\(\int x^2 \log ^3(c (a+b x)^n) \, dx\) [87]

Optimal result

Integrand size = 16, antiderivative size = 285 \[ \int x^2 \log ^3\left (c (a+b x)^n\right ) \, dx=-\frac {6 a^2 n^3 x}{b^2}+\frac {3 a n^3 (a+b x)^2}{4 b^3}-\frac {2 n^3 (a+b x)^3}{27 b^3}+\frac {6 a^2 n^2 (a+b x) \log \left (c (a+b x)^n\right )}{b^3}-\frac {3 a n^2 (a+b x)^2 \log \left (c (a+b x)^n\right )}{2 b^3}+\frac {2 n^2 (a+b x)^3 \log \left (c (a+b x)^n\right )}{9 b^3}-\frac {3 a^2 n (a+b x) \log ^2\left (c (a+b x)^n\right )}{b^3}+\frac {3 a n (a+b x)^2 \log ^2\left (c (a+b x)^n\right )}{2 b^3}-\frac {n (a+b x)^3 \log ^2\left (c (a+b x)^n\right )}{3 b^3}+\frac {a^2 (a+b x) \log ^3\left (c (a+b x)^n\right )}{b^3}-\frac {a (a+b x)^2 \log ^3\left (c (a+b x)^n\right )}{b^3}+\frac {(a+b x)^3 \log ^3\left (c (a+b x)^n\right )}{3 b^3} \]

[Out]

-6*a^2*n^3*x/b^2+3/4*a*n^3*(b*x+a)^2/b^3-2/27*n^3*(b*x+a)^3/b^3+6*a^2*n^2*(b*x+a)*ln(c*(b*x+a)^n)/b^3-3/2*a*n^
2*(b*x+a)^2*ln(c*(b*x+a)^n)/b^3+2/9*n^2*(b*x+a)^3*ln(c*(b*x+a)^n)/b^3-3*a^2*n*(b*x+a)*ln(c*(b*x+a)^n)^2/b^3+3/
2*a*n*(b*x+a)^2*ln(c*(b*x+a)^n)^2/b^3-1/3*n*(b*x+a)^3*ln(c*(b*x+a)^n)^2/b^3+a^2*(b*x+a)*ln(c*(b*x+a)^n)^3/b^3-
a*(b*x+a)^2*ln(c*(b*x+a)^n)^3/b^3+1/3*(b*x+a)^3*ln(c*(b*x+a)^n)^3/b^3

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 285, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {2448, 2436, 2333, 2332, 2437, 2342, 2341} \[ \int x^2 \log ^3\left (c (a+b x)^n\right ) \, dx=\frac {6 a^2 n^2 (a+b x) \log \left (c (a+b x)^n\right )}{b^3}+\frac {a^2 (a+b x) \log ^3\left (c (a+b x)^n\right )}{b^3}-\frac {3 a^2 n (a+b x) \log ^2\left (c (a+b x)^n\right )}{b^3}-\frac {6 a^2 n^3 x}{b^2}+\frac {2 n^2 (a+b x)^3 \log \left (c (a+b x)^n\right )}{9 b^3}-\frac {3 a n^2 (a+b x)^2 \log \left (c (a+b x)^n\right )}{2 b^3}+\frac {(a+b x)^3 \log ^3\left (c (a+b x)^n\right )}{3 b^3}-\frac {a (a+b x)^2 \log ^3\left (c (a+b x)^n\right )}{b^3}-\frac {n (a+b x)^3 \log ^2\left (c (a+b x)^n\right )}{3 b^3}+\frac {3 a n (a+b x)^2 \log ^2\left (c (a+b x)^n\right )}{2 b^3}-\frac {2 n^3 (a+b x)^3}{27 b^3}+\frac {3 a n^3 (a+b x)^2}{4 b^3} \]

[In]

Int[x^2*Log[c*(a + b*x)^n]^3,x]

[Out]

