### 3.9 $$\int \frac{\log (c x^n) (a x^m+b \log ^2(c x^n))^3}{x} \, dx$$

Optimal. Leaf size=272 $\frac{9 a^2 b n^2 x^{2 m} \log \left (c x^n\right )}{4 m^3}-\frac{9 a^2 b n x^{2 m} \log ^2\left (c x^n\right )}{4 m^2}+\frac{3 a^2 b x^{2 m} \log ^3\left (c x^n\right )}{2 m}-\frac{9 a^2 b n^3 x^{2 m}}{8 m^4}+\frac{a^3 x^{3 m} \log \left (c x^n\right )}{3 m}-\frac{a^3 n x^{3 m}}{9 m^2}+\frac{60 a b^2 n^2 x^m \log ^3\left (c x^n\right )}{m^3}-\frac{180 a b^2 n^3 x^m \log ^2\left (c x^n\right )}{m^4}+\frac{360 a b^2 n^4 x^m \log \left (c x^n\right )}{m^5}-\frac{15 a b^2 n x^m \log ^4\left (c x^n\right )}{m^2}+\frac{3 a b^2 x^m \log ^5\left (c x^n\right )}{m}-\frac{360 a b^2 n^5 x^m}{m^6}+\frac{b^3 \log ^8\left (c x^n\right )}{8 n}$

[Out]

(-360*a*b^2*n^5*x^m)/m^6 - (9*a^2*b*n^3*x^(2*m))/(8*m^4) - (a^3*n*x^(3*m))/(9*m^2) + (360*a*b^2*n^4*x^m*Log[c*
x^n])/m^5 + (9*a^2*b*n^2*x^(2*m)*Log[c*x^n])/(4*m^3) + (a^3*x^(3*m)*Log[c*x^n])/(3*m) - (180*a*b^2*n^3*x^m*Log
[c*x^n]^2)/m^4 - (9*a^2*b*n*x^(2*m)*Log[c*x^n]^2)/(4*m^2) + (60*a*b^2*n^2*x^m*Log[c*x^n]^3)/m^3 + (3*a^2*b*x^(
2*m)*Log[c*x^n]^3)/(2*m) - (15*a*b^2*n*x^m*Log[c*x^n]^4)/m^2 + (3*a*b^2*x^m*Log[c*x^n]^5)/m + (b^3*Log[c*x^n]^
8)/(8*n)

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Rubi [A]  time = 0.305071, antiderivative size = 272, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 5, integrand size = 28, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.179, Rules used = {2539, 2304, 2305, 2302, 30} $\frac{9 a^2 b n^2 x^{2 m} \log \left (c x^n\right )}{4 m^3}-\frac{9 a^2 b n x^{2 m} \log ^2\left (c x^n\right )}{4 m^2}+\frac{3 a^2 b x^{2 m} \log ^3\left (c x^n\right )}{2 m}-\frac{9 a^2 b n^3 x^{2 m}}{8 m^4}+\frac{a^3 x^{3 m} \log \left (c x^n\right )}{3 m}-\frac{a^3 n x^{3 m}}{9 m^2}+\frac{60 a b^2 n^2 x^m \log ^3\left (c x^n\right )}{m^3}-\frac{180 a b^2 n^3 x^m \log ^2\left (c x^n\right )}{m^4}+\frac{360 a b^2 n^4 x^m \log \left (c x^n\right )}{m^5}-\frac{15 a b^2 n x^m \log ^4\left (c x^n\right )}{m^2}+\frac{3 a b^2 x^m \log ^5\left (c x^n\right )}{m}-\frac{360 a b^2 n^5 x^m}{m^6}+\frac{b^3 \log ^8\left (c x^n\right )}{8 n}$

Antiderivative was successfully veriﬁed.

