {631098 x^2}{(7+2 x)^5}+\frac {347480 x^3}{(7+2 x)^5}+\frac {98000 x^4}{(7+2 x)^5}+\frac {14000 x^5}{(7+2 x)^5}+\frac {800 x^6}{(7+2 x)^5}-x \log (x)-x \log ^2(x)\right ) \, dx\\ &=-(2 \int x \log (x) \, dx)-2 \int x \log ^2(x) \, dx+1600 \int \frac {x^6}{(7+2 x)^5} \, dx+28000 \int \frac {x^5}{(7+2 x)^5} \, dx+196000 \int \frac {x^4}{(7+2 x)^5} \, dx+694960 \int \frac {x^3}{(7+2 x)^5} \, dx+953694 \int \frac {x}{(7+2 x)^5} \, dx+1262196 \int \frac {x^2}{(7+2 x)^5} \, dx\\ &=\frac {x^2}{2}+\frac {24820 x^4}{(7+2 x)^4}-x^2 \log (x)-x^2 \log ^2(x)+2 \int x \log (x) \, dx+1600 \int \left (-\frac {35}{64}+\frac {x}{32}+\frac {117649}{64 (7+2 x)^5}-\frac {50421}{32 (7+2 x)^4}+\frac {36015}{64 (7+2 x)^3}-\frac {1715}{16 (7+2 x)^2}+\frac {735}{64 (7+2 x)}\right ) \, dx+28000 \int \left (\frac {1}{32}-\frac {16807}{32 (7+2 x)^5}+\frac {12005}{32 (7+2 x)^4}-\frac {1715}{16 (7+2 x)^3}+\frac {245}{16 (7+2 x)^2}-\frac {35}{32 (7+2 x)}\right ) \, dx+196000 \int \left (\frac {2401}{16 (7+2 x)^5}-\frac {343}{4 (7+2 x)^4}+\frac {147}{8 (7+2 x)^3}-\frac {7}{4 (7+2 x)^2}+\frac {1}{16 (7+2 x)}\right ) \, dx+953694 \int \left (-\frac {7}{2 (7+2 x)^5}+\frac {1}{2 (7+2 x)^4}\right ) \, dx+1262196 \int \left (\frac {49}{4 (7+2 x)^5}-\frac {7}{2 (7+2 x)^4}+\frac {1}{4 (7+2 x)^3}\right ) \, dx\\ &=25 x^2-\frac {14885661}{4 (7+2 x)^4}+\frac {24820 x^4}{(7+2 x)^4}+\frac {2127419}{(7+2 x)^3}-\frac {908087}{2 (7+2 x)^2}+\frac {42875}{7+2 x}-x^2 \log ^2(x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.09, size = 39, normalized size = 1.39 \begin {gather*} 25 x^2-\frac {8 \left (12005+13720 x+4868 x^2+560 x^3\right )}{(7+2 x)^4}-x^2 \log ^2(x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(953694*x + 1262196*x^2 + 694960*x^3 + 196000*x^4 + 28000*x^5 + 1600*x^6 + (-33614*x - 48020*x^2 - 2
7440*x^3 - 7840*x^4 - 1120*x^5 - 64*x^6)*Log[x] + (-33614*x - 48020*x^2 - 27440*x^3 - 7840*x^4 - 1120*x^5 - 64
*x^6)*Log[x]^2)/(16807 + 24010*x + 13720*x^2 + 3920*x^3 + 560*x^4 + 32*x^5),x]

[Out]

25*x^2 - (8*(12005 + 13720*x + 4868*x^2 + 560*x^3))/(7 + 2*x)^4 - x^2*Log[x]^2

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fricas [B]  time = 0.57, size = 85, normalized size = 3.04 \begin {gather*} \frac {400 \, x^{6} + 5600 \, x^{5} + 29400 \, x^{4} + 64120 \, x^{3} - {\left (16 \, x^{6} + 224 \, x^{5} + 1176 \, x^{4} + 2744 \, x^{3} + 2401 \, x^{2}\right )} \log \relax (x)^{2} + 21081 \, x^{2} - 109760 \, x - 96040}{16 \, x^{4} + 224 \, x^{3} + 1176 \, x^{2} + 2744 \, x + 2401} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-64*x^6-1120*x^5-7840*x^4-27440*x^3-48020*x^2-33614*x)*log(x)^2+(-64*x^6-1120*x^5-7840*x^4-27440*x
^3-48020*x^2-33614*x)*log(x)+1600*x^6+28000*x^5+196000*x^4+694960*x^3+1262196*x^2+953694*x)/(32*x^5+560*x^4+39
20*x^3+13720*x^2+24010*x+16807),x, algorithm="fricas")

