\(\int \frac {a+b \log (c x^n)}{x^2 (d+e x)^7} \, dx\) [72]

Optimal result

Integrand size = 21, antiderivative size = 339 \[ \int \frac {a+b \log \left (c x^n\right )}{x^2 (d+e x)^7} \, dx=-\frac {b n}{d^7 x}+\frac {b e n}{30 d^3 (d+e x)^5}+\frac {17 b e n}{120 d^4 (d+e x)^4}+\frac {79 b e n}{180 d^5 (d+e x)^3}+\frac {53 b e n}{40 d^6 (d+e x)^2}+\frac {103 b e n}{20 d^7 (d+e x)}+\frac {103 b e n \log (x)}{20 d^8}-\frac {a+b \log \left (c x^n\right )}{d^7 x}-\frac {e \left (a+b \log \left (c x^n\right )\right )}{6 d^2 (d+e x)^6}-\frac {2 e \left (a+b \log \left (c x^n\right )\right )}{5 d^3 (d+e x)^5}-\frac {3 e \left (a+b \log \left (c x^n\right )\right )}{4 d^4 (d+e x)^4}-\frac {4 e \left (a+b \log \left (c x^n\right )\right )}{3 d^5 (d+e x)^3}-\frac {5 e \left (a+b \log \left (c x^n\right )\right )}{2 d^6 (d+e x)^2}+\frac {6 e^2 x \left (a+b \log \left (c x^n\right )\right )}{d^8 (d+e x)}+\frac {7 e \log \left (1+\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^8}-\frac {223 b e n \log (d+e x)}{20 d^8}-\frac {7 b e n \operatorname {PolyLog}\left (2,-\frac {d}{e x}\right )}{d^8} \]

[Out]

-b*n/d^7/x+1/30*b*e*n/d^3/(e*x+d)^5+17/120*b*e*n/d^4/(e*x+d)^4+79/180*b*e*n/d^5/(e*x+d)^3+53/40*b*e*n/d^6/(e*x
+d)^2+103/20*b*e*n/d^7/(e*x+d)+103/20*b*e*n*ln(x)/d^8+(-a-b*ln(c*x^n))/d^7/x-1/6*e*(a+b*ln(c*x^n))/d^2/(e*x+d)
^6-2/5*e*(a+b*ln(c*x^n))/d^3/(e*x+d)^5-3/4*e*(a+b*ln(c*x^n))/d^4/(e*x+d)^4-4/3*e*(a+b*ln(c*x^n))/d^5/(e*x+d)^3
-5/2*e*(a+b*ln(c*x^n))/d^6/(e*x+d)^2+6*e^2*x*(a+b*ln(c*x^n))/d^8/(e*x+d)+7*e*ln(1+d/e/x)*(a+b*ln(c*x^n))/d^8-2
23/20*b*e*n*ln(e*x+d)/d^8-7*b*e*n*polylog(2,-d/e/x)/d^8

Rubi [A] (verified)

Time = 0.40 (sec) , antiderivative size = 339, normalized size of antiderivative = 1.00, number of steps used = 22, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {46, 2393, 2341, 2356, 2351, 31, 2379, 2438} \[ \int \frac {a+b \log \left (c x^n\right )}{x^2 (d+e x)^7} \, dx=\frac {6 e^2 x \left (a+b \log \left (c x^n\right )\right )}{d^8 (d+e x)}+\frac {7 e \log \left (\frac {d}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d^8}-\frac {a+b \log \left (c x^n\right )}{d^7 x}-\frac {5 e \left (a+b \log \left (c x^n\right )\right )}{2 d^6 (d+e x)^2}-\frac {4 e \left (a+b \log \left (c x^n\right )\right )}{3 d^5 (d+e x)^3}-\frac {3 e \left (a+b \log \left (c x^n\right )\right )}{4 d^4 (d+e x)^4}-\frac {2 e \left (a+b \log \left (c x^n\right )\right )}{5 d^3 (d+e x)^5}-\frac {e \left (a+b \log \left (c x^n\right )\right )}{6 d^2 (d+e x)^6}-\frac {7 b e n \operatorname {PolyLog}\left (2,-\frac {d}{e x}\right )}{d^8}+\frac {103 b e n \log (x)}{20 d^8}-\frac {223 b e n \log (d+e x)}{20 d^8}+\frac {103 b e n}{20 d^7 (d+e x)}-\frac {b n}{d^7 x}+\frac {53 b e n}{40 d^6 (d+e x)^2}+\frac {79 b e n}{180 d^5 (d+e x)^3}+\frac {17 b e n}{120 d^4 (d+e x)^4}+\frac {b e n}{30 d^3 (d+e x)^5} \]

