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

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

[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)

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Rubi [A]  time = 0.110053, antiderivative size = 136, normalized size of antiderivative = 1., 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 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]

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])

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*}

Mathematica [B]  time = 0.285324, size = 335, normalized size = 2.46 \[ -\frac{900 a d^4 e^2 x^2+1200 a d^3 e^3 x^3+900 a d^2 e^4 x^4+360 a d^5 e x+60 a d^6+360 a d e^5 x^5+60 b d \left (15 d^3 e^2 x^2+20 d^2 e^3 x^3+6 d^4 e x+d^5+15 d e^4 x^4+6 e^5 x^5\right ) \log \left (c x^n\right )+1725 b d^4 e^2 n x^2+2000 b d^3 e^3 n x^3+1200 b d^2 e^4 n x^4+900 b d^4 e^2 n x^2 \log (d+e x)+1200 b d^3 e^3 n x^3 \log (d+e x)+900 b d^2 e^4 n x^4 \log (d+e x)+762 b d^5 e n x+60 b d^6 n \log (d+e x)+360 b d^5 e n x \log (d+e x)+137 b d^6 n+300 b d e^5 n x^5+360 b d e^5 n x^5 \log (d+e x)+60 b e^6 n x^6 \log (d+e x)-60 b n \log (x) (d+e x)^6}{360 d e^6 (d+e x)^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(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*Log[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])/(360*d*e^6*(d + e*x)^6)

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Maple [C]  time = 0.173, size = 1165, normalized size = 8.6 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(a+b*ln(c*x^n))/(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*(-60*a
*d^6-600*I*Pi*b*d^3*e^3*x^3*csgn(I*x^n)*csgn(I*c*x^n)^2-600*I*Pi*b*d^3*e^3*x^3*csgn(I*c*x^n)^2*csgn(I*c)-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-180*I*Pi*b*d^5*e*x
*csgn(I*c*x^n)^2*csgn(I*c)-180*I*Pi*b*d*e^5*x^5*csgn(I*x^n)*csgn(I*c*x^n)^2-180*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)-1200*a*d^3*e^3*x^3-900*a*d^4*e^2*x^2-360*a*d^5*e*x-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*csgn(I*c*x^n)^2*csgn(I*c)-180*I*Pi*b*d^5*e*x*csgn(I*x^n)*csgn(I*c*x^n)^2-137*b*d^6*n-360*ln(
e*x+d)*b*d*e^5*n*x^5-30*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)-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^2-360*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*x^n)*csgn(I*c*x^n)*csgn(I*c)+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)+600*I*Pi*b*d^3*e^3*x^3*cs
gn(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)-60*ln(c)*b*d^6+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)^3-3
60*a*d*e^5*x^5-900*a*d^2*e^4*x^4+30*I*Pi*b*d^6*csgn(I*c*x^n)^3-60*ln(e*x+d)*b*e^6*n*x^6+60*ln(-x)*b*e^6*n*x^6-
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^3-900*ln(c)*b*d^4*e^2*x^2+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)-3
60*ln(c)*b*d^5*e*x-60*ln(e*x+d)*b*d^6*n+60*ln(-x)*b*d^6*n)/d/e^6/(e*x+d)^6

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Maxima [B]  time = 1.24815, size = 509, normalized size = 3.74 \begin{align*} -\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 \left (x\right )}{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 )}} \end{align*}

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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Fricas [B]  time = 1.11362, size = 809, normalized size = 5.95 \begin{align*} \frac{60 \, b e^{6} n x^{6} \log \left (x\right ) - 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 \left (c\right )}{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 )}} \end{align*}

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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Sympy [A]  time = 125.053, size = 2018, normalized size = 14.84 \begin{align*} \text{result too large to display} \end{align*}

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]

