3.40.71 \(\int \frac {-10 x^7 \log (2)+(45 x^4+5 x^9) \log ^2(2)+(10 x^5+(-54 x^2-6 x^7) \log (2)) \log (9+x^5)+(9+x^5) \log ^2(9+x^5)}{(9+x^5) \log ^2(2)} \, dx\)

Optimal. Leaf size=21 \[ x \left (-x^2+\frac {\log \left (9+x^5\right )}{\log (2)}\right )^2 \]

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Rubi [B]  time = 10.72, antiderivative size = 568, normalized size of antiderivative = 27.05, number of steps used = 223, number of rules used = 25, integrand size = 77, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.325, Rules used = {12, 6725, 1836, 321, 293, 634, 618, 204, 628, 31, 2528, 2448, 201, 2471, 2462, 260, 2416, 2394, 2393, 2391, 2390, 2301, 2455, 2450, 2476} \begin {gather*} x^5+\frac {x \log ^2\left (x^5+9\right )}{\log ^2(2)}+\frac {10 x^3 \log (8)}{9 \log ^2(2)}-\frac {10 x^3}{3 \log (2)}+\frac {\sqrt [5]{3} \left (1+\sqrt {5}\right ) \log (8) \log \left (x^2-\frac {1}{2} 3^{2/5} \left (1-\sqrt {5}\right ) x+3^{4/5}\right )}{2 \log ^2(2)}+\frac {\sqrt [5]{3} \left (1-\sqrt {5}\right ) \log (8) \log \left (x^2-\frac {1}{2} 3^{2/5} \left (1+\sqrt {5}\right ) x+3^{4/5}\right )}{2 \log ^2(2)}-\frac {3 \sqrt [5]{3} \left (1+\sqrt {5}\right ) \log \left (x^2-\frac {1}{2} 3^{2/5} \left (1-\sqrt {5}\right ) x+3^{4/5}\right )}{2 \log (2)}-\frac {3 \sqrt [5]{3} \left (1-\sqrt {5}\right ) \log \left (x^2-\frac {1}{2} 3^{2/5} \left (1+\sqrt {5}\right ) x+3^{4/5}\right )}{2 \log (2)}-\frac {2 x^3 \log (8) \log \left (x^5+9\right )}{3 \log ^2(2)}-\frac {2 \sqrt [5]{3} \log (8) \log \left (x+3^{2/5}\right )}{\log ^2(2)}+\frac {6 \sqrt [5]{3} \log \left (x+3^{2/5}\right )}{\log (2)}-\frac {2 \sqrt [5]{3} \sqrt {10} \log (8) \tan ^{-1}\left (\frac {3^{2/5} \left (1-\sqrt {5}\right )-4 x}{3^{2/5} \sqrt {2 \left (5+\sqrt {5}\right )}}\right )}{\sqrt {5+\sqrt {5}} \log ^2(2)}+\frac {\sqrt [5]{3} \sqrt {2 \left (5+\sqrt {5}\right )} \log (8) \tan ^{-1}\left (\frac {\sqrt {\frac {1}{10} \left (5+\sqrt {5}\right )} \left (3^{2/5} \left (1+\sqrt {5}\right )-4 x\right )}{2\ 3^{2/5}}\right )}{\log ^2(2)}+\frac {6 \sqrt [5]{3} \sqrt {10} \tan ^{-1}\left (\frac {3^{2/5} \left (1-\sqrt {5}\right )-4 x}{3^{2/5} \sqrt {2 \left (5+\sqrt {5}\right )}}\right )}{\sqrt {5+\sqrt {5}} \log (2)}-\frac {3 \sqrt [5]{3} \sqrt {2 \left (5+\sqrt {5}\right )} \tan ^{-1}\left (\frac {\sqrt {\frac {1}{10} \left (5+\sqrt {5}\right )} \left (3^{2/5} \left (1+\sqrt {5}\right )-4 x\right )}{2\ 3^{2/5}}\right )}{\log (2)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-10*x^7*Log[2] + (45*x^4 + 5*x^9)*Log[2]^2 + (10*x^5 + (-54*x^2 - 6*x^7)*Log[2])*Log[9 + x^5] + (9 + x^5)
*Log[9 + x^5]^2)/((9 + x^5)*Log[2]^2),x]

