\(\int \frac {x^6 (a+b \log (c x^n))}{(d+e x)^7} \, dx\) [64]

Optimal result

Integrand size = 21, antiderivative size = 243 \[ \int \frac {x^6 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^7} \, dx=-\frac {x^6 \left (a+b \log \left (c x^n\right )\right )}{6 e (d+e x)^6}-\frac {x^5 \left (6 a+b n+6 b \log \left (c x^n\right )\right )}{30 e^2 (d+e x)^5}-\frac {x^2 \left (20 a+19 b n+20 b \log \left (c x^n\right )\right )}{40 e^5 (d+e x)^2}-\frac {x \left (20 a+29 b n+20 b \log \left (c x^n\right )\right )}{20 e^6 (d+e x)}-\frac {x^4 \left (30 a+11 b n+30 b \log \left (c x^n\right )\right )}{120 e^3 (d+e x)^4}-\frac {x^3 \left (60 a+37 b n+60 b \log \left (c x^n\right )\right )}{180 e^4 (d+e x)^3}+\frac {\left (20 a+49 b n+20 b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{20 e^7}+\frac {b n \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{e^7} \]

[Out]

-1/6*x^6*(a+b*ln(c*x^n))/e/(e*x+d)^6-1/30*x^5*(6*a+b*n+6*b*ln(c*x^n))/e^2/(e*x+d)^5-1/40*x^2*(20*a+19*b*n+20*b
*ln(c*x^n))/e^5/(e*x+d)^2-1/20*x*(20*a+29*b*n+20*b*ln(c*x^n))/e^6/(e*x+d)-1/120*x^4*(30*a+11*b*n+30*b*ln(c*x^n
))/e^3/(e*x+d)^4-1/180*x^3*(60*a+37*b*n+60*b*ln(c*x^n))/e^4/(e*x+d)^3+1/20*(20*a+49*b*n+20*b*ln(c*x^n))*ln(1+e
*x/d)/e^7+b*n*polylog(2,-e*x/d)/e^7

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2384, 2354, 2438} \[ \int \frac {x^6 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^7} \, dx=\frac {\log \left (\frac {e x}{d}+1\right ) \left (20 a+20 b \log \left (c x^n\right )+49 b n\right )}{20 e^7}-\frac {x \left (20 a+20 b \log \left (c x^n\right )+29 b n\right )}{20 e^6 (d+e x)}-\frac {x^2 \left (20 a+20 b \log \left (c x^n\right )+19 b n\right )}{40 e^5 (d+e x)^2}-\frac {x^3 \left (60 a+60 b \log \left (c x^n\right )+37 b n\right )}{180 e^4 (d+e x)^3}-\frac {x^4 \left (30 a+30 b \log \left (c x^n\right )+11 b n\right )}{120 e^3 (d+e x)^4}-\frac {x^5 \left (6 a+6 b \log \left (c x^n\right )+b n\right )}{30 e^2 (d+e x)^5}-\frac {x^6 \left (a+b \log \left (c x^n\right )\right )}{6 e (d+e x)^6}+\frac {b n \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{e^7} \]

[In]

Int[(x^6*(a + b*Log[c*x^n]))/(d + e*x)^7,x]

[Out]

-1/6*(x^6*(a + b*Log[c*x^n]))/(e*(d + e*x)^6) - (x^5*(6*a + b*n + 6*b*Log[c*x^n]))/(30*e^2*(d + e*x)^5) - (x^2
*(20*a + 19*b*n + 20*b*Log[c*x^n]))/(40*e^5*(d + e*x)^2) - (x*(20*a + 29*b*n + 20*b*Log[c*x^n]))/(20*e^6*(d +
e*x)) - (x^4*(30*a + 11*b*n + 30*b*Log[c*x^n]))/(120*e^3*(d + e*x)^4) - (x^3*(60*a + 37*b*n + 60*b*Log[c*x^n])
)/(180*e^4*(d + e*x)^3) + ((20*a + 49*b*n + 20*b*Log[c*x^n])*Log[1 + (e*x)/d])/(20*e^7) + (b*n*PolyLog[2, -((e
*x)/d)])/e^7

