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

Optimal. Leaf size=178 \[ \frac{1}{3} d^2 x^3 \left (a+b \log \left (c x^n\right )\right )^2-\frac{2}{9} b d^2 n x^3 \left (a+b \log \left (c x^n\right )\right )+\frac{1}{2} d e x^4 \left (a+b \log \left (c x^n\right )\right )^2-\frac{1}{4} b d e n x^4 \left (a+b \log \left (c x^n\right )\right )+\frac{1}{5} e^2 x^5 \left (a+b \log \left (c x^n\right )\right )^2-\frac{2}{25} b e^2 n x^5 \left (a+b \log \left (c x^n\right )\right )+\frac{2}{27} b^2 d^2 n^2 x^3+\frac{1}{16} b^2 d e n^2 x^4+\frac{2}{125} b^2 e^2 n^2 x^5 \]

[Out]

(2*b^2*d^2*n^2*x^3)/27 + (b^2*d*e*n^2*x^4)/16 + (2*b^2*e^2*n^2*x^5)/125 - (2*b*d^2*n*x^3*(a + b*Log[c*x^n]))/9
- (b*d*e*n*x^4*(a + b*Log[c*x^n]))/4 - (2*b*e^2*n*x^5*(a + b*Log[c*x^n]))/25 + (d^2*x^3*(a + b*Log[c*x^n])^2)
/3 + (d*e*x^4*(a + b*Log[c*x^n])^2)/2 + (e^2*x^5*(a + b*Log[c*x^n])^2)/5

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Rubi [A]  time = 0.218305, antiderivative size = 178, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {2353, 2305, 2304} \[ \frac{1}{3} d^2 x^3 \left (a+b \log \left (c x^n\right )\right )^2-\frac{2}{9} b d^2 n x^3 \left (a+b \log \left (c x^n\right )\right )+\frac{1}{2} d e x^4 \left (a+b \log \left (c x^n\right )\right )^2-\frac{1}{4} b d e n x^4 \left (a+b \log \left (c x^n\right )\right )+\frac{1}{5} e^2 x^5 \left (a+b \log \left (c x^n\right )\right )^2-\frac{2}{25} b e^2 n x^5 \left (a+b \log \left (c x^n\right )\right )+\frac{2}{27} b^2 d^2 n^2 x^3+\frac{1}{16} b^2 d e n^2 x^4+\frac{2}{125} b^2 e^2 n^2 x^5 \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*b^2*d^2*n^2*x^3)/27 + (b^2*d*e*n^2*x^4)/16 + (2*b^2*e^2*n^2*x^5)/125 - (2*b*d^2*n*x^3*(a + b*Log[c*x^n]))/9
- (b*d*e*n*x^4*(a + b*Log[c*x^n]))/4 - (2*b*e^2*n*x^5*(a + b*Log[c*x^n]))/25 + (d^2*x^3*(a + b*Log[c*x^n])^2)
/3 + (d*e*x^4*(a + b*Log[c*x^n])^2)/2 + (e^2*x^5*(a + b*Log[c*x^n])^2)/5

