3.73 \(\int \frac{a+b \log (c x^n)}{x^3 (d+e x)^7} \, dx\)

Optimal. Leaf size=401 \[ \frac{28 b e^2 n \text{PolyLog}\left (2,-\frac{d}{e x}\right )}{d^9}-\frac{21 e^3 x \left (a+b \log \left (c x^n\right )\right )}{d^9 (d+e x)}-\frac{28 e^2 \log \left (\frac{d}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d^9}+\frac{15 e^2 \left (a+b \log \left (c x^n\right )\right )}{2 d^7 (d+e x)^2}+\frac{10 e^2 \left (a+b \log \left (c x^n\right )\right )}{3 d^6 (d+e x)^3}+\frac{3 e^2 \left (a+b \log \left (c x^n\right )\right )}{2 d^5 (d+e x)^4}+\frac{3 e^2 \left (a+b \log \left (c x^n\right )\right )}{5 d^4 (d+e x)^5}+\frac{e^2 \left (a+b \log \left (c x^n\right )\right )}{6 d^3 (d+e x)^6}+\frac{7 e \left (a+b \log \left (c x^n\right )\right )}{d^8 x}-\frac{a+b \log \left (c x^n\right )}{2 d^7 x^2}-\frac{131 b e^2 n}{10 d^8 (d+e x)}-\frac{14 b e^2 n}{5 d^7 (d+e x)^2}-\frac{34 b e^2 n}{45 d^6 (d+e x)^3}-\frac{23 b e^2 n}{120 d^5 (d+e x)^4}-\frac{b e^2 n}{30 d^4 (d+e x)^5}-\frac{131 b e^2 n \log (x)}{10 d^9}+\frac{341 b e^2 n \log (d+e x)}{10 d^9}+\frac{7 b e n}{d^8 x}-\frac{b n}{4 d^7 x^2} \]

[Out]

-(b*n)/(4*d^7*x^2) + (7*b*e*n)/(d^8*x) - (b*e^2*n)/(30*d^4*(d + e*x)^5) - (23*b*e^2*n)/(120*d^5*(d + e*x)^4) -
(34*b*e^2*n)/(45*d^6*(d + e*x)^3) - (14*b*e^2*n)/(5*d^7*(d + e*x)^2) - (131*b*e^2*n)/(10*d^8*(d + e*x)) - (13
1*b*e^2*n*Log[x])/(10*d^9) - (a + b*Log[c*x^n])/(2*d^7*x^2) + (7*e*(a + b*Log[c*x^n]))/(d^8*x) + (e^2*(a + b*L
og[c*x^n]))/(6*d^3*(d + e*x)^6) + (3*e^2*(a + b*Log[c*x^n]))/(5*d^4*(d + e*x)^5) + (3*e^2*(a + b*Log[c*x^n]))/
(2*d^5*(d + e*x)^4) + (10*e^2*(a + b*Log[c*x^n]))/(3*d^6*(d + e*x)^3) + (15*e^2*(a + b*Log[c*x^n]))/(2*d^7*(d
+ e*x)^2) - (21*e^3*x*(a + b*Log[c*x^n]))/(d^9*(d + e*x)) - (28*e^2*Log[1 + d/(e*x)]*(a + b*Log[c*x^n]))/d^9 +
(341*b*e^2*n*Log[d + e*x])/(10*d^9) + (28*b*e^2*n*PolyLog[2, -(d/(e*x))])/d^9

