3.1.71 \(\int \frac {a+b \log (c x^n)}{x (d+e x)^7} \, dx\) [71]

Optimal. Leaf size=294 \[ -\frac {b n}{30 d^2 (d+e x)^5}-\frac {11 b n}{120 d^3 (d+e x)^4}-\frac {37 b n}{180 d^4 (d+e x)^3}-\frac {19 b n}{40 d^5 (d+e x)^2}-\frac {29 b n}{20 d^6 (d+e x)}-\frac {29 b n \log (x)}{20 d^7}+\frac {a+b \log \left (c x^n\right )}{6 d (d+e x)^6}+\frac {a+b \log \left (c x^n\right )}{5 d^2 (d+e x)^5}+\frac {a+b \log \left (c x^n\right )}{4 d^3 (d+e x)^4}+\frac {a+b \log \left (c x^n\right )}{3 d^4 (d+e x)^3}+\frac {a+b \log \left (c x^n\right )}{2 d^5 (d+e x)^2}-\frac {e x \left (a+b \log \left (c x^n\right )\right )}{d^7 (d+e x)}-\frac {\log \left (1+\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^7}+\frac {49 b n \log (d+e x)}{20 d^7}+\frac {b n \text {Li}_2\left (-\frac {d}{e x}\right )}{d^7} \]

[Out]

-1/30*b*n/d^2/(e*x+d)^5-11/120*b*n/d^3/(e*x+d)^4-37/180*b*n/d^4/(e*x+d)^3-19/40*b*n/d^5/(e*x+d)^2-29/20*b*n/d^
6/(e*x+d)-29/20*b*n*ln(x)/d^7+1/6*(a+b*ln(c*x^n))/d/(e*x+d)^6+1/5*(a+b*ln(c*x^n))/d^2/(e*x+d)^5+1/4*(a+b*ln(c*
x^n))/d^3/(e*x+d)^4+1/3*(a+b*ln(c*x^n))/d^4/(e*x+d)^3+1/2*(a+b*ln(c*x^n))/d^5/(e*x+d)^2-e*x*(a+b*ln(c*x^n))/d^
7/(e*x+d)-ln(1+d/e/x)*(a+b*ln(c*x^n))/d^7+49/20*b*n*ln(e*x+d)/d^7+b*n*polylog(2,-d/e/x)/d^7

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Rubi [A]
time = 0.48, antiderivative size = 294, normalized size of antiderivative = 1.00, number of steps used = 25, number of rules used = 7, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2389, 2379, 2438, 2351, 31, 2356, 46} \begin {gather*} \frac {b n \text {PolyLog}\left (2,-\frac {d}{e x}\right )}{d^7}-\frac {\log \left (\frac {d}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d^7}-\frac {e x \left (a+b \log \left (c x^n\right )\right )}{d^7 (d+e x)}+\frac {a+b \log \left (c x^n\right )}{2 d^5 (d+e x)^2}+\frac {a+b \log \left (c x^n\right )}{3 d^4 (d+e x)^3}+\frac {a+b \log \left (c x^n\right )}{4 d^3 (d+e x)^4}+\frac {a+b \log \left (c x^n\right )}{5 d^2 (d+e x)^5}+\frac {a+b \log \left (c x^n\right )}{6 d (d+e x)^6}+\frac {49 b n \log (d+e x)}{20 d^7}-\frac {29 b n \log (x)}{20 d^7}-\frac {29 b n}{20 d^6 (d+e x)}-\frac {19 b n}{40 d^5 (d+e x)^2}-\frac {37 b n}{180 d^4 (d+e x)^3}-\frac {11 b n}{120 d^3 (d+e x)^4}-\frac {b n}{30 d^2 (d+e x)^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/30*(b*n)/(d^2*(d + e*x)^5) - (11*b*n)/(120*d^3*(d + e*x)^4) - (37*b*n)/(180*d^4*(d + e*x)^3) - (19*b*n)/(40
*d^5*(d + e*x)^2) - (29*b*n)/(20*d^6*(d + e*x)) - (29*b*n*Log[x])/(20*d^7) + (a + b*Log[c*x^n])/(6*d*(d + e*x)
