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

Optimal. Leaf size=285 \[ \frac{6 a^2 n^2 (a+b x) \log \left (c (a+b x)^n\right )}{b^3}-\frac{3 a^2 n (a+b x) \log ^2\left (c (a+b x)^n\right )}{b^3}+\frac{a^2 (a+b x) \log ^3\left (c (a+b x)^n\right )}{b^3}-\frac{6 a^2 n^3 x}{b^2}+\frac{2 n^2 (a+b x)^3 \log \left (c (a+b x)^n\right )}{9 b^3}-\frac{3 a n^2 (a+b x)^2 \log \left (c (a+b x)^n\right )}{2 b^3}-\frac{n (a+b x)^3 \log ^2\left (c (a+b x)^n\right )}{3 b^3}+\frac{3 a n (a+b x)^2 \log ^2\left (c (a+b x)^n\right )}{2 b^3}+\frac{(a+b x)^3 \log ^3\left (c (a+b x)^n\right )}{3 b^3}-\frac{a (a+b x)^2 \log ^3\left (c (a+b x)^n\right )}{b^3}-\frac{2 n^3 (a+b x)^3}{27 b^3}+\frac{3 a n^3 (a+b x)^2}{4 b^3} \]

[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)

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Rubi [A]  time = 0.223719, antiderivative size = 285, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 7, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.438, Rules used = {2401, 2389, 2296, 2295, 2390, 2305, 2304} \[ \frac{6 a^2 n^2 (a+b x) \log \left (c (a+b x)^n\right )}{b^3}-\frac{3 a^2 n (a+b x) \log ^2\left (c (a+b x)^n\right )}{b^3}+\frac{a^2 (a+b x) \log ^3\left (c (a+b x)^n\right )}{b^3}-\frac{6 a^2 n^3 x}{b^2}+\frac{2 n^2 (a+b x)^3 \log \left (c (a+b x)^n\right )}{9 b^3}-\frac{3 a n^2 (a+b x)^2 \log \left (c (a+b x)^n\right )}{2 b^3}-\frac{n (a+b x)^3 \log ^2\left (c (a+b x)^n\right )}{3 b^3}+\frac{3 a n (a+b x)^2 \log ^2\left (c (a+b x)^n\right )}{2 b^3}+\frac{(a+b x)^3 \log ^3\left (c (a+b x)^n\right )}{3 b^3}-\frac{a (a+b x)^2 \log ^3\left (c (a+b x)^n\right )}{b^3}-\frac{2 n^3 (a+b x)^3}{27 b^3}+\frac{3 a n^3 (a+b x)^2}{4 b^3} \]

Antiderivative was successfully verified.

[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 2401

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]

Rule 2389

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 2296

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 2295

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

Rule 2390

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 2305

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 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
]

Rubi steps

\begin{align*} \int x^2 \log ^3\left (c (a+b x)^n\right ) \, dx &=\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{\operatorname{Subst}\left (\int x^2 \log ^3\left (c x^n\right ) \, dx,x,a+b x\right )}{b^3}-\frac{(2 a) \operatorname{Subst}\left (\int x \log ^3\left (c x^n\right ) \, dx,x,a+b x\right )}{b^3}+\frac{a^2 \operatorname{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 \operatorname{Subst}\left (\int x^2 \log ^2\left (c x^n\right ) \, dx,x,a+b x\right )}{b^3}+\frac{(3 a n) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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]  time = 0.0715175, size = 260, normalized size = 0.91 \[ \frac{85 a^3 n^2 \log \left (c (a+b x)^n\right )}{18 b^3}+\frac{11 a^2 n^2 x \log \left (c (a+b x)^n\right )}{3 b^2}+\frac{a^3 \log ^3\left (c (a+b x)^n\right )}{3 b^3}-\frac{11 a^3 n \log ^2\left (c (a+b x)^n\right )}{6 b^3}-\frac{a^2 n x \log ^2\left (c (a+b x)^n\right )}{b^2}-\frac{85 a^2 n^3 x}{18 b^2}-\frac{5 a n^2 x^2 \log \left (c (a+b x)^n\right )}{6 b}+\frac{2}{9} n^2 x^3 \log \left (c (a+b x)^n\right )+\frac{a n x^2 \log ^2\left (c (a+b x)^n\right )}{2 b}+\frac{1}{3} x^3 \log ^3\left (c (a+b x)^n\right )-\frac{1}{3} n x^3 \log ^2\left (c (a+b x)^n\right )+\frac{19 a n^3 x^2}{36 b}-\frac{2 n^3 x^3}{27} \]

