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*ln(c*x^n)-3*b*n*x*(a+b*ln(c*x^n))^2+x*(a+b*ln(c*x^n))^3
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Rubi [A] time = 0.02, antiderivative size = 66, normalized size of antiderivative = 1.00,
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 2295
Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]
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]
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*}
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Mathematica [A] time = 0.01, 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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fricas [B] time = 0.46, size = 198, normalized size = 3.00 \[ b^{3} n^{3} x \log \relax (x)^{3} + b^{3} x \log \relax (c)^{3} - 3 \, {\left (b^{3} n - a b^{2}\right )} x \log \relax (c)^{2} + 3 \, {\left (2 \, b^{3} n^{2} - 2 \, a b^{2} n + a^{2} b\right )} x \log \relax (c) + 3 \, {\left (b^{3} n^{2} x \log \relax (c) - {\left (b^{3} n^{3} - a b^{2} n^{2}\right )} x\right )} \log \relax (x)^{2} - {\left (6 \, b^{3} n^{3} - 6 \, a b^{2} n^{2} + 3 \, a^{2} b n - a^{3}\right )} x + 3 \, {\left (b^{3} n x \log \relax (c)^{2} - 2 \, {\left (b^{3} n^{2} - a b^{2} n\right )} x \log \relax (c) + {\left (2 \, b^{3} n^{3} - 2 \, a b^{2} n^{2} + a^{2} b n\right )} x\right )} \log \relax (x) \]
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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giac [B] time = 0.25, size = 219, normalized size = 3.32 \[ b^{3} n^{3} x \log \relax (x)^{3} - 3 \, b^{3} n^{3} x \log \relax (x)^{2} + 3 \, b^{3} n^{2} x \log \relax (c) \log \relax (x)^{2} + 6 \, b^{3} n^{3} x \log \relax (x) - 6 \, b^{3} n^{2} x \log \relax (c) \log \relax (x) + 3 \, b^{3} n x \log \relax (c)^{2} \log \relax (x) + 3 \, a b^{2} n^{2} x \log \relax (x)^{2} - 6 \, b^{3} n^{3} x + 6 \, b^{3} n^{2} x \log \relax (c) - 3 \, b^{3} n x \log \relax (c)^{2} + b^{3} x \log \relax (c)^{3} - 6 \, a b^{2} n^{2} x \log \relax (x) + 6 \, a b^{2} n x \log \relax (c) \log \relax (x) + 6 \, a b^{2} n^{2} x - 6 \, a b^{2} n x \log \relax (c) + 3 \, a b^{2} x \log \relax (c)^{2} + 3 \, a^{2} b n x \log \relax (x) - 3 \, a^{2} b n x + 3 \, a^{2} b x \log \relax (c) + a^{3} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*log(c*x^n))^3,x, 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
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maple [C] time = 0.29, size = 2641, normalized size = 40.02 \[ \text {Expression too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a+b*ln(c*x^n))^3,x)
[Out]
x*b^3*ln(x^n)^3+3/2*b^2*(I*Pi*b*csgn(I*x^n)*csgn(I*c*x^n)^2-I*Pi*b*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)-I*Pi*b*
csgn(I*c*x^n)^3+I*Pi*b*csgn(I*c)*csgn(I*c*x^n)^2+2*b*ln(c)-2*b*n+2*a)*x*ln(x^n)^2+3/4*b*(-Pi^2*b^2*csgn(I*c)^2
*csgn(I*x^n)^2*csgn(I*c*x^n)^2-4*Pi^2*b^2*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)^4+2*Pi^2*b^2*csgn(I*c)^2*csgn(I*
x^n)*csgn(I*c*x^n)^3+2*Pi^2*b^2*csgn(I*c)*csgn(I*x^n)^2*csgn(I*c*x^n)^3-4*I*Pi*a*b*csgn(I*c*x^n)^3-4*I*Pi*b^2*
csgn(I*c*x^n)^3*ln(c)+4*a^2+8*b^2*n^2+8*a*b*ln(c)-8*b^2*n*ln(c)+4*b^2*ln(c)^2-Pi^2*b^2*csgn(I*x^n)^2*csgn(I*c*
x^n)^4+2*Pi^2*b^2*csgn(I*x^n)*csgn(I*c*x^n)^5-8*a*b*n-Pi^2*b^2*csgn(I*c*x^n)^6+2*Pi^2*b^2*csgn(I*c)*csgn(I*c*x
^n)^5-Pi^2*b^2*csgn(I*c)^2*csgn(I*c*x^n)^4+4*I*Pi*a*b*csgn(I*c)*csgn(I*c*x^n)^2+4*I*Pi*b^2*n*csgn(I*x^n)*csgn(
I*c*x^n)*csgn(I*c)+4*I*Pi*b^2*n*csgn(I*c*x^n)^3+4*I*Pi*b^2*csgn(I*x^n)*csgn(I*c*x^n)^2*ln(c)+4*I*Pi*b^2*csgn(I
*c)*csgn(I*c*x^n)^2*ln(c)+4*I*Pi*a*b*csgn(I*x^n)*csgn(I*c*x^n)^2-4*I*Pi*b^2*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n
