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

Optimal. Leaf size=708 \[ -\frac{4 b n \text{PolyLog}\left (2,-d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 d^6 f^6}+\frac{4 b^2 n^2 \text{PolyLog}\left (2,-d f \sqrt{x}\right )}{9 d^6 f^6}+\frac{8 b^2 n^2 \text{PolyLog}\left (3,-d f \sqrt{x}\right )}{3 d^6 f^6}-\frac{x^2 \left (a+b \log \left (c x^n\right )\right )^2}{12 d^2 f^2}+\frac{5 b n x^2 \left (a+b \log \left (c x^n\right )\right )}{36 d^2 f^2}+\frac{x^{3/2} \left (a+b \log \left (c x^n\right )\right )^2}{9 d^3 f^3}-\frac{2 b n x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{9 d^3 f^3}-\frac{x \left (a+b \log \left (c x^n\right )\right )^2}{6 d^4 f^4}+\frac{b n x \left (a+b \log \left (c x^n\right )\right )}{9 d^4 f^4}+\frac{\sqrt{x} \left (a+b \log \left (c x^n\right )\right )^2}{3 d^5 f^5}-\frac{14 b n \sqrt{x} \left (a+b \log \left (c x^n\right )\right )}{9 d^5 f^5}-\frac{\log \left (d f \sqrt{x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{3 d^6 f^6}+\frac{2 b n \log \left (d f \sqrt{x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{9 d^6 f^6}+\frac{1}{3} x^3 \log \left (d f \sqrt{x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac{2}{9} b n x^3 \log \left (d f \sqrt{x}+1\right ) \left (a+b \log \left (c x^n\right )\right )+\frac{x^{5/2} \left (a+b \log \left (c x^n\right )\right )^2}{15 d f}-\frac{22 b n x^{5/2} \left (a+b \log \left (c x^n\right )\right )}{225 d f}-\frac{1}{18} x^3 \left (a+b \log \left (c x^n\right )\right )^2+\frac{2}{27} b n x^3 \left (a+b \log \left (c x^n\right )\right )+\frac{a b n x}{3 d^4 f^4}+\frac{b^2 n x \log \left (c x^n\right )}{3 d^4 f^4}-\frac{19 b^2 n^2 x^2}{216 d^2 f^2}+\frac{14 b^2 n^2 x^{3/2}}{81 d^3 f^3}-\frac{13 b^2 n^2 x}{27 d^4 f^4}+\frac{86 b^2 n^2 \sqrt{x}}{27 d^5 f^5}-\frac{2 b^2 n^2 \log \left (d f \sqrt{x}+1\right )}{27 d^6 f^6}+\frac{182 b^2 n^2 x^{5/2}}{3375 d f}+\frac{2}{27} b^2 n^2 x^3 \log \left (d f \sqrt{x}+1\right )-\frac{1}{27} b^2 n^2 x^3 \]

[Out]

(86*b^2*n^2*Sqrt[x])/(27*d^5*f^5) + (a*b*n*x)/(3*d^4*f^4) - (13*b^2*n^2*x)/(27*d^4*f^4) + (14*b^2*n^2*x^(3/2))
/(81*d^3*f^3) - (19*b^2*n^2*x^2)/(216*d^2*f^2) + (182*b^2*n^2*x^(5/2))/(3375*d*f) - (b^2*n^2*x^3)/27 - (2*b^2*
n^2*Log[1 + d*f*Sqrt[x]])/(27*d^6*f^6) + (2*b^2*n^2*x^3*Log[1 + d*f*Sqrt[x]])/27 + (b^2*n*x*Log[c*x^n])/(3*d^4
*f^4) - (14*b*n*Sqrt[x]*(a + b*Log[c*x^n]))/(9*d^5*f^5) + (b*n*x*(a + b*Log[c*x^n]))/(9*d^4*f^4) - (2*b*n*x^(3
/2)*(a + b*Log[c*x^n]))/(9*d^3*f^3) + (5*b*n*x^2*(a + b*Log[c*x^n]))/(36*d^2*f^2) - (22*b*n*x^(5/2)*(a + b*Log
[c*x^n]))/(225*d*f) + (2*b*n*x^3*(a + b*Log[c*x^n]))/27 + (2*b*n*Log[1 + d*f*Sqrt[x]]*(a + b*Log[c*x^n]))/(9*d
^6*f^6) - (2*b*n*x^3*Log[1 + d*f*Sqrt[x]]*(a + b*Log[c*x^n]))/9 + (Sqrt[x]*(a + b*Log[c*x^n])^2)/(3*d^5*f^5) -
(x*(a + b*Log[c*x^n])^2)/(6*d^4*f^4) + (x^(3/2)*(a + b*Log[c*x^n])^2)/(9*d^3*f^3) - (x^2*(a + b*Log[c*x^n])^2
)/(12*d^2*f^2) + (x^(5/2)*(a + b*Log[c*x^n])^2)/(15*d*f) - (x^3*(a + b*Log[c*x^n])^2)/18 - (Log[1 + d*f*Sqrt[x
]]*(a + b*Log[c*x^n])^2)/(3*d^6*f^6) + (x^3*Log[1 + d*f*Sqrt[x]]*(a + b*Log[c*x^n])^2)/3 + (4*b^2*n^2*PolyLog[
2, -(d*f*Sqrt[x])])/(9*d^6*f^6) - (4*b*n*(a + b*Log[c*x^n])*PolyLog[2, -(d*f*Sqrt[x])])/(3*d^6*f^6) + (8*b^2*n
^2*PolyLog[3, -(d*f*Sqrt[x])])/(3*d^6*f^6)

