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

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

[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

________________________________________________________________________________________

Rubi [A]  time = 0.580702, antiderivative size = 361, normalized size of antiderivative = 1.06, number of steps used = 23, 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} \[ \frac{7 b e n \text{PolyLog}\left (2,-\frac{e x}{d}\right )}{d^8}+\frac{6 e^2 x \left (a+b \log \left (c x^n\right )\right )}{d^8 (d+e x)}-\frac{7 e \left (a+b \log \left (c x^n\right )\right )^2}{2 b d^8 n}-\frac{5 e \left (a+b \log \left (c x^n\right )\right )}{2 d^6 (d+e x)^2}-\frac{4 e \left (a+b \log \left (c x^n\right )\right )}{3 d^5 (d+e x)^3}-\frac{3 e \left (a+b \log \left (c x^n\right )\right )}{4 d^4 (d+e x)^4}-\frac{2 e \left (a+b \log \left (c x^n\right )\right )}{5 d^3 (d+e x)^5}-\frac{e \left (a+b \log \left (c x^n\right )\right )}{6 d^2 (d+e x)^6}+\frac{7 e \log \left (\frac{e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d^8}-\frac{a+b \log \left (c x^n\right )}{d^7 x}+\frac{103 b e n}{20 d^7 (d+e x)}+\frac{53 b e n}{40 d^6 (d+e x)^2}+\frac{79 b e n}{180 d^5 (d+e x)^3}+\frac{17 b e n}{120 d^4 (d+e x)^4}+\frac{b e n}{30 d^3 (d+e x)^5}+\frac{103 b e n \log (x)}{20 d^8}-\frac{223 b e n \log (d+e x)}{20 d^8}-\frac{b n}{d^7 x} \]

Antiderivative was successfully verified.

[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*(a + b*Log[c*x^n])^
2)/(2*b*d^8*n) - (223*b*e*n*Log[d + e*x])/(20*d^8) + (7*e*(a + b*Log[c*x^n])*Log[1 + (e*x)/d])/d^8 + (7*b*e*n*
PolyLog[2, -((e*x)/d)])/d^8

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^2 (d+e x)^7} \, dx &=\int \left (\frac{a+b \log \left (c x^n\right )}{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 )}{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^2 \left (a+b \log \left (c x^n\right )\right )}{d^8 (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} \, dx}{d^8}+\frac{\left (7 e^2\right ) \int \frac{a+b \log \left (c x^n\right )}{d+e x} \, dx}{d^8}+\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 \left (a+b \log \left (c x^n\right )\right )^2}{2 b d^8 n}+\frac{7 e \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{e x}{d}\right )}{d^8}-\frac{(7 b e n) \int \frac{\log \left (1+\frac{e x}{d}\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 \left (a+b \log \left (c x^n\right )\right )^2}{2 b d^8 n}-\frac{6 b e n \log (d+e x)}{d^8}+\frac{7 e \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{e x}{d}\right )}{d^8}+\frac{7 b e n \text{Li}_2\left (-\frac{e x}{d}\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 \left (a+b \log \left (c x^n\right )\right )^2}{2 b d^8 n}-\frac{223 b e n \log (d+e x)}{20 d^8}+\frac{7 e \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{e x}{d}\right )}{d^8}+\frac{7 b e n \text{Li}_2\left (-\frac{e x}{d}\right )}{d^8}\\ \end{align*}

Mathematica [A]  time = 0.624457, size = 401, normalized size = 1.18 \[ -\frac{-2520 b e n \text{PolyLog}\left (2,-\frac{e x}{d}\right )+\frac{2520 a e \log \left (c x^n\right )}{n}+\frac{60 a d^6 e}{(d+e x)^6}+\frac{144 a d^5 e}{(d+e x)^5}+\frac{270 a d^4 e}{(d+e x)^4}+\frac{480 a d^3 e}{(d+e x)^3}+\frac{900 a d^2 e}{(d+e x)^2}+\frac{2160 a d e}{d+e x}-2520 a e \log \left (\frac{e x}{d}+1\right )+\frac{360 a d}{x}+\frac{60 b d^6 e \log \left (c x^n\right )}{(d+e x)^6}+\frac{144 b d^5 e \log \left (c x^n\right )}{(d+e x)^5}+\frac{270 b d^4 e \log \left (c x^n\right )}{(d+e x)^4}+\frac{480 b d^3 e \log \left (c x^n\right )}{(d+e x)^3}+\frac{900 b d^2 e \log \left (c x^n\right )}{(d+e x)^2}+\frac{2160 b d e \log \left (c x^n\right )}{d+e x}-2520 b e \log \left (c x^n\right ) \log \left (\frac{e x}{d}+1\right )+\frac{360 b d \log \left (c x^n\right )}{x}+\frac{1260 b e \log ^2\left (c x^n\right )}{n}-\frac{12 b d^5 e n}{(d+e x)^5}-\frac{51 b d^4 e n}{(d+e x)^4}-\frac{158 b d^3 e n}{(d+e x)^3}-\frac{477 b d^2 e n}{(d+e x)^2}-\frac{1854 b d e n}{d+e x}+4014 b e n \log (d+e x)+\frac{360 b d n}{x}-4014 b e n \log (x)}{360 d^8} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((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*Log[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] - 2520*b*
e*n*PolyLog[2, -((e*x)/d)])/(360*d^8)

