3.90 \(\int \frac{(d+e x)^2 (a+b \log (c x^n))^2}{x^4} \, dx\)

Optimal. Leaf size=168 \[ -\frac{d^2 \left (a+b \log \left (c x^n\right )\right )^2}{3 x^3}-\frac{2 b d^2 n \left (a+b \log \left (c x^n\right )\right )}{9 x^3}-\frac{d e \left (a+b \log \left (c x^n\right )\right )^2}{x^2}-\frac{b d e n \left (a+b \log \left (c x^n\right )\right )}{x^2}-\frac{e^2 \left (a+b \log \left (c x^n\right )\right )^2}{x}-\frac{2 b e^2 n \left (a+b \log \left (c x^n\right )\right )}{x}-\frac{2 b^2 d^2 n^2}{27 x^3}-\frac{b^2 d e n^2}{2 x^2}-\frac{2 b^2 e^2 n^2}{x} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.20804, antiderivative size = 168, 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} \[ -\frac{d^2 \left (a+b \log \left (c x^n\right )\right )^2}{3 x^3}-\frac{2 b d^2 n \left (a+b \log \left (c x^n\right )\right )}{9 x^3}-\frac{d e \left (a+b \log \left (c x^n\right )\right )^2}{x^2}-\frac{b d e n \left (a+b \log \left (c x^n\right )\right )}{x^2}-\frac{e^2 \left (a+b \log \left (c x^n\right )\right )^2}{x}-\frac{2 b e^2 n \left (a+b \log \left (c x^n\right )\right )}{x}-\frac{2 b^2 d^2 n^2}{27 x^3}-\frac{b^2 d e n^2}{2 x^2}-\frac{2 b^2 e^2 n^2}{x} \]

Antiderivative was successfully verified.

[In]

Int[((d + e*x)^2*(a + b*Log[c*x^n])^2)/x^4,x]

[Out]

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

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

Mathematica [A]  time = 0.0914908, size = 131, normalized size = 0.78 \[ -\frac{18 d^2 \left (a+b \log \left (c x^n\right )\right )^2+4 b d^2 n \left (3 a+3 b \log \left (c x^n\right )+b n\right )+54 d e x \left (a+b \log \left (c x^n\right )\right )^2+27 b d e n x \left (2 a+2 b \log \left (c x^n\right )+b n\right )+54 e^2 x^2 \left (a+b \log \left (c x^n\right )\right )^2+108 b e^2 n x^2 \left (a+b \log \left (c x^n\right )+b n\right )}{54 x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[((d + e*x)^2*(a + b*Log[c*x^n])^2)/x^4,x]

[Out]

-(18*d^2*(a + b*Log[c*x^n])^2 + 54*d*e*x*(a + b*Log[c*x^n])^2 + 54*e^2*x^2*(a + b*Log[c*x^n])^2 + 108*b*e^2*n*
x^2*(a + b*n + b*Log[c*x^n]) + 27*b*d*e*n*x*(2*a + b*n + 2*b*Log[c*x^n]) + 4*b*d^2*n*(3*a + b*n + 3*b*Log[c*x^
n]))/(54*x^3)

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Maple [C]  time = 0.251, size = 2473, normalized size = 14.7 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^2*(a+b*ln(c*x^n))^2/x^4,x)

[Out]

