\(\int \frac {x^8 (a+b \log (c x^n))}{(d+e x)^7} \, dx\) [62]

Optimal result

Integrand size = 21, antiderivative size = 329 \[ \int \frac {x^8 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^7} \, dx=\frac {28 b d n x}{e^8}-\frac {d (280 a+341 b n) x}{10 e^8}-\frac {7 b n x^2}{e^7}-\frac {28 b d x \log \left (c x^n\right )}{e^8}-\frac {x^8 \left (a+b \log \left (c x^n\right )\right )}{6 e (d+e x)^6}-\frac {x^7 \left (8 a+b n+8 b \log \left (c x^n\right )\right )}{30 e^2 (d+e x)^5}-\frac {x^6 \left (56 a+15 b n+56 b \log \left (c x^n\right )\right )}{120 e^3 (d+e x)^4}-\frac {x^5 \left (168 a+73 b n+168 b \log \left (c x^n\right )\right )}{180 e^4 (d+e x)^3}+\frac {x^2 \left (280 a+341 b n+280 b \log \left (c x^n\right )\right )}{20 e^7}-\frac {x^4 \left (840 a+533 b n+840 b \log \left (c x^n\right )\right )}{360 e^5 (d+e x)^2}-\frac {x^3 \left (840 a+743 b n+840 b \log \left (c x^n\right )\right )}{90 e^6 (d+e x)}+\frac {d^2 \left (280 a+341 b n+280 b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{10 e^9}+\frac {28 b d^2 n \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{e^9} \]

[Out]

28*b*d*n*x/e^8-1/10*d*(341*b*n+280*a)*x/e^8-7*b*n*x^2/e^7-28*b*d*x*ln(c*x^n)/e^8-1/6*x^8*(a+b*ln(c*x^n))/e/(e*
x+d)^6-1/30*x^7*(8*a+b*n+8*b*ln(c*x^n))/e^2/(e*x+d)^5-1/120*x^6*(56*a+15*b*n+56*b*ln(c*x^n))/e^3/(e*x+d)^4-1/1
80*x^5*(168*a+73*b*n+168*b*ln(c*x^n))/e^4/(e*x+d)^3+1/20*x^2*(280*a+341*b*n+280*b*ln(c*x^n))/e^7-1/360*x^4*(84
0*a+533*b*n+840*b*ln(c*x^n))/e^5/(e*x+d)^2-1/90*x^3*(840*a+743*b*n+840*b*ln(c*x^n))/e^6/(e*x+d)+1/10*d^2*(280*
a+341*b*n+280*b*ln(c*x^n))*ln(1+e*x/d)/e^9+28*b*d^2*n*polylog(2,-e*x/d)/e^9

Rubi [A] (verified)

Time = 0.66 (sec) , antiderivative size = 329, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2384, 45, 2393, 2332, 2341, 2354, 2438} \[ \int \frac {x^8 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^7} \, dx=\frac {d^2 \log \left (\frac {e x}{d}+1\right ) \left (280 a+280 b \log \left (c x^n\right )+341 b n\right )}{10 e^9}-\frac {x^3 \left (840 a+840 b \log \left (c x^n\right )+743 b n\right )}{90 e^6 (d+e x)}-\frac {x^4 \left (840 a+840 b \log \left (c x^n\right )+533 b n\right )}{360 e^5 (d+e x)^2}-\frac {x^5 \left (168 a+168 b \log \left (c x^n\right )+73 b n\right )}{180 e^4 (d+e x)^3}-\frac {x^6 \left (56 a+56 b \log \left (c x^n\right )+15 b n\right )}{120 e^3 (d+e x)^4}-\frac {x^7 \left (8 a+8 b \log \left (c x^n\right )+b n\right )}{30 e^2 (d+e x)^5}-\frac {x^8 \left (a+b \log \left (c x^n\right )\right )}{6 e (d+e x)^6}+\frac {x^2 \left (280 a+280 b \log \left (c x^n\right )+341 b n\right )}{20 e^7}-\frac {d x (280 a+341 b n)}{10 e^8}-\frac {28 b d x \log \left (c x^n\right )}{e^8}+\frac {28 b d^2 n \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{e^9}+\frac {28 b d n x}{e^8}-\frac {7 b n x^2}{e^7} \]