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)^2 \left (a+b \log \left (c x^n\right )\right )^2 \, dx &=\int \left (d^2 x^2 \left (a+b \log \left (c x^n\right )\right )^2+2 d e x^3 \left (a+b \log \left (c x^n\right )\right )^2+e^2 x^4 \left (a+b \log \left (c x^n\right )\right )^2\right ) \, dx\\ &=d^2 \int x^2 \left (a+b \log \left (c x^n\right )\right )^2 \, dx+(2 d e) \int x^3 \left (a+b \log \left (c x^n\right )\right )^2 \, dx+e^2 \int x^4 \left (a+b \log \left (c x^n\right )\right )^2 \, dx\\ &=\frac{1}{3} d^2 x^3 \left (a+b \log \left (c x^n\right )\right )^2+\frac{1}{2} d e x^4 \left (a+b \log \left (c x^n\right )\right )^2+\frac{1}{5} e^2 x^5 \left (a+b \log \left (c x^n\right )\right )^2-\frac{1}{3} \left (2 b d^2 n\right ) \int x^2 \left (a+b \log \left (c x^n\right )\right ) \, dx-(b d e n) \int x^3 \left (a+b \log \left (c x^n\right )\right ) \, dx-\frac{1}{5} \left (2 b e^2 n\right ) \int x^4 \left (a+b \log \left (c x^n\right )\right ) \, dx\\ &=\frac{2}{27} b^2 d^2 n^2 x^3+\frac{1}{16} b^2 d e n^2 x^4+\frac{2}{125} b^2 e^2 n^2 x^5-\frac{2}{9} b d^2 n x^3 \left (a+b \log \left (c x^n\right )\right )-\frac{1}{4} b d e n x^4 \left (a+b \log \left (c x^n\right )\right )-\frac{2}{25} b e^2 n x^5 \left (a+b \log \left (c x^n\right )\right )+\frac{1}{3} d^2 x^3 \left (a+b \log \left (c x^n\right )\right )^2+\frac{1}{2} d e x^4 \left (a+b \log \left (c x^n\right )\right )^2+\frac{1}{5} e^2 x^5 \left (a+b \log \left (c x^n\right )\right )^2\\ \end{align*}

Mathematica [A]  time = 0.0863446, size = 149, normalized size = 0.84 \[ \frac{1}{3} d^2 x^3 \left (a+b \log \left (c x^n\right )\right )^2+\frac{2}{27} b d^2 n x^3 \left (-3 a-3 b \log \left (c x^n\right )+b n\right )+\frac{1}{2} d e x^4 \left (a+b \log \left (c x^n\right )\right )^2+\frac{1}{16} b d e n x^4 \left (-4 a-4 b \log \left (c x^n\right )+b n\right )+\frac{1}{5} e^2 x^5 \left (a+b \log \left (c x^n\right )\right )^2+\frac{2}{125} b e^2 n x^5 \left (-5 a-5 b \log \left (c x^n\right )+b n\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*b*e^2*n*x^5*(-5*a + b*n - 5*b*Log[c*x^n]))/125 + (b*d*e*n*x^4*(-4*a + b*n - 4*b*Log[c*x^n]))/16 + (2*b*d^2*
n*x^3*(-3*a + b*n - 3*b*Log[c*x^n]))/27 + (d^2*x^3*(a + b*Log[c*x^n])^2)/3 + (d*e*x^4*(a + b*Log[c*x^n])^2)/2
+ (e^2*x^5*(a + b*Log[c*x^n])^2)/5

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Maple [C]  time = 0.306, 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^2*(e*x+d)^2*(a+b*ln(c*x^n))^2,x)

[Out]