[In]

Int[(a + b*Log[c*x^n])/(x^2*(d + e*x)^7),x]

[Out]

-((b*n)/(d^7*x)) + (b*e*n)/(30*d^3*(d + e*x)^5) + (17*b*e*n)/(120*d^4*(d + e*x)^4) + (79*b*e*n)/(180*d^5*(d +
e*x)^3) + (53*b*e*n)/(40*d^6*(d + e*x)^2) + (103*b*e*n)/(20*d^7*(d + e*x)) + (103*b*e*n*Log[x])/(20*d^8) - (a
+ b*Log[c*x^n])/(d^7*x) - (e*(a + b*Log[c*x^n]))/(6*d^2*(d + e*x)^6) - (2*e*(a + b*Log[c*x^n]))/(5*d^3*(d + e*
x)^5) - (3*e*(a + b*Log[c*x^n]))/(4*d^4*(d + e*x)^4) - (4*e*(a + b*Log[c*x^n]))/(3*d^5*(d + e*x)^3) - (5*e*(a
+ b*Log[c*x^n]))/(2*d^6*(d + e*x)^2) + (6*e^2*x*(a + b*Log[c*x^n]))/(d^8*(d + e*x)) + (7*e*Log[1 + d/(e*x)]*(a
+ b*Log[c*x^n]))/d^8 - (223*b*e*n*Log[d + e*x])/(20*d^8) - (7*b*e*n*PolyLog[2, -(d/(e*x))])/d^8

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 46

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x
)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && Lt
Q[m + n + 2, 0])

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 2351

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_) + (e_.)*(x_)^(r_.))^(q_), x_Symbol] :> Simp[x*(d + e*x^r)^(q +
1)*((a + b*Log[c*x^n])/d), x] - Dist[b*(n/d), Int[(d + e*x^r)^(q + 1), x], x] /; FreeQ[{a, b, c, d, e, n, q,
r}, x] && EqQ[r*(q + 1) + 1, 0]

Rule 2356

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_.), x_Symbol] :> Simp[(d + e*x)^(q + 1)
*((a + b*Log[c*x^n])^p/(e*(q + 1))), x] - Dist[b*n*(p/(e*(q + 1))), Int[((d + e*x)^(q + 1)*(a + b*Log[c*x^n])^
(p - 1))/x, x], x] /; FreeQ[{a, b, c, d, e, n, p, q}, x] && GtQ[p, 0] && NeQ[q, -1] && (EqQ[p, 1] || (Integers
Q[2*p, 2*q] &&  !IGtQ[q, 0]) || (EqQ[p, 2] && NeQ[q, 1]))

Rule 2379

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_)^(r_.))), x_Symbol] :> Simp[(-Log[1 +
d/(e*x^r)])*((a + b*Log[c*x^n])^p/(d*r)), x] + Dist[b*n*(p/(d*r)), Int[Log[1 + d/(e*x^r)]*((a + b*Log[c*x^n])^
(p - 1)/x), x], x] /; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[p, 0]

Rule 2393

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> Wit
h[{u = ExpandIntegrand[a + b*Log[c*x^n], (f*x)^m*(d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c,
d, e, f, m, n, q, r}, x] && IntegerQ[q] && (GtQ[q, 0] || (IntegerQ[m] && IntegerQ[r]))

