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

Optimal. Leaf size=285 \[ -\frac{7 b d n \text{PolyLog}\left (2,-\frac{e x}{d}\right )}{e^8}-\frac{x^6 \left (7 a+7 b \log \left (c x^n\right )+b n\right )}{30 e^2 (d+e x)^5}-\frac{x^5 \left (42 a+42 b \log \left (c x^n\right )+13 b n\right )}{120 e^3 (d+e x)^4}-\frac{x^4 \left (210 a+210 b \log \left (c x^n\right )+107 b n\right )}{360 e^4 (d+e x)^3}-\frac{x^3 \left (420 a+420 b \log \left (c x^n\right )+319 b n\right )}{360 e^5 (d+e x)^2}-\frac{x^2 \left (140 a+140 b \log \left (c x^n\right )+153 b n\right )}{40 e^6 (d+e x)}-\frac{d \log \left (\frac{e x}{d}+1\right ) \left (140 a+140 b \log \left (c x^n\right )+223 b n\right )}{20 e^8}-\frac{x^7 \left (a+b \log \left (c x^n\right )\right )}{6 e (d+e x)^6}+\frac{x (140 a+223 b n)}{20 e^7}+\frac{7 b x \log \left (c x^n\right )}{e^7}-\frac{7 b n x}{e^7} \]

[Out]

(-7*b*n*x)/e^7 + ((140*a + 223*b*n)*x)/(20*e^7) + (7*b*x*Log[c*x^n])/e^7 - (x^7*(a + b*Log[c*x^n]))/(6*e*(d +
e*x)^6) - (x^6*(7*a + b*n + 7*b*Log[c*x^n]))/(30*e^2*(d + e*x)^5) - (x^5*(42*a + 13*b*n + 42*b*Log[c*x^n]))/(1
20*e^3*(d + e*x)^4) - (x^2*(140*a + 153*b*n + 140*b*Log[c*x^n]))/(40*e^6*(d + e*x)) - (x^4*(210*a + 107*b*n +
210*b*Log[c*x^n]))/(360*e^4*(d + e*x)^3) - (x^3*(420*a + 319*b*n + 420*b*Log[c*x^n]))/(360*e^5*(d + e*x)^2) -
(d*(140*a + 223*b*n + 140*b*Log[c*x^n])*Log[1 + (e*x)/d])/(20*e^8) - (7*b*d*n*PolyLog[2, -((e*x)/d)])/e^8

________________________________________________________________________________________

Rubi [A]  time = 0.571391, antiderivative size = 351, normalized size of antiderivative = 1.23, 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 = {43, 2351, 2295, 2319, 44, 2314, 31, 2317, 2391} \[ -\frac{7 b d n \text{PolyLog}\left (2,-\frac{e x}{d}\right )}{e^8}+\frac{d^7 \left (a+b \log \left (c x^n\right )\right )}{6 e^8 (d+e x)^6}-\frac{7 d^6 \left (a+b \log \left (c x^n\right )\right )}{5 e^8 (d+e x)^5}+\frac{21 d^5 \left (a+b \log \left (c x^n\right )\right )}{4 e^8 (d+e x)^4}-\frac{35 d^4 \left (a+b \log \left (c x^n\right )\right )}{3 e^8 (d+e x)^3}+\frac{35 d^3 \left (a+b \log \left (c x^n\right )\right )}{2 e^8 (d+e x)^2}+\frac{21 d x \left (a+b \log \left (c x^n\right )\right )}{e^7 (d+e x)}-\frac{7 d \log \left (\frac{e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{e^8}+\frac{a x}{e^7}+\frac{b x \log \left (c x^n\right )}{e^7}-\frac{b d^6 n}{30 e^8 (d+e x)^5}+\frac{37 b d^5 n}{120 e^8 (d+e x)^4}-\frac{241 b d^4 n}{180 e^8 (d+e x)^3}+\frac{153 b d^3 n}{40 e^8 (d+e x)^2}-\frac{197 b d^2 n}{20 e^8 (d+e x)}-\frac{197 b d n \log (x)}{20 e^8}-\frac{223 b d n \log (d+e x)}{20 e^8}-\frac{b n x}{e^7} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*x)/e^7 - (b*n*x)/e^7 - (b*d^6*n)/(30*e^8*(d + e*x)^5) + (37*b*d^5*n)/(120*e^8*(d + e*x)^4) - (241*b*d^4*n)/
(180*e^8*(d + e*x)^3) + (153*b*d^3*n)/(40*e^8*(d + e*x)^2) - (197*b*d^2*n)/(20*e^8*(d + e*x)) - (197*b*d*n*Log
[x])/(20*e^8) + (b*x*Log[c*x^n])/e^7 + (d^7*(a + b*Log[c*x^n]))/(6*e^8*(d + e*x)^6) - (7*d^6*(a + b*Log[c*x^n]
))/(5*e^8*(d + e*x)^5) + (21*d^5*(a + b*Log[c*x^n]))/(4*e^8*(d + e*x)^4) - (35*d^4*(a + b*Log[c*x^n]))/(3*e^8*
(d + e*x)^3) + (35*d^3*(a + b*Log[c*x^n]))/(2*e^8*(d + e*x)^2) + (21*d*x*(a + b*Log[c*x^n]))/(e^7*(d + e*x)) -
(223*b*d*n*Log[d + e*x])/(20*e^8) - (7*d*(a + b*Log[c*x^n])*Log[1 + (e*x)/d])/e^8 - (7*b*d*n*PolyLog[2, -((e*
x)/d)])/e^8

