3.65 \(\int \frac {x^5 (a+b \log (c x^n))}{(d+e x)^7} \, dx\)

Optimal. Leaf size=136 \[ \frac {x^6 \left (a+b \log \left (c x^n\right )\right )}{6 d (d+e x)^6}-\frac {b d^4 n}{30 e^6 (d+e x)^5}+\frac {5 b d^3 n}{24 e^6 (d+e x)^4}-\frac {5 b d^2 n}{9 e^6 (d+e x)^3}-\frac {5 b n}{6 e^6 (d+e x)}+\frac {5 b d n}{6 e^6 (d+e x)^2}-\frac {b n \log (d+e x)}{6 d e^6} \]

[Out]

-1/30*b*d^4*n/e^6/(e*x+d)^5+5/24*b*d^3*n/e^6/(e*x+d)^4-5/9*b*d^2*n/e^6/(e*x+d)^3+5/6*b*d*n/e^6/(e*x+d)^2-5/6*b
*n/e^6/(e*x+d)+1/6*x^6*(a+b*ln(c*x^n))/d/(e*x+d)^6-1/6*b*n*ln(e*x+d)/d/e^6

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Rubi [A]  time = 0.11, antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2335, 43} \[ \frac {x^6 \left (a+b \log \left (c x^n\right )\right )}{6 d (d+e x)^6}-\frac {5 b d^2 n}{9 e^6 (d+e x)^3}+\frac {5 b d^3 n}{24 e^6 (d+e x)^4}-\frac {b d^4 n}{30 e^6 (d+e x)^5}-\frac {5 b n}{6 e^6 (d+e x)}+\frac {5 b d n}{6 e^6 (d+e x)^2}-\frac {b n \log (d+e x)}{6 d e^6} \]

Antiderivative was successfully verified.

[In]

Int[(x^5*(a + b*Log[c*x^n]))/(d + e*x)^7,x]

[Out]

-(b*d^4*n)/(30*e^6*(d + e*x)^5) + (5*b*d^3*n)/(24*e^6*(d + e*x)^4) - (5*b*d^2*n)/(9*e^6*(d + e*x)^3) + (5*b*d*
n)/(6*e^6*(d + e*x)^2) - (5*b*n)/(6*e^6*(d + e*x)) + (x^6*(a + b*Log[c*x^n]))/(6*d*(d + e*x)^6) - (b*n*Log[d +
e*x])/(6*d*e^6)

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 2335

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

Rubi steps

\begin {align*} \int \frac {x^5 \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 d (d+e x)^6}-\frac {(b n) \int \frac {x^5}{(d+e x)^6} \, dx}{6 d}\\ &=\frac {x^6 \left (a+b \log \left (c x^n\right )\right )}{6 d (d+e x)^6}-\frac {(b n) \int \left (-\frac {d^5}{e^5 (d+e x)^6}+\frac {5 d^4}{e^5 (d+e x)^5}-\frac {10 d^3}{e^5 (d+e x)^4}+\frac {10 d^2}{e^5 (d+e x)^3}-\frac {5 d}{e^5 (d+e x)^2}+\frac {1}{e^5 (d+e x)}\right ) \, dx}{6 d}\\ &=-\frac {b d^4 n}{30 e^6 (d+e x)^5}+\frac {5 b d^3 n}{24 e^6 (d+e x)^4}-\frac {5 b d^2 n}{9 e^6 (d+e x)^3}+\frac {5 b d n}{6 e^6 (d+e x)^2}-\frac {5 b n}{6 e^6 (d+e x)}+\frac {x^6 \left (a+b \log \left (c x^n\right )\right )}{6 d (d+e x)^6}-\frac {b n \log (d+e x)}{6 d e^6}\\ \end {align*}

