3.91 \(\int \frac{(d+e x)^2 (a+b \log (c x^n))^2}{x^5} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.205352, 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} \[ -\frac{d^2 \left (a+b \log \left (c x^n\right )\right )^2}{4 x^4}-\frac{b d^2 n \left (a+b \log \left (c x^n\right )\right )}{8 x^4}-\frac{2 d e \left (a+b \log \left (c x^n\right )\right )^2}{3 x^3}-\frac{4 b d e n \left (a+b \log \left (c x^n\right )\right )}{9 x^3}-\frac{e^2 \left (a+b \log \left (c x^n\right )\right )^2}{2 x^2}-\frac{b e^2 n \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac{b^2 d^2 n^2}{32 x^4}-\frac{4 b^2 d e n^2}{27 x^3}-\frac{b^2 e^2 n^2}{4 x^2} \]

Antiderivative was successfully verified.

[In]

Int[((d + e*x)^2*(a + b*Log[c*x^n])^2)/x^5,x]

[Out]

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

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^5} \, dx &=\int \left (\frac{d^2 \left (a+b \log \left (c x^n\right )\right )^2}{x^5}+\frac{2 d e \left (a+b \log \left (c x^n\right )\right )^2}{x^4}+\frac{e^2 \left (a+b \log \left (c x^n\right )\right )^2}{x^3}\right ) \, dx\\ &=d^2 \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{x^5} \, dx+(2 d e) \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{x^4} \, dx+e^2 \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{x^3} \, dx\\ &=-\frac{d^2 \left (a+b \log \left (c x^n\right )\right )^2}{4 x^4}-\frac{2 d e \left (a+b \log \left (c x^n\right )\right )^2}{3 x^3}-\frac{e^2 \left (a+b \log \left (c x^n\right )\right )^2}{2 x^2}+\frac{1}{2} \left (b d^2 n\right ) \int \frac{a+b \log \left (c x^n\right )}{x^5} \, dx+\frac{1}{3} (4 b d e n) \int \frac{a+b \log \left (c x^n\right )}{x^4} \, dx+\left (b e^2 n\right ) \int \frac{a+b \log \left (c x^n\right )}{x^3} \, dx\\ &=-\frac{b^2 d^2 n^2}{32 x^4}-\frac{4 b^2 d e n^2}{27 x^3}-\frac{b^2 e^2 n^2}{4 x^2}-\frac{b d^2 n \left (a+b \log \left (c x^n\right )\right )}{8 x^4}-\frac{4 b d e n \left (a+b \log \left (c x^n\right )\right )}{9 x^3}-\frac{b e^2 n \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac{d^2 \left (a+b \log \left (c x^n\right )\right )^2}{4 x^4}-\frac{2 d e \left (a+b \log \left (c x^n\right )\right )^2}{3 x^3}-\frac{e^2 \left (a+b \log \left (c x^n\right )\right )^2}{2 x^2}\\ \end{align*}

Mathematica [A]  time = 0.0917837, size = 134, normalized size = 0.75 \[ -\frac{216 d^2 \left (a+b \log \left (c x^n\right )\right )^2+27 b d^2 n \left (4 a+4 b \log \left (c x^n\right )+b n\right )+576 d e x \left (a+b \log \left (c x^n\right )\right )^2+128 b d e n x \left (3 a+3 b \log \left (c x^n\right )+b n\right )+432 e^2 x^2 \left (a+b \log \left (c x^n\right )\right )^2+216 b e^2 n x^2 \left (2 a+2 b \log \left (c x^n\right )+b n\right )}{864 x^4} \]

Antiderivative was successfully verified.

