3.59.44 \(\int \frac {-16+6 x-163840 x^3+69632 x^4-5120 x^5-1920 x^6+440 x^7-35 x^8+x^9+(-131072 x^3+16384 x^4+20480 x^5-7680 x^6+1120 x^7-76 x^8+2 x^9) \log (x)}{-32768 x^3+20480 x^4-5120 x^5+640 x^6-40 x^7+x^8} \, dx\)

Optimal. Leaf size=22 \[ -2-\frac {1}{(8-x)^4 x^2}+x+x (4+x) \log (x) \]

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Rubi [B]  time = 0.85, antiderivative size = 100, normalized size of antiderivative = 4.55, number of steps used = 20, number of rules used = 7, integrand size = 109, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.064, Rules used = {6688, 6742, 44, 43, 37, 2313, 9} \begin {gather*} \frac {60 x^4}{(8-x)^4}+\frac {x^2}{2}-\frac {1}{4096 x^2}+\left (x^2+4 x\right ) \log (x)+5 x-\frac {1}{2} (x+4)^2+\frac {15728639}{8192 (8-x)}-\frac {94371843}{4096 (8-x)^2}+\frac {31457279}{256 (8-x)^3}-\frac {15728641}{64 (8-x)^4}-\frac {1}{8192 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-16 + 6*x - 163840*x^3 + 69632*x^4 - 5120*x^5 - 1920*x^6 + 440*x^7 - 35*x^8 + x^9 + (-131072*x^3 + 16384*
x^4 + 20480*x^5 - 7680*x^6 + 1120*x^7 - 76*x^8 + 2*x^9)*Log[x])/(-32768*x^3 + 20480*x^4 - 5120*x^5 + 640*x^6 -
40*x^7 + x^8),x]

[Out]

-15728641/(64*(8 - x)^4) + 31457279/(256*(8 - x)^3) - 94371843/(4096*(8 - x)^2) + 15728639/(8192*(8 - x)) - 1/
(4096*x^2) - 1/(8192*x) + 5*x + x^2/2 + (60*x^4)/(8 - x)^4 - (4 + x)^2/2 + (4*x + x^2)*Log[x]

Rule 9

Int[(a_)*((b_) + (c_.)*(x_)), x_Symbol] :> Simp[(a*(b + c*x)^2)/(2*c), x] /; FreeQ[{a, b, c}, 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 44

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}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rule 2313