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Mathematica [B]  time = 0.29, size = 335, normalized size = 2.46 \[ -\frac {60 a d^6+360 a d^5 e x+900 a d^4 e^2 x^2+1200 a d^3 e^3 x^3+900 a d^2 e^4 x^4+360 a d e^5 x^5+60 b d \left (d^5+6 d^4 e x+15 d^3 e^2 x^2+20 d^2 e^3 x^3+15 d e^4 x^4+6 e^5 x^5\right ) \log \left (c x^n\right )+60 b d^6 n \log (d+e x)+137 b d^6 n+762 b d^5 e n x+360 b d^5 e n x \log (d+e x)+1725 b d^4 e^2 n x^2+900 b d^4 e^2 n x^2 \log (d+e x)+2000 b d^3 e^3 n x^3+1200 b d^3 e^3 n x^3 \log (d+e x)+1200 b d^2 e^4 n x^4+900 b d^2 e^4 n x^4 \log (d+e x)+60 b e^6 n x^6 \log (d+e x)+300 b d e^5 n x^5+360 b d e^5 n x^5 \log (d+e x)-60 b n \log (x) (d+e x)^6}{360 d e^6 (d+e x)^6} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^5*(a + b*Log[c*x^n]))/(d + e*x)^7,x]

[Out]

-1/360*(60*a*d^6 + 137*b*d^6*n + 360*a*d^5*e*x + 762*b*d^5*e*n*x + 900*a*d^4*e^2*x^2 + 1725*b*d^4*e^2*n*x^2 +
1200*a*d^3*e^3*x^3 + 2000*b*d^3*e^3*n*x^3 + 900*a*d^2*e^4*x^4 + 1200*b*d^2*e^4*n*x^4 + 360*a*d*e^5*x^5 + 300*b
*d*e^5*n*x^5 - 60*b*n*(d + e*x)^6*Log[x] + 60*b*d*(d^5 + 6*d^4*e*x + 15*d^3*e^2*x^2 + 20*d^2*e^3*x^3 + 15*d*e^
4*x^4 + 6*e^5*x^5)*Log[c*x^n] + 60*b*d^6*n*Log[d + e*x] + 360*b*d^5*e*n*x*Log[d + e*x] + 900*b*d^4*e^2*n*x^2*L
og[d + e*x] + 1200*b*d^3*e^3*n*x^3*Log[d + e*x] + 900*b*d^2*e^4*n*x^4*Log[d + e*x] + 360*b*d*e^5*n*x^5*Log[d +
e*x] + 60*b*e^6*n*x^6*Log[d + e*x])/(d*e^6*(d + e*x)^6)

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fricas [B]  time = 0.78, size = 361, normalized size = 2.65 \[ \frac {60 \, b e^{6} n x^{6} \log \relax (x) - 137 \, b d^{6} n - 60 \, a d^{6} - 60 \, {\left (5 \, b d e^{5} n + 6 \, a d e^{5}\right )} x^{5} - 300 \, {\left (4 \, b d^{2} e^{4} n + 3 \, a d^{2} e^{4}\right )} x^{4} - 400 \, {\left (5 \, b d^{3} e^{3} n + 3 \, a d^{3} e^{3}\right )} x^{3} - 75 \, {\left (23 \, b d^{4} e^{2} n + 12 \, a d^{4} e^{2}\right )} x^{2} - 6 \, {\left (127 \, b d^{5} e n + 60 \, a d^{5} e\right )} x - 60 \, {\left (b e^{6} n x^{6} + 6 \, b d e^{5} n x^{5} + 15 \, b d^{2} e^{4} n x^{4} + 20 \, b d^{3} e^{3} n x^{3} + 15 \, b d^{4} e^{2} n x^{2} + 6 \, b d^{5} e n x + b d^{6} n\right )} \log \left (e x + d\right ) - 60 \, {\left (6 \, b d e^{5} x^{5} + 15 \, b d^{2} e^{4} x^{4} + 20 \, b d^{3} e^{3} x^{3} + 15 \, b d^{4} e^{2} x^{2} + 6 \, b d^{5} e x + b d^{6}\right )} \log \relax (c)}{360 \, {\left (d e^{12} x^{6} + 6 \, d^{2} e^{11} x^{5} + 15 \, d^{3} e^{10} x^{4} + 20 \, d^{4} e^{9} x^{3} + 15 \, d^{5} e^{8} x^{2} + 6 \, d^{6} e^{7} x + d^{7} e^{6}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(a+b*log(c*x^n))/(e*x+d)^7,x, algorithm="fricas")

