3.51 \(\int x^2 (a+b \log (c x^n))^2 \, dx\)

Optimal. Leaf size=52 \[ \frac{1}{3} x^3 \left (a+b \log \left (c x^n\right )\right )^2-\frac{2}{9} b n x^3 \left (a+b \log \left (c x^n\right )\right )+\frac{2}{27} b^2 n^2 x^3 \]

[Out]

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

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Rubi [A]  time = 0.0367015, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {2305, 2304} \[ \frac{1}{3} x^3 \left (a+b \log \left (c x^n\right )\right )^2-\frac{2}{9} b n x^3 \left (a+b \log \left (c x^n\right )\right )+\frac{2}{27} b^2 n^2 x^3 \]

Antiderivative was successfully verified.

[In]

Int[x^2*(a + b*Log[c*x^n])^2,x]

[Out]

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

Mathematica [A]  time = 0.0188488, size = 46, normalized size = 0.88 \[ \frac{1}{3} \left (x^3 \left (a+b \log \left (c x^n\right )\right )^2+\frac{2}{9} b n x^3 \left (-3 a-3 b \log \left (c x^n\right )+b n\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*(a + b*Log[c*x^n])^2,x]

[Out]

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

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Maple [C]  time = 0.204, size = 692, normalized size = 13.3 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(a+b*ln(c*x^n))^2,x)

[Out]

1/3*b^2*x^3*ln(x^n)^2+1/9*b*x^3*(3*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2-3*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)*csgn(
I*c)-3*I*b*Pi*csgn(I*c*x^n)^3+3*I*b*Pi*csgn(I*c*x^n)^2*csgn(I*c)+6*b*ln(c)-2*b*n+6*a)*ln(x^n)+1/108*x^3*(-12*I
*Pi*b^2*n*csgn(I*c*x^n)^2*csgn(I*c)+36*I*ln(c)*Pi*b^2*csgn(I*x^n)*csgn(I*c*x^n)^2+36*I*ln(c)*Pi*b^2*csgn(I*c*x
^n)^2*csgn(I*c)+36*I*Pi*a*b*csgn(I*x^n)*csgn(I*c*x^n)^2+36*I*Pi*a*b*csgn(I*c*x^n)^2*csgn(I*c)-12*I*Pi*b^2*n*cs
gn(I*x^n)*csgn(I*c*x^n)^2+36*ln(c)^2*b^2-9*Pi^2*b^2*csgn(I*c*x^n)^4*csgn(I*c)^2-24*a*b*n+8*b^2*n^2+36*a^2+18*P
i^2*b^2*csgn(I*x^n)*csgn(I*c*x^n)^3*csgn(I*c)^2+18*Pi^2*b^2*csgn(I*x^n)^2*csgn(I*c*x^n)^3*csgn(I*c)-9*Pi^2*b^2
*csgn(I*x^n)^2*csgn(I*c*x^n)^2*csgn(I*c)^2-36*Pi^2*b^2*csgn(I*x^n)*csgn(I*c*x^n)^4*csgn(I*c)-36*I*ln(c)*Pi*b^2
*csgn(I*c*x^n)^3-36*I*Pi*a*b*csgn(I*c*x^n)^3+12*I*Pi*b^2*n*csgn(I*c*x^n)^3-36*I*ln(c)*Pi*b^2*csgn(I*x^n)*csgn(
I*c*x^n)*csgn(I*c)-36*I*Pi*a*b*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-9*Pi^2*b^2*csgn(I*c*x^n)^6+72*ln(c)*a*b-24*
ln(c)*b^2*n+12*I*Pi*b^2*n*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+18*Pi^2*b^2*csgn(I*c*x^n)^5*csgn(I*c)+18*Pi^2*b^
2*csgn(I*x^n)*csgn(I*c*x^n)^5-9*Pi^2*b^2*csgn(I*x^n)^2*csgn(I*c*x^n)^4)

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Maxima [A]  time = 1.08595, size = 96, normalized size = 1.85 \begin{align*} \frac{1}{3} \, b^{2} x^{3} \log \left (c x^{n}\right )^{2} - \frac{2}{9} \, a b n x^{3} + \frac{2}{3} \, a b x^{3} \log \left (c x^{n}\right ) + \frac{1}{3} \, a^{2} x^{3} + \frac{2}{27} \,{\left (n^{2} x^{3} - 3 \, n x^{3} \log \left (c x^{n}\right )\right )} b^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(a+b*log(c*x^n))^2,x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 0.848358, size = 247, normalized size = 4.75 \begin{align*} \frac{1}{3} \, b^{2} n^{2} x^{3} \log \left (x\right )^{2} + \frac{1}{3} \, b^{2} x^{3} \log \left (c\right )^{2} - \frac{2}{9} \,{\left (b^{2} n - 3 \, a b\right )} x^{3} \log \left (c\right ) + \frac{1}{27} \,{\left (2 \, b^{2} n^{2} - 6 \, a b n + 9 \, a^{2}\right )} x^{3} + \frac{2}{9} \,{\left (3 \, b^{2} n x^{3} \log \left (c\right ) -{\left (b^{2} n^{2} - 3 \, a b n\right )} x^{3}\right )} \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(a+b*log(c*x^n))^2,x, algorithm="fricas")

[Out]

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

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Sympy [B]  time = 1.73516, size = 143, normalized size = 2.75 \begin{align*} \frac{a^{2} x^{3}}{3} + \frac{2 a b n x^{3} \log{\left (x \right )}}{3} - \frac{2 a b n x^{3}}{9} + \frac{2 a b x^{3} \log{\left (c \right )}}{3} + \frac{b^{2} n^{2} x^{3} \log{\left (x \right )}^{2}}{3} - \frac{2 b^{2} n^{2} x^{3} \log{\left (x \right )}}{9} + \frac{2 b^{2} n^{2} x^{3}}{27} + \frac{2 b^{2} n x^{3} \log{\left (c \right )} \log{\left (x \right )}}{3} - \frac{2 b^{2} n x^{3} \log{\left (c \right )}}{9} + \frac{b^{2} x^{3} \log{\left (c \right )}^{2}}{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*(a+b*ln(c*x**n))**2,x)

[Out]

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

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Giac [B]  time = 1.18623, size = 150, normalized size = 2.88 \begin{align*} \frac{1}{3} \, b^{2} n^{2} x^{3} \log \left (x\right )^{2} - \frac{2}{9} \, b^{2} n^{2} x^{3} \log \left (x\right ) + \frac{2}{3} \, b^{2} n x^{3} \log \left (c\right ) \log \left (x\right ) + \frac{2}{27} \, b^{2} n^{2} x^{3} - \frac{2}{9} \, b^{2} n x^{3} \log \left (c\right ) + \frac{1}{3} \, b^{2} x^{3} \log \left (c\right )^{2} + \frac{2}{3} \, a b n x^{3} \log \left (x\right ) - \frac{2}{9} \, a b n x^{3} + \frac{2}{3} \, a b x^{3} \log \left (c\right ) + \frac{1}{3} \, a^{2} x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(a+b*log(c*x^n))^2,x, algorithm="giac")

[Out]

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