________________________________________________________________________________________

Rubi [A]  time = 0.636512, antiderivative size = 423, normalized size of antiderivative = 1.05, number of steps used = 24, number of rules used = 9, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {44, 2351, 2304, 2301, 2319, 2314, 31, 2317, 2391} \[ -\frac{28 b e^2 n \text{PolyLog}\left (2,-\frac{e x}{d}\right )}{d^9}-\frac{21 e^3 x \left (a+b \log \left (c x^n\right )\right )}{d^9 (d+e x)}+\frac{14 e^2 \left (a+b \log \left (c x^n\right )\right )^2}{b d^9 n}+\frac{15 e^2 \left (a+b \log \left (c x^n\right )\right )}{2 d^7 (d+e x)^2}+\frac{10 e^2 \left (a+b \log \left (c x^n\right )\right )}{3 d^6 (d+e x)^3}+\frac{3 e^2 \left (a+b \log \left (c x^n\right )\right )}{2 d^5 (d+e x)^4}+\frac{3 e^2 \left (a+b \log \left (c x^n\right )\right )}{5 d^4 (d+e x)^5}+\frac{e^2 \left (a+b \log \left (c x^n\right )\right )}{6 d^3 (d+e x)^6}-\frac{28 e^2 \log \left (\frac{e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d^9}+\frac{7 e \left (a+b \log \left (c x^n\right )\right )}{d^8 x}-\frac{a+b \log \left (c x^n\right )}{2 d^7 x^2}-\frac{131 b e^2 n}{10 d^8 (d+e x)}-\frac{14 b e^2 n}{5 d^7 (d+e x)^2}-\frac{34 b e^2 n}{45 d^6 (d+e x)^3}-\frac{23 b e^2 n}{120 d^5 (d+e x)^4}-\frac{b e^2 n}{30 d^4 (d+e x)^5}-\frac{131 b e^2 n \log (x)}{10 d^9}+\frac{341 b e^2 n \log (d+e x)}{10 d^9}+\frac{7 b e n}{d^8 x}-\frac{b n}{4 d^7 x^2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Log[c*x^n])/(x^3*(d + e*x)^7),x]

[Out]

-(b*n)/(4*d^7*x^2) + (7*b*e*n)/(d^8*x) - (b*e^2*n)/(30*d^4*(d + e*x)^5) - (23*b*e^2*n)/(120*d^5*(d + e*x)^4) -
(34*b*e^2*n)/(45*d^6*(d + e*x)^3) - (14*b*e^2*n)/(5*d^7*(d + e*x)^2) - (131*b*e^2*n)/(10*d^8*(d + e*x)) - (13
1*b*e^2*n*Log[x])/(10*d^9) - (a + b*Log[c*x^n])/(2*d^7*x^2) + (7*e*(a + b*Log[c*x^n]))/(d^8*x) + (e^2*(a + b*L
og[c*x^n]))/(6*d^3*(d + e*x)^6) + (3*e^2*(a + b*Log[c*x^n]))/(5*d^4*(d + e*x)^5) + (3*e^2*(a + b*Log[c*x^n]))/
(2*d^5*(d + e*x)^4) + (10*e^2*(a + b*Log[c*x^n]))/(3*d^6*(d + e*x)^3) + (15*e^2*(a + b*Log[c*x^n]))/(2*d^7*(d
+ e*x)^2) - (21*e^3*x*(a + b*Log[c*x^n]))/(d^9*(d + e*x)) + (14*e^2*(a + b*Log[c*x^n])^2)/(b*d^9*n) + (341*b*e
^2*n*Log[d + e*x])/(10*d^9) - (28*e^2*(a + b*Log[c*x^n])*Log[1 + (e*x)/d])/d^9 - (28*b*e^2*n*PolyLog[2, -((e*x
)/d)])/d^9

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])

Rule 2351

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 2304

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 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 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 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 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]