[Out]

1/360*(60*b*e^6*n*x^6*log(x) - 137*b*d^6*n - 60*a*d^6 - 60*(5*b*d*e^5*n + 6*a*d*e^5)*x^5 - 300*(4*b*d^2*e^4*n
+ 3*a*d^2*e^4)*x^4 - 400*(5*b*d^3*e^3*n + 3*a*d^3*e^3)*x^3 - 75*(23*b*d^4*e^2*n + 12*a*d^4*e^2)*x^2 - 6*(127*b
*d^5*e*n + 60*a*d^5*e)*x - 60*(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) - 60*(6*b*d*e^5*x^5 + 15*b*d^2*e^4*x^4 + 20*b*d^3*e^3*x^
3 + 15*b*d^4*e^2*x^2 + 6*b*d^5*e*x + b*d^6)*log(c))/(d*e^12*x^6 + 6*d^2*e^11*x^5 + 15*d^3*e^10*x^4 + 20*d^4*e^
9*x^3 + 15*d^5*e^8*x^2 + 6*d^6*e^7*x + d^7*e^6)

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giac [B]  time = 0.31, size = 388, normalized size = 2.85 \[ -\frac {60 \, b n x^{6} e^{6} \log \left (x e + d\right ) + 360 \, b d n x^{5} e^{5} \log \left (x e + d\right ) + 900 \, b d^{2} n x^{4} e^{4} \log \left (x e + d\right ) + 1200 \, b d^{3} n x^{3} e^{3} \log \left (x e + d\right ) + 900 \, b d^{4} n x^{2} e^{2} \log \left (x e + d\right ) + 360 \, b d^{5} n x e \log \left (x e + d\right ) - 60 \, b n x^{6} e^{6} \log \relax (x) + 300 \, b d n x^{5} e^{5} + 1200 \, b d^{2} n x^{4} e^{4} + 2000 \, b d^{3} n x^{3} e^{3} + 1725 \, b d^{4} n x^{2} e^{2} + 762 \, b d^{5} n x e + 60 \, b d^{6} n \log \left (x e + d\right ) + 360 \, b d x^{5} e^{5} \log \relax (c) + 900 \, b d^{2} x^{4} e^{4} \log \relax (c) + 1200 \, b d^{3} x^{3} e^{3} \log \relax (c) + 900 \, b d^{4} x^{2} e^{2} \log \relax (c) + 360 \, b d^{5} x e \log \relax (c) + 137 \, b d^{6} n + 360 \, a d x^{5} e^{5} + 900 \, a d^{2} x^{4} e^{4} + 1200 \, a d^{3} x^{3} e^{3} + 900 \, a d^{4} x^{2} e^{2} + 360 \, a d^{5} x e + 60 \, b d^{6} \log \relax (c) + 60 \, a d^{6}}{360 \, {\left (d x^{6} e^{12} + 6 \, d^{2} x^{5} e^{11} + 15 \, d^{3} x^{4} e^{10} + 20 \, d^{4} x^{3} e^{9} + 15 \, d^{5} x^{2} e^{8} + 6 \, d^{6} x e^{7} + d^{7} e^{6}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(a+b*log(c*x^n))/(e*x+d)^7,x, algorithm="giac")

[Out]

