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

Optimal. Leaf size=174 \[ -\frac {a+b \log \left (c x^n\right )}{5 e^2 (d+e x)^5}+\frac {d \left (a+b \log \left (c x^n\right )\right )}{6 e^2 (d+e x)^6}+\frac {b n \log (x)}{30 d^5 e^2}-\frac {b n \log (d+e x)}{30 d^5 e^2}+\frac {b n}{30 d^4 e^2 (d+e x)}+\frac {b n}{60 d^3 e^2 (d+e x)^2}+\frac {b n}{90 d^2 e^2 (d+e x)^3}+\frac {b n}{120 d e^2 (d+e x)^4}-\frac {b n}{30 e^2 (d+e x)^5} \]

[Out]

-1/30*b*n/e^2/(e*x+d)^5+1/120*b*n/d/e^2/(e*x+d)^4+1/90*b*n/d^2/e^2/(e*x+d)^3+1/60*b*n/d^3/e^2/(e*x+d)^2+1/30*b
*n/d^4/e^2/(e*x+d)+1/30*b*n*ln(x)/d^5/e^2+1/6*d*(a+b*ln(c*x^n))/e^2/(e*x+d)^6+1/5*(-a-b*ln(c*x^n))/e^2/(e*x+d)
^5-1/30*b*n*ln(e*x+d)/d^5/e^2

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Rubi [A]  time = 0.12, antiderivative size = 174, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {43, 2350, 12, 77} \[ -\frac {a+b \log \left (c x^n\right )}{5 e^2 (d+e x)^5}+\frac {d \left (a+b \log \left (c x^n\right )\right )}{6 e^2 (d+e x)^6}+\frac {b n}{30 d^4 e^2 (d+e x)}+\frac {b n}{60 d^3 e^2 (d+e x)^2}+\frac {b n}{90 d^2 e^2 (d+e x)^3}+\frac {b n \log (x)}{30 d^5 e^2}-\frac {b n \log (d+e x)}{30 d^5 e^2}+\frac {b n}{120 d e^2 (d+e x)^4}-\frac {b n}{30 e^2 (d+e x)^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(b*n)/(30*e^2*(d + e*x)^5) + (b*n)/(120*d*e^2*(d + e*x)^4) + (b*n)/(90*d^2*e^2*(d + e*x)^3) + (b*n)/(60*d^3*e
^2*(d + e*x)^2) + (b*n)/(30*d^4*e^2*(d + e*x)) + (b*n*Log[x])/(30*d^5*e^2) + (d*(a + b*Log[c*x^n]))/(6*e^2*(d
+ e*x)^6) - (a + b*Log[c*x^n])/(5*e^2*(d + e*x)^5) - (b*n*Log[d + e*x])/(30*d^5*e^2)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rule 2350

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> Wit
h[{u = IntHide[(f*x)^m*(d + e*x^r)^q, x]}, Dist[a + b*Log[c*x^n], u, x] - Dist[b*n, Int[SimplifyIntegrand[u/x,
x], x], x] /; ((EqQ[r, 1] || EqQ[r, 2]) && IntegerQ[m] && IntegerQ[q - 1/2]) || InverseFunctionFreeQ[u, x]] /
; FreeQ[{a, b, c, d, e, f, m, n, q, r}, x] && IntegerQ[2*q] && ((IntegerQ[m] && IntegerQ[r]) || IGtQ[q, 0])

Rubi steps

\begin {align*} \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^7} \, dx &=\frac {d \left (a+b \log \left (c x^n\right )\right )}{6 e^2 (d+e x)^6}-\frac {a+b \log \left (c x^n\right )}{5 e^2 (d+e x)^5}-(b n) \int \frac {-d-6 e x}{30 e^2 x (d+e x)^6} \, dx\\ &=\frac {d \left (a+b \log \left (c x^n\right )\right )}{6 e^2 (d+e x)^6}-\frac {a+b \log \left (c x^n\right )}{5 e^2 (d+e x)^5}-\frac {(b n) \int \frac {-d-6 e x}{x (d+e x)^6} \, dx}{30 e^2}\\ &=\frac {d \left (a+b \log \left (c x^n\right )\right )}{6 e^2 (d+e x)^6}-\frac {a+b \log \left (c x^n\right )}{5 e^2 (d+e x)^5}-\frac {(b n) \int \left (-\frac {1}{d^5 x}-\frac {5 e}{(d+e x)^6}+\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}{30 e^2}\\ &=-\frac {b n}{30 e^2 (d+e x)^5}+\frac {b n}{120 d e^2 (d+e x)^4}+\frac {b n}{90 d^2 e^2 (d+e x)^3}+\frac {b n}{60 d^3 e^2 (d+e x)^2}+\frac {b n}{30 d^4 e^2 (d+e x)}+\frac {b n \log (x)}{30 d^5 e^2}+\frac {d \left (a+b \log \left (c x^n\right )\right )}{6 e^2 (d+e x)^6}-\frac {a+b \log \left (c x^n\right )}{5 e^2 (d+e x)^5}-\frac {b n \log (d+e x)}{30 d^5 e^2}\\ \end {align*}

