3.62 \(\int \frac{(a+b \log (c x^n))^3}{x^2} \, dx\)
Optimal. Leaf size=69 \[ -\frac{6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right )}{x}-\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2}{x}-\frac{\left (a+b \log \left (c x^n\right )\right )^3}{x}-\frac{6 b^3 n^3}{x} \]
[Out]
(-6*b^3*n^3)/x - (6*b^2*n^2*(a + b*Log[c*x^n]))/x - (3*b*n*(a + b*Log[c*x^n])^2)/x - (a + b*Log[c*x^n])^3/x
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Rubi [A] time = 0.0595732, antiderivative size = 69, normalized size of antiderivative = 1.,
number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used =
{2305, 2304} \[ -\frac{6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right )}{x}-\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2}{x}-\frac{\left (a+b \log \left (c x^n\right )\right )^3}{x}-\frac{6 b^3 n^3}{x} \]
Antiderivative was successfully verified.
[In]
Int[(a + b*Log[c*x^n])^3/x^2,x]
[Out]
(-6*b^3*n^3)/x - (6*b^2*n^2*(a + b*Log[c*x^n]))/x - (3*b*n*(a + b*Log[c*x^n])^2)/x - (a + b*Log[c*x^n])^3/x
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 \frac{\left (a+b \log \left (c x^n\right )\right )^3}{x^2} \, dx &=-\frac{\left (a+b \log \left (c x^n\right )\right )^3}{x}+(3 b n) \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{x^2} \, dx\\ &=-\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2}{x}-\frac{\left (a+b \log \left (c x^n\right )\right )^3}{x}+\left (6 b^2 n^2\right ) \int \frac{a+b \log \left (c x^n\right )}{x^2} \, dx\\ &=-\frac{6 b^3 n^3}{x}-\frac{6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right )}{x}-\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2}{x}-\frac{\left (a+b \log \left (c x^n\right )\right )^3}{x}\\ \end{align*}
Mathematica [A] time = 0.018329, size = 52, normalized size = 0.75 \[ -\frac{\left (a+b \log \left (c x^n\right )\right )^3+3 b n \left (\left (a+b \log \left (c x^n\right )\right )^2+2 b n \left (a+b \log \left (c x^n\right )+b n\right )\right )}{x} \]
Antiderivative was successfully verified.
[In]
Integrate[(a + b*Log[c*x^n])^3/x^2,x]
[Out]
-(((a + b*Log[c*x^n])^3 + 3*b*n*((a + b*Log[c*x^n])^2 + 2*b*n*(a + b*n + b*Log[c*x^n])))/x)
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Maple [C] time = 0.24, size = 2674, normalized size = 38.8 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a+b*ln(c*x^n))^3/x^2,x)
[Out]
