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

Optimal. Leaf size=750 \[ \text{result too large to display} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.846206, antiderivative size = 750, normalized size of antiderivative = 1., number of steps used = 22, number of rules used = 13, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.464, Rules used = {2454, 2395, 43, 2377, 2295, 2304, 2375, 2337, 2374, 6589, 2376, 2394, 2315} \[ -\frac{4 b e^6 n \text{PolyLog}\left (2,-\frac{f \sqrt{x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 f^6}-\frac{4 b^2 e^6 n^2 \text{PolyLog}\left (2,\frac{f \sqrt{x}}{e}+1\right )}{9 f^6}+\frac{8 b^2 e^6 n^2 \text{PolyLog}\left (3,-\frac{f \sqrt{x}}{e}\right )}{3 f^6}+\frac{1}{3} x^3 \log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac{2}{9} b n x^3 \log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{e^6 \log \left (\frac{f \sqrt{x}}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{3 f^6}+\frac{2 b e^6 n \log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{9 f^6}+\frac{e^5 \sqrt{x} \left (a+b \log \left (c x^n\right )\right )^2}{3 f^5}-\frac{14 b e^5 n \sqrt{x} \left (a+b \log \left (c x^n\right )\right )}{9 f^5}-\frac{e^4 x \left (a+b \log \left (c x^n\right )\right )^2}{6 f^4}+\frac{b e^4 n x \left (a+b \log \left (c x^n\right )\right )}{9 f^4}+\frac{e^3 x^{3/2} \left (a+b \log \left (c x^n\right )\right )^2}{9 f^3}-\frac{2 b e^3 n x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{9 f^3}-\frac{e^2 x^2 \left (a+b \log \left (c x^n\right )\right )^2}{12 f^2}+\frac{5 b e^2 n x^2 \left (a+b \log \left (c x^n\right )\right )}{36 f^2}+\frac{e x^{5/2} \left (a+b \log \left (c x^n\right )\right )^2}{15 f}-\frac{22 b e n x^{5/2} \left (a+b \log \left (c x^n\right )\right )}{225 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 e^4 n x}{3 f^4}+\frac{b^2 e^4 n x \log \left (c x^n\right )}{3 f^4}+\frac{2}{27} b^2 n^2 x^3 \log \left (d \left (e+f \sqrt{x}\right )\right )+\frac{14 b^2 e^3 n^2 x^{3/2}}{81 f^3}-\frac{19 b^2 e^2 n^2 x^2}{216 f^2}+\frac{86 b^2 e^5 n^2 \sqrt{x}}{27 f^5}-\frac{13 b^2 e^4 n^2 x}{27 f^4}-\frac{2 b^2 e^6 n^2 \log \left (e+f \sqrt{x}\right )}{27 f^6}-\frac{4 b^2 e^6 n^2 \log \left (e+f \sqrt{x}\right ) \log \left (-\frac{f \sqrt{x}}{e}\right )}{9 f^6}+\frac{182 b^2 e n^2 x^{5/2}}{3375 f}-\frac{1}{27} b^2 n^2 x^3 \]

Antiderivative was successfully verified.

[In]

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

[Out]

(86*b^2*e^5*n^2*Sqrt[x])/(27*f^5) + (a*b*e^4*n*x)/(3*f^4) - (13*b^2*e^4*n^2*x)/(27*f^4) + (14*b^2*e^3*n^2*x^(3
/2))/(81*f^3) - (19*b^2*e^2*n^2*x^2)/(216*f^2) + (182*b^2*e*n^2*x^(5/2))/(3375*f) - (b^2*n^2*x^3)/27 - (2*b^2*
e^6*n^2*Log[e + f*Sqrt[x]])/(27*f^6) + (2*b^2*n^2*x^3*Log[d*(e + f*Sqrt[x])])/27 - (4*b^2*e^6*n^2*Log[e + f*Sq
rt[x]]*Log[-((f*Sqrt[x])/e)])/(9*f^6) + (b^2*e^4*n*x*Log[c*x^n])/(3*f^4) - (14*b*e^5*n*Sqrt[x]*(a + b*Log[c*x^
n]))/(9*f^5) + (b*e^4*n*x*(a + b*Log[c*x^n]))/(9*f^4) - (2*b*e^3*n*x^(3/2)*(a + b*Log[c*x^n]))/(9*f^3) + (5*b*
e^2*n*x^2*(a + b*Log[c*x^n]))/(36*f^2) - (22*b*e*n*x^(5/2)*(a + b*Log[c*x^n]))/(225*f) + (2*b*n*x^3*(a + b*Log
[c*x^n]))/27 + (2*b*e^6*n*Log[e + f*Sqrt[x]]*(a + b*Log[c*x^n]))/(9*f^6) - (2*b*n*x^3*Log[d*(e + f*Sqrt[x])]*(
a + b*Log[c*x^n]))/9 + (e^5*Sqrt[x]*(a + b*Log[c*x^n])^2)/(3*f^5) - (e^4*x*(a + b*Log[c*x^n])^2)/(6*f^4) + (e^
3*x^(3/2)*(a + b*Log[c*x^n])^2)/(9*f^3) - (e^2*x^2*(a + b*Log[c*x^n])^2)/(12*f^2) + (e*x^(5/2)*(a + b*Log[c*x^
n])^2)/(15*f) - (x^3*(a + b*Log[c*x^n])^2)/18 + (x^3*Log[d*(e + f*Sqrt[x])]*(a + b*Log[c*x^n])^2)/3 - (e^6*Log
[1 + (f*Sqrt[x])/e]*(a + b*Log[c*x^n])^2)/(3*f^6) - (4*b^2*e^6*n^2*PolyLog[2, 1 + (f*Sqrt[x])/e])/(9*f^6) - (4
*b*e^6*n*(a + b*Log[c*x^n])*PolyLog[2, -((f*Sqrt[x])/e)])/(3*f^6) + (8*b^2*e^6*n^2*PolyLog[3, -((f*Sqrt[x])/e)
])/(3*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 2375

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

Rule 2337

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.))/((d_) + (e_.)*(x_)^(r_)), x_Symbol] :> Si
mp[(f^m*Log[1 + (e*x^r)/d]*(a + b*Log[c*x^n])^p)/(e*r), x] - Dist[(b*f^m*n*p)/(e*r), Int[(Log[1 + (e*x^r)/d]*(
a + b*Log[c*x^n])^(p - 1))/x, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, r}, x] && EqQ[m, r - 1] && IGtQ[p, 0] &
& (IntegerQ[m] || GtQ[f, 0]) && NeQ[r, n]

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 2394

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

Rule 2315

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[PolyLog[2, 1 - c*x]/e, x] /; FreeQ[{c, d, e}, x] &
& EqQ[e + c*d, 0]

Rubi steps

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

Mathematica [A]  time = 0.872822, size = 1319, normalized size = 1.76 \[ \text{result too large to display} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.033, 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 ( e+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*(e+f*x^(1/2))),x)

[Out]

int(x^2*(a+b*ln(c*x^n))^2*ln(d*(e+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} + e\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*(e+f*x^(1/2))),x, algorithm="maxima")

[Out]

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

[Out]

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