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Mathematica [A]  time = 0.15, size = 160, normalized size = 0.92 \[ \frac {60 a d^6-72 a d^5 (d+e x)+60 b d^6 \log \left (c x^n\right )-72 b d^5 (d+e x) \log \left (c x^n\right )-12 b d^5 n (d+e x)+3 b d^4 n (d+e x)^2+4 b d^3 n (d+e x)^3+6 b d^2 n (d+e x)^4+12 b d n (d+e x)^5+12 b n \log (x) (d+e x)^6-12 b n (d+e x)^6 \log (d+e x)}{360 d^5 e^2 (d+e x)^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(60*a*d^6 - 72*a*d^5*(d + e*x) - 12*b*d^5*n*(d + e*x) + 3*b*d^4*n*(d + e*x)^2 + 4*b*d^3*n*(d + e*x)^3 + 6*b*d^
2*n*(d + e*x)^4 + 12*b*d*n*(d + e*x)^5 + 12*b*n*(d + e*x)^6*Log[x] + 60*b*d^6*Log[c*x^n] - 72*b*d^5*(d + e*x)*
Log[c*x^n] - 12*b*n*(d + e*x)^6*Log[d + e*x])/(360*d^5*e^2*(d + e*x)^6)

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fricas [B]  time = 0.81, size = 323, normalized size = 1.86 \[ \frac {12 \, b d e^{5} n x^{5} + 66 \, b d^{2} e^{4} n x^{4} + 148 \, b d^{3} e^{3} n x^{3} + 171 \, b d^{4} e^{2} n x^{2} + 13 \, b d^{6} n - 12 \, a d^{6} + 18 \, {\left (5 \, b d^{5} e n - 4 \, a d^{5} e\right )} x - 12 \, {\left (b e^{6} n x^{6} + 6 \, b d e^{5} n x^{5} + 15 \, b d^{2} e^{4} n x^{4} + 20 \, b d^{3} e^{3} n x^{3} + 15 \, b d^{4} e^{2} n x^{2} + 6 \, b d^{5} e n x + b d^{6} n\right )} \log \left (e x + d\right ) - 12 \, {\left (6 \, b d^{5} e x + b d^{6}\right )} \log \relax (c) + 12 \, {\left (b e^{6} n x^{6} + 6 \, b d e^{5} n x^{5} + 15 \, b d^{2} e^{4} n x^{4} + 20 \, b d^{3} e^{3} n x^{3} + 15 \, b d^{4} e^{2} n x^{2}\right )} \log \relax (x)}{360 \, {\left (d^{5} e^{8} x^{6} + 6 \, d^{6} e^{7} x^{5} + 15 \, d^{7} e^{6} x^{4} + 20 \, d^{8} e^{5} x^{3} + 15 \, d^{9} e^{4} x^{2} + 6 \, d^{10} e^{3} x + d^{11} e^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/360*(12*b*d*e^5*n*x^5 + 66*b*d^2*e^4*n*x^4 + 148*b*d^3*e^3*n*x^3 + 171*b*d^4*e^2*n*x^2 + 13*b*d^6*n - 12*a*d
^6 + 18*(5*b*d^5*e*n - 4*a*d^5*e)*x - 12*(b*e^6*n*x^6 + 6*b*d*e^5*n*x^5 + 15*b*d^2*e^4*n*x^4 + 20*b*d^3*e^3*n*
x^3 + 15*b*d^4*e^2*n*x^2 + 6*b*d^5*e*n*x + b*d^6*n)*log(e*x + d) - 12*(6*b*d^5*e*x + b*d^6)*log(c) + 12*(b*e^6
*n*x^6 + 6*b*d*e^5*n*x^5 + 15*b*d^2*e^4*n*x^4 + 20*b*d^3*e^3*n*x^3 + 15*b*d^4*e^2*n*x^2)*log(x))/(d^5*e^8*x^6
+ 6*d^6*e^7*x^5 + 15*d^7*e^6*x^4 + 20*d^8*e^5*x^3 + 15*d^9*e^4*x^2 + 6*d^10*e^3*x + d^11*e^2)