-1/360*(60*b*n*x^6*e^6*log(x*e + d) + 360*b*d*n*x^5*e^5*log(x*e + d) + 900*b*d^2*n*x^4*e^4*log(x*e + d) + 1200
*b*d^3*n*x^3*e^3*log(x*e + d) + 900*b*d^4*n*x^2*e^2*log(x*e + d) + 360*b*d^5*n*x*e*log(x*e + d) - 60*b*n*x^6*e
^6*log(x) + 300*b*d*n*x^5*e^5 + 1200*b*d^2*n*x^4*e^4 + 2000*b*d^3*n*x^3*e^3 + 1725*b*d^4*n*x^2*e^2 + 762*b*d^5
*n*x*e + 60*b*d^6*n*log(x*e + d) + 360*b*d*x^5*e^5*log(c) + 900*b*d^2*x^4*e^4*log(c) + 1200*b*d^3*x^3*e^3*log(
c) + 900*b*d^4*x^2*e^2*log(c) + 360*b*d^5*x*e*log(c) + 137*b*d^6*n + 360*a*d*x^5*e^5 + 900*a*d^2*x^4*e^4 + 120
0*a*d^3*x^3*e^3 + 900*a*d^4*x^2*e^2 + 360*a*d^5*x*e + 60*b*d^6*log(c) + 60*a*d^6)/(d*x^6*e^12 + 6*d^2*x^5*e^11
+ 15*d^3*x^4*e^10 + 20*d^4*x^3*e^9 + 15*d^5*x^2*e^8 + 6*d^6*x*e^7 + d^7*e^6)

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maple [C]  time = 0.28, size = 1165, normalized size = 8.57 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(b*ln(c*x^n)+a)/(e*x+d)^7,x)

[Out]

-1/6*b*(6*e^5*x^5+15*d*e^4*x^4+20*d^2*e^3*x^3+15*d^3*e^2*x^2+6*d^4*e*x+d^5)/(e*x+d)^6/e^6*ln(x^n)+1/360*(-300*
b*d*e^5*n*x^5-1200*b*d^2*e^4*n*x^4-2000*b*d^3*e^3*n*x^3-1725*b*d^4*e^2*n*x^2-762*b*d^5*e*n*x-60*ln(e*x+d)*b*d^
6*n+60*ln(-x)*b*d^6*n-360*a*d*e^5*x^5-900*a*d^2*e^4*x^4-1200*a*d^3*e^3*x^3-900*a*d^4*e^2*x^2-360*a*d^5*e*x-60*
a*d^6-60*ln(c)*b*d^6-137*b*d^6*n+450*I*Pi*b*d^4*e^2*x^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+180*I*Pi*b*d^5*e*x
*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+600*I*Pi*b*d^3*e^3*x^3*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-600*I*Pi*b*d^3
*e^3*x^3*csgn(I*c*x^n)^2*csgn(I*c)-450*I*Pi*b*d^4*e^2*x^2*csgn(I*x^n)*csgn(I*c*x^n)^2-450*I*Pi*b*d^4*e^2*x^2*c
sgn(I*c*x^n)^2*csgn(I*c)-180*I*Pi*b*d^5*e*x*csgn(I*c*x^n)^2*csgn(I*c)-60*ln(e*x+d)*b*e^6*n*x^6+60*ln(-x)*b*e^6
*n*x^6-180*I*Pi*b*d^5*e*x*csgn(I*x^n)*csgn(I*c*x^n)^2+180*I*Pi*b*d*e^5*x^5*csgn(I*c*x^n)^3+450*I*Pi*b*d^2*e^4*
x^4*csgn(I*c*x^n)^3+600*I*Pi*b*d^3*e^3*x^3*csgn(I*c*x^n)^3-180*I*Pi*b*d*e^5*x^5*csgn(I*x^n)*csgn(I*c*x^n)^2-18
0*I*Pi*b*d*e^5*x^5*csgn(I*c*x^n)^2*csgn(I*c)-450*I*Pi*b*d^2*e^4*x^4*csgn(I*x^n)*csgn(I*c*x^n)^2-450*I*Pi*b*d^2
*e^4*x^4*csgn(I*c*x^n)^2*csgn(I*c)-600*I*Pi*b*d^3*e^3*x^3*csgn(I*x^n)*csgn(I*c*x^n)^2+30*I*Pi*b*d^6*csgn(I*c*x
^n)^3-900*ln(c)*b*d^4*e^2*x^2-360*ln(c)*b*d^5*e*x-360*ln(c)*b*d*e^5*x^5-900*ln(c)*b*d^2*e^4*x^4-1200*ln(c)*b*d
^3*e^3*x^3+180*I*Pi*b*d*e^5*x^5*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+450*I*Pi*b*d^2*e^4*x^4*csgn(I*x^n)*csgn(I*
c*x^n)*csgn(I*c)-30*I*Pi*b*d^6*csgn(I*x^n)*csgn(I*c*x^n)^2-30*I*Pi*b*d^6*csgn(I*c*x^n)^2*csgn(I*c)-360*ln(e*x+
d)*b*d*e^5*n*x^5-900*ln(e*x+d)*b*d^2*e^4*n*x^4-1200*ln(e*x+d)*b*d^3*e^3*n*x^3-900*ln(e*x+d)*b*d^4*e^2*n*x^2-36
0*ln(e*x+d)*b*d^5*e*n*x+360*ln(-x)*b*d*e^5*n*x^5+900*ln(-x)*b*d^2*e^4*n*x^4+1200*ln(-x)*b*d^3*e^3*n*x^3+900*ln
(-x)*b*d^4*e^2*n*x^2+360*ln(-x)*b*d^5*e*n*x+450*I*Pi*b*d^4*e^2*x^2*csgn(I*c*x^n)^3+180*I*Pi*b*d^5*e*x*csgn(I*c
*x^n)^3+30*I*Pi*b*d^6*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c))/d/e^6/(e*x+d)^6

