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

Optimal. Leaf size=66 \[ 6 a b^2 n^2 x-3 b n x \left (a+b \log \left (c x^n\right )\right )^2+x \left (a+b \log \left (c x^n\right )\right )^3+6 b^3 n^2 x \log \left (c x^n\right )-6 b^3 n^3 x \]

[Out]

6*a*b^2*n^2*x - 6*b^3*n^3*x + 6*b^3*n^2*x*Log[c*x^n] - 3*b*n*x*(a + b*Log[c*x^n])^2 + x*(a + b*Log[c*x^n])^3

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Rubi [A]  time = 0.0238015, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {2296, 2295} \[ 6 a b^2 n^2 x-3 b n x \left (a+b \log \left (c x^n\right )\right )^2+x \left (a+b \log \left (c x^n\right )\right )^3+6 b^3 n^2 x \log \left (c x^n\right )-6 b^3 n^3 x \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Log[c*x^n])^3,x]

[Out]

6*a*b^2*n^2*x - 6*b^3*n^3*x + 6*b^3*n^2*x*Log[c*x^n] - 3*b*n*x*(a + b*Log[c*x^n])^2 + x*(a + b*Log[c*x^n])^3

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]

Rubi steps

\begin{align*} \int \left (a+b \log \left (c x^n\right )\right )^3 \, dx &=x \left (a+b \log \left (c x^n\right )\right )^3-(3 b n) \int \left (a+b \log \left (c x^n\right )\right )^2 \, dx\\ &=-3 b n x \left (a+b \log \left (c x^n\right )\right )^2+x \left (a+b \log \left (c x^n\right )\right )^3+\left (6 b^2 n^2\right ) \int \left (a+b \log \left (c x^n\right )\right ) \, dx\\ &=6 a b^2 n^2 x-3 b n x \left (a+b \log \left (c x^n\right )\right )^2+x \left (a+b \log \left (c x^n\right )\right )^3+\left (6 b^3 n^2\right ) \int \log \left (c x^n\right ) \, dx\\ &=6 a b^2 n^2 x-6 b^3 n^3 x+6 b^3 n^2 x \log \left (c x^n\right )-3 b n x \left (a+b \log \left (c x^n\right )\right )^2+x \left (a+b \log \left (c x^n\right )\right )^3\\ \end{align*}

Mathematica [A]  time = 0.0087649, size = 50, normalized size = 0.76 \[ x \left (\left (a+b \log \left (c x^n\right )\right )^3-3 b n \left (\left (a+b \log \left (c x^n\right )\right )^2-2 b n \left (a+b \log \left (c x^n\right )-b n\right )\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Log[c*x^n])^3,x]

[Out]

x*((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])))

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Maple [C]  time = 0.299, size = 2641, normalized size = 40. \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)

[Out]

