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

Optimal result

Integrand size = 23, antiderivative size = 178 \[ \int \frac {(d+e x)^2 \left (a+b \log \left (c x^n\right )\right )^2}{x^5} \, dx=-\frac {b^2 d^2 n^2}{32 x^4}-\frac {4 b^2 d e n^2}{27 x^3}-\frac {b^2 e^2 n^2}{4 x^2}-\frac {b d^2 n \left (a+b \log \left (c x^n\right )\right )}{8 x^4}-\frac {4 b d e n \left (a+b \log \left (c x^n\right )\right )}{9 x^3}-\frac {b e^2 n \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac {d^2 \left (a+b \log \left (c x^n\right )\right )^2}{4 x^4}-\frac {2 d e \left (a+b \log \left (c x^n\right )\right )^2}{3 x^3}-\frac {e^2 \left (a+b \log \left (c x^n\right )\right )^2}{2 x^2} \]

[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

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {2395, 2342, 2341} \[ \int \frac {(d+e x)^2 \left (a+b \log \left (c x^n\right )\right )^2}{x^5} \, dx=-\frac {d^2 \left (a+b \log \left (c x^n\right )\right )^2}{4 x^4}-\frac {b d^2 n \left (a+b \log \left (c x^n\right )\right )}{8 x^4}-\frac {2 d e \left (a+b \log \left (c x^n\right )\right )^2}{3 x^3}-\frac {4 b d e n \left (a+b \log \left (c x^n\right )\right )}{9 x^3}-\frac {e^2 \left (a+b \log \left (c x^n\right )\right )^2}{2 x^2}-\frac {b e^2 n \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac {b^2 d^2 n^2}{32 x^4}-\frac {4 b^2 d e n^2}{27 x^3}-\frac {b^2 e^2 n^2}{4 x^2} \]

[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*} \text {integral}& = \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] (verified)

Time = 0.06 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.75 \[ \int \frac {(d+e x)^2 \left (a+b \log \left (c x^n\right )\right )^2}{x^5} \, dx=-\frac {216 d^2 \left (a+b \log \left (c x^n\right )\right )^2+576 d e x \left (a+b \log \left (c x^n\right )\right )^2+432 e^2 x^2 \left (a+b \log \left (c x^n\right )\right )^2+216 b e^2 n x^2 \left (2 a+b n+2 b \log \left (c x^n\right )\right )+128 b d e n x \left (3 a+b n+3 b \log \left (c x^n\right )\right )+27 b d^2 n \left (4 a+b n+4 b \log \left (c x^n\right )\right )}{864 x^4} \]

[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

Maple [A] (verified)

Time = 0.50 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.34

method	result	size
parallelrisch	\(-\frac {432 b^{2} \ln \left (c \,x^{n}\right )^{2} e^{2} x^{2}+432 x^{2} \ln \left (c \,x^{n}\right ) b^{2} e^{2} n +216 b^{2} e^{2} n^{2} x^{2}+864 a b \ln \left (c \,x^{n}\right ) e^{2} x^{2}+432 b n \,x^{2} a \,e^{2}+576 b^{2} \ln \left (c \,x^{n}\right )^{2} d e x +384 b^{2} d e n x \ln \left (c \,x^{n}\right )+128 b^{2} d e \,n^{2} x +432 a^{2} e^{2} x^{2}+1152 a b \ln \left (c \,x^{n}\right ) d e x +384 a b d e n x +216 b^{2} \ln \left (c \,x^{n}\right )^{2} d^{2}+108 \ln \left (c \,x^{n}\right ) n \,b^{2} d^{2}+27 b^{2} d^{2} n^{2}+576 a^{2} d e x +432 a b \ln \left (c \,x^{n}\right ) d^{2}+108 b \,d^{2} n a +216 a^{2} d^{2}}{864 x^{4}}\)	\(238\)
risch	\(\text {Expression too large to display}\)	\(2475\)