(-6*a^2*n^3*x)/b^2 + (3*a*n^3*(a + b*x)^2)/(4*b^3) - (2*n^3*(a + b*x)^3)/(27*b^3) + (6*a^2*n^2*(a + b*x)*Log[c
*(a + b*x)^n])/b^3 - (3*a*n^2*(a + b*x)^2*Log[c*(a + b*x)^n])/(2*b^3) + (2*n^2*(a + b*x)^3*Log[c*(a + b*x)^n])
/(9*b^3) - (3*a^2*n*(a + b*x)*Log[c*(a + b*x)^n]^2)/b^3 + (3*a*n*(a + b*x)^2*Log[c*(a + b*x)^n]^2)/(2*b^3) - (
n*(a + b*x)^3*Log[c*(a + b*x)^n]^2)/(3*b^3) + (a^2*(a + b*x)*Log[c*(a + b*x)^n]^3)/b^3 - (a*(a + b*x)^2*Log[c*
(a + b*x)^n]^3)/b^3 + ((a + b*x)^3*Log[c*(a + b*x)^n]^3)/(3*b^3)

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 2333

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*Log[c*x^n])^p, x] - Dist[b*n*p, In
t[(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{a, b, c, n}, x] && GtQ[p, 0] && IntegerQ[2*p]

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 2342

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

Rule 2436

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

Rule 2437

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_) + (g_.)*(x_))^(q_.), x_Symbol] :> Dist[1/
e, Subst[Int[(f*(x/d))^q*(a + b*Log[c*x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p, q}, x]
&& EqQ[e*f - d*g, 0]

