3.1.58 \(\int \frac {\log (d (\frac {1}{d}+f \sqrt {x})) (a+b \log (c x^n))^2}{x^3} \, dx\) [58]

Optimal. Leaf size=555 \[ -\frac {37 b^2 d f n^2}{108 x^{3/2}}+\frac {7 b^2 d^2 f^2 n^2}{8 x}-\frac {21 b^2 d^3 f^3 n^2}{4 \sqrt {x}}+\frac {1}{4} b^2 d^4 f^4 n^2 \log \left (1+d f \sqrt {x}\right )-\frac {b^2 n^2 \log \left (1+d f \sqrt {x}\right )}{4 x^2}-\frac {1}{8} b^2 d^4 f^4 n^2 \log (x)+\frac {1}{8} b^2 d^4 f^4 n^2 \log ^2(x)-\frac {7 b d f n \left (a+b \log \left (c x^n\right )\right )}{18 x^{3/2}}+\frac {3 b d^2 f^2 n \left (a+b \log \left (c x^n\right )\right )}{4 x}-\frac {5 b d^3 f^3 n \left (a+b \log \left (c x^n\right )\right )}{2 \sqrt {x}}+\frac {1}{2} b d^4 f^4 n \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {b n \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac {1}{4} b d^4 f^4 n \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac {d f \left (a+b \log \left (c x^n\right )\right )^2}{6 x^{3/2}}+\frac {d^2 f^2 \left (a+b \log \left (c x^n\right )\right )^2}{4 x}-\frac {d^3 f^3 \left (a+b \log \left (c x^n\right )\right )^2}{2 \sqrt {x}}+\frac {1}{2} d^4 f^4 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac {\log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 x^2}-\frac {d^4 f^4 \left (a+b \log \left (c x^n\right )\right )^3}{12 b n}+b^2 d^4 f^4 n^2 \text {Li}_2\left (-d f \sqrt {x}\right )+2 b d^4 f^4 n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-d f \sqrt {x}\right )-4 b^2 d^4 f^4 n^2 \text {Li}_3\left (-d f \sqrt {x}\right ) \]

[Out]