-1/3*b^2*(3*e^2*x^2+3*d*e*x+d^2)/x^3*ln(x^n)^2-1/9*(9*I*Pi*b^2*d*e*x*csgn(I*x^n)*csgn(I*c*x^n)^2+9*I*Pi*b^2*d*
e*x*csgn(I*c*x^n)^2*csgn(I*c)-3*I*Pi*b^2*d^2*csgn(I*c*x^n)^3+3*I*Pi*b^2*d^2*csgn(I*x^n)*csgn(I*c*x^n)^2+18*ln(
c)*b^2*e^2*x^2+18*b^2*e^2*n*x^2+18*a*b*e^2*x^2-9*I*Pi*b^2*d*e*x*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+3*I*Pi*b^2
*d^2*csgn(I*c*x^n)^2*csgn(I*c)-9*I*Pi*b^2*e^2*x^2*csgn(I*c*x^n)^3-9*I*Pi*b^2*e^2*x^2*csgn(I*x^n)*csgn(I*c*x^n)
*csgn(I*c)+18*ln(c)*b^2*d*e*x+9*b^2*d*e*n*x+18*a*b*d*e*x+9*I*Pi*b^2*e^2*x^2*csgn(I*x^n)*csgn(I*c*x^n)^2+9*I*Pi
*b^2*e^2*x^2*csgn(I*c*x^n)^2*csgn(I*c)-3*I*Pi*b^2*d^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-9*I*Pi*b^2*d*e*x*csg
n(I*c*x^n)^3+6*ln(c)*b^2*d^2+2*b^2*d^2*n+6*a*b*d^2)/x^3*ln(x^n)-1/108*(36*a^2*d^2-108*I*ln(c)*Pi*b^2*d*e*x*csg
n(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-54*I*Pi*b^2*d*e*n*x*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-108*I*Pi*a*b*d*e*x*cs
gn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+108*I*Pi*a*b*e^2*x^2*csgn(I*x^n)*csgn(I*c*x^n)^2-12*I*Pi*b^2*d^2*n*csgn(I*x^
n)*csgn(I*c*x^n)*csgn(I*c)-36*I*ln(c)*Pi*b^2*d^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+108*a^2*e^2*x^2+8*b^2*d^2
*n^2-9*Pi^2*b^2*d^2*csgn(I*x^n)^2*csgn(I*c*x^n)^4+18*Pi^2*b^2*d^2*csgn(I*x^n)*csgn(I*c*x^n)^5+108*I*ln(c)*Pi*b
^2*e^2*x^2*csgn(I*c*x^n)^2*csgn(I*c)-108*I*Pi*a*b*d*e*x*csgn(I*c*x^n)^3+108*I*n*Pi*b^2*e^2*x^2*csgn(I*x^n)*csg
n(I*c*x^n)^2+24*a*b*n*d^2-108*I*ln(c)*Pi*b^2*d*e*x*csgn(I*c*x^n)^3+108*I*ln(c)*Pi*b^2*e^2*x^2*csgn(I*x^n)*csgn
(I*c*x^n)^2-108*I*Pi*a*b*e^2*x^2*csgn(I*c*x^n)^3+36*I*ln(c)*Pi*b^2*d^2*csgn(I*x^n)*csgn(I*c*x^n)^2+36*I*ln(c)*
Pi*b^2*d^2*csgn(I*c*x^n)^2*csgn(I*c)-36*I*Pi*a*b*d^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+108*I*n*Pi*b^2*e^2*x^
2*csgn(I*c*x^n)^2*csgn(I*c)-54*I*n*Pi*b^2*d*e*x*csgn(I*c*x^n)^3+108*I*Pi*a*b*e^2*x^2*csgn(I*c*x^n)^2*csgn(I*c)
-9*Pi^2*b^2*d^2*csgn(I*c*x^n)^4*csgn(I*c)^2+54*Pi^2*b^2*d*e*x*csgn(I*x^n)^2*csgn(I*c*x^n)^3*csgn(I*c)-27*Pi^2*
b^2*d*e*x*csgn(I*x^n)^2*csgn(I*c*x^n)^2*csgn(I*c)^2+108*I*Pi*a*b*d*e*x*csgn(I*c*x^n)^2*csgn(I*c)+36*ln(c)^2*b^