[In]

Integrate[((d + e*x)^2*(a + b*Log[c*x^n])^2)/x^5,x]

[Out]

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

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Maple [C]  time = 0.254, size = 2475, normalized size = 13.9 \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^5,x)

[Out]

-1/12*b^2*(6*e^2*x^2+8*d*e*x+3*d^2)/x^4*ln(x^n)^2-1/72*(-48*I*Pi*b^2*d*e*x*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)
+48*I*Pi*b^2*d*e*x*csgn(I*x^n)*csgn(I*c*x^n)^2-48*I*Pi*b^2*d*e*x*csgn(I*c*x^n)^3+18*I*Pi*b^2*d^2*csgn(I*x^n)*c
sgn(I*c*x^n)^2+72*ln(c)*b^2*e^2*x^2+36*b^2*e^2*n*x^2+72*a*b*e^2*x^2-36*I*Pi*b^2*e^2*x^2*csgn(I*x^n)*csgn(I*c*x
^n)*csgn(I*c)+36*I*Pi*b^2*e^2*x^2*csgn(I*x^n)*csgn(I*c*x^n)^2-36*I*Pi*b^2*e^2*x^2*csgn(I*c*x^n)^3+36*I*Pi*b^2*
e^2*x^2*csgn(I*c*x^n)^2*csgn(I*c)+96*ln(c)*b^2*d*e*x+32*b^2*d*e*n*x+96*a*b*d*e*x+48*I*Pi*b^2*d*e*x*csgn(I*c*x^
n)^2*csgn(I*c)+18*I*Pi*b^2*d^2*csgn(I*c*x^n)^2*csgn(I*c)-18*I*Pi*b^2*d^2*csgn(I*c*x^n)^3-18*I*Pi*b^2*d^2*csgn(
I*x^n)*csgn(I*c*x^n)*csgn(I*c)+36*ln(c)*b^2*d^2+9*b^2*d^2*n+36*a*b*d^2)/x^4*ln(x^n)-1/864*(216*a^2*d^2-192*I*P
i*b^2*d*e*n*x*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-576*I*Pi*a*b*d*e*x*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-576*I
*ln(c)*Pi*b^2*d*e*x*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+432*a^2*e^2*x^2+27*b^2*d^2*n^2-54*Pi^2*b^2*d^2*csgn(I*
x^n)^2*csgn(I*c*x^n)^4+108*Pi^2*b^2*d^2*csgn(I*x^n)*csgn(I*c*x^n)^5+108*a*b*n*d^2-54*Pi^2*b^2*d^2*csgn(I*c*x^n
)^4*csgn(I*c)^2+288*Pi^2*b^2*d*e*x*csgn(I*x^n)^2*csgn(I*c*x^n)^3*csgn(I*c)-144*Pi^2*b^2*d*e*x*csgn(I*x^n)^2*cs
gn(I*c*x^n)^2*csgn(I*c)^2+216*ln(c)^2*b^2*d^2+576*a^2*d*e*x+432*b*n*a*e^2*x^2+432*I*ln(c)*Pi*b^2*e^2*x^2*csgn(
I*x^n)*csgn(I*c*x^n)^2+432*I*ln(c)*Pi*b^2*e^2*x^2*csgn(I*c*x^n)^2*csgn(I*c)-216*I*ln(c)*Pi*b^2*d^2*csgn(I*x^n)
*csgn(I*c*x^n)*csgn(I*c)-216*I*Pi*a*b*d^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-576*I*ln(c)*Pi*b^2*d*e*x*csgn(I*
c*x^n)^3+432*I*Pi*a*b*e^2*x^2*csgn(I*c*x^n)^2*csgn(I*c)+432*I*Pi*a*b*e^2*x^2*csgn(I*x^n)*csgn(I*c*x^n)^2-54*I*
Pi*b^2*d^2*n*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-576*I*Pi*a*b*d*e*x*csgn(I*c*x^n)^3+216*I*n*Pi*b^2*e^2*x^2*csg
n(I*x^n)*csgn(I*c*x^n)^2+216*I*n*Pi*b^2*e^2*x^2*csgn(I*c*x^n)^2*csgn(I*c)-192*I*n*Pi*b^2*d*e*x*csgn(I*c*x^n)^3