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 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^6 \left (a+b \log \left (c x^n\right )\right )}{6 e (d+e x)^6}+\frac {\int \frac {x^5 \left (6 a+b n+6 b \log \left (c x^n\right )\right )}{(d+e x)^6} \, dx}{6 e} \\ & = -\frac {x^6 \left (a+b \log \left (c x^n\right )\right )}{6 e (d+e x)^6}-\frac {x^5 \left (6 a+b n+6 b \log \left (c x^n\right )\right )}{30 e^2 (d+e x)^5}+\frac {\int \frac {x^4 \left (6 b n+5 (6 a+b n)+30 b \log \left (c x^n\right )\right )}{(d+e x)^5} \, dx}{30 e^2} \\ & = -\frac {x^6 \left (a+b \log \left (c x^n\right )\right )}{6 e (d+e x)^6}-\frac {x^5 \left (6 a+b n+6 b \log \left (c x^n\right )\right )}{30 e^2 (d+e x)^5}-\frac {x^4 \left (30 a+11 b n+30 b \log \left (c x^n\right )\right )}{120 e^3 (d+e x)^4}+\frac {\int \frac {x^3 \left (30 b n+4 (6 b n+5 (6 a+b n))+120 b \log \left (c x^n\right )\right )}{(d+e x)^4} \, dx}{120 e^3} \\ & = -\frac {x^6 \left (a+b \log \left (c x^n\right )\right )}{6 e (d+e x)^6}-\frac {x^5 \left (6 a+b n+6 b \log \left (c x^n\right )\right )}{30 e^2 (d+e x)^5}-\frac {x^4 \left (30 a+11 b n+30 b \log \left (c x^n\right )\right )}{120 e^3 (d+e x)^4}-\frac {x^3 \left (60 a+37 b n+60 b \log \left (c x^n\right )\right )}{180 e^4 (d+e x)^3}+\frac {\int \frac {x^2 \left (120 b n+3 (30 b n+4 (6 b n+5 (6 a+b n)))+360 b \log \left (c x^n\right )\right )}{(d+e x)^3} \, dx}{360 e^4} \\ & = -\frac {x^6 \left (a+b \log \left (c x^n\right )\right )}{6 e (d+e x)^6}-\frac {x^5 \left (6 a+b n+6 b \log \left (c x^n\right )\right )}{30 e^2 (d+e x)^5}-\frac {x^2 \left (20 a+19 b n+20 b \log \left (c x^n\right )\right )}{40 e^5 (d+e x)^2}-\frac {x^4 \left (30 a+11 b n+30 b \log \left (c x^n\right )\right )}{120 e^3 (d+e x)^4}-\frac {x^3 \left (60 a+37 b n+60 b \log \left (c x^n\right )\right )}{180 e^4 (d+e x)^3}+\frac {\int \frac {x \left (360 b n+2 (120 b n+3 (30 b n+4 (6 b n+5 (6 a+b n))))+720 b \log \left (c x^n\right )\right )}{(d+e x)^2} \, dx}{720 e^5} \\ & = -\frac {x^6 \left (a+b \log \left (c x^n\right )\right )}{6 e (d+e x)^6}-\frac {x^5 \left (6 a+b n+6 b \log \left (c x^n\right )\right )}{30 e^2 (d+e x)^5}-\frac {x^2 \left (20 a+19 b n+20 b \log \left (c x^n\right )\right )}{40 e^5 (d+e x)^2}-\frac {x \left (20 a+29 b n+20 b \log \left (c x^n\right )\right )}{20 e^6 (d+e x)}-\frac {x^4 \left (30 a+11 b n+30 b \log \left (c x^n\right )\right )}{120 e^3 (d+e x)^4}-\frac {x^3 \left (60 a+37 b n+60 b \log \left (c x^n\right )\right )}{180 e^4 (d+e x)^3}+\frac {\int \frac {1080 b n+2 (120 b n+3 (30 b n+4 (6 b n+5 (6 a+b n))))+720 b \log \left (c x^n\right )}{d+e x} \, dx}{720 e^6} \\ & = -\frac {x^6 \left (a+b \log \left (c x^n\right )\right )}{6 e (d+e x)^6}-\frac {x^5 \left (6 a+b n+6 b \log \left (c x^n\right )\right )}{30 e^2 (d+e x)^5}-\frac {x^2 \left (20 a+19 b n+20 b \log \left (c x^n\right )\right )}{40 e^5 (d+e x)^2}-\frac {x \left (20 a+29 b n+20 b \log \left (c x^n\right )\right )}{20 e^6 (d+e x)}-\frac {x^4 \left (30 a+11 b n+30 b \log \left (c x^n\right )\right )}{120 e^3 (d+e x)^4}-\frac {x^3 \left (60 a+37 b n+60 b \log \left (c x^n\right )\right )}{180 e^4 (d+e x)^3}+\frac {\left (20 a+49 b n+20 b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{20 e^7}-\frac {(b n) \int \frac {\log \left (1+\frac {e x}{d}\right )}{x} \, dx}{e^7} \\ & = -\frac {x^6 \left (a+b \log \left (c x^n\right )\right )}{6 e (d+e x)^6}-\frac {x^5 \left (6 a+b n+6 b \log \left (c x^n\right )\right )}{30 e^2 (d+e x)^5}-\frac {x^2 \left (20 a+19 b n+20 b \log \left (c x^n\right )\right )}{40 e^5 (d+e x)^2}-\frac {x \left (20 a+29 b n+20 b \log \left (c x^n\right )\right )}{20 e^6 (d+e x)}-\frac {x^4 \left (30 a+11 b n+30 b \log \left (c x^n\right )\right )}{120 e^3 (d+e x)^4}-\frac {x^3 \left (60 a+37 b n+60 b \log \left (c x^n\right )\right )}{180 e^4 (d+e x)^3}+\frac {\left (20 a+49 b n+20 b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{20 e^7}+\frac {b n \text {Li}_2\left (-\frac {e x}{d}\right )}{e^7} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.30 (sec) , antiderivative size = 333, normalized size of antiderivative = 1.37 \[ \int \frac {x^6 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^7} \, dx=\frac {-882 b n \log (x)+\frac {-60 a d^6+432 a d^5 (d+e x)+12 b d^5 n (d+e x)-1350 a d^4 (d+e x)^2-93 b d^4 n (d+e x)^2+2400 a d^3 (d+e x)^3+326 b d^3 n (d+e x)^3-2700 a d^2 (d+e x)^4-711 b d^2 n (d+e x)^4+2160 a d (d+e x)^5+1278 b d n (d+e x)^5-60 b d^6 \log \left (c x^n\right )+432 b d^5 (d+e x) \log \left (c x^n\right )-1350 b d^4 (d+e x)^2 \log \left (c x^n\right )+2400 b d^3 (d+e x)^3 \log \left (c x^n\right )-2700 b d^2 (d+e x)^4 \log \left (c x^n\right )+2160 b d (d+e x)^5 \log \left (c x^n\right )+882 b n (d+e x)^6 \log (d+e x)+360 a (d+e x)^6 \log \left (1+\frac {e x}{d}\right )+360 b (d+e x)^6 \log \left (c x^n\right ) \log \left (1+\frac {e x}{d}\right )}{(d+e x)^6}+360 b n \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{360 e^7} \]

