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

Optimal. Leaf size=109 \[ \frac{1}{3} d x^3 \left (a+b \log \left (c x^n\right )\right )^2-\frac{2}{9} b d n x^3 \left (a+b \log \left (c x^n\right )\right )+\frac{1}{4} e x^4 \left (a+b \log \left (c x^n\right )\right )^2-\frac{1}{8} b e n x^4 \left (a+b \log \left (c x^n\right )\right )+\frac{2}{27} b^2 d n^2 x^3+\frac{1}{32} b^2 e n^2 x^4 \]

[Out]

(2*b^2*d*n^2*x^3)/27 + (b^2*e*n^2*x^4)/32 - (2*b*d*n*x^3*(a + b*Log[c*x^n]))/9 - (b*e*n*x^4*(a + b*Log[c*x^n])
)/8 + (d*x^3*(a + b*Log[c*x^n])^2)/3 + (e*x^4*(a + b*Log[c*x^n])^2)/4

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Rubi [A]  time = 0.143042, antiderivative size = 109, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {2353, 2305, 2304} \[ \frac{1}{3} d x^3 \left (a+b \log \left (c x^n\right )\right )^2-\frac{2}{9} b d n x^3 \left (a+b \log \left (c x^n\right )\right )+\frac{1}{4} e x^4 \left (a+b \log \left (c x^n\right )\right )^2-\frac{1}{8} b e n x^4 \left (a+b \log \left (c x^n\right )\right )+\frac{2}{27} b^2 d n^2 x^3+\frac{1}{32} b^2 e n^2 x^4 \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*b^2*d*n^2*x^3)/27 + (b^2*e*n^2*x^4)/32 - (2*b*d*n*x^3*(a + b*Log[c*x^n]))/9 - (b*e*n*x^4*(a + b*Log[c*x^n])
)/8 + (d*x^3*(a + b*Log[c*x^n])^2)/3 + (e*x^4*(a + b*Log[c*x^n])^2)/4

Rule 2353

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol]
:> With[{u = ExpandIntegrand[(a + b*Log[c*x^n])^p, (f*x)^m*(d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[
{a, b, c, d, e, f, m, n, p, q, r}, x] && IntegerQ[q] && (GtQ[q, 0] || (IGtQ[p, 0] && IntegerQ[m] && IntegerQ[r
]))

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 (d+e x) \left (a+b \log \left (c x^n\right )\right )^2 \, dx &=\int \left (d x^2 \left (a+b \log \left (c x^n\right )\right )^2+e x^3 \left (a+b \log \left (c x^n\right )\right )^2\right ) \, dx\\ &=d \int x^2 \left (a+b \log \left (c x^n\right )\right )^2 \, dx+e \int x^3 \left (a+b \log \left (c x^n\right )\right )^2 \, dx\\ &=\frac{1}{3} d x^3 \left (a+b \log \left (c x^n\right )\right )^2+\frac{1}{4} e x^4 \left (a+b \log \left (c x^n\right )\right )^2-\frac{1}{3} (2 b d n) \int x^2 \left (a+b \log \left (c x^n\right )\right ) \, dx-\frac{1}{2} (b e n) \int x^3 \left (a+b \log \left (c x^n\right )\right ) \, dx\\ &=\frac{2}{27} b^2 d n^2 x^3+\frac{1}{32} b^2 e n^2 x^4-\frac{2}{9} b d n x^3 \left (a+b \log \left (c x^n\right )\right )-\frac{1}{8} b e n x^4 \left (a+b \log \left (c x^n\right )\right )+\frac{1}{3} d x^3 \left (a+b \log \left (c x^n\right )\right )^2+\frac{1}{4} e x^4 \left (a+b \log \left (c x^n\right )\right )^2\\ \end{align*}

