3.1.85 \(\int x (d+e x)^2 (a+b \log (c x^n))^2 \, dx\) [85]

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

[Out]

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

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Rubi [A]
time = 0.12, antiderivative size = 178, normalized size of antiderivative = 1.00, 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 = {2395, 2342, 2341} \begin {gather*} \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 \end {gather*}

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 2341

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 2342

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 2395

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
]))

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*}

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Mathematica [A]
time = 0.07, size = 134, normalized size = 0.75 \begin {gather*} \frac {1}{864} x^2 \left (27 b e^2 n x^2 \left (-4 a+b n-4 b \log \left (c x^n\right )\right )+128 b d e n x \left (-3 a+b n-3 b \log \left (c x^n\right )\right )+216 b d^2 n \left (-2 a+b n-2 b \log \left (c x^n\right )\right )+432 d^2 \left (a+b \log \left (c x^n\right )\right )^2+576 d e x \left (a+b \log \left (c x^n\right )\right )^2+216 e^2 x^2 \left (a+b \log \left (c x^n\right )\right )^2\right ) \end {gather*}

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] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 83.60, size = 2597, normalized size = 14.59

method	result	size
risch	\(\text {Expression too large to display}\)	\(2597\)

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,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.29, size = 250, normalized size = 1.40 \begin {gather*} \frac {1}{4} \, b^{2} x^{4} e^{2} \log \left (c x^{n}\right )^{2} + \frac {2}{3} \, b^{2} d x^{3} e \log \left (c x^{n}\right )^{2} - \frac {1}{8} \, a b n x^{4} e^{2} - \frac {4}{9} \, a b d n x^{3} e + \frac {1}{2} \, a b x^{4} e^{2} \log \left (c x^{n}\right ) + \frac {4}{3} \, a b d x^{3} e \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 {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 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 {gather*}

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*x^4*e^2*log(c*x^n)^2 + 2/3*b^2*d*x^3*e*log(c*x^n)^2 - 1/8*a*b*n*x^4*e^2 - 4/9*a*b*d*n*x^3*e + 1/2*a*b*
x^4*e^2*log(c*x^n) + 4/3*a*b*d*x^3*e*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 + 1/4*a^2*x
^4*e^2 + 2/3*a^2*d*x^3*e + 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] Leaf count of result is larger than twice the leaf count of optimal. 347 vs. \(2 (160) = 320\).
time = 0.35, size = 347, normalized size = 1.95 \begin {gather*} \frac {1}{32} \, {\left (b^{2} n^{2} - 4 \, a b n + 8 \, a^{2}\right )} x^{4} e^{2} + \frac {2}{27} \, {\left (2 \, b^{2} d n^{2} - 6 \, a b d n + 9 \, a^{2} d\right )} x^{3} e + \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} x^{4} e^{2} + 8 \, b^{2} d x^{3} e + 6 \, b^{2} d^{2} x^{2}\right )} \log \left (c\right )^{2} + \frac {1}{12} \, {\left (3 \, b^{2} n^{2} x^{4} e^{2} + 8 \, b^{2} d n^{2} x^{3} e + 6 \, b^{2} d^{2} n^{2} x^{2}\right )} \log \left (x\right )^{2} - \frac {1}{72} \, {\left (9 \, {\left (b^{2} n - 4 \, a b\right )} x^{4} e^{2} + 32 \, {\left (b^{2} d n - 3 \, a b d\right )} x^{3} e + 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} n^{2} - 4 \, a b n\right )} x^{4} e^{2} + 32 \, {\left (b^{2} d n^{2} - 3 \, a b d n\right )} x^{3} e + 36 \, {\left (b^{2} d^{2} n^{2} - 2 \, a b d^{2} n\right )} x^{2} - 12 \, {\left (3 \, b^{2} n x^{4} e^{2} + 8 \, b^{2} d n x^{3} e + 6 \, b^{2} d^{2} n x^{2}\right )} \log \left (c\right )\right )} \log \left (x\right ) \end {gather*}

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

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Sympy [A]
time = 0.44, size = 308, normalized size = 1.73 \begin {gather*} \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} - \frac {a b d^{2} n x^{2}}{2} + a b d^{2} x^{2} \log {\left (c x^{n} \right )} - \frac {4 a b d e n x^{3}}{9} + \frac {4 a b d e x^{3} \log {\left (c x^{n} \right )}}{3} - \frac {a b e^{2} n x^{4}}{8} + \frac {a b e^{2} x^{4} \log {\left (c x^{n} \right )}}{2} + \frac {b^{2} d^{2} n^{2} x^{2}}{4} - \frac {b^{2} d^{2} n x^{2} \log {\left (c x^{n} \right )}}{2} + \frac {b^{2} d^{2} x^{2} \log {\left (c x^{n} \right )}^{2}}{2} + \frac {4 b^{2} d e n^{2} x^{3}}{27} - \frac {4 b^{2} d e n x^{3} \log {\left (c x^{n} \right )}}{9} + \frac {2 b^{2} d e x^{3} \log {\left (c x^{n} \right )}^{2}}{3} + \frac {b^{2} e^{2} n^{2} x^{4}}{32} - \frac {b^{2} e^{2} n x^{4} \log {\left (c x^{n} \right )}}{8} + \frac {b^{2} e^{2} x^{4} \log {\left (c x^{n} \right )}^{2}}{4} \end {gather*}

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 408 vs. \(2 (160) = 320\).
time = 2.60, size = 408, normalized size = 2.29 \begin {gather*} \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 {gather*}

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

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Mupad [B]
time = 3.62, size = 179, normalized size = 1.01 \begin {gather*} \ln \left (c\,x^n\right )\,\left (\frac {b\,\left (2\,a-b\,n\right )\,d^2\,x^2}{2}+\frac {4\,b\,\left (3\,a-b\,n\right )\,d\,e\,x^3}{9}+\frac {b\,\left (4\,a-b\,n\right )\,e^2\,x^4}{8}\right )+{\ln \left (c\,x^n\right )}^2\,\left (\frac {b^2\,d^2\,x^2}{2}+\frac {2\,b^2\,d\,e\,x^3}{3}+\frac {b^2\,e^2\,x^4}{4}\right )+\frac {d^2\,x^2\,\left (2\,a^2-2\,a\,b\,n+b^2\,n^2\right )}{4}+\frac {e^2\,x^4\,\left (8\,a^2-4\,a\,b\,n+b^2\,n^2\right )}{32}+\frac {2\,d\,e\,x^3\,\left (9\,a^2-6\,a\,b\,n+2\,b^2\,n^2\right )}{27} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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