Rule 43

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, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[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 2295

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{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 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 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{x^7 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^7} \, dx &=\int \left (\frac{a+b \log \left (c x^n\right )}{e^7}-\frac{d^7 \left (a+b \log \left (c x^n\right )\right )}{e^7 (d+e x)^7}+\frac{7 d^6 \left (a+b \log \left (c x^n\right )\right )}{e^7 (d+e x)^6}-\frac{21 d^5 \left (a+b \log \left (c x^n\right )\right )}{e^7 (d+e x)^5}+\frac{35 d^4 \left (a+b \log \left (c x^n\right )\right )}{e^7 (d+e x)^4}-\frac{35 d^3 \left (a+b \log \left (c x^n\right )\right )}{e^7 (d+e x)^3}+\frac{21 d^2 \left (a+b \log \left (c x^n\right )\right )}{e^7 (d+e x)^2}-\frac{7 d \left (a+b \log \left (c x^n\right )\right )}{e^7 (d+e x)}\right ) \, dx\\ &=\frac{\int \left (a+b \log \left (c x^n\right )\right ) \, dx}{e^7}-\frac{(7 d) \int \frac{a+b \log \left (c x^n\right )}{d+e x} \, dx}{e^7}+\frac{\left (21 d^2\right ) \int \frac{a+b \log \left (c x^n\right )}{(d+e x)^2} \, dx}{e^7}-\frac{\left (35 d^3\right ) \int \frac{a+b \log \left (c x^n\right )}{(d+e x)^3} \, dx}{e^7}+\frac{\left (35 d^4\right ) \int \frac{a+b \log \left (c x^n\right )}{(d+e x)^4} \, dx}{e^7}-\frac{\left (21 d^5\right ) \int \frac{a+b \log \left (c x^n\right )}{(d+e x)^5} \, dx}{e^7}+\frac{\left (7 d^6\right ) \int \frac{a+b \log \left (c x^n\right )}{(d+e x)^6} \, dx}{e^7}-\frac{d^7 \int \frac{a+b \log \left (c x^n\right )}{(d+e x)^7} \, dx}{e^7}\\ &=\frac{a x}{e^7}+\frac{d^7 \left (a+b \log \left (c x^n\right )\right )}{6 e^8 (d+e x)^6}-\frac{7 d^6 \left (a+b \log \left (c x^n\right )\right )}{5 e^8 (d+e x)^5}+\frac{21 d^5 \left (a+b \log \left (c x^n\right )\right )}{4 e^8 (d+e x)^4}-\frac{35 d^4 \left (a+b \log \left (c x^n\right )\right )}{3 e^8 (d+e x)^3}+\frac{35 d^3 \left (a+b \log \left (c x^n\right )\right )}{2 e^8 (d+e x)^2}+\frac{21 d x \left (a+b \log \left (c x^n\right )\right )}{e^7 (d+e x)}-\frac{7 d \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{e x}{d}\right )}{e^8}+\frac{b \int \log \left (c x^n\right ) \, dx}{e^7}+\frac{(7 b d n) \int \frac{\log \left (1+\frac{e x}{d}\right )}{x} \, dx}{e^8}-\frac{\left (35 b d^3 n\right ) \int \frac{1}{x (d+e x)^2} \, dx}{2 e^8}+\frac{\left (35 b d^4 n\right ) \int \frac{1}{x (d+e x)^3} \, dx}{3 e^8}-\frac{\left (21 b d^5 n\right ) \int \frac{1}{x (d+e x)^4} \, dx}{4 e^8}+\frac{\left (7 b d^6 n\right ) \int \frac{1}{x (d+e x)^5} \, dx}{5 e^8}-\frac{\left (b d^7 n\right ) \int \frac{1}{x (d+e x)^6} \, dx}{6 e^8}-\frac{(21 b d n) \int \frac{1}{d+e x} \, dx}{e^7}\\ &=\frac{a x}{e^7}-\frac{b n x}{e^7}+\frac{b x \log \left (c x^n\right )}{e^7}+\frac{d^7 \left (a+b \log \left (c x^n\right )\right )}{6 e^8 (d+e x)^6}-\frac{7 d^6 \left (a+b \log \left (c x^n\right )\right )}{5 e^8 (d+e x)^5}+\frac{21 d^5 \left (a+b \log \left (c x^n\right )\right )}{4 e^8 (d+e x)^4}-\frac{35 d^4 \left (a+b \log \left (c x^n\right )\right )}{3 e^8 (d+e x)^3}+\frac{35 d^3 \left (a+b \log \left (c x^n\right )\right )}{2 e^8 (d+e x)^2}+\frac{21 d x \left (a+b \log \left (c x^n\right )\right )}{e^7 (d+e x)}-\frac{21 b d n \log (d+e x)}{e^8}-\frac{7 d \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{e x}{d}\right )}{e^8}-\frac{7 b d n \text{Li}_2\left (-\frac{e x}{d}\right )}{e^8}-\frac{\left (35 b d^3 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 e^8}+\frac{\left (35 b d^4 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 e^8}-\frac{\left (21 b d^5 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}{4 e^8}+\frac{\left (7 b d^6 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 e^8}-\frac{\left (b d^7 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 e^8}\\ &=\frac{a x}{e^7}-\frac{b n x}{e^7}-\frac{b d^6 n}{30 e^8 (d+e x)^5}+\frac{37 b d^5 n}{120 e^8 (d+e x)^4}-\frac{241 b d^4 n}{180 e^8 (d+e x)^3}+\frac{153 b d^3 n}{40 e^8 (d+e x)^2}-\frac{197 b d^2 n}{20 e^8 (d+e x)}-\frac{197 b d n \log (x)}{20 e^8}+\frac{b x \log \left (c x^n\right )}{e^7}+\frac{d^7 \left (a+b \log \left (c x^n\right )\right )}{6 e^8 (d+e x)^6}-\frac{7 d^6 \left (a+b \log \left (c x^n\right )\right )}{5 e^8 (d+e x)^5}+\frac{21 d^5 \left (a+b \log \left (c x^n\right )\right )}{4 e^8 (d+e x)^4}-\frac{35 d^4 \left (a+b \log \left (c x^n\right )\right )}{3 e^8 (d+e x)^3}+\frac{35 d^3 \left (a+b \log \left (c x^n\right )\right )}{2 e^8 (d+e x)^2}+\frac{21 d x \left (a+b \log \left (c x^n\right )\right )}{e^7 (d+e x)}-\frac{223 b d n \log (d+e x)}{20 e^8}-\frac{7 d \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{e x}{d}\right )}{e^8}-\frac{7 b d n \text{Li}_2\left (-\frac{e x}{d}\right )}{e^8}\\ \end{align*}

