3.64 \(\int \frac{(a+b \log (c x^n))^3}{x^4} \, dx\)

Optimal. Leaf size=77 \[ -\frac{2 b^2 n^2 \left (a+b \log \left (c x^n\right )\right )}{9 x^3}-\frac{b n \left (a+b \log \left (c x^n\right )\right )^2}{3 x^3}-\frac{\left (a+b \log \left (c x^n\right )\right )^3}{3 x^3}-\frac{2 b^3 n^3}{27 x^3} \]

[Out]

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

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Rubi [A]  time = 0.0606733, antiderivative size = 77, 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} \[ -\frac{2 b^2 n^2 \left (a+b \log \left (c x^n\right )\right )}{9 x^3}-\frac{b n \left (a+b \log \left (c x^n\right )\right )^2}{3 x^3}-\frac{\left (a+b \log \left (c x^n\right )\right )^3}{3 x^3}-\frac{2 b^3 n^3}{27 x^3} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Log[c*x^n])^3/x^4,x]

[Out]

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

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^4} \, dx &=-\frac{\left (a+b \log \left (c x^n\right )\right )^3}{3 x^3}+(b n) \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{x^4} \, dx\\ &=-\frac{b n \left (a+b \log \left (c x^n\right )\right )^2}{3 x^3}-\frac{\left (a+b \log \left (c x^n\right )\right )^3}{3 x^3}+\frac{1}{3} \left (2 b^2 n^2\right ) \int \frac{a+b \log \left (c x^n\right )}{x^4} \, dx\\ &=-\frac{2 b^3 n^3}{27 x^3}-\frac{2 b^2 n^2 \left (a+b \log \left (c x^n\right )\right )}{9 x^3}-\frac{b n \left (a+b \log \left (c x^n\right )\right )^2}{3 x^3}-\frac{\left (a+b \log \left (c x^n\right )\right )^3}{3 x^3}\\ \end{align*}

Mathematica [A]  time = 0.0242901, size = 60, normalized size = 0.78 \[ -\frac{9 \left (a+b \log \left (c x^n\right )\right )^3+b n \left (9 \left (a+b \log \left (c x^n\right )\right )^2+2 b n \left (3 a+3 b \log \left (c x^n\right )+b n\right )\right )}{27 x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Log[c*x^n])^3/x^4,x]

[Out]

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

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Maple [C]  time = 0.249, size = 2674, normalized size = 34.7 \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^4,x)

[Out]

