\(\int \frac {x^7 (a+b \log (c x^n))}{(d+e x)^7} \, dx\) [63]

Optimal result

Integrand size = 21, antiderivative size = 285 \[ \int \frac {x^7 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^7} \, dx=-\frac {7 b n x}{e^7}+\frac {(140 a+223 b n) x}{20 e^7}+\frac {7 b x \log \left (c x^n\right )}{e^7}-\frac {x^7 \left (a+b \log \left (c x^n\right )\right )}{6 e (d+e x)^6}-\frac {x^6 \left (7 a+b n+7 b \log \left (c x^n\right )\right )}{30 e^2 (d+e x)^5}-\frac {x^5 \left (42 a+13 b n+42 b \log \left (c x^n\right )\right )}{120 e^3 (d+e x)^4}-\frac {x^2 \left (140 a+153 b n+140 b \log \left (c x^n\right )\right )}{40 e^6 (d+e x)}-\frac {x^4 \left (210 a+107 b n+210 b \log \left (c x^n\right )\right )}{360 e^4 (d+e x)^3}-\frac {x^3 \left (420 a+319 b n+420 b \log \left (c x^n\right )\right )}{360 e^5 (d+e x)^2}-\frac {d \left (140 a+223 b n+140 b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{20 e^8}-\frac {7 b d n \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{e^8} \]

[Out]

-7*b*n*x/e^7+1/20*(223*b*n+140*a)*x/e^7+7*b*x*ln(c*x^n)/e^7-1/6*x^7*(a+b*ln(c*x^n))/e/(e*x+d)^6-1/30*x^6*(7*a+
b*n+7*b*ln(c*x^n))/e^2/(e*x+d)^5-1/120*x^5*(42*a+13*b*n+42*b*ln(c*x^n))/e^3/(e*x+d)^4-1/40*x^2*(140*a+153*b*n+
140*b*ln(c*x^n))/e^6/(e*x+d)-1/360*x^4*(210*a+107*b*n+210*b*ln(c*x^n))/e^4/(e*x+d)^3-1/360*x^3*(420*a+319*b*n+
420*b*ln(c*x^n))/e^5/(e*x+d)^2-1/20*d*(140*a+223*b*n+140*b*ln(c*x^n))*ln(1+e*x/d)/e^8-7*b*d*n*polylog(2,-e*x/d
)/e^8

Rubi [A] (verified)

Time = 0.57 (sec) , antiderivative size = 285, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2384, 45, 2393, 2332, 2354, 2438} \[ \int \frac {x^7 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^7} \, dx=-\frac {d \log \left (\frac {e x}{d}+1\right ) \left (140 a+140 b \log \left (c x^n\right )+223 b n\right )}{20 e^8}-\frac {x^2 \left (140 a+140 b \log \left (c x^n\right )+153 b n\right )}{40 e^6 (d+e x)}-\frac {x^3 \left (420 a+420 b \log \left (c x^n\right )+319 b n\right )}{360 e^5 (d+e x)^2}-\frac {x^4 \left (210 a+210 b \log \left (c x^n\right )+107 b n\right )}{360 e^4 (d+e x)^3}-\frac {x^5 \left (42 a+42 b \log \left (c x^n\right )+13 b n\right )}{120 e^3 (d+e x)^4}-\frac {x^6 \left (7 a+7 b \log \left (c x^n\right )+b n\right )}{30 e^2 (d+e x)^5}-\frac {x^7 \left (a+b \log \left (c x^n\right )\right )}{6 e (d+e x)^6}+\frac {x (140 a+223 b n)}{20 e^7}+\frac {7 b x \log \left (c x^n\right )}{e^7}-\frac {7 b d n \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{e^8}-\frac {7 b n x}{e^7} \]

[In]