Mathematica [A]  time = 0.566353, size = 356, normalized size = 1.25 \[ -\frac{2520 b d n \text{PolyLog}\left (2,-\frac{e x}{d}\right )-\frac{60 a d^7}{(d+e x)^6}+\frac{504 a d^6}{(d+e x)^5}-\frac{1890 a d^5}{(d+e x)^4}+\frac{4200 a d^4}{(d+e x)^3}-\frac{6300 a d^3}{(d+e x)^2}+\frac{7560 a d^2}{d+e x}+2520 a d \log \left (\frac{e x}{d}+1\right )-360 a e x-\frac{60 b d^7 \log \left (c x^n\right )}{(d+e x)^6}+\frac{504 b d^6 \log \left (c x^n\right )}{(d+e x)^5}-\frac{1890 b d^5 \log \left (c x^n\right )}{(d+e x)^4}+\frac{4200 b d^4 \log \left (c x^n\right )}{(d+e x)^3}-\frac{6300 b d^3 \log \left (c x^n\right )}{(d+e x)^2}+\frac{7560 b d^2 \log \left (c x^n\right )}{d+e x}+2520 b d \log \left (c x^n\right ) \log \left (\frac{e x}{d}+1\right )-360 b e x \log \left (c x^n\right )+\frac{12 b d^6 n}{(d+e x)^5}-\frac{111 b d^5 n}{(d+e x)^4}+\frac{482 b d^4 n}{(d+e x)^3}-\frac{1377 b d^3 n}{(d+e x)^2}+\frac{3546 b d^2 n}{d+e x}+4014 b d n \log (d+e x)-4014 b d n \log (x)+360 b e n x}{360 e^8} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(-360*a*e*x + 360*b*e*n*x - (60*a*d^7)/(d + e*x)^6 + (504*a*d^6)/(d + e*x)^5 + (12*b*d^6*n)/(d + e*x)^5 - (18
90*a*d^5)/(d + e*x)^4 - (111*b*d^5*n)/(d + e*x)^4 + (4200*a*d^4)/(d + e*x)^3 + (482*b*d^4*n)/(d + e*x)^3 - (63
00*a*d^3)/(d + e*x)^2 - (1377*b*d^3*n)/(d + e*x)^2 + (7560*a*d^2)/(d + e*x) + (3546*b*d^2*n)/(d + e*x) - 4014*
b*d*n*Log[x] - 360*b*e*x*Log[c*x^n] - (60*b*d^7*Log[c*x^n])/(d + e*x)^6 + (504*b*d^6*Log[c*x^n])/(d + e*x)^5 -
(1890*b*d^5*Log[c*x^n])/(d + e*x)^4 + (4200*b*d^4*Log[c*x^n])/(d + e*x)^3 - (6300*b*d^3*Log[c*x^n])/(d + e*x)
^2 + (7560*b*d^2*Log[c*x^n])/(d + e*x) + 4014*b*d*n*Log[d + e*x] + 2520*a*d*Log[1 + (e*x)/d] + 2520*b*d*Log[c*
x^n]*Log[1 + (e*x)/d] + 2520*b*d*n*PolyLog[2, -((e*x)/d)])/(360*e^8)

