3.54 \(\int x \log (d (\frac{1}{d}+f \sqrt{x})) (a+b \log (c x^n))^2 \, dx\)

Optimal. Leaf size=557 \[ -\frac{2 b n \text{PolyLog}\left (2,-d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^4 f^4}+\frac{b^2 n^2 \text{PolyLog}\left (2,-d f \sqrt{x}\right )}{d^4 f^4}+\frac{4 b^2 n^2 \text{PolyLog}\left (3,-d f \sqrt{x}\right )}{d^4 f^4}+\frac{b n x \left (a+b \log \left (c x^n\right )\right )}{4 d^2 f^2}+\frac{b n \log \left (d f \sqrt{x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{2 d^4 f^4}-\frac{5 b n \sqrt{x} \left (a+b \log \left (c x^n\right )\right )}{2 d^3 f^3}-\frac{x \left (a+b \log \left (c x^n\right )\right )^2}{4 d^2 f^2}-\frac{\log \left (d f \sqrt{x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 d^4 f^4}+\frac{\sqrt{x} \left (a+b \log \left (c x^n\right )\right )^2}{2 d^3 f^3}-\frac{7 b n x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{18 d f}-\frac{1}{2} b n x^2 \log \left (d f \sqrt{x}+1\right ) \left (a+b \log \left (c x^n\right )\right )+\frac{x^{3/2} \left (a+b \log \left (c x^n\right )\right )^2}{6 d f}+\frac{1}{2} x^2 \log \left (d f \sqrt{x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2+\frac{1}{4} b n x^2 \left (a+b \log \left (c x^n\right )\right )-\frac{1}{8} x^2 \left (a+b \log \left (c x^n\right )\right )^2+\frac{a b n x}{2 d^2 f^2}+\frac{b^2 n x \log \left (c x^n\right )}{2 d^2 f^2}-\frac{7 b^2 n^2 x}{8 d^2 f^2}+\frac{21 b^2 n^2 \sqrt{x}}{4 d^3 f^3}-\frac{b^2 n^2 \log \left (d f \sqrt{x}+1\right )}{4 d^4 f^4}+\frac{37 b^2 n^2 x^{3/2}}{108 d f}+\frac{1}{4} b^2 n^2 x^2 \log \left (d f \sqrt{x}+1\right )-\frac{3}{16} b^2 n^2 x^2 \]

[Out]

(21*b^2*n^2*Sqrt[x])/(4*d^3*f^3) + (a*b*n*x)/(2*d^2*f^2) - (7*b^2*n^2*x)/(8*d^2*f^2) + (37*b^2*n^2*x^(3/2))/(1
08*d*f) - (3*b^2*n^2*x^2)/16 - (b^2*n^2*Log[1 + d*f*Sqrt[x]])/(4*d^4*f^4) + (b^2*n^2*x^2*Log[1 + d*f*Sqrt[x]])
/4 + (b^2*n*x*Log[c*x^n])/(2*d^2*f^2) - (5*b*n*Sqrt[x]*(a + b*Log[c*x^n]))/(2*d^3*f^3) + (b*n*x*(a + b*Log[c*x
^n]))/(4*d^2*f^2) - (7*b*n*x^(3/2)*(a + b*Log[c*x^n]))/(18*d*f) + (b*n*x^2*(a + b*Log[c*x^n]))/4 + (b*n*Log[1
+ d*f*Sqrt[x]]*(a + b*Log[c*x^n]))/(2*d^4*f^4) - (b*n*x^2*Log[1 + d*f*Sqrt[x]]*(a + b*Log[c*x^n]))/2 + (Sqrt[x
]*(a + b*Log[c*x^n])^2)/(2*d^3*f^3) - (x*(a + b*Log[c*x^n])^2)/(4*d^2*f^2) + (x^(3/2)*(a + b*Log[c*x^n])^2)/(6
*d*f) - (x^2*(a + b*Log[c*x^n])^2)/8 - (Log[1 + d*f*Sqrt[x]]*(a + b*Log[c*x^n])^2)/(2*d^4*f^4) + (x^2*Log[1 +
d*f*Sqrt[x]]*(a + b*Log[c*x^n])^2)/2 + (b^2*n^2*PolyLog[2, -(d*f*Sqrt[x])])/(d^4*f^4) - (2*b*n*(a + b*Log[c*x^
n])*PolyLog[2, -(d*f*Sqrt[x])])/(d^4*f^4) + (4*b^2*n^2*PolyLog[3, -(d*f*Sqrt[x])])/(d^4*f^4)