[In]

int((e*x+d)^2*(a+b*ln(c*x^n))^2/x^5,x,method=_RETURNVERBOSE)

[Out]

-1/864/x^4*(432*b^2*ln(c*x^n)^2*e^2*x^2+432*x^2*ln(c*x^n)*b^2*e^2*n+216*b^2*e^2*n^2*x^2+864*a*b*ln(c*x^n)*e^2*
x^2+432*b*n*x^2*a*e^2+576*b^2*ln(c*x^n)^2*d*e*x+384*b^2*d*e*n*x*ln(c*x^n)+128*b^2*d*e*n^2*x+432*a^2*e^2*x^2+11
52*a*b*ln(c*x^n)*d*e*x+384*a*b*d*e*n*x+216*b^2*ln(c*x^n)^2*d^2+108*ln(c*x^n)*n*b^2*d^2+27*b^2*d^2*n^2+576*a^2*
d*e*x+432*a*b*ln(c*x^n)*d^2+108*b*d^2*n*a+216*a^2*d^2)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 332 vs. \(2 (160) = 320\).

Time = 0.30 (sec) , antiderivative size = 332, normalized size of antiderivative = 1.87 \[ \int \frac {(d+e x)^2 \left (a+b \log \left (c x^n\right )\right )^2}{x^5} \, dx=-\frac {27 \, b^{2} d^{2} n^{2} + 108 \, a b d^{2} n + 216 \, a^{2} d^{2} + 216 \, {\left (b^{2} e^{2} n^{2} + 2 \, a b e^{2} n + 2 \, a^{2} e^{2}\right )} x^{2} + 72 \, {\left (6 \, b^{2} e^{2} x^{2} + 8 \, b^{2} d e x + 3 \, b^{2} d^{2}\right )} \log \left (c\right )^{2} + 72 \, {\left (6 \, b^{2} e^{2} n^{2} x^{2} + 8 \, b^{2} d e n^{2} x + 3 \, b^{2} d^{2} n^{2}\right )} \log \left (x\right )^{2} + 64 \, {\left (2 \, b^{2} d e n^{2} + 6 \, a b d e n + 9 \, a^{2} d e\right )} x + 12 \, {\left (9 \, b^{2} d^{2} n + 36 \, a b d^{2} + 36 \, {\left (b^{2} e^{2} n + 2 \, a b e^{2}\right )} x^{2} + 32 \, {\left (b^{2} d e n + 3 \, a b d e\right )} x\right )} \log \left (c\right ) + 12 \, {\left (9 \, b^{2} d^{2} n^{2} + 36 \, a b d^{2} n + 36 \, {\left (b^{2} e^{2} n^{2} + 2 \, a b e^{2} n\right )} x^{2} + 32 \, {\left (b^{2} d e n^{2} + 3 \, a b d e n\right )} x + 12 \, {\left (6 \, b^{2} e^{2} n x^{2} + 8 \, b^{2} d e n x + 3 \, b^{2} d^{2} n\right )} \log \left (c\right )\right )} \log \left (x\right )}{864 \, x^{4}} \]

[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

Sympy [A] (verification not implemented)