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> With[{u = IntHide[(d +
e*x^r)^q, x]}, Simp[u*(a + b*Log[c*x^n]), x] - Dist[b*n, Int[SimplifyIntegrand[u/x, x], x], x]] /; FreeQ[{a,
b, c, d, e, n, r}, x] && IGtQ[q, 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 {16-6 x+163840 x^3-69632 x^4+5120 x^5+1920 x^6-440 x^7+35 x^8-x^9-2 (-8+x)^5 x^3 (2+x) \log (x)}{(8-x)^5 x^3} \, dx\\ &=\int \left (-\frac {163840}{(-8+x)^5}-\frac {16}{(-8+x)^5 x^3}+\frac {6}{(-8+x)^5 x^2}+\frac {69632 x}{(-8+x)^5}-\frac {5120 x^2}{(-8+x)^5}-\frac {1920 x^3}{(-8+x)^5}+\frac {440 x^4}{(-8+x)^5}-\frac {35 x^5}{(-8+x)^5}+\frac {x^6}{(-8+x)^5}+2 (2+x) \log (x)\right ) \, dx\\ &=\frac {40960}{(8-x)^4}+2 \int (2+x) \log (x) \, dx+6 \int \frac {1}{(-8+x)^5 x^2} \, dx-16 \int \frac {1}{(-8+x)^5 x^3} \, dx-35 \int \frac {x^5}{(-8+x)^5} \, dx+440 \int \frac {x^4}{(-8+x)^5} \, dx-1920 \int \frac {x^3}{(-8+x)^5} \, dx-5120 \int \frac {x^2}{(-8+x)^5} \, dx+69632 \int \frac {x}{(-8+x)^5} \, dx+\int \frac {x^6}{(-8+x)^5} \, dx\\ &=\frac {40960}{(8-x)^4}+\frac {60 x^4}{(8-x)^4}+\left (4 x+x^2\right ) \log (x)-2 \int \frac {4+x}{2} \, dx+6 \int \left (\frac {1}{64 (-8+x)^5}-\frac {1}{256 (-8+x)^4}+\frac {3}{4096 (-8+x)^3}-\frac {1}{8192 (-8+x)^2}+\frac {5}{262144 (-8+x)}-\frac {1}{32768 x^2}-\frac {5}{262144 x}\right ) \, dx-16 \int \left (\frac {1}{512 (-8+x)^5}-\frac {3}{4096 (-8+x)^4}+\frac {3}{16384 (-8+x)^3}-\frac {5}{131072 (-8+x)^2}+\frac {15}{2097152 (-8+x)}-\frac {1}{32768 x^3}-\frac {5}{262144 x^2}-\frac {15}{2097152 x}\right ) \, dx-35 \int \left (1+\frac {32768}{(-8+x)^5}+\frac {20480}{(-8+x)^4}+\frac {5120}{(-8+x)^3}+\frac {640}{(-8+x)^2}+\frac {40}{-8+x}\right ) \, dx+440 \int \left (\frac {4096}{(-8+x)^5}+\frac {2048}{(-8+x)^4}+\frac {384}{(-8+x)^3}+\frac {32}{(-8+x)^2}+\frac {1}{-8+x}\right ) \, dx-5120 \int \left (\frac {64}{(-8+x)^5}+\frac {16}{(-8+x)^4}+\frac {1}{(-8+x)^3}\right ) \, dx+69632 \int \left (\frac {8}{(-8+x)^5}+\frac {1}{(-8+x)^4}\right ) \, dx+\int \left (40+\frac {262144}{(-8+x)^5}+\frac {196608}{(-8+x)^4}+\frac {61440}{(-8+x)^3}+\frac {10240}{(-8+x)^2}+\frac {960}{-8+x}+x\right ) \, dx\\ &=-\frac {15728641}{64 (8-x)^4}+\frac {31457279}{256 (8-x)^3}-\frac {94371843}{4096 (8-x)^2}+\frac {15728639}{8192 (8-x)}-\frac {1}{4096 x^2}-\frac {1}{8192 x}+5 x+\frac {x^2}{2}+\frac {60 x^4}{(8-x)^4}-\frac {1}{2} (4+x)^2+\left (4 x+x^2\right ) \log (x)\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.11, size = 48, normalized size = 2.18 \begin {gather*} \frac {-1+4096 x^3-2048 x^4+384 x^5-32 x^6+x^7+(-8+x)^4 x^3 (4+x) \log (x)}{(-8+x)^4 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-16 + 6*x - 163840*x^3 + 69632*x^4 - 5120*x^5 - 1920*x^6 + 440*x^7 - 35*x^8 + x^9 + (-131072*x^3 +
16384*x^4 + 20480*x^5 - 7680*x^6 + 1120*x^7 - 76*x^8 + 2*x^9)*Log[x])/(-32768*x^3 + 20480*x^4 - 5120*x^5 + 640
*x^6 - 40*x^7 + x^8),x]

[Out]

(-1 + 4096*x^3 - 2048*x^4 + 384*x^5 - 32*x^6 + x^7 + (-8 + x)^4*x^3*(4 + x)*Log[x])/((-8 + x)^4*x^2)

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fricas [B]  time = 0.73, size = 84, normalized size = 3.82 \begin {gather*} \frac {x^{7} - 32 \, x^{6} + 384 \, x^{5} - 2048 \, x^{4} + 4096 \, x^{3} + {\left (x^{8} - 28 \, x^{7} + 256 \, x^{6} - 512 \, x^{5} - 4096 \, x^{4} + 16384 \, x^{3}\right )} \log \relax (x) - 1}{x^{6} - 32 \, x^{5} + 384 \, x^{4} - 2048 \, x^{3} + 4096 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^9-76*x^8+1120*x^7-7680*x^6+20480*x^5+16384*x^4-131072*x^3)*log(x)+x^9-35*x^8+440*x^7-1920*x^6-
5120*x^5+69632*x^4-163840*x^3+6*x-16)/(x^8-40*x^7+640*x^6-5120*x^5+20480*x^4-32768*x^3),x, algorithm="fricas")