________________________________________________________________________________________

Rubi [A]  time = 0.464518, antiderivative size = 557, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 10, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.357, Rules used = {2454, 2395, 43, 2377, 2295, 2304, 2374, 6589, 2376, 2391} \[ -\frac{2 b n \text{PolyLog}\left (2,-d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^4 f^4}+\frac{b^2 n^2 \text{PolyLog}\left (2,-d f \sqrt{x}\right )}{d^4 f^4}+\frac{4 b^2 n^2 \text{PolyLog}\left (3,-d f \sqrt{x}\right )}{d^4 f^4}+\frac{b n x \left (a+b \log \left (c x^n\right )\right )}{4 d^2 f^2}+\frac{b n \log \left (d f \sqrt{x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{2 d^4 f^4}-\frac{5 b n \sqrt{x} \left (a+b \log \left (c x^n\right )\right )}{2 d^3 f^3}-\frac{x \left (a+b \log \left (c x^n\right )\right )^2}{4 d^2 f^2}-\frac{\log \left (d f \sqrt{x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 d^4 f^4}+\frac{\sqrt{x} \left (a+b \log \left (c x^n\right )\right )^2}{2 d^3 f^3}-\frac{7 b n x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{18 d f}-\frac{1}{2} b n x^2 \log \left (d f \sqrt{x}+1\right ) \left (a+b \log \left (c x^n\right )\right )+\frac{x^{3/2} \left (a+b \log \left (c x^n\right )\right )^2}{6 d f}+\frac{1}{2} x^2 \log \left (d f \sqrt{x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2+\frac{1}{4} b n x^2 \left (a+b \log \left (c x^n\right )\right )-\frac{1}{8} x^2 \left (a+b \log \left (c x^n\right )\right )^2+\frac{a b n x}{2 d^2 f^2}+\frac{b^2 n x \log \left (c x^n\right )}{2 d^2 f^2}-\frac{7 b^2 n^2 x}{8 d^2 f^2}+\frac{21 b^2 n^2 \sqrt{x}}{4 d^3 f^3}-\frac{b^2 n^2 \log \left (d f \sqrt{x}+1\right )}{4 d^4 f^4}+\frac{37 b^2 n^2 x^{3/2}}{108 d f}+\frac{1}{4} b^2 n^2 x^2 \log \left (d f \sqrt{x}+1\right )-\frac{3}{16} b^2 n^2 x^2 \]

Antiderivative was successfully verified.

[In]

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

[Out]

(21*b^2*n^2*Sqrt[x])/(4*d^3*f^3) + (a*b*n*x)/(2*d^2*f^2) - (7*b^2*n^2*x)/(8*d^2*f^2) + (37*b^2*n^2*x^(3/2))/(1
08*d*f) - (3*b^2*n^2*x^2)/16 - (b^2*n^2*Log[1 + d*f*Sqrt[x]])/(4*d^4*f^4) + (b^2*n^2*x^2*Log[1 + d*f*Sqrt[x]])
/4 + (b^2*n*x*Log[c*x^n])/(2*d^2*f^2) - (5*b*n*Sqrt[x]*(a + b*Log[c*x^n]))/(2*d^3*f^3) + (b*n*x*(a + b*Log[c*x
^n]))/(4*d^2*f^2) - (7*b*n*x^(3/2)*(a + b*Log[c*x^n]))/(18*d*f) + (b*n*x^2*(a + b*Log[c*x^n]))/4 + (b*n*Log[1
+ d*f*Sqrt[x]]*(a + b*Log[c*x^n]))/(2*d^4*f^4) - (b*n*x^2*Log[1 + d*f*Sqrt[x]]*(a + b*Log[c*x^n]))/2 + (Sqrt[x
]*(a + b*Log[c*x^n])^2)/(2*d^3*f^3) - (x*(a + b*Log[c*x^n])^2)/(4*d^2*f^2) + (x^(3/2)*(a + b*Log[c*x^n])^2)/(6
*d*f) - (x^2*(a + b*Log[c*x^n])^2)/8 - (Log[1 + d*f*Sqrt[x]]*(a + b*Log[c*x^n])^2)/(2*d^4*f^4) + (x^2*Log[1 +
d*f*Sqrt[x]]*(a + b*Log[c*x^n])^2)/2 + (b^2*n^2*PolyLog[2, -(d*f*Sqrt[x])])/(d^4*f^4) - (2*b*n*(a + b*Log[c*x^
n])*PolyLog[2, -(d*f*Sqrt[x])])/(d^4*f^4) + (4*b^2*n^2*PolyLog[3, -(d*f*Sqrt[x])])/(d^4*f^4)