[Out]

x^5 - (10*x^3)/(3*Log[2]) + (6*3^(1/5)*Sqrt[10]*ArcTan[(3^(2/5)*(1 - Sqrt[5]) - 4*x)/(3^(2/5)*Sqrt[2*(5 + Sqrt
[5])])])/(Sqrt[5 + Sqrt[5]]*Log[2]) - (3*3^(1/5)*Sqrt[2*(5 + Sqrt[5])]*ArcTan[(Sqrt[(5 + Sqrt[5])/10]*(3^(2/5)
*(1 + Sqrt[5]) - 4*x))/(2*3^(2/5))])/Log[2] + (10*x^3*Log[8])/(9*Log[2]^2) - (2*3^(1/5)*Sqrt[10]*ArcTan[(3^(2/
5)*(1 - Sqrt[5]) - 4*x)/(3^(2/5)*Sqrt[2*(5 + Sqrt[5])])]*Log[8])/(Sqrt[5 + Sqrt[5]]*Log[2]^2) + (3^(1/5)*Sqrt[
2*(5 + Sqrt[5])]*ArcTan[(Sqrt[(5 + Sqrt[5])/10]*(3^(2/5)*(1 + Sqrt[5]) - 4*x))/(2*3^(2/5))]*Log[8])/Log[2]^2 +
(6*3^(1/5)*Log[3^(2/5) + x])/Log[2] - (2*3^(1/5)*Log[8]*Log[3^(2/5) + x])/Log[2]^2 - (3*3^(1/5)*(1 + Sqrt[5])
*Log[3^(4/5) - (3^(2/5)*(1 - Sqrt[5])*x)/2 + x^2])/(2*Log[2]) + (3^(1/5)*(1 + Sqrt[5])*Log[8]*Log[3^(4/5) - (3
^(2/5)*(1 - Sqrt[5])*x)/2 + x^2])/(2*Log[2]^2) - (3*3^(1/5)*(1 - Sqrt[5])*Log[3^(4/5) - (3^(2/5)*(1 + Sqrt[5])
*x)/2 + x^2])/(2*Log[2]) + (3^(1/5)*(1 - Sqrt[5])*Log[8]*Log[3^(4/5) - (3^(2/5)*(1 + Sqrt[5])*x)/2 + x^2])/(2*
Log[2]^2) - (2*x^3*Log[8]*Log[9 + x^5])/(3*Log[2]^2) + (x*Log[9 + x^5]^2)/Log[2]^2

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 201

Int[((a_) + (b_.)*(x_)^(n_))^(-1), x_Symbol] :> Module[{r = Numerator[Rt[a/b, n]], s = Denominator[Rt[a/b, n]]
, k, u}, Simp[u = Int[(r - s*Cos[((2*k - 1)*Pi)/n]*x)/(r^2 - 2*r*s*Cos[((2*k - 1)*Pi)/n]*x + s^2*x^2), x]; (r*
Int[1/(r + s*x), x])/(a*n) + Dist[(2*r)/(a*n), Sum[u, {k, 1, (n - 1)/2}], x], x]] /; FreeQ[{a, b}, x] && IGtQ[
(n - 3)/2, 0] && PosQ[a/b]

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 260

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

Rule 293

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Module[{r = Numerator[Rt[a/b, n]], s = Denominator[Rt[a/
b, n]], k, u}, Simp[u = Int[(r*Cos[((2*k - 1)*m*Pi)/n] - s*Cos[((2*k - 1)*(m + 1)*Pi)/n]*x)/(r^2 - 2*r*s*Cos[(
(2*k - 1)*Pi)/n]*x + s^2*x^2), x]; -(((-r)^(m + 1)*Int[1/(r + s*x), x])/(a*n*s^m)) + Dist[(2*r^(m + 1))/(a*n*s
^m), Sum[u, {k, 1, (n - 1)/2}], x], x]] /; FreeQ[{a, b}, x] && IGtQ[(n - 1)/2, 0] && IGtQ[m, 0] && LtQ[m, n -
1] && PosQ[a/b]

