3.1.69 \(\int \frac {x (a+b \log (c x^n))}{(d+e x)^7} \, dx\) [69]

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

[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 = 0.09, 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 = {45, 2382, 12, 78} \begin {gather*} -\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} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/30*(b*n)/(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 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 78

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 2382

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(x_)^(m_.)*((d_) + (e_.)*(x_))^(q_), x_Symbol] :> With[{u = IntHide[
x^m*(d + e*x)^q, x]}, Dist[a + b*Log[c*x^n], u, x] - Dist[b*n, Int[SimplifyIntegrand[u/x, x], x], x]] /; FreeQ
[{a, b, c, d, e, n}, x] && ILtQ[m + q + 2, 0] && IGtQ[m, 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 = 0.10, size = 160, normalized size = 0.92 \begin {gather*} \frac {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 \left (c x^n\right )-72 b d^5 (d+e x) \log \left (c x^n\right )-12 b n (d+e x)^6 \log (d+e x)}{360 d^5 e^2 (d+e x)^6} \end {gather*}

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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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.14, size = 557, normalized size = 3.20

method	result	size
risch	\(-\frac {b \left (6 e x +d \right ) \ln \left (x^{n}\right )}{30 \left (e x +d \right )^{6} e^{2}}-\frac {12 \ln \left (c \right ) b \,d^{6}+72 \ln \left (e x +d \right ) b d \,e^{5} n \,x^{5}+180 \ln \left (e x +d \right ) b \,d^{2} e^{4} n \,x^{4}+240 \ln \left (e x +d \right ) b \,d^{3} e^{3} n \,x^{3}+180 \ln \left (e x +d \right ) b \,d^{4} e^{2} n \,x^{2}+72 \ln \left (e x +d \right ) b \,d^{5} e n x -72 \ln \left (-x \right ) b d \,e^{5} n \,x^{5}-180 \ln \left (-x \right ) b \,d^{2} e^{4} n \,x^{4}-240 \ln \left (-x \right ) b \,d^{3} e^{3} n \,x^{3}+6 i \pi b \,d^{6} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+72 a \,d^{5} e x -13 b \,d^{6} n +6 i \pi b \,d^{6} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-180 \ln \left (-x \right ) b \,d^{4} e^{2} n \,x^{2}-72 \ln \left (-x \right ) b \,d^{5} e n x +12 a \,d^{6}+12 \ln \left (e x +d \right ) b \,d^{6} n -12 \ln \left (-x \right ) b \,d^{6} n -171 b \,d^{4} e^{2} n \,x^{2}-90 b \,d^{5} e n x -12 b d \,e^{5} n \,x^{5}-66 b \,d^{2} e^{4} n \,x^{4}-148 b \,d^{3} e^{3} n \,x^{3}+36 i \pi b \,d^{5} e x \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+36 i \pi b \,d^{5} e x \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+12 \ln \left (e x +d \right ) b \,e^{6} n \,x^{6}-12 \ln \left (-x \right ) b \,e^{6} n \,x^{6}+72 \ln \left (c \right ) b \,d^{5} e x -6 i \pi b \,d^{6} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}-36 i \pi b \,d^{5} e x \mathrm {csgn}\left (i c \,x^{n}\right )^{3}-6 i \pi b \,d^{6} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )-36 i \pi b \,d^{5} e x \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{360 e^{2} d^{5} \left (e x +d \right )^{6}}\)	\(557\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.32, size = 273, normalized size = 1.57 \begin {gather*} \frac {1}{360} \, b n {\left (\frac {12 \, x^{4} e^{4} + 54 \, d x^{3} e^{3} + 94 \, d^{2} x^{2} e^{2} + 77 \, d^{3} x e + 13 \, d^{4}}{d^{4} x^{5} e^{7} + 5 \, d^{5} x^{4} e^{6} + 10 \, d^{6} x^{3} e^{5} + 10 \, d^{7} x^{2} e^{4} + 5 \, d^{8} x e^{3} + d^{9} e^{2}} - \frac {12 \, e^{\left (-2\right )} \log \left (x e + d\right )}{d^{5}} + \frac {12 \, e^{\left (-2\right )} \log \left (x\right )}{d^{5}}\right )} - \frac {{\left (6 \, x e + d\right )} b \log \left (c x^{n}\right )}{30 \, {\left (x^{6} e^{8} + 6 \, d x^{5} e^{7} + 15 \, d^{2} x^{4} e^{6} + 20 \, d^{3} x^{3} e^{5} + 15 \, d^{4} x^{2} e^{4} + 6 \, d^{5} x e^{3} + d^{6} e^{2}\right )}} - \frac {{\left (6 \, x e + d\right )} a}{30 \, {\left (x^{6} e^{8} + 6 \, d x^{5} e^{7} + 15 \, d^{2} x^{4} e^{6} + 20 \, d^{3} x^{3} e^{5} + 15 \, d^{4} x^{2} e^{4} + 6 \, d^{5} x e^{3} + d^{6} e^{2}\right )}} \end {gather*}