[In]

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

[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

Rule 45

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 2332

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rule 2341

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 2354

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 2384

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_))^(q_.), x_Symbol] :> Simp[(f*x
)^m*(d + e*x)^(q + 1)*((a + b*Log[c*x^n])/(e*(q + 1))), x] - Dist[f/(e*(q + 1)), Int[(f*x)^(m - 1)*(d + e*x)^(
q + 1)*(a*m + b*n + b*m*Log[c*x^n]), x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && ILtQ[q, -1] && GtQ[m, 0]

Rule 2393

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 2438

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*} \text {integral}& = -\frac {x^8 \left (a+b \log \left (c x^n\right )\right )}{6 e (d+e x)^6}+\frac {\int \frac {x^7 \left (8 a+b n+8 b \log \left (c x^n\right )\right )}{(d+e x)^6} \, dx}{6 e} \\ & = -\frac {x^8 \left (a+b \log \left (c x^n\right )\right )}{6 e (d+e x)^6}-\frac {x^7 \left (8 a+b n+8 b \log \left (c x^n\right )\right )}{30 e^2 (d+e x)^5}+\frac {\int \frac {x^6 \left (8 b n+7 (8 a+b n)+56 b \log \left (c x^n\right )\right )}{(d+e x)^5} \, dx}{30 e^2} \\ & = -\frac {x^8 \left (a+b \log \left (c x^n\right )\right )}{6 e (d+e x)^6}-\frac {x^7 \left (8 a+b n+8 b \log \left (c x^n\right )\right )}{30 e^2 (d+e x)^5}-\frac {x^6 \left (56 a+15 b n+56 b \log \left (c x^n\right )\right )}{120 e^3 (d+e x)^4}+\frac {\int \frac {x^5 \left (56 b n+6 (8 b n+7 (8 a+b n))+336 b \log \left (c x^n\right )\right )}{(d+e x)^4} \, dx}{120 e^3} \\ & = -\frac {x^8 \left (a+b \log \left (c x^n\right )\right )}{6 e (d+e x)^6}-\frac {x^7 \left (8 a+b n+8 b \log \left (c x^n\right )\right )}{30 e^2 (d+e x)^5}-\frac {x^6 \left (56 a+15 b n+56 b \log \left (c x^n\right )\right )}{120 e^3 (d+e x)^4}-\frac {x^5 \left (168 a+73 b n+168 b \log \left (c x^n\right )\right )}{180 e^4 (d+e x)^3}+\frac {\int \frac {x^4 \left (336 b n+5 (56 b n+6 (8 b n+7 (8 a+b n)))+1680 b \log \left (c x^n\right )\right )}{(d+e x)^3} \, dx}{360 e^4} \\ & = -\frac {x^8 \left (a+b \log \left (c x^n\right )\right )}{6 e (d+e x)^6}-\frac {x^7 \left (8 a+b n+8 b \log \left (c x^n\right )\right )}{30 e^2 (d+e x)^5}-\frac {x^6 \left (56 a+15 b n+56 b \log \left (c x^n\right )\right )}{120 e^3 (d+e x)^4}-\frac {x^5 \left (168 a+73 b n+168 b \log \left (c x^n\right )\right )}{180 e^4 (d+e x)^3}-\frac {x^4 \left (840 a+533 b n+840 b \log \left (c x^n\right )\right )}{360 e^5 (d+e x)^2}+\frac {\int \frac {x^3 \left (1680 b n+4 (336 b n+5 (56 b n+6 (8 b n+7 (8 a+b n))))+6720 b \log \left (c x^n\right )\right )}{(d+e x)^2} \, dx}{720 e^5} \\ & = -\frac {x^8 \left (a+b \log \left (c