________________________________________________________________________________________

Maple [C]  time = 0.23, size = 1584, normalized size = 5.6 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-21/8*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)/e^8*d^5/(e*x+d)^4-21/2*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2/e^8
*d^2/(e*x+d)-7/2*I*b*Pi*csgn(I*c*x^n)^2*csgn(I*c)/e^8*d*ln(e*x+d)-1/12*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I
*c)*d^7/e^8/(e*x+d)^6+7/2*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)/e^8*d*ln(e*x+d)+7*b*n/e^8*d*ln(e*x+d)*ln(
-e*x/d)-1/2*I*b*Pi*csgn(I*c*x^n)^3/e^7*x-7*b*ln(x^n)/e^8*d*ln(e*x+d)-21*b*ln(x^n)/e^8*d^2/(e*x+d)-7/5*b*ln(x^n
)/e^8*d^6/(e*x+d)^5+21/8*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2/e^8*d^5/(e*x+d)^4+21/8*I*b*Pi*csgn(I*c*x^n)^2*csgn
(I*c)/e^8*d^5/(e*x+d)^4+7/10*I*b*Pi*csgn(I*c*x^n)^3/e^8*d^6/(e*x+d)^5-35/6*I*b*Pi*csgn(I*c*x^n)^2*csgn(I*c)/e^
8*d^4/(e*x+d)^3-21/2*I*b*Pi*csgn(I*c*x^n)^2*csgn(I*c)/e^8*d^2/(e*x+d)+1/12*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2*
d^7/e^8/(e*x+d)^6+1/12*I*b*Pi*csgn(I*c*x^n)^2*csgn(I*c)*d^7/e^8/(e*x+d)^6+35/4*I*b*Pi*csgn(I*c*x^n)^2*csgn(I*c
)/e^8*d^3/(e*x+d)^2-7/10*I*b*Pi*csgn(I*c*x^n)^2*csgn(I*c)/e^8*d^6/(e*x+d)^5-1/2*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^
n)*csgn(I*c)/e^7*x-7/2*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2/e^8*d*ln(e*x+d)-b*n/e^8*d-21/8*I*b*Pi*csgn(I*c*x^n)^
3/e^8*d^5/(e*x+d)^4+1/2*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2/e^7*x+7/2*I*b*Pi*csgn(I*c*x^n)^3/e^8*d*ln(e*x+d)+35
/4*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2/e^8*d^3/(e*x+d)^2-35/6*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2/e^8*d^4/(e*x+d
)^3+7/10*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)/e^8*d^6/(e*x+d)^5+223/20*b*n/e^8*d*ln(e*x)-223/20*b*n/e^8*
d*ln(e*x+d)-197/20*b*n/e^8*d^2/(e*x+d)+153/40*b*n/e^8*d^3/(e*x+d)^2-241/180*b*n/e^8*d^4/(e*x+d)^3+37/120*b*n/e
^8*d^5/(e*x+d)^4-1/30*b*n/e^8*d^6/(e*x+d)^5+7*b*n/e^8*d*dilog(-e*x/d)+21/4*a/e^8*d^5/(e*x+d)^4-21*a/e^8*d^2/(e
*x+d)-7/5*a/e^8*d^6/(e*x+d)^5-7*a/e^8*d*ln(e*x+d)+1/6*a*d^7/e^8/(e*x+d)^6+35/2*a/e^8*d^3/(e*x+d)^2-35/3*a/e^8*
d^4/(e*x+d)^3+b*ln(c)/e^7*x+21/2*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)/e^8*d^2/(e*x+d)-7/10*I*b*Pi*csgn(I
*x^n)*csgn(I*c*x^n)^2/e^8*d^6/(e*x+d)^5+b*ln(x^n)/e^7*x-7*b*ln(c)/e^8*d*ln(e*x+d)+1/6*b*ln(c)*d^7/e^8/(e*x+d)^
6+35/2*b*ln(c)/e^8*d^3/(e*x+d)^2-35/3*b*ln(c)/e^8*d^4/(e*x+d)^3+21/4*b*ln(c)/e^8*d^5/(e*x+d)^4-21*b*ln(c)/e^8*
d^2/(e*x+d)-7/5*b*ln(c)/e^8*d^6/(e*x+d)^5+35/6*I*b*Pi*csgn(I*c*x^n)^3/e^8*d^4/(e*x+d)^3+21/2*I*b*Pi*csgn(I*c*x
^n)^3/e^8*d^2/(e*x+d)+a/e^7*x-1/12*I*b*Pi*csgn(I*c*x^n)^3*d^7/e^8/(e*x+d)^6+1/2*I*b*Pi*csgn(I*c*x^n)^2*csgn(I*
c)/e^7*x-35/4*I*b*Pi*csgn(I*c*x^n)^3/e^8*d^3/(e*x+d)^2-35/4*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)/e^8*d^3
/(e*x+d)^2+35/6*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)/e^8*d^4/(e*x+d)^3+1/6*b*ln(x^n)*d^7/e^8/(e*x+d)^6+3
5/2*b*ln(x^n)/e^8*d^3/(e*x+d)^2-35/3*b*ln(x^n)/e^8*d^4/(e*x+d)^3+21/4*b*ln(x^n)/e^8*d^5/(e*x+d)^4-b*n*x/e^7