^6) + (a + b*Log[c*x^n])/(5*d^2*(d + e*x)^5) + (a + b*Log[c*x^n])/(4*d^3*(d + e*x)^4) + (a + b*Log[c*x^n])/(3*
d^4*(d + e*x)^3) + (a + b*Log[c*x^n])/(2*d^5*(d + e*x)^2) - (e*x*(a + b*Log[c*x^n]))/(d^7*(d + e*x)) - (Log[1
+ d/(e*x)]*(a + b*Log[c*x^n]))/d^7 + (49*b*n*Log[d + e*x])/(20*d^7) + (b*n*PolyLog[2, -(d/(e*x))])/d^7

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 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 2389

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_))/(x_), x_Symbol] :> Dist[1/d, Int[(d
+ e*x)^(q + 1)*((a + b*Log[c*x^n])^p/x), x], x] - Dist[e/d, Int[(d + e*x)^q*(a + b*Log[c*x^n])^p, x], x] /; F
reeQ[{a, b, c, d, e, n}, x] && IGtQ[p, 0] && LtQ[q, -1] && IntegerQ[2*q]

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*} \int \frac {a+b \log \left (c x^n\right )}{x (d+e x)^7} \, dx &=\frac {\int \frac {a+b \log \left (c x^n\right )}{x (d+e x)^6} \, dx}{d}-\frac {e \int \frac {a+b \log \left (c x^n\right )}{(d+e x)^7} \, dx}{d}\\ &=\frac {a+b \log \left (c x^n\right )}{6 d (d+e x)^6}+\frac {\int \frac {a+b \log \left (c x^n\right )}{x (d+e x)^5} \, dx}{d^2}-\frac {e \int \frac {a+b \log \left (c x^n\right )}{(d+e x)^6} \, dx}{d^2}-\frac {(b n) \int \frac {1}{x (d+e x)^6} \, dx}{6 d}\\ &=\frac {a+b \log \left (c x^n\right )}{6 d (d+e x)^6}+\frac {a+b \log \left (c x^n\right )}{5 d^2 (d+e x)^5}+\frac {\int \frac {a+b \log \left (c x^n\right )}{x (d+e x)^4} \, dx}{d^3}-\frac {e \int \frac {a+b \log \left (c x^n\right )}{(d+e x)^5} \, dx}{d^3}-\frac {(b n) \int \frac {1}{x (d+e x)^5} \, dx}{5 d^2}-\frac {(b 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}\\ &=-\frac {b n}{30 d^2 (d+e x)^5}-\frac {b n}{24 d^3 (d+e x)^4}-\frac {b n}{18 d^4 (d+e x)^3}-\frac {b n}{12 d^5 (d+e x)^2}-\frac {b n}{6 d^6 (d+e x)}-\frac {b n \log (x)}{6 d^7}+\frac {a+b \log \left (c x^n\right )}{6 d (d+e x)^6}+\frac {a+b \log \left (c x^n\right )}{5 d^2 (d+e x)^5}+\frac {a+b \log \left (c x^n\right )}{4 d^3 (d+e x)^4}+\frac {b n \log (d+e x)}{6 d^7}+\frac {\int \frac {a+b \log \left (c x^n\right )}{x (d+e x)^3} \, dx}{d^4}-\frac {e \int \frac {a+b \log \left (c x^n\right )}{(d+e x)^4} \, dx}{d^4}-\frac {(b n) \int \frac {1}{x (d+e x)^4} \, dx}{4 d^3}-\frac {(b 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^2}\\ &=-\frac {b n}{30 d^2 (d+e x)^5}-\frac {11 b n}{120 d^3 (d+e x)^4}-\frac {11 b n}{90 d^4 (d+e x)^3}-\frac {11 b n}{60 d^5 (d+e x)^2}-\frac {11 b n}{30 d^6 (d+e x)}-\frac {11 b n \log (x)}{30 d^7}+\frac {a+b \log \left (c x^n\right )}{6 