x^n\right )\right )}{6 e (d+e x)^6}-\frac {x^7 \left (8 a+b n+8 b \log \left (c x^n\right )\right )}{30 e^2 (d+e x)^5}-\frac {x^6 \left (56 a+15 b n+56 b \log \left (c x^n\right )\right )}{120 e^3 (d+e x)^4}-\frac {x^5 \left (168 a+73 b n+168 b \log \left (c x^n\right )\right )}{180 e^4 (d+e x)^3}-\frac {x^4 \left (840 a+533 b n+840 b \log \left (c x^n\right )\right )}{360 e^5 (d+e x)^2}-\frac {x^3 \left (840 a+743 b n+840 b \log \left (c x^n\right )\right )}{90 e^6 (d+e x)}+\frac {\int \frac {x^2 \left (6720 b n+3 (1680 b n+4 (336 b n+5 (56 b n+6 (8 b n+7 (8 a+b n)))))+20160 b \log \left (c x^n\right )\right )}{d+e x} \, dx}{720 e^6} \\ & = -\frac {x^8 \left (a+b \log \left (c x^n\right )\right )}{6 e (d+e x)^6}-\frac {x^7 \left (8 a+b n+8 b \log \left (c x^n\right )\right )}{30 e^2 (d+e x)^5}-\frac {x^6 \left (56 a+15 b n+56 b \log \left (c x^n\right )\right )}{120 e^3 (d+e x)^4}-\frac {x^5 \left (168 a+73 b n+168 b \log \left (c x^n\right )\right )}{180 e^4 (d+e x)^3}-\frac {x^4 \left (840 a+533 b n+840 b \log \left (c x^n\right )\right )}{360 e^5 (d+e x)^2}-\frac {x^3 \left (840 a+743 b n+840 b \log \left (c x^n\right )\right )}{90 e^6 (d+e x)}+\frac {\int \left (-\frac {d \left (6720 b n+3 (1680 b n+4 (336 b n+5 (56 b n+6 (8 b n+7 (8 a+b n)))))+20160 b \log \left (c x^n\right )\right )}{e^2}+\frac {x \left (6720 b n+3 (1680 b n+4 (336 b n+5 (56 b n+6 (8 b n+7 (8 a+b n)))))+20160 b \log \left (c x^n\right )\right )}{e}+\frac {d^2 \left (6720 b n+3 (1680 b n+4 (336 b n+5 (56 b n+6 (8 b n+7 (8 a+b n)))))+20160 b \log \left (c x^n\right )\right )}{e^2 (d+e x)}\right ) \, dx}{720 e^6} \\ & = -\frac {x^8 \left (a+b \log \left (c x^n\right )\right )}{6 e (d+e x)^6}-\frac {x^7 \left (8 a+b n+8 b \log \left (c x^n\right )\right )}{30 e^2 (d+e x)^5}-\frac {x^6 \left (56 a+15 b n+56 b \log \left (c x^n\right )\right )}{120 e^3 (d+e x)^4}-\frac {x^5 \left (168 a+73 b n+168 b \log \left (c x^n\right )\right )}{180 e^4 (d+e x)^3}-\frac {x^4 \left (840 a+533 b n+840 b \log \left (c x^n\right )\right )}{360 e^5 (d+e x)^2}-\frac {x^3 \left (840 a+743 b n+840 b \log \left (c x^n\right )\right )}{90 e^6 (d+e x)}-\frac {d \int \left (6720 b n+3 (1680 b n+4 (336 b n+5 (56 b n+6 (8 b n+7 (8 a+b n)))))+20160 b \log \left (c x^n\right )\right ) \, dx}{720 e^8}+\frac {d^2 \int \frac {6720 b n+3 (1680 b n+4 (336 b n+5 (56 b n+6 (8 b n+7 (8 a+b n)))))+20160 b \log \left (c x^n\right )}{d+e x} \, dx}{720 e^8}+\frac {\int x \left (6720 b n+3 (1680 b n+4 (336 b n+5 (56 b n+6 (8 b n+7 (8 a+b n)))))+20160 b \log \left (c x^n\right )\right ) \, dx}{720 e^7} \\ & = -\frac {d (280 a+341 b n) x}{10 e^8}-\frac {7 b n x^2}{e^7}-\frac {x^8 \left (a+b \log \left (c x^n\right )\right )}{6 e (d+e x)^6}-\frac {x^7 \left (8 a+b n+8 b \log \left (c x^n\right )\right )}{30 