[In]

Int[(Log[c*x^n]*(a*x^m + b*Log[c*x^n]^2)^3)/x,x]

[Out]

(-360*a*b^2*n^5*x^m)/m^6 - (9*a^2*b*n^3*x^(2*m))/(8*m^4) - (a^3*n*x^(3*m))/(9*m^2) + (360*a*b^2*n^4*x^m*Log[c*
x^n])/m^5 + (9*a^2*b*n^2*x^(2*m)*Log[c*x^n])/(4*m^3) + (a^3*x^(3*m)*Log[c*x^n])/(3*m) - (180*a*b^2*n^3*x^m*Log
[c*x^n]^2)/m^4 - (9*a^2*b*n*x^(2*m)*Log[c*x^n]^2)/(4*m^2) + (60*a*b^2*n^2*x^m*Log[c*x^n]^3)/m^3 + (3*a^2*b*x^(
2*m)*Log[c*x^n]^3)/(2*m) - (15*a*b^2*n*x^m*Log[c*x^n]^4)/m^2 + (3*a*b^2*x^m*Log[c*x^n]^5)/m + (b^3*Log[c*x^n]^
8)/(8*n)

Rule 2539

Int[(Log[(c_.)*(x_)^(n_.)]^(r_.)*(Log[(c_.)*(x_)^(n_.)]^(q_)*(b_.) + (a_.)*(x_)^(m_.))^(p_.))/(x_), x_Symbol]
:> Int[ExpandIntegrand[Log[c*x^n]^r/x, (a*x^m + b*Log[c*x^n]^q)^p, x], x] /; FreeQ[{a, b, c, m, n, p, q, r}, x
] && EqQ[r, q - 1] && IGtQ[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
]

