3.46.80 \(\int \frac {e (50 x+4 x^2) \log (3+25 x+x^2)+e (-3-25 x-x^2) \log ^2(3+25 x+x^2)}{3 x^2+25 x^3+x^4} \, dx\)

Optimal. Leaf size=26 \[ i \pi +\frac {e \log ^2(3+x (25+x))}{x}+\log (3+\log (2)) \]

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Rubi [A]  time = 1.63, antiderivative size = 16, normalized size of antiderivative = 0.62, number of steps used = 60, number of rules used = 19, integrand size = 61, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {1594, 6688, 12, 14, 800, 632, 31, 2525, 6742, 2528, 2524, 2357, 2317, 2391, 2418, 2390, 2301, 2394, 2393} \begin {gather*} \frac {e \log ^2\left (x^2+25 x+3\right )}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E*(50*x + 4*x^2)*Log[3 + 25*x + x^2] + E*(-3 - 25*x - x^2)*Log[3 + 25*x + x^2]^2)/(3*x^2 + 25*x^3 + x^4),
x]

[Out]

(E*Log[3 + 25*x + x^2]^2)/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 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 31

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

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 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 1594

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.) + (c_.)*(x_)^(r_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^
(q - p) + c*x^(r - p))^n, x] /; FreeQ[{a, b, c, p, q, r}, x] && IntegerQ[n] && PosQ[q - p] && PosQ[r - p]

Rule 2301

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

Rule 2317

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

Rule 2357

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(RFx_), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*Log[c*x^
n])^p, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, n}, x] && RationalFunctionQ[RFx, x] && IGtQ[p, 0]

Rule 2390

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_) + (g_.)*(x_))^(q_.), x_Symbol] :> Dist[1/
e, Subst[Int[((f*x)/d)^q*(a + b*Log[c*x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p, q}, x]
&& EqQ[e*f - d*g, 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 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 2524

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[(Log[d + e*x]*(a + b
*Log[c*RFx^p])^n)/e, x] - Dist[(b*n*p)/e, Int[(Log[d + e*x]*(a + b*Log[c*RFx^p])^(n - 1)*D[RFx, x])/RFx, x], x
] /; FreeQ[{a, b, c, d, e, p}, x] && RationalFunctionQ[RFx, x] && IGtQ[n, 0]

Rule 2525

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)*((d_.) + (e_.)*(x_))^(m_.), x_Symbol] :> Simp[((d + e*x)^(m
+ 1)*(a + b*Log[c*RFx^p])^n)/(e*(m + 1)), x] - Dist[(b*n*p)/(e*(m + 1)), Int[SimplifyIntegrand[((d + e*x)^(m +
1)*(a + b*Log[c*RFx^p])^(n - 1)*D[RFx, x])/RFx, x], x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && RationalFunc
tionQ[RFx, x] && IGtQ[n, 0] && (EqQ[n, 1] || IntegerQ[m]) && NeQ[m, -1]