e^2 (d+e x)^5}-\frac {x^6 \left (56 a+15 b n+56 b \log \left (c x^n\right )\right )}{120 e^3 (d+e x)^4}-\frac {x^5 \left (168 a+73 b n+168 b \log \left (c x^n\right )\right )}{180 e^4 (d+e x)^3}+\frac {x^2 \left (280 a+341 b n+280 b \log \left (c x^n\right )\right )}{20 e^7}-\frac {x^4 \left (840 a+533 b n+840 b \log \left (c x^n\right )\right )}{360 e^5 (d+e x)^2}-\frac {x^3 \left (840 a+743 b n+840 b \log \left (c x^n\right )\right )}{90 e^6 (d+e x)}+\frac {d^2 \left (280 a+341 b n+280 b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{10 e^9}-\frac {(28 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 {28 b d n x}{e^8}-\frac {d (280 a+341 b n) x}{10 e^8}-\frac {7 b n x^2}{e^7}-\frac {28 b d x \log \left (c x^n\right )}{e^8}-\frac {x^8 \left (a+b \log \left (c x^n\right )\right )}{6 e (d+e x)^6}-\frac {x^7 \left (8 a+b n+8 b \log \left (c x^n\right )\right )}{30 e^2 (d+e x)^5}-\frac {x^6 \left (56 a+15 b n+56 b \log \left (c x^n\right )\right )}{120 e^3 (d+e x)^4}-\frac {x^5 \left (168 a+73 b n+168 b \log \left (c x^n\right )\right )}{180 e^4 (d+e x)^3}+\frac {x^2 \left (280 a+341 b n+280 b \log \left (c x^n\right )\right )}{20 e^7}-\frac {x^4 \left (840 a+533 b n+840 b \log \left (c x^n\right )\right )}{360 e^5 (d+e x)^2}-\frac {x^3 \left (840 a+743 b n+840 b \log \left (c x^n\right )\right )}{90 e^6 (d+e x)}+\frac {d^2 \left (280 a+341 b n+280 b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{10 e^9}+\frac {28 b d^2 n \text {Li}_2\left (-\frac {e x}{d}\right )}{e^9} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.30 (sec) , antiderivative size = 403, normalized size of antiderivative = 1.22 \[ \int \frac {x^8 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^7} \, dx=\frac {-2520 a d e x+2520 b d e n x+180 a e^2 x^2-90 b e^2 n x^2-\frac {60 a d^8}{(d+e x)^6}+\frac {576 a d^7}{(d+e x)^5}+\frac {12 b d^7 n}{(d+e x)^5}-\frac {2520 a d^6}{(d+e x)^4}-\frac {129 b d^6 n}{(d+e x)^4}+\frac {6720 a d^5}{(d+e x)^3}+\frac {668 b d^5 n}{(d+e x)^3}-\frac {12600 a d^4}{(d+e x)^2}-\frac {2358 b d^4 n}{(d+e x)^2}+\frac {20160 a d^3}{d+e x}+\frac {7884 b d^3 n}{d+e x}-12276 b d^2 n \log (x)-2520 b d e x \log \left (c x^n\right )+180 b e^2 x^2 \log \left (c x^n\right )-\frac {60 b d^8 \log \left (c x^n\right )}{(d+e x)^6}+\frac {576 b d^7 \log \left (c x^n\right )}{(d+e x)^5}-\frac {2520 b d^6 \log \left (c x^n\right )}{(d+e x)^4}+\frac {6720 b d^5 \log \left (c x^n\right )}{(d+e x)^3}-\frac {12600 b d^4 \log \left (c x^n\right )}{(d+e x)^2}+\frac {20160 b d^3 \log \left (c x^n\right )}{d+e x}+12276 b d^2 n \log (d+e x)+10080 a d^2 \log \left (1+\frac {e x}{d}\right )+10080 b d^2 \log \left (c x^n\right ) \log \left (1+\frac {e x}{d}\right )+10080 b d^2 n \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{360 e^9} \]

