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

Optimal. Leaf size=178 \[ \frac{1}{2} d^2 x^2 \left (a+b \log \left (c x^n\right )\right )^2-\frac{1}{2} b d^2 n x^2 \left (a+b \log \left (c x^n\right )\right )+\frac{2}{3} d e x^3 \left (a+b \log \left (c x^n\right )\right )^2-\frac{4}{9} b d e n x^3 \left (a+b \log \left (c x^n\right )\right )+\frac{1}{4} e^2 x^4 \left (a+b \log \left (c x^n\right )\right )^2-\frac{1}{8} b e^2 n x^4 \left (a+b \log \left (c x^n\right )\right )+\frac{1}{4} b^2 d^2 n^2 x^2+\frac{4}{27} b^2 d e n^2 x^3+\frac{1}{32} b^2 e^2 n^2 x^4 \]

[Out]

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

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Rubi [A]  time = 0.18062, antiderivative size = 178, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {2353, 2305, 2304} \[ \frac{1}{2} d^2 x^2 \left (a+b \log \left (c x^n\right )\right )^2-\frac{1}{2} b d^2 n x^2 \left (a+b \log \left (c x^n\right )\right )+\frac{2}{3} d e x^3 \left (a+b \log \left (c x^n\right )\right )^2-\frac{4}{9} b d e n x^3 \left (a+b \log \left (c x^n\right )\right )+\frac{1}{4} e^2 x^4 \left (a+b \log \left (c x^n\right )\right )^2-\frac{1}{8} b e^2 n x^4 \left (a+b \log \left (c x^n\right )\right )+\frac{1}{4} b^2 d^2 n^2 x^2+\frac{4}{27} b^2 d e n^2 x^3+\frac{1}{32} b^2 e^2 n^2 x^4 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0897678, size = 134, normalized size = 0.75 \[ \frac{1}{864} x^2 \left (432 d^2 \left (a+b \log \left (c x^n\right )\right )^2+216 b d^2 n \left (-2 a-2 b \log \left (c x^n\right )+b n\right )+576 d e x \left (a+b \log \left (c x^n\right )\right )^2+128 b d e n x \left (-3 a-3 b \log \left (c x^n\right )+b n\right )+216 e^2 x^2 \left (a+b \log \left (c x^n\right )\right )^2+27 b e^2 n x^2 \left (-4 a-4 b \log \left (c x^n\right )+b n\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x^2*(27*b*e^2*n*x^2*(-4*a + b*n - 4*b*Log[c*x^n]) + 128*b*d*e*n*x*(-3*a + b*n - 3*b*Log[c*x^n]) + 216*b*d^2*n
*(-2*a + b*n - 2*b*Log[c*x^n]) + 432*d^2*(a + b*Log[c*x^n])^2 + 576*d*e*x*(a + b*Log[c*x^n])^2 + 216*e^2*x^2*(
a + b*Log[c*x^n])^2))/864