________________________________________________________________________________________

Rubi [A]  time = 0.638287, antiderivative size = 708, normalized size of antiderivative = 1., number of steps used = 18, number of rules used = 10, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {2454, 2395, 43, 2377, 2295, 2304, 2374, 6589, 2376, 2391} \[ -\frac{4 b n \text{PolyLog}\left (2,-d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 d^6 f^6}+\frac{4 b^2 n^2 \text{PolyLog}\left (2,-d f \sqrt{x}\right )}{9 d^6 f^6}+\frac{8 b^2 n^2 \text{PolyLog}\left (3,-d f \sqrt{x}\right )}{3 d^6 f^6}-\frac{x^2 \left (a+b \log \left (c x^n\right )\right )^2}{12 d^2 f^2}+\frac{5 b n x^2 \left (a+b \log \left (c x^n\right )\right )}{36 d^2 f^2}+\frac{x^{3/2} \left (a+b \log \left (c x^n\right )\right )^2}{9 d^3 f^3}-\frac{2 b n x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{9 d^3 f^3}-\frac{x \left (a+b \log \left (c x^n\right )\right )^2}{6 d^4 f^4}+\frac{b n x \left (a+b \log \left (c x^n\right )\right )}{9 d^4 f^4}+\frac{\sqrt{x} \left (a+b \log \left (c x^n\right )\right )^2}{3 d^5 f^5}-\frac{14 b n \sqrt{x} \left (a+b \log \left (c x^n\right )\right )}{9 d^5 f^5}-\frac{\log \left (d f \sqrt{x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{3 d^6 f^6}+\frac{2 b n \log \left (d f \sqrt{x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{9 d^6 f^6}+\frac{1}{3} x^3 \log \left (d f \sqrt{x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac{2}{9} b n x^3 \log \left (d f \sqrt{x}+1\right ) \left (a+b \log \left (c x^n\right )\right )+\frac{x^{5/2} \left (a+b \log \left (c x^n\right )\right )^2}{15 d f}-\frac{22 b n x^{5/2} \left (a+b \log \left (c x^n\right )\right )}{225 d f}-\frac{1}{18} x^3 \left (a+b \log \left (c x^n\right )\right )^2+\frac{2}{27} b n x^3 \left (a+b \log \left (c x^n\right )\right )+\frac{a b n x}{3 d^4 f^4}+\frac{b^2 n x \log \left (c x^n\right )}{3 d^4 f^4}-\frac{19 b^2 n^2 x^2}{216 d^2 f^2}+\frac{14 b^2 n^2 x^{3/2}}{81 d^3 f^3}-\frac{13 b^2 n^2 x}{27 d^4 f^4}+\frac{86 b^2 n^2 \sqrt{x}}{27 d^5 f^5}-\frac{2 b^2 n^2 \log \left (d f \sqrt{x}+1\right )}{27 d^6 f^6}+\frac{182 b^2 n^2 x^{5/2}}{3375 d f}+\frac{2}{27} b^2 n^2 x^3 \log \left (d f \sqrt{x}+1\right )-\frac{1}{27} b^2 n^2 x^3 \]

Antiderivative was successfully verified.

[In]

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

[Out]