2*d^2-36*I*ln(c)*Pi*b^2*d^2*csgn(I*c*x^n)^3-36*I*Pi*a*b*d^2*csgn(I*c*x^n)^3+54*I*Pi*b^2*d*e*n*x*csgn(I*c*x^n)^
2*csgn(I*c)-108*I*Pi*a*b*e^2*x^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+108*I*ln(c)*Pi*b^2*d*e*x*csgn(I*x^n)*csgn
(I*c*x^n)^2+108*a^2*d*e*x+216*b*n*a*e^2*x^2+54*Pi^2*b^2*d*e*x*csgn(I*c*x^n)^5*csgn(I*c)-27*Pi^2*b^2*d*e*x*csgn
(I*c*x^n)^4*csgn(I*c)^2-108*I*n*Pi*b^2*e^2*x^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+54*I*n*Pi*b^2*d*e*x*csgn(I*
x^n)*csgn(I*c*x^n)^2+24*ln(c)*b^2*d^2*n+72*ln(c)*a*b*d^2+18*Pi^2*b^2*d^2*csgn(I*c*x^n)^5*csgn(I*c)-12*I*Pi*b^2
*d^2*n*csgn(I*c*x^n)^3-36*Pi^2*b^2*d^2*csgn(I*x^n)*csgn(I*c*x^n)^4*csgn(I*c)+18*Pi^2*b^2*d^2*csgn(I*x^n)*csgn(
I*c*x^n)^3*csgn(I*c)^2+216*b^2*e^2*n^2*x^2-108*Pi^2*b^2*d*e*x*csgn(I*x^n)*csgn(I*c*x^n)^4*csgn(I*c)+54*Pi^2*b^
2*d*e*x*csgn(I*x^n)*csgn(I*c*x^n)^3*csgn(I*c)^2+216*ln(c)*a*b*d*e*x+108*n*ln(c)*b^2*d*e*x+12*I*Pi*b^2*d^2*n*cs
gn(I*c*x^n)^2*csgn(I*c)-108*I*n*Pi*b^2*e^2*x^2*csgn(I*c*x^n)^3+108*I*Pi*a*b*d*e*x*csgn(I*x^n)*csgn(I*c*x^n)^2-
108*I*ln(c)*Pi*b^2*e^2*x^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+108*I*ln(c)*Pi*b^2*d*e*x*csgn(I*c*x^n)^2*csgn(I
*c)+18*Pi^2*b^2*d^2*csgn(I*x^n)^2*csgn(I*c*x^n)^3*csgn(I*c)-9*Pi^2*b^2*d^2*csgn(I*x^n)^2*csgn(I*c*x^n)^2*csgn(
I*c)^2-108*Pi^2*b^2*e^2*x^2*csgn(I*x^n)*csgn(I*c*x^n)^4*csgn(I*c)+54*Pi^2*b^2*e^2*x^2*csgn(I*x^n)*csgn(I*c*x^n
)^3*csgn(I*c)^2-27*Pi^2*b^2*d*e*x*csgn(I*x^n)^2*csgn(I*c*x^n)^4+54*Pi^2*b^2*d*e*x*csgn(I*x^n)*csgn(I*c*x^n)^5-
108*I*ln(c)*Pi*b^2*e^2*x^2*csgn(I*c*x^n)^3+108*a*b*d*e*n*x+54*b^2*d*e*n^2*x-27*Pi^2*b^2*e^2*x^2*csgn(I*c*x^n)^
6+108*ln(c)^2*b^2*e^2*x^2+36*I*Pi*a*b*d^2*csgn(I*x^n)*csgn(I*c*x^n)^2+36*I*Pi*a*b*d^2*csgn(I*c*x^n)^2*csgn(I*c
)+12*I*Pi*b^2*d^2*n*csgn(I*x^n)*csgn(I*c*x^n)^2-27*Pi^2*b^2*d*e*x*csgn(I*c*x^n)^6-27*Pi^2*b^2*e^2*x^2*csgn(I*x
^n)^2*csgn(I*c*x^n)^4+54*Pi^2*b^2*e^2*x^2*csgn(I*x^n)*csgn(I*c*x^n)^5+54*Pi^2*b^2*e^2*x^2*csgn(I*c*x^n)^5*csgn
(I*c)-27*Pi^2*b^2*e^2*x^2*csgn(I*c*x^n)^4*csgn(I*c)^2-9*Pi^2*b^2*d^2*csgn(I*c*x^n)^6+54*Pi^2*b^2*e^2*x^2*csgn(
I*x^n)^2*csgn(I*c*x^n)^3*csgn(I*c)-27*Pi^2*b^2*e^2*x^2*csgn(I*x^n)^2*csgn(I*c*x^n)^2*csgn(I*c)^2+216*n*ln(c)*b
^2*e^2*x^2+108*ln(c)^2*b^2*d*e*x+216*ln(c)*a*b*e^2*x^2)/x^3