[Out]

(400*x^6 + 5600*x^5 + 29400*x^4 + 64120*x^3 - (16*x^6 + 224*x^5 + 1176*x^4 + 2744*x^3 + 2401*x^2)*log(x)^2 + 2
1081*x^2 - 109760*x - 96040)/(16*x^4 + 224*x^3 + 1176*x^2 + 2744*x + 2401)

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giac [A]  time = 0.15, size = 54, normalized size = 1.93 \begin {gather*} -x^{2} \log \relax (x)^{2} + 25 \, x^{2} - \frac {8 \, {\left (560 \, x^{3} + 4868 \, x^{2} + 13720 \, x + 12005\right )}}{16 \, x^{4} + 224 \, x^{3} + 1176 \, x^{2} + 2744 \, x + 2401} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-64*x^6-1120*x^5-7840*x^4-27440*x^3-48020*x^2-33614*x)*log(x)^2+(-64*x^6-1120*x^5-7840*x^4-27440*x
^3-48020*x^2-33614*x)*log(x)+1600*x^6+28000*x^5+196000*x^4+694960*x^3+1262196*x^2+953694*x)/(32*x^5+560*x^4+39
20*x^3+13720*x^2+24010*x+16807),x, algorithm="giac")

[Out]

-x^2*log(x)^2 + 25*x^2 - 8*(560*x^3 + 4868*x^2 + 13720*x + 12005)/(16*x^4 + 224*x^3 + 1176*x^2 + 2744*x + 2401
)

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maple [A]  time = 0.06, size = 52, normalized size = 1.86

method	result	size
default	\(-x^{2} \ln \relax (x )^{2}+25 x^{2}-\frac {896}{\left (7+2 x \right )^{3}}-\frac {560}{7+2 x}+\frac {3136}{\left (7+2 x \right )^{4}}+\frac {2024}{\left (7+2 x \right )^{2}}\)	\(52\)
risch	\(-x^{2} \ln \relax (x )^{2}+\frac {400 x^{6}+5600 x^{5}+29400 x^{4}+64120 x^{3}+21081 x^{2}-109760 x -96040}{16 x^{4}+224 x^{3}+1176 x^{2}+2744 x +2401}\)	\(64\)
norman	\(\frac {-895230 x -315549 x^{2}+24820 x^{4}+5600 x^{5}+400 x^{6}-2401 x^{2} \ln \relax (x )^{2}-2744 x^{3} \ln \relax (x )^{2}-1176 x^{4} \ln \relax (x )^{2}-224 x^{5} \ln \relax (x )^{2}-16 x^{6} \ln \relax (x )^{2}-\frac {3133305}{4}}{\left (7+2 x \right )^{4}}\)	\(79\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-64*x^6-1120*x^5-7840*x^4-27440*x^3-48020*x^2-33614*x)*ln(x)^2+(-64*x^6-1120*x^5-7840*x^4-27440*x^3-4802
0*x^2-33614*x)*ln(x)+1600*x^6+28000*x^5+196000*x^4+694960*x^3+1262196*x^2+953694*x)/(32*x^5+560*x^4+3920*x^3+1
3720*x^2+24010*x+16807),x,method=_RETURNVERBOSE)

[Out]