-1/2*I*ln(c)*Pi*b^2*d*e*x^4*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+1/5*a^2*e^2*x^5-1/8*I*Pi*b^2*d*e*n*x^4*csgn(I*
c*x^n)^2*csgn(I*c)+1/2*I*Pi*a*b*d*e*x^4*csgn(I*x^n)*csgn(I*c*x^n)^2+1/2*I*Pi*a*b*d*e*x^4*csgn(I*c*x^n)^2*csgn(
I*c)-1/3*I*ln(c)*Pi*b^2*d^2*x^3*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-2/25*a*b*e^2*n*x^5-1/20*Pi^2*b^2*e^2*x^5*c
sgn(I*x^n)^2*csgn(I*c*x^n)^2*csgn(I*c)^2-1/5*Pi^2*b^2*e^2*x^5*csgn(I*x^n)*csgn(I*c*x^n)^4*csgn(I*c)+1/10*Pi^2*
b^2*e^2*x^5*csgn(I*x^n)*csgn(I*c*x^n)^3*csgn(I*c)^2-1/5*I*ln(c)*Pi*b^2*e^2*x^5*csgn(I*c*x^n)^3+1/25*I*Pi*b^2*e
^2*n*x^5*csgn(I*c*x^n)^3-1/5*I*Pi*a*b*e^2*x^5*csgn(I*c*x^n)^3-1/8*Pi^2*b^2*d*e*x^4*csgn(I*x^n)^2*csgn(I*c*x^n)
^4-1/2*I*ln(c)*Pi*b^2*d*e*x^4*csgn(I*c*x^n)^3+1/8*I*Pi*b^2*d*e*n*x^4*csgn(I*c*x^n)^3-1/2*I*Pi*a*b*d*e*x^4*csgn
(I*c*x^n)^3-1/8*Pi^2*b^2*d*e*x^4*csgn(I*x^n)^2*csgn(I*c*x^n)^2*csgn(I*c)^2-1/2*Pi^2*b^2*d*e*x^4*csgn(I*x^n)*cs
gn(I*c*x^n)^4*csgn(I*c)+1/4*Pi^2*b^2*d*e*x^4*csgn(I*x^n)*csgn(I*c*x^n)^3*csgn(I*c)^2+1/5*I*Pi*a*b*e^2*x^5*csgn
(I*x^n)*csgn(I*c*x^n)^2-1/9*I*Pi*b^2*d^2*n*x^3*csgn(I*x^n)*csgn(I*c*x^n)^2-1/9*I*Pi*b^2*d^2*n*x^3*csgn(I*c*x^n
)^2*csgn(I*c)+1/3*I*Pi*a*b*d^2*x^3*csgn(I*x^n)*csgn(I*c*x^n)^2+1/5*I*Pi*a*b*e^2*x^5*csgn(I*c*x^n)^2*csgn(I*c)+
1/5*I*ln(c)*Pi*b^2*e^2*x^5*csgn(I*x^n)*csgn(I*c*x^n)^2+1/3*I*ln(c)*Pi*b^2*d^2*x^3*csgn(I*x^n)*csgn(I*c*x^n)^2+
1/3*I*ln(c)*Pi*b^2*d^2*x^3*csgn(I*c*x^n)^2*csgn(I*c)+1/5*I*ln(c)*Pi*b^2*e^2*x^5*csgn(I*c*x^n)^2*csgn(I*c)-1/25
*I*Pi*b^2*e^2*n*x^5*csgn(I*x^n)*csgn(I*c*x^n)^2-1/25*I*Pi*b^2*e^2*n*x^5*csgn(I*c*x^n)^2*csgn(I*c)+1/4*Pi^2*b^2
*d*e*x^4*csgn(I*x^n)^2*csgn(I*c*x^n)^3*csgn(I*c)+1/2*a^2*d*e*x^4-1/4*ln(c)*b^2*d*e*n*x^4+ln(c)*a*b*d*e*x^4-1/4
*a*b*d*e*n*x^4+1/3*a^2*d^2*x^3+1/4*Pi^2*b^2*d*e*x^4*csgn(I*x^n)*csgn(I*c*x^n)^5+1/4*Pi^2*b^2*d*e*x^4*csgn(I*c*
x^n)^5*csgn(I*c)+1/3*I*Pi*a*b*d^2*x^3*csgn(I*c*x^n)^2*csgn(I*c)+1/900*b*(-300*I*Pi*b*d^2*x^3*csgn(I*c*x^n)^3+4
50*I*Pi*b*d*e*x^4*csgn(I*x^n)*csgn(I*c*x^n)^2+180*I*Pi*b*e^2*x^5*csgn(I*c*x^n)^2*csgn(I*c)-180*I*Pi*b*e^2*x^5*
csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+360*ln(c)*b*e^2*x^5-72*b*e^2*n*x^5+360*a*e^2*x^5+180*I*Pi*b*e^2*x^5*csgn(I
*x^n)*csgn(I*c*x^n)^2-450*I*Pi*b*d*e*x^4*csgn(I*c*x^n)^3-450*I*Pi*b*d*e*x^4*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c