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

[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

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 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^7 \left (a+b \log \left (c x^n\right )\right )}{6 e (d+e x)^6}+\frac {\int \frac {x^6 \left (7 a+b n+7 b \log \left (c x^n\right )\right )}{(d+e x)^6} \, dx}{6 e} \\ & = -\frac {x^7 \left (a+b \log \left (c x^n\right )\right )}{6 e (d+e x)^6}-\frac {x^6 \left (7 a+b n+7 b \log \left (c x^n\right )\right )}{30 e^2 (d+e x)^5}+\frac {\int \frac {x^5 \left (7 b n+6 (7 a+b n)+42 b \log \left (c x^n\right )\right )}{(d+e x)^5} \, dx}{30 e^2} \\ & = -\frac {x^7 \left (a+b \log \left (c x^n\right )\right )}{6 e (d+e x)^6}-\frac {x^6 \left (7 a+b n+7 b \log \left (c x^n\right )\right )}{30 e^2 (d+e x)^5}-\frac {x^5 \left (42 a+13 b n+42 b \log \left (c x^n\right )\right )}{120 e^3 (d+e x)^4}+\frac {\int \frac {x^4 \left (42 b n+5 (7 b n+6 (7 a+b n))+210 b \log \left (c x^n\right )\right )}{(d+e x)^4} \, dx}{120 e^3} \\ & = -\frac {x^7 \left (a+b \log \left (c x^n\right )\right )}{6 e (d+e x)^6}-\frac {x^6 \left (7 a+b n+7 b \log \left (c x^n\right )\right )}{30 e^2 (d+e x)^5}-\frac {x^5 \left (42 a+13 b n+42 b \log \left (c x^n\right )\right )}{120 e^3 (d+e x)^4}-\frac {x^4 \left (210 a+107 b n+210 b \log \left (c x^n\right )\right )}{360 e^4 (d+e x)^3}+\frac {\int \frac {x^3 \left (210 b n+4 (42 b n+5 (7 b n+6 (7 a+b n)))+840 b \log \left (c x^n\right )\right )}{(d+e x)^3} \, dx}{360 e^4} \\ & = -\frac {x^7 \left (a+b \log \left (c x^n\right )\right )}{6 e (d+e x)^6}-\frac {x^6 \left (7 a+b n+7 b \log \left (c x^n\right )\right )}{30 e^2 (d+e x)^5}-\frac {x^5 \left (42 a+13 b n+42 b \log \left (c x^n\right )\right )}{120 e^3 (d+e x)^4}-\frac {x^4 \left (210 a+107 b n+210 b \log \left (c x^n\right )\right )}{360 e^4 (d+e x)^3}-\frac {x^3 \left (420 a+319 b n+420 b \log \left (c x^n\right )\right )}{360 e^5 (d+e x)^2}+\frac {\int \frac {x^2 \left (840 b n+3 (210 b n+4 (42 b n+5 (7 b n+6 (7 a+b n))))+2520 b \log \left (c x^n\right )\right )}{(d+e x)^2} \, dx}{720 e^5} \\ & = -\frac {x^7 \left (a+b \log \left (c x^n\right )\right )}{6 e (d+e x)^6}-\frac {x^6 \left (7 a+b n+7 b \log \left (c x^n\right )\right )}{30 e^2 (d+e x)^5}-\frac {x^5 \left (42 a+13 b n+42 b \log \left (c x^n\right )\right )}{120 e^3 (d+e x)^4}-\frac {x^2 \left (140 a+153 b n+140 b \log \left (c x^n\right )\right )}{40 e^6 (d+e x)}-\frac {x^4 \left (210 a+107 b n+210 b \log \left (c x^n\right )\right )}{360 e^4 (d+e x)^3}-\frac {x^3 \left (420 a+319 b n+420 b \log \left (c x^n\right )\right )}{360 e^5 (d+e x)^2}+\frac {\int \frac {x \left (2520 b n+2 (840 b n+3 (210 b n+4 (42 b n+5 (7 b n+6 (7 a+b n)))))+5040 b \log \left (c x^n\right )\right )}{d+e x} \, dx}{720 e^6} \\ & = -\frac {x^7 \left (a+b \log \left (c x^n\right )\right )}{6 e (d+e x)^6}-\frac {x^6 \left (7 a+b n+7 b \log \left (c x^n\right )\right )}{30 e^2 (d+e x)^5}-\frac {x^5 \left (42 a+13 b n+42 b \log \left (c x^n\right )\right )}{120 e^3 (d+e x)^4}-\frac {x^2 \left (140 a+153 b n+140 b \log \left (c x^n\right )\right )}{40 e^6 (d+e x)}-\frac {x^4 \left (210 a+107 b n+210 b \log \left (c x^n\right )\right )}{360 e^4 (d+e x)^3}-\frac {x^3 \left (420 a+319 b n+420 b \log \left (c x^n\right )\right )}{360 e^5 (d+e x)^2}+\frac {\int \left (\frac {2520 b n+2 (840 b n+3 (210 b n+4 (42 b n+5 (7 b n+6 (7 a+b n)))))+5040 b \log \left (c x^n\right )}{e}-\frac {d \left (2520 b n+2 (840 b n+3 (210 b n+4 (42 b n+5 (7 b n+6 (7 a+b n)))))+5040 b \log \left (c x^n\right )\right )}{e (d+e x)}\right ) \, dx}{720 e^6} \\ & = -\frac {x^7 \left (a+b \log \left (c x^n\right )\right )}{6 e (d+e x)^6}-\frac {x^6 \left (7 a+b n+7 b \log \left (c x^n\right )\right )}{30 e^2 (d+e x)^5}-\frac {x^5 \left (42 a+13 b n+42 b \log \left (c x^n\right )\right )}{120 e^3 (d+e x)^4}-\frac {x^2 \left (140 a+153 b n+140 b \log \left (c x^n\right )\right )}{40 e^6 (d+e x)}-\frac {x^4 \left (210 a+107 b n+210 b \log \left (c x^n\right )\right )}{360 e^4 (d+e x)^3}-\frac {x^3 \left (420 a+319 b n+420 b \log \left (c x^n\right )\right )}{360 e^5 (d+e x)^2}+\frac {\int \left (2520 b n+2 (840 b n+3 (210 b n+4 (42 b n+5 (7 b n+6 (7 a+b n)))))+5040 b \log \left (c x^n\right )\right ) \, dx}{720 e^7}-\frac {d \int \frac {2520 b n+2 (840 b n+3 (210 b n+4 (42 b n+5 (7 b n+6 (7 a+b n)))))+5040 b \log \left (c x^n\right )}{d+e x} \, dx}{720 e^7} \\ & = \frac {(140 a+223 b n) x}{20 e^7}-\frac {x^7 \left (a+b \log \left (c x^n\right )\right )}{6 e (d+e x)^6}-\frac {x^6 \left (7 a+b n+7 b \log \left (c x^n\right )\right )}{30 e^2 (d+e x)^5}-\frac {x^5 \left (42 a+13 b n+42 b \log \left (c x^n\right )\right )}{120 e^3 (d+e x)^4}-\frac {x^2 \left (140 a+153 b n+140 b \log \left (c x^n\right )\right )}{40 e^6 (d+e x)}-\frac {x^4 \left (210 a+107 b n+210 b \log \left (c x^n\right )\right )}{360 e^4 (d+e x)^3}-\frac {x^3 \left (420 a+319 b n+420 b \log \left (c x^n\right )\right )}{360 e^5 (d+e x)^2}-\frac {d \left (140 a+223 b n+140 b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{20 e^8}+\frac {(7 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 {7 b n x}{e^7}+\frac {(140 a+223 b n) x}{20 e^7}+\frac {7 b x \log \left (c x^n\right )}{e^7}-\frac {x^7 \left (a+b \log \left (c x^n\right )\right )}{6 e (d+e x)^6}-\frac {x^6 \left (7 a+b n+7 b \log \left (c x^n\right )\right )}{30 e^2 (d+e x)^5}-\frac {x^5 \left (42 a+13 b n+42 b \log \left (c x^n\right )\right )}{120 e^3 (d+e x)^4}-\frac {x^2 \left (140 a+153 b n+140 b \log \left (c x^n\right )\right )}{40 e^6 (d+e x)}-\frac {x^4 \left (210 a+107 b n+210 b \log \left (c x^n\right )\right )}{360 e^4 (d+e x)^3}-\frac {x^3 \left (420 a+319 b n+420 b \log \left (c x^n\right )\right )}{360 e^5 (d+e x)^2}-\frac {d \left (140 a+223 b n+140 b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{20 e^8}-\frac {7 b d n \text {Li}_2\left (-\frac {e x}{d}\right )}{e^8} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.33 (sec) , antiderivative size = 356, normalized size of antiderivative = 1.25 \[ \int \frac {x^7 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^7} \, dx=-\frac {-360 a e x+360 b e n x-\frac {60 a d^7}{(d+e x)^6}+\frac {504 a d^6}{(d+e x)^5}+\frac {12 b d^6 n}{(d+e x)^5}-\frac {1890 a d^5}{(d+e x)^4}-\frac {111 b d^5 n}{(d+e x)^4}+\frac {4200 a d^4}{(d+e x)^3}+\frac {482 b d^4 n}{(d+e x)^3}-\frac {6300 a d^3}{(d+e x)^2}-\frac {1377 b d^3 n}{(d+e x)^2}+\frac {7560 a d^2}{d+e x}+\frac {3546 b d^2 n}{d+e x}-4014 b d n \log (x)-360 b e x \log \left (c x^n\right )-\frac {60 b d^7 \log \left (c x^n\right )}{(d+e x)^6}+\frac {504 b d^6 \log \left (c x^n\right )}{(d+e x)^5}-\frac {1890 b d^5 \log \left (c x^n\right )}{(d+e x)^4}+\frac {4200 b d^4 \log \left (c x^n\right )}{(d+e x)^3}-\frac {6300 b d^3 \log \left (c x^n\right )}{(d+e x)^2}+\frac {7560 b d^2 \log \left (c x^n\right )}{d+e x}+4014 b d n \log (d+e x)+2520 a d \log \left (1+\frac {e x}{d}\right )+2520 b d \log \left (c x^n\right ) \log \left (1+\frac {e x}{d}\right )+2520 b d n \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{360 e^8} \]