-b^3/x*ln(x^n)^3-3/2*(I*Pi*b^3*csgn(I*x^n)*csgn(I*c*x^n)^2-I*Pi*b^3*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-I*Pi*b
^3*csgn(I*c*x^n)^3+I*Pi*b^3*csgn(I*c*x^n)^2*csgn(I*c)+2*ln(c)*b^3+2*b^3*n+2*a*b^2)/x*ln(x^n)^2-3/4*(4*a^2*b+8*
b^3*n^2+8*ln(c)*a*b^2+8*n*ln(c)*b^3+2*Pi^2*b^3*csgn(I*x^n)*csgn(I*c*x^n)^3*csgn(I*c)^2-4*I*ln(c)*Pi*b^3*csgn(I
*c*x^n)^3-4*I*Pi*a*b^2*csgn(I*c*x^n)^3-4*I*n*Pi*b^3*csgn(I*c*x^n)^3+8*a*b^2*n+2*Pi^2*b^3*csgn(I*x^n)^2*csgn(I*
c*x^n)^3*csgn(I*c)-Pi^2*b^3*csgn(I*x^n)^2*csgn(I*c*x^n)^2*csgn(I*c)^2-4*Pi^2*b^3*csgn(I*x^n)*csgn(I*c*x^n)^4*c
sgn(I*c)+4*ln(c)^2*b^3-Pi^2*b^3*csgn(I*c*x^n)^6-Pi^2*b^3*csgn(I*x^n)^2*csgn(I*c*x^n)^4+2*Pi^2*b^3*csgn(I*x^n)*
csgn(I*c*x^n)^5+2*Pi^2*b^3*csgn(I*c*x^n)^5*csgn(I*c)-Pi^2*b^3*csgn(I*c*x^n)^4*csgn(I*c)^2-4*I*ln(c)*Pi*b^3*csg
n(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-4*I*Pi*a*b^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-4*I*n*Pi*b^3*csgn(I*x^n)*csg
n(I*c*x^n)*csgn(I*c)+4*I*Pi*a*b^2*csgn(I*c*x^n)^2*csgn(I*c)+4*I*n*Pi*b^3*csgn(I*x^n)*csgn(I*c*x^n)^2+4*I*n*Pi*
b^3*csgn(I*c*x^n)^2*csgn(I*c)+4*I*ln(c)*Pi*b^3*csgn(I*x^n)*csgn(I*c*x^n)^2+4*I*ln(c)*Pi*b^3*csgn(I*c*x^n)^2*cs
gn(I*c)+4*I*Pi*a*b^2*csgn(I*x^n)*csgn(I*c*x^n)^2)/x*ln(x^n)-1/8*(8*a^3+48*a*b^2*n^2+24*a^2*b*n-6*Pi^2*a*b^2*cs
gn(I*x^n)^2*csgn(I*c*x^n)^4+8*ln(c)^3*b^3-24*Pi^2*a*b^2*csgn(I*x^n)*csgn(I*c*x^n)^4*csgn(I*c)-3*I*Pi^3*b^3*csg
n(I*x^n)*csgn(I*c*x^n)^8-3*I*Pi^3*b^3*csgn(I*c*x^n)^8*csgn(I*c)+3*I*Pi^3*b^3*csgn(I*c*x^n)^7*csgn(I*c)^2-I*Pi^
3*b^3*csgn(I*c*x^n)^6*csgn(I*c)^3-12*I*ln(c)^2*Pi*b^3*csgn(I*c*x^n)^3+12*Pi^2*b^3*n*csgn(I*x^n)^2*csgn(I*c*x^n
)^3*csgn(I*c)-24*I*ln(c)*Pi*b^3*n*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-6*Pi^2*b^3*n*csgn(I*x^n)^2*csgn(I*c*x^n)
^2*csgn(I*c)^2-24*Pi^2*b^3*n*csgn(I*x^n)*csgn(I*c*x^n)^4*csgn(I*c)-6*ln(c)*Pi^2*b^3*csgn(I*x^n)^2*csgn(I*c*x^n
)^4+12*ln(c)*Pi^2*b^3*csgn(I*x^n)*csgn(I*c*x^n)^5+12*ln(c)*Pi^2*b^3*csgn(I*c*x^n)^5*csgn(I*c)+12*Pi^2*b^3*n*cs
gn(I*c*x^n)^5*csgn(I*c)-6*Pi^2*b^3*n*csgn(I*c*x^n)^4*csgn(I*c)^2-24*I*ln(c)*Pi*a*b^2*csgn(I*x^n)*csgn(I*c*x^n)
*csgn(I*c)-24*I*Pi*a*b^2*n*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+48*b^3*n^3+12*Pi^2*a*b^2*csgn(I*x^n)*csgn(I*c*x
^n)^5-24*I*Pi*b^3*n^2*csgn(I*c*x^n)^3-12*I*Pi*a^2*b*csgn(I*c*x^n)^3-I*Pi^3*b^3*csgn(I*x^n)^3*csgn(I*c*x^n)^6+3
*I*Pi^3*b^3*csgn(I*x^n)^2*csgn(I*c*x^n)^7+24*ln(c)^2*b^3*n+48*ln(c)*b^3*n^2+24*ln(c)*a^2*b+24*ln(c)^2*a*b^2+I*