d (d+e x)^6}+\frac {a+b \log \left (c x^n\right )}{5 d^2 (d+e x)^5}+\frac {a+b \log \left (c x^n\right )}{4 d^3 (d+e x)^4}+\frac {a+b \log \left (c x^n\right )}{3 d^4 (d+e x)^3}+\frac {11 b n \log (d+e x)}{30 d^7}+\frac {\int \frac {a+b \log \left (c x^n\right )}{x (d+e x)^2} \, dx}{d^5}-\frac {e \int \frac {a+b \log \left (c x^n\right )}{(d+e x)^3} \, dx}{d^5}-\frac {(b n) \int \frac {1}{x (d+e x)^3} \, dx}{3 d^4}-\frac {(b 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^3}\\ &=-\frac {b n}{30 d^2 (d+e x)^5}-\frac {11 b n}{120 d^3 (d+e x)^4}-\frac {37 b n}{180 d^4 (d+e x)^3}-\frac {37 b n}{120 d^5 (d+e x)^2}-\frac {37 b n}{60 d^6 (d+e x)}-\frac {37 b n \log (x)}{60 d^7}+\frac {a+b \log \left (c x^n\right )}{6 d (d+e x)^6}+\frac {a+b \log \left (c x^n\right )}{5 d^2 (d+e x)^5}+\frac {a+b \log \left (c x^n\right )}{4 d^3 (d+e x)^4}+\frac {a+b \log \left (c x^n\right )}{3 d^4 (d+e x)^3}+\frac {a+b \log \left (c x^n\right )}{2 d^5 (d+e x)^2}+\frac {37 b n \log (d+e x)}{60 d^7}+\frac {\int \frac {a+b \log \left (c x^n\right )}{x (d+e x)} \, dx}{d^6}-\frac {e \int \frac {a+b \log \left (c x^n\right )}{(d+e x)^2} \, dx}{d^6}-\frac {(b n) \int \frac {1}{x (d+e x)^2} \, dx}{2 d^5}-\frac {(b 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^4}\\ &=-\frac {b n}{30 d^2 (d+e x)^5}-\frac {11 b n}{120 d^3 (d+e x)^4}-\frac {37 b n}{180 d^4 (d+e x)^3}-\frac {19 b n}{40 d^5 (d+e x)^2}-\frac {19 b n}{20 d^6 (d+e x)}-\frac {19 b n \log (x)}{20 d^7}+\frac {a+b \log \left (c x^n\right )}{6 d (d+e x)^6}+\frac {a+b \log \left (c x^n\right )}{5 d^2 (d+e x)^5}+\frac {a+b \log \left (c x^n\right )}{4 d^3 (d+e x)^4}+\frac {a+b \log \left (c x^n\right )}{3 d^4 (d+e x)^3}+\frac {a+b \log \left (c x^n\right )}{2 d^5 (d+e x)^2}-\frac {e x \left (a+b \log \left (c x^n\right )\right )}{d^7 (d+e x)}+\frac {19 b n \log (d+e x)}{20 d^7}+\frac {\int \frac {a+b \log \left (c x^n\right )}{x} \, dx}{d^7}-\frac {e \int \frac {a+b \log \left (c x^n\right )}{d+e x} \, dx}{d^7}-\frac {(b 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^5}+\frac {(b e n) \int \frac {1}{d+e x} \, dx}{d^7}\\ &=-\frac {b n}{30 d^2 (d+e x)^5}-\frac {11 b n}{120 d^3 (d+e x)^4}-\frac {37 b n}{180 d^4 (d+e x)^3}-\frac {19 b n}{40 d^5 (d+e x)^2}-\frac {29 b n}{20 d^6 (d+e x)}-\frac {29 b n \log (x)}{20 d^7}+\frac {a+b \log \left (c x^n\right )}{6 d (d+e x)^6}+\frac {a+b \log \left (c x^n\right )}{5 d^2 (d+e x)^5}+\frac {a+b \log \left (c x^n\right )}{4 d^3 (d+e x)^4}+\frac {a+b \log \left (c x^n\right )}{3 d^4 (d+e x)^3}+\frac {a+b \log \left (c x^n\right )}{2 d^5 (d+e x)^2}-\frac {e x \left (a+b \log \left (c x^n\right )\right )}{d^7 (d+e x)}+\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 b d^7 n}+\frac {49 b n \log (d+e x)}{20 d^7}-\frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{d^7}+\frac {(b n) \int \frac {\log \left (1+\frac {e x}{d}\right )}{x} \, dx}{d^7}\\ &=-\frac {b n}{30 d^2 (d+e x)^5}-\frac {11 b n}{120 d^3 (d+e x)^4}-\frac {37 b n}{180 d^4 (d+e x)^3}-\frac {19 b n}{40 d^5 (d+e x)^2}-\frac {29 b n}{20 d^6 (d+e x)}-\frac {29 b n \log (x)}{20 d^7}+\frac {a+b \log \left (c x^n\right )}{6 d (d+e x)^6}+\frac {a+b \log \left (c x^n\right )}{5 d^2 (d+e x)^5}+\frac {a+b \log \left (c x^n\right )}{4 d^3 (d+e x)^4}+\frac {a+b \log \left (c x^n\right )}{3 d^4 (d+e x)^3}+\frac {a+b \log \left (c x^n\right )}{2 d^5 (d+e x)^2}-\frac {e x \left (a+b \log \left (c x^n\right )\right )}{d^7 (d+e x)}+\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 b d^7 n}+\frac {49 b n \log (d+e x)}{20 d^7}-\frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{d^7}-\frac {b n \text {Li}_2\left (-\frac {e x}{d}\right )}{d^7}\\ \end {align*}

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Mathematica [A]
time = 0.24, size = 349, normalized size = 1.19 \begin {gather*} \frac {\frac {60 a d^6}{(d+e x)^6}+\frac {72 a d^5}{(d+e x)^5}-\frac {12 b d^5 n}{(d+e x)^5}+\frac {90 a d^4}{(d+e x)^4}-\frac {33 b d^4 n}{(d+e x)^4}+\frac {120 a d^3}{(d+e x)^3}-\frac {74 b d^3 n}{(d+e x)^3}+\frac {180 a d^2}{(d+e x)^2}-\frac {171 b d^2 n}{(d+e x)^2}+\frac {360 a d}{d+e x}-\frac {522 b d n}{d+e x}-882 b n \log (x)+\frac {360 a \log \left (c x^n\right )}{n}+\frac {60 b d^6 \log \left (c x^n\right )}{(d+e x)^6}+\frac {72 b d^5 \log \left (c x^n\right )}{(d+e x)^5}+\frac {90 b d^4 \log \left (c x^n\right )}{(d+e x)^4}+\frac {120 b d^3 \log \left (c x^n\right )}{(d+e x)^3}+\frac {180 b d^2 \log \left (c x^n\right )}{(d+e x)^2}+\frac {360 b d \log \left (c x^n\right )}{d+e x}+\frac {180 b \log ^2\left (c x^n\right )}{n}+882 b n \log (d+e x)-360 a \log \left (1+\frac {e x}{d}\right )-360 b \log \left (c x^n\right ) \log \left (1+\frac {e x}{d}\right )-360 b n \text {Li}_2\left (-\frac {e x}{d}\right )}{360 d^7} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((60*a*d^6)/(d + e*x)^6 + (72*a*d^5)/(d + e*x)^5 - (12*b*d^5*n)/(d + e*x)^5 + (90*a*d^4)/(d + e*x)^4 - (33*b*d
^4*n)/(d + e*x)^4 + (120*a*d^3)/(d + e*x)^3 - (74*b*d^3*n)/(d + e*x)^3 + (180*a*d^2)/(d + e*x)^2 - (171*b*d^2*