Mathematica [A]  time = 0.0574034, size = 82, normalized size = 0.75 \[ \frac{1}{864} x^3 \left (288 d \left (a+b \log \left (c x^n\right )\right )^2+64 b d n \left (-3 a-3 b \log \left (c x^n\right )+b n\right )+216 e x \left (a+b \log \left (c x^n\right )\right )^2+27 b e n x \left (-4 a-4 b \log \left (c x^n\right )+b n\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x^3*(27*b*e*n*x*(-4*a + b*n - 4*b*Log[c*x^n]) + 64*b*d*n*(-3*a + b*n - 3*b*Log[c*x^n]) + 288*d*(a + b*Log[c*x
^n])^2 + 216*e*x*(a + b*Log[c*x^n])^2))/864

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Maple [C]  time = 0.268, size = 1622, normalized size = 14.9 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*I*ln(c)*Pi*b^2*e*x^4*csgn(I*x^n)*csgn(I*c*x^n)^2+1/4*I*ln(c)*Pi*b^2*e*x^4*csgn(I*c*x^n)^2*csgn(I*c)+1/3*I*
Pi*a*b*d*x^3*csgn(I*x^n)*csgn(I*c*x^n)^2+1/3*I*Pi*a*b*d*x^3*csgn(I*c*x^n)^2*csgn(I*c)+1/3*I*ln(c)*Pi*b^2*d*x^3
*csgn(I*x^n)*csgn(I*c*x^n)^2-1/12*Pi^2*b^2*d*x^3*csgn(I*x^n)^2*csgn(I*c*x^n)^4+1/6*Pi^2*b^2*d*x^3*csgn(I*x^n)*
csgn(I*c*x^n)^5+1/6*Pi^2*b^2*d*x^3*csgn(I*c*x^n)^5*csgn(I*c)-1/12*Pi^2*b^2*d*x^3*csgn(I*c*x^n)^4*csgn(I*c)^2+1
/72*b*(18*I*Pi*b*e*x^4*csgn(I*x^n)*csgn(I*c*x^n)^2-18*I*Pi*b*e*x^4*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-18*I*Pi
*b*e*x^4*csgn(I*c*x^n)^3+18*I*Pi*b*e*x^4*csgn(I*c*x^n)^2*csgn(I*c)+36*ln(c)*b*e*x^4-9*b*e*n*x^4+36*a*e*x^4+24*
I*Pi*b*d*x^3*csgn(I*x^n)*csgn(I*c*x^n)^2-24*I*Pi*b*d*x^3*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-24*I*Pi*b*d*x^3*c
sgn(I*c*x^n)^3+24*I*Pi*b*d*x^3*csgn(I*c*x^n)^2*csgn(I*c)+48*ln(c)*b*d*x^3-16*b*d*n*x^3+48*a*d*x^3)*ln(x^n)-1/3
*I*ln(c)*Pi*b^2*d*x^3*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+1/9*I*Pi*b^2*d*n*x^3*csgn(I*x^n)*csgn(I*c*x^n)*csgn(
I*c)-1/3*I*Pi*a*b*d*x^3*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+1/4*a^2*e*x^4-2/9*a*b*d*n*x^3-1/16*I*Pi*b^2*e*n*x^
4*csgn(I*x^n)*csgn(I*c*x^n)^2-1/16*I*Pi*b^2*e*n*x^4*csgn(I*c*x^n)^2*csgn(I*c)+1/4*I*Pi*a*b*e*x^4*csgn(I*x^n)*c
sgn(I*c*x^n)^2+1/3*I*ln(c)*Pi*b^2*d*x^3*csgn(I*c*x^n)^2*csgn(I*c)+1/12*b^2*x^3*(3*e*x+4*d)*ln(x^n)^2+1/3*a^2*d
*x^3+2/27*b^2*d*n^2*x^3+1/32*b^2*e*n^2*x^4-1/16*Pi^2*b^2*e*x^4*csgn(I*x^n)^2*csgn(I*c*x^n)^4+1/8*Pi^2*b^2*e*x^
4*csgn(I*x^n)*csgn(I*c*x^n)^5+1/8*Pi^2*b^2*e*x^4*csgn(I*c*x^n)^5*csgn(I*c)-1/16*Pi^2*b^2*e*x^4*csgn(I*c*x^n)^4
*csgn(I*c)^2-1/8*a*b*e*n*x^4+1/4*I*Pi*a*b*e*x^4*csgn(I*c*x^n)^2*csgn(I*c)-1/9*I*Pi*b^2*d*n*x^3*csgn(I*x^n)*csg
n(I*c*x^n)^2-1/9*I*Pi*b^2*d*n*x^3*csgn(I*c*x^n)^2*csgn(I*c)-1/8*ln(c)*b^2*e*n*x^4+2/3*ln(c)*a*b*d*x^3-2/9*ln(c
)*b^2*d*n*x^3+1/2*ln(c)*a*b*e*x^4+1/6*Pi^2*b^2*d*x^3*csgn(I*x^n)*csgn(I*c*x^n)^3*csgn(I*c)^2-1/4*I*Pi*a*b*e*x^
4*csgn(I*c*x^n)^3-1/4*I*ln(c)*Pi*b^2*e*x^4*csgn(I*c*x^n)^3-1/12*Pi^2*b^2*d*x^3*csgn(I*c*x^n)^6+1/8*Pi^2*b^2*e*
x^4*csgn(I*x^n)^2*csgn(I*c*x^n)^3*csgn(I*c)-1/16*Pi^2*b^2*e*x^4*csgn(I*x^n)^2*csgn(I*c*x^n)^2*csgn(I*c)^2-1/4*
Pi^2*b^2*e*x^4*csgn(I*x^n)*csgn(I*c*x^n)^4*csgn(I*c)+1/8*Pi^2*b^2*e*x^4*csgn(I*x^n)*csgn(I*c*x^n)^3*csgn(I*c)^
2+1/4*ln(c)^2*b^2*e*x^4-1/16*Pi^2*b^2*e*x^4*csgn(I*c*x^n)^6+1/3*ln(c)^2*b^2*d*x^3+1/16*I*Pi*b^2*e*n*x^4*csgn(I
*x^n)*csgn(I*c*x^n)*csgn(I*c)-1/4*I*ln(c)*Pi*b^2*e*x^4*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+1/6*Pi^2*b^2*d*x^3*
csgn(I*x^n)^2*csgn(I*c*x^n)^3*csgn(I*c)-1/12*Pi^2*b^2*d*x^3*csgn(I*x^n)^2*csgn(I*c*x^n)^2*csgn(I*c)^2-1/3*Pi^2
*b^2*d*x^3*csgn(I*x^n)*csgn(I*c*x^n)^4*csgn(I*c)+1/16*I*Pi*b^2*e*n*x^4*csgn(I*c*x^n)^3-1/3*I*ln(c)*Pi*b^2*d*x^
3*csgn(I*c*x^n)^3+1/9*I*Pi*b^2*d*n*x^3*csgn(I*c*x^n)^3-1/3*I*Pi*a*b*d*x^3*csgn(I*c*x^n)^3-1/4*I*Pi*a*b*e*x^4*c
sgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)