Rule 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 618

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 628

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 634

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 1836

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{q = Expon[Pq, x]}, With[{Pqq =
Coeff[Pq, x, q]}, Dist[1/(b*(m + q + n*p + 1)), Int[(c*x)^m*ExpandToSum[b*(m + q + n*p + 1)*(Pq - Pqq*x^q) - a
*Pqq*(m + q - n + 1)*x^(q - n), x]*(a + b*x^n)^p, x], x] + Simp[(Pqq*(c*x)^(m + q - n + 1)*(a + b*x^n)^(p + 1)
)/(b*c^(q - n + 1)*(m + q + n*p + 1)), x]] /; NeQ[m + q + n*p + 1, 0] && q - n >= 0 && (IntegerQ[2*p] || Integ
erQ[p + (q + 1)/(2*n)])] /; FreeQ[{a, b, c, m, p}, x] && PolyQ[Pq, x] && IGtQ[n, 0]

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 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 2416

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((h_.)*(x_))^(m_.)*((f_) + (g_.)*(x_)^(r_.))^(q
_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*Log[c*(d + e*x)^n])^p, (h*x)^m*(f + g*x^r)^q, x], x] /; FreeQ[{a,
b, c, d, e, f, g, h, m, n, p, q, r}, x] && IntegerQ[m] && IntegerQ[q]

Rule 2448

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

Rule 2450

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

Rule 2455

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

Rule 2462

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

Rule 2471

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*((f_) + (g_.)*(x_)^(s_))^(r_.), x_Symbol]
:> With[{t = ExpandIntegrand[(a + b*Log[c*(d + e*x^n)^p])^q, (f + g*x^s)^r, x]}, Int[t, x] /; SumQ[t]] /; Free
Q[{a, b, c, d, e, f, g, n, p, q, r, s}, x] && IntegerQ[n] && IGtQ[q, 0] && IntegerQ[r] && IntegerQ[s] && (EqQ[
q, 1] || (GtQ[r, 0] && GtQ[s, 1]) || (LtQ[s, 0] && LtQ[r, 0]))

Rule 2476

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m_.)*((f_) + (g_.)*(x_)^(s_))^(r_.),
x_Symbol] :> Int[ExpandIntegrand[(a + b*Log[c*(d + e*x^n)^p])^q, x^m*(f + g*x^s)^r, x], x] /; FreeQ[{a, b, c,
d, e, f, g, m, n, p, q, r, s}, x] && IGtQ[q, 0] && IntegerQ[m] && IntegerQ[r] && IntegerQ[s]