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Maxima [A]  time = 1.22468, size = 338, normalized size = 2.01 \begin{align*} -2 \, b^{2} e^{2}{\left (\frac{n^{2}}{x} + \frac{n \log \left (c x^{n}\right )}{x}\right )} - \frac{1}{2} \, b^{2} d e{\left (\frac{n^{2}}{x^{2}} + \frac{2 \, n \log \left (c x^{n}\right )}{x^{2}}\right )} - \frac{2}{27} \, b^{2} d^{2}{\left (\frac{n^{2}}{x^{3}} + \frac{3 \, n \log \left (c x^{n}\right )}{x^{3}}\right )} - \frac{b^{2} e^{2} \log \left (c x^{n}\right )^{2}}{x} - \frac{2 \, a b e^{2} n}{x} - \frac{2 \, a b e^{2} \log \left (c x^{n}\right )}{x} - \frac{b^{2} d e \log \left (c x^{n}\right )^{2}}{x^{2}} - \frac{a b d e n}{x^{2}} - \frac{a^{2} e^{2}}{x} - \frac{2 \, a b d e \log \left (c x^{n}\right )}{x^{2}} - \frac{b^{2} d^{2} \log \left (c x^{n}\right )^{2}}{3 \, x^{3}} - \frac{2 \, a b d^{2} n}{9 \, x^{3}} - \frac{a^{2} d e}{x^{2}} - \frac{2 \, a b d^{2} \log \left (c x^{n}\right )}{3 \, x^{3}} - \frac{a^{2} d^{2}}{3 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^2*(a+b*log(c*x^n))^2/x^4,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [B]  time = 1.0441, size = 724, normalized size = 4.31 \begin{align*} -\frac{4 \, b^{2} d^{2} n^{2} + 12 \, a b d^{2} n + 18 \, a^{2} d^{2} + 54 \,{\left (2 \, b^{2} e^{2} n^{2} + 2 \, a b e^{2} n + a^{2} e^{2}\right )} x^{2} + 18 \,{\left (3 \, b^{2} e^{2} x^{2} + 3 \, b^{2} d e x + b^{2} d^{2}\right )} \log \left (c\right )^{2} + 18 \,{\left (3 \, b^{2} e^{2} n^{2} x^{2} + 3 \, b^{2} d e n^{2} x + b^{2} d^{2} n^{2}\right )} \log \left (x\right )^{2} + 27 \,{\left (b^{2} d e n^{2} + 2 \, a b d e n + 2 \, a^{2} d e\right )} x + 6 \,{\left (2 \, b^{2} d^{2} n + 6 \, a b d^{2} + 18 \,{\left (b^{2} e^{2} n + a b e^{2}\right )} x^{2} + 9 \,{\left (b^{2} d e n + 2 \, a b d e\right )} x\right )} \log \left (c\right ) + 6 \,{\left (2 \, b^{2} d^{2} n^{2} + 6 \, a b d^{2} n + 18 \,{\left (b^{2} e^{2} n^{2} + a b e^{2} n\right )} x^{2} + 9 \,{\left (b^{2} d e n^{2} + 2 \, a b d e n\right )} x + 6 \,{\left (3 \, b^{2} e^{2} n x^{2} + 3 \, b^{2} d e n x + b^{2} d^{2} n\right )} \log \left (c\right )\right )} \log \left (x\right )}{54 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^2*(a+b*log(c*x^n))^2/x^4,x, algorithm="fricas")