Rule 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {a+b \log \left (c x^n\right )}{d^7 x^2}+\frac {e^2 \left (a+b \log \left (c x^n\right )\right )}{d^2 (d+e x)^7}+\frac {2 e^2 \left (a+b \log \left (c x^n\right )\right )}{d^3 (d+e x)^6}+\frac {3 e^2 \left (a+b \log \left (c x^n\right )\right )}{d^4 (d+e x)^5}+\frac {4 e^2 \left (a+b \log \left (c x^n\right )\right )}{d^5 (d+e x)^4}+\frac {5 e^2 \left (a+b \log \left (c x^n\right )\right )}{d^6 (d+e x)^3}+\frac {6 e^2 \left (a+b \log \left (c x^n\right )\right )}{d^7 (d+e x)^2}-\frac {7 e \left (a+b \log \left (c x^n\right )\right )}{d^7 x (d+e x)}\right ) \, dx \\ & = \frac {\int \frac {a+b \log \left (c x^n\right )}{x^2} \, dx}{d^7}-\frac {(7 e) \int \frac {a+b \log \left (c x^n\right )}{x (d+e x)} \, dx}{d^7}+\frac {\left (6 e^2\right ) \int \frac {a+b \log \left (c x^n\right )}{(d+e x)^2} \, dx}{d^7}+\frac {\left (5 e^2\right ) \int \frac {a+b \log \left (c x^n\right )}{(d+e x)^3} \, dx}{d^6}+\frac {\left (4 e^2\right ) \int \frac {a+b \log \left (c x^n\right )}{(d+e x)^4} \, dx}{d^5}+\frac {\left (3 e^2\right ) \int \frac {a+b \log \left (c x^n\right )}{(d+e x)^5} \, dx}{d^4}+\frac {\left (2 e^2\right ) \int \frac {a+b \log \left (c x^n\right )}{(d+e x)^6} \, dx}{d^3}+\frac {e^2 \int \frac {a+b \log \left (c x^n\right )}{(d+e x)^7} \, dx}{d^2} \\ & = -\frac {b n}{d^7 x}-\frac {a+b \log \left (c x^n\right )}{d^7 x}-\frac {e \left (a+b \log \left (c x^n\right )\right )}{6 d^2 (d+e x)^6}-\frac {2 e \left (a+b \log \left (c x^n\right )\right )}{5 d^3 (d+e x)^5}-\frac {3 e \left (a+b \log \left (c x^n\right )\right )}{4 d^4 (d+e x)^4}-\frac {4 e \left (a+b \log \left (c x^n\right )\right )}{3 d^5 (d+e x)^3}-\frac {5 e \left (a+b \log \left (c x^n\right )\right )}{2 d^6 (d+e x)^2}+\frac {6 e^2 x \left (a+b \log \left (c x^n\right )\right )}{d^8 (d+e x)}+\frac {7 e \log \left (1+\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^8}-\frac {(7 b e n) \int \frac {\log \left (1+\frac {d}{e x}\right )}{x} \, dx}{d^8}+\frac {(5 b e n) \int \frac {1}{x (d+e x)^2} \, dx}{2 d^6}+\frac {(4 b e n) \int \frac {1}{x (d+e x)^3} \, dx}{3 d^5}+\frac {(3 b e n) \int \frac {1}{x (d+e x)^4} \, dx}{4 d^4}+\frac {(2 b e n) \int \frac {1}{x (d+e x)^5} \, dx}{5 d^3}+\frac {(b e n) \int \frac {1}{x (d+e x)^6} \, dx}{6 d^2}-\frac {\left (6 b e^2 n\right ) \int \frac {1}{d+e x} \, dx}{d^8} \\ & = -\frac {b n}{d^7 x}-\frac {a+b \log \left (c x^n\right )}{d^7 x}-\frac {e \left (a+b \log \left (c x^n\right )\right )}{6 d^2 (d+e x)^6}-\frac {2 e \left (a+b \log \left (c x^n\right )\right )}{5 d^3 (d+e x)^5}-\frac {3 e \left (a+b \log \left (c x^n\right )\right )}{4 d^4 (d+e x)^4}-\frac {4 e \left (a+b \log \left (c x^n\right )\right )}{3 d^5 (d+e x)^3}-\frac {5 e \left (a+b \log \left (c x^n\right )\right )}{2 d^6 (d+e x)^2}+\frac {6 e^2 x \left (a+b \log \left (c x^n\right )\right )}{d^8 (d+e x)}+\frac {7 e \log \left (1+\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^8}-\frac {6 b e n \log (d+e x)}{d^8}-\frac {7 b e n \text {Li}_2\left (-\frac {d}{e