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 6725

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int \frac {-10 x^7 \log (2)+\left (45 x^4+5 x^9\right ) \log ^2(2)+\left (10 x^5+\left (-54 x^2-6 x^7\right ) \log (2)\right ) \log \left (9+x^5\right )+\left (9+x^5\right ) \log ^2\left (9+x^5\right )}{9+x^5} \, dx}{\log ^2(2)}\\ &=\frac {\int \left (\frac {5 x^4 \log (2) \left (-2 x^3+x^5 \log (2)+\log (512)\right )}{9+x^5}-\frac {2 x^2 \left (-5 x^3+27 \log (2)+x^5 \log (8)\right ) \log \left (9+x^5\right )}{9+x^5}+\log ^2\left (9+x^5\right )\right ) \, dx}{\log ^2(2)}\\ &=\frac {\int \log ^2\left (9+x^5\right ) \, dx}{\log ^2(2)}-\frac {2 \int \frac {x^2 \left (-5 x^3+27 \log (2)+x^5 \log (8)\right ) \log \left (9+x^5\right )}{9+x^5} \, dx}{\log ^2(2)}+\frac {5 \int \frac {x^4 \left (-2 x^3+x^5 \log (2)+\log (512)\right )}{9+x^5} \, dx}{\log (2)}\\ &=x^5+\frac {x \log ^2\left (9+x^5\right )}{\log ^2(2)}-\frac {2 \int \left (-5 \log \left (9+x^5\right )+\frac {45 \log \left (9+x^5\right )}{9+x^5}+x^2 \log (8) \log \left (9+x^5\right )\right ) \, dx}{\log ^2(2)}-\frac {10 \int \frac {x^5 \log \left (9+x^5\right )}{9+x^5} \, dx}{\log ^2(2)}+\frac {\int -\frac {10 x^7}{9+x^5} \, dx}{\log (2)}\\ &=x^5+\frac {x \log ^2\left (9+x^5\right )}{\log ^2(2)}+\frac {10 \int \log \left (9+x^5\right ) \, dx}{\log ^2(2)}-\frac {10 \int \left (\log \left (9+x^5\right )-\frac {9 \log \left (9+x^5\right )}{9+x^5}\right ) \, dx}{\log ^2(2)}-\frac {90 \int \frac {\log \left (9+x^5\right )}{9+x^5} \, dx}{\log ^2(2)}-\frac {10 \int \frac {x^7}{9+x^5} \, dx}{\log (2)}-\frac {(2 \log (8)) \int x^2 \log \left (9+x^5\right ) \, dx}{\log ^2(2)}\\ &=x^5-\frac {10 x^3}{3 \log (2)}+\frac {10 x \log \left (9+x^5\right )}{\log ^2(2)}-\frac {2 x^3 \log (8) \log \left (9+x^5\right )}{3 \log ^2(2)}+\frac {x \log ^2\left (9+x^5\right )}{\log ^2(2)}-\frac {10 \int \log \left (9+x^5\right ) \, dx}{\log ^2(2)}-\frac {50 \int \frac {x^5}{9+x^5} \, dx}{\log ^2(2)}+\frac {90 \int \frac {\log \left (9+x^5\right )}{9+x^5} \, dx}{\log ^2(2)}-\frac {90 \int \left (-\frac {\log \left (9+x^5\right )}{15\ 3^{3/5} \left (-3^{2/5}-x\right )}-\frac {\log \left (9+x^5\right )}{15\ 3^{3/5} \left (-3^{2/5}+\sqrt [5]{-1} x\right )}-\frac {\log \left (9+x^5\right )}{15\ 3^{3/5} \left (-3^{2/5}-(-1)^{2/5} x\right )}-\frac {\log \left (9+x^5\right )}{15\ 3^{3/5} \left (-3^{2/5}+(-1)^{3/5} x\right )}-\frac {\log \left (9+x^5\right )}{15\ 3^{3/5} \left (-3^{2/5}-(-1)^{4/5} x\right )}\right ) \, dx}{\log ^2(2)}+\frac {90 \int \frac {x^2}{9+x^5} \, dx}{\log (2)}+\frac {(10 \log (8)) \int \frac {x^7}{9+x^5} \, dx}{3 \log ^2(2)}\\ &=x^5-\frac {50 x}{\log ^2(2)}-\frac {10 x^3}{3 \log (2)}+\frac {10 x^3 \log (8)}{9 \log ^2(2)}-\frac {2 x^3 \log (8) \log \left (9+x^5\right )}{3 \log ^2(2)}+\frac {x \log ^2\left (9+x^5\right )}{\log ^2(2)}+\frac {50 \int \frac {x^5}{9+x^5} \, dx}{\log ^2(2)}+\frac {90 \int \left (-\frac {\log \left (9+x^5\right )}{15\ 3^{3/5} \left (-3^{2/5}-x\right )}-\frac {\log \left (9+x^5\right )}{15\ 3^{3/5} \left (-3^{2/5}+\sqrt [5]{-1} x\right )}-\frac {\log \left (9+x^5\right )}{15\ 3^{3/5} \left (-3^{2/5}-(-1)^{2/5} x\right )}-\frac {\log \left (9+x^5\right )}{15\ 3^{3/5} \left (-3^{2/5}+(-1)^{3/5} x\right )}-\frac {\log \left (9+x^5\right )}{15\ 3^{3/5} \left (-3^{2/5}-(-1)^{4/5} x\right )}\right ) \, dx}{\log ^2(2)}+\frac {450 \int \frac {1}{9+x^5} \, dx}{\log ^2(2)}+\frac {\left (2\ 3^{2/5}\right ) \int \frac {\log \left (9+x^5\right )}{-3^{2/5}-x} \, dx}{\log ^2(2)}+\frac {\left (2\ 3^{2/5}\right ) \int \frac {\log \left (9+x^5\right )}{-3^{2/5}+\sqrt [5]{-1} x} \, dx}{\log ^2(2)}+\frac {\left (2\ 3^{2/5}\right ) \int \frac {\log \left (9+x^5\right )}{-3^{2/5}-(-1)^{2/5} x} \, dx}{\log ^2(2)}+\frac {\left (2\ 3^{2/5}\right ) \int \frac {\log \left (9+x^5\right )}{-3^{2/5}+(-1)^{3/5} x} \, dx}{\log ^2(2)}+\frac {\left (2\ 3^{2/5}\right ) \int \frac {\log \left (9+x^5\right )}{-3^{2/5}-(-1)^{4/5} x} \, dx}{\log ^2(2)}+\frac {\left (6 \sqrt [5]{3}\right ) \int \frac {1}{3^{2/5}+x} \, dx}{\log (2)}+\frac {\left (12 \sqrt [5]{3}\right ) \int \frac {\frac {1}{4} 3^{2/5} \left (-1-\sqrt {5}\right )-\frac {1}{4} \left (1+\sqrt {5}\right ) x}{3^{4/5}-\frac {1}{2} 3^{2/5} \left (1-\sqrt {5}\right ) x+x^2} \, dx}{\log (2)}+\frac {\left (12 \sqrt [5]{3}\right ) \int \frac {\frac {1}{4} 3^{2/5} \left (-1+\sqrt {5}\right )-\frac {1}{4} \left (1-\sqrt {5}\right ) x}{3^{4/5}-\frac {1}{2} 3^{2/5} \left (1+\sqrt {5}\right ) x+x^2} \, dx}{\log (2)}-\frac {(30 \log (8)) \int \frac {x^2}{9+x^5} \, dx}{\log ^2(2)}\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [C]  time = 0.61, size = 131, normalized size = 6.24 \begin {gather*} \frac {x^5 \log ^2(2)+\frac {10}{9} x^3 \log (8)-\frac {10}{9} x^3 \, _2F_1\left (\frac {3}{5},1;\frac {8}{5};-\frac {x^5}{9}\right ) \log (8)-\frac {2}{3} x^3 \log (32)+\frac {1}{27} x^3 \, _2F_1\left (\frac {3}{5},1;\frac {8}{5};-\frac {x^5}{9}\right ) \log (1237940039285380274899124224)-9 \log ^2(2) \log \left (9+x^5\right )-\frac {2}{3} x^3 \log (8) \log \left (9+x^5\right )+\frac {1}{5} \log (32) \log (512) \log \left (9+x^5\right )+x \log ^2\left (9+x^5\right )}{\log ^2(2)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-10*x^7*Log[2] + (45*x^4 + 5*x^9)*Log[2]^2 + (10*x^5 + (-54*x^2 - 6*x^7)*Log[2])*Log[9 + x^5] + (9
+ x^5)*Log[9 + x^5]^2)/((9 + x^5)*Log[2]^2),x]

