3.1.91 \(\int \frac {(d+e x)^2 (a+b \log (c x^n))^2}{x^5} \, dx\) [91]

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

[Out]

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

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Rubi [A]
time = 0.14, antiderivative size = 178, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {2395, 2342, 2341} \begin {gather*} -\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} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/32*(b^2*d^2*n^2)/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 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 \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*}

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

Antiderivative was successfully verified.

[In]

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

[Out]

-1/864*(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]))/x^4

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.21, size = 2475, normalized size = 13.90

method	result	size
risch	\(\text {Expression too large to display}\)	\(2475\)

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,method=_RETURNVERBOSE)

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

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Maxima [A]
time = 0.30, size = 251, normalized size = 1.41 \begin {gather*} -\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 {4}{27} \, b^{2} d {\left (\frac {n^{2}}{x^{3}} + \frac {3 \, n \log \left (c x^{n}\right )}{x^{3}}\right )} e - \frac {1}{4} \, b^{2} {\left (\frac {n^{2}}{x^{2}} + \frac {2 \, n \log \left (c x^{n}\right )}{x^{2}}\right )} e^{2} - \frac {b^{2} e^{2} \log \left (c x^{n}\right )^{2}}{2 \, x^{2}} - \frac {2 \, b^{2} d e \log \left (c x^{n}\right )^{2}}{3 \, x^{3}} - \frac {a b n e^{2}}{2 \, x^{2}} - \frac {4 \, a b d n e}{9 \, x^{3}} - \frac {a b e^{2} \log \left (c x^{n}\right )}{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 {a^{2} e^{2}}{2 \, x^{2}} - \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 {gather*}

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]

-1/32*b^2*d^2*(n^2/x^4 + 4*n*log(c*x^n)/x^4) - 4/27*b^2*d*(n^2/x^3 + 3*n*log(c*x^n)/x^3)*e - 1/4*b^2*(n^2/x^2
+ 2*n*log(c*x^n)/x^2)*e^2 - 1/2*b^2*e^2*log(c*x^n)^2/x^2 - 2/3*b^2*d*e*log(c*x^n)^2/x^3 - 1/2*a*b*n*e^2/x^2 -
4/9*a*b*d*n*e/x^3 - a*b*e^2*log(c*x^n)/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 - 1/2*a^2*e^2/x^2 - 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 [A]
time = 0.35, size = 316, normalized size = 1.78 \begin {gather*} -\frac {27 \, b^{2} d^{2} n^{2} + 108 \, a b d^{2} n + 216 \, a^{2} d^{2} + 216 \, {\left (b^{2} n^{2} + 2 \, a b n + 2 \, a^{2}\right )} x^{2} e^{2} + 64 \, {\left (2 \, b^{2} d n^{2} + 6 \, a b d n + 9 \, a^{2} d\right )} x e + 72 \, {\left (6 \, b^{2} x^{2} e^{2} + 8 \, b^{2} d x e + 3 \, b^{2} d^{2}\right )} \log \left (c\right )^{2} + 72 \, {\left (6 \, b^{2} n^{2} x^{2} e^{2} + 8 \, b^{2} d n^{2} x e + 3 \, b^{2} d^{2} n^{2}\right )} \log \left (x\right )^{2} + 12 \, {\left (9 \, b^{2} d^{2} n + 36 \, a b d^{2} + 36 \, {\left (b^{2} n + 2 \, a b\right )} x^{2} e^{2} + 32 \, {\left (b^{2} d n + 3 \, a b d\right )} x e\right )} \log \left (c\right ) + 12 \, {\left (9 \, b^{2} d^{2} n^{2} + 36 \, a b d^{2} n + 36 \, {\left (b^{2} n^{2} + 2 \, a b n\right )} x^{2} e^{2} + 32 \, {\left (b^{2} d n^{2} + 3 \, a b d n\right )} x e + 12 \, {\left (6 \, b^{2} n x^{2} e^{2} + 8 \, b^{2} d n x e + 3 \, b^{2} d^{2} n\right )} \log \left (c\right )\right )} \log \left (x\right )}{864 \, x^{4}} \end {gather*}