x}\right )}{d^8}+\frac {(5 b e n) \int \left (\frac {1}{d^2 x}-\frac {e}{d (d+e x)^2}-\frac {e}{d^2 (d+e x)}\right ) \, dx}{2 d^6}+\frac {(4 b e n) \int \left (\frac {1}{d^3 x}-\frac {e}{d (d+e x)^3}-\frac {e}{d^2 (d+e x)^2}-\frac {e}{d^3 (d+e x)}\right ) \, dx}{3 d^5}+\frac {(3 b e n) \int \left (\frac {1}{d^4 x}-\frac {e}{d (d+e x)^4}-\frac {e}{d^2 (d+e x)^3}-\frac {e}{d^3 (d+e x)^2}-\frac {e}{d^4 (d+e x)}\right ) \, dx}{4 d^4}+\frac {(2 b e n) \int \left (\frac {1}{d^5 x}-\frac {e}{d (d+e x)^5}-\frac {e}{d^2 (d+e x)^4}-\frac {e}{d^3 (d+e x)^3}-\frac {e}{d^4 (d+e x)^2}-\frac {e}{d^5 (d+e x)}\right ) \, dx}{5 d^3}+\frac {(b e n) \int \left (\frac {1}{d^6 x}-\frac {e}{d (d+e x)^6}-\frac {e}{d^2 (d+e x)^5}-\frac {e}{d^3 (d+e x)^4}-\frac {e}{d^4 (d+e x)^3}-\frac {e}{d^5 (d+e x)^2}-\frac {e}{d^6 (d+e x)}\right ) \, dx}{6 d^2} \\ & = -\frac {b n}{d^7 x}+\frac {b e n}{30 d^3 (d+e x)^5}+\frac {17 b e n}{120 d^4 (d+e x)^4}+\frac {79 b e n}{180 d^5 (d+e x)^3}+\frac {53 b e n}{40 d^6 (d+e x)^2}+\frac {103 b e n}{20 d^7 (d+e x)}+\frac {103 b e n \log (x)}{20 d^8}-\frac {a+b \log \left (c x^n\right )}{d^7 x}-\frac {e \left (a+b \log \left (c x^n\right )\right )}{6 d^2 (d+e x)^6}-\frac {2 e \left (a+b \log \left (c x^n\right )\right )}{5 d^3 (d+e x)^5}-\frac {3 e \left (a+b \log \left (c x^n\right )\right )}{4 d^4 (d+e x)^4}-\frac {4 e \left (a+b \log \left (c x^n\right )\right )}{3 d^5 (d+e x)^3}-\frac {5 e \left (a+b \log \left (c x^n\right )\right )}{2 d^6 (d+e x)^2}+\frac {6 e^2 x \left (a+b \log \left (c x^n\right )\right )}{d^8 (d+e x)}+\frac {7 e \log \left (1+\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^8}-\frac {223 b e n \log (d+e x)}{20 d^8}-\frac {7 b e n \text {Li}_2\left (-\frac {d}{e x}\right )}{d^8} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.39 (sec) , antiderivative size = 401, normalized size of antiderivative = 1.18 \[ \int \frac {a+b \log \left (c x^n\right )}{x^2 (d+e x)^7} \, dx=-\frac {\frac {360 a d}{x}+\frac {360 b d n}{x}+\frac {60 a d^6 e}{(d+e x)^6}+\frac {144 a d^5 e}{(d+e x)^5}-\frac {12 b d^5 e n}{(d+e x)^5}+\frac {270 a d^4 e}{(d+e x)^4}-\frac {51 b d^4 e n}{(d+e x)^4}+\frac {480 a d^3 e}{(d+e x)^3}-\frac {158 b d^3 e n}{(d+e x)^3}+\frac {900 a d^2 e}{(d+e x)^2}-\frac {477 b d^2 e n}{(d+e x)^2}+\frac {2160 a d e}{d+e x}-\frac {1854 b d e n}{d+e x}-4014 b e n \log (x)+\frac {2520 a e \log \left (c x^n\right )}{n}+\frac {360 b d \log \left (c x^n\right )}{x}+\frac {60 b d^6 e \log \left (c x^n\right )}{(d+e x)^6}+\frac {144 b d^5 e \log \left (c x^n\right )}{(d+e x)^5}+\frac {270 b d^4 e \log \left (c x^n\right )}{(d+e x)^4}+\frac {480 b d^3 e \log \left (c x^n\right )}{(d+e x)^3}+\frac {900 b d^2 e \log \left (c x^n\right )}{(d+e x)^2}+\frac {2160 b d e \log \left (c x^n\right )}{d+e x}+\frac {1260 b e \log ^2\left (c x^n\right )}{n}+4014 b e n \log (d+e x)-2520 a e \log \left (1+\frac {e x}{d}\right )-2520 b e \log \left (c x^n\right ) \log \left (1+\frac {e x}{d}\right )-2520 b e n \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{360 d^8} \]

