\(\int \frac {3440-2592 x-872 x^2+1288 x^3-256 x^4-128 x^5+73 x^6-14 x^7+x^8+(80+96 x-84 x^2+16 x^3) \log (5)}{3600-2400 x-1040 x^2+1320 x^3-256 x^4-128 x^5+73 x^6-14 x^7+x^8} \, dx\) [1632]

Optimal result

Integrand size = 97, antiderivative size = 35 \[ \int \frac {3440-2592 x-872 x^2+1288 x^3-256 x^4-128 x^5+73 x^6-14 x^7+x^8+\left (80+96 x-84 x^2+16 x^3\right ) \log (5)}{3600-2400 x-1040 x^2+1320 x^3-256 x^4-128 x^5+73 x^6-14 x^7+x^8} \, dx=x-\frac {2-\log (5)}{(3-x) \left (5-\frac {1}{4} x \left (4 x-x^2\right )\right )} \]

[Out]

x-(2-ln(5))/(5-x*(x-1/4*x^2))/(-x+3)

Rubi [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(1979\) vs. \(2(35)=70\).

Time = 8.36 (sec) , antiderivative size = 1979, normalized size of antiderivative = 56.54, number of steps used = 20, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {2099, 2126, 2106, 2104, 836, 814, 648, 632, 210, 642} \[ \int \frac {3440-2592 x-872 x^2+1288 x^3-256 x^4-128 x^5+73 x^6-14 x^7+x^8+\left (80+96 x-84 x^2+16 x^3\right ) \log (5)}{3600-2400 x-1040 x^2+1320 x^3-256 x^4-128 x^5+73 x^6-14 x^7+x^8} \, dx =\text {Too large to display} \]

[In]

Int[(3440 - 2592*x - 872*x^2 + 1288*x^3 - 256*x^4 - 128*x^5 + 73*x^6 - 14*x^7 + x^8 + (80 + 96*x - 84*x^2 + 16
*x^3)*Log[5])/(3600 - 2400*x - 1040*x^2 + 1320*x^3 - 256*x^4 - 128*x^5 + 73*x^6 - 14*x^7 + x^8),x]

[Out]