)-300*I*Pi*b*d^2*x^3*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+900*ln(c)*b*d*e*x^4-225*b*d*e*n*x^4+900*a*d*e*x^4-180
*I*Pi*b*e^2*x^5*csgn(I*c*x^n)^3+300*I*Pi*b*d^2*x^3*csgn(I*c*x^n)^2*csgn(I*c)+300*I*Pi*b*d^2*x^3*csgn(I*x^n)*cs
gn(I*c*x^n)^2+450*I*Pi*b*d*e*x^4*csgn(I*c*x^n)^2*csgn(I*c)+600*ln(c)*b*d^2*x^3-200*b*d^2*n*x^3+600*a*d^2*x^3)*
ln(x^n)+1/25*I*Pi*b^2*e^2*n*x^5*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-1/5*I*Pi*a*b*e^2*x^5*csgn(I*x^n)*csgn(I*c*
x^n)*csgn(I*c)+1/2*I*ln(c)*Pi*b^2*d*e*x^4*csgn(I*x^n)*csgn(I*c*x^n)^2+1/3*ln(c)^2*b^2*d^2*x^3+1/5*ln(c)^2*b^2*
e^2*x^5-1/20*Pi^2*b^2*e^2*x^5*csgn(I*c*x^n)^4*csgn(I*c)^2-1/8*Pi^2*b^2*d*e*x^4*csgn(I*c*x^n)^6-1/12*Pi^2*b^2*d
^2*x^3*csgn(I*x^n)^2*csgn(I*c*x^n)^4+1/6*Pi^2*b^2*d^2*x^3*csgn(I*x^n)*csgn(I*c*x^n)^5+1/6*Pi^2*b^2*d^2*x^3*csg
n(I*c*x^n)^5*csgn(I*c)-1/12*Pi^2*b^2*d^2*x^3*csgn(I*c*x^n)^4*csgn(I*c)^2-1/20*Pi^2*b^2*e^2*x^5*csgn(I*x^n)^2*c
sgn(I*c*x^n)^4+1/10*Pi^2*b^2*e^2*x^5*csgn(I*x^n)*csgn(I*c*x^n)^5+1/10*Pi^2*b^2*e^2*x^5*csgn(I*c*x^n)^5*csgn(I*
c)+1/30*b^2*x^3*(6*e^2*x^2+15*d*e*x+10*d^2)*ln(x^n)^2+1/2*I*ln(c)*Pi*b^2*d*e*x^4*csgn(I*c*x^n)^2*csgn(I*c)-1/8
*I*Pi*b^2*d*e*n*x^4*csgn(I*x^n)*csgn(I*c*x^n)^2+2/5*ln(c)*a*b*e^2*x^5-2/25*ln(c)*b^2*e^2*n*x^5+2/27*b^2*d^2*n^
2*x^3+2/125*b^2*e^2*n^2*x^5-1/20*Pi^2*b^2*e^2*x^5*csgn(I*c*x^n)^6-1/12*Pi^2*b^2*d^2*x^3*csgn(I*c*x^n)^6+1/9*I*
Pi*b^2*d^2*n*x^3*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-1/3*I*Pi*a*b*d^2*x^3*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-
1/5*I*ln(c)*Pi*b^2*e^2*x^5*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-1/8*Pi^2*b^2*d*e*x^4*csgn(I*c*x^n)^4*csgn(I*c)^
2-2/9*ln(c)*b^2*d^2*n*x^3+2/3*ln(c)*a*b*d^2*x^3+1/2*ln(c)^2*b^2*d*e*x^4-2/9*a*b*d^2*n*x^3-1/3*I*ln(c)*Pi*b^2*d
^2*x^3*csgn(I*c*x^n)^3+1/9*I*Pi*b^2*d^2*n*x^3*csgn(I*c*x^n)^3-1/3*I*Pi*a*b*d^2*x^3*csgn(I*c*x^n)^3+1/16*b^2*d*
e*n^2*x^4+1/8*I*Pi*b^2*d*e*n*x^4*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-1/2*I*Pi*a*b*d*e*x^4*csgn(I*x^n)*csgn(I*c
*x^n)*csgn(I*c)+1/6*Pi^2*b^2*d^2*x^3*csgn(I*x^n)^2*csgn(I*c*x^n)^3*csgn(I*c)-1/12*Pi^2*b^2*d^2*x^3*csgn(I*x^n)
^2*csgn(I*c*x^n)^2*csgn(I*c)^2-1/3*Pi^2*b^2*d^2*x^3*csgn(I*x^n)*csgn(I*c*x^n)^4*csgn(I*c)+1/6*Pi^2*b^2*d^2*x^3
*csgn(I*x^n)*csgn(I*c*x^n)^3*csgn(I*c)^2+1/10*Pi^2*b^2*e^2*x^5*csgn(I*x^n)^2*csgn(I*c*x^n)^3*csgn(I*c)