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

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Fricas [A]
time = 0.38, size = 305, normalized size = 1.75 \begin {gather*} \frac {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} + 13 \, b d^{6} n - 12 \, a d^{6} + 18 \, {\left (5 \, b d^{5} n - 4 \, a d^{5}\right )} x e - 12 \, {\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 ) - 12 \, {\left (6 \, b d^{5} x e + b d^{6}\right )} \log \left (c\right ) + 12 \, {\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}\right )} \log \left (x\right )}{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 )}} \end {gather*}

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*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 + 13*b*d^6*n - 12*a*d
^6 + 18*(5*b*d^5*n - 4*a*d^5)*x*e - 12*(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) - 12*(6*b*d^5*x*e + b*d^6)*log(c) + 12*(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)*log(x))/(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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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1992 vs. \(2 (168) = 336\).
time = 84.57, size = 1992, normalized size = 11.45 \begin {gather*} \begin {cases} \tilde {\infty } \left (- \frac {a}{5 x^{5}} - \frac {b n}{25 x^{5}} - \frac {b \log {\left (c x^{n} \right )}}{5 x^{5}}\right ) & \text {for}\: d = 0 \wedge e = 0 \\\frac {\frac {a x^{2}}{2} - \frac {b n x^{2}}{4} + \frac {b x^{2} \log {\left (c x^{n} \right )}}{2}}{d^{7}} & \text {for}\: e = 0 \\\frac {- \frac {a}{5 x^{5}} - \frac {b n}{25 x^{5}} - \frac {b \log {\left (c x^{n} \right )}}{5 x^{5}}}{e^{7}} & \text {for}\: d = 0 \\- \frac {12 a d^{6}}{360 d^{11} e^{2} + 2160 d^{10} e^{3} x + 5400 d^{9} e^{4} x^{2} + 7200 d^{8} e^{5} x^{3} + 5400 d^{7} e^{6} x^{4} + 2160 d^{6} e^{7} x^{5} + 360 d^{5} e^{8} x^{6}} - \frac {72 a d^{5} e x}{360 d^{11} e^{2} + 2160 d^{10} e^{3} x + 5400 d^{9} e^{4} x^{2} + 7200 d^{8} e^{5} x^{3} + 5400 d^{7} e^{6} x^{4} + 2160 d^{6} e^{7} x^{5} + 360 d^{5} e^{8} x^{6}} - \frac {12 b d^{6} n \log {\left (\frac {d}{e} + x \right )}}{360 d^{11} e^{2} + 2160 d^{10} e^{3} x + 5400 d^{9} e^{4} x^{2} + 7200 d^{8} e^{5} x^{3} + 5400 d^{7} e^{6} x^{4} + 2160 d^{6} e^{7} x^{5} + 360 d^{5} e^{8} x^{6}} + \frac {13 b d^{6} n}{360 d^{11} e^{2} + 2160 d^{10} e^{3} x + 5400 d^{9} e^{4} x^{2} + 7200 d^{8} e^{5} x^{3} + 5400 d^{7} e^{6} x^{4} + 2160 d^{6} e^{7} x^{5} + 360 d^{5} e^{8} x^{6}} - \frac {72 b d^{5} e n x \log {\left (\frac {d}{e} + x \right )}}{360 d^{11} e^{2} + 2160 d^{10} e^{3} x + 5400 d^{9} e^{4} x^{2} + 7200 d^{8} e^{5} x^{3} + 5400 d^{7} e^{6} x^{4} + 2160 d^{6} e^{7} x^{5} + 360 d^{5} e^{8} x^{6}} + \frac {90 b d^{5} e n x}{360 d^{11} e^{2} + 2160 d^{10} e^{3} x + 5400 d^{9} e^{4} x^{2} + 7200 d^{8} e^{5} x^{3} + 5400 d^{7} e^{6} x^{4} + 2160 d^{6} e^{7} x^{5} + 360 d^{5} e^{8} x^{6}} - \frac {180 b d^{4} e^{2} n x^{2} \log {\left (\frac {d}{e} + x \right )}}{360 d^{11} e^{2} + 2160 d^{10} e^{3} x + 5400 d^{9} e^{4} x^{2} + 