Rule 2454

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 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 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 2377

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 2295

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, 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 2374

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.))/(x_), x_Symbol] :> -Sim
p[(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*Log
[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 6589

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]

Rule 2376

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 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]

Rubi steps

\begin{align*} \int x \log \left (d \left (\frac{1}{d}+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2 \, dx &=\frac{\sqrt{x} \left (a+b \log \left (c x^n\right )\right )^2}{2 d^3 f^3}-\frac{x \left (a+b \log \left (c x^n\right )\right )^2}{4 d^2 f^2}+\frac{x^{3/2} \left (a+b \log \left (c x^n\right )\right )^2}{6 d f}-\frac{1}{8} x^2 \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 d^4 f^4}+\frac{1}{2} x^2 \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^2-(2 b n) \int \left (-\frac{a+b \log \left (c x^n\right )}{4 d^2 f^2}+\frac{a+b \log \left (c x^n\right )}{2 d^3 f^3 \sqrt{x}}+\frac{\sqrt{x} \left (a+b \log \left (c x^n\right )\right )}{6 d f}-\frac{1}{8} x \left (a+b \log \left (c x^n\right )\right )-\frac{\log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 d^4 f^4 x}+\frac{1}{2} x \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )\right ) \, dx\\ &=\frac{\sqrt{x} \left (a+b \log \left (c x^n\right )\right )^2}{2 d^3 f^3}-\frac{x \left (a+b \log \left (c x^n\right )\right )^2}{4 d^2 f^2}+\frac{x^{3/2} \left (a+b \log \left (c x^n\right )\right )^2}{6 d f}-\frac{1}{8} x^2 \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 d^4 f^4}+\frac{1}{2} x^2 \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^2+\frac{1}{4} (b n) \int x \left (a+b \log \left (c x^n\right )\right ) \, dx-(b n) \int x \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx+\frac{(b n) \int \frac{\log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{x} \, dx}{d^4 f^4}-\frac{(b n) \int \frac{a+b \log \left (c x^n\right )}{\sqrt{x}} \, dx}{d^3 f^3}+\frac{(b n) \int \left (a+b \log \left (c x^n\right )\right ) \, dx}{2 d^2 f^2}-\frac{(b n) \int \sqrt{x} \left (a+b \log \left (c x^n\right )\right ) \, dx}{3 d f}\\ &=\frac{4 b^2 n^2 \sqrt{x}}{d^3 f^3}+\frac{a b n x}{2 d^2 f^2}+\frac{4 b^2 n^2 x^{3/2}}{27 d f}-\frac{1}{16} b^2 n^2 x^2-\frac{5 b n \sqrt{x} \left (a+b \log \left (c x^n\right )\right )}{2 d^3 f^3}+\frac{b n x \left (a+b \log \left (c x^n\right )\right )}{4 d^2 f^2}-\frac{7 b n x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{18 d f}+\frac{1}{4} b n x^2 \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 d^4 f^4}-\frac{1}{2} b n x^2 \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )+\frac{\sqrt{x} \left (a+b \log \left (c x^n\right )\right )^2}{2 d^3 f^3}-\frac{x \left (a+b \log \left (c x^n\right )\right )^2}{4 d^2 f^2}+\frac{x^{3/2} \left (a+b \log \left (c x^n\right )\right )^2}{6 d f}-\frac{1}{8} x^2 \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 d^4 f^4}+\frac{1}{2} x^2 \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac{2 b n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-d f \sqrt{x}\right )}{d^4 f^4}+\frac{\left (b^2 n\right ) \int \log \left (c x^n\right ) \, dx}{2 d^2 f^2}+\left (b^2 n^2\right ) \int \left (-\frac{1}{4 d^2 f^2}+\frac{1}{2 d^3 f^3 \sqrt{x}}+\frac{\sqrt{x}}{6 d f}-\frac{x}{8}-\frac{\log \left (1+d f \sqrt{x}\right )}{2 d^4 f^4 x}+\frac{1}{2} x \log \left (1+d f \sqrt{x}\right )\right ) \, dx+\frac{\left (2 b^2 n^2\right ) \int \frac{\text{Li}_2\left (-d f \sqrt{x}\right )}{x} \, dx}{d^4 f^4}\\ &=\frac{5 b^2 n^2 \sqrt{x}}{d^3 f^3}+\frac{a b n x}{2 d^2 f^2}-\frac{3 b^2 n^2 x}{4 d^2 f^2}+\frac{7 b^2 n^2 x^{3/2}}{27 d f}-\frac{1}{8} b^2 n^2 x^2+\frac{b^2 n x \log \left (c x^n\right )}{2 d^2 f^2}-\frac{5 b n \sqrt{x} \left (a+b \log \left (c x^n\right )\right )}{2 d^3 f^3}+\frac{b n x \left (a+b \log \left (c x^n\right )\right )}{4 d^2 f^2}-\frac{7 b n x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{18 d f}+\frac{1}{4} b n x^2 \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 d^4 f^4}-\frac{1}{2} b n x^2 \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )+\frac{\sqrt{x} \left (a+b \log \left (c x^n\right )\right )^2}{2 d^3 f^3}-\frac{x \left (a+b \log \left (c x^n\right )\right )^2}{4 d^2 f^2}+\frac{x^{3/2} \left (a+b \log \left (c x^n\right )\right )^2}{6 d f}-\frac{1}{8} x^2 \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 d^4 f^4}+\frac{1}{2} x^2 \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac{2 b n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-d f \sqrt{x}\right )}{d^4 f^4}+\frac{4 b^2 n^2 \text{Li}_3\left (-d f \sqrt{x}\right )}{d^4 f^4}+\frac{1}{2} \left (b^2 n^2\right ) \int x \log \left (1+d f \sqrt{x}\right ) \, dx-\frac{\left (b^2 n^2\right ) \int \frac{\log \left (1+d f \sqrt{x}\right )}{x} \, dx}{2 d^4 f^4}\\ &=\frac{5 b^2 n^2 \sqrt{x}}{d^3 f^3}+\frac{a b n x}{2 d^2 f^2}-\frac{3 b^2 n^2 x}{4 d^2 f^2}+\frac{7 b^2 n^2 x^{3/2}}{27 d f}-\frac{1}{8} b^2 n^2 x^2+\frac{b^2 n x \log \left (c x^n\right )}{2 d^2 f^2}-\frac{5 b n \sqrt{x} \left (a+b \log \left (c x^n\right )\right )}{2 d^3 f^3}+\frac{b n x \left (a+b \log \left (c x^n\right )\right )}{4 d^2 f^2}-\frac{7 b n x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{18 d f}+\frac{1}{4} b n x^2 \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 d^4 f^4}-\frac{1}{2} b n x^2 \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )+\frac{\sqrt{x} \left (a+b \log \left (c x^n\right )\right )^2}{2 d^3 f^3}-\frac{x \left (a+b \log \left (c x^n\right )\right )^2}{4 d^2 f^2}+\frac{x^{3/2} \left (a+b \log \left (c x^n\right )\right )^2}{6 d f}-\frac{1}{8} x^2 \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 d^4 f^4}+\frac{1}{2} x^2 \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^2+\frac{b^2 n^2 \text{Li}_2\left (-d f \sqrt{x}\right )}{d^4 f^4}-\frac{2 b n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-d f \sqrt{x}\right )}{d^4 f^4}+\frac{4 b^2 n^2 \text{Li}_3\left (-d f \sqrt{x}\right )}{d^4 f^4}+\left (b^2 n^2\right ) \operatorname{Subst}\left (\int x^3 \log (1+d f x) \, dx,x,\sqrt{x}\right )\\ &=\frac{5 b^2 n^2 \sqrt{x}}{d^3 f^3}+\frac{a b n x}{2 d^2 f^2}-\frac{3 b^2 n^2 x}{4 d^2 f^2}+\frac{7 b^2 n^2 x^{3/2}}{27 d f}-\frac{1}{8} b^2 n^2 x^2+\frac{1}{4} b^2 n^2 x^2 \log \left (1+d f \sqrt{x}\right )+\frac{b^2 n x \log \left (c x^n\right )}{2 d^2 f^2}-\frac{5 b n \sqrt{x} \left (a+b \log \left (c x^n\right )\right )}{2 d^3 f^3}+\frac{b n x \left (a+b \log \left (c x^n\right )\right )}{4 d^2 f^2}-\frac{7 b n x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{18 d f}+\frac{1}{4} b n x^2 \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 d^4 f^4}-\frac{1}{2} b n x^2 \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )+\frac{\sqrt{x} \left (a+b \log \left (c x^n\right )\right )^2}{2 d^3 f^3}-\frac{x \left (a+b \log \left (c x^n\right )\right )^2}{4 d^2 f^2}+\frac{x^{3/2} \left (a+b \log \left (c x^n\right )\right )^2}{6 d f}-\frac{1}{8} x^2 \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 d^4 f^4}+\frac{1}{2} x^2 \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^2+\frac{b^2 n^2 \text{Li}_2\left (-d f \sqrt{x}\right )}{d^4 f^4}-\frac{2 b n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-d f \sqrt{x}\right )}{d^4 f^4}+\frac{4 b^2 n^2 \text{Li}_3\left (-d f \sqrt{x}\right )}{d^4 f^4}-\frac{1}{4} \left (b^2 d f n^2\right ) \operatorname{Subst}\left (\int \frac{x^4}{1+d f x} \, dx,x,\sqrt{x}\right )\\ &=\frac{5 b^2 n^2 \sqrt{x}}{d^3 f^3}+\frac{a b n x}{2 d^2 f^2}-\frac{3 b^2 n^2 x}{4 d^2 f^2}+\frac{7 b^2 n^2 x^{3/2}}{27 d f}-\frac{1}{8} b^2 n^2 x^2+\frac{1}{4} b^2 n^2 x^2 \log \left (1+d f \sqrt{x}\right )+\frac{b^2 n x \log \left (c x^n\right )}{2 d^2 f^2}-\frac{5 b n \sqrt{x} \left (a+b \log \left (c x^n\right )\right )}{2 d^3 f^3}+\frac{b n x \left (a+b \log \left (c x^n\right )\right )}{4 d^2 f^2}-\frac{7 b n x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{18 d f}+\frac{1}{4} b n x^2 \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 d^4 f^4}-\frac{1}{2} b n x^2 \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )+\frac{\sqrt{x} \left (a+b \log \left (c x^n\right )\right )^2}{2 d^3 f^3}-\frac{x \left (a+b \log \left (c x^n\right )\right )^2}{4 d^2 f^2}+\frac{x^{3/2} \left (a+b \log \left (c x^n\right )\right )^2}{6 d f}-\frac{1}{8} x^2 \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 d^4 f^4}+\frac{1}{2} x^2 \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^2+\frac{b^2 n^2 \text{Li}_2\left (-d f \sqrt{x}\right )}{d^4 f^4}-\frac{2 b n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-d f \sqrt{x}\right )}{d^4 f^4}+\frac{4 b^2 n^2 \text{Li}_3\left (-d f \sqrt{x}\right )}{d^4 f^4}-\frac{1}{4} \left (b^2 d f n^2\right ) \operatorname{Subst}\left (\int \left (-\frac{1}{d^4 f^4}+\frac{x}{d^3 f^3}-\frac{x^2}{d^2 f^2}+\frac{x^3}{d f}+\frac{1}{d^4 f^4 (1+d f x)}\right ) \, dx,x,\sqrt{x}\right )\\ &=\frac{21 b^2 n^2 \sqrt{x}}{4 d^3 f^3}+\frac{a b n x}{2 d^2 f^2}-\frac{7 b^2 n^2 x}{8 d^2 f^2}+\frac{37 b^2 n^2 x^{3/2}}{108 d f}-\frac{3}{16} b^2 n^2 x^2-\frac{b^2 n^2 \log \left (1+d f \sqrt{x}\right )}{4 d^4 f^4}+\frac{1}{4} b^2 n^2 x^2 \log \left (1+d f \sqrt{x}\right )+\frac{b^2 n x \log \left (c x^n\right )}{2 d^2 f^2}-\frac{5 b n \sqrt{x} \left (a+b \log \left (c x^n\right )\right )}{2 d^3 f^3}+\frac{b n x \left (a+b \log \left (c x^n\right )\right )}{4 d^2 f^2}-\frac{7 b n x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{18 d f}+\frac{1}{4} b n x^2 \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 d^4 f^4}-\frac{1}{2} b n x^2 \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )+\frac{\sqrt{x} \left (a+b \log \left (c x^n\right )\right )^2}{2 d^3 f^3}-\frac{x \left (a+b \log \left (c x^n\right )\right )^2}{4 d^2 f^2}+\frac{x^{3/2} \left (a+b \log \left (c x^n\right )\right )^2}{6 d f}-\frac{1}{8} x^2 \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 d^4 f^4}+\frac{1}{2} x^2 \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^2+\frac{b^2 n^2 \text{Li}_2\left (-d f \sqrt{x}\right )}{d^4 f^4}-\frac{2 b n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-d f \sqrt{x}\right )}{d^4 f^4}+\frac{4 b^2 n^2 \text{Li}_3\left (-d f \sqrt{x}\right )}{d^4 f^4}\\ \end{align*}