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Maxima [A]  time = 1.23205, size = 338, normalized size = 1.9 \begin{align*} \frac{1}{5} \, b^{2} e^{2} x^{5} \log \left (c x^{n}\right )^{2} - \frac{2}{25} \, a b e^{2} n x^{5} + \frac{2}{5} \, a b e^{2} x^{5} \log \left (c x^{n}\right ) + \frac{1}{2} \, b^{2} d e x^{4} \log \left (c x^{n}\right )^{2} - \frac{1}{4} \, a b d e n x^{4} + \frac{1}{5} \, a^{2} e^{2} x^{5} + a b d e x^{4} \log \left (c x^{n}\right ) + \frac{1}{3} \, b^{2} d^{2} x^{3} \log \left (c x^{n}\right )^{2} - \frac{2}{9} \, a b d^{2} n x^{3} + \frac{1}{2} \, a^{2} d e x^{4} + \frac{2}{3} \, a b d^{2} x^{3} \log \left (c x^{n}\right ) + \frac{1}{3} \, a^{2} d^{2} x^{3} + \frac{2}{27} \,{\left (n^{2} x^{3} - 3 \, n x^{3} \log \left (c x^{n}\right )\right )} b^{2} d^{2} + \frac{1}{16} \,{\left (n^{2} x^{4} - 4 \, n x^{4} \log \left (c x^{n}\right )\right )} b^{2} d e + \frac{2}{125} \,{\left (n^{2} x^{5} - 5 \, n x^{5} \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^2*(e*x+d)^2*(a+b*log(c*x^n))^2,x, algorithm="maxima")