Rule 2528

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)*(RGx_), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*Log[c*
RFx^p])^n, RGx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, p}, x] && RationalFunctionQ[RFx, x] && RationalF
unctionQ[RGx, x] && IGtQ[n, 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 {e \left (50 x+4 x^2\right ) \log \left (3+25 x+x^2\right )+e \left (-3-25 x-x^2\right ) \log ^2\left (3+25 x+x^2\right )}{x^2 \left (3+25 x+x^2\right )} \, dx\\ &=\int \frac {e \left (\frac {2 x (25+2 x)}{3+25 x+x^2}-\log \left (3+25 x+x^2\right )\right ) \log \left (3+25 x+x^2\right )}{x^2} \, dx\\ &=e \int \frac {\left (\frac {2 x (25+2 x)}{3+25 x+x^2}-\log \left (3+25 x+x^2\right )\right ) \log \left (3+25 x+x^2\right )}{x^2} \, dx\\ &=e \int \left (\frac {2 (25+2 x) \log \left (3+25 x+x^2\right )}{x \left (3+25 x+x^2\right )}-\frac {\log ^2\left (3+25 x+x^2\right )}{x^2}\right ) \, dx\\ &=-\left (e \int \frac {\log ^2\left (3+25 x+x^2\right )}{x^2} \, dx\right )+(2 e) \int \frac {(25+2 x) \log \left (3+25 x+x^2\right )}{x \left (3+25 x+x^2\right )} \, dx\\ &=\frac {e \log ^2\left (3+25 x+x^2\right )}{x}-(2 e) \int \frac {(25+2 x) \log \left (3+25 x+x^2\right )}{x \left (3+25 x+x^2\right )} \, dx+(2 e) \int \left (\frac {25 \log \left (3+25 x+x^2\right )}{3 x}+\frac {(-619-25 x) \log \left (3+25 x+x^2\right )}{3 \left (3+25 x+x^2\right )}\right ) \, dx\\ &=\frac {e \log ^2\left (3+25 x+x^2\right )}{x}+\frac {1}{3} (2 e) \int \frac {(-619-25 x) \log \left (3+25 x+x^2\right )}{3+25 x+x^2} \, dx-(2 e) \int \left (\frac {25 \log \left (3+25 x+x^2\right )}{3 x}+\frac {(-619-25 x) \log \left (3+25 x+x^2\right )}{3 \left (3+25 x+x^2\right )}\right ) \, dx+\frac {1}{3} (50 e) \int \frac {\log \left (3+25 x+x^2\right )}{x} \, dx\\ &=\frac {50}{3} e \log (x) \log \left (3+25 x+x^2\right )+\frac {e \log ^2\left (3+25 x+x^2\right )}{x}-\frac {1}{3} (2 e) \int \frac {(-619-25 x) \log \left (3+25 x+x^2\right )}{3+25 x+x^2} \, dx+\frac {1}{3} (2 e) \int \left (\frac {\left (-25-\sqrt {613}\right ) \log \left (3+25 x+x^2\right )}{25-\sqrt {613}+2 x}+\frac {\left (-25+\sqrt {613}\right ) \log \left (3+25 x+x^2\right )}{25+\sqrt {613}+2 x}\right ) \, dx-\frac {1}{3} (50 e) \int \frac {(25+2 x) \log (x)}{3+25 x+x^2} \, dx-\frac {1}{3} (50 e) \int \frac {\log \left (3+25 x+x^2\right )}{x} \, dx\\ &=\frac {e \log ^2\left (3+25 x+x^2\right )}{x}-\frac {1}{3} (2 e) \int \left (\frac {\left (-25-\sqrt {613}\right ) \log \left (3+25 x+x^2\right )}{25-\sqrt {613}+2 x}+\frac {\left (-25+\sqrt {613}\right ) \log \left (3+25 x+x^2\right )}{25+\sqrt {613}+2 x}\right ) \, dx+\frac {1}{3} (50 e) \int \frac {(25+2 x) \log (x)}{3+25 x+x^2} \, dx-\frac {1}{3} (50 e) \int \left (\frac {2 \log (x)}{25-\sqrt {613}+2 x}+\frac {2 \log (x)}{25+\sqrt {613}+2 x}\right ) \, dx-\frac {1}{3} \left (2 \left (25-\sqrt {613}\right ) e\right ) \int \frac {\log \left (3+25 x+x^2\right )}{25+\sqrt {613}+2 x} \, dx-\frac {1}{3} \left (2 \left (25+\sqrt {613}\right ) e\right ) \int \frac {\log \left (3+25 x+x^2\right )}{25-\sqrt {613}+2 x} \, dx\\ &=-\frac {1}{3} \left (25+\sqrt {613}\right ) e \log \left (25-\sqrt {613}+2 x\right ) \log \left (3+25 x+x^2\right )-\frac {1}{3} \left (25-\sqrt {613}\right ) e \log \left (25+\sqrt {613}+2 x\right ) \log \left (3+25 x+x^2\right )+\frac {e \log ^2\left (3+25 x+x^2\right )}{x}+\frac {1}{3} (50 e) \int \left (\frac {2 \log (x)}{25-\sqrt {613}+2 x}+\frac {2 \log (x)}{25+\sqrt {613}+2 x}\right ) \, dx-\frac {1}{3} (100 e) \int \frac {\log (x)}{25-\sqrt {613}+2 x} \, dx-\frac {1}{3} (100 e) \int \frac {\log (x)}{25+\sqrt {613}+2 x} \, dx+\frac {1}{3} \left (\left (25-\sqrt {613}\right ) e\right ) \int \frac {(25+2 x) \log \left (25+\sqrt {613}+2 x\right )}{3+25 x+x^2} \, dx+\frac {1}{3} \left (2 \left (25-\sqrt {613}\right ) e\right ) \int \frac {\log \left (3+25 x+x^2\right )}{25+\sqrt {613}+2 x} \, dx+\frac {1}{3} \left (\left (25+\sqrt {613}\right ) e\right ) \int \frac {(25+2 x) \log \left (25-\sqrt {613}+2 x\right )}{3+25 x+x^2} \, dx+\frac {1}{3} \left (2 \left (25+\sqrt {613}\right ) e\right ) \int \frac {\log \left (3+25 x+x^2\right )}{25-\sqrt {613}+2 x} \, dx\\ &=-\frac {50}{3} e \log (x) \log \left (1+\frac {2 x}{25-\sqrt {613}}\right )-\frac {50}{3} e \log (x) \log \left (1+\frac {2 x}{25+\sqrt {613}}\right )+\frac {e \log ^2\left (3+25 x+x^2\right )}{x}+\frac {1}{3} (50 e) \int \frac {\log \left (1+\frac {2 x}{25-\sqrt {613}}\right )}{x} \, dx+\frac {1}{3} (50 e) \int \frac {\log \left (1+\frac {2 x}{25+\sqrt {613}}\right )}{x} \, dx+\frac {1}{3} (100 e) \int \frac {\log (x)}{25-\sqrt {613}+2 x} \, dx+\frac {1}{3} (100 e) \int \frac {\log (x)}{25+\sqrt {613}+2 x} \, dx-\frac {1}{3} \left (\left (25-\sqrt {613}\right ) e\right ) \int \frac {(25+2 x) \log \left (25+\sqrt {613}+2 x\right )}{3+25 x+x^2} \, dx+\frac {1}{3} \left (\left (25-\sqrt {613}\right ) e\right ) \int \left (\frac {2 \log \left (25+\sqrt {613}+2 x\right )}{25-\sqrt {613}+2 x}+\frac {2 \log \left (25+\sqrt {613}+2 x\right )}{25+\sqrt {613}+2 x}\right ) \, dx-\frac {1}{3} \left (\left (25+\sqrt {613}\right ) e\right ) \int \frac {(25+2 x) \log \left (25-\sqrt {613}+2 x\right )}{3+25 x+x^2} \, dx+\frac {1}{3} \left (\left (25+\sqrt {613}\right ) e\right ) \int \left (\frac {2 \log \left (25-\sqrt {613}+2 x\right )}{25-\sqrt {613}+2 x}+\frac {2 \log \left (25-\sqrt {613}+2 x\right )}{25+\sqrt {613}+2 x}\right ) \, dx\\ &=\frac {e \log ^2\left (3+25 x+x^2\right )}{x}-\frac {50}{3} e \text {Li}_2\left (-\frac {2 x}{25-\sqrt {613}}\right )-\frac {50}{3} e \text {Li}_2\left (-\frac {2 x}{25+\sqrt {613}}\right )-\frac {1}{3} (50 e) \int \frac {\log \left (1+\frac {2 x}{25-\sqrt {613}}\right )}{x} \, dx-\frac {1}{3} (50 e) \int \frac {\log \left (1+\frac {2 x}{25+\sqrt {613}}\right )}{x} \, dx-\frac {1}{3} \left (\left (25-\sqrt {613}\right ) e\right ) \int \left (\frac {2 \log \left (25+\sqrt {613}+2 x\right )}{25-\sqrt {613}+2 x}+\frac {2 \log \left (25+\sqrt {613}+2 x\right )}{25+\sqrt {613}+2 x}\right ) \, dx+\frac {1}{3} \left (2 \left (25-\sqrt {613}\right ) e\right ) \int \frac {\log \left (25+\sqrt {613}+2 x\right )}{25-\sqrt {613}+2 x} \, dx+\frac {1}{3} \left (2 \left (25-\sqrt {613}\right ) e\right ) \int \frac {\log \left (25+\sqrt {613}+2 x\right )}{25+\sqrt {613}+2 x} \, dx-\frac {1}{3} \left (\left (25+\sqrt {613}\right ) e\right ) \int \left (\frac {2 \log \left (25-\sqrt {613}+2 x\right )}{25-\sqrt {613}+2 x}+\frac {2 \log \left (25-\sqrt {613}+2 x\right )}{25+\sqrt {613}+2 x}\right ) \, dx+\frac {1}{3} \left (2 \left (25+\sqrt {613}\right ) e\right ) \int \frac {\log \left (25-\sqrt {613}+2 x\right )}{25-\sqrt {613}+2 x} \, dx+\frac {1}{3} \left (2 \left (25+\sqrt {613}\right ) e\right ) \int \frac {\log \left (25-\sqrt {613}+2 x\right )}{25+\sqrt {613}+2 x} \, dx\\ &=\frac {1}{3} \left (25-\sqrt {613}\right ) e \log \left (-\frac {25-\sqrt {613}+2 x}{2 \sqrt {613}}\right ) \log \left (25+\sqrt {613}+2 x\right )+\frac {1}{3} \left (25+\sqrt {613}\right ) e \log \left (25-\sqrt {613}+2 x\right ) \log \left (\frac {25+\sqrt {613}+2 x}{2 \sqrt {613}}\right )+\frac {e \log ^2\left (3+25 x+x^2\right )}{x}+\frac {1}{3} \left (\left (25-\sqrt {613}\right ) e\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,25+\sqrt {613}+2 x\right )-\frac {1}{3} \left (2 \left (25-\sqrt {613}\right ) e\right ) \int \frac {\log \left (\frac {2 \left (25-\sqrt {613}+2 x\right )}{2 \left (25-\sqrt {613}\right )-2 \left (25+\sqrt {613}\right )}\right )}{25+\sqrt {613}+2 x} \, dx-\frac {1}{3} \left (2 \left (25-\sqrt {613}\right ) e\right ) \int \frac {\log \left (25+\sqrt {613}+2 x\right )}{25-\sqrt {613}+2 x} \, dx-\frac {1}{3} \left (2 \left (25-\sqrt {613}\right ) e\right ) \int \frac {\log \left (25+\sqrt {613}+2 x\right )}{25+\sqrt {613}+2 x} \, dx+\frac {1}{3} \left (\left (25+\sqrt {613}\right ) e\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,25-\sqrt {613}+2 x\right )-\frac {1}{3} \left (2 \left (25+\sqrt {613}\right ) e\right ) \int \frac {\log \left (25-\sqrt {613}+2 x\right )}{25-\sqrt {613}+2 x} \, dx-\frac {1}{3} \left (2 \left (25+\sqrt {613}\right ) e\right ) \int \frac {\log \left (25-\sqrt {613}+2 x\right )}{25+\sqrt {613}+2 x} \, dx-\frac {1}{3} \left (2 \left (25+\sqrt {613}\right ) e\right ) \int \frac {\log \left (\frac {2 \left (25+\sqrt {613}+2 x\right )}{-2 \left (25-\sqrt {613}\right )+2 \left (25+\sqrt {613}\right )}\right )}{25-\sqrt {613}+2 x} \, dx\\ &=\frac {1}{6} \left (25+\sqrt {613}\right ) e \log ^2\left (25-\sqrt {613}+2 x\right )+\frac {1}{6} \left (25-\sqrt {613}\right ) e \log ^2\left (25+\sqrt {613}+2 x\right )+\frac {e \log ^2\left (3+25 x+x^2\right )}{x}-\frac {1}{3} \left (\left (25-\sqrt {613}\right ) e\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,25+\sqrt {613}+2 x\right )-\frac {1}{3} \left (\left (25-\sqrt {613}\right ) e\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {2 x}{2 \left (25-\sqrt {613}\right )-2 \left (25+\sqrt {613}\right )}\right )}{x} \, dx,x,25+\sqrt {613}+2 x\right )+\frac {1}{3} \left (2 \left (25-\sqrt {613}\right ) e\right ) \int \frac {\log \left (\frac {2 \left (25-\sqrt {613}+2 x\right )}{2 \left (25-\sqrt {613}\right )-2 \left (25+\sqrt {613}\right )}\right )}{25+\sqrt {613}+2 x} \, dx-\frac {1}{3} \left (\left (25+\sqrt {613}\right ) e\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,25-\sqrt {613}+2 x\right )-\frac {1}{3} \left (\left (25+\sqrt {613}\right ) e\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {2 x}{-2 \left (25-\sqrt {613}\right )+2 \left (25+\sqrt {613}\right )}\right )}{x} \, dx,x,25-\sqrt {613}+2 x\right )+\frac {1}{3} \left (2 \left (25+\sqrt {613}\right ) e\right ) \int \frac {\log \left (\frac {2 \left (25+\sqrt {613}+2 x\right )}{-2 \left (25-\sqrt {613}\right )+2 \left (25+\sqrt {613}\right )}\right )}{25-\sqrt {613}+2 x} \, dx\\ &=\frac {e \log ^2\left (3+25 x+x^2\right )}{x}+\frac {1}{3} \left (25+\sqrt {613}\right ) e \text {Li}_2\left (-\frac {25-\sqrt {613}+2 x}{2 \sqrt {613}}\right )+\frac {1}{3} \left (25-\sqrt {613}\right ) e \text {Li}_2\left (\frac {25+\sqrt {613}+2 x}{2 \sqrt {613}}\right )+\frac {1}{3} \left (\left (25-\sqrt {613}\right ) e\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {2 x}{2 \left (25-\sqrt {613}\right )-2 \left (25+\sqrt {613}\right )}\right )}{x} \, dx,x,25+\sqrt {613}+2 x\right )+\frac {1}{3} \left (\left (25+\sqrt {613}\right ) e\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {2 x}{-2 \left (25-\sqrt {613}\right )+2 \left (25+\sqrt {613}\right )}\right )}{x} \, dx,x,25-\sqrt {613}+2 x\right )\\ &=\frac {e \log ^2\left (3+25 x+x^2\right )}{x}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.25, size = 16, normalized size = 0.62 \begin {gather*} \frac {e \log ^2\left (3+25 x+x^2\right )}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E*(50*x + 4*x^2)*Log[3 + 25*x + x^2] + E*(-3 - 25*x - x^2)*Log[3 + 25*x + x^2]^2)/(3*x^2 + 25*x^3 +
x^4),x]