-1/3*b^3/x^3*ln(x^n)^3-1/6*(3*I*Pi*b^3*csgn(I*x^n)*csgn(I*c*x^n)^2-3*I*Pi*b^3*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I
*c)-3*I*Pi*b^3*csgn(I*c*x^n)^3+3*I*Pi*b^3*csgn(I*c*x^n)^2*csgn(I*c)+6*ln(c)*b^3+2*b^3*n+6*a*b^2)/x^3*ln(x^n)^2
-1/36*(36*a^2*b+8*b^3*n^2-36*I*ln(c)*Pi*b^3*csgn(I*c*x^n)^3-36*I*Pi*a*b^2*csgn(I*c*x^n)^3-12*I*n*Pi*b^3*csgn(I
*c*x^n)^3+72*ln(c)*a*b^2+24*n*ln(c)*b^3+18*Pi^2*b^3*csgn(I*x^n)*csgn(I*c*x^n)^3*csgn(I*c)^2+24*a*b^2*n+18*Pi^2
*b^3*csgn(I*x^n)^2*csgn(I*c*x^n)^3*csgn(I*c)-9*Pi^2*b^3*csgn(I*x^n)^2*csgn(I*c*x^n)^2*csgn(I*c)^2-36*Pi^2*b^3*
csgn(I*x^n)*csgn(I*c*x^n)^4*csgn(I*c)+36*ln(c)^2*b^3-9*Pi^2*b^3*csgn(I*c*x^n)^6+36*I*ln(c)*Pi*b^3*csgn(I*x^n)*
csgn(I*c*x^n)^2+36*I*ln(c)*Pi*b^3*csgn(I*c*x^n)^2*csgn(I*c)+36*I*Pi*a*b^2*csgn(I*x^n)*csgn(I*c*x^n)^2-9*Pi^2*b
^3*csgn(I*x^n)^2*csgn(I*c*x^n)^4+18*Pi^2*b^3*csgn(I*x^n)*csgn(I*c*x^n)^5+18*Pi^2*b^3*csgn(I*c*x^n)^5*csgn(I*c)
-9*Pi^2*b^3*csgn(I*c*x^n)^4*csgn(I*c)^2-36*I*Pi*a*b^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-12*I*n*Pi*b^3*csgn(I
*x^n)*csgn(I*c*x^n)*csgn(I*c)-36*I*ln(c)*Pi*b^3*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+36*I*Pi*a*b^2*csgn(I*c*x^n
)^2*csgn(I*c)+12*I*n*Pi*b^3*csgn(I*x^n)*csgn(I*c*x^n)^2+12*I*n*Pi*b^3*csgn(I*c*x^n)^2*csgn(I*c))/x^3*ln(x^n)-1
/216*(72*a^3+48*a*b^2*n^2+72*a^2*b*n-54*Pi^2*a*b^2*csgn(I*x^n)^2*csgn(I*c*x^n)^4+72*ln(c)^3*b^3-216*Pi^2*a*b^2
*csgn(I*x^n)*csgn(I*c*x^n)^4*csgn(I*c)+36*Pi^2*b^3*n*csgn(I*x^n)^2*csgn(I*c*x^n)^3*csgn(I*c)-18*Pi^2*b^3*n*csg
n(I*x^n)^2*csgn(I*c*x^n)^2*csgn(I*c)^2-72*Pi^2*b^3*n*csgn(I*x^n)*csgn(I*c*x^n)^4*csgn(I*c)-54*ln(c)*Pi^2*b^3*c
sgn(I*x^n)^2*csgn(I*c*x^n)^4+108*ln(c)*Pi^2*b^3*csgn(I*x^n)*csgn(I*c*x^n)^5+108*ln(c)*Pi^2*b^3*csgn(I*c*x^n)^5
*csgn(I*c)+36*Pi^2*b^3*n*csgn(I*c*x^n)^5*csgn(I*c)-18*Pi^2*b^3*n*csgn(I*c*x^n)^4*csgn(I*c)^2-216*I*ln(c)*Pi*a*
b^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+16*b^3*n^3-72*I*ln(c)*Pi*b^3*n*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+9*I
*Pi^3*b^3*csgn(I*c*x^n)^9+108*Pi^2*a*b^2*csgn(I*x^n)*csgn(I*c*x^n)^5-108*I*ln(c)^2*Pi*b^3*csgn(I*c*x^n)^3-108*
I*Pi*a^2*b*csgn(I*c*x^n)^3-24*I*Pi*b^3*n^2*csgn(I*c*x^n)^3-9*I*Pi^3*b^3*csgn(I*x^n)^3*csgn(I*c*x^n)^6-27*I*Pi^
3*b^3*csgn(I*x^n)*csgn(I*c*x^n)^8-27*I*Pi^3*b^3*csgn(I*c*x^n)^8*csgn(I*c)+27*I*Pi^3*b^3*csgn(I*c*x^n)^7*csgn(I
*c)^2-9*I*Pi^3*b^3*csgn(I*c*x^n)^6*csgn(I*c)^3+72*ln(c)^2*b^3*n+48*ln(c)*b^3*n^2+216*ln(c)*a^2*b+216*ln(c)^2*a
*b^2+108*Pi^2*a*b^2*csgn(I*c*x^n)^5*csgn(I*c)-54*Pi^2*a*b^2*csgn(I*c*x^n)^4*csgn(I*c)^2-54*ln(c)*Pi^2*b^3*csgn
(I*c*x^n)^4*csgn(I*c)^2-18*Pi^2*b^3*n*csgn(I*x^n)^2*csgn(I*c*x^n)^4+36*Pi^2*b^3*n*csgn(I*x^n)*csgn(I*c*x^n)^5-
18*Pi^2*b^3*n*csgn(I*c*x^n)^6-54*ln(c)*Pi^2*b^3*csgn(I*c*x^n)^6-54*Pi^2*a*b^2*csgn(I*c*x^n)^6+27*I*Pi^3*b^3*cs
gn(I*x^n)^2*csgn(I*c*x^n)^7-72*I*Pi*a*b^2*n*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+27*I*Pi^3*b^3*csgn(I*x^n)^3*cs
gn(I*c*x^n)^5*csgn(I*c)-27*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)^3*csg
n(I*c*x^n)^3*csgn(I*c)^3-81*I*Pi^3*b^3*csgn(I*x^n)^2*csgn(I*c*x^n)^6*csgn(I*c)+108*ln(c)*Pi^2*b^3*csgn(I*x^n)*
csgn(I*c*x^n)^3*csgn(I*c)^2+108*Pi^2*a*b^2*csgn(I*x^n)*csgn(I*c*x^n)^3*csgn(I*c)^2-54*ln(c)*Pi^2*b^3*csgn(I*x^
n)^2*csgn(I*c*x^n)^2*csgn(I*c)^2-216*ln(c)*Pi^2*b^3*csgn(I*x^n)*csgn(I*c*x^n)^4*csgn(I*c)+36*Pi^2*b^3*n*csgn(I
*x^n)*csgn(I*c*x^n)^3*csgn(I*c)^2+108*ln(c)*Pi^2*b^3*csgn(I*x^n)^2*csgn(I*c*x^n)^3*csgn(I*c)+108*Pi^2*a*b^2*cs
gn(I*x^n)^2*csgn(I*c*x^n)^3*csgn(I*c)-54*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)+72*I*n*Pi*a*b^2*csgn(I*x^n)*csgn(I*c*x^n)^2+72*I*Pi*a*b^2*n*csgn(I*c*x^n)^
2*csgn(I*c)+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*x^n)^2*csgn(I*c)-216*I*ln(c)*
Pi*a*b^2*csgn(I*c*x^n)^3+108*I*Pi*a^2*b*csgn(I*x^n)*csgn(I*c*x^n)^2+108*I*Pi*a^2*b*csgn(I*c*x^n)^2*csgn(I*c)-8
1*I*Pi^3*b^3*csgn(I*x^n)*csgn(I*c*x^n)^6*csgn(I*c)^2+27*I*Pi^3*b^3*csgn(I*x^n)*csgn(I*c*x^n)^5*csgn(I*c)^3+108
*I*ln(c)^2*Pi*b^3*csgn(I*x^n)*csgn(I*c*x^n)^2+108*I*ln(c)^2*Pi*b^3*csgn(I*c*x^n)^2*csgn(I*c)+144*ln(c)*a*b^2*n
+81*I*Pi^3*b^3*csgn(I*x^n)^2*csgn(I*c*x^n)^5*csgn(I*c)^2-27*I*Pi^3*b^3*csgn(I*x^n)^2*csgn(I*c*x^n)^4*csgn(I*c)
^3+81*I*Pi^3*b^3*csgn(I*x^n)*csgn(I*c*x^n)^7*csgn(I*c)-108*I*ln(c)^2*Pi*b^3*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c
)-72*I*ln(c)*Pi*b^3*n*csgn(I*c*x^n)^3-72*I*Pi*a*b^2*n*csgn(I*c*x^n)^3-108*I*Pi*a^2*b*csgn(I*x^n)*csgn(I*c*x^n)
*csgn(I*c)+72*I*ln(c)*Pi*b^3*n*csgn(I*c*x^n)^2*csgn(I*c)+72*I*ln(c)*Pi*b^3*n*csgn(I*x^n)*csgn(I*c*x^n)^2+216*I
*ln(c)*Pi*a*b^2*csgn(I*x^n)*csgn(I*c*x^n)^2+216*I*ln(c)*Pi*a*b^2*csgn(I*c*x^n)^2*csgn(I*c))/x^3