x - (230040*(1013623 - 31059*Sqrt[1065])*(2 - Log[5]))/(11*(264622680*2^(1/3) - 8108984*2^(1/3)*Sqrt[1065] - (
2632680 - 80776*Sqrt[1065])*(1013623 - 31059*Sqrt[1065])^(1/3) + (33077835 - 1013623*Sqrt[1065])*(20194 - 618*
Sqrt[1065])^(1/3) + 63701910*(103 - 3*Sqrt[1065])^(2/3) - 1948142*Sqrt[1065]*(103 - 3*Sqrt[1065])^(2/3))*(2^(1
/3)*(8*2^(1/3) + (103 - 3*Sqrt[1065])^(2/3)) - (103 - 3*Sqrt[1065])^(1/3)*(4 - 3*x))) - (4*(2 - Log[5]))/(11*(
3 - x)) - (18*2^(2/3)*(103 - 3*Sqrt[1065])^(5/3)*(1420 + (2^(1/3)*(1775 - 43*Sqrt[1065] + (355 - 7*Sqrt[1065])
*(206 - 6*Sqrt[1065])^(1/3))*(4 - 3*x))/(103 - 3*Sqrt[1065])^(2/3))*(2 - Log[5]))/(11*(329085 - 10097*Sqrt[106
5] - 4*(3195 - 103*Sqrt[1065])*(206 - 6*Sqrt[1065])^(1/3))*(4 - (206 - 6*Sqrt[1065])^(1/3) - (8*2^(2/3))/(103
- 3*Sqrt[1065])^(1/3) - 3*x)*((15*(45 + Sqrt[1065]))^(1/3)*(4 + (206 - 6*Sqrt[1065])^(1/3)) - 2*(8 + (206 - 6*
Sqrt[1065])^(1/3) + (8*2^(2/3))/(103 - 3*Sqrt[1065])^(1/3))*x + 6*x^2)) - (4*(2 - Log[5]))/(11*(20 - 4*x^2 + x
^3)) + (51120*(979149472182 - 30003654798*Sqrt[1065] - 4*(203636674 - 6239946*Sqrt[1065])^(1/3)*(101818337 - 3
119973*Sqrt[1065]) + (169616017219 - 5197469775*Sqrt[1065])*(206 - 6*Sqrt[1065])^(1/3))*Sqrt[6/(2^(2/3)*(10097
- 309*Sqrt[1065]) + 64*2^(1/3)*(103 - 3*Sqrt[1065])^(2/3) - 16*(103 - 3*Sqrt[1065])^(4/3))]*ArcTan[(2^(1/3)*(
103 - 3*Sqrt[1065] + 8*(206 - 6*Sqrt[1065])^(1/3)) + 2*(103 - 3*Sqrt[1065])^(2/3)*(4 - 3*x))/Sqrt[6*(2^(2/3)*(
10097 - 309*Sqrt[1065]) + 64*2^(1/3)*(103 - 3*Sqrt[1065])^(2/3) - 16*(103 - 3*Sqrt[1065])^(4/3))]]*(2 - Log[5]
))/(11*(103 - 3*Sqrt[1065])*(329085 - 10097*Sqrt[1065] - 4*(3195 - 103*Sqrt[1065])*(206 - 6*Sqrt[1065])^(1/3))
*(192*2^(1/3) + 2^(2/3)*(103 - 3*Sqrt[1065])^(4/3) + (103 - 3*Sqrt[1065])^(2/3)*(24 + (10097 - 309*Sqrt[1065])
^(1/3)))^2) - (18*2^(5/6)*(2^(1/3)*(4739 - 143*Sqrt[1065]) + (193 - 5*Sqrt[1065])*(103 - 3*Sqrt[1065])^(2/3))*
Sqrt[3/(2^(2/3)*(10097 - 309*Sqrt[1065]) + 64*2^(1/3)*(103 - 3*Sqrt[1065])^(2/3) - 16*(103 - 3*Sqrt[1065])^(4/
3))]*ArcTan[(2^(1/3)*(103 - 3*Sqrt[1065] + 8*(206 - 6*Sqrt[1065])^(1/3)) + 2*(103 - 3*Sqrt[1065])^(2/3)*(4 - 3