[Out]

1/5*b^2*e^2*x^5*log(c*x^n)^2 - 2/25*a*b*e^2*n*x^5 + 2/5*a*b*e^2*x^5*log(c*x^n) + 1/2*b^2*d*e*x^4*log(c*x^n)^2
- 1/4*a*b*d*e*n*x^4 + 1/5*a^2*e^2*x^5 + a*b*d*e*x^4*log(c*x^n) + 1/3*b^2*d^2*x^3*log(c*x^n)^2 - 2/9*a*b*d^2*n*
x^3 + 1/2*a^2*d*e*x^4 + 2/3*a*b*d^2*x^3*log(c*x^n) + 1/3*a^2*d^2*x^3 + 2/27*(n^2*x^3 - 3*n*x^3*log(c*x^n))*b^2
*d^2 + 1/16*(n^2*x^4 - 4*n*x^4*log(c*x^n))*b^2*d*e + 2/125*(n^2*x^5 - 5*n*x^5*log(c*x^n))*b^2*e^2

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Fricas [B]  time = 1.05749, size = 830, normalized size = 4.66 \begin{align*} \frac{1}{125} \,{\left (2 \, b^{2} e^{2} n^{2} - 10 \, a b e^{2} n + 25 \, a^{2} e^{2}\right )} x^{5} + \frac{1}{16} \,{\left (b^{2} d e n^{2} - 4 \, a b d e n + 8 \, a^{2} d e\right )} x^{4} + \frac{1}{27} \,{\left (2 \, b^{2} d^{2} n^{2} - 6 \, a b d^{2} n + 9 \, a^{2} d^{2}\right )} x^{3} + \frac{1}{30} \,{\left (6 \, b^{2} e^{2} x^{5} + 15 \, b^{2} d e x^{4} + 10 \, b^{2} d^{2} x^{3}\right )} \log \left (c\right )^{2} + \frac{1}{30} \,{\left (6 \, b^{2} e^{2} n^{2} x^{5} + 15 \, b^{2} d e n^{2} x^{4} + 10 \, b^{2} d^{2} n^{2} x^{3}\right )} \log \left (x\right )^{2} - \frac{1}{900} \,{\left (72 \,{\left (b^{2} e^{2} n - 5 \, a b e^{2}\right )} x^{5} + 225 \,{\left (b^{2} d e n - 4 \, a b d e\right )} x^{4} + 200 \,{\left (b^{2} d^{2} n - 3 \, a b d^{2}\right )} x^{3}\right )} \log \left (c\right ) - \frac{1}{900} \,{\left (72 \,{\left (b^{2} e^{2} n^{2} - 5 \, a b e^{2} n\right )} x^{5} + 225 \,{\left (b^{2} d e n^{2} - 4 \, a b d e n\right )} x^{4} + 200 \,{\left (b^{2} d^{2} n^{2} - 3 \, a b d^{2} n\right )} x^{3} - 60 \,{\left (6 \, b^{2} e^{2} n x^{5} + 15 \, b^{2} d e n x^{4} + 10 \, b^{2} d^{2} 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)^2*(a+b*log(c*x^n))^2,x, algorithm="fricas")

[Out]

1/125*(2*b^2*e^2*n^2 - 10*a*b*e^2*n + 25*a^2*e^2)*x^5 + 1/16*(b^2*d*e*n^2 - 4*a*b*d*e*n + 8*a^2*d*e)*x^4 + 1/2
7*(2*b^2*d^2*n^2 - 6*a*b*d^2*n + 9*a^2*d^2)*x^3 + 1/30*(6*b^2*e^2*x^5 + 15*b^2*d*e*x^4 + 10*b^2*d^2*x^3)*log(c
)^2 + 1/30*(6*b^2*e^2*n^2*x^5 + 15*b^2*d*e*n^2*x^4 + 10*b^2*d^2*n^2*x^3)*log(x)^2 - 1/900*(72*(b^2*e^2*n - 5*a
*b*e^2)*x^5 + 225*(b^2*d*e*n - 4*a*b*d*e)*x^4 + 200*(b^2*d^2*n - 3*a*b*d^2)*x^3)*log(c) - 1/900*(72*(b^2*e^2*n
^2 - 5*a*b*e^2*n)*x^5 + 225*(b^2*d*e*n^2 - 4*a*b*d*e*n)*x^4 + 200*(b^2*d^2*n^2 - 3*a*b*d^2*n)*x^3 - 60*(6*b^2*
e^2*n*x^5 + 15*b^2*d*e*n*x^4 + 10*b^2*d^2*n*x^3)*log(c))*log(x)