-37/108*b^2*d*f*n^2/x^(3/2)+7/8*b^2*d^2*f^2*n^2/x-1/8*b^2*d^4*f^4*n^2*ln(x)+1/8*b^2*d^4*f^4*n^2*ln(x)^2-7/18*b
*d*f*n*(a+b*ln(c*x^n))/x^(3/2)+3/4*b*d^2*f^2*n*(a+b*ln(c*x^n))/x-1/4*b*d^4*f^4*n*ln(x)*(a+b*ln(c*x^n))-1/6*d*f
*(a+b*ln(c*x^n))^2/x^(3/2)+1/4*d^2*f^2*(a+b*ln(c*x^n))^2/x-1/12*d^4*f^4*(a+b*ln(c*x^n))^3/b/n+1/4*b^2*d^4*f^4*
n^2*ln(1+d*f*x^(1/2))-1/4*b^2*n^2*ln(1+d*f*x^(1/2))/x^2+1/2*b*d^4*f^4*n*(a+b*ln(c*x^n))*ln(1+d*f*x^(1/2))-1/2*
b*n*(a+b*ln(c*x^n))*ln(1+d*f*x^(1/2))/x^2+1/2*d^4*f^4*(a+b*ln(c*x^n))^2*ln(1+d*f*x^(1/2))-1/2*(a+b*ln(c*x^n))^
2*ln(1+d*f*x^(1/2))/x^2+b^2*d^4*f^4*n^2*polylog(2,-d*f*x^(1/2))+2*b*d^4*f^4*n*(a+b*ln(c*x^n))*polylog(2,-d*f*x
^(1/2))-4*b^2*d^4*f^4*n^2*polylog(3,-d*f*x^(1/2))-21/4*b^2*d^3*f^3*n^2/x^(1/2)-5/2*b*d^3*f^3*n*(a+b*ln(c*x^n))
/x^(1/2)-1/2*d^3*f^3*(a+b*ln(c*x^n))^2/x^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.38, antiderivative size = 555, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 14, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.467, Rules used = {2504, 2442, 46, 2424, 2341, 2423, 2438, 2338, 2421, 6724, 2413, 12, 2339, 30} \begin {gather*} 2 b d^4 f^4 n \text {PolyLog}\left (2,-d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )+b^2 d^4 f^4 n^2 \text {PolyLog}\left (2,-d f \sqrt {x}\right )-4 b^2 d^4 f^4 n^2 \text {PolyLog}\left (3,-d f \sqrt {x}\right )-\frac {d^4 f^4 \left (a+b \log \left (c x^n\right )\right )^3}{12 b n}+\frac {1}{2} d^4 f^4 \log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2+\frac {1}{2} b d^4 f^4 n \log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{4} b d^4 f^4 n \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac {d^3 f^3 \left (a+b \log \left (c x^n\right )\right )^2}{2 \sqrt {x}}-\frac {5 b d^3 f^3 n \left (a+b \log \left (c x^n\right )\right )}{2 \sqrt {x}}+\frac {d^2 f^2 \left (a+b \log \left (c x^n\right )\right )^2}{4 x}+\frac {3 b d^2 f^2 n \left (a+b \log \left (c x^n\right )\right )}{4 x}-\frac {d f \left (a+b \log \left (c x^n\right )\right )^2}{6 x^{3/2}}-\frac {7 b d f n \left (a+b \log \left (c x^n\right )\right )}{18 x^{3/2}}-\frac {\log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 x^2}-\frac {b n \log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{2 x^2}+\frac {1}{8} b^2 d^4 f^4 n^2 \log ^2(x)+\frac {1}{4} b^2 d^4 f^4 n^2 \log \left (d f \sqrt {x}+1\right )-\frac {1}{8} b^2 d^4 f^4 n^2 \log (x)-\frac {21 b^2 d^3 f^3 n^2}{4 \sqrt {x}}+\frac {7 b^2 d^2 f^2 n^2}{8 x}-\frac {37 b^2 d f n^2}{108 x^{3/2}}-\frac {b^2 n^2 \log \left (d f \sqrt {x}+1\right )}{4 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(Log[d*(d^(-1) + f*Sqrt[x])]*(a + b*Log[c*x^n])^2)/x^3,x]

[Out]

(-37*b^2*d*f*n^2)/(108*x^(3/2)) + (7*b^2*d^2*f^2*n^2)/(8*x) - (21*b^2*d^3*f^3*n^2)/(4*Sqrt[x]) + (b^2*d^4*f^4*
n^2*Log[1 + d*f*Sqrt[x]])/4 - (b^2*n^2*Log[1 + d*f*Sqrt[x]])/(4*x^2) - (b^2*d^4*f^4*n^2*Log[x])/8 + (b^2*d^4*f
^4*n^2*Log[x]^2)/8 - (7*b*d*f*n*(a + b*Log[c*x^n]))/(18*x^(3/2)) + (3*b*d^2*f^2*n*(a + b*Log[c*x^n]))/(4*x) -
(5*b*d^3*f^3*n*(a + b*Log[c*x^n]))/(2*Sqrt[x]) + (b*d^4*f^4*n*Log[1 + d*f*Sqrt[x]]*(a + b*Log[c*x^n]))/2 - (b*
n*Log[1 + d*f*Sqrt[x]]*(a + b*Log[c*x^n]))/(2*x^2) - (b*d^4*f^4*n*Log[x]*(a + b*Log[c*x^n]))/4 - (d*f*(a + b*L
og[c*x^n])^2)/(6*x^(3/2)) + (d^2*f^2*(a + b*Log[c*x^n])^2)/(4*x) - (d^3*f^3*(a + b*Log[c*x^n])^2)/(2*Sqrt[x])
+ (d^4*f^4*Log[1 + d*f*Sqrt[x]]*(a + b*Log[c*x^n])^2)/2 - (Log[1 + d*f*Sqrt[x]]*(a + b*Log[c*x^n])^2)/(2*x^2)
- (d^4*f^4*(a + b*Log[c*x^n])^3)/(12*b*n) + b^2*d^4*f^4*n^2*PolyLog[2, -(d*f*Sqrt[x])] + 2*b*d^4*f^4*n*(a + b*
Log[c*x^n])*PolyLog[2, -(d*f*Sqrt[x])] - 4*b^2*d^4*f^4*n^2*PolyLog[3, -(d*f*Sqrt[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 30