*x))/Sqrt[6*(2^(2/3)*(10097 - 309*Sqrt[1065]) + 64*2^(1/3)*(103 - 3*Sqrt[1065])^(2/3) - 16*(103 - 3*Sqrt[1065]
)^(4/3))]]*(2 - Log[5]))/(11*(192*2^(1/3) + 2^(2/3)*(103 - 3*Sqrt[1065])^(4/3) + (103 - 3*Sqrt[1065])^(2/3)*(2
4 + (10097 - 309*Sqrt[1065])^(1/3)))) + (230040*(2^(2/3)*(101818337 - 3119973*Sqrt[1065]) + 8*(1013623 - 31059
*Sqrt[1065])*(103 - 3*Sqrt[1065])^(1/3) - 5*2^(1/3)*(1013623 - 31059*Sqrt[1065])*(103 - 3*Sqrt[1065])^(2/3))*(
2 - Log[5])*Log[2^(1/3)*(8*2^(1/3) + (103 - 3*Sqrt[1065])^(2/3)) - (103 - 3*Sqrt[1065])^(1/3)*(4 - 3*x)])/(11*
(10097 - 309*Sqrt[1065])^(1/3)*(329085 - 10097*Sqrt[1065] - 4*(3195 - 103*Sqrt[1065])*(206 - 6*Sqrt[1065])^(1/
3))*(192*2^(1/3) + 2^(2/3)*(103 - 3*Sqrt[1065])^(4/3) + (103 - 3*Sqrt[1065])^(2/3)*(24 + (10097 - 309*Sqrt[106
5])^(1/3)))^2) - (6*(2*(103 - 3*Sqrt[1065]))^(1/3)*(8*2^(1/3) - 5*2^(2/3)*(103 - 3*Sqrt[1065])^(1/3) + (103 -
3*Sqrt[1065])^(2/3))*(2 - Log[5])*Log[2^(1/3)*(8*2^(1/3) + (103 - 3*Sqrt[1065])^(2/3)) - (103 - 3*Sqrt[1065])^
(1/3)*(4 - 3*x)])/(11*(192*2^(1/3) + 2^(2/3)*(103 - 3*Sqrt[1065])^(4/3) + (103 - 3*Sqrt[1065])^(2/3)*(24 + (10
097 - 309*Sqrt[1065])^(1/3)))) - (115020*(101818337 - 3119973*Sqrt[1065])*(8*2^(1/3) - 5*2^(2/3)*(103 - 3*Sqrt
[1065])^(1/3) + (103 - 3*Sqrt[1065])^(2/3))*(2 - Log[5])*Log[2^(1/3)*(45 - Sqrt[1065])*(4 + (206 - 6*Sqrt[1065
])^(1/3)) - 103*2^(1/3)*x + 3*2^(1/3)*Sqrt[1065]*x - 8*2^(2/3)*(103 - 3*Sqrt[1065])^(1/3)*x - 8*(103 - 3*Sqrt[
1065])^(2/3)*x + 3*(103 - 3*Sqrt[1065])^(2/3)*x^2])/(11*(103 - 3*Sqrt[1065])^(1/3)*(33077835 - 1013623*Sqrt[10
65] - 4*(329085 - 10097*Sqrt[1065])*(206 - 6*Sqrt[1065])^(1/3))*(192*2^(1/3) + 2^(2/3)*(103 - 3*Sqrt[1065])^(4
/3) + (103 - 3*Sqrt[1065])^(2/3)*(24 + (10097 - 309*Sqrt[1065])^(1/3)))^2) - (3*(103 - 3*Sqrt[1065])^(2/3)*(10
- (206 - 6*Sqrt[1065])^(1/3) - (8*2^(2/3))/(103 - 3*Sqrt[1065])^(1/3))*(2 - Log[5])*Log[2^(1/3)*(45 - Sqrt[10
65])*(4 + (206 - 6*Sqrt[1065])^(1/3)) - 103*2^(1/3)*x + 3*2^(1/3)*Sqrt[1065]*x - 8*2^(2/3)*(103 - 3*Sqrt[1065]
)^(1/3)*x - 8*(103 - 3*Sqrt[1065])^(2/3)*x + 3*(103 - 3*Sqrt[1065])^(2/3)*x^2])/(11*(192*2^(1/3) + 2^(2/3)*(10
3 - 3*Sqrt[1065])^(4/3) + (103 - 3*Sqrt[1065])^(2/3)*(24 + (10097 - 309*Sqrt[1065])^(1/3))))