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Sympy [B]  time = 7.05204, size = 517, normalized size = 2.9 \begin{align*} \frac{a^{2} d^{2} x^{3}}{3} + \frac{a^{2} d e x^{4}}{2} + \frac{a^{2} e^{2} x^{5}}{5} + \frac{2 a b d^{2} n x^{3} \log{\left (x \right )}}{3} - \frac{2 a b d^{2} n x^{3}}{9} + \frac{2 a b d^{2} x^{3} \log{\left (c \right )}}{3} + a b d e n x^{4} \log{\left (x \right )} - \frac{a b d e n x^{4}}{4} + a b d e x^{4} \log{\left (c \right )} + \frac{2 a b e^{2} n x^{5} \log{\left (x \right )}}{5} - \frac{2 a b e^{2} n x^{5}}{25} + \frac{2 a b e^{2} x^{5} \log{\left (c \right )}}{5} + \frac{b^{2} d^{2} n^{2} x^{3} \log{\left (x \right )}^{2}}{3} - \frac{2 b^{2} d^{2} n^{2} x^{3} \log{\left (x \right )}}{9} + \frac{2 b^{2} d^{2} n^{2} x^{3}}{27} + \frac{2 b^{2} d^{2} n x^{3} \log{\left (c \right )} \log{\left (x \right )}}{3} - \frac{2 b^{2} d^{2} n x^{3} \log{\left (c \right )}}{9} + \frac{b^{2} d^{2} x^{3} \log{\left (c \right )}^{2}}{3} + \frac{b^{2} d e n^{2} x^{4} \log{\left (x \right )}^{2}}{2} - \frac{b^{2} d e n^{2} x^{4} \log{\left (x \right )}}{4} + \frac{b^{2} d e n^{2} x^{4}}{16} + b^{2} d e n x^{4} \log{\left (c \right )} \log{\left (x \right )} - \frac{b^{2} d e n x^{4} \log{\left (c \right )}}{4} + \frac{b^{2} d e x^{4} \log{\left (c \right )}^{2}}{2} + \frac{b^{2} e^{2} n^{2} x^{5} \log{\left (x \right )}^{2}}{5} - \frac{2 b^{2} e^{2} n^{2} x^{5} \log{\left (x \right )}}{25} + \frac{2 b^{2} e^{2} n^{2} x^{5}}{125} + \frac{2 b^{2} e^{2} n x^{5} \log{\left (c \right )} \log{\left (x \right )}}{5} - \frac{2 b^{2} e^{2} n x^{5} \log{\left (c \right )}}{25} + \frac{b^{2} e^{2} x^{5} \log{\left (c \right )}^{2}}{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**2*d**2*x**3/3 + a**2*d*e*x**4/2 + a**2*e**2*x**5/5 + 2*a*b*d**2*n*x**3*log(x)/3 - 2*a*b*d**2*n*x**3/9 + 2*a
*b*d**2*x**3*log(c)/3 + a*b*d*e*n*x**4*log(x) - a*b*d*e*n*x**4/4 + a*b*d*e*x**4*log(c) + 2*a*b*e**2*n*x**5*log
(x)/5 - 2*a*b*e**2*n*x**5/25 + 2*a*b*e**2*x**5*log(c)/5 + b**2*d**2*n**2*x**3*log(x)**2/3 - 2*b**2*d**2*n**2*x
**3*log(x)/9 + 2*b**2*d**2*n**2*x**3/27 + 2*b**2*d**2*n*x**3*log(c)*log(x)/3 - 2*b**2*d**2*n*x**3*log(c)/9 + b
**2*d**2*x**3*log(c)**2/3 + b**2*d*e*n**2*x**4*log(x)**2/2 - b**2*d*e*n**2*x**4*log(x)/4 + b**2*d*e*n**2*x**4/
16 + b**2*d*e*n*x**4*log(c)*log(x) - b**2*d*e*n*x**4*log(c)/4 + b**2*d*e*x**4*log(c)**2/2 + b**2*e**2*n**2*x**
5*log(x)**2/5 - 2*b**2*e**2*n**2*x**5*log(x)/25 + 2*b**2*e**2*n**2*x**5/125 + 2*b**2*e**2*n*x**5*log(c)*log(x)
/5 - 2*b**2*e**2*n*x**5*log(c)/25 + b**2*e**2*x**5*log(c)**2/5