[In]

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

[Out]

-1/360*(-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
- (1890*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
- (6300*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)])/e^8

Maple [C] (warning: unable to verify)

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

Time = 1.34 (sec) , antiderivative size = 511, normalized size of antiderivative = 1.79

method	result	size
risch	\(\frac {b \ln \left (x^{n}\right ) x}{e^{7}}+\frac {b \ln \left (x^{n}\right ) d^{7}}{6 e^{8} \left (e x +d \right )^{6}}-\frac {35 b \ln \left (x^{n}\right ) d^{4}}{3 e^{8} \left (e x +d \right )^{3}}-\frac {7 b \ln \left (x^{n}\right ) d \ln \left (e x +d \right )}{e^{8}}-\frac {21 b \ln \left (x^{n}\right ) d^{2}}{e^{8} \left (e x +d \right )}+\frac {35 b \ln \left (x^{n}\right ) d^{3}}{2 e^{8} \left (e x +d \right )^{2}}+\frac {21 b \ln \left (x^{n}\right ) d^{5}}{4 e^{8} \left (e x +d \right )^{4}}-\frac {7 b \ln \left (x^{n}\right ) d^{6}}{5 e^{8} \left (e x +d \right )^{5}}-\frac {b n x}{e^{7}}-\frac {b n d}{e^{8}}-\frac {223 b n d \ln \left (e x +d \right )}{20 e^{8}}-\frac {197 b n \,d^{2}}{20 e^{8} \left (e x +d \right )}+\frac {153 b n \,d^{3}}{40 e^{8} \left (e x +d \right )^{2}}-\frac {241 b n \,d^{4}}{180 e^{8} \left (e x +d \right )^{3}}+\frac {37 b n \,d^{5}}{120 e^{8} \left (e x +d \right )^{4}}-\frac {b n \,d^{6}}{30 e^{8} \left (e x +d \right )^{5}}+\frac {223 b n d \ln \left (e x \right )}{20 e^{8}}+\frac {7 b n d \ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right )}{e^{8}}+\frac {7 b n d \operatorname {dilog}\left (-\frac {e x}{d}\right )}{e^{8}}+\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 {x}{e^{7}}+\frac {d^{7}}{6 e^{8} \left (e x +d \right )^{6}}-\frac {35 d^{4}}{3 e^{8} \left (e x +d \right )^{3}}-\frac {7 d \ln \left (e x +d \right )}{e^{8}}-\frac {21 d^{2}}{e^{8} \left (e x +d \right )}+\frac {35 d^{3}}{2 e^{8} \left (e x +d \right )^{2}}+\frac {21 d^{5}}{4 e^{8} \left (e x +d \right )^{4}}-\frac {7 d^{6}}{5 e^{8} \left (e x +d \right )^{5}}\right )\)	\(511\)

[In]