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Maple [C]  time = 0.3, size = 2597, normalized size = 14.6 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*I*ln(c)*Pi*b^2*d*e*x^3*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+2/3*a^2*d*e*x^3-2/3*I*Pi*a*b*d*e*x^3*csgn(I*x^
n)*csgn(I*c*x^n)*csgn(I*c)+2/9*I*Pi*b^2*d*e*n*x^3*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-1/16*I*Pi*b^2*e^2*n*x^4*
csgn(I*x^n)*csgn(I*c*x^n)^2-1/16*I*Pi*b^2*e^2*n*x^4*csgn(I*c*x^n)^2*csgn(I*c)+1/4*I*Pi*a*b*e^2*x^4*csgn(I*x^n)
*csgn(I*c*x^n)^2-4/9*b*n*a*d*e*x^3+1/2*I*Pi*a*b*d^2*x^2*csgn(I*x^n)*csgn(I*c*x^n)^2-2/3*I*ln(c)*Pi*b^2*d*e*x^3
*csgn(I*c*x^n)^3+2/9*I*Pi*b^2*d*e*n*x^3*csgn(I*c*x^n)^3+1/3*Pi^2*b^2*d*e*x^3*csgn(I*x^n)^2*csgn(I*c*x^n)^3*csg
n(I*c)-1/2*b*n*a*d^2*x^2-1/8*b*n*a*e^2*x^4+1/4*a^2*e^2*x^4+1/2*a^2*d^2*x^2-1/6*Pi^2*b^2*d*e*x^3*csgn(I*x^n)^2*
csgn(I*c*x^n)^2*csgn(I*c)^2-2/3*Pi^2*b^2*d*e*x^3*csgn(I*x^n)*csgn(I*c*x^n)^4*csgn(I*c)+1/3*Pi^2*b^2*d*e*x^3*cs
gn(I*x^n)*csgn(I*c*x^n)^3*csgn(I*c)^2+1/2*I*ln(c)*Pi*b^2*d^2*x^2*csgn(I*x^n)*csgn(I*c*x^n)^2+1/4*I*Pi*a*b*e^2*
x^4*csgn(I*c*x^n)^2*csgn(I*c)-4/9*ln(c)*b^2*d*e*n*x^3+4/3*ln(c)*a*b*d*e*x^3-1/6*Pi^2*b^2*d*e*x^3*csgn(I*x^n)^2
*csgn(I*c*x^n)^4-1/4*I*ln(c)*Pi*b^2*e^2*x^4*csgn(I*c*x^n)^3+1/16*I*Pi*b^2*e^2*n*x^4*csgn(I*c*x^n)^3-1/4*I*Pi*a
*b*e^2*x^4*csgn(I*c*x^n)^3+1/72*b*(48*I*Pi*b*d*e*x^3*csgn(I*c*x^n)^2*csgn(I*c)+36*I*Pi*b*d^2*x^2*csgn(I*c*x^n)
^2*csgn(I*c)-18*I*Pi*b*e^2*x^4*csgn(I*c*x^n)^3+18*I*Pi*b*e^2*x^4*csgn(I*x^n)*csgn(I*c*x^n)^2+36*ln(c)*b*e^2*x^
4-9*b*e^2*n*x^4+36*a*e^2*x^4-36*I*Pi*b*d^2*x^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+36*I*Pi*b*d^2*x^2*csgn(I*x^
n)*csgn(I*c*x^n)^2-36*I*Pi*b*d^2*x^2*csgn(I*c*x^n)^3-48*I*Pi*b*d*e*x^3*csgn(I*c*x^n)^3+96*ln(c)*b*d*e*x^3-32*b
*d*e*n*x^3+96*a*d*e*x^3+18*I*Pi*b*e^2*x^4*csgn(I*c*x^n)^2*csgn(I*c)-18*I*Pi*b*e^2*x^4*csgn(I*x^n)*csgn(I*c*x^n
)*csgn(I*c)+48*I*Pi*b*d*e*x^3*csgn(I*x^n)*csgn(I*c*x^n)^2-48*I*Pi*b*d*e*x^3*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c
)+72*ln(c)*b*d^2*x^2-36*b*d^2*n*x^2+72*a*d^2*x^2)*ln(x^n)+1/2*I*ln(c)*Pi*b^2*d^2*x^2*csgn(I*c*x^n)^2*csgn(I*c)
-1/4*I*Pi*b^2*d^2*n*x^2*csgn(I*x^n)*csgn(I*c*x^n)^2-1/4*I*Pi*b^2*d^2*n*x^2*csgn(I*c*x^n)^2*csgn(I*c)-1/8*ln(c)
*b^2*e^2*n*x^4+1/2*ln(c)*a*b*e^2*x^4-1/2*ln(c)*b^2*d^2*n*x^2+ln(c)*a*b*d^2*x^2-2/3*I*Pi*a*b*d*e*x^3*csgn(I*c*x
