3.62 \(\int \frac{x^8 (a+b \log (c x^n))}{(d+e x)^7} \, dx\)

Optimal. Leaf size=329 \[ \frac{28 b d^2 n \text{PolyLog}\left (2,-\frac{e x}{d}\right )}{e^9}+\frac{d^2 \log \left (\frac{e x}{d}+1\right ) \left (280 a+280 b \log \left (c x^n\right )+341 b n\right )}{10 e^9}-\frac{x^7 \left (8 a+8 b \log \left (c x^n\right )+b n\right )}{30 e^2 (d+e x)^5}-\frac{x^6 \left (56 a+56 b \log \left (c x^n\right )+15 b n\right )}{120 e^3 (d+e x)^4}-\frac{x^5 \left (168 a+168 b \log \left (c x^n\right )+73 b n\right )}{180 e^4 (d+e x)^3}-\frac{x^4 \left (840 a+840 b \log \left (c x^n\right )+533 b n\right )}{360 e^5 (d+e x)^2}-\frac{x^3 \left (840 a+840 b \log \left (c x^n\right )+743 b n\right )}{90 e^6 (d+e x)}-\frac{x^8 \left (a+b \log \left (c x^n\right )\right )}{6 e (d+e x)^6}+\frac{x^2 \left (280 a+280 b \log \left (c x^n\right )+341 b n\right )}{20 e^7}-\frac{d x (280 a+341 b n)}{10 e^8}-\frac{28 b d x \log \left (c x^n\right )}{e^8}+\frac{28 b d n x}{e^8}-\frac{7 b n x^2}{e^7} \]

[Out]

(28*b*d*n*x)/e^8 - (d*(280*a + 341*b*n)*x)/(10*e^8) - (7*b*n*x^2)/e^7 - (28*b*d*x*Log[c*x^n])/e^8 - (x^8*(a +
b*Log[c*x^n]))/(6*e*(d + e*x)^6) - (x^7*(8*a + b*n + 8*b*Log[c*x^n]))/(30*e^2*(d + e*x)^5) - (x^6*(56*a + 15*b
*n + 56*b*Log[c*x^n]))/(120*e^3*(d + e*x)^4) - (x^5*(168*a + 73*b*n + 168*b*Log[c*x^n]))/(180*e^4*(d + e*x)^3)
+ (x^2*(280*a + 341*b*n + 280*b*Log[c*x^n]))/(20*e^7) - (x^4*(840*a + 533*b*n + 840*b*Log[c*x^n]))/(360*e^5*(
d + e*x)^2) - (x^3*(840*a + 743*b*n + 840*b*Log[c*x^n]))/(90*e^6*(d + e*x)) + (d^2*(280*a + 341*b*n + 280*b*Lo
g[c*x^n])*Log[1 + (e*x)/d])/(10*e^9) + (28*b*d^2*n*PolyLog[2, -((e*x)/d)])/e^9

________________________________________________________________________________________

Rubi [A]  time = 0.639185, antiderivative size = 394, normalized size of antiderivative = 1.2, number of steps used = 24, number of rules used = 10, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.476, Rules used = {43, 2351, 2295, 2304, 2319, 44, 2314, 31, 2317, 2391} \[ \frac{28 b d^2 n \text{PolyLog}\left (2,-\frac{e x}{d}\right )}{e^9}-\frac{d^8 \left (a+b \log \left (c x^n\right )\right )}{6 e^9 (d+e x)^6}+\frac{8 d^7 \left (a+b \log \left (c x^n\right )\right )}{5 e^9 (d+e x)^5}-\frac{7 d^6 \left (a+b \log \left (c x^n\right )\right )}{e^9 (d+e x)^4}+\frac{56 d^5 \left (a+b \log \left (c x^n\right )\right )}{3 e^9 (d+e x)^3}-\frac{35 d^4 \left (a+b \log \left (c x^n\right )\right )}{e^9 (d+e x)^2}-\frac{56 d^2 x \left (a+b \log \left (c x^n\right )\right )}{e^8 (d+e x)}+\frac{28 d^2 \log \left (\frac{e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{e^9}+\frac{x^2 \left (a+b \log \left (c x^n\right )\right )}{2 e^7}-\frac{7 a d x}{e^8}-\frac{7 b d x \log \left (c x^n\right )}{e^8}+\frac{b d^7 n}{30 e^9 (d+e x)^5}-\frac{43 b d^6 n}{120 e^9 (d+e x)^4}+\frac{167 b d^5 n}{90 e^9 (d+e x)^3}-\frac{131 b d^4 n}{20 e^9 (d+e x)^2}+\frac{219 b d^3 n}{10 e^9 (d+e x)}+\frac{219 b d^2 n \log (x)}{10 e^9}+\frac{341 b d^2 n \log (d+e x)}{10 e^9}+\frac{7 b d n x}{e^8}-\frac{b n x^2}{4 e^7} \]

Antiderivative was successfully verified.