[In]

Integrate[(x^6*(a + b*Log[c*x^n]))/(d + e*x)^7,x]

[Out]

(-882*b*n*Log[x] + (-60*a*d^6 + 432*a*d^5*(d + e*x) + 12*b*d^5*n*(d + e*x) - 1350*a*d^4*(d + e*x)^2 - 93*b*d^4
*n*(d + e*x)^2 + 2400*a*d^3*(d + e*x)^3 + 326*b*d^3*n*(d + e*x)^3 - 2700*a*d^2*(d + e*x)^4 - 711*b*d^2*n*(d +
e*x)^4 + 2160*a*d*(d + e*x)^5 + 1278*b*d*n*(d + e*x)^5 - 60*b*d^6*Log[c*x^n] + 432*b*d^5*(d + e*x)*Log[c*x^n]
- 1350*b*d^4*(d + e*x)^2*Log[c*x^n] + 2400*b*d^3*(d + e*x)^3*Log[c*x^n] - 2700*b*d^2*(d + e*x)^4*Log[c*x^n] +
2160*b*d*(d + e*x)^5*Log[c*x^n] + 882*b*n*(d + e*x)^6*Log[d + e*x] + 360*a*(d + e*x)^6*Log[1 + (e*x)/d] + 360*
b*(d + e*x)^6*Log[c*x^n]*Log[1 + (e*x)/d])/(d + e*x)^6 + 360*b*n*PolyLog[2, -((e*x)/d)])/(360*e^7)