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maxima [B]  time = 0.66, size = 377, normalized size = 2.77 \[ -\frac {1}{360} \, b n {\left (\frac {300 \, e^{4} x^{4} + 900 \, d e^{3} x^{3} + 1100 \, d^{2} e^{2} x^{2} + 625 \, d^{3} e x + 137 \, d^{4}}{e^{11} x^{5} + 5 \, d e^{10} x^{4} + 10 \, d^{2} e^{9} x^{3} + 10 \, d^{3} e^{8} x^{2} + 5 \, d^{4} e^{7} x + d^{5} e^{6}} + \frac {60 \, \log \left (e x + d\right )}{d e^{6}} - \frac {60 \, \log \relax (x)}{d e^{6}}\right )} - \frac {{\left (6 \, e^{5} x^{5} + 15 \, d e^{4} x^{4} + 20 \, d^{2} e^{3} x^{3} + 15 \, d^{3} e^{2} x^{2} + 6 \, d^{4} e x + d^{5}\right )} b \log \left (c x^{n}\right )}{6 \, {\left (e^{12} x^{6} + 6 \, d e^{11} x^{5} + 15 \, d^{2} e^{10} x^{4} + 20 \, d^{3} e^{9} x^{3} + 15 \, d^{4} e^{8} x^{2} + 6 \, d^{5} e^{7} x + d^{6} e^{6}\right )}} - \frac {{\left (6 \, e^{5} x^{5} + 15 \, d e^{4} x^{4} + 20 \, d^{2} e^{3} x^{3} + 15 \, d^{3} e^{2} x^{2} + 6 \, d^{4} e x + d^{5}\right )} a}{6 \, {\left (e^{12} x^{6} + 6 \, d e^{11} x^{5} + 15 \, d^{2} e^{10} x^{4} + 20 \, d^{3} e^{9} x^{3} + 15 \, d^{4} e^{8} x^{2} + 6 \, d^{5} e^{7} x + d^{6} e^{6}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(a+b*log(c*x^n))/(e*x+d)^7,x, algorithm="maxima")

[Out]