[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)

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 1.67 (sec) , antiderivative size = 561, normalized size of antiderivative = 1.71

method	result	size
risch	\(\frac {b \ln \left (x^{n}\right ) x^{2}}{2 e^{7}}-\frac {7 b \ln \left (x^{n}\right ) d x}{e^{8}}+\frac {8 b \ln \left (x^{n}\right ) d^{7}}{5 e^{9} \left (e x +d \right )^{5}}-\frac {b \ln \left (x^{n}\right ) d^{8}}{6 e^{9} \left (e x +d \right )^{6}}+\frac {56 b \ln \left (x^{n}\right ) d^{5}}{3 e^{9} \left (e x +d \right )^{3}}+\frac {28 b \ln \left (x^{n}\right ) d^{2} \ln \left (e x +d \right )}{e^{9}}+\frac {56 b \ln \left (x^{n}\right ) d^{3}}{e^{9} \left (e x +d \right )}-\frac {35 b \ln \left (x^{n}\right ) d^{4}}{e^{9} \left (e x +d \right )^{2}}-\frac {7 b \ln \left (x^{n}\right ) d^{6}}{e^{9} \left (e x +d \right )^{4}}-\frac {b n \,x^{2}}{4 e^{7}}+\frac {7 b d n x}{e^{8}}+\frac {29 b n \,d^{2}}{4 e^{9}}+\frac {341 b n \,d^{2} \ln \left (e x +d \right )}{10 e^{9}}+\frac {219 b n \,d^{3}}{10 e^{9} \left (e x +d \right )}-\frac {131 b n \,d^{4}}{20 e^{9} \left (e x +d \right )^{2}}+\frac {167 b n \,d^{5}}{90 e^{9} \left (e x +d \right )^{3}}-\frac {43 b n \,d^{6}}{120 e^{9} \left (e x +d \right )^{4}}+\frac {b n \,d^{7}}{30 e^{9} \left (e x +d \right )^{5}}-\frac {341 b n \,d^{2} \ln \left (e x \right )}{10 e^{9}}-\frac {28 b n \,d^{2} \ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right )}{e^{9}}-\frac {28 b n \,d^{2} \operatorname {dilog}\left (-\frac {e x}{d}\right )}{e^{9}}+\left (-\frac {i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )}{2}+\frac {i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}+\frac {i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}-\frac {i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{2}+b \ln \left (c \right )+a \right ) \left (\frac {\frac {1}{2} e \,x^{2}-7 d x}{e^{8}}+\frac {8 d^{7}}{5 e^{9} \left (e x +d \right )^{5}}-\frac {d^{8}}{6 e^{9} \left (e x +d \right )^{6}}+\frac {56 d^{5}}{3 e^{9} \left (e x +d \right )^{3}}+\frac {28 d^{2} \ln \left (e x +d \right )}{e^{9}}+\frac {56 d^{3}}{e^{9} \left (e x +d \right )}-\frac {35 d^{4}}{e^{9} \left (e x +d \right )^{2}}-\frac {7 d^{6}}{e^{9} \left (e x +d \right )^{4}}\right )\)	\(561\)

[In]

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

[Out]

1/2*b*ln(x^n)/e^7*x^2-7*b*ln(x^n)/e^8*d*x+8/5*b*ln(x^n)/e^9*d^7/(e*x+d)^5-1/6*b*ln(x^n)*d^8/e^9/(e*x+d)^6+56/3
*b*ln(x^n)/e^9*d^5/(e*x+d)^3+28*b*ln(x^n)/e^9*d^2*ln(e*x+d)+56*b*ln(x^n)/e^9*d^3/(e*x+d)-35*b*ln(x^n)/e^9*d^4/
(e*x+d)^2-7*b*ln(x^n)/e^9*d^6/(e*x+d)^4-1/4*b*n*x^2/e^7+7*b*d*n*x/e^8+29/4*b*n/e^9*d^2+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)^5-341/10*b*n/e^9*d^2*ln(e*x)-28*b*n/e^9*d^2*ln(e*x+d)*ln(-e*x/d)-28*b*n/e^9
*d^2*dilog(-e*x/d)+(-1/2*I*b*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+1/2*I*b*Pi*csgn(I*c)*csgn(I*c*x^n)^2+1/2*I
*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2-1/2*I*b*Pi*csgn(I*c*x^n)^3+b*ln(c)+a)*(1/e^8*(1/2*e*x^2-7*d*x)+8/5/e^9*d^7/(
e*x+d)^5-1/6*d^8/e^9/(e*x+d)^6+56/3/e^9*d^5/(e*x+d)^3+28/e^9*d^2*ln(e*x+d)+56/e^9*d^3/(e*x+d)-35/e^9*d^4/(e*x+
d)^2-7/e^9*d^6/(e*x+d)^4)

