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*ln(c*x^n))/d/(1+m)^2+(d*x)^(1+m)*(a+b*ln(c*x^n))^2/d/(1
+m)

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Rubi [A]  time = 0.05, antiderivative size = 81, normalized size of antiderivative = 1.00, 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 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
]

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]

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*}

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Mathematica [A]  time = 0.04, 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

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fricas [B]  time = 0.45, size = 208, normalized size = 2.57 \[ \frac {{\left ({\left (b^{2} m^{2} + 2 \, b^{2} m + b^{2}\right )} n^{2} x \log \relax (x)^{2} + {\left (b^{2} m^{2} + 2 \, b^{2} m + b^{2}\right )} x \log \relax (c)^{2} + 2 \, {\left (a b m^{2} + 2 \, a b m + a b - {\left (b^{2} m + b^{2}\right )} n\right )} x \log \relax (c) + {\left (a^{2} m^{2} + 2 \, b^{2} n^{2} + 2 \, a^{2} m + a^{2} - 2 \, {\left (a b m + a b\right )} n\right )} x + 2 \, {\left ({\left (b^{2} m^{2} + 2 \, b^{2} m + b^{2}\right )} n x \log \relax (c) - {\left ({\left (b^{2} m + b^{2}\right )} n^{2} - {\left (a b m^{2} + 2 \, a b m + a b\right )} n\right )} x\right )} \log \relax (x)\right )} e^{\left (m \log \relax (d) + m \log \relax (x)\right )}}{m^{3} + 3 \, m^{2} + 3 \, m + 1} \]

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)

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giac [B]  time = 0.41, size = 402, normalized size = 4.96 \[ \frac {b^{2} d^{m} m^{2} n^{2} x x^{m} \log \relax (x)^{2}}{m^{3} + 3 \, m^{2} + 3 \, m + 1} + \frac {2 \, b^{2} d^{m} m n^{2} x x^{m} \log \relax (x)^{2}}{m^{3} + 3 \, m^{2} + 3 \, m + 1} - \frac {2 \, b^{2} d^{m} m n^{2} x x^{m} \log \relax (x)}{m^{3} + 3 \, m^{2} + 3 \, m + 1} + \frac {2 \, b^{2} d^{m} m n x x^{m} \log \relax (c) \log \relax (x)}{m^{2} + 2 \, m + 1} + \frac {b^{2} d^{m} n^{2} x x^{m} \log \relax (x)^{2}}{m^{3} + 3 \, m^{2} + 3 \, m + 1} + \frac {2 \, a b d^{m} m n x x^{m} \log \relax (x)}{m^{2} + 2 \, m + 1} - \frac {2 \, b^{2} d^{m} n^{2} x x^{m} \log \relax (x)}{m^{3} + 3 \, m^{2} + 3 \, m + 1} + \frac {2 \, b^{2} d^{m} n x x^{m} \log \relax (c) \log \relax (x)}{m^{2} + 2 \, m + 1} + \frac {2 \, b^{2} d^{m} n^{2} x x^{m}}{m^{3} + 3 \, m^{2} + 3 \, m + 1} - \frac {2 \, b^{2} d^{m} n x x^{m} \log \relax (c)}{m^{2} + 2 \, m + 1} + \frac {2 \, a b d^{m} n x x^{m} \log \relax (x)}{m^{2} + 2 \, m + 1} - \frac {2 \, a b d^{m} n x x^{m}}{m^{2} + 2 \, m + 1} + \frac {\left (d x\right )^{m} b^{2} x \log \relax (c)^{2}}{m + 1} + \frac {2 \, \left (d x\right )^{m} a b x \log \relax (c)}{m + 1} + \frac {\left (d x\right )^{m} a^{2} x}{m + 1} \]

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)