(86*b^2*n^2*Sqrt[x])/(27*d^5*f^5) + (a*b*n*x)/(3*d^4*f^4) - (13*b^2*n^2*x)/(27*d^4*f^4) + (14*b^2*n^2*x^(3/2))
/(81*d^3*f^3) - (19*b^2*n^2*x^2)/(216*d^2*f^2) + (182*b^2*n^2*x^(5/2))/(3375*d*f) - (b^2*n^2*x^3)/27 - (2*b^2*
n^2*Log[1 + d*f*Sqrt[x]])/(27*d^6*f^6) + (2*b^2*n^2*x^3*Log[1 + d*f*Sqrt[x]])/27 + (b^2*n*x*Log[c*x^n])/(3*d^4
*f^4) - (14*b*n*Sqrt[x]*(a + b*Log[c*x^n]))/(9*d^5*f^5) + (b*n*x*(a + b*Log[c*x^n]))/(9*d^4*f^4) - (2*b*n*x^(3
/2)*(a + b*Log[c*x^n]))/(9*d^3*f^3) + (5*b*n*x^2*(a + b*Log[c*x^n]))/(36*d^2*f^2) - (22*b*n*x^(5/2)*(a + b*Log
[c*x^n]))/(225*d*f) + (2*b*n*x^3*(a + b*Log[c*x^n]))/27 + (2*b*n*Log[1 + d*f*Sqrt[x]]*(a + b*Log[c*x^n]))/(9*d
^6*f^6) - (2*b*n*x^3*Log[1 + d*f*Sqrt[x]]*(a + b*Log[c*x^n]))/9 + (Sqrt[x]*(a + b*Log[c*x^n])^2)/(3*d^5*f^5) -
(x*(a + b*Log[c*x^n])^2)/(6*d^4*f^4) + (x^(3/2)*(a + b*Log[c*x^n])^2)/(9*d^3*f^3) - (x^2*(a + b*Log[c*x^n])^2
)/(12*d^2*f^2) + (x^(5/2)*(a + b*Log[c*x^n])^2)/(15*d*f) - (x^3*(a + b*Log[c*x^n])^2)/18 - (Log[1 + d*f*Sqrt[x
]]*(a + b*Log[c*x^n])^2)/(3*d^6*f^6) + (x^3*Log[1 + d*f*Sqrt[x]]*(a + b*Log[c*x^n])^2)/3 + (4*b^2*n^2*PolyLog[
2, -(d*f*Sqrt[x])])/(9*d^6*f^6) - (4*b*n*(a + b*Log[c*x^n])*PolyLog[2, -(d*f*Sqrt[x])])/(3*d^6*f^6) + (8*b^2*n
^2*PolyLog[3, -(d*f*Sqrt[x])])/(3*d^6*f^6)