n)/(d + e*x)^2 + (360*a*d)/(d + e*x) - (522*b*d*n)/(d + e*x) - 882*b*n*Log[x] + (360*a*Log[c*x^n])/n + (60*b*d
^6*Log[c*x^n])/(d + e*x)^6 + (72*b*d^5*Log[c*x^n])/(d + e*x)^5 + (90*b*d^4*Log[c*x^n])/(d + e*x)^4 + (120*b*d^
3*Log[c*x^n])/(d + e*x)^3 + (180*b*d^2*Log[c*x^n])/(d + e*x)^2 + (360*b*d*Log[c*x^n])/(d + e*x) + (180*b*Log[c
*x^n]^2)/n + 882*b*n*Log[d + e*x] - 360*a*Log[1 + (e*x)/d] - 360*b*Log[c*x^n]*Log[1 + (e*x)/d] - 360*b*n*PolyL
og[2, -((e*x)/d)])/(360*d^7)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.13, size = 1427, normalized size = 4.85

method	result	size
risch	\(\text {Expression too large to display}\)	\(1427\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b*n/d^7*ln(e*x+d)*ln(-e*x/d)-1/4*I*b*Pi*csgn(I*c*x^n)^3/d^5/(e*x+d)^2-1/2*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2/d
^7*ln(e*x+d)+1/2*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2/d^6/(e*x+d)-1/12*I*b*Pi*csgn(I*c*x^n)^3/d/(e*x+d)^6-1/2*I*
b*Pi*csgn(I*c*x^n)^3/d^7*ln(x)+1/2*I*b*Pi*csgn(I*c*x^n)^3/d^7*ln(e*x+d)+1/2*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2
/d^7*ln(x)+1/8*I*b*Pi*csgn(I*c)*csgn(I*c*x^n)^2/d^3/(e*x+d)^4+1/4*I*b*Pi*csgn(I*c)*csgn(I*c*x^n)^2/d^5/(e*x+d)
^2+1/6*I*b*Pi*csgn(I*c)*csgn(I*c*x^n)^2/d^4/(e*x+d)^3-1/2*I*b*Pi*csgn(I*c)*csgn(I*c*x^n)^2/d^7*ln(e*x+d)+1/2*I
*b*Pi*csgn(I*c)*csgn(I*c*x^n)^2/d^7*ln(x)-1/2*I*b*Pi*csgn(I*c*x^n)^3/d^6/(e*x+d)-1/6*I*b*Pi*csgn(I*c*x^n)^3/d^
4/(e*x+d)^3-1/8*I*b*Pi*csgn(I*c*x^n)^3/d^3/(e*x+d)^4-1/10*I*b*Pi*csgn(I*c*x^n)^3/d^2/(e*x+d)^5+a/d^7*ln(x)-a/d
^7*ln(e*x+d)+a/d^6/(e*x+d)+1/2*a/d^5/(e*x+d)^2+1/3*a/d^4/(e*x+d)^3+1/4*a/d^3/(e*x+d)^4+1/5*a/d^2/(e*x+d)^5+1/6
*a/d/(e*x+d)^6+1/10*I*b*Pi*csgn(I*c)*csgn(I*c*x^n)^2/d^2/(e*x+d)^5-1/2*b*n/d^7*ln(x)^2+b*n/d^7*dilog(-e*x/d)+b
*ln(c)/d^6/(e*x+d)+1/2*b*ln(c)/d^5/(e*x+d)^2+1/3*b*ln(c)/d^4/(e*x+d)^3+1/4*b*ln(c)/d^3/(e*x+d)^4+1/5*b*ln(c)/d
^2/(e*x+d)^5+1/6*b*ln(c)/d/(e*x+d)^6-b*ln(c)/d^7*ln(e*x+d)+b*ln(c)/d^7*ln(x)-1/12*I*b*Pi*csgn(I*c)*csgn(I*x^n)
*csgn(I*c*x^n)/d/(e*x+d)^6-1/8*I*b*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)/d^3/(e*x+d)^4-1/10*I*b*Pi*csgn(I*c)*
csgn(I*x^n)*csgn(I*c*x^n)/d^2/(e*x+d)^5-b*ln(x^n)/d^7*ln(e*x+d)+b*ln(x^n)/d^6/(e*x+d)+1/2*b*ln(x^n)/d^5/(e*x+d
)^2+1/3*b*ln(x^n)/d^4/(e*x+d)^3+1/4*b*ln(x^n)/d^3/(e*x+d)^4+1/5*b*ln(x^n)/d^2/(e*x+d)^5+1/6*b*ln(x^n)/d/(e*x+d
)^6+b*ln(x^n)/d^7*ln(x)-1/4*I*b*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)/d^5/(e*x+d)^2-1/2*I*b*Pi*csgn(I*c)*csgn
(I*x^n)*csgn(I*c*x^n)/d^6/(e*x+d)-1/6*I*b*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)/d^4/(e*x+d)^3+1/2*I*b*Pi*csgn