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{1}{60} \, a{\left (\frac{1260 \, d^{2} e^{5} x^{5} + 5250 \, d^{3} e^{4} x^{4} + 9100 \, d^{4} e^{3} x^{3} + 8085 \, d^{5} e^{2} x^{2} + 3654 \, d^{6} e x + 669 \, d^{7}}{e^{14} x^{6} + 6 \, d e^{13} x^{5} + 15 \, d^{2} e^{12} x^{4} + 20 \, d^{3} e^{11} x^{3} + 15 \, d^{4} e^{10} x^{2} + 6 \, d^{5} e^{9} x + d^{6} e^{8}} - \frac{60 \, x}{e^{7}} + \frac{420 \, d \log \left (e x + d\right )}{e^{8}}\right )} + b \int \frac{x^{7} \log \left (c\right ) + x^{7} \log \left (x^{n}\right )}{e^{7} x^{7} + 7 \, d e^{6} x^{6} + 21 \, d^{2} e^{5} x^{5} + 35 \, d^{3} e^{4} x^{4} + 35 \, d^{4} e^{3} x^{3} + 21 \, d^{5} e^{2} x^{2} + 7 \, d^{6} e x + d^{7}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/60*a*((1260*d^2*e^5*x^5 + 5250*d^3*e^4*x^4 + 9100*d^4*e^3*x^3 + 8085*d^5*e^2*x^2 + 3654*d^6*e*x + 669*d^7)/
(e^14*x^6 + 6*d*e^13*x^5 + 15*d^2*e^12*x^4 + 20*d^3*e^11*x^3 + 15*d^4*e^10*x^2 + 6*d^5*e^9*x + d^6*e^8) - 60*x
/e^7 + 420*d*log(e*x + d)/e^8) + b*integrate((x^7*log(c) + x^7*log(x^n))/(e^7*x^7 + 7*d*e^6*x^6 + 21*d^2*e^5*x
^5 + 35*d^3*e^4*x^4 + 35*d^4*e^3*x^3 + 21*d^5*e^2*x^2 + 7*d^6*e*x + d^7), x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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