Time = 0.43 (sec) , antiderivative size = 309, normalized size of antiderivative = 1.74 \[ \int \frac {(d+e x)^2 \left (a+b \log \left (c x^n\right )\right )^2}{x^5} \, dx=- \frac {a^{2} d^{2}}{4 x^{4}} - \frac {2 a^{2} d e}{3 x^{3}} - \frac {a^{2} e^{2}}{2 x^{2}} - \frac {a b d^{2} n}{8 x^{4}} - \frac {a b d^{2} \log {\left (c x^{n} \right )}}{2 x^{4}} - \frac {4 a b d e n}{9 x^{3}} - \frac {4 a b d e \log {\left (c x^{n} \right )}}{3 x^{3}} - \frac {a b e^{2} n}{2 x^{2}} - \frac {a b e^{2} \log {\left (c x^{n} \right )}}{x^{2}} - \frac {b^{2} d^{2} n^{2}}{32 x^{4}} - \frac {b^{2} d^{2} n \log {\left (c x^{n} \right )}}{8 x^{4}} - \frac {b^{2} d^{2} \log {\left (c x^{n} \right )}^{2}}{4 x^{4}} - \frac {4 b^{2} d e n^{2}}{27 x^{3}} - \frac {4 b^{2} d e n \log {\left (c x^{n} \right )}}{9 x^{3}} - \frac {2 b^{2} d e \log {\left (c x^{n} \right )}^{2}}{3 x^{3}} - \frac {b^{2} e^{2} n^{2}}{4 x^{2}} - \frac {b^{2} e^{2} n \log {\left (c x^{n} \right )}}{2 x^{2}} - \frac {b^{2} e^{2} \log {\left (c x^{n} \right )}^{2}}{2 x^{2}} \]

[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)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.41 \[ \int \frac {(d+e x)^2 \left (a+b \log \left (c x^n\right )\right )^2}{x^5} \, dx=-\frac {1}{4} \, b^{2} e^{2} {\left (\frac {n^{2}}{x^{2}} + \frac {2 \, n \log \left (c x^{n}\right )}{x^{2}}\right )} - \frac {4}{27} \, b^{2} d e {\left (\frac {n^{2}}{x^{3}} + \frac {3 \, n \log \left (c x^{n}\right )}{x^{3}}\right )} - \frac {1}{32} \, b^{2} d^{2} {\left (\frac {n^{2}}{x^{4}} + \frac {4 \, n \log \left (c x^{n}\right )}{x^{4}}\right )} - \frac {b^{2} e^{2} \log \left (c x^{n}\right )^{2}}{2 \, x^{2}} - \frac {a b e^{2} n}{2 \, x^{2}} - \frac {a b e^{2} \log \left (c x^{n}\right )}{x^{2}} - \frac {2 \, b^{2} d e \log \left (c x^{n}\right )^{2}}{3 \, x^{3}} - \frac {4 \, a b d e n}{9 \, x^{3}} - \frac {a^{2} e^{2}}{2 \, x^{2}} - \frac {4 \, a b d e \log \left (c x^{n}\right )}{3 \, x^{3}} - \frac {b^{2} d^{2} \log \left (c x^{n}\right )^{2}}{4 \, x^{4}} - \frac {a b d^{2} n}{8 \, x^{4}} - \frac {2 \, a^{2} d e}{3 \, x^{3}} - \frac {a b d^{2} \log \left (c x^{n}\right )}{2 \, x^{4}} - \frac {a^{2} d^{2}}{4 \, x^{4}} \]

[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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 355 vs. \(2 (160) = 320\).

Time = 0.31 (sec) , antiderivative size = 355, normalized size of antiderivative = 1.99 \[ \int \frac {(d+e x)^2 \left (a+b \log \left (c x^n\right )\right )^2}{x^5} \, dx=-\frac {{\left (6 \, b^{2} e^{2} n^{2} x^{2} + 8 \, b^{2} d e n^{2} x + 3 \, b^{2} d^{2} n^{2}\right )} \log \left (x\right )^{2}}{12 \, x^{4}} - \frac {{\left (36 \, b^{2} e^{2} n^{2} x^{2} + 72 \, b^{2} e^{2} n x^{2} \log \left (c\right ) + 32 \, b^{2} d e n^{2} x + 72 \, a b e^{2} n x^{2} + 96 \, b^{2} d e n x \log \left (c\right ) + 9 \, b^{2} d^{2} n^{2} + 96 \, a b d e n x + 36 \, b^{2} d^{2} n \log \left (c\right ) + 36 \, a b d^{2} n\right )} \log \left (x\right )}{72 \, x^{4}} - \frac {216 \, b^{2} e^{2} n^{2} x^{2} + 432 \, b^{2} e^{2} n x^{2} \log \left (c\right ) + 432 \, b^{2} e^{2} x^{2} \log \left (c\right )^{2} + 128 \, b^{2} d e n^{2} x + 432 \, a b e^{2} n x^{2} + 384 \, b^{2} d e n x \log \left (c\right ) + 864 \, a b e^{2} x^{2} \log \left (c\right ) + 576 \, b^{2} d e x \log \left (c\right )^{2} + 27 \, b^{2} d^{2} n^{2} + 384 \, a b d e n x + 432 \, a^{2} e^{2} x^{2} + 108 \, b^{2} d^{2} n \log \left (c\right ) + 1152 \, a b d e x \log \left (c\right ) + 216 \, b^{2} d^{2} \log \left (c\right )^{2} + 108 \, a b d^{2} n + 576 \, a^{2} d e x + 432 \, a b d^{2} \log \left (c\right ) + 216 \, a^{2} d^{2}}{864 \, x^{4}} \]

