3.65.53 \(\int \frac {-102-426 x-585 x^2-301 x^3-65 x^4-5 x^5+(10+12 x+7 x^2+x^3) \log (5+x)}{100+420 x+580 x^2+300 x^3+65 x^4+5 x^5} \, dx\)

Optimal. Leaf size=25 \[ 5-x+\frac {\log (5+x)}{5 \left (-3-x+\frac {1}{1+x}\right )} \]

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Rubi [B]  time = 1.55, antiderivative size = 570, normalized size of antiderivative = 22.80, number of steps used = 66, number of rules used = 16, integrand size = 71, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.225, Rules used = {6688, 12, 6742, 740, 800, 632, 31, 822, 1646, 1628, 2418, 2395, 36, 2394, 2393, 2391} \begin {gather*} \frac {51 (3 x+4)}{70 \left (x^2+4 x+2\right )}-\frac {213 (4 x+3)}{35 \left (x^2+4 x+2\right )}+\frac {117 (13 x+8)}{14 \left (x^2+4 x+2\right )}-\frac {43 (22 x+13)}{5 \left (x^2+4 x+2\right )}+\frac {13 (75 x+44)}{7 \left (x^2+4 x+2\right )}-\frac {2 (128 x+75)}{7 \left (x^2+4 x+2\right )}-x-\frac {\left (2-\sqrt {2}\right ) \log (x+5)}{20 \left (x-\sqrt {2}+2\right )}-\frac {\left (2+\sqrt {2}\right ) \log (x+5)}{20 \left (x+\sqrt {2}+2\right )}-\frac {\left (2-\sqrt {2}\right ) \log (x+5)}{20 \left (3+\sqrt {2}\right )}-\frac {\left (2+\sqrt {2}\right ) \log (x+5)}{20 \left (3-\sqrt {2}\right )}+\frac {4}{35} \log (x+5)+\frac {51 \left (8+9 \sqrt {2}\right ) \log \left (x-\sqrt {2}+2\right )}{1960}+\frac {\left (2-\sqrt {2}\right ) \log \left (x-\sqrt {2}+2\right )}{20 \left (3+\sqrt {2}\right )}-\frac {213}{490} \left (10-\sqrt {2}\right ) \log \left (x-\sqrt {2}+2\right )+\frac {117}{392} \left (100-59 \sqrt {2}\right ) \log \left (x-\sqrt {2}+2\right )-\frac {43}{70} \left (125-86 \sqrt {2}\right ) \log \left (x-\sqrt {2}+2\right )+\frac {13}{196} \left (1152-811 \sqrt {2}\right ) \log \left (x-\sqrt {2}+2\right )-\frac {1}{49} \left (1244-879 \sqrt {2}\right ) \log \left (x-\sqrt {2}+2\right )-\frac {1}{49} \left (1244+879 \sqrt {2}\right ) \log \left (x+\sqrt {2}+2\right )+\frac {13}{196} \left (1152+811 \sqrt {2}\right ) \log \left (x+\sqrt {2}+2\right )-\frac {43}{70} \left (125+86 \sqrt {2}\right ) \log \left (x+\sqrt {2}+2\right )+\frac {117}{392} \left (100+59 \sqrt {2}\right ) \log \left (x+\sqrt {2}+2\right )-\frac {213}{490} \left (10+\sqrt {2}\right ) \log \left (x+\sqrt {2}+2\right )+\frac {\left (2+\sqrt {2}\right ) \log \left (x+\sqrt {2}+2\right )}{20 \left (3-\sqrt {2}\right )}+\frac {51 \left (8-9 \sqrt {2}\right ) \log \left (x+\sqrt {2}+2\right )}{1960} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-102 - 426*x - 585*x^2 - 301*x^3 - 65*x^4 - 5*x^5 + (10 + 12*x + 7*x^2 + x^3)*Log[5 + x])/(100 + 420*x +
580*x^2 + 300*x^3 + 65*x^4 + 5*x^5),x]

[Out]