Maple [C] (warning: unable to verify)

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

Time = 1.30 (sec) , antiderivative size = 466, normalized size of antiderivative = 1.92

method	result	size
risch	\(-\frac {b \ln \left (x^{n}\right ) d^{6}}{6 e^{7} \left (e x +d \right )^{6}}+\frac {20 b \ln \left (x^{n}\right ) d^{3}}{3 e^{7} \left (e x +d \right )^{3}}+\frac {b \ln \left (x^{n}\right ) \ln \left (e x +d \right )}{e^{7}}+\frac {6 b \ln \left (x^{n}\right ) d}{e^{7} \left (e x +d \right )}-\frac {15 b \ln \left (x^{n}\right ) d^{2}}{2 e^{7} \left (e x +d \right )^{2}}-\frac {15 b \ln \left (x^{n}\right ) d^{4}}{4 e^{7} \left (e x +d \right )^{4}}+\frac {6 b \ln \left (x^{n}\right ) d^{5}}{5 e^{7} \left (e x +d \right )^{5}}+\frac {71 b n d}{20 e^{7} \left (e x +d \right )}+\frac {49 b n \ln \left (e x +d \right )}{20 e^{7}}-\frac {79 b n \,d^{2}}{40 e^{7} \left (e x +d \right )^{2}}+\frac {163 b n \,d^{3}}{180 e^{7} \left (e x +d \right )^{3}}-\frac {31 b n \,d^{4}}{120 e^{7} \left (e x +d \right )^{4}}+\frac {b n \,d^{5}}{30 e^{7} \left (e x +d \right )^{5}}-\frac {49 b n \ln \left (e x \right )}{20 e^{7}}-\frac {b n \ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right )}{e^{7}}-\frac {b n \operatorname {dilog}\left (-\frac {e x}{d}\right )}{e^{7}}+\left (-\frac {i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )}{2}+\frac {i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}+\frac {i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}-\frac {i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{2}+b \ln \left (c \right )+a \right ) \left (-\frac {d^{6}}{6 e^{7} \left (e x +d \right )^{6}}+\frac {20 d^{3}}{3 e^{7} \left (e x +d \right )^{3}}+\frac {\ln \left (e x +d \right )}{e^{7}}+\frac {6 d}{e^{7} \left (e x +d \right )}-\frac {15 d^{2}}{2 e^{7} \left (e x +d \right )^{2}}-\frac {15 d^{4}}{4 e^{7} \left (e x +d \right )^{4}}+\frac {6 d^{5}}{5 e^{7} \left (e x +d \right )^{5}}\right )\)	\(466\)

[In]

int(x^6*(a+b*ln(c*x^n))/(e*x+d)^7,x,method=_RETURNVERBOSE)

[Out]

-1/6*b*ln(x^n)/e^7*d^6/(e*x+d)^6+20/3*b*ln(x^n)/e^7*d^3/(e*x+d)^3+b*ln(x^n)/e^7*ln(e*x+d)+6*b*ln(x^n)/e^7*d/(e
*x+d)-15/2*b*ln(x^n)/e^7*d^2/(e*x+d)^2-15/4*b*ln(x^n)/e^7*d^4/(e*x+d)^4+6/5*b*ln(x^n)/e^7*d^5/(e*x+d)^5+71/20*
b*n/e^7*d/(e*x+d)+49/20*b*n/e^7*ln(e*x+d)-79/40*b*n/e^7*d^2/(e*x+d)^2+163/180*b*n/e^7*d^3/(e*x+d)^3-31/120*b*n
/e^7*d^4/(e*x+d)^4+1/30*b*n/e^7*d^5/(e*x+d)^5-49/20*b*n/e^7*ln(e*x)-b*n/e^7*ln(e*x+d)*ln(-e*x/d)-b*n/e^7*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)*(-1/6/e^7*d^6/(e*x+d)^6+20/3/e^7*d^3/(e*x+d)^3+1
/e^7*ln(e*x+d)+6/e^7*d/(e*x+d)-15/2/e^7*d^2/(e*x+d)^2-15/4/e^7*d^4/(e*x+d)^4+6/5/e^7*d^5/(e*x+d)^5)

