3.3.31 \(\int \frac {-9-63 x+x^2+7 x^3+(84 x^3-28 x^4) \log (1+7 x)+(6 x^2+38 x^3-28 x^4) \log ^2(1+7 x)+28 x^5 \log ^3(1+7 x)+(3 x^4+21 x^5) \log ^4(1+7 x)}{x^2+7 x^3} \, dx\)

Optimal. Leaf size=28 \[ \frac {\left (-3+x-(x-x (1-\log (1+7 x)))^2\right )^2}{x} \]

________________________________________________________________________________________

Rubi [B]  time = 1.13, antiderivative size = 193, normalized size of antiderivative = 6.89, number of steps used = 67, number of rules used = 19, integrand size = 101, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {1593, 6742, 14, 2418, 2389, 2295, 2395, 43, 2390, 2301, 2401, 2296, 2305, 2304, 2411, 2346, 2302, 30, 2330} \begin {gather*} x^2-2 x^2 \log (7 x+1)-\frac {1}{49} (7 x+1)^2+\frac {9 x}{7}+\frac {9}{x}+\frac {1}{343} (7 x+1)^3 \log ^4(7 x+1)-\frac {3}{343} (7 x+1)^2 \log ^4(7 x+1)+\frac {3}{343} (7 x+1) \log ^4(7 x+1)-\frac {1}{343} \log ^4(7 x+1)-\frac {2}{49} (7 x+1)^2 \log ^2(7 x+1)+\frac {46}{49} (7 x+1) \log ^2(7 x+1)-\frac {44}{49} \log ^2(7 x+1)+\frac {2}{49} (7 x+1)^2 \log (7 x+1)-\frac {4}{49} (7 x+1) \log (7 x+1)+\frac {2}{49} \log (7 x+1) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-9 - 63*x + x^2 + 7*x^3 + (84*x^3 - 28*x^4)*Log[1 + 7*x] + (6*x^2 + 38*x^3 - 28*x^4)*Log[1 + 7*x]^2 + 28*
x^5*Log[1 + 7*x]^3 + (3*x^4 + 21*x^5)*Log[1 + 7*x]^4)/(x^2 + 7*x^3),x]

[Out]

9/x + (9*x)/7 + x^2 - (1 + 7*x)^2/49 + (2*Log[1 + 7*x])/49 - 2*x^2*Log[1 + 7*x] - (4*(1 + 7*x)*Log[1 + 7*x])/4
9 + (2*(1 + 7*x)^2*Log[1 + 7*x])/49 - (44*Log[1 + 7*x]^2)/49 + (46*(1 + 7*x)*Log[1 + 7*x]^2)/49 - (2*(1 + 7*x)
^2*Log[1 + 7*x]^2)/49 - Log[1 + 7*x]^4/343 + (3*(1 + 7*x)*Log[1 + 7*x]^4)/343 - (3*(1 + 7*x)^2*Log[1 + 7*x]^4)
/343 + ((1 + 7*x)^3*Log[1 + 7*x]^4)/343

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 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 1593

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

Rule 2295

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rule 2296

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*Log[c*x^n])^p, x] - Dist[b*n*p, In
t[(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{a, b, c, n}, x] && GtQ[p, 0] && IntegerQ[2*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 2302

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/(x_), x_Symbol] :> Dist[1/(b*n), Subst[Int[x^p, x], x, a + b*L
og[c*x^n]], x] /; FreeQ[{a, b, c, n, p}, x]

Rule 2304

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Log[c*x^
n]))/(d*(m + 1)), x] - Simp[(b*n*(d*x)^(m + 1))/(d*(m + 1)^2), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1
]

Rule 2305

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

Rule 2330

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> With[{u = Expand
Integrand[(a + b*Log[c*x^n])^p, (d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, n, p, q, r}
, x] && IntegerQ[q] && (GtQ[q, 0] || (IGtQ[p, 0] && IntegerQ[r]))

Rule 2346

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_.))/(x_), x_Symbol] :> Dist[d, Int[((d
+ e*x)^(q - 1)*(a + b*Log[c*x^n])^p)/x, x], x] + Dist[e, Int[(d + e*x)^(q - 1)*(a + b*Log[c*x^n])^p, x], x] /
; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[p, 0] && GtQ[q, 0] && IntegerQ[2*q]