(I*c)*csgn(I*x^n)*csgn(I*c*x^n)/d^7*ln(e*x+d)-1/2*I*b*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)/d^7*ln(x)+1/12*I*
b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2/d/(e*x+d)^6+1/8*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2/d^3/(e*x+d)^4+1/4*I*b*Pi*c
sgn(I*x^n)*csgn(I*c*x^n)^2/d^5/(e*x+d)^2+1/6*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2/d^4/(e*x+d)^3+1/10*I*b*Pi*csgn
(I*x^n)*csgn(I*c*x^n)^2/d^2/(e*x+d)^5+1/12*I*b*Pi*csgn(I*c)*csgn(I*c*x^n)^2/d/(e*x+d)^6+1/2*I*b*Pi*csgn(I*c)*c
sgn(I*c*x^n)^2/d^6/(e*x+d)-49/20*b*n*ln(x)/d^7+49/20*b*n*ln(e*x+d)/d^7-1/30*b*n/d^2/(e*x+d)^5-11/120*b*n/d^3/(
e*x+d)^4-37/180*b*n/d^4/(e*x+d)^3-19/40*b*n/d^5/(e*x+d)^2-29/20*b*n/d^6/(e*x+d)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/60*a*((60*x^5*e^5 + 330*d*x^4*e^4 + 740*d^2*x^3*e^3 + 855*d^3*x^2*e^2 + 522*d^4*x*e + 147*d^5)/(d^6*x^6*e^6
+ 6*d^7*x^5*e^5 + 15*d^8*x^4*e^4 + 20*d^9*x^3*e^3 + 15*d^10*x^2*e^2 + 6*d^11*x*e + d^12) - 60*log(x*e + d)/d^7
+ 60*log(x)/d^7) + b*integrate((log(c) + log(x^n))/(x^8*e^7 + 7*d*x^7*e^6 + 21*d^2*x^6*e^5 + 35*d^3*x^5*e^4 +
35*d^4*x^4*e^3 + 21*d^5*x^3*e^2 + 7*d^6*x^2*e + d^7*x), x)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]
time = 159.26, size = 1518, normalized size = 5.16 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a*e*Piecewise((x/d**7, Eq(e, 0)), (-1/(6*e*(d + e*x)**6), True))/d - a*e*Piecewise((x/d**6, Eq(e, 0)), (-1/(5
*e*(d + e*x)**5), True))/d**2 - a*e*Piecewise((x/d**5, Eq(e, 0)), (-1/(4*e*(d + e*x)**4), True))/d**3 - a*e*Pi
ecewise((x/d**4, Eq(e, 0)), (-1/(3*e*(d + e*x)**3), True))/d**4 - a*e*Piecewise((x/d**3, Eq(e, 0)), (-1/(2*e*(
d + e*x)**2), True))/d**5 - a*e*Piecewise((x/d**2, Eq(e, 0)), (-1/(d*e + e**2*x), True))/d**6 - a*e*Piecewise(
(x/d, Eq(e, 0)), (log(d + e*x)/e, True))/d**7 + a*log(x)/d**7 + b*e**6*n*Piecewise((-1/(e**7*x), Eq(d, 0)), (-
137*d**4/(360*d**5*e**6 + 1800*d**4*e**7*x + 3600*d**3*e**8*x**2 + 3600*d**2*e**9*x**3 + 1800*d*e**10*x**4 + 3
60*e**11*x**5) - 625*d**3*e*x/(360*d**5*e**6 + 1800*d**4*e**7*x + 3600*d**3*e**8*x**2 + 3600*d**2*e**9*x**3 +
1800*d*e**10*x**4 + 360*e**11*x**5) - 1100*d**2*e**2*x**2/(360*d**5*e**6 + 1800*d**4*e**7*x + 3600*d**3*e**8*x
**2 + 3600*d**2*e**9*x**3 + 1800*d*e**10*x**4 + 360*e**11*x**5) - 900*d*e**3*x**3/(360*d**5*e**6 + 1800*d**4*e
**7*x + 3600*d**3*e**8*x**2 + 3600*d**2*e**9*x**3 + 1800*d*e**10*x**4 + 360*e**11*x**5) - 300*e**4*x**4/(360*d
**5*e**6 + 1800*d**4*e**7*x + 3600*d**3*e**8*x**2 + 3600*d**2*e**9*x**3 + 1800*d*e**10*x**4 + 360*e**11*x**5)
- log(d + e*x)/(6*d*e**6), True))/d**6 - b*e**6*Piecewise((1/(e**7*x), Eq(d, 0)), (-1/(6*d*(d/x + e)**6), True