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^2 \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}{3 d^5 f^5}-\frac{x \left (a+b \log \left (c x^n\right )\right )^2}{6 d^4 f^4}+\frac{x^{3/2} \left (a+b \log \left (c x^n\right )\right )^2}{9 d^3 f^3}-\frac{x^2 \left (a+b \log \left (c x^n\right )\right )^2}{12 d^2 f^2}+\frac{x^{5/2} \left (a+b \log \left (c x^n\right )\right )^2}{15 d f}-\frac{1}{18} x^3 \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}{3 d^6 f^6}+\frac{1}{3} x^3 \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 )}{6 d^4 f^4}+\frac{a+b \log \left (c x^n\right )}{3 d^5 f^5 \sqrt{x}}+\frac{\sqrt{x} \left (a+b \log \left (c x^n\right )\right )}{9 d^3 f^3}-\frac{x \left (a+b \log \left (c x^n\right )\right )}{12 d^2 f^2}+\frac{x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{15 d f}-\frac{1}{18} x^2 \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 )}{3 d^6 f^6 x}+\frac{1}{3} x^2 \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}{3 d^5 f^5}-\frac{x \left (a+b \log \left (c x^n\right )\right )^2}{6 d^4 f^4}+\frac{x^{3/2} \left (a+b \log \left (c x^n\right )\right )^2}{9 d^3 f^3}-\frac{x^2 \left (a+b \log \left (c x^n\right )\right )^2}{12 d^2 f^2}+\frac{x^{5/2} \left (a+b \log \left (c x^n\right )\right )^2}{15 d f}-\frac{1}{18} x^3 \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}{3 d^6 f^6}+\frac{1}{3} x^3 \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^2+\frac{1}{9} (b n) \int x^2 \left (a+b \log \left (c x^n\right )\right ) \, dx-\frac{1}{3} (2 b n) \int x^2 \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx+\frac{(2 b n) \int \frac{\log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{x} \, dx}{3 d^6 f^6}-\frac{(2 b n) \int \frac{a+b \log \left (c x^n\right )}{\sqrt{x}} \, dx}{3 d^5 f^5}+\frac{(b n) \int \left (a+b \log \left (c x^n\right )\right ) \, dx}{3 d^4 f^4}-\frac{(2 b n) \int \sqrt{x} \left (a+b \log \left (c x^n\right )\right ) \, dx}{9 d^3 f^3}+\frac{(b n) \int x \left (a+b \log \left (c x^n\right )\right ) \, dx}{6 d^2 f^2}-\frac{(2 b n) \int x^{3/2} \left (a+b \log \left (c x^n\right )\right ) \, dx}{15 d f}\\ &=\frac{8 b^2 n^2 \sqrt{x}}{3 d^5 f^5}+\frac{a b n x}{3 d^4 f^4}+\frac{8 b^2 n^2 x^{3/2}}{81 d^3 f^3}-\frac{b^2 n^2 x^2}{24 d^2 f^2}+\frac{8 b^2 n^2 x^{5/2}}{375 d f}-\frac{1}{81} b^2 n^2 x^3-\frac{14 b n \sqrt{x} \left (a+b \log \left (c x^n\right )\right )}{9 d^5 f^5}+\frac{b n x \left (a+b \log \left (c x^n\right )\right )}{9 d^4 f^4}-\frac{2 b n x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{9 d^3 f^3}+\frac{5 b n x^2 \left (a+b \log \left (c x^n\right )\right )}{36 d^2 f^2}-\frac{22 b n x^{5/2} \left (a+b \log \left (c x^n\right )\right )}{225 d f}+\frac{2}{27} b n x^3 \left (a+b \log \left (c x^n\right )\right )+\frac{2 b n \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{9 d^6 f^6}-\frac{2}{9} b n x^3 \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}{3 d^5 f^5}-\frac{x \left (a+b \log \left (c x^n\right )\right )^2}{6 d^4 f^4}+\frac{x^{3/2} \left (a+b \log \left (c x^n\right )\right )^2}{9 d^3 f^3}-\frac{x^2 \left (a+b \log \left (c x^n\right )\right )^2}{12 d^2 f^2}+\frac{x^{5/2} \left (a+b \log \left (c x^n\right )\right )^2}{15 d f}-\frac{1}{18} x^3 \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}{3 d^6 f^6}+\frac{1}{3} x^3 \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac{4 b n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-d f \sqrt{x}\right )}{3 d^6 f^6}+\frac{\left (b^2 n\right ) \int \log \left (c x^n\right ) \, dx}{3 d^4 f^4}+\frac{1}{3} \left (2 b^2 n^2\right ) \int \left (-\frac{1}{6 d^4 f^4}+\frac{1}{3 d^5 f^5 \sqrt{x}}+\frac{\sqrt{x}}{9 d^3 f^3}-\frac{x}{12 d^2 f^2}+\frac{x^{3/2}}{15 d f}-\frac{x^2}{18}-\frac{\log \left (1+d f \sqrt{x}\right )}{3 d^6 f^6 x}+\frac{1}{3} x^2 \log \left (1+d f \sqrt{x}\right )\right ) \, dx+\frac{\left (4 b^2 n^2\right ) \int \frac{\text{Li}_2\left (-d f \sqrt{x}\right )}{x} \, dx}{3 d^6 f^6}\\ &=\frac{28 b^2 n^2 \sqrt{x}}{9 d^5 f^5}+\frac{a b n x}{3 d^4 f^4}-\frac{4 b^2 n^2 x}{9 d^4 f^4}+\frac{4 b^2 n^2 x^{3/2}}{27 d^3 f^3}-\frac{5 b^2 n^2 x^2}{72 d^2 f^2}+\frac{44 b^2 n^2 x^{5/2}}{1125 d f}-\frac{2}{81} b^2 n^2 x^3+\frac{b^2 n x \log \left (c x^n\right )}{3 d^4 f^4}-\frac{14 b n \sqrt{x} \left (a+b \log \left (c x^n\right )\right )}{9 d^5 f^5}+\frac{b n x \left (a+b \log \left (c x^n\right )\right )}{9 d^4 f^4}-\frac{2 b n x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{9 d^3 f^3}+\frac{5 b n x^2 \left (a+b \log \left (c x^n\right )\right )}{36 d^2 f^2}-\frac{22 b n x^{5/2} \left (a+b \log \left (c x^n\right )\right )}{225 d f}+\frac{2}{27} b n x^3 \left (a+b \log \left (c x^n\right )\right )+\frac{2 b n \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{9 d^6 f^6}-\frac{2}{9} b n x^3 \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}{3 d^5 f^5}-\frac{x \left (a+b \log \left (c x^n\right )\right )^2}{6 d^4 f^4}+\frac{x^{3/2} \left (a+b \log \left (c x^n\right )\right )^2}{9 d^3 f^3}-\frac{x^2 \left (a+b \log \left (c x^n\right )\right )^2}{12 d^2 f^2}+\frac{x^{5/2} \left (a+b \log \left (c x^n\right )\right )^2}{15 d f}-\frac{1}{18} x^3 \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}{3 d^6 f^6}+\frac{1}{3} x^3 \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac{4 b n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-d f \sqrt{x}\right )}{3 d^6 f^6}+\frac{8 b^2 n^2 \text{Li}_3\left (-d f \sqrt{x}\right )}{3 d^6 f^6}+\frac{1}{9} \left (2 b^2 n^2\right ) \int x^2 \log \left (1+d f \sqrt{x}\right ) \, dx-\frac{\left (2 b^2 n^2\right ) \int \frac{\log \left (1+d f \sqrt{x}\right )}{x} \, dx}{9 d^6 f^6}\\ &=\frac{28 b^2 n^2 \sqrt{x}}{9 d^5 f^5}+\frac{a b n x}{3 d^4 f^4}-\frac{4 b^2 n^2 x}{9 d^4 f^4}+\frac{4 b^2 n^2 x^{3/2}}{27 d^3 f^3}-\frac{5 b^2 n^2 x^2}{72 d^2 f^2}+\frac{44 b^2 n^2 x^{5/2}}{1125 d f}-\frac{2}{81} b^2 n^2 x^3+\frac{b^2 n x \log \left (c x^n\right )}{3 d^4 f^4}-\frac{14 b n \sqrt{x} \left (a+b \log \left (c x^n\right )\right )}{9 d^5 f^5}+\frac{b n x \left (a+b \log \left (c x^n\right )\right )}{9 d^4 f^4}-\frac{2 b n x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{9 d^3 f^3}+\frac{5 b n x^2 \left (a+b \log \left (c x^n\right )\right )}{36 d^2 f^2}-\frac{22 b n x^{5/2} \left (a+b \log \left (c x^n\right )\right )}{225 d f}+\frac{2}{27} b n x^3 \left (a+b \log \left (c x^n\right )\right )+\frac{2 b n \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{9 d^6 f^6}-\frac{2}{9} b n x^3 \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}{3 d^5 f^5}-\frac{x \left (a+b \log \left (c x^n\right )\right )^2}{6 d^4 f^4}+\frac{x^{3/2} \left (a+b \log \left (c x^n\right )\right )^2}{9 d^3 f^3}-\frac{x^2 \left (a+b \log \left (c x^n\right )\right )^2}{12 d^2 f^2}+\frac{x^{5/2} \left (a+b \log \left (c x^n\right )\right )^2}{15 d f}-\frac{1}{18} x^3 \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}{3 d^6 f^6}+\frac{1}{3} x^3 \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^2+\frac{4 b^2 n^2 \text{Li}_2\left (-d f \sqrt{x}\right )}{9 d^6 f^6}-\frac{4 b n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-d f \sqrt{x}\right )}{3 d^6 f^6}+\frac{8 b^2 n^2 \text{Li}_3\left (-d f \sqrt{x}\right )}{3 d^6 f^6}+\frac{1}{9} \left (4 b^2 n^2\right ) \operatorname{Subst}\left (\int x^5 \log (1+d f x) \, dx,x,\sqrt{x}\right )\\ &=\frac{28 b^2 n^2 \sqrt{x}}{9 d^5 f^5}+\frac{a b n x}{3 d^4 f^4}-\frac{4 b^2 n^2 x}{9 d^4 f^4}+\frac{4 b^2 n^2 x^{3/2}}{27 d^3 f^3}-\frac{5 b^2 n^2 x^2}{72 d^2 f^2}+\frac{44 b^2 n^2 x^{5/2}}{1125 d f}-\frac{2}{81} b^2 n^2 x^3+\frac{2}{27} b^2 n^2 x^3 \log \left (1+d f \sqrt{x}\right )+\frac{b^2 n x \log \left (c x^n\right )}{3 d^4 f^4}-\frac{14 b n \sqrt{x} \left (a+b \log \left (c x^n\right )\right )}{9 d^5 f^5}+\frac{b n x \left (a+b \log \left (c x^n\right )\right )}{9 d^4 f^4}-\frac{2 b n x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{9 d^3 f^3}+\frac{5 b n x^2 \left (a+b \log \left (c x^n\right )\right )}{36 d^2 f^2}-\frac{22 b n x^{5/2} \left (a+b \log \left (c x^n\right )\right )}{225 d f}+\frac{2}{27} b n x^3 \left (a+b \log \left (c x^n\right )\right )+\frac{2 b n \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{9 d^6 f^6}-\frac{2}{9} b n x^3 \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}{3 d^5 f^5}-\frac{x \left (a+b \log \left (c x^n\right )\right )^2}{6 d^4 f^4}+\frac{x^{3/2} \left (a+b \log \left (c x^n\right )\right )^2}{9 d^3 f^3}-\frac{x^2 \left (a+b \log \left (c x^n\right )\right )^2}{12 d^2 f^2}+\frac{x^{5/2} \left (a+b \log \left (c x^n\right )\right )^2}{15 d f}-\frac{1}{18} x^3 \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}{3 d^6 f^6}+\frac{1}{3} x^3 \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^2+\frac{4 b^2 n^2 \text{Li}_2\left (-d f \sqrt{x}\right )}{9 d^6 f^6}-\frac{4 b n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-d f \sqrt{x}\right )}{3 d^6 f^6}+\frac{8 b^2 n^2 \text{Li}_3\left (-d f \sqrt{x}\right )}{3 d^6 f^6}-\frac{1}{27} \left (2 b^2 d f n^2\right ) \operatorname{Subst}\left (\int \frac{x^6}{1+d f x} \, dx,x,\sqrt{x}\right )\\ &=\frac{28 b^2 n^2 \sqrt{x}}{9 d^5 f^5}+\frac{a b n x}{3 d^4 f^4}-\frac{4 b^2 n^2 x}{9 d^4 f^4}+\frac{4 b^2 n^2 x^{3/2}}{27 d^3 f^3}-\frac{5 b^2 n^2 x^2}{72 d^2 f^2}+\frac{44 b^2 n^2 x^{5/2}}{1125 d f}-\frac{2}{81} b^2 n^2 x^3+\frac{2}{27} b^2 n^2 x^3 \log \left (1+d f \sqrt{x}\right )+\frac{b^2 n x \log \left (c x^n\right )}{3 d^4 f^4}-\frac{14 b n \sqrt{x} \left (a+b \log \left (c x^n\right )\right )}{9 d^5 f^5}+\frac{b n x \left (a+b \log \left (c x^n\right )\right )}{9 d^4 f^4}-\frac{2 b n x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{9 d^3 f^3}+\frac{5 b n x^2 \left (a+b \log \left (c x^n\right )\right )}{36 d^2 f^2}-\frac{22 b n x^{5/2} \left (a+b \log \left (c x^n\right )\right )}{225 d f}+\frac{2}{27} b n x^3 \left (a+b \log \left (c x^n\right )\right )+\frac{2 b n \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{9 d^6 f^6}-\frac{2}{9} b n x^3 \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}{3 d^5 f^5}-\frac{x \left (a+b \log \left (c x^n\right )\right )^2}{6 d^4 f^4}+\frac{x^{3/2} \left (a+b \log \left (c x^n\right )\right )^2}{9 d^3 f^3}-\frac{x^2 \left (a+b \log \left (c x^n\right )\right )^2}{12 d^2 f^2}+\frac{x^{5/2} \left (a+b \log \left (c x^n\right )\right )^2}{15 d f}-\frac{1}{18} x^3 \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}{3 d^6 f^6}+\frac{1}{3} x^3 \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^2+\frac{4 b^2 n^2 \text{Li}_2\left (-d f \sqrt{x}\right )}{9 d^6 f^6}-\frac{4 b n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-d f \sqrt{x}\right )}{3 d^6 f^6}+\frac{8 b^2 n^2 \text{Li}_3\left (-d f \sqrt{x}\right )}{3 d^6 f^6}-\frac{1}{27} \left (2 b^2 d f n^2\right ) \operatorname{Subst}\left (\int \left (-\frac{1}{d^6 f^6}+\frac{x}{d^5 f^5}-\frac{x^2}{d^4 f^4}+\frac{x^3}{d^3 f^3}-\frac{x^4}{d^2 f^2}+\frac{x^5}{d f}+\frac{1}{d^6 f^6 (1+d f x)}\right ) \, dx,x,\sqrt{x}\right )\\ &=\frac{86 b^2 n^2 \sqrt{x}}{27 d^5 f^5}+\frac{a b n x}{3 d^4 f^4}-\frac{13 b^2 n^2 x}{27 d^4 f^4}+\frac{14 b^2 n^2 x^{3/2}}{81 d^3 f^3}-\frac{19 b^2 n^2 x^2}{216 d^2 f^2}+\frac{182 b^2 n^2 x^{5/2}}{3375 d f}-\frac{1}{27} b^2 n^2 x^3-\frac{2 b^2 n^2 \log \left (1+d f \sqrt{x}\right )}{27 d^6 f^6}+\frac{2}{27} b^2 n^2 x^3 \log \left (1+d f \sqrt{x}\right )+\frac{b^2 n x \log \left (c x^n\right )}{3 d^4 f^4}-\frac{14 b n \sqrt{x} \left (a+b \log \left (c x^n\right )\right )}{9 d^5 f^5}+\frac{b n x \left (a+b \log \left (c x^n\right )\right )}{9 d^4 f^4}-\frac{2 b n x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{9 d^3 f^3}+\frac{5 b n x^2 \left (a+b \log \left (c x^n\right )\right )}{36 d^2 f^2}-\frac{22 b n x^{5/2} \left (a+b \log \left (c x^n\right )\right )}{225 d f}+\frac{2}{27} b n x^3 \left (a+b \log \left (c x^n\right )\right )+\frac{2 b n \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{9 d^6 f^6}-\frac{2}{9} b n x^3 \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}{3 d^5 f^5}-\frac{x \left (a+b \log \left (c x^n\right )\right )^2}{6 d^4 f^4}+\frac{x^{3/2} \left (a+b \log \left (c x^n\right )\right )^2}{9 d^3 f^3}-\frac{x^2 \left (a+b \log \left (c x^n\right )\right )^2}{12 d^2 f^2}+\frac{x^{5/2} \left (a+b \log \left (c x^n\right )\right )^2}{15 d f}-\frac{1}{18} x^3 \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}{3 d^6 f^6}+\frac{1}{3} x^3 \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^2+\frac{4 b^2 n^2 \text{Li}_2\left (-d f \sqrt{x}\right )}{9 d^6 f^6}-\frac{4 b n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-d f \sqrt{x}\right )}{3 d^6 f^6}+\frac{8 b^2 n^2 \text{Li}_3\left (-d f \sqrt{x}\right )}{3 d^6 f^6}\\ \end{align*}