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Maxima [A]  time = 1.10039, size = 184, normalized size = 2.39 \begin{align*} -\frac{1}{27} \,{\left (2 \, n{\left (\frac{n^{2}}{x^{3}} + \frac{3 \, n \log \left (c x^{n}\right )}{x^{3}}\right )} + \frac{9 \, n \log \left (c x^{n}\right )^{2}}{x^{3}}\right )} b^{3} - \frac{2}{9} \, a b^{2}{\left (\frac{n^{2}}{x^{3}} + \frac{3 \, n \log \left (c x^{n}\right )}{x^{3}}\right )} - \frac{b^{3} \log \left (c x^{n}\right )^{3}}{3 \, x^{3}} - \frac{a b^{2} \log \left (c x^{n}\right )^{2}}{x^{3}} - \frac{a^{2} b n}{3 \, x^{3}} - \frac{a^{2} b \log \left (c x^{n}\right )}{x^{3}} - \frac{a^{3}}{3 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))^3/x^4,x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 0.788247, size = 443, normalized size = 5.75 \begin{align*} -\frac{9 \, b^{3} n^{3} \log \left (x\right )^{3} + 2 \, b^{3} n^{3} + 9 \, b^{3} \log \left (c\right )^{3} + 6 \, a b^{2} n^{2} + 9 \, a^{2} b n + 9 \, a^{3} + 9 \,{\left (b^{3} n + 3 \, a b^{2}\right )} \log \left (c\right )^{2} + 9 \,{\left (b^{3} n^{3} + 3 \, b^{3} n^{2} \log \left (c\right ) + 3 \, a b^{2} n^{2}\right )} \log \left (x\right )^{2} + 3 \,{\left (2 \, b^{3} n^{2} + 6 \, a b^{2} n + 9 \, a^{2} b\right )} \log \left (c\right ) + 3 \,{\left (2 \, b^{3} n^{3} + 9 \, b^{3} n \log \left (c\right )^{2} + 6 \, a b^{2} n^{2} + 9 \, a^{2} b n + 6 \,{\left (b^{3} n^{2} + 3 \, a b^{2} n\right )} \log \left (c\right )\right )} \log \left (x\right )}{27 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))^3/x^4,x, algorithm="fricas")