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

Rule 46

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}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && Lt
Q[m + n + 2, 0])

Rule 2338

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 2339

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 2341

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 2413

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.) + Log[(f_.)*(x_)^(r_.)]*(e_.))*((g_.)*(x_))^(m_.), x_Sy
mbol] :> With[{u = IntHide[(g*x)^m*(a + b*Log[c*x^n])^p, x]}, Dist[d + e*Log[f*x^r], u, x] - Dist[e*r, Int[Sim
plifyIntegrand[u/x, x], x], x]] /; FreeQ[{a, b, c, d, e, f, g, m, n, p, r}, x] &&  !(EqQ[p, 1] && EqQ[a, 0] &&
NeQ[d, 0])

Rule 2421

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.))/(x_), x_Symbol] :> Simp
[(-PolyLog[2, (-d)*f*x^m])*((a + b*Log[c*x^n])^p/m), x] + Dist[b*n*(p/m), Int[PolyLog[2, (-d)*f*x^m]*((a + b*L
og[c*x^n])^(p - 1)/x), x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[p, 0] && EqQ[d*e, 1]

Rule 2423

Int[Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))^(r_.)]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((g_.)*(x_))^(q_.), x_Sym
bol] :> With[{u = IntHide[(g*x)^q*Log[d*(e + f*x^m)^r], x]}, Dist[a + b*Log[c*x^n], u, x] - Dist[b*n, Int[Dist
[1/x, u, x], x], x]] /; FreeQ[{a, b, c, d, e, f, g, r, m, n, q}, x] && (IntegerQ[(q + 1)/m] || (RationalQ[m] &
& RationalQ[q])) && NeQ[q, -1]

Rule 2424

Int[Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((g_.)*(x_))^(q_.), x_Sym
bol] :> With[{u = IntHide[(g*x)^q*Log[d*(e + f*x^m)], x]}, Dist[(a + b*Log[c*x^n])^p, u, x] - Dist[b*n*p, Int[
Dist[(a + b*Log[c*x^n])^(p - 1)/x, u, x], x], x]] /; FreeQ[{a, b, c, d, e, f, g, m, n, q}, x] && IGtQ[p, 0] &&
RationalQ[m] && RationalQ[q] && NeQ[q, -1] && (EqQ[p, 1] || (FractionQ[m] && IntegerQ[(q + 1)/m]) || (IGtQ[q,
0] && IntegerQ[(q + 1)/m] && EqQ[d*e, 1]))