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giac [B]  time = 0.34, size = 352, normalized size = 2.02 \[ -\frac {12 \, b n x^{6} e^{6} \log \left (x e + d\right ) + 72 \, b d n x^{5} e^{5} \log \left (x e + d\right ) + 180 \, b d^{2} n x^{4} e^{4} \log \left (x e + d\right ) + 240 \, b d^{3} n x^{3} e^{3} \log \left (x e + d\right ) + 180 \, b d^{4} n x^{2} e^{2} \log \left (x e + d\right ) + 72 \, b d^{5} n x e \log \left (x e + d\right ) - 12 \, b n x^{6} e^{6} \log \relax (x) - 72 \, b d n x^{5} e^{5} \log \relax (x) - 180 \, b d^{2} n x^{4} e^{4} \log \relax (x) - 240 \, b d^{3} n x^{3} e^{3} \log \relax (x) - 180 \, b d^{4} n x^{2} e^{2} \log \relax (x) - 12 \, b d n x^{5} e^{5} - 66 \, b d^{2} n x^{4} e^{4} - 148 \, b d^{3} n x^{3} e^{3} - 171 \, b d^{4} n x^{2} e^{2} - 90 \, b d^{5} n x e + 12 \, b d^{6} n \log \left (x e + d\right ) + 72 \, b d^{5} x e \log \relax (c) - 13 \, b d^{6} n + 72 \, a d^{5} x e + 12 \, b d^{6} \log \relax (c) + 12 \, a d^{6}}{360 \, {\left (d^{5} x^{6} e^{8} + 6 \, d^{6} x^{5} e^{7} + 15 \, d^{7} x^{4} e^{6} + 20 \, d^{8} x^{3} e^{5} + 15 \, d^{9} x^{2} e^{4} + 6 \, d^{10} x e^{3} + d^{11} e^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/360*(12*b*n*x^6*e^6*log(x*e + d) + 72*b*d*n*x^5*e^5*log(x*e + d) + 180*b*d^2*n*x^4*e^4*log(x*e + d) + 240*b
*d^3*n*x^3*e^3*log(x*e + d) + 180*b*d^4*n*x^2*e^2*log(x*e + d) + 72*b*d^5*n*x*e*log(x*e + d) - 12*b*n*x^6*e^6*
log(x) - 72*b*d*n*x^5*e^5*log(x) - 180*b*d^2*n*x^4*e^4*log(x) - 240*b*d^3*n*x^3*e^3*log(x) - 180*b*d^4*n*x^2*e
^2*log(x) - 12*b*d*n*x^5*e^5 - 66*b*d^2*n*x^4*e^4 - 148*b*d^3*n*x^3*e^3 - 171*b*d^4*n*x^2*e^2 - 90*b*d^5*n*x*e
+ 12*b*d^6*n*log(x*e + d) + 72*b*d^5*x*e*log(c) - 13*b*d^6*n + 72*a*d^5*x*e + 12*b*d^6*log(c) + 12*a*d^6)/(d^
5*x^6*e^8 + 6*d^6*x^5*e^7 + 15*d^7*x^4*e^6 + 20*d^8*x^3*e^5 + 15*d^9*x^2*e^4 + 6*d^10*x*e^3 + d^11*e^2)