Antiderivative was successfully verified.

[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

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Maple [C]  time = 0.744, size = 5345, normalized size = 18.8 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

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Maxima [A]  time = 1.48307, size = 290, normalized size = 1.02 \begin{align*} \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 )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [A]  time = 2.08624, size = 756, normalized size = 2.65 \begin{align*} -\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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Sympy [A]  time = 7.31348, size = 517, normalized size = 1.81 \begin{align*} \begin{cases} \frac{a^{3} n^{3} \log{\left (a + b x \right )}^{3}}{3 b^{3}} - \frac{11 a^{3} n^{3} \log{\left (a + b x \right )}^{2}}{6 b^{3}} + \frac{85 a^{3} n^{3} \log{\left (a + b x \right )}}{18 b^{3}} + \frac{a^{3} n^{2} \log{\left (c \right )} \log{\left (a + b x \right )}^{2}}{b^{3}} - \frac{11 a^{3} n^{2} \log{\left (c \right )} \log{\left (a + b x \right )}}{3 b^{3}} + \frac{a^{3} n \log{\left (c \right )}^{2} \log{\left (a + b x \right )}}{b^{3}} - \frac{a^{2} n^{3} x \log{\left (a + b x \right )}^{2}}{b^{2}} + \frac{11 a^{2} n^{3} x \log{\left (a + b x \right )}}{3 b^{2}} - \frac{85 a^{2} n^{3} x}{18 b^{2}} - \frac{2 a^{2} n^{2} x \log{\left (c \right )} \log{\left (a + b x \right )}}{b^{2}} + \frac{11 a^{2} n^{2} x \log{\left (c \right )}}{3 b^{2}} - \frac{a^{2} n x \log{\left (c \right )}^{2}}{b^{2}} + \frac{a n^{3} x^{2} \log{\left (a + b x \right )}^{2}}{2 b} - \frac{5 a n^{3} x^{2} \log{\left (a + b x \right )}}{6 b} + \frac{19 a n^{3} x^{2}}{36 b} + \frac{a n^{2} x^{2} \log{\left (c \right )} \log{\left (a + b x \right )}}{b} - \frac{5 a n^{2} x^{2} \log{\left (c \right )}}{6 b} + \frac{a n x^{2} \log{\left (c \right )}^{2}}{2 b} + \frac{n^{3} x^{3} \log{\left (a + b x \right )}^{3}}{3} - \frac{n^{3} x^{3} \log{\left (a + b x \right )}^{2}}{3} + \frac{2 n^{3} x^{3} \log{\left (a + b x \right )}}{9} - \frac{2 n^{3} x^{3}}{27} + n^{2} x^{3} \log{\left (c \right )} \log{\left (a + b x \right )}^{2} - \frac{2 n^{2} x^{3} \log{\left (c \right )} \log{\left (a + b x \right )}}{3} + \frac{2 n^{2} x^{3} \log{\left (c \right )}}{9} + n x^{3} \log{\left (c \right )}^{2} \log{\left (a + b x \right )} - \frac{n x^{3} \log{\left (c \right )}^{2}}{3} + \frac{x^{3} \log{\left (c \right )}^{3}}{3} & \text{for}\: b \neq 0 \\\frac{x^{3} \log{\left (a^{n} c \right )}^{3}}{3} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.2271, size = 845, normalized size = 2.96 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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