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Maxima [A]  time = 1.12608, size = 204, normalized size = 1.87 \begin{align*} \frac{1}{4} \, b^{2} e x^{4} \log \left (c x^{n}\right )^{2} - \frac{1}{8} \, a b e n x^{4} + \frac{1}{2} \, a b e x^{4} \log \left (c x^{n}\right ) + \frac{1}{3} \, b^{2} d x^{3} \log \left (c x^{n}\right )^{2} - \frac{2}{9} \, a b d n x^{3} + \frac{1}{4} \, a^{2} e x^{4} + \frac{2}{3} \, a b d x^{3} \log \left (c x^{n}\right ) + \frac{1}{3} \, a^{2} d x^{3} + \frac{2}{27} \,{\left (n^{2} x^{3} - 3 \, n x^{3} \log \left (c x^{n}\right )\right )} b^{2} d + \frac{1}{32} \,{\left (n^{2} x^{4} - 4 \, n x^{4} \log \left (c x^{n}\right )\right )} b^{2} e \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*b^2*e*x^4*log(c*x^n)^2 - 1/8*a*b*e*n*x^4 + 1/2*a*b*e*x^4*log(c*x^n) + 1/3*b^2*d*x^3*log(c*x^n)^2 - 2/9*a*b
*d*n*x^3 + 1/4*a^2*e*x^4 + 2/3*a*b*d*x^3*log(c*x^n) + 1/3*a^2*d*x^3 + 2/27*(n^2*x^3 - 3*n*x^3*log(c*x^n))*b^2*
d + 1/32*(n^2*x^4 - 4*n*x^4*log(c*x^n))*b^2*e