Rubi steps

\begin{align*} \int \frac{a+b \log \left (c x^n\right )}{x^3 (d+e x)^7} \, dx &=\int \left (\frac{a+b \log \left (c x^n\right )}{d^7 x^3}-\frac{7 e \left (a+b \log \left (c x^n\right )\right )}{d^8 x^2}+\frac{28 e^2 \left (a+b \log \left (c x^n\right )\right )}{d^9 x}-\frac{e^3 \left (a+b \log \left (c x^n\right )\right )}{d^3 (d+e x)^7}-\frac{3 e^3 \left (a+b \log \left (c x^n\right )\right )}{d^4 (d+e x)^6}-\frac{6 e^3 \left (a+b \log \left (c x^n\right )\right )}{d^5 (d+e x)^5}-\frac{10 e^3 \left (a+b \log \left (c x^n\right )\right )}{d^6 (d+e x)^4}-\frac{15 e^3 \left (a+b \log \left (c x^n\right )\right )}{d^7 (d+e x)^3}-\frac{21 e^3 \left (a+b \log \left (c x^n\right )\right )}{d^8 (d+e x)^2}-\frac{28 e^3 \left (a+b \log \left (c x^n\right )\right )}{d^9 (d+e x)}\right ) \, dx\\ &=\frac{\int \frac{a+b \log \left (c x^n\right )}{x^3} \, dx}{d^7}-\frac{(7 e) \int \frac{a+b \log \left (c x^n\right )}{x^2} \, dx}{d^8}+\frac{\left (28 e^2\right ) \int \frac{a+b \log \left (c x^n\right )}{x} \, dx}{d^9}-\frac{\left (28 e^3\right ) \int \frac{a+b \log \left (c x^n\right )}{d+e x} \, dx}{d^9}-\frac{\left (21 e^3\right ) \int \frac{a+b \log \left (c x^n\right )}{(d+e x)^2} \, dx}{d^8}-\frac{\left (15 e^3\right ) \int \frac{a+b \log \left (c x^n\right )}{(d+e x)^3} \, dx}{d^7}-\frac{\left (10 e^3\right ) \int \frac{a+b \log \left (c x^n\right )}{(d+e x)^4} \, dx}{d^6}-\frac{\left (6 e^3\right ) \int \frac{a+b \log \left (c x^n\right )}{(d+e x)^5} \, dx}{d^5}-\frac{\left (3 e^3\right ) \int \frac{a+b \log \left (c x^n\right )}{(d+e x)^6} \, dx}{d^4}-\frac{e^3 \int \frac{a+b \log \left (c x^n\right )}{(d+e x)^7} \, dx}{d^3}\\ &=-\frac{b n}{4 d^7 x^2}+\frac{7 b e n}{d^8 x}-\frac{a+b \log \left (c x^n\right )}{2 d^7 x^2}+\frac{7 e \left (a+b \log \left (c x^n\right )\right )}{d^8 x}+\frac{e^2 \left (a+b \log \left (c x^n\right )\right )}{6 d^3 (d+e x)^6}+\frac{3 e^2 \left (a+b \log \left (c x^n\right )\right )}{5 d^4 (d+e x)^5}+\frac{3 e^2 \left (a+b \log \left (c x^n\right )\right )}{2 d^5 (d+e x)^4}+\frac{10 e^2 \left (a+b \log \left (c x^n\right )\right )}{3 d^6 (d+e x)^3}+\frac{15 e^2 \left (a+b \log \left (c x^n\right )\right )}{2 d^7 (d+e x)^2}-\frac{21 e^3 x \left (a+b \log \left (c x^n\right )\right )}{d^9 (d+e x)}+\frac{14 e^2 \left (a+b \log \left (c x^n\right )\right )^2}{b d^9 n}-\frac{28 e^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{e x}{d}\right )}{d^9}+\frac{\left (28 b e^2 n\right ) \int \frac{\log \left (1+\frac{e x}{d}\right )}{x} \, dx}{d^9}-\frac{\left (15 b e^2 n\right ) \int \frac{1}{x (d+e x)^2} \, dx}{2 d^7}-\frac{\left (10 b e^2 n\right ) \int \frac{1}{x (d+e x)^3} \, dx}{3 d^6}-\frac{\left (3 b e^2 n\right ) \int \frac{1}{x (d+e x)^4} \, dx}{2 d^5}-\frac{\left (3 b e^2 n\right ) \int \frac{1}{x (d+e x)^5} \, dx}{5 d^4}-\frac{\left (b e^2 n\right ) \int \frac{1}{x (d+e x)^6} \, dx}{6 d^3}+\frac{\left (21 b e^3 n\right ) \int \frac{1}{d+e x} \, dx}{d^9}\\ &=-\frac{b n}{4 d^7 x^2}+\frac{7 b e n}{d^8 x}-\frac{a+b \log \left (c x^n\right )}{2 d^7 x^2}+\frac{7 e \left (a+b \log \left (c x^n\right )\right )}{d^8 x}+\frac{e^2 \left (a+b \log \left (c x^n\right )\right )}{6 d^3 (d+e x)^6}+\frac{3 e^2 \left (a+b \log \left (c x^n\right )\right )}{5 d^4 (d+e x)^5}+\frac{3 e^2 \left (a+b \log \left (c x^n\right )\right )}{2 d^5 (d+e x)^4}+\frac{10 e^2 \left (a+b \log \left (c x^n\right )\right )}{3 d^6 (d+e x)^3}+\frac{15 e^2 \left (a+b \log \left (c x^n\right )\right )}{2 d^7 (d+e x)^2}-\frac{21 e^3 x \left (a+b \log \left (c x^n\right )\right )}{d^9 (d+e x)}+\frac{14 e^2 \left (a+b \log \left (c x^n\right )\right )^2}{b d^9 n}+\frac{21 b e^2 n \log (d+e x)}{d^9}-\frac{28 e^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{e x}{d}\right )}{d^9}-\frac{28 b e^2 n \text{Li}_2\left (-\frac{e x}{d}\right )}{d^9}-\frac{\left (15 b e^2 n\right ) \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^7}-\frac{\left (10 b e^2 n\right ) \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^6}-\frac{\left (3 b e^2 n\right ) \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}{2 d^5}-\frac{\left (3 b e^2 n\right ) \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^4}-\frac{\left (b e^2 n\right ) \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^3}\\ &=-\frac{b n}{4 d^7 x^2}+\frac{7 b e n}{d^8 x}-\frac{b e^2 n}{30 d^4 (d+e x)^5}-\frac{23 b e^2 n}{120 d^5 (d+e x)^4}-\frac{34 b e^2 n}{45 d^6 (d+e x)^3}-\frac{14 b e^2 n}{5 d^7 (d+e x)^2}-\frac{131 b e^2 n}{10 d^8 (d+e x)}-\frac{131 b e^2 n \log (x)}{10 d^9}-\frac{a+b \log \left (c x^n\right )}{2 d^7 x^2}+\frac{7 e \left (a+b \log \left (c x^n\right )\right )}{d^8 x}+\frac{e^2 \left (a+b \log \left (c x^n\right )\right )}{6 d^3 (d+e x)^6}+\frac{3 e^2 \left (a+b \log \left (c x^n\right )\right )}{5 d^4 (d+e x)^5}+\frac{3 e^2 \left (a+b \log \left (c x^n\right )\right )}{2 d^5 (d+e x)^4}+\frac{10 e^2 \left (a+b \log \left (c x^n\right )\right )}{3 d^6 (d+e x)^3}+\frac{15 e^2 \left (a+b \log \left (c x^n\right )\right )}{2 d^7 (d+e x)^2}-\frac{21 e^3 x \left (a+b \log \left (c x^n\right )\right )}{d^9 (d+e x)}+\frac{14 e^2 \left (a+b \log \left (c x^n\right )\right )^2}{b d^9 n}+\frac{341 b e^2 n \log (d+e x)}{10 d^9}-\frac{28 e^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{e x}{d}\right )}{d^9}-\frac{28 b e^2 n \text{Li}_2\left (-\frac{e x}{d}\right )}{d^9}\\ \end{align*}