7200 d^{8} e^{5} x^{3} + 5400 d^{7} e^{6} x^{4} + 2160 d^{6} e^{7} x^{5} + 360 d^{5} e^{8} x^{6}} + \frac {171 b d^{4} e^{2} n x^{2}}{360 d^{11} e^{2} + 2160 d^{10} e^{3} x + 5400 d^{9} e^{4} x^{2} + 7200 d^{8} e^{5} x^{3} + 5400 d^{7} e^{6} x^{4} + 2160 d^{6} e^{7} x^{5} + 360 d^{5} e^{8} x^{6}} + \frac {180 b d^{4} e^{2} x^{2} \log {\left (c x^{n} \right )}}{360 d^{11} e^{2} + 2160 d^{10} e^{3} x + 5400 d^{9} e^{4} x^{2} + 7200 d^{8} e^{5} x^{3} + 5400 d^{7} e^{6} x^{4} + 2160 d^{6} e^{7} x^{5} + 360 d^{5} e^{8} x^{6}} - \frac {240 b d^{3} e^{3} n x^{3} \log {\left (\frac {d}{e} + x \right )}}{360 d^{11} e^{2} + 2160 d^{10} e^{3} x + 5400 d^{9} e^{4} x^{2} + 7200 d^{8} e^{5} x^{3} + 5400 d^{7} e^{6} x^{4} + 2160 d^{6} e^{7} x^{5} + 360 d^{5} e^{8} x^{6}} + \frac {148 b d^{3} e^{3} n x^{3}}{360 d^{11} e^{2} + 2160 d^{10} e^{3} x + 5400 d^{9} e^{4} x^{2} + 7200 d^{8} e^{5} x^{3} + 5400 d^{7} e^{6} x^{4} + 2160 d^{6} e^{7} x^{5} + 360 d^{5} e^{8} x^{6}} + \frac {240 b d^{3} e^{3} x^{3} \log {\left (c x^{n} \right )}}{360 d^{11} e^{2} + 2160 d^{10} e^{3} x + 5400 d^{9} e^{4} x^{2} + 7200 d^{8} e^{5} x^{3} + 5400 d^{7} e^{6} x^{4} + 2160 d^{6} e^{7} x^{5} + 360 d^{5} e^{8} x^{6}} - \frac {180 b d^{2} e^{4} n x^{4} \log {\left (\frac {d}{e} + x \right )}}{360 d^{11} e^{2} + 2160 d^{10} e^{3} x + 5400 d^{9} e^{4} x^{2} + 7200 d^{8} e^{5} x^{3} + 5400 d^{7} e^{6} x^{4} + 2160 d^{6} e^{7} x^{5} + 360 d^{5} e^{8} x^{6}} + \frac {66 b d^{2} e^{4} n x^{4}}{360 d^{11} e^{2} + 2160 d^{10} e^{3} x + 5400 d^{9} e^{4} x^{2} + 7200 d^{8} e^{5} x^{3} + 5400 d^{7} e^{6} x^{4} + 2160 d^{6} e^{7} x^{5} + 360 d^{5} e^{8} x^{6}} + \frac {180 b d^{2} e^{4} x^{4} \log {\left (c x^{n} \right )}}{360 d^{11} e^{2} + 2160 d^{10} e^{3} x + 5400 d^{9} e^{4} x^{2} + 7200 d^{8} e^{5} x^{3} + 5400 d^{7} e^{6} x^{4} + 2160 d^{6} e^{7} x^{5} + 360 d^{5} e^{8} x^{6}} - \frac {72 b d e^{5} n x^{5} \log {\left (\frac {d}{e} + x \right )}}{360 d^{11} e^{2} + 2160 d^{10} e^{3} x + 5400 d^{9} e^{4} x^{2} + 7200 d^{8} e^{5} x^{3} + 5400 d^{7} e^{6} x^{4} + 2160 d^{6} e^{7} x^{5} + 360 d^{5} e^{8} x^{6}} + \frac {12 b d e^{5} n x^{5}}{360 d^{11} e^{2} + 2160 d^{10} e^{3} x + 5400 d^{9} e^{4} x^{2} + 7200 d^{8} e^{5} x^{3} + 5400 d^{7} e^{6} x^{4} + 2160 d^{6} e^{7} x^{5} + 360 d^{5} e^{8} x^{6}} + \frac {72 b d e^{5} x^{5} \log {\left (c x^{n} \right )}}{360 d^{11} e^{2} + 2160 d^{10} e^{3} x + 5400 d^{9} e^{4} x^{2} + 7200 d^{8} e^{5} x^{3} + 5400 d^{7} e^{6} x^{4} + 2160 d^{6} e^{7} x^{5} + 360 d^{5} e^{8} x^{6}} - \frac {12 b e^{6} n x^{6} \log {\left (\frac {d}{e} + x \right )}}{360 d^{11} e^{2} + 2160 d^{10} e^{3} x + 5400 d^{9} e^{4} x^{2} + 7200 d^{8} e^{5} x^{3} + 5400 d^{7} e^{6} x^{4} + 2160 d^{6} e^{7} x^{5} + 360 d^{5} e^{8} x^{6}} + \frac {12 b e^{6} x^{6} \log {\left (c x^{n} \right )}}{360 d^{11} e^{2} + 2160 d^{10} e^{3} x + 5400 d^{9} e^{4} x^{2} + 7200 d^{8} e^{5} x^{3} + 5400 d^{7} e^{6} x^{4} + 2160 d^{6} e^{7} x^{5} + 360 d^{5} e^{8} x^{6}} & \text {otherwise} \end {cases} \end {gather*}