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

[Out]

b*ln(x^n)/e^7*x+1/6*b*ln(x^n)/e^8*d^7/(e*x+d)^6-35/3*b*ln(x^n)/e^8*d^4/(e*x+d)^3-7*b*ln(x^n)/e^8*d*ln(e*x+d)-2
1*b*ln(x^n)/e^8*d^2/(e*x+d)+35/2*b*ln(x^n)/e^8*d^3/(e*x+d)^2+21/4*b*ln(x^n)/e^8*d^5/(e*x+d)^4-7/5*b*ln(x^n)/e^
8*d^6/(e*x+d)^5-b*n*x/e^7-b*n/e^8*d-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+223/20*b*n/e^8*
d*ln(e*x)+7*b*n/e^8*d*ln(e*x+d)*ln(-e*x/d)+7*b*n/e^8*d*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)*(x/e^7+1/6/e^8*d^7/(e*x+d)^6-35/3/e^8*d^4/(e*x+d)^3-7/e^8*d*ln(e*x+d)-21/e^8*d^2/(e*x+d)+35/2/e^8*
d^3/(e*x+d)^2+21/4/e^8*d^5/(e*x+d)^4-7/5/e^8*d^6/(e*x+d)^5)

Fricas [F]

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

[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 [A] (verification not implemented)

Time = 112.19 (sec) , antiderivative size = 1632, normalized size of antiderivative = 5.73 \[ \int \frac {x^7 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^7} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-a*d**7*Piecewise((x/d**7, Eq(e, 0)), (-1/(6*e*(d + e*x)**6), True))/e**7 + 7*a*d**6*Piecewise((x/d**6, Eq(e,
0)), (-1/(5*e*(d + e*x)**5), True))/e**7 - 21*a*d**5*Piecewise((x/d**5, Eq(e, 0)), (-1/(4*e*(d + e*x)**4), Tru
e))/e**7 + 35*a*d**4*Piecewise((x/d**4, Eq(e, 0)), (-1/(3*e*(d + e*x)**3), True))/e**7 - 35*a*d**3*Piecewise((
x/d**3, Eq(e, 0)), (-1/(2*e*(d + e*x)**2), True))/e**7 + 21*a*d**2*Piecewise((x/d**2, Eq(e, 0)), (-1/(d*e + e*
*2*x), True))/e**7 - 7*a*d*Piecewise((x/d, Eq(e, 0)), (log(d + e*x)/e, True))/e**7 + a*x/e**7 + b*d**7*n*Piece
wise((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 + 180
0*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**7 - b*d*
*7*Piecewise((x/d**7, Eq(e, 0)), (-1/(6*e*(d + e*x)**6), True))*log(c*x**n)/e**7 - 7*b*d**6*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**7 + 7*b*d**6*Piecewise((x/d**6, Eq(e, 0)), (-1/(5*e*(d
+ e*x)**5), True))*log(c*x**n)/e**7 + 21*b*d**5*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**7 - 21*b*d**5*Piecewise((x/d**5, Eq(e, 0)), (-1/(4*e*(d + e*x
)**4), True))*log(c*x**n)/e**7 - 35*b*d**4*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**7 + 35*b*d**4*Piecewise((x/d**4, Eq(e, 0)), (-1/(3*e*(d + e*x)**3), True))*log(c*x**n)/e**
7 + 35*b*d**3*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**7 - 35*b*d**3*Piecewise((x/d**3, Eq(e, 0)), (-1/(2*e*(d + e*x)**2), True))*log(c*x**n)/e*
*7 - 21*b*d**2*n*Piecewise((x/d**2, Eq(e, 0)), (-log(x)/(d*e) + log(d/e + x)/(d*e), True))/e**7 + 21*b*d**2*Pi
ecewise((x/d**2, Eq(e, 0)), (-1/(d*e + e**2*x), True))*log(c*x**n)/e**7 + 7*b*d*n*Piecewise((x/d, Eq(e, 0)), (
Piecewise((-polylog(2, e*x*exp_polar(I*pi)/d), (Abs(x) < 1) & (1/Abs(x) < 1)), (log(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), (-meij
erg(((), (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**7 - 7*b*d*Piecewise((x/d, Eq(e, 0)), (log(d + e*x)/e, True))*log(c*x**n)/
e**7 - b*n*x/e**7 + b*x*log(c*x**n)/e**7

Maxima [F]

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

[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)

Giac [F]

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

[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)

Mupad [F(-1)]

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

[In]

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

[Out]

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