-x + (51*(4 + 3*x))/(70*(2 + 4*x + x^2)) - (213*(3 + 4*x))/(35*(2 + 4*x + x^2)) + (117*(8 + 13*x))/(14*(2 + 4*
x + x^2)) - (43*(13 + 22*x))/(5*(2 + 4*x + x^2)) + (13*(44 + 75*x))/(7*(2 + 4*x + x^2)) - (2*(75 + 128*x))/(7*
(2 + 4*x + x^2)) + (4*Log[5 + x])/35 - ((2 + Sqrt[2])*Log[5 + x])/(20*(3 - Sqrt[2])) - ((2 - Sqrt[2])*Log[5 +
x])/(20*(3 + Sqrt[2])) - ((2 - Sqrt[2])*Log[5 + x])/(20*(2 - Sqrt[2] + x)) - ((2 + Sqrt[2])*Log[5 + x])/(20*(2
+ Sqrt[2] + x)) - ((1244 - 879*Sqrt[2])*Log[2 - Sqrt[2] + x])/49 + (13*(1152 - 811*Sqrt[2])*Log[2 - Sqrt[2] +
x])/196 - (43*(125 - 86*Sqrt[2])*Log[2 - Sqrt[2] + x])/70 + (117*(100 - 59*Sqrt[2])*Log[2 - Sqrt[2] + x])/392
- (213*(10 - Sqrt[2])*Log[2 - Sqrt[2] + x])/490 + ((2 - Sqrt[2])*Log[2 - Sqrt[2] + x])/(20*(3 + Sqrt[2])) + (
51*(8 + 9*Sqrt[2])*Log[2 - Sqrt[2] + x])/1960 + (51*(8 - 9*Sqrt[2])*Log[2 + Sqrt[2] + x])/1960 + ((2 + Sqrt[2]
)*Log[2 + Sqrt[2] + x])/(20*(3 - Sqrt[2])) - (213*(10 + Sqrt[2])*Log[2 + Sqrt[2] + x])/490 + (117*(100 + 59*Sq
rt[2])*Log[2 + Sqrt[2] + x])/392 - (43*(125 + 86*Sqrt[2])*Log[2 + Sqrt[2] + x])/70 + (13*(1152 + 811*Sqrt[2])*
Log[2 + Sqrt[2] + x])/196 - ((1244 + 879*Sqrt[2])*Log[2 + Sqrt[2] + x])/49

Rule 12

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

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 36

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

Rule 632

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[
(c*d - e*(b/2 - q/2))/q, Int[1/(b/2 - q/2 + c*x), x], x] - Dist[(c*d - e*(b/2 + q/2))/q, Int[1/(b/2 + q/2 + c*
x), 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 740

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

Rule 800

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 822

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 1628

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegra
nd[(d + e*x)^m*Pq*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rule 1646

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = Polynomi
alQuotient[(d + e*x)^m*Pq, a + b*x + c*x^2, x], f = Coeff[PolynomialRemainder[(d + e*x)^m*Pq, a + b*x + c*x^2,
x], x, 0], g = Coeff[PolynomialRemainder[(d + e*x)^m*Pq, a + b*x + c*x^2, x], x, 1]}, Simp[((b*f - 2*a*g + (2
*c*f - b*g)*x)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)), x] + Dist[1/((p + 1)*(b^2 - 4*a*c)), Int[(d
+ e*x)^m*(a + b*x + c*x^2)^(p + 1)*ExpandToSum[((p + 1)*(b^2 - 4*a*c)*Q)/(d + e*x)^m - ((2*p + 3)*(2*c*f - b*
g))/(d + e*x)^m, x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2
- b*d*e + a*e^2, 0] && LtQ[p, -1] && ILtQ[m, 0]

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 2393

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Dist[1/g, Subst[Int[(a +
b*Log[1 + (c*e*x)/g])/x, x], x, f + g*x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && EqQ[g
+ c*(e*f - d*g), 0]

Rule 2394

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Simp[(Log[(e*(f +
g*x))/(e*f - d*g)]*(a + b*Log[c*(d + e*x)^n]))/g, x] - Dist[(b*e*n)/g, Int[Log[(e*(f + g*x))/(e*f - d*g)]/(d +
e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g, 0]

Rule 2395

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))*((f_.) + (g_.)*(x_))^(q_.), x_Symbol] :> Simp[((f + g
*x)^(q + 1)*(a + b*Log[c*(d + e*x)^n]))/(g*(q + 1)), x] - Dist[(b*e*n)/(g*(q + 1)), Int[(f + g*x)^(q + 1)/(d +
e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, q}, x] && NeQ[e*f - d*g, 0] && NeQ[q, -1]

Rule 2418

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*(RFx_), x_Symbol] :> With[{u = ExpandIntegrand[
(a + b*Log[c*(d + e*x)^n])^p, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, n}, x] && RationalFunct
ionQ[RFx, x] && IntegerQ[p]