Rule 2448

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

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {a^2 \log ^3\left (c (a+b x)^n\right )}{b^2}-\frac {2 a (a+b x) \log ^3\left (c (a+b x)^n\right )}{b^2}+\frac {(a+b x)^2 \log ^3\left (c (a+b x)^n\right )}{b^2}\right ) \, dx \\ & = \frac {\int (a+b x)^2 \log ^3\left (c (a+b x)^n\right ) \, dx}{b^2}-\frac {(2 a) \int (a+b x) \log ^3\left (c (a+b x)^n\right ) \, dx}{b^2}+\frac {a^2 \int \log ^3\left (c (a+b x)^n\right ) \, dx}{b^2} \\ & = \frac {\text {Subst}\left (\int x^2 \log ^3\left (c x^n\right ) \, dx,x,a+b x\right )}{b^3}-\frac {(2 a) \text {Subst}\left (\int x \log ^3\left (c x^n\right ) \, dx,x,a+b x\right )}{b^3}+\frac {a^2 \text {Subst}\left (\int \log ^3\left (c x^n\right ) \, dx,x,a+b x\right )}{b^3} \\ & = \frac {a^2 (a+b x) \log ^3\left (c (a+b x)^n\right )}{b^3}-\frac {a (a+b x)^2 \log ^3\left (c (a+b x)^n\right )}{b^3}+\frac {(a+b x)^3 \log ^3\left (c (a+b x)^n\right )}{3 b^3}-\frac {n \text {Subst}\left (\int x^2 \log ^2\left (c x^n\right ) \, dx,x,a+b x\right )}{b^3}+\frac {(3 a n) \text {Subst}\left (\int x \log ^2\left (c x^n\right ) \, dx,x,a+b x\right )}{b^3}-\frac {\left (3 a^2 n\right ) \text {Subst}\left (\int \log ^2\left (c x^n\right ) \, dx,x,a+b x\right )}{b^3} \\ & = -\frac {3 a^2 n (a+b x) \log ^2\left (c (a+b x)^n\right )}{b^3}+\frac {3 a n (a+b x)^2 \log ^2\left (c (a+b x)^n\right )}{2 b^3}-\frac {n (a+b x)^3 \log ^2\left (c (a+b x)^n\right )}{3 b^3}+\frac {a^2 (a+b x) \log ^3\left (c (a+b x)^n\right )}{b^3}-\frac {a (a+b x)^2 \log ^3\left (c (a+b x)^n\right )}{b^3}+\frac {(a+b x)^3 \log ^3\left (c (a+b x)^n\right )}{3 b^3}+\frac {\left (2 n^2\right ) \text {Subst}\left (\int x^2 \log \left (c x^n\right ) \, dx,x,a+b x\right )}{3 b^3}-\frac {\left (3 a n^2\right ) \text {Subst}\left (\int x \log \left (c x^n\right ) \, dx,x,a+b x\right )}{b^3}+\frac {\left (6 a^2 n^2\right ) \text {Subst}\left (\int \log \left (c x^n\right ) \, dx,x,a+b x\right )}{b^3} \\ & = -\frac {6 a^2 n^3 x}{b^2}+\frac {3 a n^3 (a+b x)^2}{4 b^3}-\frac {2 n^3 (a+b x)^3}{27 b^3}+\frac {6 a^2 n^2 (a+b x) \log \left (c (a+b x)^n\right )}{b^3}-\frac {3 a n^2 (a+b x)^2 \log \left (c (a+b x)^n\right )}{2 b^3}+\frac {2 n^2 (a+b x)^3 \log \left (c (a+b x)^n\right )}{9 b^3}-\frac {3 a^2 n (a+b x) \log ^2\left (c (a+b x)^n\right )}{b^3}+\frac {3 a n (a+b x)^2 \log ^2\left (c (a+b x)^n\right )}{2 b^3}-\frac {n (a+b x)^3 \log ^2\left (c (a+b x)^n\right )}{3 b^3}+\frac {a^2 (a+b x) \log ^3\left (c (a+b x)^n\right )}{b^3}-\frac {a (a+b x)^2 \log ^3\left (c (a+b x)^n\right )}{b^3}+\frac {(a+b x)^3 \log ^3\left (c (a+b x)^n\right )}{3 b^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 260, normalized size of antiderivative = 0.91 \[ \int x^2 \log ^3\left (c (a+b x)^n\right ) \, dx=-\frac {85 a^2 n^3 x}{18 b^2}+\frac {19 a n^3 x^2}{36 b}-\frac {2 n^3 x^3}{27}+\frac {85 a^3 n^2 \log \left (c (a+b x)^n\right )}{18 b^3}+\frac {11 a^2 n^2 x \log \left (c (a+b x)^n\right )}{3 b^2}-\frac {5 a n^2 x^2 \log \left (c (a+b x)^n\right )}{6 b}+\frac {2}{9} n^2 x^3 \log \left (c (a+b x)^n\right )-\frac {11 a^3 n \log ^2\left (c (a+b x)^n\right )}{6 b^3}-\frac {a^2 n x \log ^2\left (c (a+b x)^n\right )}{b^2}+\frac {a n x^2 \log ^2\left (c (a+b x)^n\right )}{2 b}-\frac {1}{3} n x^3 \log ^2\left (c (a+b x)^n\right )+\frac {a^3 \log ^3\left (c (a+b x)^n\right )}{3 b^3}+\frac {1}{3} x^3 \log ^3\left (c (a+b x)^n\right ) \]

[In]

Integrate[x^2*Log[c*(a + b*x)^n]^3,x]

[Out]

(-85*a^2*n^3*x)/(18*b^2) + (19*a*n^3*x^2)/(36*b) - (2*n^3*x^3)/27 + (85*a^3*n^2*Log[c*(a + b*x)^n])/(18*b^3) +
(11*a^2*n^2*x*Log[c*(a + b*x)^n])/(3*b^2) - (5*a*n^2*x^2*Log[c*(a + b*x)^n])/(6*b) + (2*n^2*x^3*Log[c*(a + b*
x)^n])/9 - (11*a^3*n*Log[c*(a + b*x)^n]^2)/(6*b^3) - (a^2*n*x*Log[c*(a + b*x)^n]^2)/b^2 + (a*n*x^2*Log[c*(a +
b*x)^n]^2)/(2*b) - (n*x^3*Log[c*(a + b*x)^n]^2)/3 + (a^3*Log[c*(a + b*x)^n]^3)/(3*b^3) + (x^3*Log[c*(a + b*x)^
n]^3)/3

