3.151 \(\int (d x)^m (a+b \log (c x^n))^2 \, dx\)

Optimal. Leaf size=81 \[ \frac{(d x)^{m+1} \left (a+b \log \left (c x^n\right )\right )^2}{d (m+1)}-\frac{2 b n (d x)^{m+1} \left (a+b \log \left (c x^n\right )\right )}{d (m+1)^2}+\frac{2 b^2 n^2 (d x)^{m+1}}{d (m+1)^3} \]

[Out]

(2*b^2*n^2*(d*x)^(1 + m))/(d*(1 + m)^3) - (2*b*n*(d*x)^(1 + m)*(a + b*Log[c*x^n]))/(d*(1 + m)^2) + ((d*x)^(1 +
m)*(a + b*Log[c*x^n])^2)/(d*(1 + m))

________________________________________________________________________________________

Rubi [A]  time = 0.0461994, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {2305, 2304} \[ \frac{(d x)^{m+1} \left (a+b \log \left (c x^n\right )\right )^2}{d (m+1)}-\frac{2 b n (d x)^{m+1} \left (a+b \log \left (c x^n\right )\right )}{d (m+1)^2}+\frac{2 b^2 n^2 (d x)^{m+1}}{d (m+1)^3} \]

Antiderivative was successfully verified.

[In]

Int[(d*x)^m*(a + b*Log[c*x^n])^2,x]

[Out]

(2*b^2*n^2*(d*x)^(1 + m))/(d*(1 + m)^3) - (2*b*n*(d*x)^(1 + m)*(a + b*Log[c*x^n]))/(d*(1 + m)^2) + ((d*x)^(1 +
m)*(a + b*Log[c*x^n])^2)/(d*(1 + m))

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

Mathematica [A]  time = 0.0319102, size = 76, normalized size = 0.94 \[ \frac{x (d x)^m \left (a^2 (m+1)^2+2 b (m+1) (a m+a-b n) \log \left (c x^n\right )-2 a b (m+1) n+b^2 (m+1)^2 \log ^2\left (c x^n\right )+2 b^2 n^2\right )}{(m+1)^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(d*x)^m*(a + b*Log[c*x^n])^2,x]

[Out]

(x*(d*x)^m*(a^2*(1 + m)^2 - 2*a*b*(1 + m)*n + 2*b^2*n^2 + 2*b*(1 + m)*(a + a*m - b*n)*Log[c*x^n] + b^2*(1 + m)
^2*Log[c*x^n]^2))/(1 + m)^3