Mathematica [A]  time = 0.398044, size = 769, normalized size = 1.38 \[ \frac{432 b n \text{PolyLog}\left (2,-d f \sqrt{x}\right ) \left (-2 a-2 b \log \left (c x^n\right )+b n\right )+1728 b^2 n^2 \text{PolyLog}\left (3,-d f \sqrt{x}\right )-54 a^2 d^4 f^4 x^2+72 a^2 d^3 f^3 x^{3/2}+216 a^2 d^4 f^4 x^2 \log \left (d f \sqrt{x}+1\right )-108 a^2 d^2 f^2 x+216 a^2 d f \sqrt{x}-216 a^2 \log \left (d f \sqrt{x}+1\right )-108 a b d^4 f^4 x^2 \log \left (c x^n\right )+432 a b d^4 f^4 x^2 \log \left (c x^n\right ) \log \left (d f \sqrt{x}+1\right )+144 a b d^3 f^3 x^{3/2} \log \left (c x^n\right )-216 a b d^2 f^2 x \log \left (c x^n\right )+432 a b d f \sqrt{x} \log \left (c x^n\right )-432 a b \log \left (c x^n\right ) \log \left (d f \sqrt{x}+1\right )+108 a b d^4 f^4 n x^2-168 a b d^3 f^3 n x^{3/2}-216 a b d^4 f^4 n x^2 \log \left (d f \sqrt{x}+1\right )+324 a b d^2 f^2 n x-1080 a b d f n \sqrt{x}+216 a b n \log \left (d f \sqrt{x}+1\right )-54 b^2 d^4 f^4 x^2 \log ^2\left (c x^n\right )+216 b^2 d^4 f^4 x^2 \log ^2\left (c x^n\right ) \log \left (d f \sqrt{x}+1\right )+72 b^2 d^3 f^3 x^{3/2} \log ^2\left (c x^n\right )-108 b^2 d^2 f^2 x \log ^2\left (c x^n\right )+108 b^2 d^4 f^4 n x^2 \log \left (c x^n\right )-216 b^2 d^4 f^4 n x^2 \log \left (c x^n\right ) \log \left (d f \sqrt{x}+1\right )-168 b^2 d^3 f^3 n x^{3/2} \log \left (c x^n\right )+324 b^2 d^2 f^2 n x \log \left (c x^n\right )+216 b^2 d f \sqrt{x} \log ^2\left (c x^n\right )-216 b^2 \log ^2\left (c x^n\right ) \log \left (d f \sqrt{x}+1\right )-1080 b^2 d f n \sqrt{x} \log \left (c x^n\right )+216 b^2 n \log \left (c x^n\right ) \log \left (d f \sqrt{x}+1\right )-81 b^2 d^4 f^4 n^2 x^2+148 b^2 d^3 f^3 n^2 x^{3/2}+108 b^2 d^4 f^4 n^2 x^2 \log \left (d f \sqrt{x}+1\right )-378 b^2 d^2 f^2 n^2 x+2268 b^2 d f n^2 \sqrt{x}-108 b^2 n^2 \log \left (d f \sqrt{x}+1\right )}{432 d^4 f^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(216*a^2*d*f*Sqrt[x] - 1080*a*b*d*f*n*Sqrt[x] + 2268*b^2*d*f*n^2*Sqrt[x] - 108*a^2*d^2*f^2*x + 324*a*b*d^2*f^2