Piecewise((zoo*(-a/x - b*n*log(x)/x - b*n/x - b*log(c)/x), Eq(d, 0) & Eq(e, 0)), ((a*x**6/6 + b*n*x**6*log(x)/
6 - b*n*x**6/36 + b*x**6*log(c)/6)/d**7, Eq(e, 0)), ((-a/x - b*n*log(x)/x - b*n/x - b*log(c)/x)/e**7, Eq(d, 0)
), (-60*a*d**6/(360*d**7*e**6 + 2160*d**6*e**7*x + 5400*d**5*e**8*x**2 + 7200*d**4*e**9*x**3 + 5400*d**3*e**10
*x**4 + 2160*d**2*e**11*x**5 + 360*d*e**12*x**6) - 360*a*d**5*e*x/(360*d**7*e**6 + 2160*d**6*e**7*x + 5400*d**
5*e**8*x**2 + 7200*d**4*e**9*x**3 + 5400*d**3*e**10*x**4 + 2160*d**2*e**11*x**5 + 360*d*e**12*x**6) - 900*a*d*
*4*e**2*x**2/(360*d**7*e**6 + 2160*d**6*e**7*x + 5400*d**5*e**8*x**2 + 7200*d**4*e**9*x**3 + 5400*d**3*e**10*x
**4 + 2160*d**2*e**11*x**5 + 360*d*e**12*x**6) - 1200*a*d**3*e**3*x**3/(360*d**7*e**6 + 2160*d**6*e**7*x + 540
0*d**5*e**8*x**2 + 7200*d**4*e**9*x**3 + 5400*d**3*e**10*x**4 + 2160*d**2*e**11*x**5 + 360*d*e**12*x**6) - 900
*a*d**2*e**4*x**4/(360*d**7*e**6 + 2160*d**6*e**7*x + 5400*d**5*e**8*x**2 + 7200*d**4*e**9*x**3 + 5400*d**3*e*
*10*x**4 + 2160*d**2*e**11*x**5 + 360*d*e**12*x**6) - 360*a*d*e**5*x**5/(360*d**7*e**6 + 2160*d**6*e**7*x + 54
00*d**5*e**8*x**2 + 7200*d**4*e**9*x**3 + 5400*d**3*e**10*x**4 + 2160*d**2*e**11*x**5 + 360*d*e**12*x**6) - 60
*b*d**6*n*log(d/e + x)/(360*d**7*e**6 + 2160*d**6*e**7*x + 5400*d**5*e**8*x**2 + 7200*d**4*e**9*x**3 + 5400*d*
*3*e**10*x**4 + 2160*d**2*e**11*x**5 + 360*d*e**12*x**6) - 137*b*d**6*n/(360*d**7*e**6 + 2160*d**6*e**7*x + 54
00*d**5*e**8*x**2 + 7200*d**4*e**9*x**3 + 5400*d**3*e**10*x**4 + 2160*d**2*e**11*x**5 + 360*d*e**12*x**6) - 36
0*b*d**5*e*n*x*log(d/e + x)/(360*d**7*e**6 + 2160*d**6*e**7*x + 5400*d**5*e**8*x**2 + 7200*d**4*e**9*x**3 + 54
00*d**3*e**10*x**4 + 2160*d**2*e**11*x**5 + 360*d*e**12*x**6) - 762*b*d**5*e*n*x/(360*d**7*e**6 + 2160*d**6*e*
*7*x + 5400*d**5*e**8*x**2 + 7200*d**4*e**9*x**3 + 5400*d**3*e**10*x**4 + 2160*d**2*e**11*x**5 + 360*d*e**12*x
**6) - 900*b*d**4*e**2*n*x**2*log(d/e + x)/(360*d**7*e**6 + 2160*d**6*e**7*x + 5400*d**5*e**8*x**2 + 7200*d**4
*e**9*x**3 + 5400*d**3*e**10*x**4 + 2160*d**2*e**11*x**5 + 360*d*e**12*x**6) - 1725*b*d**4*e**2*n*x**2/(360*d*
*7*e**6 + 2160*d**6*e**7*x + 5400*d**5*e**8*x**2 + 7200*d**4*e**9*x**3 + 5400*d**3*e**10*x**4 + 2160*d**2*e**1