Rule 2438

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 2442

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 2504

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

Rule 6724

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

Rubi steps

\begin {align*} \int \frac {\log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x^3} \, dx &=-\frac {d f \left (a+b \log \left (c x^n\right )\right )^2}{6 x^{3/2}}+\frac {d^2 f^2 \left (a+b \log \left (c x^n\right )\right )^2}{4 x}-\frac {d^3 f^3 \left (a+b \log \left (c x^n\right )\right )^2}{2 \sqrt {x}}+\frac {1}{2} d^4 f^4 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac {\log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 x^2}-\frac {1}{4} d^4 f^4 \log (x) \left (a+b \log \left (c x^n\right )\right )^2-(2 b n) \int \left (-\frac {d f \left (a+b \log \left (c x^n\right )\right )}{6 x^{5/2}}+\frac {d^2 f^2 \left (a+b \log \left (c x^n\right )\right )}{4 x^2}-\frac {d^3 f^3 \left (a+b \log \left (c x^n\right )\right )}{2 x^{3/2}}-\frac {\log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 x^3}+\frac {d^4 f^4 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 x}-\frac {d^4 f^4 \log (x) \left (a+b \log \left (c x^n\right )\right )}{4 x}\right ) \, dx\\ &=-\frac {d f \left (a+b \log \left (c x^n\right )\right )^2}{6 x^{3/2}}+\frac {d^2 f^2 \left (a+b \log \left (c x^n\right )\right )^2}{4 x}-\frac {d^3 f^3 \left (a+b \log \left (c x^n\right )\right )^2}{2 \sqrt {x}}+\frac {1}{2} d^4 f^4 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac {\log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 x^2}-\frac {1}{4} d^4 f^4 \log (x) \left (a+b \log \left (c x^n\right )\right )^2+(b n) \int \frac {\log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx+\frac {1}{3} (b d f n) \int \frac {a+b \log \left (c x^n\right )}{x^{5/2}} \, dx-\frac {1}{2} \left (b d^2 f^2 n\right ) \int \frac {a+b \log \left (c x^n\right )}{x^2} \, dx+\left (b d^3 f^3 n\right ) \int \frac {a+b \log \left (c x^n\right )}{x^{3/2}} \, dx+\frac {1}{2} \left (b d^4 f^4 n\right ) \int \frac {\log (x) \left (a+b \log \left (c x^n\right )\right )}{x} \, dx-\left (b d^4 f^4 n\right ) \int \frac {\log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{x} \, dx\\ &=-\frac {4 b^2 d f n^2}{27 x^{3/2}}+\frac {b^2 d^2 f^2 n^2}{2 x}-\frac {4 b^2 d^3 f^3 n^2}{\sqrt {x}}-\frac {7 b d f n \left (a+b \log \left (c x^n\right )\right )}{18 x^{3/2}}+\frac {3 b d^2 f^2 n \left (a+b \log \left (c x^n\right )\right )}{4 x}-\frac {5 b d^3 f^3 n \left (a+b \log \left (c x^n\right )\right )}{2 \sqrt {x}}+\frac {1}{2} b d^4 f^4 n \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {b n \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac {1}{4} b d^4 f^4 n \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac {d f \left (a+b \log \left (c x^n\right )\right )^2}{6 x^{3/2}}+\frac {d^2 f^2 \left (a+b \log \left (c x^n\right )\right )^2}{4 x}-\frac {d^3 f^3 \left (a+b \log \left (c x^n\right )\right )^2}{2 \sqrt {x}}+\frac {1}{2} d^4 f^4 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac {\log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 x^2}+2 b d^4 f^4 n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-d f \sqrt {x}\right )-\frac {1}{2} \left (b d^4 f^4 n\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 b n x} \, dx-\left (b^2 n^2\right ) \int \left (-\frac {d f}{6 x^{5/2}}+\frac {d^2 f^2}{4 x^2}-\frac {d^3 f^3}{2 x^{3/2}}-\frac {\log \left (1+d f \sqrt {x}\right )}{2 x^3}+\frac {d^4 f^4 \log \left (1+d f \sqrt {x}\right )}{2 x}-\frac {d^4 f^4 \log (x)}{4 