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*ln(c*x^n))/x-3*b*n*(a+b*ln(c*x^n))^2/x-(a+b*ln(c*x^n))^3/x

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Rubi [A]  time = 0.06, antiderivative size = 69, normalized size of antiderivative = 1.00, 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 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]

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*}

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Mathematica [A]  time = 0.02, 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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fricas [B]  time = 0.41, size = 180, normalized size = 2.61 \[ -\frac {b^{3} n^{3} \log \relax (x)^{3} + 6 \, b^{3} n^{3} + b^{3} \log \relax (c)^{3} + 6 \, a b^{2} n^{2} + 3 \, a^{2} b n + a^{3} + 3 \, {\left (b^{3} n + a b^{2}\right )} \log \relax (c)^{2} + 3 \, {\left (b^{3} n^{3} + b^{3} n^{2} \log \relax (c) + a b^{2} n^{2}\right )} \log \relax (x)^{2} + 3 \, {\left (2 \, b^{3} n^{2} + 2 \, a b^{2} n + a^{2} b\right )} \log \relax (c) + 3 \, {\left (2 \, b^{3} n^{3} + b^{3} n \log \relax (c)^{2} + 2 \, a b^{2} n^{2} + a^{2} b n + 2 \, {\left (b^{3} n^{2} + a b^{2} n\right )} \log \relax (c)\right )} \log \relax (x)}{x} \]

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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giac [B]  time = 0.28, size = 197, normalized size = 2.86 \[ -\frac {b^{3} n^{3} \log \relax (x)^{3}}{x} - \frac {3 \, {\left (b^{3} n^{3} + b^{3} n^{2} \log \relax (c) + a b^{2} n^{2}\right )} \log \relax (x)^{2}}{x} - \frac {3 \, {\left (2 \, b^{3} n^{3} + 2 \, b^{3} n^{2} \log \relax (c) + b^{3} n \log \relax (c)^{2} + 2 \, a b^{2} n^{2} + 2 \, a b^{2} n \log \relax (c) + a^{2} b n\right )} \log \relax (x)}{x} - \frac {6 \, b^{3} n^{3} + 6 \, b^{3} n^{2} \log \relax (c) + 3 \, b^{3} n \log \relax (c)^{2} + b^{3} \log \relax (c)^{3} + 6 \, a b^{2} n^{2} + 6 \, a b^{2} n \log \relax (c) + 3 \, a b^{2} \log \relax (c)^{2} + 3 \, a^{2} b n + 3 \, a^{2} b \log \relax (c) + a^{3}}{x} \]

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

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maple [C]  time = 0.28, size = 2674, normalized size = 38.75 \[ \text {Expression too large to display} \]

