3.1.65 \(\int \frac {x^5 (a+b \log (c x^n))}{(d+e x)^7} \, dx\) [65]

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

[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 = 0.08, 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 = {2373, 45} \begin {gather*} \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} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/30*(b*d^4*n)/(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 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 2373

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] Leaf count is larger than twice the leaf count of optimal. \(335\) vs. \(2(136)=272\).
time = 0.20, size = 335, normalized size = 2.46 \begin {gather*} -\frac {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 \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)+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} \end {gather*}

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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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.16, size = 1165, normalized size = 8.57

method	result	size
risch	\(\text {Expression too large to display}\)	\(1165\)

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,method=_RETURNVERBOSE)

[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*l
n(c)*b*d^6-600*I*Pi*b*d^3*e^3*x^3*csgn(I*c)*csgn(I*c*x^n)^2-450*I*Pi*b*d^4*e^2*x^2*csgn(I*c)*csgn(I*c*x^n)^2-4
50*I*Pi*b*d^4*e^2*x^2*csgn(I*x^n)*csgn(I*c*x^n)^2-360*ln(e*x+d)*b*d*e^5*n*x^5-900*ln(e*x+d)*b*d^2*e^4*n*x^4-12
00*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^3-360*a*d*e^5*x^5-900*a*d^2*e^4*x^4-1200*a*d^3*e^3*x^3-90
0*a*d^4*e^2*x^2-360*a*d^5*e*x-137*b*d^6*n-600*I*Pi*b*d^3*e^3*x^3*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+900*ln(-x)*b*d^4*e^2*n*x^2+360*ln(-x)*b*d^5*e*n*x-3
0*I*Pi*b*d^6*csgn(I*x^n)*csgn(I*c*x^n)^2-180*I*Pi*b*d*e^5*x^5*csgn(I*x^n)*csgn(I*c*x^n)^2-450*I*Pi*b*d^2*e^4*x
^4*csgn(I*c)*csgn(I*c*x^n)^2+600*I*Pi*b*d^3*e^3*x^3*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+450*I*Pi*b*d^4*e^2*x^2
*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+180*I*Pi*b*d^5*e*x*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)-60*a*d^6-180*I*Pi*
b*d^5*e*x*csgn(I*c)*csgn(I*c*x^n)^2-180*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)*csgn(I*c*x^n)^2-60*ln(e*x+d)*b*d^6*n+60*ln(-x)*b*d^6*n-1725*b*d^4*e^2*n*x^2-762*b*d^5*e*n*x-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^3-450*I*Pi*b*d^2*e^4*x^4*csgn(I*x^n)*csgn(I*c*x^n)^2-30*I*Pi*b*d^6
*csgn(I*c)*csgn(I*c*x^n)^2-60*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+30*I*Pi*b*d^6*
csgn(I*c*x^n)^3-900*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^2-360*ln(c)*b*d^5*e*x+6
00*I*Pi*b*d^3*e^3*x^3*csgn(I*c*x^n)^3+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*c)*csgn(I*x^n)*csgn(I*c*x^n)+180*I*Pi*b*d*e^5*x^5*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+4
50*I*Pi*b*d^2*e^4*x^4*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n))/d/e^6/(e*x+d)^6

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 348 vs. \(2 (123) = 246\).
time = 0.30, size = 348, normalized size = 2.56 \begin {gather*} -\frac {1}{360} \, b n {\left (\frac {60 \, e^{\left (-6\right )} \log \left (x e + d\right )}{d} - \frac {60 \, e^{\left (-6\right )} \log \left (x\right )}{d} + \frac {300 \, x^{4} e^{4} + 900 \, d x^{3} e^{3} + 1100 \, d^{2} x^{2} e^{2} + 625 \, d^{3} x e + 137 \, d^{4}}{x^{5} e^{11} + 5 \, d x^{4} e^{10} + 10 \, d^{2} x^{3} e^{9} + 10 \, d^{3} x^{2} e^{8} + 5 \, d^{4} x e^{7} + d^{5} e^{6}}\right )} - \frac {{\left (6 \, x^{5} e^{5} + 15 \, d x^{4} e^{4} + 20 \, d^{2} x^{3} e^{3} + 15 \, d^{3} x^{2} e^{2} + 6 \, d^{4} x e + d^{5}\right )} b \log \left (c x^{n}\right )}{6 \, {\left (x^{6} e^{12} + 6 \, d x^{5} e^{11} + 15 \, d^{2} x^{4} e^{10} + 20 \, d^{3} x^{3} e^{9} + 15 \, d^{4} x^{2} e^{8} + 6 \, d^{5} x e^{7} + d^{6} e^{6}\right )}} - \frac {{\left (6 \, x^{5} e^{5} + 15 \, d x^{4} e^{4} + 20 \, d^{2} x^{3} e^{3} + 15 \, d^{3} x^{2} e^{2} + 6 \, d^{4} x e + d^{5}\right )} a}{6 \, {\left (x^{6} e^{12} + 6 \, d x^{5} e^{11} + 15 \, d^{2} x^{4} e^{10} + 20 \, d^{3} x^{3} e^{9} + 15 \, d^{4} x^{2} e^{8} + 6 \, d^{5} x e^{7} + d^{6} e^{6}\right )}} \end {gather*}