[Out]

(x^5*Log[2]^2 + (10*x^3*Log[8])/9 - (10*x^3*Hypergeometric2F1[3/5, 1, 8/5, -1/9*x^5]*Log[8])/9 - (2*x^3*Log[32
])/3 + (x^3*Hypergeometric2F1[3/5, 1, 8/5, -1/9*x^5]*Log[1237940039285380274899124224])/27 - 9*Log[2]^2*Log[9
+ x^5] - (2*x^3*Log[8]*Log[9 + x^5])/3 + (Log[32]*Log[512]*Log[9 + x^5])/5 + x*Log[9 + x^5]^2)/Log[2]^2

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fricas [A]  time = 0.73, size = 37, normalized size = 1.76 \begin {gather*} \frac {x^{5} \log \relax (2)^{2} - 2 \, x^{3} \log \relax (2) \log \left (x^{5} + 9\right ) + x \log \left (x^{5} + 9\right )^{2}}{\log \relax (2)^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^5+9)*log(x^5+9)^2+((-6*x^7-54*x^2)*log(2)+10*x^5)*log(x^5+9)+(5*x^9+45*x^4)*log(2)^2-10*x^7*log(
2))/(x^5+9)/log(2)^2,x, algorithm="fricas")

[Out]

(x^5*log(2)^2 - 2*x^3*log(2)*log(x^5 + 9) + x*log(x^5 + 9)^2)/log(2)^2

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giac [A]  time = 0.78, size = 37, normalized size = 1.76 \begin {gather*} \frac {x^{5} \log \relax (2)^{2} - 2 \, x^{3} \log \relax (2) \log \left (x^{5} + 9\right ) + x \log \left (x^{5} + 9\right )^{2}}{\log \relax (2)^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^5+9)*log(x^5+9)^2+((-6*x^7-54*x^2)*log(2)+10*x^5)*log(x^5+9)+(5*x^9+45*x^4)*log(2)^2-10*x^7*log(
2))/(x^5+9)/log(2)^2,x, algorithm="giac")

[Out]

(x^5*log(2)^2 - 2*x^3*log(2)*log(x^5 + 9) + x*log(x^5 + 9)^2)/log(2)^2

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maple [A]  time = 0.06, size = 34, normalized size = 1.62

method	result	size
risch	\(x^{5}-\frac {2 x^{3} \ln \left (x^{5}+9\right )}{\ln \relax (2)}+\frac {x \ln \left (x^{5}+9\right )^{2}}{\ln \relax (2)^{2}}\)	\(34\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^5+9)*ln(x^5+9)^2+((-6*x^7-54*x^2)*ln(2)+10*x^5)*ln(x^5+9)+(5*x^9+45*x^4)*ln(2)^2-10*x^7*ln(2))/(x^5+9)
/ln(2)^2,x,method=_RETURNVERBOSE)

[Out]

x^5-2/ln(2)*x^3*ln(x^5+9)+1/ln(2)^2*x*ln(x^5+9)^2

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^5+9)*log(x^5+9)^2+((-6*x^7-54*x^2)*log(2)+10*x^5)*log(x^5+9)+(5*x^9+45*x^4)*log(2)^2-10*x^7*log(
2))/(x^5+9)/log(2)^2,x, algorithm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: Error executing code in Maxima: sign: argument cannot be imaginary
; found sqrt(sqrt(5)-5)

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mupad [B]  time = 0.40, size = 22, normalized size = 1.05 \begin {gather*} \frac {x\,{\left (\ln \left (x^5+9\right )-x^2\,\ln \relax (2)\right )}^2}{{\ln \relax (2)}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(x^5 + 9)*(log(2)*(54*x^2 + 6*x^7) - 10*x^5) - log(x^5 + 9)^2*(x^5 + 9) + 10*x^7*log(2) - log(2)^2*(4
5*x^4 + 5*x^9))/(log(2)^2*(x^5 + 9)),x)

[Out]

(x*(log(x^5 + 9) - x^2*log(2))^2)/log(2)^2

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sympy [B]  time = 0.20, size = 32, normalized size = 1.52 \begin {gather*} x^{5} - \frac {2 x^{3} \log {\left (x^{5} + 9 \right )}}{\log {\relax (2 )}} + \frac {x \log {\left (x^{5} + 9 \right )}^{2}}{\log {\relax (2 )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x**5+9)*ln(x**5+9)**2+((-6*x**7-54*x**2)*ln(2)+10*x**5)*ln(x**5+9)+(5*x**9+45*x**4)*ln(2)**2-10*x*
*7*ln(2))/(x**5+9)/ln(2)**2,x)

[Out]

x**5 - 2*x**3*log(x**5 + 9)/log(2) + x*log(x**5 + 9)**2/log(2)**2

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