^n)^3+1/2*I*Pi*a*b*d^2*x^2*csgn(I*c*x^n)^2*csgn(I*c)+1/4*I*ln(c)*Pi*b^2*e^2*x^4*csgn(I*x^n)*csgn(I*c*x^n)^2+1/
4*I*ln(c)*Pi*b^2*e^2*x^4*csgn(I*c*x^n)^2*csgn(I*c)+1/12*b^2*x^2*(3*e^2*x^2+8*d*e*x+6*d^2)*ln(x^n)^2-1/16*Pi^2*
b^2*e^2*x^4*csgn(I*c*x^n)^4*csgn(I*c)^2-1/6*Pi^2*b^2*d*e*x^3*csgn(I*c*x^n)^6+1/4*Pi^2*b^2*d^2*x^2*csgn(I*c*x^n
)^5*csgn(I*c)-1/8*Pi^2*b^2*d^2*x^2*csgn(I*c*x^n)^4*csgn(I*c)^2-1/16*Pi^2*b^2*e^2*x^4*csgn(I*x^n)^2*csgn(I*c*x^
n)^4+1/8*Pi^2*b^2*e^2*x^4*csgn(I*x^n)*csgn(I*c*x^n)^5+1/8*Pi^2*b^2*e^2*x^4*csgn(I*c*x^n)^5*csgn(I*c)-1/8*Pi^2*
b^2*d^2*x^2*csgn(I*x^n)^2*csgn(I*c*x^n)^4+1/4*Pi^2*b^2*d^2*x^2*csgn(I*x^n)*csgn(I*c*x^n)^5-2/9*I*Pi*b^2*d*e*n*
x^3*csgn(I*x^n)*csgn(I*c*x^n)^2-1/4*I*ln(c)*Pi*b^2*e^2*x^4*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+2/3*I*ln(c)*Pi*
b^2*d*e*x^3*csgn(I*x^n)*csgn(I*c*x^n)^2+2/3*ln(c)^2*b^2*d*e*x^3+1/3*Pi^2*b^2*d*e*x^3*csgn(I*x^n)*csgn(I*c*x^n)
^5+1/3*Pi^2*b^2*d*e*x^3*csgn(I*c*x^n)^5*csgn(I*c)-1/6*Pi^2*b^2*d*e*x^3*csgn(I*c*x^n)^4*csgn(I*c)^2+1/4*ln(c)^2
*b^2*e^2*x^4+1/2*ln(c)^2*b^2*d^2*x^2+1/4*b^2*d^2*n^2*x^2+1/32*b^2*e^2*n^2*x^4-1/8*Pi^2*b^2*d^2*x^2*csgn(I*c*x^
n)^6+2/3*I*ln(c)*Pi*b^2*d*e*x^3*csgn(I*c*x^n)^2*csgn(I*c)-1/2*I*ln(c)*Pi*b^2*d^2*x^2*csgn(I*x^n)*csgn(I*c*x^n)
*csgn(I*c)-1/4*I*Pi*a*b*e^2*x^4*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-2/9*I*Pi*b^2*d*e*n*x^3*csgn(I*c*x^n)^2*csg
n(I*c)+2/3*I*Pi*a*b*d*e*x^3*csgn(I*x^n)*csgn(I*c*x^n)^2+2/3*I*Pi*a*b*d*e*x^3*csgn(I*c*x^n)^2*csgn(I*c)+1/4*I*P
i*b^2*d^2*n*x^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-1/2*I*Pi*a*b*d^2*x^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+1
/16*I*Pi*b^2*e^2*n*x^4*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-1/4*Pi^2*b^2*e^2*x^4*csgn(I*x^n)*csgn(I*c*x^n)^4*cs
gn(I*c)+1/4*Pi^2*b^2*d^2*x^2*csgn(I*x^n)^2*csgn(I*c*x^n)^3*csgn(I*c)-1/8*Pi^2*b^2*d^2*x^2*csgn(I*x^n)^2*csgn(I
*c*x^n)^2*csgn(I*c)^2-1/2*Pi^2*b^2*d^2*x^2*csgn(I*x^n)*csgn(I*c*x^n)^4*csgn(I*c)+1/4*Pi^2*b^2*d^2*x^2*csgn(I*x
^n)*csgn(I*c*x^n)^3*csgn(I*c)^2+4/27*b^2*d*e*n^2*x^3-1/2*I*ln(c)*Pi*b^2*d^2*x^2*csgn(I*c*x^n)^3+1/4*I*Pi*b^2*d
^2*n*x^2*csgn(I*c*x^n)^3-1/2*I*Pi*a*b*d^2*x^2*csgn(I*c*x^n)^3+1/8*Pi^2*b^2*e^2*x^4*csgn(I*x^n)*csgn(I*c*x^n)^3