[Out]

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

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Sympy [B]  time = 3.78093, size = 313, normalized size = 4.06 \begin{align*} - \frac{a^{3}}{3 x^{3}} - \frac{a^{2} b n \log{\left (x \right )}}{x^{3}} - \frac{a^{2} b n}{3 x^{3}} - \frac{a^{2} b \log{\left (c \right )}}{x^{3}} - \frac{a b^{2} n^{2} \log{\left (x \right )}^{2}}{x^{3}} - \frac{2 a b^{2} n^{2} \log{\left (x \right )}}{3 x^{3}} - \frac{2 a b^{2} n^{2}}{9 x^{3}} - \frac{2 a b^{2} n \log{\left (c \right )} \log{\left (x \right )}}{x^{3}} - \frac{2 a b^{2} n \log{\left (c \right )}}{3 x^{3}} - \frac{a b^{2} \log{\left (c \right )}^{2}}{x^{3}} - \frac{b^{3} n^{3} \log{\left (x \right )}^{3}}{3 x^{3}} - \frac{b^{3} n^{3} \log{\left (x \right )}^{2}}{3 x^{3}} - \frac{2 b^{3} n^{3} \log{\left (x \right )}}{9 x^{3}} - \frac{2 b^{3} n^{3}}{27 x^{3}} - \frac{b^{3} n^{2} \log{\left (c \right )} \log{\left (x \right )}^{2}}{x^{3}} - \frac{2 b^{3} n^{2} \log{\left (c \right )} \log{\left (x \right )}}{3 x^{3}} - \frac{2 b^{3} n^{2} \log{\left (c \right )}}{9 x^{3}} - \frac{b^{3} n \log{\left (c \right )}^{2} \log{\left (x \right )}}{x^{3}} - \frac{b^{3} n \log{\left (c \right )}^{2}}{3 x^{3}} - \frac{b^{3} \log{\left (c \right )}^{3}}{3 x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*ln(c*x**n))**3/x**4,x)