[In]

Int[(x^8*(a + b*Log[c*x^n]))/(d + e*x)^7,x]

[Out]

(-7*a*d*x)/e^8 + (7*b*d*n*x)/e^8 - (b*n*x^2)/(4*e^7) + (b*d^7*n)/(30*e^9*(d + e*x)^5) - (43*b*d^6*n)/(120*e^9*
(d + e*x)^4) + (167*b*d^5*n)/(90*e^9*(d + e*x)^3) - (131*b*d^4*n)/(20*e^9*(d + e*x)^2) + (219*b*d^3*n)/(10*e^9
*(d + e*x)) + (219*b*d^2*n*Log[x])/(10*e^9) - (7*b*d*x*Log[c*x^n])/e^8 + (x^2*(a + b*Log[c*x^n]))/(2*e^7) - (d
^8*(a + b*Log[c*x^n]))/(6*e^9*(d + e*x)^6) + (8*d^7*(a + b*Log[c*x^n]))/(5*e^9*(d + e*x)^5) - (7*d^6*(a + b*Lo
g[c*x^n]))/(e^9*(d + e*x)^4) + (56*d^5*(a + b*Log[c*x^n]))/(3*e^9*(d + e*x)^3) - (35*d^4*(a + b*Log[c*x^n]))/(
e^9*(d + e*x)^2) - (56*d^2*x*(a + b*Log[c*x^n]))/(e^8*(d + e*x)) + (341*b*d^2*n*Log[d + e*x])/(10*e^9) + (28*d
^2*(a + b*Log[c*x^n])*Log[1 + (e*x)/d])/e^9 + (28*b*d^2*n*PolyLog[2, -((e*x)/d)])/e^9