Mathematica [A]  time = 0.543131, size = 486, normalized size = 1.21 \[ \frac{-10080 b e^2 n \text{PolyLog}\left (2,-\frac{e x}{d}\right )+\frac{10080 a e^2 \log \left (c x^n\right )}{n}+\frac{60 a d^6 e^2}{(d+e x)^6}+\frac{216 a d^5 e^2}{(d+e x)^5}+\frac{540 a d^4 e^2}{(d+e x)^4}+\frac{1200 a d^3 e^2}{(d+e x)^3}+\frac{2700 a d^2 e^2}{(d+e x)^2}-\frac{180 a d^2}{x^2}+\frac{7560 a d e^2}{d+e x}-10080 a e^2 \log \left (\frac{e x}{d}+1\right )+\frac{2520 a d e}{x}+\frac{60 b d^6 e^2 \log \left (c x^n\right )}{(d+e x)^6}+\frac{216 b d^5 e^2 \log \left (c x^n\right )}{(d+e x)^5}+\frac{540 b d^4 e^2 \log \left (c x^n\right )}{(d+e x)^4}+\frac{1200 b d^3 e^2 \log \left (c x^n\right )}{(d+e x)^3}+\frac{2700 b d^2 e^2 \log \left (c x^n\right )}{(d+e x)^2}-\frac{180 b d^2 \log \left (c x^n\right )}{x^2}+\frac{7560 b d e^2 \log \left (c x^n\right )}{d+e x}-10080 b e^2 \log \left (c x^n\right ) \log \left (\frac{e x}{d}+1\right )+\frac{2520 b d e \log \left (c x^n\right )}{x}+\frac{5040 b e^2 \log ^2\left (c x^n\right )}{n}-\frac{12 b d^5 e^2 n}{(d+e x)^5}-\frac{69 b d^4 e^2 n}{(d+e x)^4}-\frac{272 b d^3 e^2 n}{(d+e x)^3}-\frac{1008 b d^2 e^2 n}{(d+e x)^2}-\frac{90 b d^2 n}{x^2}-\frac{4716 b d e^2 n}{d+e x}+12276 b e^2 n \log (d+e x)+\frac{2520 b d e n}{x}-12276 b e^2 n \log (x)}{360 d^9} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Log[c*x^n])/(x^3*(d + e*x)^7),x]