Fricas [F]

\[ \int \frac {x^6 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^7} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} x^{6}}{{\left (e x + d\right )}^{7}} \,d x } \]

[In]

integrate(x^6*(a+b*log(c*x^n))/(e*x+d)^7,x, algorithm="fricas")

[Out]

integral((b*x^6*log(c*x^n) + a*x^6)/(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 = 80.64 (sec) , antiderivative size = 1588, normalized size of antiderivative = 6.53 \[ \int \frac {x^6 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^7} \, dx=\text {Too large to display} \]

[In]

integrate(x**6*(a+b*ln(c*x**n))/(e*x+d)**7,x)

[Out]

a*d**6*Piecewise((x/d**7, Eq(e, 0)), (-1/(6*e*(d + e*x)**6), True))/e**6 - 6*a*d**5*Piecewise((x/d**6, Eq(e, 0
)), (-1/(5*e*(d + e*x)**5), True))/e**6 + 15*a*d**4*Piecewise((x/d**5, Eq(e, 0)), (-1/(4*e*(d + e*x)**4), True
))/e**6 - 20*a*d**3*Piecewise((x/d**4, Eq(e, 0)), (-1/(3*e*(d + e*x)**3), True))/e**6 + 15*a*d**2*Piecewise((x
/d**3, Eq(e, 0)), (-1/(2*e*(d + e*x)**2), True))/e**6 - 6*a*d*Piecewise((x/d**2, Eq(e, 0)), (-1/(d*e + e**2*x)
, True))/e**6 + a*Piecewise((x/d, Eq(e, 0)), (log(d + e*x)/e, True))/e**6 - b*d**6*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**6 + b*d**6*Piecewise((x/d**7
, Eq(e, 0)), (-1/(6*e*(d + e*x)**6), True))*log(c*x**n)/e**6 + 6*b*d**5*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**6 - 6*b*d**5*Piecewise((x/d**6, Eq(e, 0)), (-1/(5*e*(d + e*x)**5), True))*l
og(c*x**n)/e**6 - 15*b*d**4*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**6 + 15*b*d**4*Piecewise((x/d**5, Eq(e, 0)), (-1/(4*e*(d + e*x)**4), True))*log(c*
x**n)/e**6 + 20*b*d**3*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**
6 - 20*b*d**3*Piecewise((x/d**4, Eq(e, 0)), (-1/(3*e*(d + e*x)**3), True))*log(c*x**n)/e**6 - 15*b*d**2*n*Piec
ewise((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*
*6 + 15*b*d**2*Piecewise((x/d**3, Eq(e, 0)), (-1/(2*e*(d + e*x)**2), True))*log(c*x**n)/e**6 + 6*b*d*n*Piecewi
se((x/d**2, Eq(e, 0)), (-log(x)/(d*e) + log(d/e + x)/(d*e), True))/e**6 - 6*b*d*Piecewise((x/d**2, Eq(e, 0)),
(-1/(d*e + e**2*x), True))*log(c*x**n)/e**6 - b*n*Piecewise((x/d, Eq(e, 0)), (Piecewise((-polylog(2, e*x*exp_p
olar(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), (-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**6 + b*Piecewise((x/d, Eq(e, 0)), (log(d + e*x)/e, True))*log(c*x**n)/e**6

Maxima [F(-1)]

Timed out. \[ \int \frac {x^6 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^7} \, dx=\text {Timed out} \]

[In]

integrate(x^6*(a+b*log(c*x^n))/(e*x+d)^7,x, algorithm="maxima")

[Out]

Timed out

Giac [F]

\[ \int \frac {x^6 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^7} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} x^{6}}{{\left (e x + d\right )}^{7}} \,d x } \]

[In]

integrate(x^6*(a+b*log(c*x^n))/(e*x+d)^7,x, algorithm="giac")

[Out]

integrate((b*log(c*x^n) + a)*x^6/(e*x + d)^7, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {x^6 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^7} \, dx=\int \frac {x^6\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{{\left (d+e\,x\right )}^7} \,d x \]

[In]

int((x^6*(a + b*log(c*x^n)))/(d + e*x)^7,x)

[Out]

int((x^6*(a + b*log(c*x^n)))/(d + e*x)^7, x)