3.1.53 \(\int x^2 \log (d (\frac {1}{d}+f \sqrt {x})) (a+b \log (c x^n))^2 \, dx\) [53]

Optimal. Leaf size=708 \[ \frac {86 b^2 n^2 \sqrt {x}}{27 d^5 f^5}+\frac {a b n x}{3 d^4 f^4}-\frac {13 b^2 n^2 x}{27 d^4 f^4}+\frac {14 b^2 n^2 x^{3/2}}{81 d^3 f^3}-\frac {19 b^2 n^2 x^2}{216 d^2 f^2}+\frac {182 b^2 n^2 x^{5/2}}{3375 d f}-\frac {1}{27} b^2 n^2 x^3-\frac {2 b^2 n^2 \log \left (1+d f \sqrt {x}\right )}{27 d^6 f^6}+\frac {2}{27} b^2 n^2 x^3 \log \left (1+d f \sqrt {x}\right )+\frac {b^2 n x \log \left (c x^n\right )}{3 d^4 f^4}-\frac {14 b n \sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{9 d^5 f^5}+\frac {b n x \left (a+b \log \left (c x^n\right )\right )}{9 d^4 f^4}-\frac {2 b n x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{9 d^3 f^3}+\frac {5 b n x^2 \left (a+b \log \left (c x^n\right )\right )}{36 d^2 f^2}-\frac {22 b n x^{5/2} \left (a+b \log \left (c x^n\right )\right )}{225 d f}+\frac {2}{27} b n x^3 \left (a+b \log \left (c x^n\right )\right )+\frac {2 b n \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{9 d^6 f^6}-\frac {2}{9} b n x^3 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )+\frac {\sqrt {x} \left (a+b \log \left (c x^n\right )\right )^2}{3 d^5 f^5}-\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{6 d^4 f^4}+\frac {x^{3/2} \left (a+b \log \left (c x^n\right )\right )^2}{9 d^3 f^3}-\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^2}{12 d^2 f^2}+\frac {x^{5/2} \left (a+b \log \left (c x^n\right )\right )^2}{15 d f}-\frac {1}{18} x^3 \left (a+b \log \left (c x^n\right )\right )^2-\frac {\log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{3 d^6 f^6}+\frac {1}{3} x^3 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2+\frac {4 b^2 n^2 \text {Li}_2\left (-d f \sqrt {x}\right )}{9 d^6 f^6}-\frac {4 b n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-d f \sqrt {x}\right )}{3 d^6 f^6}+\frac {8 b^2 n^2 \text {Li}_3\left (-d f \sqrt {x}\right )}{3 d^6 f^6} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.45, antiderivative size = 708, normalized size of antiderivative = 1.00, 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 = {2504, 2442, 45, 2424, 2332, 2341, 2421, 6724, 2423, 2438} \begin {gather*} -\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 {\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 {\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 {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 {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^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^{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}{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 {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 {2 b^2 n^2 \log \left (d f \sqrt {x}+1\right )}{27 d^6 f^6}+\frac {86 b^2 n^2 \sqrt {x}}{27 d^5 f^5}-\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 {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 \end {gather*}

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 45

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 2332

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

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Mathematica [A]
time = 0.34, size = 995, normalized size = 1.41 \begin {gather*} \frac {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 \left (1+d f \sqrt {x}\right )+18000 a b n \log \left (1+d f \sqrt {x}\right )-6000 b^2 n^2 \log \left (1+d f \sqrt {x}\right )+27000 a^2 d^6 f^6 x^3 \log \left (1+d f \sqrt {x}\right )-18000 a b d^6 f^6 n x^3 \log \left (1+d f \sqrt {x}\right )+6000 b^2 d^6 f^6 n^2 x^3 \log \left (1+d f \sqrt {x}\right )+54000 a b d f \sqrt {x} \log \left (c x^n\right )-126000 b^2 d f n \sqrt {x} \log \left (c x^n\right )-27000 a b d^2 f^2 x \log \left (c x^n\right )+36000 b^2 d^2 f^2 n x \log \left (c x^n\right )+18000 a b d^3 f^3 x^{3/2} \log \left (c x^n\right )-18000 b^2 d^3 f^3 n x^{3/2} \log \left (c x^n\right )-13500 a b d^4 f^4 x^2 \log \left (c x^n\right )+11250 b^2 d^4 f^4 n x^2 \log \left (c x^n\right )+10800 a b d^5 f^5 x^{5/2} \log \left (c x^n\right )-7920 b^2 d^5 f^5 n x^{5/2} \log \left (c x^n\right )-9000 a b d^6 f^6 x^3 \log \left (c x^n\right )+6000 b^2 d^6 f^6 n x^3 \log \left (c x^n\right )-54000 a b \log \left (1+d f \sqrt {x}\right ) \log \left (c x^n\right )+18000 b^2 n \log \left (1+d f \sqrt {x}\right ) \log \left (c x^n\right )+54000 a b d^6 f^6 x^3 \log \left (1+d f \sqrt {x}\right ) \log \left (c x^n\right )-18000 b^2 d^6 f^6 n x^3 \log \left (1+d f \sqrt {x}\right ) \log \left (c x^n\right )+27000 b^2 d f \sqrt {x} \log ^2\left (c x^n\right )-13500 b^2 d^2 f^2 x \log ^2\left (c x^n\right )+9000 b^2 d^3 f^3 x^{3/2} \log ^2\left (c x^n\right )-6750 b^2 d^4 f^4 x^2 \log ^2\left (c x^n\right )+5400 b^2 d^5 f^5 x^{5/2} \log ^2\left (c x^n\right )-4500 b^2 d^6 f^6 x^3 \log ^2\left (c x^n\right )-27000 b^2 \log \left (1+d f \sqrt {x}\right ) \log ^2\left (c x^n\right )+27000 b^2 d^6 f^6 x^3 \log \left (1+d f \sqrt {x}\right ) \log ^2\left (c x^n\right )+36000 b n \left (-3 a+b n-3 b \log \left (c x^n\right )\right ) \text {Li}_2\left (-d f \sqrt {x}\right )+216000 b^2 n^2 \text {Li}_3\left (-d f \sqrt {x}\right )}{81000 d^6 f^6} \end {gather*}

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

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 = 0.00, 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(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 = 0.00, 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(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(-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(x**2*(a+b*ln(c*x**n))**2*ln(d*(1/d+f*x**(1/2))),x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 4370 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(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)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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