x}\right ) \, dx-\left (2 b^2 d^4 f^4 n^2\right ) \int \frac {\text {Li}_2\left (-d f \sqrt {x}\right )}{x} \, dx\\ &=-\frac {7 b^2 d f n^2}{27 x^{3/2}}+\frac {3 b^2 d^2 f^2 n^2}{4 x}-\frac {5 b^2 d^3 f^3 n^2}{\sqrt {x}}-\frac {7 b d f n \left (a+b \log \left (c x^n\right )\right )}{18 x^{3/2}}+\frac {3 b d^2 f^2 n \left (a+b \log \left (c x^n\right )\right )}{4 x}-\frac {5 b d^3 f^3 n \left (a+b \log \left (c x^n\right )\right )}{2 \sqrt {x}}+\frac {1}{2} b d^4 f^4 n \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {b n \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac {1}{4} b d^4 f^4 n \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac {d f \left (a+b \log \left (c x^n\right )\right )^2}{6 x^{3/2}}+\frac {d^2 f^2 \left (a+b \log \left (c x^n\right )\right )^2}{4 x}-\frac {d^3 f^3 \left (a+b \log \left (c x^n\right )\right )^2}{2 \sqrt {x}}+\frac {1}{2} d^4 f^4 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac {\log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 x^2}+2 b d^4 f^4 n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-d f \sqrt {x}\right )-4 b^2 d^4 f^4 n^2 \text {Li}_3\left (-d f \sqrt {x}\right )-\frac {1}{4} \left (d^4 f^4\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x} \, dx+\frac {1}{2} \left (b^2 n^2\right ) \int \frac {\log \left (1+d f \sqrt {x}\right )}{x^3} \, dx+\frac {1}{4} \left (b^2 d^4 f^4 n^2\right ) \int \frac {\log (x)}{x} \, dx-\frac {1}{2} \left (b^2 d^4 f^4 n^2\right ) \int \frac {\log \left (1+d f \sqrt {x}\right )}{x} \, dx\\ &=-\frac {7 b^2 d f n^2}{27 x^{3/2}}+\frac {3 b^2 d^2 f^2 n^2}{4 x}-\frac {5 b^2 d^3 f^3 n^2}{\sqrt {x}}+\frac {1}{8} b^2 d^4 f^4 n^2 \log ^2(x)-\frac {7 b d f n \left (a+b \log \left (c x^n\right )\right )}{18 x^{3/2}}+\frac {3 b d^2 f^2 n \left (a+b \log \left (c x^n\right )\right )}{4 x}-\frac {5 b d^3 f^3 n \left (a+b \log \left (c x^n\right )\right )}{2 \sqrt {x}}+\frac {1}{2} b d^4 f^4 n \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {b n \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac {1}{4} b d^4 f^4 n \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac {d f \left (a+b \log \left (c x^n\right )\right )^2}{6 x^{3/2}}+\frac {d^2 f^2 \left (a+b \log \left (c x^n\right )\right )^2}{4 x}-\frac {d^3 f^3 \left (a+b \log \left (c x^n\right )\right )^2}{2 \sqrt {x}}+\frac {1}{2} d^4 f^4 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac {\log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 x^2}+b^2 d^4 f^4 n^2 \text {Li}_2\left (-d f \sqrt {x}\right )+2 b d^4 f^4 n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-d f \sqrt {x}\right )-4 b^2 d^4 f^4 n^2 \text {Li}_3\left (-d f \sqrt {x}\right )-\frac {\left (d^4 f^4\right ) \text {Subst}\left (\int x^2 \, dx,x,a+b \log \left (c x^n\right )\right )}{4 b n}+\left (b^2 n^2\right ) \text {Subst}\left (\int \frac {\log (1+d f x)}{x^5} \, dx,x,\sqrt {x}\right )\\ &=-\frac {7 b^2 d f n^2}{27 x^{3/2}}+\frac {3 b^2 d^2 f^2 n^2}{4 x}-\frac {5 b^2 d^3 f^3 n^2}{\sqrt {x}}-\frac {b^2 n^2 \log \left (1+d f \sqrt {x}\right )}{4 x^2}+\frac {1}{8} b^2 d^4 f^4 n^2 \log ^2(x)-\frac {7 b d f n \left (a+b \log \left (c x^n\right )\right )}{18 x^{3/2}}+\frac {3 b d^2 f^2 n \left (a+b \log \left (c x^n\right )\right )}{4 x}-\frac {5 b d^3 f^3 n \left (a+b \log \left (c x^n\right )\right )}{2 \sqrt {x}}+\frac {1}{2} b d^4 f^4 n \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {b n \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac {1}{4} b d^4 f^4 n \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac {d f \left (a+b \log \left (c x^n\right )\right )^2}{6 x^{3/2}}+\frac {d^2 f^2 \left (a+b \log \left (c x^n\right )\right )^2}{4 x}-\frac {d^3 f^3 \left (a+b \log \left (c x^n\right )\right )^2}{2 \sqrt {x}}+\frac {1}{2} d^4 f^4 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac {\log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 x^2}-\frac {d^4 f^4 \left (a+b \log \left (c x^n\right )\right )^3}{12 b n}+b^2 d^4 f^4 n^2 \text {Li}_2\left (-d f \sqrt {x}\right )+2 b d^4 f^4 n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-d f \sqrt {x}\right )-4 b^2 d^4 f^4 n^2 \text {Li}_3\left (-d f \sqrt {x}\right )+\frac {1}{4} \left (b^2 d f n^2\right ) \text {Subst}\left (\int \frac {1}{x^4 (1+d f x)} \, dx,x,\sqrt {x}\right )\\ &=-\frac {7 b^2 d f n^2}{27 x^{3/2}}+\frac {3 b^2 d^2 f^2 n^2}{4 x}-\frac {5 b^2 d^3 f^3 n^2}{\sqrt {x}}-\frac {b^2 n^2 \log \left (1+d f \sqrt {x}\right )}{4 x^2}+\frac {1}{8} b^2 d^4 f^4 n^2 \log ^2(x)-\frac {7 b d f n \left (a+b \log \left (c x^n\right )\right )}{18 x^{3/2}}+\frac {3 b d^2 f^2 n \left (a+b \log \left (c x^n\right )\right )}{4 x}-\frac {5 b d^3 f^3 n \left (a+b \log \left (c x^n\right )\right )}{2 \sqrt {x}}+\frac {1}{2} b d^4 f^4 n \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {b n \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac {1}{4} b d^4 f^4 n \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac {d f \left (a+b \log \left (c x^n\right )\right )^2}{6 x^{3/2}}+\frac {d^2 f^2 \left (a+b \log \left (c x^n\right )\right )^2}{4 x}-\frac {d^3 f^3 \left (a+b \log \left (c x^n\right )\right )^2}{2 \sqrt {x}}+\frac {1}{2} d^4 f^4 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac {\log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 x^2}-\frac {d^4 f^4 \left (a+b \log \left (c x^n\right )\right )^3}{12 b n}+b^2 d^4 f^4 n^2 \text {Li}_2\left (-d f \sqrt {x}\right )+2 b d^4 f^4 n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-d f \sqrt {x}\right )-4 b^2 d^4 f^4 n^2 \text {Li}_3\left (-d f \sqrt {x}\right )+\frac {1}{4} \left (b^2 d f n^2\right ) \text {Subst}\left (\int \left (\frac {1}{x^4}-\frac {d f}{x^3}+\frac {d^2 f^2}{x^2}-\frac {d^3 f^3}{x}+\frac {d^4 f^4}{1+d f x}\right ) \, dx,x,\sqrt {x}\right )\\ &=-\frac {37 b^2 d f n^2}{108 x^{3/2}}+\frac {7 b^2 d^2 f^2 n^2}{8 x}-\frac {21 b^2 d^3 f^3 n^2}{4 \sqrt {x}}+\frac {1}{4} b^2 d^4 f^4 n^2 \log \left (1+d f \sqrt {x}\right )-\frac {b^2 n^2 \log \left (1+d f \sqrt {x}\right )}{4 x^2}-\frac {1}{8} b^2 d^4 f^4 n^2 \log (x)+\frac {1}{8} b^2 d^4 f^4 n^2 \log ^2(x)-\frac {7 b d f n \left (a+b \log \left (c x^n\right )\right )}{18 x^{3/2}}+\frac {3 b d^2 f^2 n \left (a+b \log \left (c x^n\right )\right )}{4 x}-\frac {5 b d^3 f^3 n \left (a+b \log \left (c x^n\right )\right )}{2 \sqrt {x}}+\frac {1}{2} b d^4 f^4 n \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {b n \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac {1}{4} b d^4 f^4 n \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac {d f \left (a+b \log \left (c x^n\right )\right )^2}{6 x^{3/2}}+\frac {d^2 f^2 \left (a+b \log \left (c x^n\right )\right )^2}{4 x}-\frac {d^3 f^3 \left (a+b \log \left (c x^n\right )\right )^2}{2 \sqrt {x}}+\frac {1}{2} d^4 f^4 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac {\log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 x^2}-\frac {d^4 f^4 \left (a+b \log \left (c x^n\right )\right )^3}{12 b n}+b^2 d^4 f^4 n^2 \text {Li}_2\left (-d f \sqrt {x}\right )+2 b d^4 f^4 n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-d f \sqrt {x}\right )-4 b^2 d^4 f^4 n^2 \text {Li}_3\left (-d f \sqrt {x}\right )\\ \end {align*}