)*ln(c)-4*I*Pi*a*b*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)-4*I*Pi*b^2*n*csgn(I*x^n)*csgn(I*c*x^n)^2-4*I*Pi*b^2*n*c
sgn(I*c*x^n)^2*csgn(I*c))*x*ln(x^n)+1/8*(8*a^3-24*I*Pi*a*b^2*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*ln(c)-6*Pi^2*
a*b^2*csgn(I*c*x^n)^6+24*I*Pi*b^3*n*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*ln(c)+24*I*Pi*a*b^2*n*csgn(I*c)*csgn(I
*x^n)*csgn(I*c*x^n)-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*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*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*csgn(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*csg
n(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*P
i^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*csgn(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*csgn(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*csgn(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)^2-12*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)*csgn(I*x^n)^2*csgn(I*c*x^n)^3+6*Pi^2*b^3*n*csgn(I*c)^2*c
sgn(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*csg
n(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*csgn(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)-24*I*Pi*b^3*n*csgn(I*c)*csgn(I*c*x^n)^2*ln(c)-24*I*Pi*b^3*n*csgn(I*x^n)*csgn(I*c*
x^n)^2*ln(c)+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*csg
n(I*c)*csgn(I*c*x^n)^2+24*I*Pi*b^3*n*csgn(I*c*x^n)^3*ln(c)+24*I*Pi*a*b^2*n*csgn(I*c*x^n)^3-24*I*Pi*a*b^2*n*csg
n(I*x^n)*csgn(I*c*x^n)^2-24*I*Pi*a*b^2*n*csgn(I*c)*csgn(I*c*x^n)^2)*x
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maxima [A] time = 0.59, size = 113, normalized size = 1.71 \[ b^{3} x \log \left (c x^{n}\right )^{3} + 3 \, a b^{2} x \log \left (c x^{n}\right )^{2} - 3 \, a^{2} b n x + 3 \, a^{2} b x \log \left (c x^{n}\right ) + 6 \, {\left (n^{2} x - n x \log \left (c x^{n}\right )\right )} a b^{2} - 3 \, {\left (n x \log \left (c x^{n}\right )^{2} + 2 \, {\left (n^{2} x - n x \log \left (c x^{n}\right )\right )} n\right )} b^{3} + a^{3} x \]
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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mupad [B] time = 3.66, size = 94, normalized size = 1.42 \[ x\,\left (a^3-3\,a^2\,b\,n+6\,a\,b^2\,n^2-6\,b^3\,n^3\right )+x\,\ln \left (c\,x^n\right )\,\left (3\,a^2\,b-6\,a\,b^2\,n+6\,b^3\,n^2\right )+b^3\,x\,{\ln \left (c\,x^n\right )}^3+3\,b^2\,x\,{\ln \left (c\,x^n\right )}^2\,\left (a-b\,n\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a + b*log(c*x^n))^3,x)
[Out]
x*(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) + b^3*x*log(c*x
^n)^3 + 3*b^2*x*log(c*x^n)^2*(a - b*n)
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sympy [B] time = 1.04, size = 270, normalized size = 4.09 \[ a^{3} x + 3 a^{2} b n x \log {\relax (x )} - 3 a^{2} b n x + 3 a^{2} b x \log {\relax (c )} + 3 a b^{2} n^{2} x \log {\relax (x )}^{2} - 6 a b^{2} n^{2} x \log {\relax (x )} + 6 a b^{2} n^{2} x + 6 a b^{2} n x \log {\relax (c )} \log {\relax (x )} - 6 a b^{2} n x \log {\relax (c )} + 3 a b^{2} x \log {\relax (c )}^{2} + b^{3} n^{3} x \log {\relax (x )}^{3} - 3 b^{3} n^{3} x \log {\relax (x )}^{2} + 6 b^{3} n^{3} x \log {\relax (x )} - 6 b^{3} n^{3} x + 3 b^{3} n^{2} x \log {\relax (c )} \log {\relax (x )}^{2} - 6 b^{3} n^{2} x \log {\relax (c )} \log {\relax (x )} + 6 b^{3} n^{2} x \log {\relax (c )} + 3 b^{3} n x \log {\relax (c )}^{2} \log {\relax (x )} - 3 b^{3} n x \log {\relax (c )}^{2} + b^{3} x \log {\relax (c )}^{3} \]
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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