3.16.28 \(\int \frac {-65536 x^2-163840 x^3-204800 x^4-163840 x^5-92160 x^6-37888 x^7-11520 x^8-2560 x^9-400 x^{10}-40 x^{11}-2 x^{12}+(32768+146440 x+265716 x^2+287097 x^3+210432 x^4+111344 x^5+43704 x^6+12806 x^7+2760 x^8+420 x^9+41 x^{10}+2 x^{11}) \log (5)}{(32768 x+81920 x^2+102400 x^3+81920 x^4+46080 x^5+18944 x^6+5760 x^7+1280 x^8+200 x^9+20 x^{10}+x^{11}) \log (5)} \, dx\)

Optimal. Leaf size=30 \[ x+x \left (1-\frac {1}{\left (4+(2+x)^2\right )^2}\right )^2-\frac {x^2}{\log (5)}+\log (x) \]

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Rubi [A]  time = 0.73, antiderivative size = 45, normalized size of antiderivative = 1.50, number of steps used = 21, number of rules used = 6, integrand size = 173, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.035, Rules used = {12, 2074, 638, 614, 617, 204} \begin {gather*} -\frac {2 x}{\left (x^2+4 x+8\right )^2}+\frac {x}{\left (x^2+4 x+8\right )^4}-\frac {x^2}{\log (5)}+\frac {x \log (25)}{\log (5)}+\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-65536*x^2 - 163840*x^3 - 204800*x^4 - 163840*x^5 - 92160*x^6 - 37888*x^7 - 11520*x^8 - 2560*x^9 - 400*x^
10 - 40*x^11 - 2*x^12 + (32768 + 146440*x + 265716*x^2 + 287097*x^3 + 210432*x^4 + 111344*x^5 + 43704*x^6 + 12
806*x^7 + 2760*x^8 + 420*x^9 + 41*x^10 + 2*x^11)*Log[5])/((32768*x + 81920*x^2 + 102400*x^3 + 81920*x^4 + 4608
0*x^5 + 18944*x^6 + 5760*x^7 + 1280*x^8 + 200*x^9 + 20*x^10 + x^11)*Log[5]),x]

[Out]

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

Rule 12

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

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 614

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b + 2*c*x)*(a + b*x + c*x^2)^(p + 1))/((p +
1)*(b^2 - 4*a*c)), x] - Dist[(2*c*(2*p + 3))/((p + 1)*(b^2 - 4*a*c)), Int[(a + b*x + c*x^2)^(p + 1), x], x] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && NeQ[p, -3/2] && IntegerQ[4*p]

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 638

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b*d - 2*a*e + (2*c*d -
b*e)*x)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)), x] - Dist[((2*p + 3)*(2*c*d - b*e))/((p + 1)*(b^2
- 4*a*c)), Int[(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] && NeQ[b^
2 - 4*a*c, 0] && LtQ[p, -1] && NeQ[p, -3/2]