Maple [A] (verified)

Time = 0.83 (sec) , antiderivative size = 261, normalized size of antiderivative = 0.92

method	result	size
parallelrisch	\(\frac {36 x^{3} \ln \left (c \left (b x +a \right )^{n}\right )^{3} b^{3}-36 x^{3} \ln \left (c \left (b x +a \right )^{n}\right )^{2} b^{3} n +24 x^{3} \ln \left (c \left (b x +a \right )^{n}\right ) b^{3} n^{2}-8 b^{3} n^{3} x^{3}+54 x^{2} \ln \left (c \left (b x +a \right )^{n}\right )^{2} a \,b^{2} n -90 x^{2} \ln \left (c \left (b x +a \right )^{n}\right ) a \,b^{2} n^{2}+57 a \,b^{2} n^{3} x^{2}+906 \ln \left (b x +a \right ) a^{3} n^{3}-108 x \ln \left (c \left (b x +a \right )^{n}\right )^{2} a^{2} b n +396 x \ln \left (c \left (b x +a \right )^{n}\right ) a^{2} b \,n^{2}-510 a^{2} b \,n^{3} x +36 \ln \left (c \left (b x +a \right )^{n}\right )^{3} a^{3}-198 \ln \left (c \left (b x +a \right )^{n}\right )^{2} a^{3} n -396 \ln \left (c \left (b x +a \right )^{n}\right ) a^{3} n^{2}+510 a^{3} n^{3}}{108 b^{3}}\)	\(261\)
risch	\(\text {Expression too large to display}\)	\(5345\)

[In]

int(x^2*ln(c*(b*x+a)^n)^3,x,method=_RETURNVERBOSE)

[Out]

1/108*(36*x^3*ln(c*(b*x+a)^n)^3*b^3-36*x^3*ln(c*(b*x+a)^n)^2*b^3*n+24*x^3*ln(c*(b*x+a)^n)*b^3*n^2-8*b^3*n^3*x^
3+54*x^2*ln(c*(b*x+a)^n)^2*a*b^2*n-90*x^2*ln(c*(b*x+a)^n)*a*b^2*n^2+57*a*b^2*n^3*x^2+906*ln(b*x+a)*a^3*n^3-108
*x*ln(c*(b*x+a)^n)^2*a^2*b*n+396*x*ln(c*(b*x+a)^n)*a^2*b*n^2-510*a^2*b*n^3*x+36*ln(c*(b*x+a)^n)^3*a^3-198*ln(c
*(b*x+a)^n)^2*a^3*n-396*ln(c*(b*x+a)^n)*a^3*n^2+510*a^3*n^3)/b^3

Fricas [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 341, normalized size of antiderivative = 1.20 \[ \int x^2 \log ^3\left (c (a+b x)^n\right ) \, dx=-\frac {8 \, b^{3} n^{3} x^{3} - 36 \, b^{3} x^{3} \log \left (c\right )^{3} - 57 \, a b^{2} n^{3} x^{2} + 510 \, a^{2} b n^{3} x - 36 \, {\left (b^{3} n^{3} x^{3} + a^{3} n^{3}\right )} \log \left (b x + a\right )^{3} + 18 \, {\left (2 \, b^{3} n^{3} x^{3} - 3 \, a b^{2} n^{3} x^{2} + 6 \, a^{2} b n^{3} x + 11 \, a^{3} n^{3} - 6 \, {\left (b^{3} n^{2} x^{3} + a^{3} n^{2}\right )} \log \left (c\right )\right )} \log \left (b x + a\right )^{2} + 18 \, {\left (2 \, b^{3} n x^{3} - 3 \, a b^{2} n x^{2} + 6 \, a^{2} b n x\right )} \log \left (c\right )^{2} - 6 \, {\left (4 \, b^{3} n^{3} x^{3} - 15 \, a b^{2} n^{3} x^{2} + 66 \, a^{2} b n^{3} x + 85 \, a^{3} n^{3} + 18 \, {\left (b^{3} n x^{3} + a^{3} n\right )} \log \left (c\right )^{2} - 6 \, {\left (2 \, b^{3} n^{2} x^{3} - 3 \, a b^{2} n^{2} x^{2} + 6 \, a^{2} b n^{2} x + 11 \, a^{3} n^{2}\right )} \log \left (c\right )\right )} \log \left (b x + a\right ) - 6 \, {\left (4 \, b^{3} n^{2} x^{3} - 15 \, a b^{2} n^{2} x^{2} + 66 \, a^{2} b n^{2} x\right )} \log \left (c\right )}{108 \, b^{3}} \]