[In]

Integrate[(a + b*Log[c*x^n])/(x^2*(d + e*x)^7),x]

[Out]

-1/360*((360*a*d)/x + (360*b*d*n)/x + (60*a*d^6*e)/(d + e*x)^6 + (144*a*d^5*e)/(d + e*x)^5 - (12*b*d^5*e*n)/(d
+ e*x)^5 + (270*a*d^4*e)/(d + e*x)^4 - (51*b*d^4*e*n)/(d + e*x)^4 + (480*a*d^3*e)/(d + e*x)^3 - (158*b*d^3*e*
n)/(d + e*x)^3 + (900*a*d^2*e)/(d + e*x)^2 - (477*b*d^2*e*n)/(d + e*x)^2 + (2160*a*d*e)/(d + e*x) - (1854*b*d*
e*n)/(d + e*x) - 4014*b*e*n*Log[x] + (2520*a*e*Log[c*x^n])/n + (360*b*d*Log[c*x^n])/x + (60*b*d^6*e*Log[c*x^n]
)/(d + e*x)^6 + (144*b*d^5*e*Log[c*x^n])/(d + e*x)^5 + (270*b*d^4*e*Log[c*x^n])/(d + e*x)^4 + (480*b*d^3*e*Log
[c*x^n])/(d + e*x)^3 + (900*b*d^2*e*Log[c*x^n])/(d + e*x)^2 + (2160*b*d*e*Log[c*x^n])/(d + e*x) + (1260*b*e*Lo
g[c*x^n]^2)/n + 4014*b*e*n*Log[d + e*x] - 2520*a*e*Log[1 + (e*x)/d] - 2520*b*e*Log[c*x^n]*Log[1 + (e*x)/d] - 2
520*b*e*n*PolyLog[2, -((e*x)/d)])/d^8

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 2.68 (sec) , antiderivative size = 508, normalized size of antiderivative = 1.50

method	result	size
risch	\(-\frac {b \ln \left (x^{n}\right ) e}{6 d^{2} \left (e x +d \right )^{6}}+\frac {7 b \ln \left (x^{n}\right ) e \ln \left (e x +d \right )}{d^{8}}-\frac {6 b \ln \left (x^{n}\right ) e}{d^{7} \left (e x +d \right )}-\frac {5 b \ln \left (x^{n}\right ) e}{2 d^{6} \left (e x +d \right )^{2}}-\frac {4 b \ln \left (x^{n}\right ) e}{3 d^{5} \left (e x +d \right )^{3}}-\frac {3 b \ln \left (x^{n}\right ) e}{4 d^{4} \left (e x +d \right )^{4}}-\frac {2 b \ln \left (x^{n}\right ) e}{5 d^{3} \left (e x +d \right )^{5}}-\frac {b \ln \left (x^{n}\right )}{d^{7} x}-\frac {7 b \ln \left (x^{n}\right ) e \ln \left (x \right )}{d^{8}}+\frac {7 b n e \ln \left (x \right )^{2}}{2 d^{8}}-\frac {7 b n e \ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right )}{d^{8}}-\frac {7 b n e \operatorname {dilog}\left (-\frac {e x}{d}\right )}{d^{8}}+\frac {103 b e n}{20 d^{7} \left (e x +d \right )}-\frac {223 b e n \ln \left (e x +d \right )}{20 d^{8}}+\frac {53 b e n}{40 d^{6} \left (e x +d \right )^{2}}+\frac {79 b e n}{180 d^{5} \left (e x +d \right )^{3}}+\frac {17 b e n}{120 d^{4} \left (e x +d \right )^{4}}+\frac {b e n}{30 d^{3} \left (e x +d \right )^{5}}-\frac {b n}{d^{7} x}+\frac {223 b e n \ln \left (x \right )}{20 d^{8}}+\left (-\frac {i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )}{2}+\frac {i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}+\frac {i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}-\frac {i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{2}+b \ln \left (c \right )+a \right ) \left (-\frac {e}{6 d^{2} \left (e x +d \right )^{6}}+\frac {7 e \ln \left (e x +d \right )}{d^{8}}-\frac {6 e}{d^{7} \left (e x +d \right )}-\frac {5 e}{2 d^{6} \left (e x +d \right )^{2}}-\frac {4 e}{3 d^{5} \left (e x +d \right )^{3}}-\frac {3 e}{4 d^{4} \left (e x +d \right )^{4}}-\frac {2 e}{5 d^{3} \left (e x +d \right )^{5}}-\frac {1}{d^{7} x}-\frac {7 e \ln \left (x \right )}{d^{8}}\right )\)	\(508\)