[Out]

((-180*a*d^2)/x^2 - (90*b*d^2*n)/x^2 + (2520*a*d*e)/x + (2520*b*d*e*n)/x + (60*a*d^6*e^2)/(d + e*x)^6 + (216*a
*d^5*e^2)/(d + e*x)^5 - (12*b*d^5*e^2*n)/(d + e*x)^5 + (540*a*d^4*e^2)/(d + e*x)^4 - (69*b*d^4*e^2*n)/(d + e*x
)^4 + (1200*a*d^3*e^2)/(d + e*x)^3 - (272*b*d^3*e^2*n)/(d + e*x)^3 + (2700*a*d^2*e^2)/(d + e*x)^2 - (1008*b*d^
2*e^2*n)/(d + e*x)^2 + (7560*a*d*e^2)/(d + e*x) - (4716*b*d*e^2*n)/(d + e*x) - 12276*b*e^2*n*Log[x] + (10080*a
*e^2*Log[c*x^n])/n - (180*b*d^2*Log[c*x^n])/x^2 + (2520*b*d*e*Log[c*x^n])/x + (60*b*d^6*e^2*Log[c*x^n])/(d + e
*x)^6 + (216*b*d^5*e^2*Log[c*x^n])/(d + e*x)^5 + (540*b*d^4*e^2*Log[c*x^n])/(d + e*x)^4 + (1200*b*d^3*e^2*Log[
c*x^n])/(d + e*x)^3 + (2700*b*d^2*e^2*Log[c*x^n])/(d + e*x)^2 + (7560*b*d*e^2*Log[c*x^n])/(d + e*x) + (5040*b*
e^2*Log[c*x^n]^2)/n + 12276*b*e^2*n*Log[d + e*x] - 10080*a*e^2*Log[1 + (e*x)/d] - 10080*b*e^2*Log[c*x^n]*Log[1
+ (e*x)/d] - 10080*b*e^2*n*PolyLog[2, -((e*x)/d)])/(360*d^9)

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Maple [C]  time = 0.199, size = 1939, normalized size = 4.8 \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^3/(e*x+d)^7,x)

[Out]