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 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 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^8 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^7} \, dx &=\int \left (-\frac{7 d \left (a+b \log \left (c x^n\right )\right )}{e^8}+\frac{x \left (a+b \log \left (c x^n\right )\right )}{e^7}+\frac{d^8 \left (a+b \log \left (c x^n\right )\right )}{e^8 (d+e x)^7}-\frac{8 d^7 \left (a+b \log \left (c x^n\right )\right )}{e^8 (d+e x)^6}+\frac{28 d^6 \left (a+b \log \left (c x^n\right )\right )}{e^8 (d+e x)^5}-\frac{56 d^5 \left (a+b \log \left (c x^n\right )\right )}{e^8 (d+e x)^4}+\frac{70 d^4 \left (a+b \log \left (c x^n\right )\right )}{e^8 (d+e x)^3}-\frac{56 d^3 \left (a+b \log \left (c x^n\right )\right )}{e^8 (d+e x)^2}+\frac{28 d^2 \left (a+b \log \left (c x^n\right )\right )}{e^8 (d+e x)}\right ) \, dx\\ &=-\frac{(7 d) \int \left (a+b \log \left (c x^n\right )\right ) \, dx}{e^8}+\frac{\left (28 d^2\right ) \int \frac{a+b \log \left (c x^n\right )}{d+e x} \, dx}{e^8}-\frac{\left (56 d^3\right ) \int \frac{a+b \log \left (c x^n\right )}{(d+e x)^2} \, dx}{e^8}+\frac{\left (70 d^4\right ) \int \frac{a+b \log \left (c x^n\right )}{(d+e x)^3} \, dx}{e^8}-\frac{\left (56 d^5\right ) \int \frac{a+b \log \left (c x^n\right )}{(d+e x)^4} \, dx}{e^8}+\frac{\left (28 d^6\right ) \int \frac{a+b \log \left (c x^n\right )}{(d+e x)^5} \, dx}{e^8}-\frac{\left (8 d^7\right ) \int \frac{a+b \log \left (c x^n\right )}{(d+e x)^6} \, dx}{e^8}+\frac{d^8 \int \frac{a+b \log \left (c x^n\right )}{(d+e x)^7} \, dx}{e^8}+\frac{\int x \left (a+b \log \left (c x^n\right )\right ) \, dx}{e^7}\\ &=-\frac{7 a d x}{e^8}-\frac{b n x^2}{4 e^7}+\frac{x^2 \left (a+b \log \left (c x^n\right )\right )}{2 e^7}-\frac{d^8 \left (a+b \log \left (c x^n\right )\right )}{6 e^9 (d+e x)^6}+\frac{8 d^7 \left (a+b \log \left (c x^n\right )\right )}{5 e^9 (d+e x)^5}-\frac{7 d^6 \left (a+b \log \left (c x^n\right )\right )}{e^9 (d+e x)^4}+\frac{56 d^5 \left (a+b \log \left (c x^n\right )\right )}{3 e^9 (d+e x)^3}-\frac{35 d^4 \left (a+b \log \left (c x^n\right )\right )}{e^9 (d+e x)^2}-\frac{56 d^2 x \left (a+b \log \left (c x^n\right )\right )}{e^8 (d+e x)}+\frac{28 d^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{e x}{d}\right )}{e^9}-\frac{(7 b d) \int \log \left (c x^n\right ) \, dx}{e^8}-\frac{\left (28 b d^2 n\right ) \int \frac{\log \left (1+\frac{e x}{d}\right )}{x} \, dx}{e^9}+\frac{\left (35 b d^4 n\right ) \int \frac{1}{x (d+e x)^2} \, dx}{e^9}-\frac{\left (56 b d^5 n\right ) \int \frac{1}{x (d+e x)^3} \, dx}{3 e^9}+\frac{\left (7 b d^6 n\right ) \int \frac{1}{x (d+e x)^4} \, dx}{e^9}-\frac{\left (8 b d^7 n\right ) \int \frac{1}{x (d+e x)^5} \, dx}{5 e^9}+\frac{\left (b d^8 n\right ) \int \frac{1}{x (d+e x)^6} \, dx}{6 e^9}+\frac{\left (56 b d^2 n\right ) \int \frac{1}{d+e x} \, dx}{e^8}\\ &=-\frac{7 a d x}{e^8}+\frac{7 b d n x}{e^8}-\frac{b n x^2}{4 e^7}-\frac{7 b d x \log \left (c x^n\right )}{e^8}+\frac{x^2 \left (a+b \log \left (c x^n\right )\right )}{2 e^7}-\frac{d^8 \left (a+b \log \left (c x^n\right )\right )}{6 e^9 (d+e x)^6}+\frac{8 d^7 \left (a+b \log \left (c x^n\right )\right )}{5 e^9 (d+e x)^5}-\frac{7 d^6 \left (a+b \log \left (c x^n\right )\right )}{e^9 (d+e x)^4}+\frac{56 d^5 \left (a+b \log \left (c x^n\right )\right )}{3 e^9 (d+e x)^3}-\frac{35 d^4 \left (a+b \log \left (c x^n\right )\right )}{e^9 (d+e x)^2}-\frac{56 d^2 x \left (a+b \log \left (c x^n\right )\right )}{e^8 (d+e x)}+\frac{56 b d^2 n \log (d+e x)}{e^9}+\frac{28 d^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{e x}{d}\right )}{e^9}+\frac{28 b d^2 n \text{Li}_2\left (-\frac{e x}{d}\right )}{e^9}+\frac{\left (35 b d^4 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}{e^9}-\frac{\left (56 b d^5 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^9}+\frac{\left (7 b d^6 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}{e^9}-\frac{\left (8 b d^7 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^9}+\frac{\left (b d^8 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^9}\\ &=-\frac{7 a d x}{e^8}+\frac{7 b d n x}{e^8}-\frac{b n x^2}{4 e^7}+\frac{b d^7 n}{30 e^9 (d+e x)^5}-\frac{43 b d^6 n}{120 e^9 (d+e x)^4}+\frac{167 b d^5 n}{90 e^9 (d+e x)^3}-\frac{131 b d^4 n}{20 e^9 (d+e x)^2}+\frac{219 b d^3 n}{10 e^9 (d+e x)}+\frac{219 b d^2 n \log (x)}{10 e^9}-\frac{7 b d x \log \left (c x^n\right )}{e^8}+\frac{x^2 \left (a+b \log \left (c x^n\right )\right )}{2 e^7}-\frac{d^8 \left (a+b \log \left (c x^n\right )\right )}{6 e^9 (d+e x)^6}+\frac{8 d^7 \left (a+b \log \left (c x^n\right )\right )}{5 e^9 (d+e x)^5}-\frac{7 d^6 \left (a+b \log \left (c x^n\right )\right )}{e^9 (d+e x)^4}+\frac{56 d^5 \left (a+b \log \left (c x^n\right )\right )}{3 e^9 (d+e x)^3}-\frac{35 d^4 \left (a+b \log \left (c x^n\right )\right )}{e^9 (d+e x)^2}-\frac{56 d^2 x \left (a+b \log \left (c x^n\right )\right )}{e^8 (d+e x)}+\frac{341 b d^2 n \log (d+e x)}{10 e^9}+\frac{28 d^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{e x}{d}\right )}{e^9}+\frac{28 b d^2 n \text{Li}_2\left (-\frac{e x}{d}\right )}{e^9}\\ \end{align*}