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 2302

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

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

\begin{align*} \int \frac{\log \left (c x^n\right ) \left (a x^m+b \log ^2\left (c x^n\right )\right )^3}{x} \, dx &=\int \left (a^3 x^{-1+3 m} \log \left (c x^n\right )+3 a^2 b x^{-1+2 m} \log ^3\left (c x^n\right )+3 a b^2 x^{-1+m} \log ^5\left (c x^n\right )+\frac{b^3 \log ^7\left (c x^n\right )}{x}\right ) \, dx\\ &=a^3 \int x^{-1+3 m} \log \left (c x^n\right ) \, dx+\left (3 a^2 b\right ) \int x^{-1+2 m} \log ^3\left (c x^n\right ) \, dx+\left (3 a b^2\right ) \int x^{-1+m} \log ^5\left (c x^n\right ) \, dx+b^3 \int \frac{\log ^7\left (c x^n\right )}{x} \, dx\\ &=-\frac{a^3 n x^{3 m}}{9 m^2}+\frac{a^3 x^{3 m} \log \left (c x^n\right )}{3 m}+\frac{3 a^2 b x^{2 m} \log ^3\left (c x^n\right )}{2 m}+\frac{3 a b^2 x^m \log ^5\left (c x^n\right )}{m}+\frac{b^3 \operatorname{Subst}\left (\int x^7 \, dx,x,\log \left (c x^n\right )\right )}{n}-\frac{\left (9 a^2 b n\right ) \int x^{-1+2 m} \log ^2\left (c x^n\right ) \, dx}{2 m}-\frac{\left (15 a b^2 n\right ) \int x^{-1+m} \log ^4\left (c x^n\right ) \, dx}{m}\\ &=-\frac{a^3 n x^{3 m}}{9 m^2}+\frac{a^3 x^{3 m} \log \left (c x^n\right )}{3 m}-\frac{9 a^2 b n x^{2 m} \log ^2\left (c x^n\right )}{4 m^2}+\frac{3 a^2 b x^{2 m} \log ^3\left (c x^n\right )}{2 m}-\frac{15 a b^2 n x^m \log ^4\left (c x^n\right )}{m^2}+\frac{3 a b^2 x^m \log ^5\left (c x^n\right )}{m}+\frac{b^3 \log ^8\left (c x^n\right )}{8 n}+\frac{\left (9 a^2 b n^2\right ) \int x^{-1+2 m} \log \left (c x^n\right ) \, dx}{2 m^2}+\frac{\left (60 a b^2 n^2\right ) \int x^{-1+m} \log ^3\left (c x^n\right ) \, dx}{m^2}\\ &=-\frac{9 a^2 b n^3 x^{2 m}}{8 m^4}-\frac{a^3 n x^{3 m}}{9 m^2}+\frac{9 a^2 b n^2 x^{2 m} \log \left (c x^n\right )}{4 m^3}+\frac{a^3 x^{3 m} \log \left (c x^n\right )}{3 m}-\frac{9 a^2 b n x^{2 m} \log ^2\left (c x^n\right )}{4 m^2}+\frac{60 a b^2 n^2 x^m \log ^3\left (c x^n\right )}{m^3}+\frac{3 a^2 b x^{2 m} \log ^3\left (c x^n\right )}{2 m}-\frac{15 a b^2 n x^m \log ^4\left (c x^n\right )}{m^2}+\frac{3 a b^2 x^m \log ^5\left (c x^n\right )}{m}+\frac{b^3 \log ^8\left (c x^n\right )}{8 n}-\frac{\left (180 a b^2 n^3\right ) \int x^{-1+m} \log ^2\left (c x^n\right ) \, dx}{m^3}\\ &=-\frac{9 a^2 b n^3 x^{2 m}}{8 m^4}-\frac{a^3 n x^{3 m}}{9 m^2}+\frac{9 a^2 b n^2 x^{2 m} \log \left (c x^n\right )}{4 m^3}+\frac{a^3 x^{3 m} \log \left (c x^n\right )}{3 m}-\frac{180 a b^2 n^3 x^m \log ^2\left (c x^n\right )}{m^4}-\frac{9 a^2 b n x^{2 m} \log ^2\left (c x^n\right )}{4 m^2}+\frac{60 a b^2 n^2 x^m \log ^3\left (c x^n\right )}{m^3}+\frac{3 a^2 b x^{2 m} \log ^3\left (c x^n\right )}{2 m}-\frac{15 a b^2 n x^m \log ^4\left (c x^n\right )}{m^2}+\frac{3 a b^2 x^m \log ^5\left (c x^n\right )}{m}+\frac{b^3 \log ^8\left (c x^n\right )}{8 n}+\frac{\left (360 a b^2 n^4\right ) \int x^{-1+m} \log \left (c x^n\right ) \, dx}{m^4}\\ &=-\frac{360 a b^2 n^5 x^m}{m^6}-\frac{9 a^2 b n^3 x^{2 m}}{8 m^4}-\frac{a^3 n x^{3 m}}{9 m^2}+\frac{360 a b^2 n^4 x^m \log \left (c x^n\right )}{m^5}+\frac{9 a^2 b n^2 x^{2 m} \log \left (c x^n\right )}{4 m^3}+\frac{a^3 x^{3 m} \log \left (c x^n\right )}{3 m}-\frac{180 a b^2 n^3 x^m \log ^2\left (c x^n\right )}{m^4}-\frac{9 a^2 b n x^{2 m} \log ^2\left (c x^n\right )}{4 m^2}+\frac{60 a b^2 n^2 x^m \log ^3\left (c x^n\right )}{m^3}+\frac{3 a^2 b x^{2 m} \log ^3\left (c x^n\right )}{2 m}-\frac{15 a b^2 n x^m \log ^4\left (c x^n\right )}{m^2}+\frac{3 a b^2 x^m \log ^5\left (c x^n\right )}{m}+\frac{b^3 \log ^8\left (c x^n\right )}{8 n}\\ \end{align*}