Rule 2389

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.), x_Symbol] :> Dist[1/e, Subst[Int[(a + b*Log[c*
x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, n, p}, 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 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 2401

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

Rule 2411

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + (g_.)*(x_))^(q_.)*((h_.) + (i_.)*(x_))
^(r_.), x_Symbol] :> Dist[1/e, Subst[Int[((g*x)/e)^q*((e*h - d*i)/e + (i*x)/e)^r*(a + b*Log[c*x^n])^p, x], x,
d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, n, p, q, r}, x] && EqQ[e*f - d*g, 0] && (IGtQ[p, 0] || IGtQ[
r, 0]) && IntegerQ[2*r]

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 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 {-9-63 x+x^2+7 x^3+\left (84 x^3-28 x^4\right ) \log (1+7 x)+\left (6 x^2+38 x^3-28 x^4\right ) \log ^2(1+7 x)+28 x^5 \log ^3(1+7 x)+\left (3 x^4+21 x^5\right ) \log ^4(1+7 x)}{x^2 (1+7 x)} \, dx\\ &=\int \left (\frac {-9+x^2}{x^2}-\frac {28 (-3+x) x \log (1+7 x)}{1+7 x}-2 (-3+2 x) \log ^2(1+7 x)+\frac {28 x^3 \log ^3(1+7 x)}{1+7 x}+3 x^2 \log ^4(1+7 x)\right ) \, dx\\ &=-\left (2 \int (-3+2 x) \log ^2(1+7 x) \, dx\right )+3 \int x^2 \log ^4(1+7 x) \, dx-28 \int \frac {(-3+x) x \log (1+7 x)}{1+7 x} \, dx+28 \int \frac {x^3 \log ^3(1+7 x)}{1+7 x} \, dx+\int \frac {-9+x^2}{x^2} \, dx\\ &=-\left (2 \int \left (-\frac {23}{7} \log ^2(1+7 x)+\frac {2}{7} (1+7 x) \log ^2(1+7 x)\right ) \, dx\right )+3 \int \left (\frac {1}{49} \log ^4(1+7 x)-\frac {2}{49} (1+7 x) \log ^4(1+7 x)+\frac {1}{49} (1+7 x)^2 \log ^4(1+7 x)\right ) \, dx+4 \operatorname {Subst}\left (\int \frac {\left (-\frac {1}{7}+\frac {x}{7}\right )^3 \log ^3(x)}{x} \, dx,x,1+7 x\right )-28 \int \left (-\frac {22}{49} \log (1+7 x)+\frac {1}{7} x \log (1+7 x)+\frac {22 \log (1+7 x)}{49 (1+7 x)}\right ) \, dx+\int \left (1-\frac {9}{x^2}\right ) \, dx\\ &=\frac {9}{x}+x+\frac {3}{49} \int \log ^4(1+7 x) \, dx+\frac {3}{49} \int (1+7 x)^2 \log ^4(1+7 x) \, dx-\frac {6}{49} \int (1+7 x) \log ^4(1+7 x) \, dx-\frac {4}{7} \int (1+7 x) \log ^2(1+7 x) \, dx+\frac {4}{7} \operatorname {Subst}\left (\int \left (-\frac {1}{7}+\frac {x}{7}\right )^2 \log ^3(x) \, dx,x,1+7 x\right )-\frac {4}{7} \operatorname {Subst}\left (\int \frac {\left (-\frac {1}{7}+\frac {x}{7}\right )^2 \log ^3(x)}{x} \, dx,x,1+7 x\right )-4 \int x \log (1+7 x) \, dx+\frac {46}{7} \int \log ^2(1+7 x) \, dx+\frac {88}{7} \int \log (1+7 x) \, dx-\frac {88}{7} \int \frac {\log (1+7 x)}{1+7 x} \, dx\\ &=\frac {9}{x}+x-2 x^2 \log (1+7 x)+\frac {3}{343} \operatorname {Subst}\left (\int \log ^4(x) \, dx,x,1+7 x\right )+\frac {3}{343} \operatorname {Subst}\left (\int x^2 \log ^4(x) \, dx,x,1+7 x\right )-\frac {6}{343} \operatorname {Subst}\left (\int x \log ^4(x) \, dx,x,1+7 x\right )-\frac {4}{49} \operatorname {Subst}\left (\int x \log ^2(x) \, dx,x,1+7 x\right )-\frac {4}{49} \operatorname {Subst}\left (\int \left (-\frac {1}{7}+\frac {x}{7}\right ) \log ^3(x) \, dx,x,1+7 x\right )+\frac {4}{49} \operatorname {Subst}\left (\int \frac {\left (-\frac {1}{7}+\frac {x}{7}\right ) \log ^3(x)}{x} \, dx,x,1+7 x\right )+\frac {4}{7} \operatorname {Subst}\left (\int \left (\frac {\log ^3(x)}{49}-\frac {2}{49} x \log ^3(x)+\frac {1}{49} x^2 \log ^3(x)\right ) \, dx,x,1+7 x\right )+\frac {46}{49} \operatorname {Subst}\left (\int \log ^2(x) \, dx,x,1+7 x\right )+\frac {88}{49} \operatorname {Subst}(\int \log (x) \, dx,x,1+7 x)-\frac {88}{49} \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,1+7 x\right )+14 \int \frac {x^2}{1+7 x} \, dx\\ &=\frac {9}{x}-\frac {81 x}{7}-2 x^2 \log (1+7 x)+\frac {88}{49} (1+7 x) \log (1+7 x)-\frac {44}{49} \log ^2(1+7 x)+\frac {46}{49} (1+7 x) \log ^2(1+7 x)-\frac {2}{49} (1+7 x)^2 \log ^2(1+7 x)+\frac {3}{343} (1+7 x) \log ^4(1+7 x)-\frac {3}{343} (1+7 x)^2 \log ^4(1+7 x)+\frac {1}{343} (1+7 x)^3 \log ^4(1+7 x)+2 \left (\frac {4}{343} \operatorname {Subst}\left (\int \log ^3(x) \, dx,x,1+7 x\right )\right )-\frac {4}{343} \operatorname {Subst}\left (\int \frac {\log ^3(x)}{x} \, dx,x,1+7 x\right )-\frac {8}{343} \operatorname {Subst}\left (\int x \log ^3(x) \, dx,x,1+7 x\right )-\frac {12}{343} \operatorname {Subst}\left (\int \log ^3(x) \, dx,x,1+7 x\right )+\frac {12}{343} \operatorname {Subst}\left (\int x \log ^3(x) \, dx,x,1+7 x\right )+\frac {4}{49} \operatorname {Subst}(\int x \log (x) \, dx,x,1+7 x)-\frac {4}{49} \operatorname {Subst}\left (\int \left (-\frac {1}{7} \log ^3(x)+\frac {1}{7} x \log ^3(x)\right ) \, dx,x,1+7 x\right )-\frac {92}{49} \operatorname {Subst}(\int \log (x) \, dx,x,1+7 x)+14 \int \left (-\frac {1}{49}+\frac {x}{7}+\frac {1}{49 (1+7 x)}\right ) \, dx\\ &=\frac {9}{x}+\frac {9 x}{7}+x^2-\frac {1}{49} (1+7 x)^2+\frac {2}{49} \log (1+7 x)-2 x^2 \log (1+7 x)-\frac {4}{49} (1+7 x) \log (1+7 x)+\frac {2}{49} (1+7 x)^2 \log (1+7 x)-\frac {44}{49} \log ^2(1+7 x)+\frac {46}{49} (1+7 x) \log ^2(1+7 x)-\frac {2}{49} (1+7 x)^2 \log ^2(1+7 x)-\frac {12}{343} (1+7 x) \log ^3(1+7 x)+\frac {2}{343} (1+7 x)^2 \log ^3(1+7 x)+\frac {3}{343} (1+7 x) \log ^4(1+7 x)-\frac {3}{343} (1+7 x)^2 \log ^4(1+7 x)+\frac {1}{343} (1+7 x)^3 \log ^4(1+7 x)-\frac {4}{343} \operatorname {Subst}\left (\int x^3 \, dx,x,\log (1+7 x)\right )+\frac {4}{343} \operatorname {Subst}\left (\int \log ^3(x) \, dx,x,1+7 x\right )-\frac {4}{343} \operatorname {Subst}\left (\int x \log ^3(x) \, dx,x,1+7 x\right )+2 \left (\frac {4}{343} (1+7 x) \log ^3(1+7 x)-\frac {12}{343} \operatorname {Subst}\left (\int \log ^2(x) \, dx,x,1+7 x\right )\right )+\frac {12}{343} \operatorname {Subst}\left (\int x \log ^2(x) \, dx,x,1+7 x\right )-\frac {18}{343} \operatorname {Subst}\left (\int x \log ^2(x) \, dx,x,1+7 x\right )+\frac {36}{343} \operatorname {Subst}\left (\int \log ^2(x) \, dx,x,1+7 x\right )\\ &=\frac {9}{x}+\frac {9 x}{7}+x^2-\frac {1}{49} (1+7 x)^2+\frac {2}{49} \log (1+7 x)-2 x^2 \log (1+7 x)-\frac {4}{49} (1+7 x) \log (1+7 x)+\frac {2}{49} (1+7 x)^2 \log (1+7 x)-\frac {44}{49} \log ^2(1+7 x)+\frac {358}{343} (1+7 x) \log ^2(1+7 x)-\frac {17}{343} (1+7 x)^2 \log ^2(1+7 x)-\frac {8}{343} (1+7 x) \log ^3(1+7 x)-\frac {1}{343} \log ^4(1+7 x)+\frac {3}{343} (1+7 x) \log ^4(1+7 x)-\frac {3}{343} (1+7 x)^2 \log ^4(1+7 x)+\frac {1}{343} (1+7 x)^3 \log ^4(1+7 x)+\frac {6}{343} \operatorname {Subst}\left (\int x \log ^2(x) \, dx,x,1+7 x\right )-\frac {12}{343} \operatorname {Subst}(\int x \log (x) \, dx,x,1+7 x)-\frac {12}{343} \operatorname {Subst}\left (\int \log ^2(x) \, dx,x,1+7 x\right )+\frac {18}{343} \operatorname {Subst}(\int x \log (x) \, dx,x,1+7 x)+2 \left (-\frac {12}{343} (1+7 x) \log ^2(1+7 x)+\frac {4}{343} (1+7 x) \log ^3(1+7 x)+\frac {24}{343} \operatorname {Subst}(\int \log (x) \, dx,x,1+7 x)\right )-\frac {72}{343} \operatorname {Subst}(\int \log (x) \, dx,x,1+7 x)\\ &=\frac {9}{x}+\frac {135 x}{49}+x^2-\frac {17}{686} (1+7 x)^2+\frac {2}{49} \log (1+7 x)-2 x^2 \log (1+7 x)-\frac {100}{343} (1+7 x) \log (1+7 x)+\frac {17}{343} (1+7 x)^2 \log (1+7 x)-\frac {44}{49} \log ^2(1+7 x)+\frac {346}{343} (1+7 x) \log ^2(1+7 x)-\frac {2}{49} (1+7 x)^2 \log ^2(1+7 x)-\frac {8}{343} (1+7 x) \log ^3(1+7 x)-\frac {1}{343} \log ^4(1+7 x)+\frac {3}{343} (1+7 x) \log ^4(1+7 x)-\frac {3}{343} (1+7 x)^2 \log ^4(1+7 x)+\frac {1}{343} (1+7 x)^3 \log ^4(1+7 x)+2 \left (-\frac {24 x}{49}+\frac {24}{343} (1+7 x) \log (1+7 x)-\frac {12}{343} (1+7 x) \log ^2(1+7 x)+\frac {4}{343} (1+7 x) \log ^3(1+7 x)\right )-\frac {6}{343} \operatorname {Subst}(\int x \log (x) \, dx,x,1+7 x)+\frac {24}{343} \operatorname {Subst}(\int \log (x) \, dx,x,1+7 x)\\ &=\frac {9}{x}+\frac {111 x}{49}+x^2-\frac {1}{49} (1+7 x)^2+\frac {2}{49} \log (1+7 x)-2 x^2 \log (1+7 x)-\frac {76}{343} (1+7 x) \log (1+7 x)+\frac {2}{49} (1+7 x)^2 \log (1+7 x)-\frac {44}{49} \log ^2(1+7 x)+\frac {346}{343} (1+7 x) \log ^2(1+7 x)-\frac {2}{49} (1+7 x)^2 \log ^2(1+7 x)-\frac {8}{343} (1+7 x) \log ^3(1+7 x)-\frac {1}{343} \log ^4(1+7 x)+\frac {3}{343} (1+7 x) \log ^4(1+7 x)-\frac {3}{343} (1+7 x)^2 \log ^4(1+7 x)+\frac {1}{343} (1+7 x)^3 \log ^4(1+7 x)+2 \left (-\frac {24 x}{49}+\frac {24}{343} (1+7 x) \log (1+7 x)-\frac {12}{343} (1+7 x) \log ^2(1+7 x)+\frac {4}{343} (1+7 x) \log ^3(1+7 x)\right )\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.03, size = 33, normalized size = 1.18 \begin {gather*} \frac {9}{x}+x-2 (-3+x) x \log ^2(1+7 x)+x^3 \log ^4(1+7 x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-9 - 63*x + x^2 + 7*x^3 + (84*x^3 - 28*x^4)*Log[1 + 7*x] + (6*x^2 + 38*x^3 - 28*x^4)*Log[1 + 7*x]^2
+ 28*x^5*Log[1 + 7*x]^3 + (3*x^4 + 21*x^5)*Log[1 + 7*x]^4)/(x^2 + 7*x^3),x]