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]

Piecewise((zoo*(-a/(5*x**5) - b*n/(25*x**5) - b*log(c*x**n)/(5*x**5)), Eq(d, 0) & Eq(e, 0)), ((a*x**2/2 - b*n*
x**2/4 + b*x**2*log(c*x**n)/2)/d**7, Eq(e, 0)), ((-a/(5*x**5) - b*n/(25*x**5) - b*log(c*x**n)/(5*x**5))/e**7,
Eq(d, 0)), (-12*a*d**6/(360*d**11*e**2 + 2160*d**10*e**3*x + 5400*d**9*e**4*x**2 + 7200*d**8*e**5*x**3 + 5400*
d**7*e**6*x**4 + 2160*d**6*e**7*x**5 + 360*d**5*e**8*x**6) - 72*a*d**5*e*x/(360*d**11*e**2 + 2160*d**10*e**3*x
+ 5400*d**9*e**4*x**2 + 7200*d**8*e**5*x**3 + 5400*d**7*e**6*x**4 + 2160*d**6*e**7*x**5 + 360*d**5*e**8*x**6)
- 12*b*d**6*n*log(d/e + x)/(360*d**11*e**2 + 2160*d**10*e**3*x + 5400*d**9*e**4*x**2 + 7200*d**8*e**5*x**3 +
5400*d**7*e**6*x**4 + 2160*d**6*e**7*x**5 + 360*d**5*e**8*x**6) + 13*b*d**6*n/(360*d**11*e**2 + 2160*d**10*e**
3*x + 5400*d**9*e**4*x**2 + 7200*d**8*e**5*x**3 + 5400*d**7*e**6*x**4 + 2160*d**6*e**7*x**5 + 360*d**5*e**8*x*
*6) - 72*b*d**5*e*n*x*log(d/e + x)/(360*d**11*e**2 + 2160*d**10*e**3*x + 5400*d**9*e**4*x**2 + 7200*d**8*e**5*
x**3 + 5400*d**7*e**6*x**4 + 2160*d**6*e**7*x**5 + 360*d**5*e**8*x**6) + 90*b*d**5*e*n*x/(360*d**11*e**2 + 216
0*d**10*e**3*x + 5400*d**9*e**4*x**2 + 7200*d**8*e**5*x**3 + 5400*d**7*e**6*x**4 + 2160*d**6*e**7*x**5 + 360*d
**5*e**8*x**6) - 180*b*d**4*e**2*n*x**2*log(d/e + x)/(360*d**11*e**2 + 2160*d**10*e**3*x + 5400*d**9*e**4*x**2
+ 7200*d**8*e**5*x**3 + 5400*d**7*e**6*x**4 + 2160*d**6*e**7*x**5 + 360*d**5*e**8*x**6) + 171*b*d**4*e**2*n*x
**2/(360*d**11*e**2 + 2160*d**10*e**3*x + 5400*d**9*e**4*x**2 + 7200*d**8*e**5*x**3 + 5400*d**7*e**6*x**4 + 21
60*d**6*e**7*x**5 + 360*d**5*e**8*x**6) + 180*b*d**4*e**2*x**2*log(c*x**n)/(360*d**11*e**2 + 2160*d**10*e**3*x