Mathematica [A]  time = 0.223594, size = 230, normalized size = 0.85 $\frac{a x^m \log \left (c x^n\right ) \left (4 a^2 m^4 x^{2 m}+27 a b m^2 n^2 x^m+4320 b^2 n^4\right )}{12 m^5}-\frac{a n x^m \left (8 a^2 m^4 x^{2 m}+81 a b m^2 n^2 x^m+25920 b^2 n^4\right )}{72 m^6}-\frac{15 a b^2 n x^m \log ^4\left (c x^n\right )}{m^2}+\frac{3 a b^2 x^m \log ^5\left (c x^n\right )}{m}+\frac{3 a b x^m \log ^3\left (c x^n\right ) \left (a m^2 x^m+40 b n^2\right )}{2 m^3}-\frac{9 a b n x^m \log ^2\left (c x^n\right ) \left (a m^2 x^m+80 b n^2\right )}{4 m^4}+\frac{b^3 \log ^8\left (c x^n\right )}{8 n}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(Log[c*x^n]*(a*x^m + b*Log[c*x^n]^2)^3)/x,x]

[Out]

-(a*n*x^m*(25920*b^2*n^4 + 81*a*b*m^2*n^2*x^m + 8*a^2*m^4*x^(2*m)))/(72*m^6) + (a*x^m*(4320*b^2*n^4 + 27*a*b*m
^2*n^2*x^m + 4*a^2*m^4*x^(2*m))*Log[c*x^n])/(12*m^5) - (9*a*b*n*x^m*(80*b*n^2 + a*m^2*x^m)*Log[c*x^n]^2)/(4*m^
4) + (3*a*b*x^m*(40*b*n^2 + a*m^2*x^m)*Log[c*x^n]^3)/(2*m^3) - (15*a*b^2*n*x^m*Log[c*x^n]^4)/m^2 + (3*a*b^2*x^
m*Log[c*x^n]^5)/m + (b^3*Log[c*x^n]^8)/(8*n)

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Maple [C]  time = 5.05, size = 61910, normalized size = 227.6 \begin{align*} \text{output too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.16194, size = 1558, normalized size = 5.73 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/72*(9*b^3*m^6*n^7*log(x)^8 + 72*b^3*m^6*n^6*log(c)*log(x)^7 + 252*b^3*m^6*n^5*log(c)^2*log(x)^6 + 504*b^3*m^
6*n^4*log(c)^3*log(x)^5 + 630*b^3*m^6*n^3*log(c)^4*log(x)^4 + 504*b^3*m^6*n^2*log(c)^5*log(x)^3 + 252*b^3*m^6*
n*log(c)^6*log(x)^2 + 72*b^3*m^6*log(c)^7*log(x) + 8*(3*a^3*m^5*n*log(x) + 3*a^3*m^5*log(c) - a^3*m^4*n)*x^(3*
m) + 27*(4*a^2*b*m^5*n^3*log(x)^3 + 4*a^2*b*m^5*log(c)^3 - 6*a^2*b*m^4*n*log(c)^2 + 6*a^2*b*m^3*n^2*log(c) - 3
*a^2*b*m^2*n^3 + 6*(2*a^2*b*m^5*n^2*log(c) - a^2*b*m^4*n^3)*log(x)^2 + 6*(2*a^2*b*m^5*n*log(c)^2 - 2*a^2*b*m^4
*n^2*log(c) + a^2*b*m^3*n^3)*log(x))*x^(2*m) + 216*(a*b^2*m^5*n^5*log(x)^5 + a*b^2*m^5*log(c)^5 - 5*a*b^2*m^4*
n*log(c)^4 + 20*a*b^2*m^3*n^2*log(c)^3 - 60*a*b^2*m^2*n^3*log(c)^2 + 120*a*b^2*m*n^4*log(c) - 120*a*b^2*n^5 +
5*(a*b^2*m^5*n^4*log(c) - a*b^2*m^4*n^5)*log(x)^4 + 10*(a*b^2*m^5*n^3*log(c)^2 - 2*a*b^2*m^4*n^4*log(c) + 2*a*
b^2*m^3*n^5)*log(x)^3 + 10*(a*b^2*m^5*n^2*log(c)^3 - 3*a*b^2*m^4*n^3*log(c)^2 + 6*a*b^2*m^3*n^4*log(c) - 6*a*b
^2*m^2*n^5)*log(x)^2 + 5*(a*b^2*m^5*n*log(c)^4 - 4*a*b^2*m^4*n^2*log(c)^3 + 12*a*b^2*m^3*n^3*log(c)^2 - 24*a*b
^2*m^2*n^4*log(c) + 24*a*b^2*m*n^5)*log(x))*x^m)/m^6