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Fricas [B]  time = 1.02166, size = 514, normalized size = 4.72 \begin{align*} \frac{1}{32} \,{\left (b^{2} e n^{2} - 4 \, a b e n + 8 \, a^{2} e\right )} x^{4} + \frac{1}{27} \,{\left (2 \, b^{2} d n^{2} - 6 \, a b d n + 9 \, a^{2} d\right )} x^{3} + \frac{1}{12} \,{\left (3 \, b^{2} e x^{4} + 4 \, b^{2} d x^{3}\right )} \log \left (c\right )^{2} + \frac{1}{12} \,{\left (3 \, b^{2} e n^{2} x^{4} + 4 \, b^{2} d n^{2} x^{3}\right )} \log \left (x\right )^{2} - \frac{1}{72} \,{\left (9 \,{\left (b^{2} e n - 4 \, a b e\right )} x^{4} + 16 \,{\left (b^{2} d n - 3 \, a b d\right )} x^{3}\right )} \log \left (c\right ) - \frac{1}{72} \,{\left (9 \,{\left (b^{2} e n^{2} - 4 \, a b e n\right )} x^{4} + 16 \,{\left (b^{2} d n^{2} - 3 \, a b d n\right )} x^{3} - 12 \,{\left (3 \, b^{2} e n x^{4} + 4 \, b^{2} d n x^{3}\right )} \log \left (c\right )\right )} \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/32*(b^2*e*n^2 - 4*a*b*e*n + 8*a^2*e)*x^4 + 1/27*(2*b^2*d*n^2 - 6*a*b*d*n + 9*a^2*d)*x^3 + 1/12*(3*b^2*e*x^4
+ 4*b^2*d*x^3)*log(c)^2 + 1/12*(3*b^2*e*n^2*x^4 + 4*b^2*d*n^2*x^3)*log(x)^2 - 1/72*(9*(b^2*e*n - 4*a*b*e)*x^4
+ 16*(b^2*d*n - 3*a*b*d)*x^3)*log(c) - 1/72*(9*(b^2*e*n^2 - 4*a*b*e*n)*x^4 + 16*(b^2*d*n^2 - 3*a*b*d*n)*x^3 -
12*(3*b^2*e*n*x^4 + 4*b^2*d*n*x^3)*log(c))*log(x)

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Sympy [B]  time = 3.83273, size = 309, normalized size = 2.83 \begin{align*} \frac{a^{2} d x^{3}}{3} + \frac{a^{2} e x^{4}}{4} + \frac{2 a b d n x^{3} \log{\left (x \right )}}{3} - \frac{2 a b d n x^{3}}{9} + \frac{2 a b d x^{3} \log{\left (c \right )}}{3} + \frac{a b e n x^{4} \log{\left (x \right )}}{2} - \frac{a b e n x^{4}}{8} + \frac{a b e x^{4} \log{\left (c \right )}}{2} + \frac{b^{2} d n^{2} x^{3} \log{\left (x \right )}^{2}}{3} - \frac{2 b^{2} d n^{2} x^{3} \log{\left (x \right )}}{9} + \frac{2 b^{2} d n^{2} x^{3}}{27} + \frac{2 b^{2} d n x^{3} \log{\left (c \right )} \log{\left (x \right )}}{3} - \frac{2 b^{2} d n x^{3} \log{\left (c \right )}}{9} + \frac{b^{2} d x^{3} \log{\left (c \right )}^{2}}{3} + \frac{b^{2} e n^{2} x^{4} \log{\left (x \right )}^{2}}{4} - \frac{b^{2} e n^{2} x^{4} \log{\left (x \right )}}{8} + \frac{b^{2} e n^{2} x^{4}}{32} + \frac{b^{2} e n x^{4} \log{\left (c \right )} \log{\left (x \right )}}{2} - \frac{b^{2} e n x^{4} \log{\left (c \right )}}{8} + \frac{b^{2} e x^{4} \log{\left (c \right )}^{2}}{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**2*d*x**3/3 + a**2*e*x**4/4 + 2*a*b*d*n*x**3*log(x)/3 - 2*a*b*d*n*x**3/9 + 2*a*b*d*x**3*log(c)/3 + a*b*e*n*x
**4*log(x)/2 - a*b*e*n*x**4/8 + a*b*e*x**4*log(c)/2 + b**2*d*n**2*x**3*log(x)**2/3 - 2*b**2*d*n**2*x**3*log(x)
/9 + 2*b**2*d*n**2*x**3/27 + 2*b**2*d*n*x**3*log(c)*log(x)/3 - 2*b**2*d*n*x**3*log(c)/9 + b**2*d*x**3*log(c)**
2/3 + b**2*e*n**2*x**4*log(x)**2/4 - b**2*e*n**2*x**4*log(x)/8 + b**2*e*n**2*x**4/32 + b**2*e*n*x**4*log(c)*lo
g(x)/2 - b**2*e*n*x**4*log(c)/8 + b**2*e*x**4*log(c)**2/4

