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

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

[Out]

-(b*n)/(30*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

________________________________________________________________________________________

Rubi [A]  time = 0.726049, antiderivative size = 316, normalized size of antiderivative = 1.07, number of steps used = 27, number of rules used = 9, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {2347, 2344, 2301, 2317, 2391, 2314, 31, 2319, 44} \[ -\frac{b n \text{PolyLog}\left (2,-\frac{e x}{d}\right )}{d^7}-\frac{\log \left (\frac{e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d^7}-\frac{e x \left (a+b \log \left (c x^n\right )\right )}{d^7 (d+e x)}+\frac{a+b \log \left (c x^n\right )}{2 d^5 (d+e x)^2}+\frac{a+b \log \left (c x^n\right )}{3 d^4 (d+e x)^3}+\frac{a+b \log \left (c x^n\right )}{4 d^3 (d+e x)^4}+\frac{a+b \log \left (c x^n\right )}{5 d^2 (d+e x)^5}+\frac{\left (a+b \log \left (c x^n\right )\right )^2}{2 b d^7 n}+\frac{a+b \log \left (c x^n\right )}{6 d (d+e x)^6}-\frac{29 b n}{20 d^6 (d+e x)}-\frac{19 b n}{40 d^5 (d+e x)^2}-\frac{37 b n}{180 d^4 (d+e x)^3}-\frac{11 b n}{120 d^3 (d+e x)^4}-\frac{b n}{30 d^2 (d+e x)^5}+\frac{49 b n \log (d+e x)}{20 d^7}-\frac{29 b n \log (x)}{20 d^7} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(b*n)/(30*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)) + (a + b*Lo
g[c*x^n])^2/(2*b*d^7*n) + (49*b*n*Log[d + e*x])/(20*d^7) - ((a + b*Log[c*x^n])*Log[1 + (e*x)/d])/d^7 - (b*n*Po
lyLog[2, -((e*x)/d)])/d^7

Rule 2347

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 2344

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

Rule 2301

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

Rule 2317

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

Rule 2391

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]

Rule 2314

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 31

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

Rule 2319

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 44

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] && L
tQ[m + n + 2, 0])

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*}

Mathematica [A]  time = 0.352559, size = 349, normalized size = 1.19 \[ \frac{-360 b n \text{PolyLog}\left (2,-\frac{e x}{d}\right )+\frac{360 a \log \left (c x^n\right )}{n}+\frac{60 a d^6}{(d+e x)^6}+\frac{72 a d^5}{(d+e x)^5}+\frac{90 a d^4}{(d+e x)^4}+\frac{120 a d^3}{(d+e x)^3}+\frac{180 a d^2}{(d+e x)^2}+\frac{360 a d}{d+e x}-360 a \log \left (\frac{e x}{d}+1\right )+\frac{60 b d^6 \log \left (c x^n\right )}{(d+e x)^6}+\frac{72 b d^5 \log \left (c x^n\right )}{(d+e x)^5}+\frac{90 b d^4 \log \left (c x^n\right )}{(d+e x)^4}+\frac{120 b d^3 \log \left (c x^n\right )}{(d+e x)^3}+\frac{180 b d^2 \log \left (c x^n\right )}{(d+e x)^2}+\frac{360 b d \log \left (c x^n\right )}{d+e x}-360 b \log \left (c x^n\right ) \log \left (\frac{e x}{d}+1\right )+\frac{180 b \log ^2\left (c x^n\right )}{n}-\frac{12 b d^5 n}{(d+e x)^5}-\frac{33 b d^4 n}{(d+e x)^4}-\frac{74 b d^3 n}{(d+e x)^3}-\frac{171 b d^2 n}{(d+e x)^2}-\frac{522 b d n}{d+e x}+882 b n \log (d+e x)-882 b n \log (x)}{360 d^7} \]

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)