x*b^3*ln(x^n)^3+3/2*b^2*(I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2-I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-I*b*Pi*
csgn(I*c*x^n)^3+I*b*Pi*csgn(I*c*x^n)^2*csgn(I*c)+2*b*ln(c)-2*b*n+2*a)*x*ln(x^n)^2+3/4*b*(4*I*ln(c)*Pi*b^2*csgn
(I*x^n)*csgn(I*c*x^n)^2+4*ln(c)^2*b^2-Pi^2*b^2*csgn(I*c*x^n)^4*csgn(I*c)^2-8*a*b*n+4*I*Pi*b^2*n*csgn(I*c*x^n)^
3+8*b^2*n^2+4*a^2-4*I*Pi*b^2*n*csgn(I*x^n)*csgn(I*c*x^n)^2-4*I*Pi*b^2*n*csgn(I*c*x^n)^2*csgn(I*c)+2*Pi^2*b^2*c
sgn(I*x^n)*csgn(I*c*x^n)^3*csgn(I*c)^2+2*Pi^2*b^2*csgn(I*x^n)^2*csgn(I*c*x^n)^3*csgn(I*c)-Pi^2*b^2*csgn(I*x^n)
^2*csgn(I*c*x^n)^2*csgn(I*c)^2-4*Pi^2*b^2*csgn(I*x^n)*csgn(I*c*x^n)^4*csgn(I*c)+4*I*ln(c)*Pi*b^2*csgn(I*c*x^n)
^2*csgn(I*c)+4*I*Pi*a*b*csgn(I*x^n)*csgn(I*c*x^n)^2+4*I*Pi*a*b*csgn(I*c*x^n)^2*csgn(I*c)-4*I*Pi*a*b*csgn(I*c*x
^n)^3-4*I*ln(c)*Pi*b^2*csgn(I*c*x^n)^3-4*I*ln(c)*Pi*b^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-4*I*Pi*a*b*csgn(I*
x^n)*csgn(I*c*x^n)*csgn(I*c)-Pi^2*b^2*csgn(I*c*x^n)^6+8*ln(c)*a*b-8*ln(c)*b^2*n+2*Pi^2*b^2*csgn(I*c*x^n)^5*csg
n(I*c)+2*Pi^2*b^2*csgn(I*x^n)*csgn(I*c*x^n)^5-Pi^2*b^2*csgn(I*x^n)^2*csgn(I*c*x^n)^4+4*I*Pi*b^2*n*csgn(I*x^n)*
csgn(I*c*x^n)*csgn(I*c))*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*csgn(I*x^n)^2*csgn(I*c*x^n)
^4+24*I*ln(c)*Pi*b^3*n*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+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*csgn(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*cs
gn(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+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)^3-12*Pi^2*b^3*n*csgn(I*x^n)^2*csgn(I*c*x^n)^3*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*csgn(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)-48*b^3*n^3+24*I*Pi*a*b^2*n*csgn(I*x^n)*csgn(I*c*x^n)*csg
n(I*c)+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*P
i*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*csgn(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*csgn(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)^2*csgn(I*c)-24*I*Pi*
a*b^2*n*csgn(I*c*x^n)^2*csgn(I*c)-24*I*Pi*a*b^2*n*csgn(I*x^n)*csgn(I*c*x^n)^2-24*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 = 1.21959, size = 153, normalized size = 2.32 \begin{align*} b^{3} x \log \left (c x^{n}\right )^{3} + 3 \, a b^{2} x \log \left (c x^{n}\right )^{2} - 3 \, a^{2} b n x + 3 \, a^{2} b x \log \left (c x^{n}\right ) + 6 \,{\left (n^{2} x - n x \log \left (c x^{n}\right )\right )} a b^{2} - 3 \,{\left (n x \log \left (c x^{n}\right )^{2} + 2 \,{\left (n^{2} x - n x \log \left (c x^{n}\right )\right )} n\right )} b^{3} + 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, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 0.899181, size = 435, normalized size = 6.59 \begin{align*} b^{3} n^{3} x \log \left (x\right )^{3} + b^{3} x \log \left (c\right )^{3} - 3 \,{\left (b^{3} n - a b^{2}\right )} x \log \left (c\right )^{2} + 3 \,{\left (2 \, b^{3} n^{2} - 2 \, a b^{2} n + a^{2} b\right )} x \log \left (c\right ) + 3 \,{\left (b^{3} n^{2} x \log \left (c\right ) -{\left (b^{3} n^{3} - a b^{2} n^{2}\right )} x\right )} \log \left (x\right )^{2} -{\left (6 \, b^{3} n^{3} - 6 \, a b^{2} n^{2} + 3 \, a^{2} b n - a^{3}\right )} x + 3 \,{\left (b^{3} n x \log \left (c\right )^{2} - 2 \,{\left (b^{3} n^{2} - a b^{2} n\right )} x \log \left (c\right ) +{\left (2 \, b^{3} n^{3} - 2 \, a b^{2} n^{2} + a^{2} b n\right )} x\right )} \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 1.39415, size = 270, normalized size = 4.09 \begin{align*} a^{3} x + 3 a^{2} b n x \log{\left (x \right )} - 3 a^{2} b n x + 3 a^{2} b x \log{\left (c \right )} + 3 a b^{2} n^{2} x \log{\left (x \right )}^{2} - 6 a b^{2} n^{2} x \log{\left (x \right )} + 6 a b^{2} n^{2} x + 6 a b^{2} n x \log{\left (c \right )} \log{\left (x \right )} - 6 a b^{2} n x \log{\left (c \right )} + 3 a b^{2} x \log{\left (c \right )}^{2} + b^{3} n^{3} x \log{\left (x \right )}^{3} - 3 b^{3} n^{3} x \log{\left (x \right )}^{2} + 6 b^{3} n^{3} x \log{\left (x \right )} - 6 b^{3} n^{3} x + 3 b^{3} n^{2} x \log{\left (c \right )} \log{\left (x \right )}^{2} - 6 b^{3} n^{2} x \log{\left (c \right )} \log{\left (x \right )} + 6 b^{3} n^{2} x \log{\left (c \right )} + 3 b^{3} n x \log{\left (c \right )}^{2} \log{\left (x \right )} - 3 b^{3} n x \log{\left (c \right )}^{2} + b^{3} x \log{\left (c \right )}^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.22994, size = 296, normalized size = 4.48 \begin{align*} b^{3} n^{3} x \log \left (x\right )^{3} - 3 \, b^{3} n^{3} x \log \left (x\right )^{2} + 3 \, b^{3} n^{2} x \log \left (c\right ) \log \left (x\right )^{2} + 6 \, b^{3} n^{3} x \log \left (x\right ) - 6 \, b^{3} n^{2} x \log \left (c\right ) \log \left (x\right ) + 3 \, b^{3} n x \log \left (c\right )^{2} \log \left (x\right ) + 3 \, a b^{2} n^{2} x \log \left (x\right )^{2} - 6 \, b^{3} n^{3} x + 6 \, b^{3} n^{2} x \log \left (c\right ) - 3 \, b^{3} n x \log \left (c\right )^{2} + b^{3} x \log \left (c\right )^{3} - 6 \, a b^{2} n^{2} x \log \left (x\right ) + 6 \, a b^{2} n x \log \left (c\right ) \log \left (x\right ) + 6 \, a b^{2} n^{2} x - 6 \, a b^{2} n x \log \left (c\right ) + 3 \, a b^{2} x \log \left (c\right )^{2} + 3 \, a^{2} b n x \log \left (x\right ) - 3 \, a^{2} b n x + 3 \, a^{2} b x \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, algorithm="giac")

[Out]

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