[Out]

9/x + x - 2*(-3 + x)*x*Log[1 + 7*x]^2 + x^3*Log[1 + 7*x]^4

________________________________________________________________________________________

fricas [A]  time = 0.62, size = 40, normalized size = 1.43 \begin {gather*} \frac {x^{4} \log \left (7 \, x + 1\right )^{4} - 2 \, {\left (x^{3} - 3 \, x^{2}\right )} \log \left (7 \, x + 1\right )^{2} + x^{2} + 9}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((21*x^5+3*x^4)*log(7*x+1)^4+28*x^5*log(7*x+1)^3+(-28*x^4+38*x^3+6*x^2)*log(7*x+1)^2+(-28*x^4+84*x^3
)*log(7*x+1)+7*x^3+x^2-63*x-9)/(7*x^3+x^2),x, algorithm="fricas")

[Out]

(x^4*log(7*x + 1)^4 - 2*(x^3 - 3*x^2)*log(7*x + 1)^2 + x^2 + 9)/x

________________________________________________________________________________________

giac [A]  time = 0.52, size = 36, normalized size = 1.29 \begin {gather*} x^{3} \log \left (7 \, x + 1\right )^{4} - 2 \, {\left (x^{2} - 3 \, x\right )} \log \left (7 \, x + 1\right )^{2} + x + \frac {9}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((21*x^5+3*x^4)*log(7*x+1)^4+28*x^5*log(7*x+1)^3+(-28*x^4+38*x^3+6*x^2)*log(7*x+1)^2+(-28*x^4+84*x^3
)*log(7*x+1)+7*x^3+x^2-63*x-9)/(7*x^3+x^2),x, algorithm="giac")

[Out]