Mathematica [A]  time = 0.481012, size = 403, normalized size = 1.22 \[ \frac{10080 b d^2 n \text{PolyLog}\left (2,-\frac{e x}{d}\right )-\frac{60 a d^8}{(d+e x)^6}+\frac{576 a d^7}{(d+e x)^5}-\frac{2520 a d^6}{(d+e x)^4}+\frac{6720 a d^5}{(d+e x)^3}-\frac{12600 a d^4}{(d+e x)^2}+\frac{20160 a d^3}{d+e x}+10080 a d^2 \log \left (\frac{e x}{d}+1\right )-2520 a d e x+180 a e^2 x^2-\frac{60 b d^8 \log \left (c x^n\right )}{(d+e x)^6}+\frac{576 b d^7 \log \left (c x^n\right )}{(d+e x)^5}-\frac{2520 b d^6 \log \left (c x^n\right )}{(d+e x)^4}+\frac{6720 b d^5 \log \left (c x^n\right )}{(d+e x)^3}-\frac{12600 b d^4 \log \left (c x^n\right )}{(d+e x)^2}+\frac{20160 b d^3 \log \left (c x^n\right )}{d+e x}+10080 b d^2 \log \left (c x^n\right ) \log \left (\frac{e x}{d}+1\right )-2520 b d e x \log \left (c x^n\right )+180 b e^2 x^2 \log \left (c x^n\right )+\frac{12 b d^7 n}{(d+e x)^5}-\frac{129 b d^6 n}{(d+e x)^4}+\frac{668 b d^5 n}{(d+e x)^3}-\frac{2358 b d^4 n}{(d+e x)^2}+\frac{7884 b d^3 n}{d+e x}+12276 b d^2 n \log (d+e x)-12276 b d^2 n \log (x)+2520 b d e n x-90 b e^2 n x^2}{360 e^9} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^8*(a + b*Log[c*x^n]))/(d + e*x)^7,x]

[Out]

(-2520*a*d*e*x + 2520*b*d*e*n*x + 180*a*e^2*x^2 - 90*b*e^2*n*x^2 - (60*a*d^8)/(d + e*x)^6 + (576*a*d^7)/(d + e
*x)^5 + (12*b*d^7*n)/(d + e*x)^5 - (2520*a*d^6)/(d + e*x)^4 - (129*b*d^6*n)/(d + e*x)^4 + (6720*a*d^5)/(d + e*
x)^3 + (668*b*d^5*n)/(d + e*x)^3 - (12600*a*d^4)/(d + e*x)^2 - (2358*b*d^4*n)/(d + e*x)^2 + (20160*a*d^3)/(d +
e*x) + (7884*b*d^3*n)/(d + e*x) - 12276*b*d^2*n*Log[x] - 2520*b*d*e*x*Log[c*x^n] + 180*b*e^2*x^2*Log[c*x^n] -
(60*b*d^8*Log[c*x^n])/(d + e*x)^6 + (576*b*d^7*Log[c*x^n])/(d + e*x)^5 - (2520*b*d^6*Log[c*x^n])/(d + e*x)^4
+ (6720*b*d^5*Log[c*x^n])/(d + e*x)^3 - (12600*b*d^4*Log[c*x^n])/(d + e*x)^2 + (20160*b*d^3*Log[c*x^n])/(d + e
*x) + 12276*b*d^2*n*Log[d + e*x] + 10080*a*d^2*Log[1 + (e*x)/d] + 10080*b*d^2*Log[c*x^n]*Log[1 + (e*x)/d] + 10
080*b*d^2*n*PolyLog[2, -((e*x)/d)])/(360*e^9)