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 {-102-426 x-585 x^2-301 x^3-65 x^4-5 x^5+\left (10+12 x+7 x^2+x^3\right ) \log (5+x)}{5 (5+x) \left (2+4 x+x^2\right )^2} \, dx\\ &=\frac {1}{5} \int \frac {-102-426 x-585 x^2-301 x^3-65 x^4-5 x^5+\left (10+12 x+7 x^2+x^3\right ) \log (5+x)}{(5+x) \left (2+4 x+x^2\right )^2} \, dx\\ &=\frac {1}{5} \int \left (-\frac {102}{(5+x) \left (2+4 x+x^2\right )^2}-\frac {426 x}{(5+x) \left (2+4 x+x^2\right )^2}-\frac {585 x^2}{(5+x) \left (2+4 x+x^2\right )^2}-\frac {301 x^3}{(5+x) \left (2+4 x+x^2\right )^2}-\frac {65 x^4}{(5+x) \left (2+4 x+x^2\right )^2}-\frac {5 x^5}{(5+x) \left (2+4 x+x^2\right )^2}+\frac {\left (2+2 x+x^2\right ) \log (5+x)}{\left (2+4 x+x^2\right )^2}\right ) \, dx\\ &=\frac {1}{5} \int \frac {\left (2+2 x+x^2\right ) \log (5+x)}{\left (2+4 x+x^2\right )^2} \, dx-13 \int \frac {x^4}{(5+x) \left (2+4 x+x^2\right )^2} \, dx-\frac {102}{5} \int \frac {1}{(5+x) \left (2+4 x+x^2\right )^2} \, dx-\frac {301}{5} \int \frac {x^3}{(5+x) \left (2+4 x+x^2\right )^2} \, dx-\frac {426}{5} \int \frac {x}{(5+x) \left (2+4 x+x^2\right )^2} \, dx-117 \int \frac {x^2}{(5+x) \left (2+4 x+x^2\right )^2} \, dx-\int \frac {x^5}{(5+x) \left (2+4 x+x^2\right )^2} \, dx\\ &=\frac {51 (4+3 x)}{70 \left (2+4 x+x^2\right )}-\frac {213 (3+4 x)}{35 \left (2+4 x+x^2\right )}+\frac {117 (8+13 x)}{14 \left (2+4 x+x^2\right )}-\frac {43 (13+22 x)}{5 \left (2+4 x+x^2\right )}+\frac {13 (44+75 x)}{7 \left (2+4 x+x^2\right )}-\frac {2 (75+128 x)}{7 \left (2+4 x+x^2\right )}+\frac {1}{8} \int \frac {-\frac {1760}{7}-\frac {2832 x}{7}+32 x^2-8 x^3}{(5+x) \left (2+4 x+x^2\right )} \, dx+\frac {1}{5} \int \left (-\frac {2 x \log (5+x)}{\left (2+4 x+x^2\right )^2}+\frac {\log (5+x)}{2+4 x+x^2}\right ) \, dx+\frac {51}{140} \int \frac {22+6 x}{(5+x) \left (2+4 x+x^2\right )} \, dx+\frac {213}{140} \int \frac {-40-16 x}{(5+x) \left (2+4 x+x^2\right )} \, dx+\frac {13}{8} \int \frac {\frac {520}{7}+\frac {824 x}{7}-8 x^2}{(5+x) \left (2+4 x+x^2\right )} \, dx+\frac {301}{40} \int \frac {-\frac {160}{7}-\frac {232 x}{7}}{(5+x) \left (2+4 x+x^2\right )} \, dx+\frac {117}{8} \int \frac {\frac {60}{7}+\frac {52 x}{7}}{(5+x) \left (2+4 x+x^2\right )} \, dx\\ &=\frac {51 (4+3 x)}{70 \left (2+4 x+x^2\right )}-\frac {213 (3+4 x)}{35 \left (2+4 x+x^2\right )}+\frac {117 (8+13 x)}{14 \left (2+4 x+x^2\right )}-\frac {43 (13+22 x)}{5 \left (2+4 x+x^2\right )}+\frac {13 (44+75 x)}{7 \left (2+4 x+x^2\right )}-\frac {2 (75+128 x)}{7 \left (2+4 x+x^2\right )}+\frac {1}{8} \int \left (-8+\frac {25000}{49 (5+x)}-\frac {32 (365+622 x)}{49 \left (2+4 x+x^2\right )}\right ) \, dx+\frac {1}{5} \int \frac {\log (5+x)}{2+4 x+x^2} \, dx+\frac {51}{140} \int \left (-\frac {8}{7 (5+x)}+\frac {2 (17+4 x)}{7 \left (2+4 x+x^2\right )}\right ) \, dx-\frac {2}{5} \int \frac {x \log (5+x)}{\left (2+4 x+x^2\right )^2} \, dx+\frac {213}{140} \int \left (\frac {40}{7 (5+x)}-\frac {8 (9+5 x)}{7 \left (2+4 x+x^2\right )}\right ) \, dx+\frac {13}{8} \int \left (-\frac {5000}{49 (5+x)}+\frac {8 (341+576 x)}{49 \left (2+4 x+x^2\right )}\right ) \, dx+\frac {301}{40} \int \left (\frac {1000}{49 (5+x)}-\frac {8 (78+125 x)}{49 \left (2+4 x+x^2\right )}\right ) \, dx+\frac {117}{8} \int \left (-\frac {200}{49 (5+x)}+\frac {4 (41+50 x)}{49 \left (2+4 x+x^2\right )}\right ) \, dx\\ &=-x+\frac {51 (4+3 x)}{70 \left (2+4 x+x^2\right )}-\frac {213 (3+4 x)}{35 \left (2+4 x+x^2\right )}+\frac {117 (8+13 x)}{14 \left (2+4 x+x^2\right )}-\frac {43 (13+22 x)}{5 \left (2+4 x+x^2\right )}+\frac {13 (44+75 x)}{7 \left (2+4 x+x^2\right )}-\frac {2 (75+128 x)}{7 \left (2+4 x+x^2\right )}+\frac {4}{35} \log (5+x)-\frac {4}{49} \int \frac {365+622 x}{2+4 x+x^2} \, dx+\frac {51}{490} \int \frac {17+4 x}{2+4 x+x^2} \, dx+\frac {1}{5} \int \left (-\frac {\log (5+x)}{\sqrt {2} \left (-4+2 \sqrt {2}-2 x\right )}-\frac {\log (5+x)}{\sqrt {2} \left (4+2 \sqrt {2}+2 x\right )}\right ) \, dx+\frac {13}{49} \int \frac {341+576 x}{2+4 x+x^2} \, dx-\frac {2}{5} \int \left (\frac {\left (-4+2 \sqrt {2}\right ) \log (5+x)}{4 \left (-4+2 \sqrt {2}-2 x\right )^2}-\frac {\log (5+x)}{2 \sqrt {2} \left (-4+2 \sqrt {2}-2 x\right )}+\frac {\left (-4-2 \sqrt {2}\right ) \log (5+x)}{4 \left (4+2 \sqrt {2}+2 x\right )^2}-\frac {\log (5+x)}{2 \sqrt {2} \left (4+2 \sqrt {2}+2 x\right )}\right ) \, dx+\frac {117}{98} \int \frac {41+50 x}{2+4 x+x^2} \, dx-\frac {43}{35} \int \frac {78+125 x}{2+4 x+x^2} \, dx-\frac {426}{245} \int \frac {9+5 x}{2+4 x+x^2} \, dx\\ &=-x+\frac {51 (4+3 x)}{70 \left (2+4 x+x^2\right )}-\frac {213 (3+4 x)}{35 \left (2+4 x+x^2\right )}+\frac {117 (8+13 x)}{14 \left (2+4 x+x^2\right )}-\frac {43 (13+22 x)}{5 \left (2+4 x+x^2\right )}+\frac {13 (44+75 x)}{7 \left (2+4 x+x^2\right )}-\frac {2 (75+128 x)}{7 \left (2+4 x+x^2\right )}+\frac {4}{35} \log (5+x)-\frac {1}{49} \left (1244-879 \sqrt {2}\right ) \int \frac {1}{2-\sqrt {2}+x} \, dx+\frac {1}{196} \left (13 \left (1152-811 \sqrt {2}\right )\right ) \int \frac {1}{2-\sqrt {2}+x} \, dx-\frac {1}{70} \left (43 \left (125-86 \sqrt {2}\right )\right ) \int \frac {1}{2-\sqrt {2}+x} \, dx+\frac {1}{392} \left (117 \left (100-59 \sqrt {2}\right )\right ) \int \frac {1}{2-\sqrt {2}+x} \, dx+\frac {\left (51 \left (8-9 \sqrt {2}\right )\right ) \int \frac {1}{2+\sqrt {2}+x} \, dx}{1960}-\frac {1}{490} \left (213 \left (10-\sqrt {2}\right )\right ) \int \frac {1}{2-\sqrt {2}+x} \, dx-\frac {1}{5} \left (-2+\sqrt {2}\right ) \int \frac {\log (5+x)}{\left (-4+2 \sqrt {2}-2 x\right )^2} \, dx+\frac {1}{5} \left (2+\sqrt {2}\right ) \int \frac {\log (5+x)}{\left (4+2 \sqrt {2}+2 x\right )^2} \, dx-\frac {1}{490} \left (213 \left (10+\sqrt {2}\right )\right ) \int \frac {1}{2+\sqrt {2}+x} \, dx+\frac {\left (51 \left (8+9 \sqrt {2}\right )\right ) \int \frac {1}{2-\sqrt {2}+x} \, dx}{1960}+\frac {1}{392} \left (117 \left (100+59 \sqrt {2}\right )\right ) \int \frac {1}{2+\sqrt {2}+x} \, dx-\frac {1}{70} \left (43 \left (125+86 \sqrt {2}\right )\right ) \int \frac {1}{2+\sqrt {2}+x} \, dx+\frac {1}{196} \left (13 \left (1152+811 \sqrt {2}\right )\right ) \int \frac {1}{2+\sqrt {2}+x} \, dx-\frac {1}{49} \left (1244+879 \sqrt {2}\right ) \int \frac {1}{2+\sqrt {2}+x} \, dx\\ &=-x+\frac {51 (4+3 x)}{70 \left (2+4 x+x^2\right )}-\frac {213 (3+4 x)}{35 \left (2+4 x+x^2\right )}+\frac {117 (8+13 x)}{14 \left (2+4 x+x^2\right )}-\frac {43 (13+22 x)}{5 \left (2+4 x+x^2\right )}+\frac {13 (44+75 x)}{7 \left (2+4 x+x^2\right )}-\frac {2 (75+128 x)}{7 \left (2+4 x+x^2\right )}+\frac {4}{35} \log (5+x)-\frac {\left (2-\sqrt {2}\right ) \log (5+x)}{20 \left (2-\sqrt {2}+x\right )}-\frac {\left (2+\sqrt {2}\right ) \log (5+x)}{20 \left (2+\sqrt {2}+x\right )}-\frac {1}{49} \left (1244-879 \sqrt {2}\right ) \log \left (2-\sqrt {2}+x\right )+\frac {13}{196} \left (1152-811 \sqrt {2}\right ) \log \left (2-\sqrt {2}+x\right )-\frac {43}{70} \left (125-86 \sqrt {2}\right ) \log \left (2-\sqrt {2}+x\right )+\frac {117}{392} \left (100-59 \sqrt {2}\right ) \log \left (2-\sqrt {2}+x\right )-\frac {213}{490} \left (10-\sqrt {2}\right ) \log \left (2-\sqrt {2}+x\right )+\frac {51 \left (8+9 \sqrt {2}\right ) \log \left (2-\sqrt {2}+x\right )}{1960}+\frac {51 \left (8-9 \sqrt {2}\right ) \log \left (2+\sqrt {2}+x\right )}{1960}-\frac {213}{490} \left (10+\sqrt {2}\right ) \log \left (2+\sqrt {2}+x\right )+\frac {117}{392} \left (100+59 \sqrt {2}\right ) \log \left (2+\sqrt {2}+x\right )-\frac {43}{70} \left (125+86 \sqrt {2}\right ) \log \left (2+\sqrt {2}+x\right )+\frac {13}{196} \left (1152+811 \sqrt {2}\right ) \log \left (2+\sqrt {2}+x\right )-\frac {1}{49} \left (1244+879 \sqrt {2}\right ) \log \left (2+\sqrt {2}+x\right )-\frac {1}{10} \left (2-\sqrt {2}\right ) \int \frac {1}{\left (-4+2 \sqrt {2}-2 x\right ) (5+x)} \, dx+\frac {1}{10} \left (2+\sqrt {2}\right ) \int \frac {1}{(5+x) \left (4+2 \sqrt {2}+2 x\right )} \, dx\\ &=-x+\frac {51 (4+3 x)}{70 \left (2+4 x+x^2\right )}-\frac {213 (3+4 x)}{35 \left (2+4 x+x^2\right )}+\frac {117 (8+13 x)}{14 \left (2+4 x+x^2\right )}-\frac {43 (13+22 x)}{5 \left (2+4 x+x^2\right )}+\frac {13 (44+75 x)}{7 \left (2+4 x+x^2\right )}-\frac {2 (75+128 x)}{7 \left (2+4 x+x^2\right )}+\frac {4}{35} \log (5+x)-\frac {\left (2-\sqrt {2}\right ) \log (5+x)}{20 \left (2-\sqrt {2}+x\right )}-\frac {\left (2+\sqrt {2}\right ) \log (5+x)}{20 \left (2+\sqrt {2}+x\right )}-\frac {1}{49} \left (1244-879 \sqrt {2}\right ) \log \left (2-\sqrt {2}+x\right )+\frac {13}{196} \left (1152-811 \sqrt {2}\right ) \log \left (2-\sqrt {2}+x\right )-\frac {43}{70} \left (125-86 \sqrt {2}\right ) \log \left (2-\sqrt {2}+x\right )+\frac {117}{392} \left (100-59 \sqrt {2}\right ) \log \left (2-\sqrt {2}+x\right )-\frac {213}{490} \left (10-\sqrt {2}\right ) \log \left (2-\sqrt {2}+x\right )+\frac {51 \left (8+9 \sqrt {2}\right ) \log \left (2-\sqrt {2}+x\right )}{1960}+\frac {51 \left (8-9 \sqrt {2}\right ) \log \left (2+\sqrt {2}+x\right )}{1960}-\frac {213}{490} \left (10+\sqrt {2}\right ) \log \left (2+\sqrt {2}+x\right )+\frac {117}{392} \left (100+59 \sqrt {2}\right ) \log \left (2+\sqrt {2}+x\right )-\frac {43}{70} \left (125+86 \sqrt {2}\right ) \log \left (2+\sqrt {2}+x\right )+\frac {13}{196} \left (1152+811 \sqrt {2}\right ) \log \left (2+\sqrt {2}+x\right )-\frac {1}{49} \left (1244+879 \sqrt {2}\right ) \log \left (2+\sqrt {2}+x\right )-\frac {\left (2-\sqrt {2}\right ) \int \frac {1}{5+x} \, dx}{20 \left (3+\sqrt {2}\right )}-\frac {\left (2-\sqrt {2}\right ) \int \frac {1}{-4+2 \sqrt {2}-2 x} \, dx}{10 \left (3+\sqrt {2}\right )}--\frac {\left (-2-\sqrt {2}\right ) \int \frac {1}{4+2 \sqrt {2}+2 x} \, dx}{5 \left (-6+2 \sqrt {2}\right )}+\frac {\left (2+\sqrt {2}\right ) \int \frac {1}{5+x} \, dx}{10 \left (-6+2 \sqrt {2}\right )}\\ &=-x+\frac {51 (4+3 x)}{70 \left (2+4 x+x^2\right )}-\frac {213 (3+4 x)}{35 \left (2+4 x+x^2\right )}+\frac {117 (8+13 x)}{14 \left (2+4 x+x^2\right )}-\frac {43 (13+22 x)}{5 \left (2+4 x+x^2\right )}+\frac {13 (44+75 x)}{7 \left (2+4 x+x^2\right )}-\frac {2 (75+128 x)}{7 \left (2+4 x+x^2\right )}+\frac {4}{35} \log (5+x)+\frac {\left (1+\sqrt {2}\right ) \log (5+x)}{10 \left (2-3 \sqrt {2}\right )}-\frac {\left (2-\sqrt {2}\right ) \log (5+x)}{20 \left (3+\sqrt {2}\right )}-\frac {\left (2-\sqrt {2}\right ) \log (5+x)}{20 \left (2-\sqrt {2}+x\right )}-\frac {\left (2+\sqrt {2}\right ) \log (5+x)}{20 \left (2+\sqrt {2}+x\right )}-\frac {1}{49} \left (1244-879 \sqrt {2}\right ) \log \left (2-\sqrt {2}+x\right )+\frac {13}{196} \left (1152-811 \sqrt {2}\right ) \log \left (2-\sqrt {2}+x\right )-\frac {43}{70} \left (125-86 \sqrt {2}\right ) \log \left (2-\sqrt {2}+x\right )+\frac {117}{392} \left (100-59 \sqrt {2}\right ) \log \left (2-\sqrt {2}+x\right )-\frac {213}{490} \left (10-\sqrt {2}\right ) \log \left (2-\sqrt {2}+x\right )+\frac {\left (2-\sqrt {2}\right ) \log \left (2-\sqrt {2}+x\right )}{20 \left (3+\sqrt {2}\right )}+\frac {51 \left (8+9 \sqrt {2}\right ) \log \left (2-\sqrt {2}+x\right )}{1960}+\frac {51 \left (8-9 \sqrt {2}\right ) \log \left (2+\sqrt {2}+x\right )}{1960}-\frac {\left (1+\sqrt {2}\right ) \log \left (2+\sqrt {2}+x\right )}{10 \left (2-3 \sqrt {2}\right )}-\frac {213}{490} \left (10+\sqrt {2}\right ) \log \left (2+\sqrt {2}+x\right )+\frac {117}{392} \left (100+59 \sqrt {2}\right ) \log \left (2+\sqrt {2}+x\right )-\frac {43}{70} \left (125+86 \sqrt {2}\right ) \log \left (2+\sqrt {2}+x\right )+\frac {13}{196} \left (1152+811 \sqrt {2}\right ) \log \left (2+\sqrt {2}+x\right )-\frac {1}{49} \left (1244+879 \sqrt {2}\right ) \log \left (2+\sqrt {2}+x\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.52, size = 29, normalized size = 1.16 \begin {gather*} \frac {1}{5} \left (-5 (5+x)-\frac {(1+x) \log (5+x)}{2+4 x+x^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-102 - 426*x - 585*x^2 - 301*x^3 - 65*x^4 - 5*x^5 + (10 + 12*x + 7*x^2 + x^3)*Log[5 + x])/(100 + 42
0*x + 580*x^2 + 300*x^3 + 65*x^4 + 5*x^5),x]