Fricas [F]

\[ \int \frac {x^8 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^7} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} x^{8}}{{\left (e x + d\right )}^{7}} \,d x } \]

[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)

Sympy [A] (verification not implemented)

Time = 149.76 (sec) , antiderivative size = 1686, normalized size of antiderivative = 5.12 \[ \int \frac {x^8 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^7} \, dx=\text {Too large to display} \]

[In]

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

[Out]

a*d**8*Piecewise((x/d**7, Eq(e, 0)), (-1/(6*e*(d + e*x)**6), True))/e**8 - 8*a*d**7*Piecewise((x/d**6, Eq(e, 0
)), (-1/(5*e*(d + e*x)**5), True))/e**8 + 28*a*d**6*Piecewise((x/d**5, Eq(e, 0)), (-1/(4*e*(d + e*x)**4), True
))/e**8 - 56*a*d**5*Piecewise((x/d**4, Eq(e, 0)), (-1/(3*e*(d + e*x)**3), True))/e**8 + 70*a*d**4*Piecewise((x
/d**3, Eq(e, 0)), (-1/(2*e*(d + e*x)**2), True))/e**8 - 56*a*d**3*Piecewise((x/d**2, Eq(e, 0)), (-1/(d*e + e**
2*x), True))/e**8 + 28*a*d**2*Piecewise((x/d, Eq(e, 0)), (log(d + e*x)/e, True))/e**8 - 7*a*d*x/e**8 + a*x**2/
(2*e**7) - b*d**8*n*Piecewise((x/d**7, Eq(e, 0)), (-137*d**4/(360*d**10*e + 1800*d**9*e**2*x + 3600*d**8*e**3*
x**2 + 3600*d**7*e**4*x**3 + 1800*d**6*e**5*x**4 + 360*d**5*e**6*x**5) - 385*d**3*e*x/(360*d**10*e + 1800*d**9
*e**2*x + 3600*d**8*e**3*x**2 + 3600*d**7*e**4*x**3 + 1800*d**6*e**5*x**4 + 360*d**5*e**6*x**5) - 470*d**2*e**
2*x**2/(360*d**10*e + 1800*d**9*e**2*x + 3600*d**8*e**3*x**2 + 3600*d**7*e**4*x**3 + 1800*d**6*e**5*x**4 + 360
*d**5*e**6*x**5) - 270*d*e**3*x**3/(360*d**10*e + 1800*d**9*e**2*x + 3600*d**8*e**3*x**2 + 3600*d**7*e**4*x**3
+ 1800*d**6*e**5*x**4 + 360*d**5*e**6*x**5) - 60*e**4*x**4/(360*d**10*e + 1800*d**9*e**2*x + 3600*d**8*e**3*x
**2 + 3600*d**7*e**4*x**3 + 1800*d**6*e**5*x**4 + 360*d**5*e**6*x**5) - log(x)/(6*d**6*e) + log(d/e + x)/(6*d*
*6*e), True))/e**8 + b*d**8*Piecewise((x/d**7, Eq(e, 0)), (-1/(6*e*(d + e*x)**6), True))*log(c*x**n)/e**8 + 8*
b*d**7*n*Piecewise((x/d**6, Eq(e, 0)), (-25*d**3/(60*d**8*e + 240*d**7*e**2*x + 360*d**6*e**3*x**2 + 240*d**5*
e**4*x**3 + 60*d**4*e**5*x**4) - 52*d**2*e*x/(60*d**8*e + 240*d**7*e**2*x + 360*d**6*e**3*x**2 + 240*d**5*e**4
*x**3 + 60*d**4*e**5*x**4) - 42*d*e**2*x**2/(60*d**8*e + 240*d**7*e**2*x + 360*d**6*e**3*x**2 + 240*d**5*e**4*
x**3 + 60*d**4*e**5*x**4) - 12*e**3*x**3/(60*d**8*e + 240*d**7*e**2*x + 360*d**6*e**3*x**2 + 240*d**5*e**4*x**
3 + 60*d**4*e**5*x**4) - log(x)/(5*d**5*e) + log(d/e + x)/(5*d**5*e), True))/e**8 - 8*b*d**7*Piecewise((x/d**6