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*I*Pi*a*
b^2*csgn(I*c*x^n)^3+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*b*a^2+4*ln(c)^2*b^3-Pi^2*b^3*csgn(I*c*x^n)^4*csgn(I*c)^2-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)+8*a*b^2*n+8*ln(c)*a*b^2+8*n*ln(c)*
b^3-Pi^2*b^3*csgn(I*c*x^n)^6+8*b^3*n^2-4*I*n*Pi*b^3*csgn(I*c*x^n)^3-4*Pi^2*b^3*csgn(I*x^n)*csgn(I*c*x^n)^4*csg
n(I*c)+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*ln(c)*Pi*b^3*cs
gn(I*x^n)*csgn(I*c*x^n)^2+4*I*ln(c)*Pi*b^3*csgn(I*c*x^n)^2*csgn(I*c)+4*I*Pi*a*b^2*csgn(I*x^n)*csgn(I*c*x^n)^2+
4*I*Pi*a*b^2*csgn(I*c*x^n)^2*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)*csgn(I*c*x^n)*csgn(I*c)-4*I*ln(c)*Pi*b^3*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+4*I*n*Pi*b^3*csgn(I*c*x^n)^2*c
sgn(I*c)+4*I*n*Pi*b^3*csgn(I*x^n)*csgn(I*c*x^n)^2)/x*ln(x^n)-1/8*(8*a^3-24*I*Pi*a*b^2*csgn(I*c)*csgn(I*x^n)*cs
gn(I*c*x^n)*ln(c)-6*Pi^2*a*b^2*csgn(I*c*x^n)^6-6*Pi^2*b^3*csgn(I*c*x^n)^6*ln(c)+48*a*b^2*n*ln(c)+8*b^3*ln(c)^3
+24*a*b^2*ln(c)^2+24*a^2*b*ln(c)+24*b^3*n*ln(c)^2+48*b^3*n^2*ln(c)+I*Pi^3*b^3*csgn(I*c*x^n)^9+48*a*b^2*n^2+24*
a^2*b*n-6*Pi^2*b^3*n*csgn(I*c*x^n)^6+48*b^3*n^3-24*I*Pi*a*b^2*n*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-24*I*n*ln(
c)*Pi*b^3*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+24*I*Pi*a*b^2*csgn(I*c)*csgn(I*c*x^n)^2*ln(c)-12*I*Pi*a^2*b*csgn
(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+24*I*ln(c)*Pi*b^3*n*csgn(I*x^n)*csgn(I*c*x^n)^2+24*I*n*ln(c)*Pi*b^3*csgn(I*c*x
^n)^2*csgn(I*c)-24*I*Pi*a*b^2*csgn(I*c*x^n)^3*ln(c)+12*I*Pi*a^2*b*csgn(I*x^n)*csgn(I*c*x^n)^2-6*Pi^2*b^3*n*csg
n(I*x^n)^2*csgn(I*c*x^n)^4+12*Pi^2*b^3*n*csgn(I*c)*csgn(I*c*x^n)^5-6*Pi^2*b^3*n*csgn(I*c)^2*csgn(I*c*x^n)^4+12
*Pi^2*b^3*n*csgn(I*x^n)*csgn(I*c*x^n)^5+9*I*Pi^3*b^3*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)^7-9*I*Pi^3*b^3*csgn(I
*c)^2*csgn(I*x^n)*csgn(I*c*x^n)^6+3*I*Pi^3*b^3*csgn(I*c)^3*csgn(I*x^n)*csgn(I*c*x^n)^5+12*I*Pi*b^3*csgn(I*x^n)
*csgn(I*c*x^n)^2*ln(c)^2+12*I*Pi*b^3*csgn(I*c)*csgn(I*c*x^n)^2*ln(c)^2-6*Pi^2*b^3*csgn(I*c)^2*csgn(I*c*x^n)^4*
ln(c)-6*Pi^2*a*b^2*csgn(I*x^n)^2*csgn(I*c*x^n)^4+12*Pi^2*a*b^2*csgn(I*x^n)*csgn(I*c*x^n)^5-24*I*Pi*b^3*n^2*csg
n(I*c*x^n)^3+12*Pi^2*a*b^2*csgn(I*c)*csgn(I*c*x^n)^5-6*Pi^2*a*b^2*csgn(I*c)^2*csgn(I*c*x^n)^4-6*Pi^2*b^3*csgn(
I*x^n)^2*csgn(I*c*x^n)^4*ln(c)+12*Pi^2*b^3*csgn(I*x^n)*csgn(I*c*x^n)^5*ln(c)+12*Pi^2*b^3*csgn(I*c)*csgn(I*c*x^
n)^5*ln(c)-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*c
sgn(I*c*x^n)^7-3*I*Pi^3*b^3*csgn(I*x^n)*csgn(I*c*x^n)^8-3*I*Pi^3*b^3*csgn(I*c)*csgn(I*c*x^n)^8+3*I*Pi^3*b^3*cs
gn(I*c)^2*csgn(I*c*x^n)^7-I*Pi^3*b^3*csgn(I*c)^3*csgn(I*c*x^n)^6-12*I*Pi*b^3*csgn(I*c*x^n)^3*ln(c)^2-24*I*ln(c
)*Pi*b^3*n*csgn(I*c*x^n)^3-24*I*Pi*a*b^2*n*csgn(I*c*x^n)^3+24*I*Pi*a*b^2*n*csgn(I*x^n)*csgn(I*c*x^n)^2+24*I*Pi
*a*b^2*n*csgn(I*c*x^n)^2*csgn(I*c)-12*I*Pi*b^3*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*ln(c)^2+24*I*Pi*a*b^2*csgn(
I*x^n)*csgn(I*c*x^n)^2*ln(c)+12*Pi^2*a*b^2*csgn(I*c)^2*csgn(I*x^n)*csgn(I*c*x^n)^3+12*Pi^2*b^3*n*csgn(I*c)*csg
n(I*x^n)^2*csgn(I*c*x^n)^3-6*Pi^2*b^3*n*csgn(I*c)^2*csgn(I*x^n)^2*csgn(I*c*x^n)^2-24*Pi^2*b^3*n*csgn(I*c)*csgn
(I*x^n)*csgn(I*c*x^n)^4+12*Pi^2*b^3*n*csgn(I*c)^2*csgn(I*x^n)*csgn(I*c*x^n)^3-6*Pi^2*b^3*csgn(I*c)^2*csgn(I*x^
n)^2*csgn(I*c*x^n)^2*ln(c)-24*Pi^2*b^3*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)^4*ln(c)+12*Pi^2*b^3*csgn(I*c)^2*csg
n(I*x^n)*csgn(I*c*x^n)^3*ln(c)+12*Pi^2*a*b^2*csgn(I*c)*csgn(I*x^n)^2*csgn(I*c*x^n)^3-6*Pi^2*a*b^2*csgn(I*c)^2*
csgn(I*x^n)^2*csgn(I*c*x^n)^2-24*Pi^2*a*b^2*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)^4+12*Pi^2*b^3*csgn(I*c)*csgn(I
*x^n)^2*csgn(I*c*x^n)^3*ln(c)-24*I*Pi*b^3*n^2*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+12*I*Pi*a^2*b*csgn(I*c)*csgn
(I*c*x^n)^2+I*Pi^3*b^3*csgn(I*c)^3*csgn(I*x^n)^3*csgn(I*c*x^n)^3+3*I*Pi^3*b^3*csgn(I*c)*csgn(I*x^n)^3*csgn(I*c
*x^n)^5-3*I*Pi^3*b^3*csgn(I*c)^2*csgn(I*x^n)^3*csgn(I*c*x^n)^4-9*I*Pi^3*b^3*csgn(I*c)*csgn(I*x^n)^2*csgn(I*c*x
^n)^6+9*I*Pi^3*b^3*csgn(I*c)^2*csgn(I*x^n)^2*csgn(I*c*x^n)^5-3*I*Pi^3*b^3*csgn(I*c)^3*csgn(I*x^n)^2*csgn(I*c*x
^n)^4+24*I*Pi*b^3*n^2*csgn(I*x^n)*csgn(I*c*x^n)^2+24*I*Pi*b^3*n^2*csgn(I*c)*csgn(I*c*x^n)^2)/x