*csgn(I*c)^2+1/8*Pi^2*b^2*e^2*x^4*csgn(I*x^n)^2*csgn(I*c*x^n)^3*csgn(I*c)-1/16*Pi^2*b^2*e^2*x^4*csgn(I*x^n)^2*
csgn(I*c*x^n)^2*csgn(I*c)^2-1/16*Pi^2*b^2*e^2*x^4*csgn(I*c*x^n)^6

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Maxima [A]  time = 1.06772, size = 338, normalized size = 1.9 \begin{align*} \frac{1}{4} \, b^{2} e^{2} x^{4} \log \left (c x^{n}\right )^{2} - \frac{1}{8} \, a b e^{2} n x^{4} + \frac{1}{2} \, a b e^{2} x^{4} \log \left (c x^{n}\right ) + \frac{2}{3} \, b^{2} d e x^{3} \log \left (c x^{n}\right )^{2} - \frac{4}{9} \, a b d e n x^{3} + \frac{1}{4} \, a^{2} e^{2} x^{4} + \frac{4}{3} \, a b d e x^{3} \log \left (c x^{n}\right ) + \frac{1}{2} \, b^{2} d^{2} x^{2} \log \left (c x^{n}\right )^{2} - \frac{1}{2} \, a b d^{2} n x^{2} + \frac{2}{3} \, a^{2} d e x^{3} + a b d^{2} x^{2} \log \left (c x^{n}\right ) + \frac{1}{2} \, a^{2} d^{2} x^{2} + \frac{1}{4} \,{\left (n^{2} x^{2} - 2 \, n x^{2} \log \left (c x^{n}\right )\right )} b^{2} d^{2} + \frac{4}{27} \,{\left (n^{2} x^{3} - 3 \, n x^{3} \log \left (c x^{n}\right )\right )} b^{2} d e + \frac{1}{32} \,{\left (n^{2} x^{4} - 4 \, n x^{4} \log \left (c x^{n}\right )\right )} b^{2} e^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 1.03039, size = 803, normalized size = 4.51 \begin{align*} \frac{1}{32} \,{\left (b^{2} e^{2} n^{2} - 4 \, a b e^{2} n + 8 \, a^{2} e^{2}\right )} x^{4} + \frac{2}{27} \,{\left (2 \, b^{2} d e n^{2} - 6 \, a b d e n + 9 \, a^{2} d e\right )} x^{3} + \frac{1}{4} \,{\left (b^{2} d^{2} n^{2} - 2 \, a b d^{2} n + 2 \, a^{2} d^{2}\right )} x^{2} + \frac{1}{12} \,{\left (3 \, b^{2} e^{2} x^{4} + 8 \, b^{2} d e x^{3} + 6 \, b^{2} d^{2} x^{2}\right )} \log \left (c\right )^{2} + \frac{1}{12} \,{\left (3 \, b^{2} e^{2} n^{2} x^{4} + 8 \, b^{2} d e n^{2} x^{3} + 6 \, b^{2} d^{2} n^{2} x^{2}\right )} \log \left (x\right )^{2} - \frac{1}{72} \,{\left (9 \,{\left (b^{2} e^{2} n - 4 \, a b e^{2}\right )} x^{4} + 32 \,{\left (b^{2} d e n - 3 \, a b d e\right )} x^{3} + 36 \,{\left (b^{2} d^{2} n - 2 \, a b d^{2}\right )} x^{2}\right )} \log \left (c\right ) - \frac{1}{72} \,{\left (9 \,{\left (b^{2} e^{2} n^{2} - 4 \, a b e^{2} n\right )} x^{4} + 32 \,{\left (b^{2} d e n^{2} - 3 \, a b d e n\right )} x^{3} + 36 \,{\left (b^{2} d^{2} n^{2} - 2 \, a b d^{2} n\right )} x^{2} - 12 \,{\left (3 \, b^{2} e^{2} n x^{4} + 8 \, b^{2} d e n x^{3} + 6 \, b^{2} d^{2} n x^{2}\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*(e*x+d)^2*(a+b*log(c*x^n))^2,x, algorithm="fricas")

[Out]

1/32*(b^2*e^2*n^2 - 4*a*b*e^2*n + 8*a^2*e^2)*x^4 + 2/27*(2*b^2*d*e*n^2 - 6*a*b*d*e*n + 9*a^2*d*e)*x^3 + 1/4*(b
^2*d^2*n^2 - 2*a*b*d^2*n + 2*a^2*d^2)*x^2 + 1/12*(3*b^2*e^2*x^4 + 8*b^2*d*e*x^3 + 6*b^2*d^2*x^2)*log(c)^2 + 1/
12*(3*b^2*e^2*n^2*x^4 + 8*b^2*d*e*n^2*x^3 + 6*b^2*d^2*n^2*x^2)*log(x)^2 - 1/72*(9*(b^2*e^2*n - 4*a*b*e^2)*x^4
+ 32*(b^2*d*e*n - 3*a*b*d*e)*x^3 + 36*(b^2*d^2*n - 2*a*b*d^2)*x^2)*log(c) - 1/72*(9*(b^2*e^2*n^2 - 4*a*b*e^2*n
)*x^4 + 32*(b^2*d*e*n^2 - 3*a*b*d*e*n)*x^3 + 36*(b^2*d^2*n^2 - 2*a*b*d^2*n)*x^2 - 12*(3*b^2*e^2*n*x^4 + 8*b^2*
d*e*n*x^3 + 6*b^2*d^2*n*x^2)*log(c))*log(x)

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Sympy [B]  time = 5.7159, size = 510, normalized size = 2.87 \begin{align*} \frac{a^{2} d^{2} x^{2}}{2} + \frac{2 a^{2} d e x^{3}}{3} + \frac{a^{2} e^{2} x^{4}}{4} + a b d^{2} n x^{2} \log{\left (x \right )} - \frac{a b d^{2} n x^{2}}{2} + a b d^{2} x^{2} \log{\left (c \right )} + \frac{4 a b d e n x^{3} \log{\left (x \right )}}{3} - \frac{4 a b d e n x^{3}}{9} + \frac{4 a b d e x^{3} \log{\left (c \right )}}{3} + \frac{a b e^{2} n x^{4} \log{\left (x \right )}}{2} - \frac{a b e^{2} n x^{4}}{8} + \frac{a b e^{2} x^{4} \log{\left (c \right )}}{2} + \frac{b^{2} d^{2} n^{2} x^{2} \log{\left (x \right )}^{2}}{2} - \frac{b^{2} d^{2} n^{2} x^{2} \log{\left (x \right )}}{2} + \frac{b^{2} d^{2} n^{2} x^{2}}{4} + b^{2} d^{2} n x^{2} \log{\left (c \right )} \log{\left (x \right )} - \frac{b^{2} d^{2} n x^{2} \log{\left (c \right )}}{2} + \frac{b^{2} d^{2} x^{2} \log{\left (c \right )}^{2}}{2} + \frac{2 b^{2} d e n^{2} x^{3} \log{\left (x \right )}^{2}}{3} - \frac{4 b^{2} d e n^{2} x^{3} \log{\left (x \right )}}{9} + \frac{4 b^{2} d e n^{2} x^{3}}{27} + \frac{4 b^{2} d e n x^{3} \log{\left (c \right )} \log{\left (x \right )}}{3} - \frac{4 b^{2} d e n x^{3} \log{\left (c \right )}}{9} + \frac{2 