[In]

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

[Out]

-1/108*(8*b^3*n^3*x^3 - 36*b^3*x^3*log(c)^3 - 57*a*b^2*n^3*x^2 + 510*a^2*b*n^3*x - 36*(b^3*n^3*x^3 + a^3*n^3)*
log(b*x + a)^3 + 18*(2*b^3*n^3*x^3 - 3*a*b^2*n^3*x^2 + 6*a^2*b*n^3*x + 11*a^3*n^3 - 6*(b^3*n^2*x^3 + a^3*n^2)*
log(c))*log(b*x + a)^2 + 18*(2*b^3*n*x^3 - 3*a*b^2*n*x^2 + 6*a^2*b*n*x)*log(c)^2 - 6*(4*b^3*n^3*x^3 - 15*a*b^2
*n^3*x^2 + 66*a^2*b*n^3*x + 85*a^3*n^3 + 18*(b^3*n*x^3 + a^3*n)*log(c)^2 - 6*(2*b^3*n^2*x^3 - 3*a*b^2*n^2*x^2
+ 6*a^2*b*n^2*x + 11*a^3*n^2)*log(c))*log(b*x + a) - 6*(4*b^3*n^2*x^3 - 15*a*b^2*n^2*x^2 + 66*a^2*b*n^2*x)*log
(c))/b^3

Sympy [A] (verification not implemented)

Time = 1.37 (sec) , antiderivative size = 269, normalized size of antiderivative = 0.94 \[ \int x^2 \log ^3\left (c (a+b x)^n\right ) \, dx=\begin {cases} \frac {85 a^{3} n^{2} \log {\left (c \left (a + b x\right )^{n} \right )}}{18 b^{3}} - \frac {11 a^{3} n \log {\left (c \left (a + b x\right )^{n} \right )}^{2}}{6 b^{3}} + \frac {a^{3} \log {\left (c \left (a + b x\right )^{n} \right )}^{3}}{3 b^{3}} - \frac {85 a^{2} n^{3} x}{18 b^{2}} + \frac {11 a^{2} n^{2} x \log {\left (c \left (a + b x\right )^{n} \right )}}{3 b^{2}} - \frac {a^{2} n x \log {\left (c \left (a + b x\right )^{n} \right )}^{2}}{b^{2}} + \frac {19 a n^{3} x^{2}}{36 b} - \frac {5 a n^{2} x^{2} \log {\left (c \left (a + b x\right )^{n} \right )}}{6 b} + \frac {a n x^{2} \log {\left (c \left (a + b x\right )^{n} \right )}^{2}}{2 b} - \frac {2 n^{3} x^{3}}{27} + \frac {2 n^{2} x^{3} \log {\left (c \left (a + b x\right )^{n} \right )}}{9} - \frac {n x^{3} \log {\left (c \left (a + b x\right )^{n} \right )}^{2}}{3} + \frac {x^{3} \log {\left (c \left (a + b x\right )^{n} \right )}^{3}}{3} & \text {for}\: b \neq 0 \\\frac {x^{3} \log {\left (a^{n} c \right )}^{3}}{3} & \text {otherwise} \end {cases} \]

[In]

integrate(x**2*ln(c*(b*x+a)**n)**3,x)

[Out]