[In]

int((a+b*ln(c*x^n))/x^2/(e*x+d)^7,x,method=_RETURNVERBOSE)

[Out]

-1/6*b*ln(x^n)/d^2/(e*x+d)^6*e+7*b*ln(x^n)/d^8*e*ln(e*x+d)-6*b*ln(x^n)/d^7*e/(e*x+d)-5/2*b*ln(x^n)/d^6/(e*x+d)
^2*e-4/3*b*ln(x^n)/d^5/(e*x+d)^3*e-3/4*b*ln(x^n)/d^4/(e*x+d)^4*e-2/5*b*ln(x^n)/d^3/(e*x+d)^5*e-b*ln(x^n)/d^7/x
-7*b*ln(x^n)/d^8*e*ln(x)+7/2*b*n/d^8*e*ln(x)^2-7*b*n/d^8*e*ln(e*x+d)*ln(-e*x/d)-7*b*n/d^8*e*dilog(-e*x/d)+103/
20*b*e*n/d^7/(e*x+d)-223/20*b*e*n*ln(e*x+d)/d^8+53/40*b*e*n/d^6/(e*x+d)^2+79/180*b*e*n/d^5/(e*x+d)^3+17/120*b*
e*n/d^4/(e*x+d)^4+1/30*b*e*n/d^3/(e*x+d)^5-b*n/d^7/x+223/20*b*e*n*ln(x)/d^8+(-1/2*I*b*Pi*csgn(I*c)*csgn(I*x^n)
*csgn(I*c*x^n)+1/2*I*b*Pi*csgn(I*c)*csgn(I*c*x^n)^2+1/2*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2-1/2*I*b*Pi*csgn(I*c
*x^n)^3+b*ln(c)+a)*(-1/6/d^2/(e*x+d)^6*e+7/d^8*e*ln(e*x+d)-6/d^7*e/(e*x+d)-5/2/d^6/(e*x+d)^2*e-4/3/d^5/(e*x+d)
^3*e-3/4/d^4/(e*x+d)^4*e-2/5/d^3/(e*x+d)^5*e-1/d^7/x-7/d^8*e*ln(x))

Fricas [F]

\[ \int \frac {a+b \log \left (c x^n\right )}{x^2 (d+e x)^7} \, dx=\int { \frac {b \log \left (c x^{n}\right ) + a}{{\left (e x + d\right )}^{7} x^{2}} \,d x } \]

[In]

integrate((a+b*log(c*x^n))/x^2/(e*x+d)^7,x, algorithm="fricas")

[Out]

integral((b*log(c*x^n) + a)/(e^7*x^9 + 7*d*e^6*x^8 + 21*d^2*e^5*x^7 + 35*d^3*e^4*x^6 + 35*d^4*e^3*x^5 + 21*d^5
*e^2*x^4 + 7*d^6*e*x^3 + d^7*x^2), x)

Sympy [A] (verification not implemented)

Time = 146.27 (sec) , antiderivative size = 1685, normalized size of antiderivative = 4.97 \[ \int \frac {a+b \log \left (c x^n\right )}{x^2 (d+e x)^7} \, dx=\text {Too large to display} \]

[In]

integrate((a+b*ln(c*x**n))/x**2/(e*x+d)**7,x)

[Out]