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maple [C]  time = 0.25, size = 2126, normalized size = 26.25 \[ \text {Expression too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.71, size = 132, normalized size = 1.63 \[ -\frac {2 \, a b d^{m} n x x^{m}}{{\left (m + 1\right )}^{2}} - 2 \, {\left (\frac {d^{m} n x x^{m} \log \left (c x^{n}\right )}{{\left (m + 1\right )}^{2}} - \frac {d^{m} n^{2} x x^{m}}{{\left (m + 1\right )}^{3}}\right )} b^{2} + \frac {\left (d x\right )^{m + 1} b^{2} \log \left (c x^{n}\right )^{2}}{d {\left (m + 1\right )}} + \frac {2 \, \left (d x\right )^{m + 1} a b \log \left (c x^{n}\right )}{d {\left (m + 1\right )}} + \frac {\left (d x\right )^{m + 1} a^{2}}{d {\left (m + 1\right )}} \]

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]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (d\,x\right )}^m\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2 \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x)^m*(a + b*log(c*x^n))^2,x)

[Out]

int((d*x)^m*(a + b*log(c*x^n))^2, x)

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sympy [A]  time = 27.73, size = 891, normalized size = 11.00 \[ \begin {cases} \frac {a^{2} d^{m} m^{2} x x^{m}}{m^{3} + 3 m^{2} + 3 m + 1} + \frac {2 a^{2} d^{m} m x x^{m}}{m^{3} + 3 m^{2} + 3 m + 1} + \frac {a^{2} d^{m} x x^{m}}{m^{3} + 3 m^{2} + 3 m + 1} + \frac {2 a b d^{m} m^{2} n x x^{m} \log {\relax (x )}}{m^{3} + 3 m^{2} + 3 m + 1} + \frac {2 a b d^{m} m^{2} x x^{m} \log {\relax (c )}}{m^{3} + 3 m^{2} + 3 m + 1} + \frac {4 a b d^{m} m n x x^{m} \log {\relax (x )}}{m^{3} + 3 m^{2} + 3 m + 1} - \frac {2 a b d^{m} m n x x^{m}}{m^{3} + 3 m^{2} + 3 m + 1} + \frac {4 a b d^{m} m x x^{m} \log {\relax (c )}}{m^{3} + 3 m^{2} + 3 m + 1} + \frac {2 a b d^{m} n x x^{m} \log {\relax (x )}}{m^{3} + 3 m^{2} + 3 m + 1} - \frac {2 a b d^{m} n x x^{m}}{m^{3} + 3 m^{2} + 3 m + 1} + \frac {2 a b d^{m} x x^{m} \log {\relax (c )}}{m^{3} + 3 m^{2} + 3 m + 1} + \frac {b^{2} d^{m} m^{2} n^{2} x x^{m} \log {\relax (x )}^{2}}{m^{3} + 3 m^{2} + 3 m + 1} + \frac {2 b^{2} d^{m} m^{2} n x x^{m} \log {\relax (c )} \log {\relax (x )}}{m^{3} + 3 m^{2} + 3 m + 1} + \frac {b^{2} d^{m} m^{2} x x^{m} \log {\relax (c )}^{2}}{m^{3} + 3 m^{2} + 3 m + 1} + \frac {2 b^{2} d^{m} m n^{2} x x^{m} \log {\relax (x )}^{2}}{m^{3} + 3 m^{2} + 3 m + 1} - \frac {2 b^{2} d^{m} m n^{2} x x^{m} \log {\relax (x )}}{m^{3} + 3 m^{2} + 3 m + 1} + \frac {4 b^{2} d^{m} m n x x^{m} \log {\relax (c )} \log {\relax (x )}}{m^{3} + 3 m^{2} + 3 m + 