Piecewise((85*a**3*n**2*log(c*(a + b*x)**n)/(18*b**3) - 11*a**3*n*log(c*(a + b*x)**n)**2/(6*b**3) + a**3*log(c
*(a + b*x)**n)**3/(3*b**3) - 85*a**2*n**3*x/(18*b**2) + 11*a**2*n**2*x*log(c*(a + b*x)**n)/(3*b**2) - a**2*n*x
*log(c*(a + b*x)**n)**2/b**2 + 19*a*n**3*x**2/(36*b) - 5*a*n**2*x**2*log(c*(a + b*x)**n)/(6*b) + a*n*x**2*log(
c*(a + b*x)**n)**2/(2*b) - 2*n**3*x**3/27 + 2*n**2*x**3*log(c*(a + b*x)**n)/9 - n*x**3*log(c*(a + b*x)**n)**2/
3 + x**3*log(c*(a + b*x)**n)**3/3, Ne(b, 0)), (x**3*log(a**n*c)**3/3, True))

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 215, normalized size of antiderivative = 0.75 \[ \int x^2 \log ^3\left (c (a+b x)^n\right ) \, dx=\frac {1}{3} \, x^{3} \log \left ({\left (b x + a\right )}^{n} c\right )^{3} + \frac {1}{6} \, b n {\left (\frac {6 \, a^{3} \log \left (b x + a\right )}{b^{4}} - \frac {2 \, b^{2} x^{3} - 3 \, a b x^{2} + 6 \, a^{2} x}{b^{3}}\right )} \log \left ({\left (b x + a\right )}^{n} c\right )^{2} - \frac {1}{108} \, b n {\left (\frac {{\left (8 \, b^{3} x^{3} - 36 \, a^{3} \log \left (b x + a\right )^{3} - 57 \, a b^{2} x^{2} - 198 \, a^{3} \log \left (b x + a\right )^{2} + 510 \, a^{2} b x - 510 \, a^{3} \log \left (b x + a\right )\right )} n^{2}}{b^{4}} - \frac {6 \, {\left (4 \, b^{3} x^{3} - 15 \, a b^{2} x^{2} - 18 \, a^{3} \log \left (b x + a\right )^{2} + 66 \, a^{2} b x - 66 \, a^{3} \log \left (b x + a\right )\right )} n \log \left ({\left (b x + a\right )}^{n} c\right )}{b^{4}}\right )} \]

[In]

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

[Out]

1/3*x^3*log((b*x + a)^n*c)^3 + 1/6*b*n*(6*a^3*log(b*x + a)/b^4 - (2*b^2*x^3 - 3*a*b*x^2 + 6*a^2*x)/b^3)*log((b
*x + a)^n*c)^2 - 1/108*b*n*((8*b^3*x^3 - 36*a^3*log(b*x + a)^3 - 57*a*b^2*x^2 - 198*a^3*log(b*x + a)^2 + 510*a
^2*b*x - 510*a^3*log(b*x + a))*n^2/b^4 - 6*(4*b^3*x^3 - 15*a*b^2*x^2 - 18*a^3*log(b*x + a)^2 + 66*a^2*b*x - 66
*a^3*log(b*x + a))*n*log((b*x + a)^n*c)/b^4)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 626 vs. \(2 (271) = 542\).