[Out]

(-5*(5 + x) - ((1 + x)*Log[5 + x])/(2 + 4*x + x^2))/5

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fricas [A]  time = 0.72, size = 34, normalized size = 1.36 \begin {gather*} -\frac {5 \, x^{3} + 20 \, x^{2} + {\left (x + 1\right )} \log \left (x + 5\right ) + 10 \, x}{5 \, {\left (x^{2} + 4 \, x + 2\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^3+7*x^2+12*x+10)*log(5+x)-5*x^5-65*x^4-301*x^3-585*x^2-426*x-102)/(5*x^5+65*x^4+300*x^3+580*x^2+
420*x+100),x, algorithm="fricas")

[Out]

-1/5*(5*x^3 + 20*x^2 + (x + 1)*log(x + 5) + 10*x)/(x^2 + 4*x + 2)

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giac [A]  time = 0.18, size = 23, normalized size = 0.92 \begin {gather*} -x - \frac {{\left (x + 1\right )} \log \left (x + 5\right )}{5 \, {\left (x^{2} + 4 \, x + 2\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^3+7*x^2+12*x+10)*log(5+x)-5*x^5-65*x^4-301*x^3-585*x^2-426*x-102)/(5*x^5+65*x^4+300*x^3+580*x^2+
420*x+100),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.10, size = 24, normalized size = 0.96