________________________________________________________________________________________

Maple [C]  time = 0.194, size = 1650, 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^2/(e*x+d)^7,x)

[Out]

7/2*I*b*Pi*csgn(I*c*x^n)^2*csgn(I*c)/d^8*e*ln(e*x+d)-7/2*I*b*Pi*csgn(I*c*x^n)^2*csgn(I*c)/d^8*e*ln(x)+3*I*b*Pi
*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)/d^7*e/(e*x+d)-7/2*I*b*Pi*csgn(I*c*x^n)^3/d^8*e*ln(e*x+d)+1/12*I*b*Pi*csgn
(I*c*x^n)^3*e/d^2/(e*x+d)^6+2/3*I*b*Pi*csgn(I*c*x^n)^3*e/d^5/(e*x+d)^3-a/d^7/x+5/4*I*b*Pi*csgn(I*c*x^n)^3*e/d^
6/(e*x+d)^2+7/2*I*b*Pi*csgn(I*c*x^n)^3/d^8*e*ln(x)-1/2*I*b*Pi*csgn(I*c*x^n)^2*csgn(I*c)/d^7/x+3*I*b*Pi*csgn(I*
c*x^n)^3/d^7*e/(e*x+d)-7*b*n/d^8*e*ln(e*x+d)*ln(-e*x/d)-1/6*a*e/d^2/(e*x+d)^6-6*a/d^7*e/(e*x+d)+1/2*I*b*Pi*csg
n(I*c*x^n)^3/d^7/x-3/8*I*b*Pi*csgn(I*c*x^n)^2*csgn(I*c)/d^4*e/(e*x+d)^4-1/12*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^
2*e/d^2/(e*x+d)^6+7/2*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2/d^8*e*ln(e*x+d)-7/2*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^
2/d^8*e*ln(x)-1/5*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2/d^3*e/(e*x+d)^5-3*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2/d^7*
e/(e*x+d)-3*I*b*Pi*csgn(I*c*x^n)^2*csgn(I*c)/d^7*e/(e*x+d)-1/5*I*b*Pi*csgn(I*c*x^n)^2*csgn(I*c)/d^3*e/(e*x+d)^
5-2/3*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2*e/d^5/(e*x+d)^3-5/4*I*b*Pi*csgn(I*c*x^n)^2*csgn(I*c)*e/d^6/(e*x+d)^2+
1/2*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)/d^7/x+3/8*I*b*Pi*csgn(I*c*x^n)^3/d^4*e/(e*x+d)^4+1/5*I*b*Pi*csg
n(I*c*x^n)^3/d^3*e/(e*x+d)^5-1/2*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2/d^7/x-1/12*I*b*Pi*csgn(I*c*x^n)^2*csgn(I*c
)*e/d^2/(e*x+d)^6-2/3*I*b*Pi*csgn(I*c*x^n)^2*csgn(I*c)*e/d^5/(e*x+d)^3+3/8*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)*cs
gn(I*c)/d^4*e/(e*x+d)^4+1/5*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)/d^3*e/(e*x+d)^5-5/4*I*b*Pi*csgn(I*x^n)*
csgn(I*c*x^n)^2*e/d^6/(e*x+d)^2+7/2*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)/d^8*e*ln(x)+5/4*I*b*Pi*csgn(I*x
^n)*csgn(I*c*x^n)*csgn(I*c)*e/d^6/(e*x+d)^2-3/8*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2/d^4*e/(e*x+d)^4+1/12*I*b*Pi
*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*e/d^2/(e*x+d)^6-7/2*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)/d^8*e*ln(e
*x+d)-b*n/d^7/x-7*a/d^8*e*ln(x)-b*ln(c)/d^7/x-5/2*a*e/d^6/(e*x+d)^2-4/3*a*e/d^5/(e*x+d)^3-3/4*a/d^4*e/(e*x+d)^
4-2/5*a/d^3*e/(e*x+d)^5+7*a/d^8*e*ln(e*x+d)+2/3*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*e/d^5/(e*x+d)^3-b*l
n(x^n)/d^7/x-1/6*b*ln(c)*e/d^2/(e*x+d)^6-6*b*ln(c)/d^7*e/(e*x+d)-5/2*b*ln(c)*e/d^6/(e*x+d)^2-4/3*b*ln(c)*e/d^5
/(e*x+d)^3-4/3*b*ln(x^n)*e/d^5/(e*x+d)^3-3/4*b*ln(x^n)/d^4*e/(e*x+d)^4-2/5*b*ln(x^n)/d^3*e/(e*x+d)^5-7*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*dilog(-e*x/d)+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)-3/4*b*ln(c)/d^4*e/(e*x+d)^4
-2/5*b*ln(c)/d^3*e/(e*x+d)^5+7*b*ln(c)/d^8*e*ln(e*x+d)-7*b*ln(c)/d^8*e*ln(x)+223/20*b*e*n*ln(x)/d^8-223/20*b*e
*n*ln(e*x+d)/d^8+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)*e/d^6/(e*x+d)^2-1/6*b*ln(
x^n)*e/d^2/(e*x+d)^6

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{1}{60} \, a{\left (\frac{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} - \frac{420 \, e \log \left (e x + d\right )}{d^{8}} + \frac{420 \, e \log \left (x\right )}{d^{8}}\right )} + b \int \frac{\log \left (c\right ) + \log \left (x^{n}\right )}{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}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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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^{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\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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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**2/(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^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)