a*e**2*Piecewise((x/d**7, Eq(e, 0)), (-1/(6*e*(d + e*x)**6), True))/d**2 + 2*a*e**2*Piecewise((x/d**6, Eq(e, 0
)), (-1/(5*e*(d + e*x)**5), True))/d**3 + 3*a*e**2*Piecewise((x/d**5, Eq(e, 0)), (-1/(4*e*(d + e*x)**4), True)
)/d**4 + 4*a*e**2*Piecewise((x/d**4, Eq(e, 0)), (-1/(3*e*(d + e*x)**3), True))/d**5 + 5*a*e**2*Piecewise((x/d*
*3, Eq(e, 0)), (-1/(2*e*(d + e*x)**2), True))/d**6 + 6*a*e**2*Piecewise((x/d**2, Eq(e, 0)), (-1/(d*e + e**2*x)
, True))/d**7 - a/(d**7*x) + 7*a*e**2*Piecewise((x/d, Eq(e, 0)), (log(d + e*x)/e, True))/d**8 - 7*a*e*log(x)/d
**8 - b*e**2*n*Piecewise((x/d**7, Eq(e, 0)), (-137*d**4/(360*d**10*e + 1800*d**9*e**2*x + 3600*d**8*e**3*x**2
+ 3600*d**7*e**4*x**3 + 1800*d**6*e**5*x**4 + 360*d**5*e**6*x**5) - 385*d**3*e*x/(360*d**10*e + 1800*d**9*e**2
*x + 3600*d**8*e**3*x**2 + 3600*d**7*e**4*x**3 + 1800*d**6*e**5*x**4 + 360*d**5*e**6*x**5) - 470*d**2*e**2*x**
2/(360*d**10*e + 1800*d**9*e**2*x + 3600*d**8*e**3*x**2 + 3600*d**7*e**4*x**3 + 1800*d**6*e**5*x**4 + 360*d**5
*e**6*x**5) - 270*d*e**3*x**3/(360*d**10*e + 1800*d**9*e**2*x + 3600*d**8*e**3*x**2 + 3600*d**7*e**4*x**3 + 18
00*d**6*e**5*x**4 + 360*d**5*e**6*x**5) - 60*e**4*x**4/(360*d**10*e + 1800*d**9*e**2*x + 3600*d**8*e**3*x**2 +
3600*d**7*e**4*x**3 + 1800*d**6*e**5*x**4 + 360*d**5*e**6*x**5) - log(x)/(6*d**6*e) + log(d/e + x)/(6*d**6*e)
, True))/d**2 + b*e**2*Piecewise((x/d**7, Eq(e, 0)), (-1/(6*e*(d + e*x)**6), True))*log(c*x**n)/d**2 - 2*b*e**
2*n*Piecewise((x/d**6, Eq(e, 0)), (-25*d**3/(60*d**8*e + 240*d**7*e**2*x + 360*d**6*e**3*x**2 + 240*d**5*e**4*
x**3 + 60*d**4*e**5*x**4) - 52*d**2*e*x/(60*d**8*e + 240*d**7*e**2*x + 360*d**6*e**3*x**2 + 240*d**5*e**4*x**3
+ 60*d**4*e**5*x**4) - 42*d*e**2*x**2/(60*d**8*e + 240*d**7*e**2*x + 360*d**6*e**3*x**2 + 240*d**5*e**4*x**3
+ 60*d**4*e**5*x**4) - 12*e**3*x**3/(60*d**8*e + 240*d**7*e**2*x + 360*d**6*e**3*x**2 + 240*d**5*e**4*x**3 + 6
0*d**4*e**5*x**4) - log(x)/(5*d**5*e) + log(d/e + x)/(5*d**5*e), True))/d**3 + 2*b*e**2*Piecewise((x/d**6, Eq(
e, 0)), (-1/(5*e*(d + e*x)**5), True))*log(c*x**n)/d**3 - 3*b*e**2*n*Piecewise((x/d**5, Eq(e, 0)), (-11*d**2/(
24*d**6*e + 72*d**5*e**2*x + 72*d**4*e**3*x**2 + 24*d**3*e**4*x**3) - 15*d*e*x/(24*d**6*e + 72*d**5*e**2*x + 7
2*d**4*e**3*x**2 + 24*d**3*e**4*x**3) - 6*e**2*x**2/(24*d**6*e + 72*d**5*e**2*x + 72*d**4*e**3*x**2 + 24*d**3*
e**4*x**3) - log(x)/(4*d**4*e) + log(d/e + x)/(4*d**4*e), True))/d**4 + 3*b*e**2*Piecewise((x/d**5, Eq(e, 0)),
(-1/(4*e*(d + e*x)**4), True))*log(c*x**n)/d**4 - 4*b*e**2*n*Piecewise((x/d**4, Eq(e, 0)), (-3*d/(6*d**4*e +
12*d**3*e**2*x + 6*d**2*e**3*x**2) - 2*e*x/(6*d**4*e + 12*d**3*e**2*x + 6*d**2*e**3*x**2) - log(x)/(3*d**3*e)
+ log(d/e + x)/(3*d**3*e), True))/d**5 + 4*b*e**2*Piecewise((x/d**4, Eq(e, 0)), (-1/(3*e*(d + e*x)**3), True))
*log(c*x**n)/d**5 - 5*b*e**2*n*Piecewise((x/d**3, Eq(e, 0)), (-1/(2*d**2*e + 2*d*e**2*x) - log(x)/(2*d**2*e) +
log(d/e + x)/(2*d**2*e), True))/d**6 + 5*b*e**2*Piecewise((x/d**3, Eq(e, 0)), (-1/(2*e*(d + e*x)**2), True))*
log(c*x**n)/d**6 - 6*b*e**2*n*Piecewise((x/d**2, Eq(e, 0)), (-log(x)/(d*e) + log(d/e + x)/(d*e), True))/d**7 +
6*b*e**2*Piecewise((x/d**2, Eq(e, 0)), (-1/(d*e + e**2*x), True))*log(c*x**n)/d**7 - b*n/(d**7*x) - b*log(c*x
**n)/(d**7*x) - 7*b*e**2*n*Piecewise((x/d, Eq(e, 0)), (Piecewise((-polylog(2, e*x*exp_polar(I*pi)/d), (Abs(x)
< 1) & (1/Abs(x) < 1)), (log(d)*log(x) - polylog(2, e*x*exp_polar(I*pi)/d), Abs(x) < 1), (-log(d)*log(1/x) - p
olylog(2, e*x*exp_polar(I*pi)/d), 1/Abs(x) < 1), (-meijerg(((), (1, 1)), ((0, 0), ()), x)*log(d) + meijerg(((1
, 1), ()), ((), (0, 0)), x)*log(d) - polylog(2, e*x*exp_polar(I*pi)/d), True))/e, True))/d**8 + 7*b*e**2*Piece
wise((x/d, Eq(e, 0)), (log(d + e*x)/e, True))*log(c*x**n)/d**8 + 7*b*e*n*log(x)**2/(2*d**8) - 7*b*e*log(x)*log
(c*x**n)/d**8