Rule 210

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

Rule 632

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

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 648

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

Rule 814

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Int[Exp
andIntegrand[(d + e*x)^m*((f + g*x)/(a + b*x + c*x^2)), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 -
4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[m]

Rule 836

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

Rule 2099

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]

Rule 2104

Int[((e_.) + (f_.)*(x_))^(m_.)*((a_) + (b_.)*(x_) + (d_.)*(x_)^3)^(p_), x_Symbol] :> With[{r = Rt[-9*a*d^2 + S
qrt[3]*d*Sqrt[4*b^3*d + 27*a^2*d^2], 3]}, Dist[1/d^(2*p), Int[(e + f*x)^m*Simp[18^(1/3)*b*(d/(3*r)) - r/18^(1/
3) + d*x, x]^p*Simp[b*(d/3) + 12^(1/3)*b^2*(d^2/(3*r^2)) + r^2/(3*12^(1/3)) - d*(2^(1/3)*b*(d/(3^(1/3)*r)) - r
/18^(1/3))*x + d^2*x^2, x]^p, x], x]] /; FreeQ[{a, b, d, e, f, m}, x] && NeQ[4*b^3 + 27*a^2*d, 0] && ILtQ[p, 0
]

Rule 2106

Int[(P3_)^(p_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> With[{a = Coeff[P3, x, 0], b = Coeff[P3, x, 1], c = C
oeff[P3, x, 2], d = Coeff[P3, x, 3]}, Subst[Int[((3*d*e - c*f)/(3*d) + f*x)^m*Simp[(2*c^3 - 9*b*c*d + 27*a*d^2
)/(27*d^2) - (c^2 - 3*b*d)*(x/(3*d)) + d*x^3, x]^p, x], x, x + c/(3*d)] /; NeQ[c, 0]] /; FreeQ[{e, f, m, p}, x
] && PolyQ[P3, x, 3]