, Eq(e, 0)), (-1/(5*e*(d + e*x)**5), True))*log(c*x**n)/e**8 - 28*b*d**6*n*Piecewise((x/d**5, Eq(e, 0)), (-11*
d**2/(24*d**6*e + 72*d**5*e**2*x + 72*d**4*e**3*x**2 + 24*d**3*e**4*x**3) - 15*d*e*x/(24*d**6*e + 72*d**5*e**2
*x + 72*d**4*e**3*x**2 + 24*d**3*e**4*x**3) - 6*e**2*x**2/(24*d**6*e + 72*d**5*e**2*x + 72*d**4*e**3*x**2 + 24
*d**3*e**4*x**3) - log(x)/(4*d**4*e) + log(d/e + x)/(4*d**4*e), True))/e**8 + 28*b*d**6*Piecewise((x/d**5, Eq(
e, 0)), (-1/(4*e*(d + e*x)**4), True))*log(c*x**n)/e**8 + 56*b*d**5*n*Piecewise((x/d**4, Eq(e, 0)), (-3*d/(6*d
**4*e + 12*d**3*e**2*x + 6*d**2*e**3*x**2) - 2*e*x/(6*d**4*e + 12*d**3*e**2*x + 6*d**2*e**3*x**2) - log(x)/(3*
d**3*e) + log(d/e + x)/(3*d**3*e), True))/e**8 - 56*b*d**5*Piecewise((x/d**4, Eq(e, 0)), (-1/(3*e*(d + e*x)**3
), True))*log(c*x**n)/e**8 - 70*b*d**4*n*Piecewise((x/d**3, Eq(e, 0)), (-1/(2*d**2*e + 2*d*e**2*x) - log(x)/(2
*d**2*e) + log(d/e + x)/(2*d**2*e), True))/e**8 + 70*b*d**4*Piecewise((x/d**3, Eq(e, 0)), (-1/(2*e*(d + e*x)**
2), True))*log(c*x**n)/e**8 + 56*b*d**3*n*Piecewise((x/d**2, Eq(e, 0)), (-log(x)/(d*e) + log(d/e + x)/(d*e), T
rue))/e**8 - 56*b*d**3*Piecewise((x/d**2, Eq(e, 0)), (-1/(d*e + e**2*x), True))*log(c*x**n)/e**8 - 28*b*d**2*n
*Piecewise((x/d, Eq(e, 0)), (Piecewise((-polylog(2, e*x*exp_polar(I*pi)/d), (Abs(x) < 1) & (1/Abs(x) < 1)), (l
og(d)*log(x) - polylog(2, e*x*exp_polar(I*pi)/d), Abs(x) < 1), (-log(d)*log(1/x) - polylog(2, e*x*exp_polar(I*
pi)/d), 1/Abs(x) < 1), (-meijerg(((), (1, 1)), ((0, 0), ()), x)*log(d) + meijerg(((1, 1), ()), ((), (0, 0)), x
)*log(d) - polylog(2, e*x*exp_polar(I*pi)/d), True))/e, True))/e**8 + 28*b*d**2*Piecewise((x/d, Eq(e, 0)), (lo
g(d + e*x)/e, True))*log(c*x**n)/e**8 + 7*b*d*n*x/e**8 - 7*b*d*x*log(c*x**n)/e**8 - b*n*x**2/(4*e**7) + b*x**2
*log(c*x**n)/(2*e**7)

Maxima [F]

\[ \int \frac {x^8 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^7} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} x^{8}}{{\left (e x + d\right )}^{7}} \,d x } \]

[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)

Giac [F]

\[ \int \frac {x^8 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^7} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} x^{8}}{{\left (e x + d\right )}^{7}} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {x^8 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^7} \, dx=\int \frac {x^8\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{{\left (d+e\,x\right )}^7} \,d x \]

[In]

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

[Out]

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