________________________________________________________________________________________

Maple [C]  time = 0.197, size = 2126, normalized size = 26.3 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b^2/(1+m)*x*exp(1/2*m*(-I*csgn(I*d*x)^3*Pi+I*csgn(I*d*x)^2*csgn(I*d)*Pi+I*csgn(I*d*x)^2*csgn(I*x)*Pi-I*csgn(I*
d*x)*csgn(I*d)*csgn(I*x)*Pi+2*ln(x)+2*ln(d)))*ln(x^n)^2-b*(-I*Pi*b*csgn(I*x^n)*csgn(I*c*x^n)^2*m+I*Pi*b*csgn(I
*x^n)*csgn(I*c*x^n)*csgn(I*c)*m+I*Pi*b*csgn(I*c*x^n)^3*m-I*Pi*b*csgn(I*c*x^n)^2*csgn(I*c)*m-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)*m-2*b*ln(c)-2*a*m+2*b*n-2*a)/(1+m)^2*x*exp(1/2*m*(-I*csgn(I*d*x)^3*Pi+I*csgn(I*d*x)^2*csgn(I*d
)*Pi+I*csgn(I*d*x)^2*csgn(I*x)*Pi-I*csgn(I*d*x)*csgn(I*d)*csgn(I*x)*Pi+2*ln(x)+2*ln(d)))*ln(x^n)+1/4*(-8*a*b*m
*n+4*I*Pi*b^2*m*n*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-4*I*Pi*ln(c)*b^2*m^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)
+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-Pi^2*b^2*m^2*
csgn(I*c*x^n)^6-2*Pi^2*b^2*m*csgn(I*c*x^n)^6-8*a*b*n+4*a^2*m^2+8*b^2*n^2+8*a^2*m+4*a^2+8*ln(c)^2*b^2*m+4*ln(c)
^2*b^2*m^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)+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+4*I*Pi*
a*b*m^2*csgn(I*x^n)*csgn(I*c*x^n)^2+4*I*Pi*a*b*m^2*csgn(I*c*x^n)^2*csgn(I*c)-4*I*Pi*b^2*m*n*csgn(I*x^n)*csgn(I
*c*x^n)^2-4*I*Pi*b^2*m*n*csgn(I*c*x^n)^2*csgn(I*c)-2*Pi^2*b^2*m*csgn(I*x^n)^2*csgn(I*c*x^n)^4+8*ln(c)*a*b*m^2+
16*ln(c)*a*b*m-8*ln(c)*b^2*m*n+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*csgn(I*x^n)^2*csgn(I*c*x^n)^4-8*I*Pi*a*b*m*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-8*I*Pi*ln(c)*b^2*m*csgn(I
*x^n)*csgn(I*c*x^n)*csgn(I*c)-4*I*Pi*a*b*m^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+2*Pi^2*b^2*m^2*csgn(I*x^n)*cs
gn(I*c*x^n)^5+4*Pi^2*b^2*m*csgn(I*x^n)*csgn(I*c*x^n)^5+2*Pi^2*b^2*m^2*csgn(I*c*x^n)^5*csgn(I*c)+4*Pi^2*b^2*m*c
sgn(I*c*x^n)^5*csgn(I*c)-Pi^2*b^2*m^2*csgn(I*c*x^n)^4*csgn(I*c)^2-2*Pi^2*b^2*m*csgn(I*c*x^n)^4*csgn(I*c)^2-Pi^
2*b^2*m^2*csgn(I*x^n)^2*csgn(I*c*x^n)^4-8*Pi^2*b^2*m*csgn(I*x^n)*csgn(I*c*x^n)^4*csgn(I*c)+2*Pi^2*b^2*m^2*csgn
(I*x^n)*csgn(I*c*x^n)^3*csgn(I*c)^2+4*Pi^2*b^2*m*csgn(I*x^n)*csgn(I*c*x^n)^3*csgn(I*c)^2+4*I*Pi*b^2*n*csgn(I*x
^n)*csgn(I*c*x^n)*csgn(I*c)+8*I*Pi*ln(c)*b^2*m*csgn(I*c*x^n)^2*csgn(I*c)+8*I*Pi*ln(c)*b^2*m*csgn(I*x^n)*csgn(I
*c*x^n)^2+4*I*Pi*b^2*n*csgn(I*c*x^n)^3+4*I*Pi*ln(c)*b^2*m^2*csgn(I*x^n)*csgn(I*c*x^n)^2-8*I*Pi*a*b*m*csgn(I*c*
x^n)^3+2*Pi^2*b^2*m^2*csgn(I*x^n)^2*csgn(I*c*x^n)^3*csgn(I*c)+4*Pi^2*b^2*m*csgn(I*x^n)^2*csgn(I*c*x^n)^3*csgn(
I*c)-Pi^2*b^2*m^2*csgn(I*x^n)^2*csgn(I*c*x^n)^2*csgn(I*c)^2-2*Pi^2*b^2*m*csgn(I*x^n)^2*csgn(I*c*x^n)^2*csgn(I*
c)^2-4*Pi^2*b^2*m^2*csgn(I*x^n)*csgn(I*c*x^n)^4*csgn(I*c)-4*I*Pi*ln(c)*b^2*m^2*csgn(I*c*x^n)^3-8*I*Pi*ln(c)*b^
2*m*csgn(I*c*x^n)^3-4*I*Pi*a*b*m^2*csgn(I*c*x^n)^3+4*I*Pi*b^2*m*n*csgn(I*c*x^n)^3+8*I*Pi*a*b*m*csgn(I*x^n)*csg
n(I*c*x^n)^2+8*I*Pi*a*b*m*csgn(I*c*x^n)^2*csgn(I*c)+4*I*Pi*ln(c)*b^2*m^2*csgn(I*c*x^n)^2*csgn(I*c)-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))/(1+m)^3*x*exp(1/2*m*(-I*csgn(I*d*x)^3*Pi
+I*csgn(I*d*x)^2*csgn(I*d)*Pi+I*csgn(I*d*x)^2*csgn(I*x)*Pi-I*csgn(I*d*x)*csgn(I*d)*csgn(I*x)*Pi+2*ln(x)+2*ln(d
)))