b^{2} d e x^{3} \log{\left (c \right )}^{2}}{3} + \frac{b^{2} e^{2} n^{2} x^{4} \log{\left (x \right )}^{2}}{4} - \frac{b^{2} e^{2} n^{2} x^{4} \log{\left (x \right )}}{8} + \frac{b^{2} e^{2} n^{2} x^{4}}{32} + \frac{b^{2} e^{2} n x^{4} \log{\left (c \right )} \log{\left (x \right )}}{2} - \frac{b^{2} e^{2} n x^{4} \log{\left (c \right )}}{8} + \frac{b^{2} e^{2} x^{4} \log{\left (c \right )}^{2}}{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.27532, size = 551, normalized size = 3.1 \begin{align*} \frac{1}{4} \, b^{2} n^{2} x^{4} e^{2} \log \left (x\right )^{2} + \frac{2}{3} \, b^{2} d n^{2} x^{3} e \log \left (x\right )^{2} - \frac{1}{8} \, b^{2} n^{2} x^{4} e^{2} \log \left (x\right ) - \frac{4}{9} \, b^{2} d n^{2} x^{3} e \log \left (x\right ) + \frac{1}{2} \, b^{2} n x^{4} e^{2} \log \left (c\right ) \log \left (x\right ) + \frac{4}{3} \, b^{2} d n x^{3} e \log \left (c\right ) \log \left (x\right ) + \frac{1}{2} \, b^{2} d^{2} n^{2} x^{2} \log \left (x\right )^{2} + \frac{1}{32} \, b^{2} n^{2} x^{4} e^{2} + \frac{4}{27} \, b^{2} d n^{2} x^{3} e - \frac{1}{8} \, b^{2} n x^{4} e^{2} \log \left (c\right ) - \frac{4}{9} \, b^{2} d n x^{3} e \log \left (c\right ) + \frac{1}{4} \, b^{2} x^{4} e^{2} \log \left (c\right )^{2} + \frac{2}{3} \, b^{2} d x^{3} e \log \left (c\right )^{2} - \frac{1}{2} \, b^{2} d^{2} n^{2} x^{2} \log \left (x\right ) + \frac{1}{2} \, a b n x^{4} e^{2} \log \left (x\right ) + \frac{4}{3} \, a b d n x^{3} e \log \left (x\right ) + b^{2} d^{2} n x^{2} \log \left (c\right ) \log \left (x\right ) + \frac{1}{4} \, b^{2} d^{2} n^{2} x^{2} - \frac{1}{8} \, a b n x^{4} e^{2} - \frac{4}{9} \, a b d n x^{3} e - \frac{1}{2} \, b^{2} d^{2} n x^{2} \log \left (c\right ) + \frac{1}{2} \, a b x^{4} e^{2} \log \left (c\right ) + \frac{4}{3} \, a b d x^{3} e \log \left (c\right ) + \frac{1}{2} \, b^{2} d^{2} x^{2} \log \left (c\right )^{2} + a b d^{2} n x^{2} \log \left (x\right ) - \frac{1}{2} \, a b d^{2} n x^{2} + \frac{1}{4} \, a^{2} x^{4} e^{2} + \frac{2}{3} \, a^{2} d x^{3} e + a b d^{2} x^{2} \log \left (c\right ) + \frac{1}{2} \, a^{2} d^{2} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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