-54*I*Pi*b^2*d^2*n*csgn(I*c*x^n)^3+288*Pi^2*b^2*d*e*x*csgn(I*c*x^n)^5*csgn(I*c)-144*Pi^2*b^2*d*e*x*csgn(I*c*x^
n)^4*csgn(I*c)^2+108*ln(c)*b^2*d^2*n+432*ln(c)*a*b*d^2+108*Pi^2*b^2*d^2*csgn(I*c*x^n)^5*csgn(I*c)-216*Pi^2*b^2
*d^2*csgn(I*x^n)*csgn(I*c*x^n)^4*csgn(I*c)+108*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^2-576*Pi^2*b^2*d*e*x*csgn(I*x^n)*csgn(I*c*x^n)^4*csgn(I*c)+288*Pi^2*b^2*d*e*x*csgn(I*x^n)*csgn(I*c*x^n
)^3*csgn(I*c)^2+192*I*n*Pi*b^2*d*e*x*csgn(I*x^n)*csgn(I*c*x^n)^2+576*I*Pi*a*b*d*e*x*csgn(I*x^n)*csgn(I*c*x^n)^
2-432*I*ln(c)*Pi*b^2*e^2*x^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+576*I*ln(c)*Pi*b^2*d*e*x*csgn(I*c*x^n)^2*csgn
(I*c)+1152*ln(c)*a*b*d*e*x+384*n*ln(c)*b^2*d*e*x+108*Pi^2*b^2*d^2*csgn(I*x^n)^2*csgn(I*c*x^n)^3*csgn(I*c)-54*P
i^2*b^2*d^2*csgn(I*x^n)^2*csgn(I*c*x^n)^2*csgn(I*c)^2-432*Pi^2*b^2*e^2*x^2*csgn(I*x^n)*csgn(I*c*x^n)^4*csgn(I*
c)+216*Pi^2*b^2*e^2*x^2*csgn(I*x^n)*csgn(I*c*x^n)^3*csgn(I*c)^2-144*Pi^2*b^2*d*e*x*csgn(I*x^n)^2*csgn(I*c*x^n)
^4+288*Pi^2*b^2*d*e*x*csgn(I*x^n)*csgn(I*c*x^n)^5+54*I*Pi*b^2*d^2*n*csgn(I*x^n)*csgn(I*c*x^n)^2+54*I*Pi*b^2*d^
2*n*csgn(I*c*x^n)^2*csgn(I*c)-216*I*Pi*a*b*d^2*csgn(I*c*x^n)^3-216*I*n*Pi*b^2*e^2*x^2*csgn(I*c*x^n)^3-432*I*ln
(c)*Pi*b^2*e^2*x^2*csgn(I*c*x^n)^3-432*I*Pi*a*b*e^2*x^2*csgn(I*c*x^n)^3+384*a*b*d*e*n*x+128*b^2*d*e*n^2*x-108*
Pi^2*b^2*e^2*x^2*csgn(I*c*x^n)^6+432*ln(c)^2*b^2*e^2*x^2+216*I*ln(c)*Pi*b^2*d^2*csgn(I*x^n)*csgn(I*c*x^n)^2+57
6*I*Pi*a*b*d*e*x*csgn(I*c*x^n)^2*csgn(I*c)+192*I*Pi*b^2*d*e*n*x*csgn(I*c*x^n)^2*csgn(I*c)-216*I*ln(c)*Pi*b^2*d
^2*csgn(I*c*x^n)^3-144*Pi^2*b^2*d*e*x*csgn(I*c*x^n)^6-108*Pi^2*b^2*e^2*x^2*csgn(I*x^n)^2*csgn(I*c*x^n)^4+216*P
i^2*b^2*e^2*x^2*csgn(I*x^n)*csgn(I*c*x^n)^5+216*Pi^2*b^2*e^2*x^2*csgn(I*c*x^n)^5*csgn(I*c)-108*Pi^2*b^2*e^2*x^
2*csgn(I*c*x^n)^4*csgn(I*c)^2-54*Pi^2*b^2*d^2*csgn(I*c*x^n)^6+576*I*ln(c)*Pi*b^2*d*e*x*csgn(I*x^n)*csgn(I*c*x^
n)^2-216*I*n*Pi*b^2*e^2*x^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-432*I*Pi*a*b*e^2*x^2*csgn(I*x^n)*csgn(I*c*x^n)
*csgn(I*c)+216*Pi^2*b^2*e^2*x^2*csgn(I*x^n)^2*csgn(I*c*x^n)^3*csgn(I*c)-108*Pi^2*b^2*e^2*x^2*csgn(I*x^n)^2*csg
n(I*c*x^n)^2*csgn(I*c)^2+216*I*ln(c)*Pi*b^2*d^2*csgn(I*c*x^n)^2*csgn(I*c)+216*I*Pi*a*b*d^2*csgn(I*x^n)*csgn(I*
c*x^n)^2+216*I*Pi*a*b*d^2*csgn(I*c*x^n)^2*csgn(I*c)+432*n*ln(c)*b^2*e^2*x^2+576*ln(c)^2*b^2*d*e*x+864*ln(c)*a*
b*e^2*x^2)/x^4