Mathematica [A]  time = 0.563197, size = 995, normalized size = 1.41 \[ \frac{-4500 a^2 d^6 x^3 f^6-3000 b^2 d^6 n^2 x^3 f^6+6000 a b d^6 n x^3 f^6-4500 b^2 d^6 x^3 \log ^2\left (c x^n\right ) f^6+27000 b^2 d^6 x^3 \log \left (d \sqrt{x} f+1\right ) \log ^2\left (c x^n\right ) f^6+27000 a^2 d^6 x^3 \log \left (d \sqrt{x} f+1\right ) f^6+6000 b^2 d^6 n^2 x^3 \log \left (d \sqrt{x} f+1\right ) f^6-18000 a b d^6 n x^3 \log \left (d \sqrt{x} f+1\right ) f^6-9000 a b d^6 x^3 \log \left (c x^n\right ) f^6+6000 b^2 d^6 n x^3 \log \left (c x^n\right ) f^6+54000 a b d^6 x^3 \log \left (d \sqrt{x} f+1\right ) \log \left (c x^n\right ) f^6-18000 b^2 d^6 n x^3 \log \left (d \sqrt{x} f+1\right ) \log \left (c x^n\right ) f^6+5400 a^2 d^5 x^{5/2} f^5+4368 b^2 d^5 n^2 x^{5/2} f^5-7920 a b d^5 n x^{5/2} f^5+5400 b^2 d^5 x^{5/2} \log ^2\left (c x^n\right ) f^5+10800 a b d^5 x^{5/2} \log \left (c x^n\right ) f^5-7920 b^2 d^5 n x^{5/2} \log \left (c x^n\right ) f^5-6750 a^2 d^4 x^2 f^4-7125 b^2 d^4 n^2 x^2 f^4+11250 a b d^4 n x^2 f^4-6750 b^2 d^4 x^2 \log ^2\left (c x^n\right ) f^4-13500 a b d^4 x^2 \log \left (c x^n\right ) f^4+11250 b^2 d^4 n x^2 \log \left (c x^n\right ) f^4+9000 b^2 d^3 x^{3/2} \log ^2\left (c x^n\right ) f^3+9000 a^2 d^3 x^{3/2} f^3+14000 b^2 d^3 n^2 x^{3/2} f^3-18000 a b d^3 n x^{3/2} f^3+18000 a b d^3 x^{3/2} \log \left (c x^n\right ) f^3-18000 b^2 d^3 n x^{3/2} \log \left (c x^n\right ) f^3-13500 b^2 d^2 x \log ^2\left (c x^n\right ) f^2-13500 a^2 d^2 x f^2-39000 b^2 d^2 n^2 x f^2+36000 a b d^2 n x f^2-27000 a b d^2 x \log \left (c x^n\right ) f^2+36000 b^2 d^2 n x \log \left (c x^n\right ) f^2+27000 b^2 d \sqrt{x} \log ^2\left (c x^n\right ) f+54000 a b d \sqrt{x} \log \left (c x^n\right ) f-126000 b^2 d n \sqrt{x} \log \left (c x^n\right ) f+258000 b^2 d n^2 \sqrt{x} f+27000 a^2 d \sqrt{x} f-126000 a b d n \sqrt{x} f-27000 b^2 \log \left (d \sqrt{x} f+1\right ) \log ^2\left (c x^n\right )-27000 a^2 \log \left (d \sqrt{x} f+1\right )-6000 b^2 n^2 \log \left (d \sqrt{x} f+1\right )+18000 a b n \log \left (d \sqrt{x} f+1\right )-54000 a b \log \left (d \sqrt{x} f+1\right ) \log \left (c x^n\right )+18000 b^2 n \log \left (d \sqrt{x} f+1\right ) \log \left (c x^n\right )+36000 b n \left (-3 a+b n-3 b \log \left (c x^n\right )\right ) \text{PolyLog}\left (2,-d f \sqrt{x}\right )+216000 b^2 n^2 \text{PolyLog}\left (3,-d f \sqrt{x}\right )}{81000 d^6 f^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(27000*a^2*d*f*Sqrt[x] - 126000*a*b*d*f*n*Sqrt[x] + 258000*b^2*d*f*n^2*Sqrt[x] - 13500*a^2*d^2*f^2*x + 36000*a