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Mathematica [A]
time = 0.34, size = 881, normalized size = 1.59 \begin {gather*} -\frac {36 a^2 d f \sqrt {x}+84 a b d f n \sqrt {x}+74 b^2 d f n^2 \sqrt {x}-54 a^2 d^2 f^2 x-162 a b d^2 f^2 n x-189 b^2 d^2 f^2 n^2 x+108 a^2 d^3 f^3 x^{3/2}+540 a b d^3 f^3 n x^{3/2}+1134 b^2 d^3 f^3 n^2 x^{3/2}+108 a^2 \log \left (1+d f \sqrt {x}\right )+108 a b n \log \left (1+d f \sqrt {x}\right )+54 b^2 n^2 \log \left (1+d f \sqrt {x}\right )-108 a^2 d^4 f^4 x^2 \log \left (1+d f \sqrt {x}\right )-108 a b d^4 f^4 n x^2 \log \left (1+d f \sqrt {x}\right )-54 b^2 d^4 f^4 n^2 x^2 \log \left (1+d f \sqrt {x}\right )+54 a^2 d^4 f^4 x^2 \log (x)+54 a b d^4 f^4 n x^2 \log (x)+27 b^2 d^4 f^4 n^2 x^2 \log (x)-54 a b d^4 f^4 n x^2 \log ^2(x)-27 b^2 d^4 f^4 n^2 x^2 \log ^2(x)+18 b^2 d^4 f^4 n^2 x^2 \log ^3(x)+72 a b d f \sqrt {x} \log \left (c x^n\right )+84 b^2 d f n \sqrt {x} \log \left (c x^n\right )-108 a b d^2 f^2 x \log \left (c x^n\right )-162 b^2 d^2 f^2 n x \log \left (c x^n\right )+216 a b d^3 f^3 x^{3/2} \log \left (c x^n\right )+540 b^2 d^3 f^3 n x^{3/2} \log \left (c x^n\right )+216 a b \log \left (1+d f \sqrt {x}\right ) \log \left (c x^n\right )+108 b^2 n \log \left (1+d f \sqrt {x}\right ) \log \left (c x^n\right )-216 a b d^4 f^4 x^2 \log \left (1+d f \sqrt {x}\right ) \log \left (c x^n\right )-108 b^2 d^4 f^4 n x^2 \log \left (1+d f \sqrt {x}\right ) \log \left (c x^n\right )+108 a b d^4 f^4 x^2 \log (x) \log \left (c x^n\right )+54 b^2 d^4 f^4 n x^2 \log (x) \log \left (c x^n\right )-54 b^2 d^4 f^4 n x^2 \log ^2(x) \log \left (c x^n\right )+36 b^2 d f \sqrt {x} \log ^2\left (c x^n\right )-54 b^2 d^2 f^2 x \log ^2\left (c x^n\right )+108 b^2 d^3 f^3 x^{3/2} \log ^2\left (c x^n\right )+108 b^2 \log \left (1+d f \sqrt {x}\right ) \log ^2\left (c x^n\right )-108 b^2 d^4 f^4 x^2 \log \left (1+d f \sqrt {x}\right ) \log ^2\left (c x^n\right )+54 b^2 d^4 f^4 x^2 \log (x) \log ^2\left (c x^n\right )-216 b d^4 f^4 n x^2 \left (2 a+b n+2 b \log \left (c x^n\right )\right ) \text {Li}_2\left (-d f \sqrt {x}\right )+864 b^2 d^4 f^4 n^2 x^2 \text {Li}_3\left (-d f \sqrt {x}\right )}{216 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(Log[d*(d^(-1) + f*Sqrt[x])]*(a + b*Log[c*x^n])^2)/x^3,x]