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Giac [B]  time = 1.29531, size = 551, normalized size = 3.1 \begin{align*} \frac{1}{5} \, b^{2} n^{2} x^{5} e^{2} \log \left (x\right )^{2} + \frac{1}{2} \, b^{2} d n^{2} x^{4} e \log \left (x\right )^{2} - \frac{2}{25} \, b^{2} n^{2} x^{5} e^{2} \log \left (x\right ) - \frac{1}{4} \, b^{2} d n^{2} x^{4} e \log \left (x\right ) + \frac{2}{5} \, b^{2} n x^{5} e^{2} \log \left (c\right ) \log \left (x\right ) + b^{2} d n x^{4} e \log \left (c\right ) \log \left (x\right ) + \frac{1}{3} \, b^{2} d^{2} n^{2} x^{3} \log \left (x\right )^{2} + \frac{2}{125} \, b^{2} n^{2} x^{5} e^{2} + \frac{1}{16} \, b^{2} d n^{2} x^{4} e - \frac{2}{25} \, b^{2} n x^{5} e^{2} \log \left (c\right ) - \frac{1}{4} \, b^{2} d n x^{4} e \log \left (c\right ) + \frac{1}{5} \, b^{2} x^{5} e^{2} \log \left (c\right )^{2} + \frac{1}{2} \, b^{2} d x^{4} e \log \left (c\right )^{2} - \frac{2}{9} \, b^{2} d^{2} n^{2} x^{3} \log \left (x\right ) + \frac{2}{5} \, a b n x^{5} e^{2} \log \left (x\right ) + a b d n x^{4} e \log \left (x\right ) + \frac{2}{3} \, b^{2} d^{2} n x^{3} \log \left (c\right ) \log \left (x\right ) + \frac{2}{27} \, b^{2} d^{2} n^{2} x^{3} - \frac{2}{25} \, a b n x^{5} e^{2} - \frac{1}{4} \, a b d n x^{4} e - \frac{2}{9} \, b^{2} d^{2} n x^{3} \log \left (c\right ) + \frac{2}{5} \, a b x^{5} e^{2} \log \left (c\right ) + a b d x^{4} e \log \left (c\right ) + \frac{1}{3} \, b^{2} d^{2} x^{3} \log \left (c\right )^{2} + \frac{2}{3} \, a b d^{2} n x^{3} \log \left (x\right ) - \frac{2}{9} \, a b d^{2} n x^{3} + \frac{1}{5} \, a^{2} x^{5} e^{2} + \frac{1}{2} \, a^{2} d x^{4} e + \frac{2}{3} \, a b d^{2} x^{3} \log \left (c\right ) + \frac{1}{3} \, a^{2} d^{2} x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*b^2*n^2*x^5*e^2*log(x)^2 + 1/2*b^2*d*n^2*x^4*e*log(x)^2 - 2/25*b^2*n^2*x^5*e^2*log(x) - 1/4*b^2*d*n^2*x^4*
e*log(x) + 2/5*b^2*n*x^5*e^2*log(c)*log(x) + b^2*d*n*x^4*e*log(c)*log(x) + 1/3*b^2*d^2*n^2*x^3*log(x)^2 + 2/12
5*b^2*n^2*x^5*e^2 + 1/16*b^2*d*n^2*x^4*e - 2/25*b^2*n*x^5*e^2*log(c) - 1/4*b^2*d*n*x^4*e*log(c) + 1/5*b^2*x^5*
e^2*log(c)^2 + 1/2*b^2*d*x^4*e*log(c)^2 - 2/9*b^2*d^2*n^2*x^3*log(x) + 2/5*a*b*n*x^5*e^2*log(x) + a*b*d*n*x^4*
e*log(x) + 2/3*b^2*d^2*n*x^3*log(c)*log(x) + 2/27*b^2*d^2*n^2*x^3 - 2/25*a*b*n*x^5*e^2 - 1/4*a*b*d*n*x^4*e - 2
/9*b^2*d^2*n*x^3*log(c) + 2/5*a*b*x^5*e^2*log(c) + a*b*d*x^4*e*log(c) + 1/3*b^2*d^2*x^3*log(c)^2 + 2/3*a*b*d^2
*n*x^3*log(x) - 2/9*a*b*d^2*n*x^3 + 1/5*a^2*x^5*e^2 + 1/2*a^2*d*x^4*e + 2/3*a*b*d^2*x^3*log(c) + 1/3*a^2*d^2*x
^3