________________________________________________________________________________________

Maple [C]  time = 0.181, size = 1427, normalized size = 4.9 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2/d^7*ln(e*x+d)+1/10*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2/d^2/(e*x+d)^5+b*
ln(x^n)/d^7*ln(x)+1/3*b*ln(x^n)/d^4/(e*x+d)^3+1/4*b*ln(x^n)/d^3/(e*x+d)^4+b*ln(c)/d^6/(e*x+d)+1/5*a/d^2/(e*x+d
)^5+1/6*a/d/(e*x+d)^6-a/d^7*ln(e*x+d)+a/d^7*ln(x)-1/10*I*b*Pi*csgn(I*c*x^n)^3/d^2/(e*x+d)^5-1/12*I*b*Pi*csgn(I
*c*x^n)^3/d/(e*x+d)^6-1/6*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)^4-1/4*I*
b*Pi*csgn(I*c*x^n)^3/d^5/(e*x+d)^2+1/2*I*b*Pi*csgn(I*c*x^n)^3/d^7*ln(e*x+d)-1/2*I*b*Pi*csgn(I*c*x^n)^3/d^6/(e*
x+d)+1/2*b*ln(c)/d^5/(e*x+d)^2+b*ln(c)/d^7*ln(x)-b*ln(c)/d^7*ln(e*x+d)+1/6*b*ln(x^n)/d/(e*x+d)^6-1/4*I*b*Pi*cs
gn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)/d^5/(e*x+d)^2-1/12*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)/d/(e*x+d)^6+1/
2*I*b*Pi*csgn(I*c*x^n)^2*csgn(I*c)/d^6/(e*x+d)+1/8*I*b*Pi*csgn(I*c*x^n)^2*csgn(I*c)/d^3/(e*x+d)^4-1/2*I*b*Pi*c
sgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)/d^7*ln(x)-1/10*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)/d^2/(e*x+d)^5+b*n
/d^7*ln(e*x+d)*ln(-e*x/d)+1/10*I*b*Pi*csgn(I*c*x^n)^2*csgn(I*c)/d^2/(e*x+d)^5+1/4*I*b*Pi*csgn(I*x^n)*csgn(I*c*
x^n)^2/d^5/(e*x+d)^2-1/2*I*b*Pi*csgn(I*c*x^n)^3/d^7*ln(x)+1/12*I*b*Pi*csgn(I*c*x^n)^2*csgn(I*c)/d/(e*x+d)^6+1/
2*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)/d^7*ln(e*x+d)+a/d^6/(e*x+d)+1/2*a/d^5/(e*x+d)^2+1/2*I*b*Pi*csgn(I
*x^n)*csgn(I*c*x^n)^2/d^6/(e*x+d)+1/6*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2/d^4/(e*x+d)^3-1/2*I*b*Pi*csgn(I*x^n)*
csgn(I*c*x^n)*csgn(I*c)/d^6/(e*x+d)-1/6*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)/d^4/(e*x+d)^3-1/8*I*b*Pi*cs
gn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)/d^3/(e*x+d)^4+1/4*I*b*Pi*csgn(I*c*x^n)^2*csgn(I*c)/d^5/(e*x+d)^2+1/2*I*b*Pi*
csgn(I*x^n)*csgn(I*c*x^n)^2/d^7*ln(x)+1/3*a/d^4/(e*x+d)^3+1/4*a/d^3/(e*x+d)^4-1/2*I*b*Pi*csgn(I*c*x^n)^2*csgn(
I*c)/d^7*ln(e*x+d)+1/6*I*b*Pi*csgn(I*c*x^n)^2*csgn(I*c)/d^4/(e*x+d)^3-1/30*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)+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-b*ln(x^n)/d^7*ln(e*x+d)+1/5*b*ln(x^n
)/d^2/(e*x+d)^5+1/2*b*ln(x^n)/d^5/(e*x+d)^2+1/2*I*b*Pi*csgn(I*c*x^n)^2*csgn(I*c)/d^7*ln(x)-1/2*b*n/d^7*ln(x)^2
+b*n/d^7*dilog(-e*x/d)+b*ln(x^n)/d^6/(e*x+d)+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)^6-49/20*b*n*ln(x)/d^7+49/20*b*n*ln(e*x+d)/d^7

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{60} \, a{\left (\frac{60 \, e^{5} x^{5} + 330 \, d e^{4} x^{4} + 740 \, d^{2} e^{3} x^{3} + 855 \, d^{3} e^{2} x^{2} + 522 \, d^{4} e x + 147 \, d^{5}}{d^{6} e^{6} x^{6} + 6 \, d^{7} e^{5} x^{5} + 15 \, d^{8} e^{4} x^{4} + 20 \, d^{9} e^{3} x^{3} + 15 \, d^{10} e^{2} x^{2} + 6 \, d^{11} e x + d^{12}} - \frac{60 \, \log \left (e x + d\right )}{d^{7}} + \frac{60 \, \log \left (x\right )}{d^{7}}\right )} + b \int \frac{\log \left (c\right ) + \log \left (x^{n}\right )}{e^{7} x^{8} + 7 \, d e^{6} x^{7} + 21 \, d^{2} e^{5} x^{6} + 35 \, d^{3} e^{4} x^{5} + 35 \, d^{4} e^{3} x^{4} + 21 \, d^{5} e^{2} x^{3} + 7 \, d^{6} e x^{2} + d^{7} x}\,{d x} \end{align*}

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b \log \left (c x^{n}\right ) + a}{e^{7} x^{8} + 7 \, d e^{6} x^{7} + 21 \, d^{2} e^{5} x^{6} + 35 \, d^{3} e^{4} x^{5} + 35 \, d^{4} e^{3} x^{4} + 21 \, d^{5} e^{2} x^{3} + 7 \, d^{6} e x^{2} + d^{7} x}, x\right ) \end{align*}

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)/(e^7*x^8 + 7*d*e^6*x^7 + 21*d^2*e^5*x^6 + 35*d^3*e^4*x^5 + 35*d^4*e^3*x^4 + 21*d^5
*e^2*x^3 + 7*d^6*e*x^2 + d^7*x), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{b \log \left (c x^{n}\right ) + a}{{\left (e x + d\right )}^{7} x}\,{d x} \end{align*}

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)/((e*x + d)^7*x), x)