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [B]  time = 1.04582, size = 478, normalized size = 5.9 \begin{align*} \frac{{\left ({\left (b^{2} m^{2} + 2 \, b^{2} m + b^{2}\right )} n^{2} x \log \left (x\right )^{2} +{\left (b^{2} m^{2} + 2 \, b^{2} m + b^{2}\right )} x \log \left (c\right )^{2} + 2 \,{\left (a b m^{2} + 2 \, a b m + a b -{\left (b^{2} m + b^{2}\right )} n\right )} x \log \left (c\right ) +{\left (a^{2} m^{2} + 2 \, b^{2} n^{2} + 2 \, a^{2} m + a^{2} - 2 \,{\left (a b m + a b\right )} n\right )} x + 2 \,{\left ({\left (b^{2} m^{2} + 2 \, b^{2} m + b^{2}\right )} n x \log \left (c\right ) -{\left ({\left (b^{2} m + b^{2}\right )} n^{2} -{\left (a b m^{2} + 2 \, a b m + a b\right )} n\right )} x\right )} \log \left (x\right )\right )} e^{\left (m \log \left (d\right ) + m \log \left (x\right )\right )}}{m^{3} + 3 \, m^{2} + 3 \, m + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((b^2*m^2 + 2*b^2*m + b^2)*n^2*x*log(x)^2 + (b^2*m^2 + 2*b^2*m + b^2)*x*log(c)^2 + 2*(a*b*m^2 + 2*a*b*m + a*b
- (b^2*m + b^2)*n)*x*log(c) + (a^2*m^2 + 2*b^2*n^2 + 2*a^2*m + a^2 - 2*(a*b*m + a*b)*n)*x + 2*((b^2*m^2 + 2*b^
2*m + b^2)*n*x*log(c) - ((b^2*m + b^2)*n^2 - (a*b*m^2 + 2*a*b*m + a*b)*n)*x)*log(x))*e^(m*log(d) + m*log(x))/(
m^3 + 3*m^2 + 3*m + 1)

________________________________________________________________________________________

Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError

________________________________________________________________________________________

Giac [B]  time = 1.30953, size = 543, normalized size = 6.7 \begin{align*} \frac{b^{2} d^{m} m^{2} n^{2} x x^{m} \log \left (x\right )^{2}}{m^{3} + 3 \, m^{2} + 3 \, m + 1} + \frac{2 \, b^{2} d^{m} m n^{2} x x^{m} \log \left (x\right )^{2}}{m^{3} + 3 \, m^{2} + 3 \, m + 1} - \frac{2 \, b^{2} d^{m} m n^{2} x x^{m} \log \left (x\right )}{m^{3} + 3 \, m^{2} + 3 \, m + 1} + \frac{2 \, b^{2} d^{m} m n x x^{m} \log \left (c\right ) \log \left (x\right )}{m^{2} + 2 \, m + 1} + \frac{b^{2} d^{m} n^{2} x x^{m} \log \left (x\right )^{2}}{m^{3} + 3 \, m^{2} + 3 \, m + 1} + \frac{2 \, a b d^{m} m n x x^{m} \log \left (x\right )}{m^{2} + 2 \, m + 1} - \frac{2 \, b^{2} d^{m} n^{2} x x^{m} \log \left (x\right )}{m^{3} + 3 \, m^{2} + 3 \, m + 1} + \frac{2 \, b^{2} d^{m} n x x^{m} \log \left (c\right ) \log \left (x\right )}{m^{2} + 2 \, m + 1} + \frac{2 \, b^{2} d^{m} n^{2} x x^{m}}{m^{3} + 3 \, m^{2} + 3 \, m + 1} - \frac{2 \, b^{2} d^{m} n x x^{m} \log \left (c\right )}{m^{2} + 2 \, m + 1} + \frac{2 \, a b d^{m} n x x^{m} \log \left (x\right )}{m^{2} + 2 \, m + 1} - \frac{2 \, a b d^{m} n x x^{m}}{m^{2} + 2 \, m + 1} + \frac{\left (d x\right )^{m} b^{2} x \log \left (c\right )^{2}}{m + 1} + \frac{2 \, \left (d x\right )^{m} a b x \log \left (c\right )}{m + 1} + \frac{\left (d x\right )^{m} a^{2} x}{m + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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