[Out]

(x^7 - 32*x^6 + 384*x^5 - 2048*x^4 + 4096*x^3 + (x^8 - 28*x^7 + 256*x^6 - 512*x^5 - 4096*x^4 + 16384*x^3)*log(
x) - 1)/(x^6 - 32*x^5 + 384*x^4 - 2048*x^3 + 4096*x^2)

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giac [B]  time = 0.16, size = 55, normalized size = 2.50 \begin {gather*} {\left (x^{2} + 4 \, x\right )} \log \relax (x) + x + \frac {x^{3} - 30 \, x^{2} + 320 \, x - 1280}{8192 \, {\left (x^{4} - 32 \, x^{3} + 384 \, x^{2} - 2048 \, x + 4096\right )}} - \frac {x + 2}{8192 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^9-76*x^8+1120*x^7-7680*x^6+20480*x^5+16384*x^4-131072*x^3)*log(x)+x^9-35*x^8+440*x^7-1920*x^6-
5120*x^5+69632*x^4-163840*x^3+6*x-16)/(x^8-40*x^7+640*x^6-5120*x^5+20480*x^4-32768*x^3),x, algorithm="giac")

[Out]

(x^2 + 4*x)*log(x) + x + 1/8192*(x^3 - 30*x^2 + 320*x - 1280)/(x^4 - 32*x^3 + 384*x^2 - 2048*x + 4096) - 1/819
2*(x + 2)/x^2

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maple [B]  time = 0.06, size = 52, normalized size = 2.36

method	result	size
default	\(4 x \ln \relax (x )+x +x^{2} \ln \relax (x )-\frac {1}{64 \left (-8+x \right )^{4}}+\frac {1}{256 \left (-8+x \right )^{3}}-\frac {3}{4096 \left (-8+x \right )^{2}}+\frac {1}{-65536+8192 x}-\frac {1}{4096 x^{2}}-\frac {1}{8192 x}\)	\(52\)
risch	\(\left (x^{2}+4 x \right ) \ln \relax (x )+\frac {x^{7}-32 x^{6}+384 x^{5}-2048 x^{4}+4096 x^{3}-1}{x^{2} \left (x^{4}-32 x^{3}+384 x^{2}-2048 x +4096\right )}\)	\(61\)
norman	\(\frac {-1+x^{7}+x^{8} \ln \relax (x )-\frac {20480 x^{3}}{3}-\frac {80 x^{6}}{3}+\frac {640 x^{5}}{3}+\frac {65536 x^{2}}{3}-\frac {16384 x^{3} \ln \relax (x )}{3}-\frac {2560 x^{5} \ln \relax (x )}{3}+\frac {800 x^{6} \ln \relax (x )}{3}+\frac {131072 x^{2} \ln \relax (x )}{3}-28 x^{7} \ln \relax (x )}{x^{2} \left (-8+x \right )^{4}}-\frac {32 \ln \relax (x )}{3}\)	\(81\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*x^9-76*x^8+1120*x^7-7680*x^6+20480*x^5+16384*x^4-131072*x^3)*ln(x)+x^9-35*x^8+440*x^7-1920*x^6-5120*x^
5+69632*x^4-163840*x^3+6*x-16)/(x^8-40*x^7+640*x^6-5120*x^5+20480*x^4-32768*x^3),x,method=_RETURNVERBOSE)

[Out]

4*x*ln(x)+x+x^2*ln(x)-1/64/(-8+x)^4+1/256/(-8+x)^3-3/4096/(-8+x)^2+1/8192/(-8+x)-1/4096/x^2-1/8192/x