[Out]

-1/54*(4*b^2*d^2*n^2 + 12*a*b*d^2*n + 18*a^2*d^2 + 54*(2*b^2*e^2*n^2 + 2*a*b*e^2*n + a^2*e^2)*x^2 + 18*(3*b^2*
e^2*x^2 + 3*b^2*d*e*x + b^2*d^2)*log(c)^2 + 18*(3*b^2*e^2*n^2*x^2 + 3*b^2*d*e*n^2*x + b^2*d^2*n^2)*log(x)^2 +
27*(b^2*d*e*n^2 + 2*a*b*d*e*n + 2*a^2*d*e)*x + 6*(2*b^2*d^2*n + 6*a*b*d^2 + 18*(b^2*e^2*n + a*b*e^2)*x^2 + 9*(
b^2*d*e*n + 2*a*b*d*e)*x)*log(c) + 6*(2*b^2*d^2*n^2 + 6*a*b*d^2*n + 18*(b^2*e^2*n^2 + a*b*e^2*n)*x^2 + 9*(b^2*
d*e*n^2 + 2*a*b*d*e*n)*x + 6*(3*b^2*e^2*n*x^2 + 3*b^2*d*e*n*x + b^2*d^2*n)*log(c))*log(x))/x^3

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Sympy [B]  time = 3.90587, size = 479, normalized size = 2.85 \begin{align*} - \frac{a^{2} d^{2}}{3 x^{3}} - \frac{a^{2} d e}{x^{2}} - \frac{a^{2} e^{2}}{x} - \frac{2 a b d^{2} n \log{\left (x \right )}}{3 x^{3}} - \frac{2 a b d^{2} n}{9 x^{3}} - \frac{2 a b d^{2} \log{\left (c \right )}}{3 x^{3}} - \frac{2 a b d e n \log{\left (x \right )}}{x^{2}} - \frac{a b d e n}{x^{2}} - \frac{2 a b d e \log{\left (c \right )}}{x^{2}} - \frac{2 a b e^{2} n \log{\left (x \right )}}{x} - \frac{2 a b e^{2} n}{x} - \frac{2 a b e^{2} \log{\left (c \right )}}{x} - \frac{b^{2} d^{2} n^{2} \log{\left (x \right )}^{2}}{3 x^{3}} - \frac{2 b^{2} d^{2} n^{2} \log{\left (x \right )}}{9 x^{3}} - \frac{2 b^{2} d^{2} n^{2}}{27 x^{3}} - \frac{2 b^{2} d^{2} n \log{\left (c \right )} \log{\left (x \right )}}{3 x^{3}} - \frac{2 b^{2} d^{2} n \log{\left (c \right )}}{9 x^{3}} - \frac{b^{2} d^{2} \log{\left (c \right )}^{2}}{3 x^{3}} - \frac{b^{2} d e n^{2} \log{\left (x \right )}^{2}}{x^{2}} - \frac{b^{2} d e n^{2} \log{\left (x \right )}}{x^{2}} - \frac{b^{2} d e n^{2}}{2 x^{2}} - \frac{2 b^{2} d e n \log{\left (c \right )} \log{\left (x \right )}}{x^{2}} - \frac{b^{2} d e n \log{\left (c \right )}}{x^{2}} - \frac{b^{2} d e \log{\left (c \right )}^{2}}{x^{2}} - \frac{b^{2} e^{2} n^{2} \log{\left (x \right )}^{2}}{x} - \frac{2 b^{2} e^{2} n^{2} \log{\left (x \right )}}{x} - \frac{2 b^{2} e^{2} n^{2}}{x} - \frac{2 b^{2} e^{2} n \log{\left (c \right )} \log{\left (x \right )}}{x} - \frac{2 b^{2} e^{2} n \log{\left (c \right )}}{x} - \frac{b^{2} e^{2} \log{\left (c \right )}^{2}}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**2*(a+b*ln(c*x**n))**2/x**4,x)