Rule 2126

Int[(Pm_)*(Qn_)^(p_), x_Symbol] :> With[{m = Expon[Pm, x], n = Expon[Qn, x]}, Simp[Coeff[Pm, x, m]*(Qn^(p + 1)
/(n*(p + 1)*Coeff[Qn, x, n])), x] + Dist[1/(n*Coeff[Qn, x, n]), Int[ExpandToSum[n*Coeff[Qn, x, n]*Pm - Coeff[P
m, x, m]*D[Qn, x], x]*Qn^p, x], x] /; EqQ[m, n - 1]] /; FreeQ[p, x] && PolyQ[Pm, x] && PolyQ[Qn, x] && NeQ[p,
-1]

Rubi steps \begin{align*} \text {integral}& = \int \left (1+\frac {4 (-2+\log (5))}{11 (-3+x)^2}-\frac {4 \left (-20-36 x+3 x^2\right ) (-2+\log (5))}{11 \left (20-4 x^2+x^3\right )^2}-\frac {4 (2+x) (-2+\log (5))}{11 \left (20-4 x^2+x^3\right )}\right ) \, dx \\ & = x-\frac {4 (2-\log (5))}{11 (3-x)}+\frac {1}{11} (4 (2-\log (5))) \int \frac {-20-36 x+3 x^2}{\left (20-4 x^2+x^3\right )^2} \, dx+\frac {1}{11} (4 (2-\log (5))) \int \frac {2+x}{20-4 x^2+x^3} \, dx \\ & = x-\frac {4 (2-\log (5))}{11 (3-x)}-\frac {4 (2-\log (5))}{11 \left (20-4 x^2+x^3\right )}+\frac {1}{33} (4 (2-\log (5))) \int \frac {-60-84 x}{\left (20-4 x^2+x^3\right )^2} \, dx+\frac {1}{11} (4 (2-\log (5))) \text {Subst}\left (\int \frac {\frac {10}{3}+x}{\frac {412}{27}-\frac {16 x}{3}+x^3} \, dx,x,-\frac {4}{3}+x\right ) \\ & = x-\frac {4 (2-\log (5))}{11 (3-x)}-\frac {4 (2-\log (5))}{11 \left (20-4 x^2+x^3\right )}+\frac {1}{33} (4 (2-\log (5))) \text {Subst}\left (\int \frac {-172-84 x}{\left (\frac {412}{27}-\frac {16 x}{3}+x^3\right )^2} \, dx,x,-\frac {4}{3}+x\right )+\frac {1}{11} (4 (2-\log (5))) \text {Subst}\left (\int \frac {\frac {10}{3}+x}{\left (\frac {1}{3} \sqrt [3]{\frac {2}{103-3 \sqrt {1065}}} \left (8 \sqrt [3]{2}+\left (103-3 \sqrt {1065}\right )^{2/3}\right )+x\right ) \left (\frac {1}{9} \left (-16+\left (206-6 \sqrt {1065}\right )^{2/3}+\frac {128 \sqrt [3]{2}}{\left (103-3 \sqrt {1065}\right )^{2/3}}\right )-\frac {1}{3} \left (\sqrt [3]{206-6 \sqrt {1065}}+\frac {8\ 2^{2/3}}{\sqrt [3]{103-3 \sqrt {1065}}}\right ) x+x^2\right )} \, dx,x,-\frac {4}{3}+x\right ) \\ & = x-\frac {4 (2-\log (5))}{11 (3-x)}-\frac {4 (2-\log (5))}{11 \left (20-4 x^2+x^3\right )}+\frac {1}{33} (4 (2-\log (5))) \text {Subst}\left (\int \frac {-172-84 x}{\left (\frac {1}{3} \sqrt [3]{\frac {2}{103-3 \sqrt {1065}}} \left (8 \sqrt [3]{2}+\left (103-3 \sqrt {1065}\right )^{2/3}\right )+x\right )^2 \left (\frac {1}{9} \left (-16+\left (206-6 \sqrt {1065}\right )^{2/3}+\frac {128 \sqrt [3]{2}}{\left (103-3 \sqrt {1065}\right )^{2/3}}\right )-\frac {1}{3} \left (\sqrt [3]{206-6 \sqrt {1065}}+\frac {8\ 2^{2/3}}{\sqrt [3]{103-3 \sqrt {1065}}}\right ) x+x^2\right )^2} \, dx,x,-\frac {4}{3}+x\right )+\frac {1}{11} (4 (2-\log (5))) \text {Subst}\left (\int \left (\frac {9 \left (103-3 \sqrt {1065}\right )^{2/3} \left (-8 2^{2/3}+10 \sqrt [3]{103-3 \sqrt {1065}}-\sqrt [3]{2} \left (103-3 \sqrt {1065}\right )^{2/3}\right )}{2 \left (192 \sqrt [3]{2}+2^{2/3} \left (103-3 \sqrt {1065}\right )^{4/3}+\left (103-3 \sqrt {1065}\right )^{2/3} \left (24+\sqrt [3]{10097-309 \sqrt {1065}}\right )\right ) \left (8\ 2^{2/3}+\sqrt [3]{2} \left (103-3 \sqrt {1065}\right )^{2/3}+3 \sqrt [3]{103-3 \sqrt {1065}} x\right )}+\frac {9 \left (103-3 \sqrt {1065}\right )^{2/3} \left (4 \sqrt [3]{2} \left (547-15 \sqrt {1065}\right )-16 \left (103-3 \sqrt {1065}\right )^{2/3}+2^{2/3} \sqrt [3]{103-3 \sqrt {1065}} \left (263-3 \sqrt {1065}\right )+3 \left (8\ 2^{2/3} \sqrt [3]{103-3 \sqrt {1065}}-10 \left (103-3 \sqrt {1065}\right )^{2/3}+\sqrt [3]{2} \left (103-3 \sqrt {1065}\right )\right ) x\right )}{2 \left (192 \sqrt [3]{2}+2^{2/3} \left (103-3 \sqrt {1065}\right )^{4/3}+\left (103-3 \sqrt {1065}\right )^{2/3} \left (24+\sqrt [3]{10097-309 \sqrt {1065}}\right )\right ) \left (128 \sqrt [3]{2}-16 \left (103-3 \sqrt {1065}\right )^{2/3}+2^{2/3} \left (103-3 \sqrt {1065}\right )^{4/3}-3 \sqrt [3]{2} \left (103-3 \sqrt {1065}+8 \sqrt [3]{206-6 \sqrt {1065}}\right ) x+9 \left (103-3 \sqrt {1065}\right )^{2/3} x^2\right )}\right ) \, dx,x,-\frac {4}{3}+x\right ) \\ & = x-\frac {4 (2-\log (5))}{11 (3-x)}-\frac {18\ 2^{2/3} \left (103-3 \sqrt {1065}\right )^{5/3} \left (1420+\frac {\sqrt [3]{2} \left (1775-43 \sqrt {1065}+\left (355-7 \sqrt {1065}\right ) \sqrt [3]{206-6 \sqrt {1065}}\right ) (4-3 x)}{\left (103-3 \sqrt {1065}\right )^{2/3}}\right ) (2-\log (5))}{11 \left (329085-10097 \sqrt {1065}-4 \left (3195-103 \sqrt {1065}\right ) \sqrt [3]{206-6 \sqrt {1065}}\right ) \left (4-\sqrt [3]{206-6 \sqrt {1065}}-\frac {8\ 2^{2/3}}{\sqrt [3]{103-3 \sqrt {1065}}}-3 x\right ) \left (\sqrt [3]{15 \left (45+\sqrt {1065}\right )} \left (4+\sqrt [3]{206-6 \sqrt {1065}}\right )-2 \left (8+\sqrt [3]{206-6 \sqrt {1065}}+\frac {8\ 2^{2/3}}{\sqrt [3]{103-3 \sqrt {1065}}}\right ) x+6 x^2\right )}-\frac {4 (2-\log (5))}{11 \left (20-4 x^2+x^3\right )}-\frac {6 \sqrt [3]{2 \left (103-3 \sqrt {1065}\right )} \left (8 \sqrt [3]{2}-5\ 2^{2/3} \sqrt [3]{103-3 \sqrt {1065}}+\left (103-3 \sqrt {1065}\right )^{2/3}\right ) (2-\log (5)) \log \left (\sqrt [3]{2} \left (8 \sqrt [3]{2}+\left (103-3 \sqrt {1065}\right )^{2/3}\right )-\sqrt [3]{103-3 \sqrt {1065}} (4-3 x)\right )}{11 \left (192 \sqrt [3]{2}+2^{2/3} \left (103-3 \sqrt {1065}\right )^{4/3}+\left (103-3 \sqrt {1065}\right )^{2/3} \left (24+\sqrt [3]{10097-309 \sqrt {1065}}\right )\right )}+\frac {\left (\left (103-3 \sqrt {1065}\right )^{5/3} (2-\log (5))\right ) \text {Subst}\left (\int \frac {16 \left (355-\frac {16 \sqrt [3]{2} \left (29465-893 \sqrt {1065}\right )}{\left (103-3 \sqrt {1065}\right )^{5/3}}-\frac {2^{2/3} \left (160105-4877 \sqrt {1065}\right )}{\left (103-3 \sqrt {1065}\right )^{4/3}}\right )-\frac {24 \sqrt [3]{2} \left (1775-43 \sqrt {1065}+\left (355-7 \sqrt {1065}\right ) \sqrt [3]{206-6 \sqrt {1065}}\right ) x}{\left (103-3 \sqrt {1065}\right )^{2/3}}}{\left (\frac {1}{3} \sqrt [3]{\frac {2}{103-3 \sqrt {1065}}} \left (8 \sqrt [3]{2}+\left (103-3 \sqrt {1065}\right )^{2/3}\right )+x\right )^2 \left (\frac {1}{9} \left (-16+\left (206-6 \sqrt {1065}\right )^{2/3}+\frac {128 \sqrt [3]{2}}{\left (103-3 \sqrt {1065}\right )^{2/3}}\right )+\frac {1}{3} \left (-\sqrt [3]{206-6 \sqrt {1065}}-\frac {8\ 2^{2/3}}{\sqrt [3]{103-3 \sqrt {1065}}}\right ) x+x^2\right )} \, dx,x,-\frac {4}{3}+x\right )}{22 \sqrt [3]{2} \left (329085-10097 \sqrt {1065}-4 \left (3195-103 \sqrt {1065}\right ) \sqrt [3]{206-6 \sqrt {1065}}\right )}+\frac {\left (18 \left (103-3 \sqrt {1065}\right )^{2/3} (2-\log (5))\right ) \text {Subst}\left (\int \frac {4 \sqrt [3]{2} \left (547-15 \sqrt {1065}\right )-16 \left (103-3 \sqrt {1065}\right )^{2/3}+2^{2/3} \sqrt [3]{103-3 \sqrt {1065}} \left (263-3 \sqrt {1065}\right )+3 \left (8\ 2^{2/3} \sqrt [3]{103-3 \sqrt {1065}}-10 \left (103-3 \sqrt {1065}\right )^{2/3}+\sqrt [3]{2} \left (103-3 \sqrt {1065}\right )\right ) x}{128 \sqrt [3]{2}-16 \left (103-3 \sqrt {1065}\right )^{2/3}+2^{2/3} \left (103-3 \sqrt {1065}\right )^{4/3}-3 \sqrt [3]{2} \left (103-3 \sqrt {1065}+8 \sqrt [3]{206-6 \sqrt {1065}}\right ) x+9 \left (103-3 \sqrt {1065}\right )^{2/3} x^2} \, dx,x,-\frac {4}{3}+x\right )}{11 \left (192 \sqrt [3]{2}+2^{2/3} \left (103-3 \sqrt {1065}\right )^{4/3}+\left (103-3 \sqrt {1065}\right )^{2/3} \left (24+\sqrt [3]{10097-309 \sqrt {1065}}\right )\right )} \\ & = \text {Too large to display} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.03 \[ \int \frac {3440-2592 x-872 x^2+1288 x^3-256 x^4-128 x^5+73 x^6-14 x^7+x^8+\left (80+96 x-84 x^2+16 x^3\right ) \log (5)}{3600-2400 x-1040 x^2+1320 x^3-256 x^4-128 x^5+73 x^6-14 x^7+x^8} \, dx=x+\frac {4 (17182+229489 \log (5)-119040 \log (25))}{8591 \left (-60+20 x+12 x^2-7 x^3+x^4\right )} \]