Rule 2074

Int[(P_)^(p_)*(Q_)^(q_.), x_Symbol] :> With[{PP = Factor[P]}, Int[ExpandIntegrand[PP^p*Q^q, x], x] /;  !SumQ[N
onfreeFactors[PP, x]]] /; FreeQ[q, x] && PolyQ[P, x] && PolyQ[Q, x] && IntegerQ[p] && NeQ[P, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int \frac {-65536 x^2-163840 x^3-204800 x^4-163840 x^5-92160 x^6-37888 x^7-11520 x^8-2560 x^9-400 x^{10}-40 x^{11}-2 x^{12}+\left (32768+146440 x+265716 x^2+287097 x^3+210432 x^4+111344 x^5+43704 x^6+12806 x^7+2760 x^8+420 x^9+41 x^{10}+2 x^{11}\right ) \log (5)}{32768 x+81920 x^2+102400 x^3+81920 x^4+46080 x^5+18944 x^6+5760 x^7+1280 x^8+200 x^9+20 x^{10}+x^{11}} \, dx}{\log (5)}\\ &=\frac {\int \left (-2 x+\frac {\log (5)}{x}+\frac {16 (4+x) \log (5)}{\left (8+4 x+x^2\right )^5}-\frac {7 \log (5)}{\left (8+4 x+x^2\right )^4}-\frac {16 (4+x) \log (5)}{\left (8+4 x+x^2\right )^3}+\frac {6 \log (5)}{\left (8+4 x+x^2\right )^2}+\log (25)\right ) \, dx}{\log (5)}\\ &=-\frac {x^2}{\log (5)}+\frac {x \log (25)}{\log (5)}+\log (x)+6 \int \frac {1}{\left (8+4 x+x^2\right )^2} \, dx-7 \int \frac {1}{\left (8+4 x+x^2\right )^4} \, dx+16 \int \frac {4+x}{\left (8+4 x+x^2\right )^5} \, dx-16 \int \frac {4+x}{\left (8+4 x+x^2\right )^3} \, dx\\ &=\frac {x}{\left (8+4 x+x^2\right )^4}-\frac {7 (2+x)}{24 \left (8+4 x+x^2\right )^3}-\frac {2 x}{\left (8+4 x+x^2\right )^2}+\frac {3 (2+x)}{4 \left (8+4 x+x^2\right )}-\frac {x^2}{\log (5)}+\frac {x \log (25)}{\log (5)}+\log (x)+\frac {3}{4} \int \frac {1}{8+4 x+x^2} \, dx-\frac {35}{24} \int \frac {1}{\left (8+4 x+x^2\right )^3} \, dx-6 \int \frac {1}{\left (8+4 x+x^2\right )^2} \, dx+7 \int \frac {1}{\left (8+4 x+x^2\right )^4} \, dx\\ &=\frac {x}{\left (8+4 x+x^2\right )^4}-\frac {2 x}{\left (8+4 x+x^2\right )^2}-\frac {35 (2+x)}{384 \left (8+4 x+x^2\right )^2}-\frac {x^2}{\log (5)}+\frac {x \log (25)}{\log (5)}+\log (x)-\frac {35}{128} \int \frac {1}{\left (8+4 x+x^2\right )^2} \, dx-\frac {3}{8} \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {x}{2}\right )-\frac {3}{4} \int \frac {1}{8+4 x+x^2} \, dx+\frac {35}{24} \int \frac {1}{\left (8+4 x+x^2\right )^3} \, dx\\ &=\frac {x}{\left (8+4 x+x^2\right )^4}-\frac {2 x}{\left (8+4 x+x^2\right )^2}-\frac {35 (2+x)}{1024 \left (8+4 x+x^2\right )}+\frac {3}{8} \tan ^{-1}\left (\frac {2+x}{2}\right )-\frac {x^2}{\log (5)}+\frac {x \log (25)}{\log (5)}+\log (x)-\frac {35 \int \frac {1}{8+4 x+x^2} \, dx}{1024}+\frac {35}{128} \int \frac {1}{\left (8+4 x+x^2\right )^2} \, dx+\frac {3}{8} \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {x}{2}\right )\\ &=\frac {x}{\left (8+4 x+x^2\right )^4}-\frac {2 x}{\left (8+4 x+x^2\right )^2}-\frac {x^2}{\log (5)}+\frac {x \log (25)}{\log (5)}+\log (x)+\frac {35 \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {x}{2}\right )}{2048}+\frac {35 \int \frac {1}{8+4 x+x^2} \, dx}{1024}\\ &=\frac {x}{\left (8+4 x+x^2\right )^4}-\frac {2 x}{\left (8+4 x+x^2\right )^2}-\frac {35 \tan ^{-1}\left (\frac {2+x}{2}\right )}{2048}-\frac {x^2}{\log (5)}+\frac {x \log (25)}{\log (5)}+\log (x)-\frac {35 \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {x}{2}\right )}{2048}\\ &=\frac {x}{\left (8+4 x+x^2\right )^4}-\frac {2 x}{\left (8+4 x+x^2\right )^2}-\frac {x^2}{\log (5)}+\frac {x \log (25)}{\log (5)}+\log (x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.08, size = 45, normalized size = 1.50 \begin {gather*} \frac {x \left (-x+\left (\frac {1}{\left (8+4 x+x^2\right )^4}-\frac {2}{\left (8+4 x+x^2\right )^2}\right ) \log (5)+\log (25)\right )+\log (5) \log (x)}{\log (5)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-65536*x^2 - 163840*x^3 - 204800*x^4 - 163840*x^5 - 92160*x^6 - 37888*x^7 - 11520*x^8 - 2560*x^9 -
400*x^10 - 40*x^11 - 2*x^12 + (32768 + 146440*x + 265716*x^2 + 287097*x^3 + 210432*x^4 + 111344*x^5 + 43704*x^
6 + 12806*x^7 + 2760*x^8 + 420*x^9 + 41*x^10 + 2*x^11)*Log[5])/((32768*x + 81920*x^2 + 102400*x^3 + 81920*x^4
+ 46080*x^5 + 18944*x^6 + 5760*x^7 + 1280*x^8 + 200*x^9 + 20*x^10 + x^11)*Log[5]),x]