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*n^2 + 2*a*b*n + 2*a^2)*x^2*e^2 + 64*(2*b^2*d*n
^2 + 6*a*b*d*n + 9*a^2*d)*x*e + 72*(6*b^2*x^2*e^2 + 8*b^2*d*x*e + 3*b^2*d^2)*log(c)^2 + 72*(6*b^2*n^2*x^2*e^2
+ 8*b^2*d*n^2*x*e + 3*b^2*d^2*n^2)*log(x)^2 + 12*(9*b^2*d^2*n + 36*a*b*d^2 + 36*(b^2*n + 2*a*b)*x^2*e^2 + 32*(
b^2*d*n + 3*a*b*d)*x*e)*log(c) + 12*(9*b^2*d^2*n^2 + 36*a*b*d^2*n + 36*(b^2*n^2 + 2*a*b*n)*x^2*e^2 + 32*(b^2*d
*n^2 + 3*a*b*d*n)*x*e + 12*(6*b^2*n*x^2*e^2 + 8*b^2*d*n*x*e + 3*b^2*d^2*n)*log(c))*log(x))/x^4

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Sympy [A]
time = 0.57, size = 309, normalized size = 1.74 \begin {gather*} - \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}{8 x^{4}} - \frac {a b d^{2} \log {\left (c x^{n} \right )}}{2 x^{4}} - \frac {4 a b d e n}{9 x^{3}} - \frac {4 a b d e \log {\left (c x^{n} \right )}}{3 x^{3}} - \frac {a b e^{2} n}{2 x^{2}} - \frac {a b e^{2} \log {\left (c x^{n} \right )}}{x^{2}} - \frac {b^{2} d^{2} n^{2}}{32 x^{4}} - \frac {b^{2} d^{2} n \log {\left (c x^{n} \right )}}{8 x^{4}} - \frac {b^{2} d^{2} \log {\left (c x^{n} \right )}^{2}}{4 x^{4}} - \frac {4 b^{2} d e n^{2}}{27 x^{3}} - \frac {4 b^{2} d e n \log {\left (c x^{n} \right )}}{9 x^{3}} - \frac {2 b^{2} d e \log {\left (c x^{n} \right )}^{2}}{3 x^{3}} - \frac {b^{2} e^{2} n^{2}}{4 x^{2}} - \frac {b^{2} e^{2} n \log {\left (c x^{n} \right )}}{2 x^{2}} - \frac {b^{2} e^{2} \log {\left (c x^{n} \right )}^{2}}{2 x^{2}} \end {gather*}

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 366 vs. \(2 (160) = 320\).
time = 3.43, size = 366, normalized size = 2.06 \begin {gather*} -\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 {gather*}

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

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Mupad [B]
time = 3.67, size = 188, normalized size = 1.06 \begin {gather*} -\frac {x\,\left (48\,d\,e\,a^2+32\,d\,e\,a\,b\,n+\frac {32\,d\,e\,b^2\,n^2}{3}\right )+x^2\,\left (36\,a^2\,e^2+36\,a\,b\,e^2\,n+18\,b^2\,e^2\,n^2\right )+18\,a^2\,d^2+\frac {9\,b^2\,d^2\,n^2}{4}+9\,a\,b\,d^2\,n}{72\,x^4}-\frac {{\ln \left (c\,x^n\right )}^2\,\left (\frac {b^2\,d^2}{4}+\frac {2\,b^2\,d\,e\,x}{3}+\frac {b^2\,e^2\,x^2}{2}\right )}{x^4}-\frac {\ln \left (c\,x^n\right )\,\left (\frac {3\,b\,\left (4\,a+b\,n\right )\,d^2}{4}+\frac {8\,b\,\left (3\,a+b\,n\right )\,d\,e\,x}{3}+3\,b\,\left (2\,a+b\,n\right )\,e^2\,x^2\right )}{6\,x^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + b*log(c*x^n))^2*(d + e*x)^2)/x^5,x)

[Out]

- (x*(48*a^2*d*e + (32*b^2*d*e*n^2)/3 + 32*a*b*d*e*n) + x^2*(36*a^2*e^2 + 18*b^2*e^2*n^2 + 36*a*b*e^2*n) + 18*
a^2*d^2 + (9*b^2*d^2*n^2)/4 + 9*a*b*d^2*n)/(72*x^4) - (log(c*x^n)^2*((b^2*d^2)/4 + (b^2*e^2*x^2)/2 + (2*b^2*d*
e*x)/3))/x^4 - (log(c*x^n)*((3*b*d^2*(4*a + b*n))/4 + 3*b*e^2*x^2*(2*a + b*n) + (8*b*d*e*x*(3*a + b*n))/3))/(6
*x^4)

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