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Maple [C]  time = 0.24, size = 1768, normalized size = 5.4 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^8*(a+b*ln(c*x^n))/(e*x+d)^7,x)

[Out]

-28/3*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)/e^9*d^5/(e*x+d)^3+7/2*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I
*c)/e^9*d^6/(e*x+d)^4-28*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)/e^9*d^3/(e*x+d)-28*b*n/e^9*d^2*ln(e*x+d)*l
n(-e*x/d)-7*b*ln(x^n)/e^8*d*x-1/6*b*ln(x^n)*d^8/e^9/(e*x+d)^6-1/4*I*b*Pi*csgn(I*c*x^n)^3/e^7*x^2-1/4*I*b*Pi*cs
gn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)/e^7*x^2-1/12*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2*d^8/e^9/(e*x+d)^6-7/2*I*b*Pi
*csgn(I*c*x^n)^2*csgn(I*c)/e^8*d*x+4/5*I*b*Pi*csgn(I*c*x^n)^2*csgn(I*c)/e^9*d^7/(e*x+d)^5+28*I*b*Pi*csgn(I*x^n
)*csgn(I*c*x^n)^2/e^9*d^3/(e*x+d)+14*I*b*Pi*csgn(I*c*x^n)^2*csgn(I*c)/e^9*d^2*ln(e*x+d)-1/12*I*b*Pi*csgn(I*c*x
^n)^2*csgn(I*c)*d^8/e^9/(e*x+d)^6-35/2*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2/e^9*d^4/(e*x+d)^2+28/3*I*b*Pi*csgn(I
*x^n)*csgn(I*c*x^n)^2/e^9*d^5/(e*x+d)^3-35/2*I*b*Pi*csgn(I*c*x^n)^2*csgn(I*c)/e^9*d^4/(e*x+d)^2+28/3*I*b*Pi*cs
gn(I*c*x^n)^2*csgn(I*c)/e^9*d^5/(e*x+d)^3+14*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2/e^9*d^2*ln(e*x+d)+4/5*I*b*Pi*c
sgn(I*x^n)*csgn(I*c*x^n)^2/e^9*d^7/(e*x+d)^5-7/2*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2/e^8*d*x+28*I*b*Pi*csgn(I*c
*x^n)^2*csgn(I*c)/e^9*d^3/(e*x+d)-7/2*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2/e^9*d^6/(e*x+d)^4-7/2*I*b*Pi*csgn(I*c
*x^n)^2*csgn(I*c)/e^9*d^6/(e*x+d)^4-7*b*ln(c)/e^8*d*x+28*b*ln(c)/e^9*d^2*ln(e*x+d)-1/6*b*ln(c)*d^8/e^9/(e*x+d)
^6+7/2*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)/e^8*d*x-14*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)/e^9*d^
2*ln(e*x+d)+1/4*I*b*Pi*csgn(I*c*x^n)^2*csgn(I*c)/e^7*x^2+1/12*I*b*Pi*csgn(I*c*x^n)^3*d^8/e^9/(e*x+d)^6+1/2*b*l
n(x^n)/e^7*x^2-341/10*b*n/e^9*d^2*ln(e*x)+341/10*b*n/e^9*d^2*ln(e*x+d)+219/10*b*n/e^9*d^3/(e*x+d)-131/20*b*n/e
^9*d^4/(e*x+d)^2+167/90*b*n/e^9*d^5/(e*x+d)^3-43/120*b*n/e^9*d^6/(e*x+d)^4+1/30*b*n/e^9*d^7/(e*x+d)^5-28*b*n/e
^9*d^2*dilog(-e*x/d)+1/2*a/e^7*x^2-28*I*b*Pi*csgn(I*c*x^n)^3/e^9*d^3/(e*x+d)-4/5*I*b*Pi*csgn(I*c*x^n)^3/e^9*d^
7/(e*x+d)^5-28/3*I*b*Pi*csgn(I*c*x^n)^3/e^9*d^5/(e*x+d)^3-7*b*ln(c)/e^9*d^6/(e*x+d)^4+56*b*ln(c)/e^9*d^3/(e*x+
d)+8/5*b*ln(c)/e^9*d^7/(e*x+d)^5-14*I*b*Pi*csgn(I*c*x^n)^3/e^9*d^2*ln(e*x+d)+7/2*I*b*Pi*csgn(I*c*x^n)^3/e^8*d*
x+28*b*ln(x^n)/e^9*d^2*ln(e*x+d)+56/3*b*ln(c)/e^9*d^5/(e*x+d)^3-35*b*ln(c)/e^9*d^4/(e*x+d)^2+35/2*I*b*Pi*csgn(
I*c*x^n)^3/e^9*d^4/(e*x+d)^2+7/2*I*b*Pi*csgn(I*c*x^n)^3/e^9*d^6/(e*x+d)^4-4/5*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)
*csgn(I*c)/e^9*d^7/(e*x+d)^5+1/12*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*d^8/e^9/(e*x+d)^6+1/4*I*b*Pi*csgn
(I*x^n)*csgn(I*c*x^n)^2/e^7*x^2+35/2*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)/e^9*d^4/(e*x+d)^2+56*a/e^9*d^3
/(e*x+d)+8/5*a/e^9*d^7/(e*x+d)^5-7*a/e^8*d*x+28*a/e^9*d^2*ln(e*x+d)-1/6*a*d^8/e^9/(e*x+d)^6-35*a/e^9*d^4/(e*x+
d)^2+8/5*b*ln(x^n)/e^9*d^7/(e*x+d)^5+29/4*b*n/e^9*d^2+56/3*b*ln(x^n)/e^9*d^5/(e*x+d)^3-7*b*ln(x^n)/e^9*d^6/(e*
x+d)^4+56*b*ln(x^n)/e^9*d^3/(e*x+d)+56/3*a/e^9*d^5/(e*x+d)^3-7*a/e^9*d^6/(e*x+d)^4+1/2*b*ln(c)/e^7*x^2-35*b*ln
(x^n)/e^9*d^4/(e*x+d)^2-1/4*b*n*x^2/e^7+7*b*d*n*x/e^8