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Maxima [A]  time = 1.13299, size = 339, normalized size = 1.9 \begin{align*} -\frac{1}{4} \, b^{2} e^{2}{\left (\frac{n^{2}}{x^{2}} + \frac{2 \, n \log \left (c x^{n}\right )}{x^{2}}\right )} - \frac{4}{27} \, b^{2} d e{\left (\frac{n^{2}}{x^{3}} + \frac{3 \, n \log \left (c x^{n}\right )}{x^{3}}\right )} - \frac{1}{32} \, b^{2} d^{2}{\left (\frac{n^{2}}{x^{4}} + \frac{4 \, n \log \left (c x^{n}\right )}{x^{4}}\right )} - \frac{b^{2} e^{2} \log \left (c x^{n}\right )^{2}}{2 \, x^{2}} - \frac{a b e^{2} n}{2 \, x^{2}} - \frac{a b e^{2} \log \left (c x^{n}\right )}{x^{2}} - \frac{2 \, b^{2} d e \log \left (c x^{n}\right )^{2}}{3 \, x^{3}} - \frac{4 \, a b d e n}{9 \, x^{3}} - \frac{a^{2} e^{2}}{2 \, x^{2}} - \frac{4 \, a b d e \log \left (c x^{n}\right )}{3 \, x^{3}} - \frac{b^{2} d^{2} \log \left (c x^{n}\right )^{2}}{4 \, x^{4}} - \frac{a b d^{2} n}{8 \, x^{4}} - \frac{2 \, a^{2} d e}{3 \, x^{3}} - \frac{a b d^{2} \log \left (c x^{n}\right )}{2 \, x^{4}} - \frac{a^{2} d^{2}}{4 \, x^{4}} \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^5,x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 1.04074, size = 756, normalized size = 4.25 \begin{align*} -\frac{27 \, b^{2} d^{2} n^{2} + 108 \, a b d^{2} n + 216 \, a^{2} d^{2} + 216 \,{\left (b^{2} e^{2} n^{2} + 2 \, a b e^{2} n + 2 \, a^{2} e^{2}\right )} x^{2} + 72 \,{\left (6 \, b^{2} e^{2} x^{2} + 8 \, b^{2} d e x + 3 \, b^{2} d^{2}\right )} \log \left (c\right )^{2} + 72 \,{\left (6 \, b^{2} e^{2} n^{2} x^{2} + 8 \, b^{2} d e n^{2} x + 3 \, b^{2} d^{2} n^{2}\right )} \log \left (x\right )^{2} + 64 \,{\left (2 \, b^{2} d e n^{2} + 6 \, a b d e n + 9 \, a^{2} d e\right )} x + 12 \,{\left (9 \, b^{2} d^{2} n + 36 \, a b d^{2} + 36 \,{\left (b^{2} e^{2} n + 2 \, a b e^{2}\right )} x^{2} + 32 \,{\left (b^{2} d e n + 3 \, a b d e\right )} x\right )} \log \left (c\right ) + 12 \,{\left (9 \, b^{2} d^{2} n^{2} + 36 \, a b d^{2} n + 36 \,{\left (b^{2} e^{2} n^{2} + 2 \, a b e^{2} n\right )} x^{2} + 32 \,{\left (b^{2} d e n^{2} + 3 \, a b d e n\right )} x + 12 \,{\left (6 \, b^{2} e^{2} n x^{2} + 8 \, b^{2} d e n x + 3 \, b^{2} d^{2} n\right )} \log \left (c\right )\right )} \log \left (x\right )}{864 \, x^{4}} \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^5,x, algorithm="fricas")