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Giac [B]  time = 1.34929, size = 339, normalized size = 3.11 \begin{align*} \frac{1}{4} \, b^{2} n^{2} x^{4} e \log \left (x\right )^{2} - \frac{1}{8} \, b^{2} n^{2} x^{4} e \log \left (x\right ) + \frac{1}{2} \, b^{2} n x^{4} e \log \left (c\right ) \log \left (x\right ) + \frac{1}{3} \, b^{2} d n^{2} x^{3} \log \left (x\right )^{2} + \frac{1}{32} \, b^{2} n^{2} x^{4} e - \frac{1}{8} \, b^{2} n x^{4} e \log \left (c\right ) + \frac{1}{4} \, b^{2} x^{4} e \log \left (c\right )^{2} - \frac{2}{9} \, b^{2} d n^{2} x^{3} \log \left (x\right ) + \frac{1}{2} \, a b n x^{4} e \log \left (x\right ) + \frac{2}{3} \, b^{2} d n x^{3} \log \left (c\right ) \log \left (x\right ) + \frac{2}{27} \, b^{2} d n^{2} x^{3} - \frac{1}{8} \, a b n x^{4} e - \frac{2}{9} \, b^{2} d n x^{3} \log \left (c\right ) + \frac{1}{2} \, a b x^{4} e \log \left (c\right ) + \frac{1}{3} \, b^{2} d x^{3} \log \left (c\right )^{2} + \frac{2}{3} \, a b d n x^{3} \log \left (x\right ) - \frac{2}{9} \, a b d n x^{3} + \frac{1}{4} \, a^{2} x^{4} e + \frac{2}{3} \, a b d x^{3} \log \left (c\right ) + \frac{1}{3} \, a^{2} d x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*b^2*n^2*x^4*e*log(x)^2 - 1/8*b^2*n^2*x^4*e*log(x) + 1/2*b^2*n*x^4*e*log(c)*log(x) + 1/3*b^2*d*n^2*x^3*log(
x)^2 + 1/32*b^2*n^2*x^4*e - 1/8*b^2*n*x^4*e*log(c) + 1/4*b^2*x^4*e*log(c)^2 - 2/9*b^2*d*n^2*x^3*log(x) + 1/2*a
*b*n*x^4*e*log(x) + 2/3*b^2*d*n*x^3*log(c)*log(x) + 2/27*b^2*d*n^2*x^3 - 1/8*a*b*n*x^4*e - 2/9*b^2*d*n*x^3*log
(c) + 1/2*a*b*x^4*e*log(c) + 1/3*b^2*d*x^3*log(c)^2 + 2/3*a*b*d*n*x^3*log(x) - 2/9*a*b*d*n*x^3 + 1/4*a^2*x^4*e
+ 2/3*a*b*d*x^3*log(c) + 1/3*a^2*d*x^3