*n*x - 378*b^2*d^2*f^2*n^2*x + 72*a^2*d^3*f^3*x^(3/2) - 168*a*b*d^3*f^3*n*x^(3/2) + 148*b^2*d^3*f^3*n^2*x^(3/2
) - 54*a^2*d^4*f^4*x^2 + 108*a*b*d^4*f^4*n*x^2 - 81*b^2*d^4*f^4*n^2*x^2 - 216*a^2*Log[1 + d*f*Sqrt[x]] + 216*a
*b*n*Log[1 + d*f*Sqrt[x]] - 108*b^2*n^2*Log[1 + d*f*Sqrt[x]] + 216*a^2*d^4*f^4*x^2*Log[1 + d*f*Sqrt[x]] - 216*
a*b*d^4*f^4*n*x^2*Log[1 + d*f*Sqrt[x]] + 108*b^2*d^4*f^4*n^2*x^2*Log[1 + d*f*Sqrt[x]] + 432*a*b*d*f*Sqrt[x]*Lo
g[c*x^n] - 1080*b^2*d*f*n*Sqrt[x]*Log[c*x^n] - 216*a*b*d^2*f^2*x*Log[c*x^n] + 324*b^2*d^2*f^2*n*x*Log[c*x^n] +
144*a*b*d^3*f^3*x^(3/2)*Log[c*x^n] - 168*b^2*d^3*f^3*n*x^(3/2)*Log[c*x^n] - 108*a*b*d^4*f^4*x^2*Log[c*x^n] +
108*b^2*d^4*f^4*n*x^2*Log[c*x^n] - 432*a*b*Log[1 + d*f*Sqrt[x]]*Log[c*x^n] + 216*b^2*n*Log[1 + d*f*Sqrt[x]]*Lo
g[c*x^n] + 432*a*b*d^4*f^4*x^2*Log[1 + d*f*Sqrt[x]]*Log[c*x^n] - 216*b^2*d^4*f^4*n*x^2*Log[1 + d*f*Sqrt[x]]*Lo
g[c*x^n] + 216*b^2*d*f*Sqrt[x]*Log[c*x^n]^2 - 108*b^2*d^2*f^2*x*Log[c*x^n]^2 + 72*b^2*d^3*f^3*x^(3/2)*Log[c*x^
n]^2 - 54*b^2*d^4*f^4*x^2*Log[c*x^n]^2 - 216*b^2*Log[1 + d*f*Sqrt[x]]*Log[c*x^n]^2 + 216*b^2*d^4*f^4*x^2*Log[1
+ d*f*Sqrt[x]]*Log[c*x^n]^2 + 432*b*n*(-2*a + b*n - 2*b*Log[c*x^n])*PolyLog[2, -(d*f*Sqrt[x])] + 1728*b^2*n^2
*PolyLog[3, -(d*f*Sqrt[x])])/(432*d^4*f^4)

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Maple [F]  time = 0.02, size = 0, normalized size = 0. \begin{align*} \int x \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) ^{2}\ln \left ( d \left ({d}^{-1}+f\sqrt{x} \right ) \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} x \log \left ({\left (f \sqrt{x} + \frac{1}{d}\right )} d\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (b^{2} x \log \left (c x^{n}\right )^{2} + 2 \, a b x \log \left (c x^{n}\right ) + a^{2} x\right )} \log \left (d f \sqrt{x} + 1\right ), x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} x \log \left ({\left (f \sqrt{x} + \frac{1}{d}\right )} d\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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