))*log(c*x**n)/d**6 - 6*b*e**5*n*Piecewise((-1/(e**6*x), Eq(d, 0)), (-25*d**3/(60*d**4*e**5 + 240*d**3*e**6*x
+ 360*d**2*e**7*x**2 + 240*d*e**8*x**3 + 60*e**9*x**4) - 88*d**2*e*x/(60*d**4*e**5 + 240*d**3*e**6*x + 360*d**
2*e**7*x**2 + 240*d*e**8*x**3 + 60*e**9*x**4) - 108*d*e**2*x**2/(60*d**4*e**5 + 240*d**3*e**6*x + 360*d**2*e**
7*x**2 + 240*d*e**8*x**3 + 60*e**9*x**4) - 48*e**3*x**3/(60*d**4*e**5 + 240*d**3*e**6*x + 360*d**2*e**7*x**2 +
240*d*e**8*x**3 + 60*e**9*x**4) - log(d + e*x)/(5*d*e**5), True))/d**6 + 6*b*e**5*Piecewise((1/(e**6*x), Eq(d
, 0)), (-1/(5*d*(d/x + e)**5), True))*log(c*x**n)/d**6 + 15*b*e**4*n*Piecewise((-1/(e**5*x), Eq(d, 0)), (-11*d
**2/(24*d**3*e**4 + 72*d**2*e**5*x + 72*d*e**6*x**2 + 24*e**7*x**3) - 27*d*e*x/(24*d**3*e**4 + 72*d**2*e**5*x
+ 72*d*e**6*x**2 + 24*e**7*x**3) - 18*e**2*x**2/(24*d**3*e**4 + 72*d**2*e**5*x + 72*d*e**6*x**2 + 24*e**7*x**3
) - log(d + e*x)/(4*d*e**4), True))/d**6 - 15*b*e**4*Piecewise((1/(e**5*x), Eq(d, 0)), (-1/(4*d*(d/x + e)**4),
True))*log(c*x**n)/d**6 - 20*b*e**3*n*Piecewise((-1/(e**4*x), Eq(d, 0)), (-3*d/(6*d**2*e**3 + 12*d*e**4*x + 6
*e**5*x**2) - 4*e*x/(6*d**2*e**3 + 12*d*e**4*x + 6*e**5*x**2) - log(d + e*x)/(3*d*e**3), True))/d**6 + 20*b*e*
*3*Piecewise((1/(e**4*x), Eq(d, 0)), (-1/(3*d*(d/x + e)**3), True))*log(c*x**n)/d**6 + 15*b*e**2*n*Piecewise((
-1/(e**3*x), Eq(d, 0)), (-1/(2*d*e**2 + 2*e**3*x) - log(d + e*x)/(2*d*e**2), True))/d**6 - 15*b*e**2*Piecewise
((1/(e**3*x), Eq(d, 0)), (-1/(2*d*(d/x + e)**2), True))*log(c*x**n)/d**6 - 6*b*e*n*Piecewise((-1/(e**2*x), Eq(
d, 0)), (-log(d**2 + d*e*x)/(d*e), True))/d**6 + 6*b*e*Piecewise((1/(e**2*x), Eq(d, 0)), (-1/(d**2/x + d*e), T
rue))*log(c*x**n)/d**6 + b*n*Piecewise((-1/(e*x), Eq(d, 0)), (Piecewise((polylog(2, d*exp_polar(I*pi)/(e*x)),
(Abs(x) < 1) & (1/Abs(x) < 1)), (log(e)*log(x) + polylog(2, d*exp_polar(I*pi)/(e*x)), Abs(x) < 1), (-log(e)*lo
g(1/x) + polylog(2, d*exp_polar(I*pi)/(e*x)), 1/Abs(x) < 1), (-meijerg(((), (1, 1)), ((0, 0), ()), x)*log(e) +
meijerg(((1, 1), ()), ((), (0, 0)), x)*log(e) + polylog(2, d*exp_polar(I*pi)/(e*x)), True))/d, True))/d**6 -
b*Piecewise((1/(e*x), Eq(d, 0)), (log(d/x + e)/d, True))*log(c*x**n)/d**6

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {a+b\,\ln \left (c\,x^n\right )}{x\,{\left (d+e\,x\right )}^7} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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