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maple [C]  time = 0.25, size = 557, normalized size = 3.20 \[ -\frac {\left (6 e x +d \right ) b \ln \left (x^{n}\right )}{30 \left (e x +d \right )^{6} e^{2}}-\frac {-6 i \pi b \,d^{6} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}-12 b d \,e^{5} n \,x^{5}-66 b \,d^{2} e^{4} n \,x^{4}-148 b \,d^{3} e^{3} n \,x^{3}-171 b \,d^{4} e^{2} n \,x^{2}-90 b \,d^{5} e n x +12 b \,d^{6} n \ln \left (e x +d \right )-12 b \,d^{6} n \ln \left (-x \right )+72 a \,d^{5} e x +12 a \,d^{6}+12 b \,d^{6} \ln \relax (c )-13 b \,d^{6} n +12 b \,e^{6} n \,x^{6} \ln \left (e x +d \right )-12 b \,e^{6} n \,x^{6} \ln \left (-x \right )-36 i \pi b \,d^{5} e x \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )+36 i \pi b \,d^{5} e x \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+36 i \pi b \,d^{5} e x \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+72 b \,d^{5} e x \ln \relax (c )-36 i \pi b \,d^{5} e x \mathrm {csgn}\left (i c \,x^{n}\right )^{3}-6 i \pi b \,d^{6} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )+72 b d \,e^{5} n \,x^{5} \ln \left (e x +d \right )+180 b \,d^{2} e^{4} n \,x^{4} \ln \left (e x +d \right )+240 b \,d^{3} e^{3} n \,x^{3} \ln \left (e x +d \right )+180 b \,d^{4} e^{2} n \,x^{2} \ln \left (e x +d \right )+72 b \,d^{5} e n x \ln \left (e x +d \right )-72 b d \,e^{5} n \,x^{5} \ln \left (-x \right )-180 b \,d^{2} e^{4} n \,x^{4} \ln \left (-x \right )-240 b \,d^{3} e^{3} n \,x^{3} \ln \left (-x \right )-180 b \,d^{4} e^{2} n \,x^{2} \ln \left (-x \right )-72 b \,d^{5} e n x \ln \left (-x \right )+6 i \pi b \,d^{6} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+6 i \pi b \,d^{6} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{360 \left (e x +d \right )^{6} d^{5} e^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/30*b*(6*e*x+d)/(e*x+d)^6/e^2*ln(x^n)-1/360*(-12*b*d*e^5*n*x^5-66*b*d^2*e^4*n*x^4-148*b*d^3*e^3*n*x^3-171*b*
d^4*e^2*n*x^2-90*b*d^5*e*n*x+12*b*d^6*n*ln(e*x+d)-12*b*d^6*n*ln(-x)+72*a*d^5*e*x+12*a*d^6+12*b*d^6*ln(c)-13*b*
d^6*n+36*I*Pi*b*d^5*e*x*csgn(I*c)*csgn(I*c*x^n)^2+12*b*e^6*n*x^6*ln(e*x+d)-12*b*e^6*n*x^6*ln(-x)+36*I*Pi*b*d^5
*e*x*csgn(I*x^n)*csgn(I*c*x^n)^2+72*b*d^5*e*x*ln(c)-6*I*Pi*b*d^6*csgn(I*c*x^n)^3-36*I*Pi*b*d^5*e*x*csgn(I*c)*c
sgn(I*x^n)*csgn(I*c*x^n)+6*I*Pi*b*d^6*csgn(I*x^n)*csgn(I*c*x^n)^2+6*I*Pi*b*d^6*csgn(I*c)*csgn(I*c*x^n)^2+72*b*
d*e^5*n*x^5*ln(e*x+d)+180*b*d^2*e^4*n*x^4*ln(e*x+d)+240*b*d^3*e^3*n*x^3*ln(e*x+d)+180*b*d^4*e^2*n*x^2*ln(e*x+d
)+72*b*d^5*e*n*x*ln(e*x+d)-72*b*d*e^5*n*x^5*ln(-x)-180*b*d^2*e^4*n*x^4*ln(-x)-240*b*d^3*e^3*n*x^3*ln(-x)-180*b
*d^4*e^2*n*x^2*ln(-x)-72*b*d^5*e*n*x*ln(-x)-36*I*Pi*b*d^5*e*x*csgn(I*c*x^n)^3-6*I*Pi*b*d^6*csgn(I*c)*csgn(I*x^
n)*csgn(I*c*x^n))/e^2/d^5/(e*x+d)^6