Time = 0.32 (sec) , antiderivative size = 626, normalized size of antiderivative = 2.20 \[ \int x^2 \log ^3\left (c (a+b x)^n\right ) \, dx=\frac {{\left (b x + a\right )}^{3} n^{3} \log \left (b x + a\right )^{3}}{3 \, b^{3}} - \frac {{\left (b x + a\right )}^{2} a n^{3} \log \left (b x + a\right )^{3}}{b^{3}} + \frac {{\left (b x + a\right )} a^{2} n^{3} \log \left (b x + a\right )^{3}}{b^{3}} - \frac {{\left (b x + a\right )}^{3} n^{3} \log \left (b x + a\right )^{2}}{3 \, b^{3}} + \frac {3 \, {\left (b x + a\right )}^{2} a n^{3} \log \left (b x + a\right )^{2}}{2 \, b^{3}} - \frac {3 \, {\left (b x + a\right )} a^{2} n^{3} \log \left (b x + a\right )^{2}}{b^{3}} + \frac {{\left (b x + a\right )}^{3} n^{2} \log \left (b x + a\right )^{2} \log \left (c\right )}{b^{3}} - \frac {3 \, {\left (b x + a\right )}^{2} a n^{2} \log \left (b x + a\right )^{2} \log \left (c\right )}{b^{3}} + \frac {3 \, {\left (b x + a\right )} a^{2} n^{2} \log \left (b x + a\right )^{2} \log \left (c\right )}{b^{3}} + \frac {2 \, {\left (b x + a\right )}^{3} n^{3} \log \left (b x + a\right )}{9 \, b^{3}} - \frac {3 \, {\left (b x + a\right )}^{2} a n^{3} \log \left (b x + a\right )}{2 \, b^{3}} + \frac {6 \, {\left (b x + a\right )} a^{2} n^{3} \log \left (b x + a\right )}{b^{3}} - \frac {2 \, {\left (b x + a\right )}^{3} n^{2} \log \left (b x + a\right ) \log \left (c\right )}{3 \, b^{3}} + \frac {3 \, {\left (b x + a\right )}^{2} a n^{2} \log \left (b x + a\right ) \log \left (c\right )}{b^{3}} - \frac {6 \, {\left (b x + a\right )} a^{2} n^{2} \log \left (b x + a\right ) \log \left (c\right )}{b^{3}} + \frac {{\left (b x + a\right )}^{3} n \log \left (b x + a\right ) \log \left (c\right )^{2}}{b^{3}} - \frac {3 \, {\left (b x + a\right )}^{2} a n \log \left (b x + a\right ) \log \left (c\right )^{2}}{b^{3}} + \frac {3 \, {\left (b x + a\right )} a^{2} n \log \left (b x + a\right ) \log \left (c\right )^{2}}{b^{3}} - \frac {2 \, {\left (b x + a\right )}^{3} n^{3}}{27 \, b^{3}} + \frac {3 \, {\left (b x + a\right )}^{2} a n^{3}}{4 \, b^{3}} - \frac {6 \, {\left (b x + a\right )} a^{2} n^{3}}{b^{3}} + \frac {2 \, {\left (b x + a\right )}^{3} n^{2} \log \left (c\right )}{9 \, b^{3}} - \frac {3 \, {\left (b x + a\right )}^{2} a n^{2} \log \left (c\right )}{2 \, b^{3}} + \frac {6 \, {\left (b x + a\right )} a^{2} n^{2} \log \left (c\right )}{b^{3}} - \frac {{\left (b x + a\right )}^{3} n \log \left (c\right )^{2}}{3 \, b^{3}} + \frac {3 \, {\left (b x + a\right )}^{2} a n \log \left (c\right )^{2}}{2 \, b^{3}} - \frac {3 \, {\left (b x + a\right )} a^{2} n \log \left (c\right )^{2}}{b^{3}} + \frac {{\left (b x + a\right )}^{3} \log \left (c\right )^{3}}{3 \, b^{3}} - \frac {{\left (b x + a\right )}^{2} a \log \left (c\right )^{3}}{b^{3}} + \frac {{\left (b x + a\right )} a^{2} \log \left (c\right )^{3}}{b^{3}} \]

[In]

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

[Out]