[Out]

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

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fricas [B]  time = 0.66, size = 182, normalized size = 6.07 \begin {gather*} -\frac {x^{10} + 16 \, x^{9} + 128 \, x^{8} + 640 \, x^{7} + 2176 \, x^{6} + 5120 \, x^{5} + 8192 \, x^{4} + 8192 \, x^{3} - {\left (x^{8} + 16 \, x^{7} + 128 \, x^{6} + 640 \, x^{5} + 2176 \, x^{4} + 5120 \, x^{3} + 8192 \, x^{2} + 8192 \, x + 4096\right )} \log \relax (5) \log \relax (x) + 4096 \, x^{2} - {\left (2 \, x^{9} + 32 \, x^{8} + 256 \, x^{7} + 1280 \, x^{6} + 4350 \, x^{5} + 10224 \, x^{4} + 16320 \, x^{3} + 16256 \, x^{2} + 8065 \, x\right )} \log \relax (5)}{{\left (x^{8} + 16 \, x^{7} + 128 \, x^{6} + 640 \, x^{5} + 2176 \, x^{4} + 5120 \, x^{3} + 8192 \, x^{2} + 8192 \, x + 4096\right )} \log \relax (5)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^11+41*x^10+420*x^9+2760*x^8+12806*x^7+43704*x^6+111344*x^5+210432*x^4+287097*x^3+265716*x^2+14
6440*x+32768)*log(5)-2*x^12-40*x^11-400*x^10-2560*x^9-11520*x^8-37888*x^7-92160*x^6-163840*x^5-204800*x^4-1638
40*x^3-65536*x^2)/(x^11+20*x^10+200*x^9+1280*x^8+5760*x^7+18944*x^6+46080*x^5+81920*x^4+102400*x^3+81920*x^2+3
2768*x)/log(5),x, algorithm="fricas")

[Out]

-(x^10 + 16*x^9 + 128*x^8 + 640*x^7 + 2176*x^6 + 5120*x^5 + 8192*x^4 + 8192*x^3 - (x^8 + 16*x^7 + 128*x^6 + 64
0*x^5 + 2176*x^4 + 5120*x^3 + 8192*x^2 + 8192*x + 4096)*log(5)*log(x) + 4096*x^2 - (2*x^9 + 32*x^8 + 256*x^7 +
1280*x^6 + 4350*x^5 + 10224*x^4 + 16320*x^3 + 16256*x^2 + 8065*x)*log(5))/((x^8 + 16*x^7 + 128*x^6 + 640*x^5
+ 2176*x^4 + 5120*x^3 + 8192*x^2 + 8192*x + 4096)*log(5))

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giac [B]  time = 2.22, size = 67, normalized size = 2.23 \begin {gather*} -\frac {x^{2} - 2 \, x \log \relax (5) - \log \relax (5) \log \left ({\left | x \right |}\right ) + \frac {2 \, x^{5} \log \relax (5) + 16 \, x^{4} \log \relax (5) + 64 \, x^{3} \log \relax (5) + 128 \, x^{2} \log \relax (5) + 127 \, x \log \relax (5)}{{\left (x^{2} + 4 \, x + 8\right )}^{4}}}{\log \relax (5)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^11+41*x^10+420*x^9+2760*x^8+12806*x^7+43704*x^6+111344*x^5+210432*x^4+287097*x^3+265716*x^2+14
6440*x+32768)*log(5)-2*x^12-40*x^11-400*x^10-2560*x^9-11520*x^8-37888*x^7-92160*x^6-163840*x^5-204800*x^4-1638
40*x^3-65536*x^2)/(x^11+20*x^10+200*x^9+1280*x^8+5760*x^7+18944*x^6+46080*x^5+81920*x^4+102400*x^3+81920*x^2+3
2768*x)/log(5),x, algorithm="giac")