method	result	size
risch	\(-\frac {\left (x +1\right ) \ln \left (5+x \right )}{5 \left (x^{2}+4 x +2\right )}-x\)	\(24\)
norman	\(\frac {-\frac {7 x^{2}}{2}-\frac {\ln \left (5+x \right )}{10}+\frac {\ln \left (5+x \right ) x^{2}}{20}-x^{3}+1}{x^{2}+4 x +2}-\frac {\ln \left (5+x \right )}{20}\)	\(46\)
derivativedivides	\(-\frac {\ln \left (5+x \right ) \left (7 \sqrt {2}\, \ln \left (\frac {-2+\sqrt {2}-x}{3+\sqrt {2}}\right ) \left (5+x \right )^{2}-7 \sqrt {2}\, \ln \left (\frac {2+\sqrt {2}+x}{-3+\sqrt {2}}\right ) \left (5+x \right )^{2}-42 \sqrt {2}\, \ln \left (\frac {-2+\sqrt {2}-x}{3+\sqrt {2}}\right ) \left (5+x \right )+42 \sqrt {2}\, \ln \left (\frac {2+\sqrt {2}+x}{-3+\sqrt {2}}\right ) \left (5+x \right )+49 \sqrt {2}\, \ln \left (\frac {-2+\sqrt {2}-x}{3+\sqrt {2}}\right )-49 \sqrt {2}\, \ln \left (\frac {2+\sqrt {2}+x}{-3+\sqrt {2}}\right )+16 \left (5+x \right )^{2}-340-68 x \right )}{140 \left (\left (5+x \right )^{2}-23-6 x \right )}+\frac {\sqrt {2}\, \ln \left (5+x \right ) \ln \left (\frac {-2+\sqrt {2}-x}{3+\sqrt {2}}\right )}{20}-\frac {\sqrt {2}\, \ln \left (5+x \right ) \ln \left (\frac {2+\sqrt {2}+x}{-3+\sqrt {2}}\right )}{20}-5-x +\frac {4 \ln \left (5+x \right )}{35}\)	\(234\)
default	\(-\frac {\ln \left (5+x \right ) \left (7 \sqrt {2}\, \ln \left (\frac {-2+\sqrt {2}-x}{3+\sqrt {2}}\right ) \left (5+x \right )^{2}-7 \sqrt {2}\, \ln \left (\frac {2+\sqrt {2}+x}{-3+\sqrt {2}}\right ) \left (5+x \right )^{2}-42 \sqrt {2}\, \ln \left (\frac {-2+\sqrt {2}-x}{3+\sqrt {2}}\right ) \left (5+x \right )+42 \sqrt {2}\, \ln \left (\frac {2+\sqrt {2}+x}{-3+\sqrt {2}}\right ) \left (5+x \right )+49 \sqrt {2}\, \ln \left (\frac {-2+\sqrt {2}-x}{3+\sqrt {2}}\right )-49 \sqrt {2}\, \ln \left (\frac {2+\sqrt {2}+x}{-3+\sqrt {2}}\right )+16 \left (5+x \right )^{2}-340-68 x \right )}{140 \left (\left (5+x \right )^{2}-23-6 x \right )}+\frac {\sqrt {2}\, \ln \left (5+x \right ) \ln \left (\frac {-2+\sqrt {2}-x}{3+\sqrt {2}}\right )}{20}-\frac {\sqrt {2}\, \ln \left (5+x \right ) \ln \left (\frac {2+\sqrt {2}+x}{-3+\sqrt {2}}\right )}{20}-5-x +\frac {4 \ln \left (5+x \right )}{35}\)	\(234\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^3+7*x^2+12*x+10)*ln(5+x)-5*x^5-65*x^4-301*x^3-585*x^2-426*x-102)/(5*x^5+65*x^4+300*x^3+580*x^2+420*x+1
00),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [B]  time = 0.53, size = 138, normalized size = 5.52 \begin {gather*} -x - \frac {{\left (4 \, x^{2} + 23 \, x + 15\right )} \log \left (x + 5\right )}{35 \, {\left (x^{2} + 4 \, x + 2\right )}} - \frac {2 \, {\left (128 \, x + 75\right )}}{7 \, {\left (x^{2} + 4 \, x + 2\right )}} + \frac {13 \, {\left (75 \, x + 44\right )}}{7 \, {\left (x^{2} + 4 \, x + 2\right )}} - \frac {43 \, {\left (22 \, x + 13\right )}}{5 \, {\left (x^{2} + 4 \, x + 2\right )}} + \frac {117 \, {\left (13 \, x + 8\right )}}{14 \, {\left (x^{2} + 4 \, x + 2\right )}} - \frac {213 \, {\left (4 \, x + 3\right )}}{35 \, {\left (x^{2} + 4 \, x + 2\right )}} + \frac {51 \, {\left (3 \, x + 4\right )}}{70 \, {\left (x^{2} + 4 \, x + 2\right )}} + \frac {4}{35} \, \log \left (x + 5\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^3+7*x^2+12*x+10)*log(5+x)-5*x^5-65*x^4-301*x^3-585*x^2-426*x-102)/(5*x^5+65*x^4+300*x^3+580*x^2+
420*x+100),x, algorithm="maxima")

[Out]

-x - 1/35*(4*x^2 + 23*x + 15)*log(x + 5)/(x^2 + 4*x + 2) - 2/7*(128*x + 75)/(x^2 + 4*x + 2) + 13/7*(75*x + 44)
/(x^2 + 4*x + 2) - 43/5*(22*x + 13)/(x^2 + 4*x + 2) + 117/14*(13*x + 8)/(x^2 + 4*x + 2) - 213/35*(4*x + 3)/(x^
2 + 4*x + 2) + 51/70*(3*x + 4)/(x^2 + 4*x + 2) + 4/35*log(x + 5)

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mupad [B]  time = 4.18, size = 25, normalized size = 1.00 \begin {gather*} -x-\frac {\ln \left (x+5\right )\,\left (\frac {x}{5}+\frac {1}{5}\right )}{x^2+4\,x+2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(426*x + 585*x^2 + 301*x^3 + 65*x^4 + 5*x^5 - log(x + 5)*(12*x + 7*x^2 + x^3 + 10) + 102)/(420*x + 580*x^
2 + 300*x^3 + 65*x^4 + 5*x^5 + 100),x)

[Out]

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

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sympy [A]  time = 0.18, size = 20, normalized size = 0.80 \begin {gather*} - x + \frac {\left (- x - 1\right ) \log {\left (x + 5 \right )}}{5 x^{2} + 20 x + 10} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x**3+7*x**2+12*x+10)*ln(5+x)-5*x**5-65*x**4-301*x**3-585*x**2-426*x-102)/(5*x**5+65*x**4+300*x**3+
580*x**2+420*x+100),x)

[Out]

-x + (-x - 1)*log(x + 5)/(5*x**2 + 20*x + 10)

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