x^3*log(7*x + 1)^4 - 2*(x^2 - 3*x)*log(7*x + 1)^2 + x + 9/x

________________________________________________________________________________________

maple [A]  time = 0.07, size = 41, normalized size = 1.46

method	result	size
risch	\(x^{3} \ln \left (7 x +1\right )^{4}+\left (-2 x^{2}+6 x \right ) \ln \left (7 x +1\right )^{2}+\frac {x^{2}+9}{x}\)	\(41\)
derivativedivides	\(\frac {\ln \left (7 x +1\right )^{4} \left (7 x +1\right )^{3}}{343}-\frac {3 \ln \left (7 x +1\right )^{4} \left (7 x +1\right )^{2}}{343}+\frac {3 \ln \left (7 x +1\right )^{4} \left (7 x +1\right )}{343}-\frac {2 \ln \left (7 x +1\right )^{2} \left (7 x +1\right )^{2}}{49}+\frac {46 \ln \left (7 x +1\right )^{2} \left (7 x +1\right )}{49}-\frac {\ln \left (7 x +1\right )^{4}}{343}-\frac {44 \ln \left (7 x +1\right )^{2}}{49}+\frac {9}{x}+x +\frac {1}{7}\)	\(110\)
default	\(\frac {\ln \left (7 x +1\right )^{4} \left (7 x +1\right )^{3}}{343}-\frac {3 \ln \left (7 x +1\right )^{4} \left (7 x +1\right )^{2}}{343}+\frac {3 \ln \left (7 x +1\right )^{4} \left (7 x +1\right )}{343}-\frac {2 \ln \left (7 x +1\right )^{2} \left (7 x +1\right )^{2}}{49}+\frac {46 \ln \left (7 x +1\right )^{2} \left (7 x +1\right )}{49}-\frac {\ln \left (7 x +1\right )^{4}}{343}-\frac {44 \ln \left (7 x +1\right )^{2}}{49}+\frac {9}{x}+x +\frac {1}{7}\)	\(110\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((21*x^5+3*x^4)*ln(7*x+1)^4+28*x^5*ln(7*x+1)^3+(-28*x^4+38*x^3+6*x^2)*ln(7*x+1)^2+(-28*x^4+84*x^3)*ln(7*x+
1)+7*x^3+x^2-63*x-9)/(7*x^3+x^2),x,method=_RETURNVERBOSE)

[Out]

x^3*ln(7*x+1)^4+(-2*x^2+6*x)*ln(7*x+1)^2+(x^2+9)/x

________________________________________________________________________________________

maxima [B]  time = 0.43, size = 395, normalized size = 14.11 \begin {gather*} \frac {1}{9261} \, {\left (27 \, \log \left (7 \, x + 1\right )^{4} - 36 \, \log \left (7 \, x + 1\right )^{3} + 36 \, \log \left (7 \, x + 1\right )^{2} - 24 \, \log \left (7 \, x + 1\right ) + 8\right )} {\left (7 \, x + 1\right )}^{3} + \frac {4}{9261} \, {\left (9 \, \log \left (7 \, x + 1\right )^{3} - 9 \, \log \left (7 \, x + 1\right )^{2} + 6 \, \log \left (7 \, x + 1\right ) - 2\right )} {\left (7 \, x + 1\right )}^{3} - \frac {1}{343} \, \log \left (7 \, x + 1\right )^{4} - \frac {3}{686} \, {\left (2 \, \log \left (7 \, x + 1\right )^{4} - 4 \, \log \left (7 \, x + 1\right )^{3} + 6 \, \log \left (7 \, x + 1\right )^{2} - 6 \, \log \left (7 \, x + 1\right ) + 3\right )} {\left (7 \, x + 1\right )}^{2} - \frac {3}{686} \, {\left (4 \, \log \left (7 \, x + 1\right )^{3} - 6 \, \log \left (7 \, x + 1\right )^{2} + 6 \, \log \left (7 \, x + 1\right ) - 3\right )} {\left (7 \, x + 1\right )}^{2} - \frac {1}{49} \, {\left (2 \, \log \left (7 \, x + 1\right )^{2} - 2 \, \log \left (7 \, x + 1\right ) + 1\right )} {\left (7 \, x + 1\right )}^{2} + \frac {3}{343} \, {\left (\log \left (7 \, x + 1\right )^{4} - 4 \, \log \left (7 \, x + 1\right )^{3} + 12 \, \log \left (7 \, x + 1\right )^{2} - 24 \, \log \left (7 \, x + 1\right ) + 24\right )} {\left (7 \, x + 1\right )} + \frac {12}{343} \, {\left (\log \left (7 \, x + 1\right )^{3} - 3 \, \log \left (7 \, x + 1\right )^{2} + 6 \, \log \left (7 \, x + 1\right ) - 6\right )} {\left (7 \, x + 1\right )} + \frac {46}{49} \, {\left (\log \left (7 \, x + 1\right )^{2} - 2 \, \log \left (7 \, x + 1\right ) + 2\right )} {\left (7 \, x + 1\right )} + x^{2} - \frac {2}{49} \, {\left (49 \, x^{2} - 14 \, x + 2 \, \log \left (7 \, x + 1\right )\right )} \log \left (7 \, x + 1\right ) + \frac {12}{7} \, {\left (7 \, x - \log \left (7 \, x + 1\right )\right )} \log \left (7 \, x + 1\right ) + \frac {44}{49} \, \log \left (7 \, x + 1\right )^{2} - \frac {83}{7} \, x + \frac {9}{x} + \frac {90}{49} \, \log \left (7 \, x + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((21*x^5+3*x^4)*log(7*x+1)^4+28*x^5*log(7*x+1)^3+(-28*x^4+38*x^3+6*x^2)*log(7*x+1)^2+(-28*x^4+84*x^3
)*log(7*x+1)+7*x^3+x^2-63*x-9)/(7*x^3+x^2),x, algorithm="maxima")