-x^2*ln(x)^2+25*x^2-896/(7+2*x)^3-560/(7+2*x)+3136/(7+2*x)^4+2024/(7+2*x)^2

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maxima [B]  time = 0.42, size = 515, normalized size = 18.39 \begin {gather*} 25 \, x^{2} + \frac {1715 \, {\left (32 \, x^{3} + 168 \, x^{2} + 392 \, x + 343\right )} \log \relax (x)}{4 \, {\left (16 \, x^{4} + 224 \, x^{3} + 1176 \, x^{2} + 2744 \, x + 2401\right )}} + \frac {12005 \, {\left (24 \, x^{2} + 56 \, x + 49\right )} \log \relax (x)}{24 \, {\left (16 \, x^{4} + 224 \, x^{3} + 1176 \, x^{2} + 2744 \, x + 2401\right )}} + \frac {16807 \, {\left (8 \, x + 7\right )} \log \relax (x)}{24 \, {\left (16 \, x^{4} + 224 \, x^{3} + 1176 \, x^{2} + 2744 \, x + 2401\right )}} - \frac {76832 \, x^{3} + 8 \, {\left (16 \, x^{6} + 224 \, x^{5} + 1176 \, x^{4} + 2744 \, x^{3} + 2401 \, x^{2}\right )} \log \relax (x)^{2} + 701092 \, x^{2} - 392 \, {\left (24 \, x^{4} + 56 \, x^{3} + 49 \, x^{2}\right )} \log \relax (x) + 2151296 \, x + 2235331}{8 \, {\left (16 \, x^{4} + 224 \, x^{3} + 1176 \, x^{2} + 2744 \, x + 2401\right )}} - \frac {42875 \, {\left (960 \, x^{3} + 8400 \, x^{2} + 25480 \, x + 26411\right )}}{24 \, {\left (16 \, x^{4} + 224 \, x^{3} + 1176 \, x^{2} + 2744 \, x + 2401\right )}} + \frac {8575 \, {\left (640 \, x^{3} + 5880 \, x^{2} + 18424 \, x + 19551\right )}}{8 \, {\left (16 \, x^{4} + 224 \, x^{3} + 1176 \, x^{2} + 2744 \, x + 2401\right )}} + \frac {42875 \, {\left (384 \, x^{3} + 3024 \, x^{2} + 8624 \, x + 8575\right )}}{12 \, {\left (16 \, x^{4} + 224 \, x^{3} + 1176 \, x^{2} + 2744 \, x + 2401\right )}} - \frac {43435 \, {\left (32 \, x^{3} + 168 \, x^{2} + 392 \, x + 343\right )}}{4 \, {\left (16 \, x^{4} + 224 \, x^{3} + 1176 \, x^{2} + 2744 \, x + 2401\right )}} + \frac {1715 \, {\left (72 \, x^{2} + 378 \, x + 539\right )}}{24 \, {\left (8 \, x^{3} + 84 \, x^{2} + 294 \, x + 343\right )}} - \frac {105183 \, {\left (24 \, x^{2} + 56 \, x + 49\right )}}{8 \, {\left (16 \, x^{4} + 224 \, x^{3} + 1176 \, x^{2} + 2744 \, x + 2401\right )}} - \frac {343 \, {\left (8 \, x^{2} + 70 \, x + 49\right )}}{48 \, {\left (8 \, x^{3} + 84 \, x^{2} + 294 \, x + 343\right )}} - \frac {1715 \, {\left (8 \, x^{2} - 14 \, x - 49\right )}}{48 \, {\left (8 \, x^{3} + 84 \, x^{2} + 294 \, x + 343\right )}} - \frac {158949 \, {\left (8 \, x + 7\right )}}{8 \, {\left (16 \, x^{4} + 224 \, x^{3} + 1176 \, x^{2} + 2744 \, x + 2401\right )}} - \frac {147}{2} \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-64*x^6-1120*x^5-7840*x^4-27440*x^3-48020*x^2-33614*x)*log(x)^2+(-64*x^6-1120*x^5-7840*x^4-27440*x
^3-48020*x^2-33614*x)*log(x)+1600*x^6+28000*x^5+196000*x^4+694960*x^3+1262196*x^2+953694*x)/(32*x^5+560*x^4+39
20*x^3+13720*x^2+24010*x+16807),x, algorithm="maxima")

[Out]