[In]

integrate((e*x+d)^2*(a+b*log(c*x^n))^2/x^5,x, algorithm="giac")

[Out]

-1/12*(6*b^2*e^2*n^2*x^2 + 8*b^2*d*e*n^2*x + 3*b^2*d^2*n^2)*log(x)^2/x^4 - 1/72*(36*b^2*e^2*n^2*x^2 + 72*b^2*e
^2*n*x^2*log(c) + 32*b^2*d*e*n^2*x + 72*a*b*e^2*n*x^2 + 96*b^2*d*e*n*x*log(c) + 9*b^2*d^2*n^2 + 96*a*b*d*e*n*x
+ 36*b^2*d^2*n*log(c) + 36*a*b*d^2*n)*log(x)/x^4 - 1/864*(216*b^2*e^2*n^2*x^2 + 432*b^2*e^2*n*x^2*log(c) + 43
2*b^2*e^2*x^2*log(c)^2 + 128*b^2*d*e*n^2*x + 432*a*b*e^2*n*x^2 + 384*b^2*d*e*n*x*log(c) + 864*a*b*e^2*x^2*log(
c) + 576*b^2*d*e*x*log(c)^2 + 27*b^2*d^2*n^2 + 384*a*b*d*e*n*x + 432*a^2*e^2*x^2 + 108*b^2*d^2*n*log(c) + 1152
*a*b*d*e*x*log(c) + 216*b^2*d^2*log(c)^2 + 108*a*b*d^2*n + 576*a^2*d*e*x + 432*a*b*d^2*log(c) + 216*a^2*d^2)/x
^4

Mupad [B] (verification not implemented)

Time = 0.62 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.06 \[ \int \frac {(d+e x)^2 \left (a+b \log \left (c x^n\right )\right )^2}{x^5} \, dx=-\frac {x\,\left (48\,d\,e\,a^2+32\,d\,e\,a\,b\,n+\frac {32\,d\,e\,b^2\,n^2}{3}\right )+x^2\,\left (36\,a^2\,e^2+36\,a\,b\,e^2\,n+18\,b^2\,e^2\,n^2\right )+18\,a^2\,d^2+\frac {9\,b^2\,d^2\,n^2}{4}+9\,a\,b\,d^2\,n}{72\,x^4}-\frac {{\ln \left (c\,x^n\right )}^2\,\left (\frac {b^2\,d^2}{4}+\frac {2\,b^2\,d\,e\,x}{3}+\frac {b^2\,e^2\,x^2}{2}\right )}{x^4}-\frac {\ln \left (c\,x^n\right )\,\left (\frac {3\,b\,\left (4\,a+b\,n\right )\,d^2}{4}+\frac {8\,b\,\left (3\,a+b\,n\right )\,d\,e\,x}{3}+3\,b\,\left (2\,a+b\,n\right )\,e^2\,x^2\right )}{6\,x^4} \]

[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)