[Out]

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

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Giac [B]  time = 1.36316, size = 494, normalized size = 2.94 \begin{align*} -\frac{54 \, b^{2} n^{2} x^{2} e^{2} \log \left (x\right )^{2} + 54 \, b^{2} d n^{2} x e \log \left (x\right )^{2} + 108 \, b^{2} n^{2} x^{2} e^{2} \log \left (x\right ) + 54 \, b^{2} d n^{2} x e \log \left (x\right ) + 108 \, b^{2} n x^{2} e^{2} \log \left (c\right ) \log \left (x\right ) + 108 \, b^{2} d n x e \log \left (c\right ) \log \left (x\right ) + 18 \, b^{2} d^{2} n^{2} \log \left (x\right )^{2} + 108 \, b^{2} n^{2} x^{2} e^{2} + 27 \, b^{2} d n^{2} x e + 108 \, b^{2} n x^{2} e^{2} \log \left (c\right ) + 54 \, b^{2} d n x e \log \left (c\right ) + 54 \, b^{2} x^{2} e^{2} \log \left (c\right )^{2} + 54 \, b^{2} d x e \log \left (c\right )^{2} + 12 \, b^{2} d^{2} n^{2} \log \left (x\right ) + 108 \, a b n x^{2} e^{2} \log \left (x\right ) + 108 \, a b d n x e \log \left (x\right ) + 36 \, b^{2} d^{2} n \log \left (c\right ) \log \left (x\right ) + 4 \, b^{2} d^{2} n^{2} + 108 \, a b n x^{2} e^{2} + 54 \, a b d n x e + 12 \, b^{2} d^{2} n \log \left (c\right ) + 108 \, a b x^{2} e^{2} \log \left (c\right ) + 108 \, a b d x e \log \left (c\right ) + 18 \, b^{2} d^{2} \log \left (c\right )^{2} + 36 \, a b d^{2} n \log \left (x\right ) + 12 \, a b d^{2} n + 54 \, a^{2} x^{2} e^{2} + 54 \, a^{2} d x e + 36 \, a b d^{2} \log \left (c\right ) + 18 \, a^{2} d^{2}}{54 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^2*(a+b*log(c*x^n))^2/x^4,x, algorithm="giac")

[Out]

-1/54*(54*b^2*n^2*x^2*e^2*log(x)^2 + 54*b^2*d*n^2*x*e*log(x)^2 + 108*b^2*n^2*x^2*e^2*log(x) + 54*b^2*d*n^2*x*e
*log(x) + 108*b^2*n*x^2*e^2*log(c)*log(x) + 108*b^2*d*n*x*e*log(c)*log(x) + 18*b^2*d^2*n^2*log(x)^2 + 108*b^2*
n^2*x^2*e^2 + 27*b^2*d*n^2*x*e + 108*b^2*n*x^2*e^2*log(c) + 54*b^2*d*n*x*e*log(c) + 54*b^2*x^2*e^2*log(c)^2 +
54*b^2*d*x*e*log(c)^2 + 12*b^2*d^2*n^2*log(x) + 108*a*b*n*x^2*e^2*log(x) + 108*a*b*d*n*x*e*log(x) + 36*b^2*d^2
*n*log(c)*log(x) + 4*b^2*d^2*n^2 + 108*a*b*n*x^2*e^2 + 54*a*b*d*n*x*e + 12*b^2*d^2*n*log(c) + 108*a*b*x^2*e^2*
log(c) + 108*a*b*d*x*e*log(c) + 18*b^2*d^2*log(c)^2 + 36*a*b*d^2*n*log(x) + 12*a*b*d^2*n + 54*a^2*x^2*e^2 + 54
*a^2*d*x*e + 36*a*b*d^2*log(c) + 18*a^2*d^2)/x^3