[Out]

-1/864*(27*b^2*d^2*n^2 + 108*a*b*d^2*n + 216*a^2*d^2 + 216*(b^2*e^2*n^2 + 2*a*b*e^2*n + 2*a^2*e^2)*x^2 + 72*(6
*b^2*e^2*x^2 + 8*b^2*d*e*x + 3*b^2*d^2)*log(c)^2 + 72*(6*b^2*e^2*n^2*x^2 + 8*b^2*d*e*n^2*x + 3*b^2*d^2*n^2)*lo
g(x)^2 + 64*(2*b^2*d*e*n^2 + 6*a*b*d*e*n + 9*a^2*d*e)*x + 12*(9*b^2*d^2*n + 36*a*b*d^2 + 36*(b^2*e^2*n + 2*a*b
*e^2)*x^2 + 32*(b^2*d*e*n + 3*a*b*d*e)*x)*log(c) + 12*(9*b^2*d^2*n^2 + 36*a*b*d^2*n + 36*(b^2*e^2*n^2 + 2*a*b*
e^2*n)*x^2 + 32*(b^2*d*e*n^2 + 3*a*b*d*e*n)*x + 12*(6*b^2*e^2*n*x^2 + 8*b^2*d*e*n*x + 3*b^2*d^2*n)*log(c))*log
(x))/x^4

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Sympy [B]  time = 5.57175, size = 512, normalized size = 2.88 \begin{align*} - \frac{a^{2} d^{2}}{4 x^{4}} - \frac{2 a^{2} d e}{3 x^{3}} - \frac{a^{2} e^{2}}{2 x^{2}} - \frac{a b d^{2} n \log{\left (x \right )}}{2 x^{4}} - \frac{a b d^{2} n}{8 x^{4}} - \frac{a b d^{2} \log{\left (c \right )}}{2 x^{4}} - \frac{4 a b d e n \log{\left (x \right )}}{3 x^{3}} - \frac{4 a b d e n}{9 x^{3}} - \frac{4 a b d e \log{\left (c \right )}}{3 x^{3}} - \frac{a b e^{2} n \log{\left (x \right )}}{x^{2}} - \frac{a b e^{2} n}{2 x^{2}} - \frac{a b e^{2} \log{\left (c \right )}}{x^{2}} - \frac{b^{2} d^{2} n^{2} \log{\left (x \right )}^{2}}{4 x^{4}} - \frac{b^{2} d^{2} n^{2} \log{\left (x \right )}}{8 x^{4}} - \frac{b^{2} d^{2} n^{2}}{32 x^{4}} - \frac{b^{2} d^{2} n \log{\left (c \right )} \log{\left (x \right )}}{2 x^{4}} - \frac{b^{2} d^{2} n \log{\left (c \right )}}{8 x^{4}} - \frac{b^{2} d^{2} \log{\left (c \right )}^{2}}{4 x^{4}} - \frac{2 b^{2} d e n^{2} \log{\left (x \right )}^{2}}{3 x^{3}} - \frac{4 b^{2} d e n^{2} \log{\left (x \right )}}{9 x^{3}} - \frac{4 b^{2} d e n^{2}}{27 x^{3}} - \frac{4 b^{2} d e n \log{\left (c \right )} \log{\left (x \right )}}{3 x^{3}} - \frac{4 b^{2} d e n \log{\left (c \right )}}{9 x^{3}} - \frac{2 b^{2} d e \log{\left (c \right )}^{2}}{3 x^{3}} - \frac{b^{2} e^{2} n^{2} \log{\left (x \right )}^{2}}{2 x^{2}} - \frac{b^{2} e^{2} n^{2} \log{\left (x \right )}}{2 x^{2}} - \frac{b^{2} e^{2} n^{2}}{4 x^{2}} - \frac{b^{2} e^{2} n \log{\left (c \right )} \log{\left (x \right )}}{x^{2}} - \frac{b^{2} e^{2} n \log{\left (c \right )}}{2 x^{2}} - \frac{b^{2} e^{2} \log{\left (c \right )}^{2}}{2 x^{2}} \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**5,x)