1} - \frac {2 b^{2} d^{m} m n x x^{m} \log {\relax (c )}}{m^{3} + 3 m^{2} + 3 m + 1} + \frac {2 b^{2} d^{m} m x x^{m} \log {\relax (c )}^{2}}{m^{3} + 3 m^{2} + 3 m + 1} + \frac {b^{2} d^{m} n^{2} x x^{m} \log {\relax (x )}^{2}}{m^{3} + 3 m^{2} + 3 m + 1} - \frac {2 b^{2} d^{m} n^{2} x x^{m} \log {\relax (x )}}{m^{3} + 3 m^{2} + 3 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 {\relax (c )} \log {\relax (x )}}{m^{3} + 3 m^{2} + 3 m + 1} - \frac {2 b^{2} d^{m} n x x^{m} \log {\relax (c )}}{m^{3} + 3 m^{2} + 3 m + 1} + \frac {b^{2} d^{m} x x^{m} \log {\relax (c )}^{2}}{m^{3} + 3 m^{2} + 3 m + 1} & \text {for}\: m \neq -1 \\\frac {\begin {cases} \frac {a^{2} \log {\left (c x^{n} \right )} + a b \log {\left (c x^{n} \right )}^{2} + \frac {b^{2} \log {\left (c x^{n} \right )}^{3}}{3}}{n} & \text {for}\: n \neq 0 \\\left (a^{2} + 2 a b \log {\relax (c )} + b^{2} \log {\relax (c )}^{2}\right ) \log {\relax (x )} & \text {otherwise} \end {cases}}{d} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a**2*d**m*m**2*x*x**m/(m**3 + 3*m**2 + 3*m + 1) + 2*a**2*d**m*m*x*x**m/(m**3 + 3*m**2 + 3*m + 1) +
a**2*d**m*x*x**m/(m**3 + 3*m**2 + 3*m + 1) + 2*a*b*d**m*m**2*n*x*x**m*log(x)/(m**3 + 3*m**2 + 3*m + 1) + 2*a*b
*d**m*m**2*x*x**m*log(c)/(m**3 + 3*m**2 + 3*m + 1) + 4*a*b*d**m*m*n*x*x**m*log(x)/(m**3 + 3*m**2 + 3*m + 1) -
2*a*b*d**m*m*n*x*x**m/(m**3 + 3*m**2 + 3*m + 1) + 4*a*b*d**m*m*x*x**m*log(c)/(m**3 + 3*m**2 + 3*m + 1) + 2*a*b
*d**m*n*x*x**m*log(x)/(m**3 + 3*m**2 + 3*m + 1) - 2*a*b*d**m*n*x*x**m/(m**3 + 3*m**2 + 3*m + 1) + 2*a*b*d**m*x
*x**m*log(c)/(m**3 + 3*m**2 + 3*m + 1) + 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**2*n*x*x**m*log(c)*log(x)/(m**3 + 3*m**2 + 3*m + 1) + b**2*d**m*m**2*x*x**m*log(c)**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) + 4*b**2*d**m*m*n*x*x**m*log(c)*log(x)/(m**3 + 3*m**2 + 3*m + 1) - 2*b**2*d**m*m*n*
x*x**m*log(c)/(m**3 + 3*m**2 + 3*m + 1) + 2*b**2*d**m*m*x*x**m*log(c)**2/(m**3 + 3*m**2 + 3*m + 1) + b**2*d**m
*n**2*x*x**m*log(x)**2/(m**3 + 3*m**2 + 3*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**2*x*x**m/(m**3 + 3*m**2 + 3*m + 1) + 2*b**2*d**m*n*x*x**m*log(c)*log(x)/(m**3 + 3*m**2 + 3*m +
1) - 2*b**2*d**m*n*x*x**m*log(c)/(m**3 + 3*m**2 + 3*m + 1) + b**2*d**m*x*x**m*log(c)**2/(m**3 + 3*m**2 + 3*m +
1), Ne(m, -1)), (Piecewise(((a**2*log(c*x**n) + a*b*log(c*x**n)**2 + b**2*log(c*x**n)**3/3)/n, Ne(n, 0)), ((a
**2 + 2*a*b*log(c) + b**2*log(c)**2)*log(x), True))/d, True))

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