-3/10*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)/d^4*e^2/(e*x+d)^5-1/12*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)*csgn(
I*c)*e^2/d^3/(e*x+d)^6+14*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)/d^9*e^2*ln(e*x+d)-21/2*I*b*Pi*csgn(I*x^n)
*csgn(I*c*x^n)*csgn(I*c)/d^8*e^2/(e*x+d)+5/3*I*b*Pi*csgn(I*c*x^n)^2*csgn(I*c)*e^2/d^6/(e*x+d)^3-7/2*I*b*Pi*csg
n(I*x^n)*csgn(I*c*x^n)*csgn(I*c)/d^8*e/x-14*b*n/d^9*e^2*ln(x)^2+28*b*n/d^9*e^2*dilog(-e*x/d)+28*b*n/d^9*e^2*ln
(e*x+d)*ln(-e*x/d)+3/2*b*ln(x^n)/d^5*e^2/(e*x+d)^4+15/4*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2*e^2/d^7/(e*x+d)^2+1
/12*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2*e^2/d^3/(e*x+d)^6-14*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2/d^9*e^2*ln(e*x+
d)+21/2*I*b*Pi*csgn(I*c*x^n)^2*csgn(I*c)/d^8*e^2/(e*x+d)+1/4*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)/d^7/x^
2+5/3*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2*e^2/d^6/(e*x+d)^3-1/2*b*ln(x^n)/d^7/x^2+1/12*I*b*Pi*csgn(I*c*x^n)^2*c
sgn(I*c)*e^2/d^3/(e*x+d)^6+14*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2/d^9*e^2*ln(x)+3/10*I*b*Pi*csgn(I*c*x^n)^2*csg
n(I*c)/d^4*e^2/(e*x+d)^5-1/4*I*b*Pi*csgn(I*c*x^n)^2*csgn(I*c)/d^7/x^2-21/2*I*b*Pi*csgn(I*c*x^n)^3/d^8*e^2/(e*x
+d)-3/4*I*b*Pi*csgn(I*c*x^n)^3/d^5*e^2/(e*x+d)^4+3/10*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2/d^4*e^2/(e*x+d)^5-3/1
0*I*b*Pi*csgn(I*c*x^n)^3/d^4*e^2/(e*x+d)^5-1/12*I*b*Pi*csgn(I*c*x^n)^3*e^2/d^3/(e*x+d)^6+21/2*I*b*Pi*csgn(I*x^
n)*csgn(I*c*x^n)^2/d^8*e^2/(e*x+d)+3/4*I*b*Pi*csgn(I*c*x^n)^2*csgn(I*c)/d^5*e^2/(e*x+d)^4+15/4*I*b*Pi*csgn(I*c
*x^n)^2*csgn(I*c)*e^2/d^7/(e*x+d)^2+7/2*I*b*Pi*csgn(I*c*x^n)^2*csgn(I*c)/d^8*e/x-14*I*b*Pi*csgn(I*c*x^n)^2*csg
n(I*c)/d^9*e^2*ln(e*x+d)+15/2*a*e^2/d^7/(e*x+d)^2+10/3*a*e^2/d^6/(e*x+d)^3+3/2*a/d^5*e^2/(e*x+d)^4+3/4*I*b*Pi*
csgn(I*x^n)*csgn(I*c*x^n)^2/d^5*e^2/(e*x+d)^4+14*I*b*Pi*csgn(I*c*x^n)^2*csgn(I*c)/d^9*e^2*ln(x)+7/2*I*b*Pi*csg
n(I*x^n)*csgn(I*c*x^n)^2/d^8*e/x+1/4*I*b*Pi*csgn(I*c*x^n)^3/d^7/x^2-15/4*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)*csgn
(I*c)*e^2/d^7/(e*x+d)^2-5/3*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*e^2/d^6/(e*x+d)^3-15/4*I*b*Pi*csgn(I*c*
x^n)^3*e^2/d^7/(e*x+d)^2-5/3*I*b*Pi*csgn(I*c*x^n)^3*e^2/d^6/(e*x+d)^3-1/4*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2/d
^7/x^2-1/4*b*n/d^7/x^2+14*I*b*Pi*csgn(I*c*x^n)^3/d^9*e^2*ln(e*x+d)-7/2*I*b*Pi*csgn(I*c*x^n)^3/d^8*e/x-14*I*b*P
i*csgn(I*c*x^n)^3/d^9*e^2*ln(x)-3/4*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)/d^5*e^2/(e*x+d)^4-14*I*b*Pi*csg
n(I*x^n)*csgn(I*c*x^n)*csgn(I*c)/d^9*e^2*ln(x)-1/2*a/d^7/x^2+3/5*a/d^4*e^2/(e*x+d)^5+1/6*a*e^2/d^3/(e*x+d)^6+7
*a/d^8*e/x-28*a/d^9*e^2*ln(e*x+d)+28*a/d^9*e^2*ln(x)+21*a/d^8*e^2/(e*x+d)+341/10*b*e^2*n*ln(e*x+d)/d^9-1/2*b*l
n(c)/d^7/x^2+10/3*b*ln(c)*e^2/d^6/(e*x+d)^3+3/2*b*ln(c)/d^5*e^2/(e*x+d)^4+3/5*b*ln(c)/d^4*e^2/(e*x+d)^5+1/6*b*
ln(c)*e^2/d^3/(e*x+d)^6+7*b*ln(c)/d^8*e/x-28*b*ln(c)/d^9*e^2*ln(e*x+d)+28*b*ln(c)/d^9*e^2*ln(x)+7*b*e*n/d^8/x-
1/30*b*e^2*n/d^4/(e*x+d)^5-23/120*b*e^2*n/d^5/(e*x+d)^4-34/45*b*e^2*n/d^6/(e*x+d)^3-14/5*b*e^2*n/d^7/(e*x+d)^2
-131/10*b*e^2*n/d^8/(e*x+d)-28*b*ln(x^n)/d^9*e^2*ln(e*x+d)+21*b*ln(x^n)/d^8*e^2/(e*x+d)+15/2*b*ln(x^n)*e^2/d^7
/(e*x+d)^2+10/3*b*ln(x^n)*e^2/d^6/(e*x+d)^3-341/10*b*e^2*n*ln(x)/d^9+21*b*ln(c)/d^8*e^2/(e*x+d)+15/2*b*ln(c)*e
^2/d^7/(e*x+d)^2+3/5*b*ln(x^n)/d^4*e^2/(e*x+d)^5+1/6*b*ln(x^n)*e^2/d^3/(e*x+d)^6+28*b*ln(x^n)/d^9*e^2*ln(x)+7*
b*ln(x^n)/d^8*e/x