[Out]

-a**3/(3*x**3) - a**2*b*n*log(x)/x**3 - a**2*b*n/(3*x**3) - a**2*b*log(c)/x**3 - a*b**2*n**2*log(x)**2/x**3 -
2*a*b**2*n**2*log(x)/(3*x**3) - 2*a*b**2*n**2/(9*x**3) - 2*a*b**2*n*log(c)*log(x)/x**3 - 2*a*b**2*n*log(c)/(3*
x**3) - a*b**2*log(c)**2/x**3 - b**3*n**3*log(x)**3/(3*x**3) - b**3*n**3*log(x)**2/(3*x**3) - 2*b**3*n**3*log(
x)/(9*x**3) - 2*b**3*n**3/(27*x**3) - b**3*n**2*log(c)*log(x)**2/x**3 - 2*b**3*n**2*log(c)*log(x)/(3*x**3) - 2
*b**3*n**2*log(c)/(9*x**3) - b**3*n*log(c)**2*log(x)/x**3 - b**3*n*log(c)**2/(3*x**3) - b**3*log(c)**3/(3*x**3
)

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Giac [B]  time = 1.17249, size = 275, normalized size = 3.57 \begin{align*} -\frac{b^{3} n^{3} \log \left (x\right )^{3}}{3 \, x^{3}} - \frac{{\left (b^{3} n^{3} + 3 \, b^{3} n^{2} \log \left (c\right ) + 3 \, a b^{2} n^{2}\right )} \log \left (x\right )^{2}}{3 \, x^{3}} - \frac{{\left (2 \, b^{3} n^{3} + 6 \, b^{3} n^{2} \log \left (c\right ) + 9 \, b^{3} n \log \left (c\right )^{2} + 6 \, a b^{2} n^{2} + 18 \, a b^{2} n \log \left (c\right ) + 9 \, a^{2} b n\right )} \log \left (x\right )}{9 \, x^{3}} - \frac{2 \, b^{3} n^{3} + 6 \, b^{3} n^{2} \log \left (c\right ) + 9 \, b^{3} n \log \left (c\right )^{2} + 9 \, b^{3} \log \left (c\right )^{3} + 6 \, a b^{2} n^{2} + 18 \, a b^{2} n \log \left (c\right ) + 27 \, a b^{2} \log \left (c\right )^{2} + 9 \, a^{2} b n + 27 \, a^{2} b \log \left (c\right ) + 9 \, a^{3}}{27 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))^3/x^4,x, algorithm="giac")

[Out]

-1/3*b^3*n^3*log(x)^3/x^3 - 1/3*(b^3*n^3 + 3*b^3*n^2*log(c) + 3*a*b^2*n^2)*log(x)^2/x^3 - 1/9*(2*b^3*n^3 + 6*b
^3*n^2*log(c) + 9*b^3*n*log(c)^2 + 6*a*b^2*n^2 + 18*a*b^2*n*log(c) + 9*a^2*b*n)*log(x)/x^3 - 1/27*(2*b^3*n^3 +
6*b^3*n^2*log(c) + 9*b^3*n*log(c)^2 + 9*b^3*log(c)^3 + 6*a*b^2*n^2 + 18*a*b^2*n*log(c) + 27*a*b^2*log(c)^2 +
9*a^2*b*n + 27*a^2*b*log(c) + 9*a^3)/x^3