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 331 vs. \(2 (123) = 246\).
time = 0.36, size = 331, normalized size = 2.43 \begin {gather*} \frac {60 \, b n x^{6} e^{6} \log \left (x\right ) - 137 \, b d^{6} n - 60 \, a d^{6} - 60 \, {\left (5 \, b d n + 6 \, a d\right )} x^{5} e^{5} - 300 \, {\left (4 \, b d^{2} n + 3 \, a d^{2}\right )} x^{4} e^{4} - 400 \, {\left (5 \, b d^{3} n + 3 \, a d^{3}\right )} x^{3} e^{3} - 75 \, {\left (23 \, b d^{4} n + 12 \, a d^{4}\right )} x^{2} e^{2} - 6 \, {\left (127 \, b d^{5} n + 60 \, a d^{5}\right )} x e - 60 \, {\left (b n x^{6} e^{6} + 6 \, b d n x^{5} e^{5} + 15 \, b d^{2} n x^{4} e^{4} + 20 \, b d^{3} n x^{3} e^{3} + 15 \, b d^{4} n x^{2} e^{2} + 6 \, b d^{5} n x e + b d^{6} n\right )} \log \left (x e + d\right ) - 60 \, {\left (6 \, b d x^{5} e^{5} + 15 \, b d^{2} x^{4} e^{4} + 20 \, b d^{3} x^{3} e^{3} + 15 \, b d^{4} x^{2} e^{2} + 6 \, b d^{5} x e + b d^{6}\right )} \log \left (c\right )}{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 {gather*}

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*n*x^6*e^6*log(x) - 137*b*d^6*n - 60*a*d^6 - 60*(5*b*d*n + 6*a*d)*x^5*e^5 - 300*(4*b*d^2*n + 3*a*d^
2)*x^4*e^4 - 400*(5*b*d^3*n + 3*a*d^3)*x^3*e^3 - 75*(23*b*d^4*n + 12*a*d^4)*x^2*e^2 - 6*(127*b*d^5*n + 60*a*d^
5)*x*e - 60*(b*n*x^6*e^6 + 6*b*d*n*x^5*e^5 + 15*b*d^2*n*x^4*e^4 + 20*b*d^3*n*x^3*e^3 + 15*b*d^4*n*x^2*e^2 + 6*
b*d^5*n*x*e + b*d^6*n)*log(x*e + d) - 60*(6*b*d*x^5*e^5 + 15*b*d^2*x^4*e^4 + 20*b*d^3*x^3*e^3 + 15*b*d^4*x^2*e
^2 + 6*b*d^5*x*e + b*d^6)*log(c))/(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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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1911 vs. \(2 (133) = 266\).
time = 85.69, size = 1911, normalized size = 14.05 \begin {gather*} \begin {cases} \tilde {\infty } \left (- \frac {a}{x} - \frac {b n}{x} - \frac {b \log {\left (c x^{n} \right )}}{x}\right ) & \text {for}\: d = 0 \wedge e = 0 \\\frac {- \frac {a}{x} - \frac {b n}{x} - \frac {b \log {\left (c x^{n} \right )}}{x}}{e^{7}} & \text {for}\: d = 0 \\\frac {\frac {a x^{6}}{6} - \frac {b n x^{6}}{36} + \frac {b x^{6} \log {\left (c x^{n} \right )}}{6}}{d^{7}} & \text {for}\: e = 0 \\- \frac {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}} - \frac {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}} - \frac {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}} - \frac {1200 a d^{3} e^{3} 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^{10} x^{4} + 2160 d^{2} e^{11} x^{5} + 360 d e^{12} x^{6}} - \frac {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}} - \frac {360 a d e^{5} 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}} - \frac {60 b d^{6} n \log {\left (\frac {d}{e} + x \right )}}{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}} - \frac {137 b d^{6} n}{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}} - \frac {360 b d^{5} e n x \log {\left (\frac {d}{e} + x \right )}}{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}} - \frac {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}} - \frac {900 b d^{4} e^{2} n x^{2} \log {\left (\frac {d}{e} + x \right )}}{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}} - \frac {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^{11} x^{5} + 360 d e^{12} x^{6}} - \frac {1200 b d^{3} e^{3} n x^{3} \log {\left (\frac {d}{e} + x \right )}}{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}} - \frac {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^{10} x^{4} + 2160 d^{2} e^{11} x^{5} + 360 d e^{12} x^{6}} - \frac {900 b d^{2} e^{4} n x^{4} \log {\left (\frac {d}{e} + x \right )}}{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}} - \frac {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}} - \frac {360 b d e^{5} n x^{5} \log {\left (\frac {d}{e} + x \right )}}{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}} - \frac {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}} - \frac {60 b e^{6} n x^{6} \log {\left (\frac {d}{e} + x \right )}}{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}} + \frac {60 b e^{6} x^{6} \log {\left (c x^{n} \right )}}{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}} & \text {otherwise} \end {cases} \end {gather*}

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/x - b*log(c*x**n)/x), Eq(d, 0) & Eq(e, 0)), ((-a/x - b*n/x - b*log(c*x**n)/x)/e**7,
Eq(d, 0)), ((a*x**6/6 - b*n*x**6/36 + b*x**6*log(c*x**n)/6)/d**7, Eq(e, 0)), (-60*a*d**6/(360*d**7*e**6 + 216
0*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 + 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**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 + 3
60*d*e**12*x**6) - 360*a*d*e**5*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*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 + 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**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 + 5400*d**3*e**10*x**4 + 2160*d**2*e**1
1*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**11*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 + 21
60*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**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)/(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) - 30
0*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(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**1
2*x**6) + 60*b*e**6*x**6*log(c*x**n)/(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))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 388 vs. \(2 (123) = 246\).
time = 3.29, size = 388, normalized size = 2.85 \begin {gather*} -\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 {gather*}

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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Mupad [B]
time = 4.48, size = 341, normalized size = 2.51 \begin {gather*} -\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} \end {gather*}

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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