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{30} \, a{\left (\frac{840 \, e^{7} x^{7} + 4620 \, d e^{6} x^{6} + 10360 \, d^{2} e^{5} x^{5} + 11970 \, d^{3} e^{4} x^{4} + 7308 \, d^{4} e^{3} x^{3} + 2058 \, d^{5} e^{2} x^{2} + 120 \, d^{6} e x - 15 \, d^{7}}{d^{8} e^{6} x^{8} + 6 \, d^{9} e^{5} x^{7} + 15 \, d^{10} e^{4} x^{6} + 20 \, d^{11} e^{3} x^{5} + 15 \, d^{12} e^{2} x^{4} + 6 \, d^{13} e x^{3} + d^{14} x^{2}} - \frac{840 \, e^{2} \log \left (e x + d\right )}{d^{9}} + \frac{840 \, e^{2} \log \left (x\right )}{d^{9}}\right )} + b \int \frac{\log \left (c\right ) + \log \left (x^{n}\right )}{e^{7} x^{10} + 7 \, d e^{6} x^{9} + 21 \, d^{2} e^{5} x^{8} + 35 \, d^{3} e^{4} x^{7} + 35 \, d^{4} e^{3} x^{6} + 21 \, d^{5} e^{2} x^{5} + 7 \, d^{6} e x^{4} + d^{7} x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))/x^3/(e*x+d)^7,x, algorithm="maxima")

[Out]

1/30*a*((840*e^7*x^7 + 4620*d*e^6*x^6 + 10360*d^2*e^5*x^5 + 11970*d^3*e^4*x^4 + 7308*d^4*e^3*x^3 + 2058*d^5*e^
2*x^2 + 120*d^6*e*x - 15*d^7)/(d^8*e^6*x^8 + 6*d^9*e^5*x^7 + 15*d^10*e^4*x^6 + 20*d^11*e^3*x^5 + 15*d^12*e^2*x
^4 + 6*d^13*e*x^3 + d^14*x^2) - 840*e^2*log(e*x + d)/d^9 + 840*e^2*log(x)/d^9) + b*integrate((log(c) + log(x^n
))/(e^7*x^10 + 7*d*e^6*x^9 + 21*d^2*e^5*x^8 + 35*d^3*e^4*x^7 + 35*d^4*e^3*x^6 + 21*d^5*e^2*x^5 + 7*d^6*e*x^4 +
d^7*x^3), 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^{10} + 7 \, d e^{6} x^{9} + 21 \, d^{2} e^{5} x^{8} + 35 \, d^{3} e^{4} x^{7} + 35 \, d^{4} e^{3} x^{6} + 21 \, d^{5} e^{2} x^{5} + 7 \, d^{6} e x^{4} + d^{7} x^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))/x^3/(e*x+d)^7,x, algorithm="fricas")

[Out]

integral((b*log(c*x^n) + a)/(e^7*x^10 + 7*d*e^6*x^9 + 21*d^2*e^5*x^8 + 35*d^3*e^4*x^7 + 35*d^4*e^3*x^6 + 21*d^
5*e^2*x^5 + 7*d^6*e*x^4 + d^7*x^3), 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**3/(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^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))/x^3/(e*x+d)^7,x, algorithm="giac")

[Out]

integrate((b*log(c*x^n) + a)/((e*x + d)^7*x^3), x)