[Out]

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

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Giac [B]  time = 1.30862, size = 494, normalized size = 2.78 \begin{align*} -\frac{432 \, b^{2} n^{2} x^{2} e^{2} \log \left (x\right )^{2} + 576 \, b^{2} d n^{2} x e \log \left (x\right )^{2} + 432 \, b^{2} n^{2} x^{2} e^{2} \log \left (x\right ) + 384 \, b^{2} d n^{2} x e \log \left (x\right ) + 864 \, b^{2} n x^{2} e^{2} \log \left (c\right ) \log \left (x\right ) + 1152 \, b^{2} d n x e \log \left (c\right ) \log \left (x\right ) + 216 \, b^{2} d^{2} n^{2} \log \left (x\right )^{2} + 216 \, b^{2} n^{2} x^{2} e^{2} + 128 \, b^{2} d n^{2} x e + 432 \, b^{2} n x^{2} e^{2} \log \left (c\right ) + 384 \, b^{2} d n x e \log \left (c\right ) + 432 \, b^{2} x^{2} e^{2} \log \left (c\right )^{2} + 576 \, b^{2} d x e \log \left (c\right )^{2} + 108 \, b^{2} d^{2} n^{2} \log \left (x\right ) + 864 \, a b n x^{2} e^{2} \log \left (x\right ) + 1152 \, a b d n x e \log \left (x\right ) + 432 \, b^{2} d^{2} n \log \left (c\right ) \log \left (x\right ) + 27 \, b^{2} d^{2} n^{2} + 432 \, a b n x^{2} e^{2} + 384 \, a b d n x e + 108 \, b^{2} d^{2} n \log \left (c\right ) + 864 \, a b x^{2} e^{2} \log \left (c\right ) + 1152 \, a b d x e \log \left (c\right ) + 216 \, b^{2} d^{2} \log \left (c\right )^{2} + 432 \, a b d^{2} n \log \left (x\right ) + 108 \, a b d^{2} n + 432 \, a^{2} x^{2} e^{2} + 576 \, a^{2} d x e + 432 \, a b d^{2} \log \left (c\right ) + 216 \, a^{2} d^{2}}{864 \, x^{4}} \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^5,x, algorithm="giac")

[Out]

-1/864*(432*b^2*n^2*x^2*e^2*log(x)^2 + 576*b^2*d*n^2*x*e*log(x)^2 + 432*b^2*n^2*x^2*e^2*log(x) + 384*b^2*d*n^2
*x*e*log(x) + 864*b^2*n*x^2*e^2*log(c)*log(x) + 1152*b^2*d*n*x*e*log(c)*log(x) + 216*b^2*d^2*n^2*log(x)^2 + 21
6*b^2*n^2*x^2*e^2 + 128*b^2*d*n^2*x*e + 432*b^2*n*x^2*e^2*log(c) + 384*b^2*d*n*x*e*log(c) + 432*b^2*x^2*e^2*lo
g(c)^2 + 576*b^2*d*x*e*log(c)^2 + 108*b^2*d^2*n^2*log(x) + 864*a*b*n*x^2*e^2*log(x) + 1152*a*b*d*n*x*e*log(x)
+ 432*b^2*d^2*n*log(c)*log(x) + 27*b^2*d^2*n^2 + 432*a*b*n*x^2*e^2 + 384*a*b*d*n*x*e + 108*b^2*d^2*n*log(c) +
864*a*b*x^2*e^2*log(c) + 1152*a*b*d*x*e*log(c) + 216*b^2*d^2*log(c)^2 + 432*a*b*d^2*n*log(x) + 108*a*b*d^2*n +
432*a^2*x^2*e^2 + 576*a^2*d*x*e + 432*a*b*d^2*log(c) + 216*a^2*d^2)/x^4