*b*d^2*f^2*n*x - 39000*b^2*d^2*f^2*n^2*x + 9000*a^2*d^3*f^3*x^(3/2) - 18000*a*b*d^3*f^3*n*x^(3/2) + 14000*b^2*
d^3*f^3*n^2*x^(3/2) - 6750*a^2*d^4*f^4*x^2 + 11250*a*b*d^4*f^4*n*x^2 - 7125*b^2*d^4*f^4*n^2*x^2 + 5400*a^2*d^5
*f^5*x^(5/2) - 7920*a*b*d^5*f^5*n*x^(5/2) + 4368*b^2*d^5*f^5*n^2*x^(5/2) - 4500*a^2*d^6*f^6*x^3 + 6000*a*b*d^6
*f^6*n*x^3 - 3000*b^2*d^6*f^6*n^2*x^3 - 27000*a^2*Log[1 + d*f*Sqrt[x]] + 18000*a*b*n*Log[1 + d*f*Sqrt[x]] - 60
00*b^2*n^2*Log[1 + d*f*Sqrt[x]] + 27000*a^2*d^6*f^6*x^3*Log[1 + d*f*Sqrt[x]] - 18000*a*b*d^6*f^6*n*x^3*Log[1 +
d*f*Sqrt[x]] + 6000*b^2*d^6*f^6*n^2*x^3*Log[1 + d*f*Sqrt[x]] + 54000*a*b*d*f*Sqrt[x]*Log[c*x^n] - 126000*b^2*
d*f*n*Sqrt[x]*Log[c*x^n] - 27000*a*b*d^2*f^2*x*Log[c*x^n] + 36000*b^2*d^2*f^2*n*x*Log[c*x^n] + 18000*a*b*d^3*f
^3*x^(3/2)*Log[c*x^n] - 18000*b^2*d^3*f^3*n*x^(3/2)*Log[c*x^n] - 13500*a*b*d^4*f^4*x^2*Log[c*x^n] + 11250*b^2*
d^4*f^4*n*x^2*Log[c*x^n] + 10800*a*b*d^5*f^5*x^(5/2)*Log[c*x^n] - 7920*b^2*d^5*f^5*n*x^(5/2)*Log[c*x^n] - 9000
*a*b*d^6*f^6*x^3*Log[c*x^n] + 6000*b^2*d^6*f^6*n*x^3*Log[c*x^n] - 54000*a*b*Log[1 + d*f*Sqrt[x]]*Log[c*x^n] +
18000*b^2*n*Log[1 + d*f*Sqrt[x]]*Log[c*x^n] + 54000*a*b*d^6*f^6*x^3*Log[1 + d*f*Sqrt[x]]*Log[c*x^n] - 18000*b^
2*d^6*f^6*n*x^3*Log[1 + d*f*Sqrt[x]]*Log[c*x^n] + 27000*b^2*d*f*Sqrt[x]*Log[c*x^n]^2 - 13500*b^2*d^2*f^2*x*Log
[c*x^n]^2 + 9000*b^2*d^3*f^3*x^(3/2)*Log[c*x^n]^2 - 6750*b^2*d^4*f^4*x^2*Log[c*x^n]^2 + 5400*b^2*d^5*f^5*x^(5/
2)*Log[c*x^n]^2 - 4500*b^2*d^6*f^6*x^3*Log[c*x^n]^2 - 27000*b^2*Log[1 + d*f*Sqrt[x]]*Log[c*x^n]^2 + 27000*b^2*
d^6*f^6*x^3*Log[1 + d*f*Sqrt[x]]*Log[c*x^n]^2 + 36000*b*n*(-3*a + b*n - 3*b*Log[c*x^n])*PolyLog[2, -(d*f*Sqrt[
x])] + 216000*b^2*n^2*PolyLog[3, -(d*f*Sqrt[x])])/(81000*d^6*f^6)

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Maple [F]  time = 0.019, size = 0, normalized size = 0. \begin{align*} \int{x}^{2} \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^2*(a+b*ln(c*x^n))^2*ln(d*(1/d+f*x^(1/2))),x)

[Out]

int(x^2*(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^{2} \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^2*(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^2*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^{2} \log \left (c x^{n}\right )^{2} + 2 \, a b x^{2} \log \left (c x^{n}\right ) + a^{2} x^{2}\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^2*(a+b*log(c*x^n))^2*log(d*(1/d+f*x^(1/2))),x, algorithm="fricas")

[Out]

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