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maxima [A]  time = 0.61, size = 133, normalized size = 1.93 \[ -\frac {b^{3} \log \left (c x^{n}\right )^{3}}{x} - 3 \, {\left (2 \, n {\left (\frac {n^{2}}{x} + \frac {n \log \left (c x^{n}\right )}{x}\right )} + \frac {n \log \left (c x^{n}\right )^{2}}{x}\right )} b^{3} - 6 \, a b^{2} {\left (\frac {n^{2}}{x} + \frac {n \log \left (c x^{n}\right )}{x}\right )} - \frac {3 \, a b^{2} \log \left (c x^{n}\right )^{2}}{x} - \frac {3 \, a^{2} b n}{x} - \frac {3 \, a^{2} b \log \left (c x^{n}\right )}{x} - \frac {a^{3}}{x} \]

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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mupad [B]  time = 3.37, size = 104, normalized size = 1.51 \[ -\frac {a^3+3\,a^2\,b\,n+6\,a\,b^2\,n^2+6\,b^3\,n^3}{x}-\frac {\ln \left (c\,x^n\right )\,\left (3\,a^2\,b+6\,a\,b^2\,n+6\,b^3\,n^2\right )}{x}-\frac {b^3\,{\ln \left (c\,x^n\right )}^3}{x}-\frac {3\,b^2\,{\ln \left (c\,x^n\right )}^2\,\left (a+b\,n\right )}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*log(c*x^n))^3/x^2,x)

[Out]

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

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sympy [B]  time = 1.03, size = 272, normalized size = 3.94 \[ - \frac {a^{3}}{x} - \frac {3 a^{2} b n \log {\relax (x )}}{x} - \frac {3 a^{2} b n}{x} - \frac {3 a^{2} b \log {\relax (c )}}{x} - \frac {3 a b^{2} n^{2} \log {\relax (x )}^{2}}{x} - \frac {6 a b^{2} n^{2} \log {\relax (x )}}{x} - \frac {6 a b^{2} n^{2}}{x} - \frac {6 a b^{2} n \log {\relax (c )} \log {\relax (x )}}{x} - \frac {6 a b^{2} n \log {\relax (c )}}{x} - \frac {3 a b^{2} \log {\relax (c )}^{2}}{x} - \frac {b^{3} n^{3} \log {\relax (x )}^{3}}{x} - \frac {3 b^{3} n^{3} \log {\relax (x )}^{2}}{x} - \frac {6 b^{3} n^{3} \log {\relax (x )}}{x} - \frac {6 b^{3} n^{3}}{x} - \frac {3 b^{3} n^{2} \log {\relax (c )} \log {\relax (x )}^{2}}{x} - \frac {6 b^{3} n^{2} \log {\relax (c )} \log {\relax (x )}}{x} - \frac {6 b^{3} n^{2} \log {\relax (c )}}{x} - \frac {3 b^{3} n \log {\relax (c )}^{2} \log {\relax (x )}}{x} - \frac {3 b^{3} n \log {\relax (c )}^{2}}{x} - \frac {b^{3} \log {\relax (c )}^{3}}{x} \]

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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