[Out]

-1/216*(36*a^2*d*f*Sqrt[x] + 84*a*b*d*f*n*Sqrt[x] + 74*b^2*d*f*n^2*Sqrt[x] - 54*a^2*d^2*f^2*x - 162*a*b*d^2*f^
2*n*x - 189*b^2*d^2*f^2*n^2*x + 108*a^2*d^3*f^3*x^(3/2) + 540*a*b*d^3*f^3*n*x^(3/2) + 1134*b^2*d^3*f^3*n^2*x^(
3/2) + 108*a^2*Log[1 + d*f*Sqrt[x]] + 108*a*b*n*Log[1 + d*f*Sqrt[x]] + 54*b^2*n^2*Log[1 + d*f*Sqrt[x]] - 108*a
^2*d^4*f^4*x^2*Log[1 + d*f*Sqrt[x]] - 108*a*b*d^4*f^4*n*x^2*Log[1 + d*f*Sqrt[x]] - 54*b^2*d^4*f^4*n^2*x^2*Log[
1 + d*f*Sqrt[x]] + 54*a^2*d^4*f^4*x^2*Log[x] + 54*a*b*d^4*f^4*n*x^2*Log[x] + 27*b^2*d^4*f^4*n^2*x^2*Log[x] - 5
4*a*b*d^4*f^4*n*x^2*Log[x]^2 - 27*b^2*d^4*f^4*n^2*x^2*Log[x]^2 + 18*b^2*d^4*f^4*n^2*x^2*Log[x]^3 + 72*a*b*d*f*
Sqrt[x]*Log[c*x^n] + 84*b^2*d*f*n*Sqrt[x]*Log[c*x^n] - 108*a*b*d^2*f^2*x*Log[c*x^n] - 162*b^2*d^2*f^2*n*x*Log[
c*x^n] + 216*a*b*d^3*f^3*x^(3/2)*Log[c*x^n] + 540*b^2*d^3*f^3*n*x^(3/2)*Log[c*x^n] + 216*a*b*Log[1 + d*f*Sqrt[
x]]*Log[c*x^n] + 108*b^2*n*Log[1 + d*f*Sqrt[x]]*Log[c*x^n] - 216*a*b*d^4*f^4*x^2*Log[1 + d*f*Sqrt[x]]*Log[c*x^
n] - 108*b^2*d^4*f^4*n*x^2*Log[1 + d*f*Sqrt[x]]*Log[c*x^n] + 108*a*b*d^4*f^4*x^2*Log[x]*Log[c*x^n] + 54*b^2*d^
4*f^4*n*x^2*Log[x]*Log[c*x^n] - 54*b^2*d^4*f^4*n*x^2*Log[x]^2*Log[c*x^n] + 36*b^2*d*f*Sqrt[x]*Log[c*x^n]^2 - 5
4*b^2*d^2*f^2*x*Log[c*x^n]^2 + 108*b^2*d^3*f^3*x^(3/2)*Log[c*x^n]^2 + 108*b^2*Log[1 + d*f*Sqrt[x]]*Log[c*x^n]^
2 - 108*b^2*d^4*f^4*x^2*Log[1 + d*f*Sqrt[x]]*Log[c*x^n]^2 + 54*b^2*d^4*f^4*x^2*Log[x]*Log[c*x^n]^2 - 216*b*d^4
*f^4*n*x^2*(2*a + b*n + 2*b*Log[c*x^n])*PolyLog[2, -(d*f*Sqrt[x])] + 864*b^2*d^4*f^4*n^2*x^2*PolyLog[3, -(d*f*
Sqrt[x])])/x^2

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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \ln \left (c \,x^{n}\right )\right )^{2} \ln \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right )}{x^{3}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*ln(c*x^n))^2*ln(d*(1/d+f*x^(1/2)))/x^3,x)

[Out]

int((a+b*ln(c*x^n))^2*ln(d*(1/d+f*x^(1/2)))/x^3,x)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))^2*log(d*(1/d+f*x^(1/2)))/x^3,x, algorithm="maxima")

[Out]

integrate((b*log(c*x^n) + a)^2*log((f*sqrt(x) + 1/d)*d)/x^3, x)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))^2*log(d*(1/d+f*x^(1/2)))/x^3,x, algorithm="fricas")

[Out]

integral((b^2*log(c*x^n)^2 + 2*a*b*log(c*x^n) + a^2)*log(d*f*sqrt(x) + 1)/x^3, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*ln(c*x**n))**2*ln(d*(1/d+f*x**(1/2)))/x**3,x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 3005 deep

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))^2*log(d*(1/d+f*x^(1/2)))/x^3,x, algorithm="giac")

[Out]

integrate((b*log(c*x^n) + a)^2*log((f*sqrt(x) + 1/d)*d)/x^3, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\ln \left (d\,\left (f\,\sqrt {x}+\frac {1}{d}\right )\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2}{x^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(d*(f*x^(1/2) + 1/d))*(a + b*log(c*x^n))^2)/x^3,x)

[Out]

int((log(d*(f*x^(1/2) + 1/d))*(a + b*log(c*x^n))^2)/x^3, x)

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