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maxima [A]  time = 0.60, size = 294, normalized size = 1.69 \[ \frac {1}{360} \, b n {\left (\frac {12 \, e^{4} x^{4} + 54 \, d e^{3} x^{3} + 94 \, d^{2} e^{2} x^{2} + 77 \, d^{3} e x + 13 \, d^{4}}{d^{4} e^{7} x^{5} + 5 \, d^{5} e^{6} x^{4} + 10 \, d^{6} e^{5} x^{3} + 10 \, d^{7} e^{4} x^{2} + 5 \, d^{8} e^{3} x + d^{9} e^{2}} - \frac {12 \, \log \left (e x + d\right )}{d^{5} e^{2}} + \frac {12 \, \log \relax (x)}{d^{5} e^{2}}\right )} - \frac {{\left (6 \, e x + d\right )} b \log \left (c x^{n}\right )}{30 \, {\left (e^{8} x^{6} + 6 \, d e^{7} x^{5} + 15 \, d^{2} e^{6} x^{4} + 20 \, d^{3} e^{5} x^{3} + 15 \, d^{4} e^{4} x^{2} + 6 \, d^{5} e^{3} x + d^{6} e^{2}\right )}} - \frac {{\left (6 \, e x + d\right )} a}{30 \, {\left (e^{8} x^{6} + 6 \, d e^{7} x^{5} + 15 \, d^{2} e^{6} x^{4} + 20 \, d^{3} e^{5} x^{3} + 15 \, d^{4} e^{4} x^{2} + 6 \, d^{5} e^{3} x + d^{6} e^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/360*b*n*((12*e^4*x^4 + 54*d*e^3*x^3 + 94*d^2*e^2*x^2 + 77*d^3*e*x + 13*d^4)/(d^4*e^7*x^5 + 5*d^5*e^6*x^4 + 1
0*d^6*e^5*x^3 + 10*d^7*e^4*x^2 + 5*d^8*e^3*x + d^9*e^2) - 12*log(e*x + d)/(d^5*e^2) + 12*log(x)/(d^5*e^2)) - 1
/30*(6*e*x + d)*b*log(c*x^n)/(e^8*x^6 + 6*d*e^7*x^5 + 15*d^2*e^6*x^4 + 20*d^3*e^5*x^3 + 15*d^4*e^4*x^2 + 6*d^5
*e^3*x + d^6*e^2) - 1/30*(6*e*x + d)*a/(e^8*x^6 + 6*d*e^7*x^5 + 15*d^2*e^6*x^4 + 20*d^3*e^5*x^3 + 15*d^4*e^4*x
^2 + 6*d^5*e^3*x + d^6*e^2)

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mupad [B]  time = 4.04, size = 251, normalized size = 1.44 \[ \frac {\frac {13\,b\,d\,n}{12}-x\,\left (6\,a\,e-\frac {15\,b\,e\,n}{2}\right )-a\,d+\frac {57\,b\,e^2\,n\,x^2}{4\,d}+\frac {37\,b\,e^3\,n\,x^3}{3\,d^2}+\frac {11\,b\,e^4\,n\,x^4}{2\,d^3}+\frac {b\,e^5\,n\,x^5}{d^4}}{30\,d^6\,e^2+180\,d^5\,e^3\,x+450\,d^4\,e^4\,x^2+600\,d^3\,e^5\,x^3+450\,d^2\,e^6\,x^4+180\,d\,e^7\,x^5+30\,e^8\,x^6}-\frac {\ln \left (c\,x^n\right )\,\left (\frac {b\,d}{30\,e^2}+\frac {b\,x}{5\,e}\right )}{d^6+6\,d^5\,e\,x+15\,d^4\,e^2\,x^2+20\,d^3\,e^3\,x^3+15\,d^2\,e^4\,x^4+6\,d\,e^5\,x^5+e^6\,x^6}-\frac {b\,n\,\mathrm {atanh}\left (\frac {2\,e\,x}{d}+1\right )}{15\,d^5\,e^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((13*b*d*n)/12 - x*(6*a*e - (15*b*e*n)/2) - a*d + (57*b*e^2*n*x^2)/(4*d) + (37*b*e^3*n*x^3)/(3*d^2) + (11*b*e^
4*n*x^4)/(2*d^3) + (b*e^5*n*x^5)/d^4)/(30*d^6*e^2 + 30*e^8*x^6 + 180*d^5*e^3*x + 180*d*e^7*x^5 + 450*d^4*e^4*x
^2 + 600*d^3*e^5*x^3 + 450*d^2*e^6*x^4) - (log(c*x^n)*((b*d)/(30*e^2) + (b*x)/(5*e)))/(d^6 + e^6*x^6 + 6*d*e^5
*x^5 + 15*d^4*e^2*x^2 + 20*d^3*e^3*x^3 + 15*d^2*e^4*x^4 + 6*d^5*e*x) - (b*n*atanh((2*e*x)/d + 1))/(15*d^5*e^2)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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