1/3*(b*x + a)^3*n^3*log(b*x + a)^3/b^3 - (b*x + a)^2*a*n^3*log(b*x + a)^3/b^3 + (b*x + a)*a^2*n^3*log(b*x + a)
^3/b^3 - 1/3*(b*x + a)^3*n^3*log(b*x + a)^2/b^3 + 3/2*(b*x + a)^2*a*n^3*log(b*x + a)^2/b^3 - 3*(b*x + a)*a^2*n
^3*log(b*x + a)^2/b^3 + (b*x + a)^3*n^2*log(b*x + a)^2*log(c)/b^3 - 3*(b*x + a)^2*a*n^2*log(b*x + a)^2*log(c)/
b^3 + 3*(b*x + a)*a^2*n^2*log(b*x + a)^2*log(c)/b^3 + 2/9*(b*x + a)^3*n^3*log(b*x + a)/b^3 - 3/2*(b*x + a)^2*a
*n^3*log(b*x + a)/b^3 + 6*(b*x + a)*a^2*n^3*log(b*x + a)/b^3 - 2/3*(b*x + a)^3*n^2*log(b*x + a)*log(c)/b^3 + 3
*(b*x + a)^2*a*n^2*log(b*x + a)*log(c)/b^3 - 6*(b*x + a)*a^2*n^2*log(b*x + a)*log(c)/b^3 + (b*x + a)^3*n*log(b
*x + a)*log(c)^2/b^3 - 3*(b*x + a)^2*a*n*log(b*x + a)*log(c)^2/b^3 + 3*(b*x + a)*a^2*n*log(b*x + a)*log(c)^2/b
^3 - 2/27*(b*x + a)^3*n^3/b^3 + 3/4*(b*x + a)^2*a*n^3/b^3 - 6*(b*x + a)*a^2*n^3/b^3 + 2/9*(b*x + a)^3*n^2*log(
c)/b^3 - 3/2*(b*x + a)^2*a*n^2*log(c)/b^3 + 6*(b*x + a)*a^2*n^2*log(c)/b^3 - 1/3*(b*x + a)^3*n*log(c)^2/b^3 +
3/2*(b*x + a)^2*a*n*log(c)^2/b^3 - 3*(b*x + a)*a^2*n*log(c)^2/b^3 + 1/3*(b*x + a)^3*log(c)^3/b^3 - (b*x + a)^2
*a*log(c)^3/b^3 + (b*x + a)*a^2*log(c)^3/b^3

Mupad [B] (verification not implemented)

Time = 1.36 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.60 \[ \int x^2 \log ^3\left (c (a+b x)^n\right ) \, dx={\ln \left (c\,{\left (a+b\,x\right )}^n\right )}^3\,\left (\frac {x^3}{3}+\frac {a^3}{3\,b^3}\right )-\frac {2\,n^3\,x^3}{27}-{\ln \left (c\,{\left (a+b\,x\right )}^n\right )}^2\,\left (\frac {n\,x^3}{3}+\frac {11\,a^3\,n}{6\,b^3}-\frac {a\,n\,x^2}{2\,b}+\frac {a^2\,n\,x}{b^2}\right )+\frac {\ln \left (c\,{\left (a+b\,x\right )}^n\right )\,\left (\frac {2\,b\,n^2\,x^3}{3}-\frac {5\,a\,n^2\,x^2}{2}+\frac {11\,a^2\,n^2\,x}{b}\right )}{3\,b}+\frac {85\,a^3\,n^3\,\ln \left (a+b\,x\right )}{18\,b^3}+\frac {19\,a\,n^3\,x^2}{36\,b}-\frac {85\,a^2\,n^3\,x}{18\,b^2} \]

[In]

int(x^2*log(c*(a + b*x)^n)^3,x)

[Out]

log(c*(a + b*x)^n)^3*(x^3/3 + a^3/(3*b^3)) - (2*n^3*x^3)/27 - log(c*(a + b*x)^n)^2*((n*x^3)/3 + (11*a^3*n)/(6*
b^3) - (a*n*x^2)/(2*b) + (a^2*n*x)/b^2) + (log(c*(a + b*x)^n)*((2*b*n^2*x^3)/3 - (5*a*n^2*x^2)/2 + (11*a^2*n^2
*x)/b))/(3*b) + (85*a^3*n^3*log(a + b*x))/(18*b^3) + (19*a*n^3*x^2)/(36*b) - (85*a^2*n^3*x)/(18*b^2)