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maxima [B]  time = 0.42, size = 628, normalized size = 28.55 \begin {gather*} \frac {1}{2} \, x^{2} + 5 \, x + \frac {7680 \, {\left (x^{3} - 12 \, x^{2} + 64 \, x - 128\right )} \log \relax (x)}{x^{4} - 32 \, x^{3} + 384 \, x^{2} - 2048 \, x + 4096} - \frac {10240 \, {\left (3 \, x^{2} - 16 \, x + 32\right )} \log \relax (x)}{3 \, {\left (x^{4} - 32 \, x^{3} + 384 \, x^{2} - 2048 \, x + 4096\right )}} - \frac {16384 \, {\left (x - 2\right )} \log \relax (x)}{3 \, {\left (x^{4} - 32 \, x^{3} + 384 \, x^{2} - 2048 \, x + 4096\right )}} - \frac {15 \, x^{5} - 420 \, x^{4} + 4160 \, x^{3} - 16000 \, x^{2} + 12288 \, x + 16384}{16384 \, {\left (x^{6} - 32 \, x^{5} + 384 \, x^{4} - 2048 \, x^{3} + 4096 \, x^{2}\right )}} + \frac {15 \, x^{4} - 420 \, x^{3} + 4160 \, x^{2} - 16000 \, x + 12288}{16384 \, {\left (x^{5} - 32 \, x^{4} + 384 \, x^{3} - 2048 \, x^{2} + 4096 \, x\right )}} - \frac {3 \, x^{6} - 72 \, x^{5} + 384 \, x^{4} + 36864 \, x^{3} - 737280 \, x^{2} - 6 \, {\left (x^{6} - 28 \, x^{5}\right )} \log \relax (x) + 4980736 \, x - 11534336}{6 \, {\left (x^{4} - 32 \, x^{3} + 384 \, x^{2} - 2048 \, x + 4096\right )}} + \frac {4480 \, {\left (15 \, x^{3} - 300 \, x^{2} + 2080 \, x - 4928\right )}}{3 \, {\left (x^{4} - 32 \, x^{3} + 384 \, x^{2} - 2048 \, x + 4096\right )}} - \frac {2048 \, {\left (5 \, x^{3} - 105 \, x^{2} + 752 \, x - 1824\right )}}{x^{4} - 32 \, x^{3} + 384 \, x^{2} - 2048 \, x + 4096} - \frac {14080 \, {\left (3 \, x^{3} - 54 \, x^{2} + 352 \, x - 800\right )}}{3 \, {\left (x^{4} - 32 \, x^{3} + 384 \, x^{2} - 2048 \, x + 4096\right )}} + \frac {1920 \, {\left (x^{3} - 12 \, x^{2} + 64 \, x - 128\right )}}{x^{4} - 32 \, x^{3} + 384 \, x^{2} - 2048 \, x + 4096} + \frac {640 \, {\left (9 \, x^{2} - 108 \, x + 352\right )}}{x^{3} - 24 \, x^{2} + 192 \, x - 512} + \frac {2560 \, {\left (3 \, x^{2} - 16 \, x + 32\right )}}{3 \, {\left (x^{4} - 32 \, x^{3} + 384 \, x^{2} - 2048 \, x + 4096\right )}} + \frac {64 \, {\left (3 \, x^{2} - 60 \, x + 352\right )}}{3 \, {\left (x^{3} - 24 \, x^{2} + 192 \, x - 512\right )}} - \frac {640 \, {\left (x^{2} + 4 \, x - 32\right )}}{3 \, {\left (x^{3} - 24 \, x^{2} + 192 \, x - 512\right )}} + \frac {64 \, {\left (x^{2} - 20 \, x + 32\right )}}{3 \, {\left (x^{3} - 24 \, x^{2} + 192 \, x - 512\right )}} - \frac {69632 \, {\left (x - 2\right )}}{3 \, {\left (x^{4} - 32 \, x^{3} + 384 \, x^{2} - 2048 \, x + 4096\right )}} + \frac {32768 \, \log \relax (x)}{x^{4} - 32 \, x^{3} + 384 \, x^{2} - 2048 \, x + 4096} + \frac {40960}{x^{4} - 32 \, x^{3} + 384 \, x^{2} - 2048 \, x + 4096} + 256 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^9-76*x^8+1120*x^7-7680*x^6+20480*x^5+16384*x^4-131072*x^3)*log(x)+x^9-35*x^8+440*x^7-1920*x^6-
5120*x^5+69632*x^4-163840*x^3+6*x-16)/(x^8-40*x^7+640*x^6-5120*x^5+20480*x^4-32768*x^3),x, algorithm="maxima")

[Out]