25*x^2 + 1715/4*(32*x^3 + 168*x^2 + 392*x + 343)*log(x)/(16*x^4 + 224*x^3 + 1176*x^2 + 2744*x + 2401) + 12005/
24*(24*x^2 + 56*x + 49)*log(x)/(16*x^4 + 224*x^3 + 1176*x^2 + 2744*x + 2401) + 16807/24*(8*x + 7)*log(x)/(16*x
^4 + 224*x^3 + 1176*x^2 + 2744*x + 2401) - 1/8*(76832*x^3 + 8*(16*x^6 + 224*x^5 + 1176*x^4 + 2744*x^3 + 2401*x
^2)*log(x)^2 + 701092*x^2 - 392*(24*x^4 + 56*x^3 + 49*x^2)*log(x) + 2151296*x + 2235331)/(16*x^4 + 224*x^3 + 1
176*x^2 + 2744*x + 2401) - 42875/24*(960*x^3 + 8400*x^2 + 25480*x + 26411)/(16*x^4 + 224*x^3 + 1176*x^2 + 2744
*x + 2401) + 8575/8*(640*x^3 + 5880*x^2 + 18424*x + 19551)/(16*x^4 + 224*x^3 + 1176*x^2 + 2744*x + 2401) + 428
75/12*(384*x^3 + 3024*x^2 + 8624*x + 8575)/(16*x^4 + 224*x^3 + 1176*x^2 + 2744*x + 2401) - 43435/4*(32*x^3 + 1
68*x^2 + 392*x + 343)/(16*x^4 + 224*x^3 + 1176*x^2 + 2744*x + 2401) + 1715/24*(72*x^2 + 378*x + 539)/(8*x^3 +
84*x^2 + 294*x + 343) - 105183/8*(24*x^2 + 56*x + 49)/(16*x^4 + 224*x^3 + 1176*x^2 + 2744*x + 2401) - 343/48*(
8*x^2 + 70*x + 49)/(8*x^3 + 84*x^2 + 294*x + 343) - 1715/48*(8*x^2 - 14*x - 49)/(8*x^3 + 84*x^2 + 294*x + 343)
- 158949/8*(8*x + 7)/(16*x^4 + 224*x^3 + 1176*x^2 + 2744*x + 2401) - 147/2*log(x)

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mupad [B]  time = 7.04, size = 52, normalized size = 1.86 \begin {gather*} 25\,x^2-x^2\,{\ln \relax (x)}^2-\frac {280\,x^3+2434\,x^2+6860\,x+\frac {12005}{2}}{x^4+14\,x^3+\frac {147\,x^2}{2}+\frac {343\,x}{2}+\frac {2401}{16}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((953694*x - log(x)*(33614*x + 48020*x^2 + 27440*x^3 + 7840*x^4 + 1120*x^5 + 64*x^6) + 1262196*x^2 + 694960
*x^3 + 196000*x^4 + 28000*x^5 + 1600*x^6 - log(x)^2*(33614*x + 48020*x^2 + 27440*x^3 + 7840*x^4 + 1120*x^5 + 6
4*x^6))/(24010*x + 13720*x^2 + 3920*x^3 + 560*x^4 + 32*x^5 + 16807),x)

[Out]

25*x^2 - x^2*log(x)^2 - (6860*x + 2434*x^2 + 280*x^3 + 12005/2)/((343*x)/2 + (147*x^2)/2 + 14*x^3 + x^4 + 2401
/16)

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sympy [B]  time = 0.20, size = 49, normalized size = 1.75 \begin {gather*} - x^{2} \log {\relax (x )}^{2} + 25 x^{2} + \frac {- 4480 x^{3} - 38944 x^{2} - 109760 x - 96040}{16 x^{4} + 224 x^{3} + 1176 x^{2} + 2744 x + 2401} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-64*x**6-1120*x**5-7840*x**4-27440*x**3-48020*x**2-33614*x)*ln(x)**2+(-64*x**6-1120*x**5-7840*x**4
-27440*x**3-48020*x**2-33614*x)*ln(x)+1600*x**6+28000*x**5+196000*x**4+694960*x**3+1262196*x**2+953694*x)/(32*
x**5+560*x**4+3920*x**3+13720*x**2+24010*x+16807),x)

[Out]

-x**2*log(x)**2 + 25*x**2 + (-4480*x**3 - 38944*x**2 - 109760*x - 96040)/(16*x**4 + 224*x**3 + 1176*x**2 + 274
4*x + 2401)

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