[Out]

(E*Log[3 + 25*x + x^2]^2)/x

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fricas [A]  time = 0.65, size = 17, normalized size = 0.65 \begin {gather*} \frac {e \log \left (x^{2} + 25 \, x + 3\right )^{2}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^2-25*x-3)*exp(1)*log(x^2+25*x+3)^2+(4*x^2+50*x)*exp(1)*log(x^2+25*x+3))/(x^4+25*x^3+3*x^2),x, a
lgorithm="fricas")

[Out]

e*log(x^2 + 25*x + 3)^2/x

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giac [A]  time = 0.24, size = 17, normalized size = 0.65 \begin {gather*} \frac {e \log \left (x^{2} + 25 \, x + 3\right )^{2}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^2-25*x-3)*exp(1)*log(x^2+25*x+3)^2+(4*x^2+50*x)*exp(1)*log(x^2+25*x+3))/(x^4+25*x^3+3*x^2),x, a
lgorithm="giac")

[Out]

e*log(x^2 + 25*x + 3)^2/x

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maple [A]  time = 0.07, size = 18, normalized size = 0.69

method	result	size
norman	\(\frac {{\mathrm e} \ln \left (x^{2}+25 x +3\right )^{2}}{x}\)	\(18\)
risch	\(\frac {{\mathrm e} \ln \left (x^{2}+25 x +3\right )^{2}}{x}\)	\(18\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-x^2-25*x-3)*exp(1)*ln(x^2+25*x+3)^2+(4*x^2+50*x)*exp(1)*ln(x^2+25*x+3))/(x^4+25*x^3+3*x^2),x,method=_RE
TURNVERBOSE)

[Out]

exp(1)*ln(x^2+25*x+3)^2/x

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maxima [A]  time = 0.50, size = 17, normalized size = 0.65 \begin {gather*} \frac {e \log \left (x^{2} + 25 \, x + 3\right )^{2}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^2-25*x-3)*exp(1)*log(x^2+25*x+3)^2+(4*x^2+50*x)*exp(1)*log(x^2+25*x+3))/(x^4+25*x^3+3*x^2),x, a
lgorithm="maxima")

[Out]

e*log(x^2 + 25*x + 3)^2/x

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mupad [B]  time = 0.48, size = 17, normalized size = 0.65 \begin {gather*} \frac {\mathrm {e}\,{\ln \left (x^2+25\,x+3\right )}^2}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(1)*log(25*x + x^2 + 3)*(50*x + 4*x^2) - exp(1)*log(25*x + x^2 + 3)^2*(25*x + x^2 + 3))/(3*x^2 + 25*x^
3 + x^4),x)

[Out]

(exp(1)*log(25*x + x^2 + 3)^2)/x

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sympy [A]  time = 0.17, size = 15, normalized size = 0.58 \begin {gather*} \frac {e \log {\left (x^{2} + 25 x + 3 \right )}^{2}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x**2-25*x-3)*exp(1)*ln(x**2+25*x+3)**2+(4*x**2+50*x)*exp(1)*ln(x**2+25*x+3))/(x**4+25*x**3+3*x**2
),x)

[Out]

E*log(x**2 + 25*x + 3)**2/x

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