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError

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Giac [B]  time = 1.34126, size = 1034, normalized size = 3.8 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*b^3*n^7*log(x)^8 + b^3*n^6*log(c)*log(x)^7 + 7/2*b^3*n^5*log(c)^2*log(x)^6 + 7*b^3*n^4*log(c)^3*log(x)^5 +
35/4*b^3*n^3*log(c)^4*log(x)^4 + 7*b^3*n^2*log(c)^5*log(x)^3 + 3*a*b^2*n^5*x^m*log(x)^5/m + 7/2*b^3*n*log(c)^
6*log(x)^2 + 15*a*b^2*n^4*x^m*log(c)*log(x)^4/m + b^3*log(c)^7*log(x) + 30*a*b^2*n^3*x^m*log(c)^2*log(x)^3/m -
15*a*b^2*n^5*x^m*log(x)^4/m^2 + 30*a*b^2*n^2*x^m*log(c)^3*log(x)^2/m - 60*a*b^2*n^4*x^m*log(c)*log(x)^3/m^2 +
15*a*b^2*n*x^m*log(c)^4*log(x)/m - 90*a*b^2*n^3*x^m*log(c)^2*log(x)^2/m^2 + 3/2*a^2*b*n^3*x^(2*m)*log(x)^3/m
+ 60*a*b^2*n^5*x^m*log(x)^3/m^3 + 3*a*b^2*x^m*log(c)^5/m - 60*a*b^2*n^2*x^m*log(c)^3*log(x)/m^2 + 9/2*a^2*b*n^
2*x^(2*m)*log(c)*log(x)^2/m + 180*a*b^2*n^4*x^m*log(c)*log(x)^2/m^3 - 15*a*b^2*n*x^m*log(c)^4/m^2 + 9/2*a^2*b*
n*x^(2*m)*log(c)^2*log(x)/m + 180*a*b^2*n^3*x^m*log(c)^2*log(x)/m^3 - 9/4*a^2*b*n^3*x^(2*m)*log(x)^2/m^2 - 180
*a*b^2*n^5*x^m*log(x)^2/m^4 + 3/2*a^2*b*x^(2*m)*log(c)^3/m + 60*a*b^2*n^2*x^m*log(c)^3/m^3 - 9/2*a^2*b*n^2*x^(
2*m)*log(c)*log(x)/m^2 - 360*a*b^2*n^4*x^m*log(c)*log(x)/m^4 - 9/4*a^2*b*n*x^(2*m)*log(c)^2/m^2 - 180*a*b^2*n^
3*x^m*log(c)^2/m^4 + 1/3*a^3*n*x^(3*m)*log(x)/m + 9/4*a^2*b*n^3*x^(2*m)*log(x)/m^3 + 360*a*b^2*n^5*x^m*log(x)/
m^5 + 1/3*a^3*x^(3*m)*log(c)/m + 9/4*a^2*b*n^2*x^(2*m)*log(c)/m^3 + 360*a*b^2*n^4*x^m*log(c)/m^5 - 1/9*a^3*n*x
^(3*m)/m^2 - 9/8*a^2*b*n^3*x^(2*m)/m^4 - 360*a*b^2*n^5*x^m/m^6