Pi^3*b^3*csgn(I*c*x^n)^9+12*Pi^2*a*b^2*csgn(I*c*x^n)^5*csgn(I*c)-6*Pi^2*a*b^2*csgn(I*c*x^n)^4*csgn(I*c)^2-6*ln
(c)*Pi^2*b^3*csgn(I*c*x^n)^4*csgn(I*c)^2-6*Pi^2*b^3*n*csgn(I*x^n)^2*csgn(I*c*x^n)^4+12*Pi^2*b^3*n*csgn(I*x^n)*
csgn(I*c*x^n)^5-6*Pi^2*b^3*n*csgn(I*c*x^n)^6-6*ln(c)*Pi^2*b^3*csgn(I*c*x^n)^6-6*Pi^2*a*b^2*csgn(I*c*x^n)^6+12*
ln(c)*Pi^2*b^3*csgn(I*x^n)*csgn(I*c*x^n)^3*csgn(I*c)^2+12*Pi^2*a*b^2*csgn(I*x^n)*csgn(I*c*x^n)^3*csgn(I*c)^2-6
*ln(c)*Pi^2*b^3*csgn(I*x^n)^2*csgn(I*c*x^n)^2*csgn(I*c)^2-24*ln(c)*Pi^2*b^3*csgn(I*x^n)*csgn(I*c*x^n)^4*csgn(I
*c)+12*Pi^2*b^3*n*csgn(I*x^n)*csgn(I*c*x^n)^3*csgn(I*c)^2+12*ln(c)*Pi^2*b^3*csgn(I*x^n)^2*csgn(I*c*x^n)^3*csgn
(I*c)-12*I*ln(c)^2*Pi*b^3*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+12*Pi^2*a*b^2*csgn(I*x^n)^2*csgn(I*c*x^n)^3*csgn
(I*c)-6*Pi^2*a*b^2*csgn(I*x^n)^2*csgn(I*c*x^n)^2*csgn(I*c)^2-24*I*Pi*b^3*n^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*
c)+12*I*Pi*a^2*b*csgn(I*c*x^n)^2*csgn(I*c)+I*Pi^3*b^3*csgn(I*x^n)^3*csgn(I*c*x^n)^3*csgn(I*c)^3+3*I*Pi^3*b^3*c
sgn(I*x^n)*csgn(I*c*x^n)^5*csgn(I*c)^3+12*I*ln(c)^2*Pi*b^3*csgn(I*x^n)*csgn(I*c*x^n)^2+12*I*ln(c)^2*Pi*b^3*csg
n(I*c*x^n)^2*csgn(I*c)-24*I*ln(c)*Pi*a*b^2*csgn(I*c*x^n)^3+12*I*Pi*a^2*b*csgn(I*x^n)*csgn(I*c*x^n)^2-3*I*Pi^3*
b^3*csgn(I*x^n)^2*csgn(I*c*x^n)^4*csgn(I*c)^3+9*I*Pi^3*b^3*csgn(I*x^n)*csgn(I*c*x^n)^7*csgn(I*c)-9*I*Pi^3*b^3*
csgn(I*x^n)*csgn(I*c*x^n)^6*csgn(I*c)^2+24*I*Pi*b^3*n^2*csgn(I*x^n)*csgn(I*c*x^n)^2+3*I*Pi^3*b^3*csgn(I*x^n)^3
*csgn(I*c*x^n)^5*csgn(I*c)+24*I*Pi*b^3*n^2*csgn(I*c*x^n)^2*csgn(I*c)-3*I*Pi^3*b^3*csgn(I*x^n)^3*csgn(I*c*x^n)^
4*csgn(I*c)^2-9*I*Pi^3*b^3*csgn(I*x^n)^2*csgn(I*c*x^n)^6*csgn(I*c)+9*I*Pi^3*b^3*csgn(I*x^n)^2*csgn(I*c*x^n)^5*
csgn(I*c)^2+48*ln(c)*a*b^2*n+24*I*ln(c)*Pi*a*b^2*csgn(I*x^n)*csgn(I*c*x^n)^2+24*I*ln(c)*Pi*a*b^2*csgn(I*c*x^n)
^2*csgn(I*c)-24*I*ln(c)*Pi*b^3*n*csgn(I*c*x^n)^3+24*I*n*Pi*a*b^2*csgn(I*x^n)*csgn(I*c*x^n)^2-24*I*Pi*a*b^2*n*c
sgn(I*c*x^n)^3+24*I*Pi*a*b^2*n*csgn(I*c*x^n)^2*csgn(I*c)+24*I*ln(c)*Pi*b^3*n*csgn(I*c*x^n)^2*csgn(I*c)+24*I*ln
(c)*Pi*b^3*n*csgn(I*x^n)*csgn(I*c*x^n)^2-12*I*Pi*a^2*b*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c))/x