[Out]

1/9261*(27*log(7*x + 1)^4 - 36*log(7*x + 1)^3 + 36*log(7*x + 1)^2 - 24*log(7*x + 1) + 8)*(7*x + 1)^3 + 4/9261*
(9*log(7*x + 1)^3 - 9*log(7*x + 1)^2 + 6*log(7*x + 1) - 2)*(7*x + 1)^3 - 1/343*log(7*x + 1)^4 - 3/686*(2*log(7
*x + 1)^4 - 4*log(7*x + 1)^3 + 6*log(7*x + 1)^2 - 6*log(7*x + 1) + 3)*(7*x + 1)^2 - 3/686*(4*log(7*x + 1)^3 -
6*log(7*x + 1)^2 + 6*log(7*x + 1) - 3)*(7*x + 1)^2 - 1/49*(2*log(7*x + 1)^2 - 2*log(7*x + 1) + 1)*(7*x + 1)^2
+ 3/343*(log(7*x + 1)^4 - 4*log(7*x + 1)^3 + 12*log(7*x + 1)^2 - 24*log(7*x + 1) + 24)*(7*x + 1) + 12/343*(log
(7*x + 1)^3 - 3*log(7*x + 1)^2 + 6*log(7*x + 1) - 6)*(7*x + 1) + 46/49*(log(7*x + 1)^2 - 2*log(7*x + 1) + 2)*(
7*x + 1) + x^2 - 2/49*(49*x^2 - 14*x + 2*log(7*x + 1))*log(7*x + 1) + 12/7*(7*x - log(7*x + 1))*log(7*x + 1) +
44/49*log(7*x + 1)^2 - 83/7*x + 9/x + 90/49*log(7*x + 1)

________________________________________________________________________________________

mupad [B]  time = 0.41, size = 43, normalized size = 1.54 \begin {gather*} x-2\,x^2\,{\ln \left (7\,x+1\right )}^2+x^3\,{\ln \left (7\,x+1\right )}^4+\frac {9}{x}+6\,x\,{\ln \left (7\,x+1\right )}^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(7*x + 1)^4*(3*x^4 + 21*x^5) - 63*x + 28*x^5*log(7*x + 1)^3 + log(7*x + 1)^2*(6*x^2 + 38*x^3 - 28*x^4)
+ log(7*x + 1)*(84*x^3 - 28*x^4) + x^2 + 7*x^3 - 9)/(x^2 + 7*x^3),x)

[Out]

x - 2*x^2*log(7*x + 1)^2 + x^3*log(7*x + 1)^4 + 9/x + 6*x*log(7*x + 1)^2

________________________________________________________________________________________

sympy [A]  time = 0.19, size = 32, normalized size = 1.14 \begin {gather*} x^{3} \log {\left (7 x + 1 \right )}^{4} + x + \left (- 2 x^{2} + 6 x\right ) \log {\left (7 x + 1 \right )}^{2} + \frac {9}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((21*x**5+3*x**4)*ln(7*x+1)**4+28*x**5*ln(7*x+1)**3+(-28*x**4+38*x**3+6*x**2)*ln(7*x+1)**2+(-28*x**4
+84*x**3)*ln(7*x+1)+7*x**3+x**2-63*x-9)/(7*x**3+x**2),x)

[Out]

x**3*log(7*x + 1)**4 + x + (-2*x**2 + 6*x)*log(7*x + 1)**2 + 9/x

________________________________________________________________________________________