[Out]

-(x^2 - 2*x*log(5) - log(5)*log(abs(x)) + (2*x^5*log(5) + 16*x^4*log(5) + 64*x^3*log(5) + 128*x^2*log(5) + 127
*x*log(5))/(x^2 + 4*x + 8)^4)/log(5)

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maple [A]  time = 0.16, size = 59, normalized size = 1.97

method	result	size
default	\(\frac {-x^{2}+2 x \ln \relax (5)+\frac {\ln \relax (5) \left (-2 x^{5}-16 x^{4}-64 x^{3}-128 x^{2}-127 x \right )}{\left (x^{2}+4 x +8\right )^{4}}+\ln \relax (5) \ln \relax (x )}{\ln \relax (5)}\)	\(59\)
risch	\(2 x -\frac {x^{2}}{\ln \relax (5)}+\frac {-2 x^{5} \ln \relax (5)-16 x^{4} \ln \relax (5)-64 x^{3} \ln \relax (5)-128 x^{2} \ln \relax (5)-127 x \ln \relax (5)}{\ln \relax (5) \left (x^{8}+16 x^{7}+128 x^{6}+640 x^{5}+2176 x^{4}+5120 x^{3}+8192 x^{2}+8192 x +4096\right )}+\ln \relax (x )\)	\(95\)
norman	\(\frac {-\frac {128 \left (2 \ln \relax (5)-11\right ) x^{7}}{\ln \relax (5)}-\frac {128 \left (22 \ln \relax (5)-111\right ) x^{6}}{\ln \relax (5)}-\frac {10 \left (1613 \ln \relax (5)-7680\right ) x^{5}}{\ln \relax (5)}-\frac {16 \left (3713 \ln \relax (5)-16896\right ) x^{4}}{\ln \relax (5)}-\frac {64 \left (2305 \ln \relax (5)-10112\right ) x^{3}}{\ln \relax (5)}-\frac {2176 \left (113 \ln \relax (5)-480\right ) x^{2}}{\ln \relax (5)}-\frac {\left (254079 \ln \relax (5)-1048576\right ) x}{\ln \relax (5)}-\frac {x^{10}}{\ln \relax (5)}+\frac {2 \left (-8+\ln \relax (5)\right ) x^{9}}{\ln \relax (5)}-\frac {131072 \left (\ln \relax (5)-4\right )}{\ln \relax (5)}}{\left (x^{2}+4 x +8\right )^{4}}+\ln \relax (x )\)	\(151\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*x^11+41*x^10+420*x^9+2760*x^8+12806*x^7+43704*x^6+111344*x^5+210432*x^4+287097*x^3+265716*x^2+146440*x
+32768)*ln(5)-2*x^12-40*x^11-400*x^10-2560*x^9-11520*x^8-37888*x^7-92160*x^6-163840*x^5-204800*x^4-163840*x^3-
65536*x^2)/(x^11+20*x^10+200*x^9+1280*x^8+5760*x^7+18944*x^6+46080*x^5+81920*x^4+102400*x^3+81920*x^2+32768*x)
/ln(5),x,method=_RETURNVERBOSE)

[Out]

1/ln(5)*(-x^2+2*x*ln(5)+ln(5)*(-2*x^5-16*x^4-64*x^3-128*x^2-127*x)/(x^2+4*x+8)^4+ln(5)*ln(x))