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Maxima [A] time = 1.15879, size = 180, normalized size = 2.61 \begin{align*} -\frac{b^{3} \log \left (c x^{n}\right )^{3}}{x} - 3 \,{\left (2 \, n{\left (\frac{n^{2}}{x} + \frac{n \log \left (c x^{n}\right )}{x}\right )} + \frac{n \log \left (c x^{n}\right )^{2}}{x}\right )} b^{3} - 6 \, a b^{2}{\left (\frac{n^{2}}{x} + \frac{n \log \left (c x^{n}\right )}{x}\right )} - \frac{3 \, a b^{2} \log \left (c x^{n}\right )^{2}}{x} - \frac{3 \, a^{2} b n}{x} - \frac{3 \, a^{2} b \log \left (c x^{n}\right )}{x} - \frac{a^{3}}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*log(c*x^n))^3/x^2,x, algorithm="maxima")
[Out]
-b^3*log(c*x^n)^3/x - 3*(2*n*(n^2/x + n*log(c*x^n)/x) + n*log(c*x^n)^2/x)*b^3 - 6*a*b^2*(n^2/x + n*log(c*x^n)/
x) - 3*a*b^2*log(c*x^n)^2/x - 3*a^2*b*n/x - 3*a^2*b*log(c*x^n)/x - a^3/x
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Fricas [B] time = 0.881548, size = 406, normalized size = 5.88 \begin{align*} -\frac{b^{3} n^{3} \log \left (x\right )^{3} + 6 \, b^{3} n^{3} + b^{3} \log \left (c\right )^{3} + 6 \, a b^{2} n^{2} + 3 \, a^{2} b n + a^{3} + 3 \,{\left (b^{3} n + a b^{2}\right )} \log \left (c\right )^{2} + 3 \,{\left (b^{3} n^{3} + b^{3} n^{2} \log \left (c\right ) + a b^{2} n^{2}\right )} \log \left (x\right )^{2} + 3 \,{\left (2 \, b^{3} n^{2} + 2 \, a b^{2} n + a^{2} b\right )} \log \left (c\right ) + 3 \,{\left (2 \, b^{3} n^{3} + b^{3} n \log \left (c\right )^{2} + 2 \, a b^{2} n^{2} + a^{2} b n + 2 \,{\left (b^{3} n^{2} + a b^{2} n\right )} \log \left (c\right )\right )} \log \left (x\right )}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*log(c*x^n))^3/x^2,x, algorithm="fricas")
[Out]
-(b^3*n^3*log(x)^3 + 6*b^3*n^3 + b^3*log(c)^3 + 6*a*b^2*n^2 + 3*a^2*b*n + a^3 + 3*(b^3*n + a*b^2)*log(c)^2 + 3
*(b^3*n^3 + b^3*n^2*log(c) + a*b^2*n^2)*log(x)^2 + 3*(2*b^3*n^2 + 2*a*b^2*n + a^2*b)*log(c) + 3*(2*b^3*n^3 + b
^3*n*log(c)^2 + 2*a*b^2*n^2 + a^2*b*n + 2*(b^3*n^2 + a*b^2*n)*log(c))*log(x))/x
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Sympy [B] time = 1.601, size = 272, normalized size = 3.94 \begin{align*} - \frac{a^{3}}{x} - \frac{3 a^{2} b n \log{\left (x \right )}}{x} - \frac{3 a^{2} b n}{x} - \frac{3 a^{2} b \log{\left (c \right )}}{x} - \frac{3 a b^{2} n^{2} \log{\left (x \right )}^{2}}{x} - \frac{6 a b^{2} n^{2} \log{\left (x \right )}}{x} - \frac{6 a b^{2} n^{2}}{x} - \frac{6 a b^{2} n \log{\left (c \right )} \log{\left (x \right )}}{x} - \frac{6 a b^{2} n \log{\left (c \right )}}{x} - \frac{3 a b^{2} \log{\left (c \right )}^{2}}{x} - \frac{b^{3} n^{3} \log{\left (x \right )}^{3}}{x} - \frac{3 b^{3} n^{3} \log{\left (x \right )}^{2}}{x} - \frac{6 b^{3} n^{3} \log{\left (x \right )}}{x} - \frac{6 b^{3} n^{3}}{x} - \frac{3 b^{3} n^{2} \log{\left (c \right )} \log{\left (x \right )}^{2}}{x} - \frac{6 b^{3} n^{2} \log{\left (c \right )} \log{\left (x \right )}}{x} - \frac{6 b^{3} n^{2} \log{\left (c \right )}}{x} - \frac{3 b^{3} n \log{\left (c \right )}^{2} \log{\left (x \right )}}{x} - \frac{3 b^{3} n \log{\left (c \right )}^{2}}{x} - \frac{b^{3} \log{\left (c \right )}^{3}}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*ln(c*x**n))**3/x**2,x)
[Out]
-a**3/x - 3*a**2*b*n*log(x)/x - 3*a**2*b*n/x - 3*a**2*b*log(c)/x - 3*a*b**2*n**2*log(x)**2/x - 6*a*b**2*n**2*l
og(x)/x - 6*a*b**2*n**2/x - 6*a*b**2*n*log(c)*log(x)/x - 6*a*b**2*n*log(c)/x - 3*a*b**2*log(c)**2/x - b**3*n**
3*log(x)**3/x - 3*b**3*n**3*log(x)**2/x - 6*b**3*n**3*log(x)/x - 6*b**3*n**3/x - 3*b**3*n**2*log(c)*log(x)**2/
x - 6*b**3*n**2*log(c)*log(x)/x - 6*b**3*n**2*log(c)/x - 3*b**3*n*log(c)**2*log(x)/x - 3*b**3*n*log(c)**2/x -
b**3*log(c)**3/x
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Giac [B] time = 1.21943, size = 266, normalized size = 3.86 \begin{align*} -\frac{b^{3} n^{3} \log \left (x\right )^{3}}{x} - \frac{3 \,{\left (b^{3} n^{3} + b^{3} n^{2} \log \left (c\right ) + a b^{2} n^{2}\right )} \log \left (x\right )^{2}}{x} - \frac{3 \,{\left (2 \, b^{3} n^{3} + 2 \, b^{3} n^{2} \log \left (c\right ) + b^{3} n \log \left (c\right )^{2} + 2 \, a b^{2} n^{2} + 2 \, a b^{2} n \log \left (c\right ) + a^{2} b n\right )} \log \left (x\right )}{x} - \frac{6 \, b^{3} n^{3} + 6 \, b^{3} n^{2} \log \left (c\right ) + 3 \, b^{3} n \log \left (c\right )^{2} + b^{3} \log \left (c\right )^{3} + 6 \, a b^{2} n^{2} + 6 \, a b^{2} n \log \left (c\right ) + 3 \, a b^{2} \log \left (c\right )^{2} + 3 \, a^{2} b n + 3 \, a^{2} b \log \left (c\right ) + a^{3}}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*log(c*x^n))^3/x^2,x, algorithm="giac")
[Out]
-b^3*n^3*log(x)^3/x - 3*(b^3*n^3 + b^3*n^2*log(c) + a*b^2*n^2)*log(x)^2/x - 3*(2*b^3*n^3 + 2*b^3*n^2*log(c) +
b^3*n*log(c)^2 + 2*a*b^2*n^2 + 2*a*b^2*n*log(c) + a^2*b*n)*log(x)/x - (6*b^3*n^3 + 6*b^3*n^2*log(c) + 3*b^3*n*
log(c)^2 + b^3*log(c)^3 + 6*a*b^2*n^2 + 6*a*b^2*n*log(c) + 3*a*b^2*log(c)^2 + 3*a^2*b*n + 3*a^2*b*log(c) + a^3
)/x