Maxima [F]

\[ \int \frac {a+b \log \left (c x^n\right )}{x^2 (d+e x)^7} \, dx=\int { \frac {b \log \left (c x^{n}\right ) + a}{{\left (e x + d\right )}^{7} x^{2}} \,d x } \]

[In]

integrate((a+b*log(c*x^n))/x^2/(e*x+d)^7,x, algorithm="maxima")

[Out]

-1/60*a*((420*e^6*x^6 + 2310*d*e^5*x^5 + 5180*d^2*e^4*x^4 + 5985*d^3*e^3*x^3 + 3654*d^4*e^2*x^2 + 1029*d^5*e*x
+ 60*d^6)/(d^7*e^6*x^7 + 6*d^8*e^5*x^6 + 15*d^9*e^4*x^5 + 20*d^10*e^3*x^4 + 15*d^11*e^2*x^3 + 6*d^12*e*x^2 +
d^13*x) - 420*e*log(e*x + d)/d^8 + 420*e*log(x)/d^8) + b*integrate((log(c) + log(x^n))/(e^7*x^9 + 7*d*e^6*x^8
+ 21*d^2*e^5*x^7 + 35*d^3*e^4*x^6 + 35*d^4*e^3*x^5 + 21*d^5*e^2*x^4 + 7*d^6*e*x^3 + d^7*x^2), x)

Giac [F]

\[ \int \frac {a+b \log \left (c x^n\right )}{x^2 (d+e x)^7} \, dx=\int { \frac {b \log \left (c x^{n}\right ) + a}{{\left (e x + d\right )}^{7} x^{2}} \,d x } \]

[In]

integrate((a+b*log(c*x^n))/x^2/(e*x+d)^7,x, algorithm="giac")

[Out]

integrate((b*log(c*x^n) + a)/((e*x + d)^7*x^2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {a+b \log \left (c x^n\right )}{x^2 (d+e x)^7} \, dx=\int \frac {a+b\,\ln \left (c\,x^n\right )}{x^2\,{\left (d+e\,x\right )}^7} \,d x \]

[In]

int((a + b*log(c*x^n))/(x^2*(d + e*x)^7),x)

[Out]

int((a + b*log(c*x^n))/(x^2*(d + e*x)^7), x)