1/2*x^2 + 5*x + 7680*(x^3 - 12*x^2 + 64*x - 128)*log(x)/(x^4 - 32*x^3 + 384*x^2 - 2048*x + 4096) - 10240/3*(3*
x^2 - 16*x + 32)*log(x)/(x^4 - 32*x^3 + 384*x^2 - 2048*x + 4096) - 16384/3*(x - 2)*log(x)/(x^4 - 32*x^3 + 384*
x^2 - 2048*x + 4096) - 1/16384*(15*x^5 - 420*x^4 + 4160*x^3 - 16000*x^2 + 12288*x + 16384)/(x^6 - 32*x^5 + 384
*x^4 - 2048*x^3 + 4096*x^2) + 1/16384*(15*x^4 - 420*x^3 + 4160*x^2 - 16000*x + 12288)/(x^5 - 32*x^4 + 384*x^3
- 2048*x^2 + 4096*x) - 1/6*(3*x^6 - 72*x^5 + 384*x^4 + 36864*x^3 - 737280*x^2 - 6*(x^6 - 28*x^5)*log(x) + 4980
736*x - 11534336)/(x^4 - 32*x^3 + 384*x^2 - 2048*x + 4096) + 4480/3*(15*x^3 - 300*x^2 + 2080*x - 4928)/(x^4 -
32*x^3 + 384*x^2 - 2048*x + 4096) - 2048*(5*x^3 - 105*x^2 + 752*x - 1824)/(x^4 - 32*x^3 + 384*x^2 - 2048*x + 4
096) - 14080/3*(3*x^3 - 54*x^2 + 352*x - 800)/(x^4 - 32*x^3 + 384*x^2 - 2048*x + 4096) + 1920*(x^3 - 12*x^2 +
64*x - 128)/(x^4 - 32*x^3 + 384*x^2 - 2048*x + 4096) + 640*(9*x^2 - 108*x + 352)/(x^3 - 24*x^2 + 192*x - 512)
+ 2560/3*(3*x^2 - 16*x + 32)/(x^4 - 32*x^3 + 384*x^2 - 2048*x + 4096) + 64/3*(3*x^2 - 60*x + 352)/(x^3 - 24*x^
2 + 192*x - 512) - 640/3*(x^2 + 4*x - 32)/(x^3 - 24*x^2 + 192*x - 512) + 64/3*(x^2 - 20*x + 32)/(x^3 - 24*x^2
+ 192*x - 512) - 69632/3*(x - 2)/(x^4 - 32*x^3 + 384*x^2 - 2048*x + 4096) + 32768*log(x)/(x^4 - 32*x^3 + 384*x
^2 - 2048*x + 4096) + 40960/(x^4 - 32*x^3 + 384*x^2 - 2048*x + 4096) + 256*log(x)

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mupad [B]  time = 4.18, size = 40, normalized size = 1.82 \begin {gather*} x+\ln \relax (x)\,\left (x^2+4\,x\right )-\frac {1}{x^6-32\,x^5+384\,x^4-2048\,x^3+4096\,x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(6*x + log(x)*(16384*x^4 - 131072*x^3 + 20480*x^5 - 7680*x^6 + 1120*x^7 - 76*x^8 + 2*x^9) - 163840*x^3 +
69632*x^4 - 5120*x^5 - 1920*x^6 + 440*x^7 - 35*x^8 + x^9 - 16)/(32768*x^3 - 20480*x^4 + 5120*x^5 - 640*x^6 + 4
0*x^7 - x^8),x)

[Out]

x + log(x)*(4*x + x^2) - 1/(4096*x^2 - 2048*x^3 + 384*x^4 - 32*x^5 + x^6)

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sympy [A]  time = 0.20, size = 36, normalized size = 1.64 \begin {gather*} x + \left (x^{2} + 4 x\right ) \log {\relax (x )} - \frac {1}{x^{6} - 32 x^{5} + 384 x^{4} - 2048 x^{3} + 4096 x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x**9-76*x**8+1120*x**7-7680*x**6+20480*x**5+16384*x**4-131072*x**3)*ln(x)+x**9-35*x**8+440*x**7-
1920*x**6-5120*x**5+69632*x**4-163840*x**3+6*x-16)/(x**8-40*x**7+640*x**6-5120*x**5+20480*x**4-32768*x**3),x)

[Out]

x + (x**2 + 4*x)*log(x) - 1/(x**6 - 32*x**5 + 384*x**4 - 2048*x**3 + 4096*x**2)

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