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

Optimal. Leaf size=52 \[ \frac{1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^2-\frac{1}{2} b n x^2 \left (a+b \log \left (c x^n\right )\right )+\frac{1}{4} b^2 n^2 x^2 \]

[Out]

(b^2*n^2*x^2)/4 - (b*n*x^2*(a + b*Log[c*x^n]))/2 + (x^2*(a + b*Log[c*x^n])^2)/2

________________________________________________________________________________________

Rubi [A]  time = 0.0230684, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {2305, 2304} \[ \frac{1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^2-\frac{1}{2} b n x^2 \left (a+b \log \left (c x^n\right )\right )+\frac{1}{4} b^2 n^2 x^2 \]

Antiderivative was successfully verified.

[In]

Int[x*(a + b*Log[c*x^n])^2,x]

[Out]

(b^2*n^2*x^2)/4 - (b*n*x^2*(a + b*Log[c*x^n]))/2 + (x^2*(a + b*Log[c*x^n])^2)/2

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

Mathematica [A]  time = 0.0130538, size = 41, normalized size = 0.79 \[ \frac{1}{4} x^2 \left (2 \left (a+b \log \left (c x^n\right )\right )^2+b n \left (-2 a-2 b \log \left (c x^n\right )+b n\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x*(a + b*Log[c*x^n])^2,x]

[Out]

(x^2*(b*n*(-2*a + b*n - 2*b*Log[c*x^n]) + 2*(a + b*Log[c*x^n])^2))/4

________________________________________________________________________________________

Maple [C]  time = 0.213, 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*(a+b*ln(c*x^n))^2,x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.11882, size = 95, normalized size = 1.83 \begin{align*} \frac{1}{2} \, b^{2} x^{2} \log \left (c x^{n}\right )^{2} - \frac{1}{2} \, a b n x^{2} + a b x^{2} \log \left (c x^{n}\right ) + \frac{1}{2} \, a^{2} x^{2} + \frac{1}{4} \,{\left (n^{2} x^{2} - 2 \, n x^{2} \log \left (c x^{n}\right )\right )} b^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*b^2*x^2*log(c*x^n)^2 - 1/2*a*b*n*x^2 + a*b*x^2*log(c*x^n) + 1/2*a^2*x^2 + 1/4*(n^2*x^2 - 2*n*x^2*log(c*x^n
))*b^2

________________________________________________________________________________________

Fricas [B]  time = 0.87917, size = 243, normalized size = 4.67 \begin{align*} \frac{1}{2} \, b^{2} n^{2} x^{2} \log \left (x\right )^{2} + \frac{1}{2} \, b^{2} x^{2} \log \left (c\right )^{2} - \frac{1}{2} \,{\left (b^{2} n - 2 \, a b\right )} x^{2} \log \left (c\right ) + \frac{1}{4} \,{\left (b^{2} n^{2} - 2 \, a b n + 2 \, a^{2}\right )} x^{2} + \frac{1}{2} \,{\left (2 \, b^{2} n x^{2} \log \left (c\right ) -{\left (b^{2} n^{2} - 2 \, a b n\right )} x^{2}\right )} \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*b^2*n^2*x^2*log(x)^2 + 1/2*b^2*x^2*log(c)^2 - 1/2*(b^2*n - 2*a*b)*x^2*log(c) + 1/4*(b^2*n^2 - 2*a*b*n + 2*
a^2)*x^2 + 1/2*(2*b^2*n*x^2*log(c) - (b^2*n^2 - 2*a*b*n)*x^2)*log(x)

________________________________________________________________________________________

Sympy [B]  time = 1.02012, size = 126, normalized size = 2.42 \begin{align*} \frac{a^{2} x^{2}}{2} + a b n x^{2} \log{\left (x \right )} - \frac{a b n x^{2}}{2} + a b x^{2} \log{\left (c \right )} + \frac{b^{2} n^{2} x^{2} \log{\left (x \right )}^{2}}{2} - \frac{b^{2} n^{2} x^{2} \log{\left (x \right )}}{2} + \frac{b^{2} n^{2} x^{2}}{4} + b^{2} n x^{2} \log{\left (c \right )} \log{\left (x \right )} - \frac{b^{2} n x^{2} \log{\left (c \right )}}{2} + \frac{b^{2} x^{2} \log{\left (c \right )}^{2}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**2*x**2/2 + a*b*n*x**2*log(x) - a*b*n*x**2/2 + a*b*x**2*log(c) + b**2*n**2*x**2*log(x)**2/2 - b**2*n**2*x**2
*log(x)/2 + b**2*n**2*x**2/4 + b**2*n*x**2*log(c)*log(x) - b**2*n*x**2*log(c)/2 + b**2*x**2*log(c)**2/2

________________________________________________________________________________________

Giac [B]  time = 1.15036, size = 146, normalized size = 2.81 \begin{align*} \frac{1}{2} \, b^{2} n^{2} x^{2} \log \left (x\right )^{2} - \frac{1}{2} \, b^{2} n^{2} x^{2} \log \left (x\right ) + b^{2} n x^{2} \log \left (c\right ) \log \left (x\right ) + \frac{1}{4} \, b^{2} n^{2} x^{2} - \frac{1}{2} \, b^{2} n x^{2} \log \left (c\right ) + \frac{1}{2} \, b^{2} x^{2} \log \left (c\right )^{2} + a b n x^{2} \log \left (x\right ) - \frac{1}{2} \, a b n x^{2} + a b x^{2} \log \left (c\right ) + \frac{1}{2} \, a^{2} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*b^2*n^2*x^2*log(x)^2 - 1/2*b^2*n^2*x^2*log(x) + b^2*n*x^2*log(c)*log(x) + 1/4*b^2*n^2*x^2 - 1/2*b^2*n*x^2*
log(c) + 1/2*b^2*x^2*log(c)^2 + a*b*n*x^2*log(x) - 1/2*a*b*n*x^2 + a*b*x^2*log(c) + 1/2*a^2*x^2