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maxima [B]  time = 0.56, size = 96, normalized size = 3.20 \begin {gather*} -\frac {x^{2} - 2 \, x \log \relax (5) - \log \relax (5) \log \relax (x) + \frac {2 \, x^{5} \log \relax (5) + 16 \, x^{4} \log \relax (5) + 64 \, x^{3} \log \relax (5) + 128 \, x^{2} \log \relax (5) + 127 \, x \log \relax (5)}{x^{8} + 16 \, x^{7} + 128 \, x^{6} + 640 \, x^{5} + 2176 \, x^{4} + 5120 \, x^{3} + 8192 \, x^{2} + 8192 \, x + 4096}}{\log \relax (5)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^11+41*x^10+420*x^9+2760*x^8+12806*x^7+43704*x^6+111344*x^5+210432*x^4+287097*x^3+265716*x^2+14
6440*x+32768)*log(5)-2*x^12-40*x^11-400*x^10-2560*x^9-11520*x^8-37888*x^7-92160*x^6-163840*x^5-204800*x^4-1638
40*x^3-65536*x^2)/(x^11+20*x^10+200*x^9+1280*x^8+5760*x^7+18944*x^6+46080*x^5+81920*x^4+102400*x^3+81920*x^2+3
2768*x)/log(5),x, algorithm="maxima")

[Out]

-(x^2 - 2*x*log(5) - log(5)*log(x) + (2*x^5*log(5) + 16*x^4*log(5) + 64*x^3*log(5) + 128*x^2*log(5) + 127*x*lo
g(5))/(x^8 + 16*x^7 + 128*x^6 + 640*x^5 + 2176*x^4 + 5120*x^3 + 8192*x^2 + 8192*x + 4096))/log(5)

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mupad [B]  time = 1.29, size = 96, normalized size = 3.20 \begin {gather*} \ln \relax (x)-\frac {x^2}{\ln \relax (5)}-\frac {2\,x^5+16\,x^4+64\,x^3+128\,x^2+127\,x}{x^8+16\,x^7+128\,x^6+640\,x^5+2176\,x^4+5120\,x^3+8192\,x^2+8192\,x+4096}+x\,\left (\frac {40}{\ln \relax (5)}+\frac {\ln \left (25\right )-40}{\ln \relax (5)}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(65536*x^2 - log(5)*(146440*x + 265716*x^2 + 287097*x^3 + 210432*x^4 + 111344*x^5 + 43704*x^6 + 12806*x^7
+ 2760*x^8 + 420*x^9 + 41*x^10 + 2*x^11 + 32768) + 163840*x^3 + 204800*x^4 + 163840*x^5 + 92160*x^6 + 37888*x
^7 + 11520*x^8 + 2560*x^9 + 400*x^10 + 40*x^11 + 2*x^12)/(log(5)*(32768*x + 81920*x^2 + 102400*x^3 + 81920*x^4
+ 46080*x^5 + 18944*x^6 + 5760*x^7 + 1280*x^8 + 200*x^9 + 20*x^10 + x^11)),x)

[Out]

log(x) - x^2/log(5) - (127*x + 128*x^2 + 64*x^3 + 16*x^4 + 2*x^5)/(8192*x + 8192*x^2 + 5120*x^3 + 2176*x^4 + 6
40*x^5 + 128*x^6 + 16*x^7 + x^8 + 4096) + x*(40/log(5) + (log(25) - 40)/log(5))

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sympy [B]  time = 1.12, size = 75, normalized size = 2.50 \begin {gather*} - \frac {x^{2}}{\log {\relax (5 )}} + 2 x - \frac {2 x^{5} + 16 x^{4} + 64 x^{3} + 128 x^{2} + 127 x}{x^{8} + 16 x^{7} + 128 x^{6} + 640 x^{5} + 2176 x^{4} + 5120 x^{3} + 8192 x^{2} + 8192 x + 4096} + \log {\relax (x )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x**11+41*x**10+420*x**9+2760*x**8+12806*x**7+43704*x**6+111344*x**5+210432*x**4+287097*x**3+2657
16*x**2+146440*x+32768)*ln(5)-2*x**12-40*x**11-400*x**10-2560*x**9-11520*x**8-37888*x**7-92160*x**6-163840*x**
5-204800*x**4-163840*x**3-65536*x**2)/(x**11+20*x**10+200*x**9+1280*x**8+5760*x**7+18944*x**6+46080*x**5+81920
*x**4+102400*x**3+81920*x**2+32768*x)/ln(5),x)

[Out]

-x**2/log(5) + 2*x - (2*x**5 + 16*x**4 + 64*x**3 + 128*x**2 + 127*x)/(x**8 + 16*x**7 + 128*x**6 + 640*x**5 + 2
176*x**4 + 5120*x**3 + 8192*x**2 + 8192*x + 4096) + log(x)

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