1*x**5 + 360*d*e**12*x**6) - 1200*b*d**3*e**3*n*x**3*log(d/e + x)/(360*d**7*e**6 + 2160*d**6*e**7*x + 5400*d**
5*e**8*x**2 + 7200*d**4*e**9*x**3 + 5400*d**3*e**10*x**4 + 2160*d**2*e**11*x**5 + 360*d*e**12*x**6) - 2000*b*d
**3*e**3*n*x**3/(360*d**7*e**6 + 2160*d**6*e**7*x + 5400*d**5*e**8*x**2 + 7200*d**4*e**9*x**3 + 5400*d**3*e**1
0*x**4 + 2160*d**2*e**11*x**5 + 360*d*e**12*x**6) - 900*b*d**2*e**4*n*x**4*log(d/e + x)/(360*d**7*e**6 + 2160*
d**6*e**7*x + 5400*d**5*e**8*x**2 + 7200*d**4*e**9*x**3 + 5400*d**3*e**10*x**4 + 2160*d**2*e**11*x**5 + 360*d*
e**12*x**6) - 1200*b*d**2*e**4*n*x**4/(360*d**7*e**6 + 2160*d**6*e**7*x + 5400*d**5*e**8*x**2 + 7200*d**4*e**9
*x**3 + 5400*d**3*e**10*x**4 + 2160*d**2*e**11*x**5 + 360*d*e**12*x**6) - 360*b*d*e**5*n*x**5*log(d/e + x)/(36
0*d**7*e**6 + 2160*d**6*e**7*x + 5400*d**5*e**8*x**2 + 7200*d**4*e**9*x**3 + 5400*d**3*e**10*x**4 + 2160*d**2*
e**11*x**5 + 360*d*e**12*x**6) - 300*b*d*e**5*n*x**5/(360*d**7*e**6 + 2160*d**6*e**7*x + 5400*d**5*e**8*x**2 +
7200*d**4*e**9*x**3 + 5400*d**3*e**10*x**4 + 2160*d**2*e**11*x**5 + 360*d*e**12*x**6) + 60*b*e**6*n*x**6*log(
x)/(360*d**7*e**6 + 2160*d**6*e**7*x + 5400*d**5*e**8*x**2 + 7200*d**4*e**9*x**3 + 5400*d**3*e**10*x**4 + 2160
*d**2*e**11*x**5 + 360*d*e**12*x**6) - 60*b*e**6*n*x**6*log(d/e + x)/(360*d**7*e**6 + 2160*d**6*e**7*x + 5400*
d**5*e**8*x**2 + 7200*d**4*e**9*x**3 + 5400*d**3*e**10*x**4 + 2160*d**2*e**11*x**5 + 360*d*e**12*x**6) + 60*b*
e**6*x**6*log(c)/(360*d**7*e**6 + 2160*d**6*e**7*x + 5400*d**5*e**8*x**2 + 7200*d**4*e**9*x**3 + 5400*d**3*e**
10*x**4 + 2160*d**2*e**11*x**5 + 360*d*e**12*x**6), True))

________________________________________________________________________________________

Giac [B]  time = 1.38406, size = 524, normalized size = 3.85 \begin{align*} -\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 \left (x\right ) + 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 \left (c\right ) + 900 \, b d^{2} x^{4} e^{4} \log \left (c\right ) + 1200 \, b d^{3} x^{3} e^{3} \log \left (c\right ) + 900 \, b d^{4} x^{2} e^{2} \log \left (c\right ) + 360 \, b d^{5} x e \log \left (c\right ) + 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 \left (c\right ) + 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 )}} \end{align*}

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)