[In]

Integrate[(3440 - 2592*x - 872*x^2 + 1288*x^3 - 256*x^4 - 128*x^5 + 73*x^6 - 14*x^7 + x^8 + (80 + 96*x - 84*x^
2 + 16*x^3)*Log[5])/(3600 - 2400*x - 1040*x^2 + 1320*x^3 - 256*x^4 - 128*x^5 + 73*x^6 - 14*x^7 + x^8),x]

[Out]

x + (4*(17182 + 229489*Log[5] - 119040*Log[25]))/(8591*(-60 + 20*x + 12*x^2 - 7*x^3 + x^4))

Maple [A] (verified)

Time = 0.08 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.26

method	result	size
norman	\(\frac {x^{5}-412-37 x^{3}+104 x^{2}+80 x -4 \ln \left (5\right )}{x^{4}-7 x^{3}+12 x^{2}+20 x -60}\)	\(44\)
gosper	\(-\frac {-x^{5}+37 x^{3}-104 x^{2}+4 \ln \left (5\right )-80 x +412}{x^{4}-7 x^{3}+12 x^{2}+20 x -60}\)	\(47\)
parallelrisch	\(-\frac {-x^{5}+37 x^{3}-104 x^{2}+4 \ln \left (5\right )-80 x +412}{x^{4}-7 x^{3}+12 x^{2}+20 x -60}\)	\(47\)
risch	\(x -\frac {4 \ln \left (5\right )}{x^{4}-7 x^{3}+12 x^{2}+20 x -60}+\frac {8}{x^{4}-7 x^{3}+12 x^{2}+20 x -60}\)	\(49\)
default	\(x -\frac {4 \left (\left (2-\ln \left (5\right )\right ) x^{2}+\left (\ln \left (5\right )-2\right ) x -6+3 \ln \left (5\right )\right )}{11 \left (x^{3}-4 x^{2}+20\right )}-\frac {\frac {4 \ln \left (5\right )}{11}-\frac {8}{11}}{-3+x}\)	\(52\)

[In]

int(((16*x^3-84*x^2+96*x+80)*ln(5)+x^8-14*x^7+73*x^6-128*x^5-256*x^4+1288*x^3-872*x^2-2592*x+3440)/(x^8-14*x^7
+73*x^6-128*x^5-256*x^4+1320*x^3-1040*x^2-2400*x+3600),x,method=_RETURNVERBOSE)

[Out]

(x^5-412-37*x^3+104*x^2+80*x-4*ln(5))/(x^4-7*x^3+12*x^2+20*x-60)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.37 \[ \int \frac {3440-2592 x-872 x^2+1288 x^3-256 x^4-128 x^5+73 x^6-14 x^7+x^8+\left (80+96 x-84 x^2+16 x^3\right ) \log (5)}{3600-2400 x-1040 x^2+1320 x^3-256 x^4-128 x^5+73 x^6-14 x^7+x^8} \, dx=\frac {x^{5} - 7 \, x^{4} + 12 \, x^{3} + 20 \, x^{2} - 60 \, x - 4 \, \log \left (5\right ) + 8}{x^{4} - 7 \, x^{3} + 12 \, x^{2} + 20 \, x - 60} \]

[In]

integrate(((16*x^3-84*x^2+96*x+80)*log(5)+x^8-14*x^7+73*x^6-128*x^5-256*x^4+1288*x^3-872*x^2-2592*x+3440)/(x^8
-14*x^7+73*x^6-128*x^5-256*x^4+1320*x^3-1040*x^2-2400*x+3600),x, algorithm="fricas")

[Out]

(x^5 - 7*x^4 + 12*x^3 + 20*x^2 - 60*x - 4*log(5) + 8)/(x^4 - 7*x^3 + 12*x^2 + 20*x - 60)

Sympy [A] (verification not implemented)