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{30} \, a{\left (\frac{1680 \, d^{3} e^{5} x^{5} + 7350 \, d^{4} e^{4} x^{4} + 13160 \, d^{5} e^{3} x^{3} + 11970 \, d^{6} e^{2} x^{2} + 5508 \, d^{7} e x + 1023 \, d^{8}}{e^{15} x^{6} + 6 \, d e^{14} x^{5} + 15 \, d^{2} e^{13} x^{4} + 20 \, d^{3} e^{12} x^{3} + 15 \, d^{4} e^{11} x^{2} + 6 \, d^{5} e^{10} x + d^{6} e^{9}} + \frac{840 \, d^{2} \log \left (e x + d\right )}{e^{9}} + \frac{15 \,{\left (e x^{2} - 14 \, d x\right )}}{e^{8}}\right )} + b \int \frac{x^{8} \log \left (c\right ) + x^{8} \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^8*(a+b*log(c*x^n))/(e*x+d)^7,x, algorithm="maxima")

[Out]

1/30*a*((1680*d^3*e^5*x^5 + 7350*d^4*e^4*x^4 + 13160*d^5*e^3*x^3 + 11970*d^6*e^2*x^2 + 5508*d^7*e*x + 1023*d^8
)/(e^15*x^6 + 6*d*e^14*x^5 + 15*d^2*e^13*x^4 + 20*d^3*e^12*x^3 + 15*d^4*e^11*x^2 + 6*d^5*e^10*x + d^6*e^9) + 8
40*d^2*log(e*x + d)/e^9 + 15*(e*x^2 - 14*d*x)/e^8) + b*integrate((x^8*log(c) + x^8*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)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b x^{8} \log \left (c x^{n}\right ) + a x^{8}}{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^8*(a+b*log(c*x^n))/(e*x+d)^7,x, algorithm="fricas")

[Out]

integral((b*x^8*log(c*x^n) + a*x^8)/(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)

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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(x**8*(a+b*ln(c*x**n))/(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{{\left (b \log \left (c x^{n}\right ) + a\right )} x^{8}}{{\left (e x + d\right )}^{7}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8*(a+b*log(c*x^n))/(e*x+d)^7,x, algorithm="giac")

[Out]

integrate((b*log(c*x^n) + a)*x^8/(e*x + d)^7, x)