-1/360*b*n*((300*e^4*x^4 + 900*d*e^3*x^3 + 1100*d^2*e^2*x^2 + 625*d^3*e*x + 137*d^4)/(e^11*x^5 + 5*d*e^10*x^4
+ 10*d^2*e^9*x^3 + 10*d^3*e^8*x^2 + 5*d^4*e^7*x + d^5*e^6) + 60*log(e*x + d)/(d*e^6) - 60*log(x)/(d*e^6)) - 1/
6*(6*e^5*x^5 + 15*d*e^4*x^4 + 20*d^2*e^3*x^3 + 15*d^3*e^2*x^2 + 6*d^4*e*x + d^5)*b*log(c*x^n)/(e^12*x^6 + 6*d*
e^11*x^5 + 15*d^2*e^10*x^4 + 20*d^3*e^9*x^3 + 15*d^4*e^8*x^2 + 6*d^5*e^7*x + d^6*e^6) - 1/6*(6*e^5*x^5 + 15*d*
e^4*x^4 + 20*d^2*e^3*x^3 + 15*d^3*e^2*x^2 + 6*d^4*e*x + d^5)*a/(e^12*x^6 + 6*d*e^11*x^5 + 15*d^2*e^10*x^4 + 20
*d^3*e^9*x^3 + 15*d^4*e^8*x^2 + 6*d^5*e^7*x + d^6*e^6)

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mupad [B]  time = 4.48, size = 341, normalized size = 2.51 \[ -\frac {x^5\,\left (6\,a\,e^5+5\,b\,e^5\,n\right )+x\,\left (6\,a\,d^4\,e+\frac {127\,b\,d^4\,e\,n}{10}\right )+a\,d^5+x^3\,\left (20\,a\,d^2\,e^3+\frac {100\,b\,d^2\,e^3\,n}{3}\right )+x^2\,\left (15\,a\,d^3\,e^2+\frac {115\,b\,d^3\,e^2\,n}{4}\right )+x^4\,\left (15\,a\,d\,e^4+20\,b\,d\,e^4\,n\right )+\frac {137\,b\,d^5\,n}{60}}{6\,d^6\,e^6+36\,d^5\,e^7\,x+90\,d^4\,e^8\,x^2+120\,d^3\,e^9\,x^3+90\,d^2\,e^{10}\,x^4+36\,d\,e^{11}\,x^5+6\,e^{12}\,x^6}-\frac {\ln \left (c\,x^n\right )\,\left (\frac {b\,d^5}{6\,e^6}+\frac {b\,x^5}{e}+\frac {10\,b\,d^2\,x^3}{3\,e^3}+\frac {5\,b\,d^3\,x^2}{2\,e^4}+\frac {5\,b\,d\,x^4}{2\,e^2}+\frac {b\,d^4\,x}{e^5}\right )}{d^6+6\,d^5\,e\,x+15\,d^4\,e^2\,x^2+20\,d^3\,e^3\,x^3+15\,d^2\,e^4\,x^4+6\,d\,e^5\,x^5+e^6\,x^6}-\frac {b\,n\,\mathrm {atanh}\left (\frac {2\,e\,x}{d}+1\right )}{3\,d\,e^6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^5*(a + b*log(c*x^n)))/(d + e*x)^7,x)

[Out]

- (x^5*(6*a*e^5 + 5*b*e^5*n) + x*(6*a*d^4*e + (127*b*d^4*e*n)/10) + a*d^5 + x^3*(20*a*d^2*e^3 + (100*b*d^2*e^3
*n)/3) + x^2*(15*a*d^3*e^2 + (115*b*d^3*e^2*n)/4) + x^4*(15*a*d*e^4 + 20*b*d*e^4*n) + (137*b*d^5*n)/60)/(6*d^6
*e^6 + 6*e^12*x^6 + 36*d^5*e^7*x + 36*d*e^11*x^5 + 90*d^4*e^8*x^2 + 120*d^3*e^9*x^3 + 90*d^2*e^10*x^4) - (log(
c*x^n)*((b*d^5)/(6*e^6) + (b*x^5)/e + (10*b*d^2*x^3)/(3*e^3) + (5*b*d^3*x^2)/(2*e^4) + (5*b*d*x^4)/(2*e^2) + (
b*d^4*x)/e^5))/(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))/(3*d*e^6)

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

[Out]

Timed out

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