Time = 0.75 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.74 \[ \int \frac {3440-2592 x-872 x^2+1288 x^3-256 x^4-128 x^5+73 x^6-14 x^7+x^8+\left (80+96 x-84 x^2+16 x^3\right ) \log (5)}{3600-2400 x-1040 x^2+1320 x^3-256 x^4-128 x^5+73 x^6-14 x^7+x^8} \, dx=x + \frac {8 - 4 \log {\left (5 \right )}}{x^{4} - 7 x^{3} + 12 x^{2} + 20 x - 60} \]

[In]

integrate(((16*x**3-84*x**2+96*x+80)*ln(5)+x**8-14*x**7+73*x**6-128*x**5-256*x**4+1288*x**3-872*x**2-2592*x+34
40)/(x**8-14*x**7+73*x**6-128*x**5-256*x**4+1320*x**3-1040*x**2-2400*x+3600),x)

[Out]

x + (8 - 4*log(5))/(x**4 - 7*x**3 + 12*x**2 + 20*x - 60)

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.80 \[ \int \frac {3440-2592 x-872 x^2+1288 x^3-256 x^4-128 x^5+73 x^6-14 x^7+x^8+\left (80+96 x-84 x^2+16 x^3\right ) \log (5)}{3600-2400 x-1040 x^2+1320 x^3-256 x^4-128 x^5+73 x^6-14 x^7+x^8} \, dx=x - \frac {4 \, {\left (\log \left (5\right ) - 2\right )}}{x^{4} - 7 \, x^{3} + 12 \, x^{2} + 20 \, x - 60} \]

[In]

integrate(((16*x^3-84*x^2+96*x+80)*log(5)+x^8-14*x^7+73*x^6-128*x^5-256*x^4+1288*x^3-872*x^2-2592*x+3440)/(x^8
-14*x^7+73*x^6-128*x^5-256*x^4+1320*x^3-1040*x^2-2400*x+3600),x, algorithm="maxima")

[Out]

x - 4*(log(5) - 2)/(x^4 - 7*x^3 + 12*x^2 + 20*x - 60)

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.80 \[ \int \frac {3440-2592 x-872 x^2+1288 x^3-256 x^4-128 x^5+73 x^6-14 x^7+x^8+\left (80+96 x-84 x^2+16 x^3\right ) \log (5)}{3600-2400 x-1040 x^2+1320 x^3-256 x^4-128 x^5+73 x^6-14 x^7+x^8} \, dx=x - \frac {4 \, {\left (\log \left (5\right ) - 2\right )}}{x^{4} - 7 \, x^{3} + 12 \, x^{2} + 20 \, x - 60} \]

[In]

integrate(((16*x^3-84*x^2+96*x+80)*log(5)+x^8-14*x^7+73*x^6-128*x^5-256*x^4+1288*x^3-872*x^2-2592*x+3440)/(x^8
-14*x^7+73*x^6-128*x^5-256*x^4+1320*x^3-1040*x^2-2400*x+3600),x, algorithm="giac")

[Out]

x - 4*(log(5) - 2)/(x^4 - 7*x^3 + 12*x^2 + 20*x - 60)

Mupad [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.80 \[ \int \frac {3440-2592 x-872 x^2+1288 x^3-256 x^4-128 x^5+73 x^6-14 x^7+x^8+\left (80+96 x-84 x^2+16 x^3\right ) \log (5)}{3600-2400 x-1040 x^2+1320 x^3-256 x^4-128 x^5+73 x^6-14 x^7+x^8} \, dx=x-\frac {\ln \left (625\right )-8}{x^4-7\,x^3+12\,x^2+20\,x-60} \]

[In]

int((2592*x - log(5)*(96*x - 84*x^2 + 16*x^3 + 80) + 872*x^2 - 1288*x^3 + 256*x^4 + 128*x^5 - 73*x^6 + 14*x^7
- x^8 - 3440)/(2400*x + 1040*x^2 - 1320*x^3 + 256*x^4 + 128*x^5 - 73*x^6 + 14*x^7 - x^8 - 3600),x)

[Out]

x - (log(625) - 8)/(20*x + 12*x^2 - 7*x^3 + x^4 - 60)