+ 5400*d**9*e**4*x**2 + 7200*d**8*e**5*x**3 + 5400*d**7*e**6*x**4 + 2160*d**6*e**7*x**5 + 360*d**5*e**8*x**6)
- 240*b*d**3*e**3*n*x**3*log(d/e + x)/(360*d**11*e**2 + 2160*d**10*e**3*x + 5400*d**9*e**4*x**2 + 7200*d**8*e
**5*x**3 + 5400*d**7*e**6*x**4 + 2160*d**6*e**7*x**5 + 360*d**5*e**8*x**6) + 148*b*d**3*e**3*n*x**3/(360*d**11
*e**2 + 2160*d**10*e**3*x + 5400*d**9*e**4*x**2 + 7200*d**8*e**5*x**3 + 5400*d**7*e**6*x**4 + 2160*d**6*e**7*x
**5 + 360*d**5*e**8*x**6) + 240*b*d**3*e**3*x**3*log(c*x**n)/(360*d**11*e**2 + 2160*d**10*e**3*x + 5400*d**9*e
**4*x**2 + 7200*d**8*e**5*x**3 + 5400*d**7*e**6*x**4 + 2160*d**6*e**7*x**5 + 360*d**5*e**8*x**6) - 180*b*d**2*
e**4*n*x**4*log(d/e + x)/(360*d**11*e**2 + 2160*d**10*e**3*x + 5400*d**9*e**4*x**2 + 7200*d**8*e**5*x**3 + 540
0*d**7*e**6*x**4 + 2160*d**6*e**7*x**5 + 360*d**5*e**8*x**6) + 66*b*d**2*e**4*n*x**4/(360*d**11*e**2 + 2160*d*
*10*e**3*x + 5400*d**9*e**4*x**2 + 7200*d**8*e**5*x**3 + 5400*d**7*e**6*x**4 + 2160*d**6*e**7*x**5 + 360*d**5*
e**8*x**6) + 180*b*d**2*e**4*x**4*log(c*x**n)/(360*d**11*e**2 + 2160*d**10*e**3*x + 5400*d**9*e**4*x**2 + 7200
*d**8*e**5*x**3 + 5400*d**7*e**6*x**4 + 2160*d**6*e**7*x**5 + 360*d**5*e**8*x**6) - 72*b*d*e**5*n*x**5*log(d/e
+ x)/(360*d**11*e**2 + 2160*d**10*e**3*x + 5400*d**9*e**4*x**2 + 7200*d**8*e**5*x**3 + 5400*d**7*e**6*x**4 +
2160*d**6*e**7*x**5 + 360*d**5*e**8*x**6) + 12*b*d*e**5*n*x**5/(360*d**11*e**2 + 2160*d**10*e**3*x + 5400*d**9
*e**4*x**2 + 7200*d**8*e**5*x**3 + 5400*d**7*e**6*x**4 + 2160*d**6*e**7*x**5 + 360*d**5*e**8*x**6) + 72*b*d*e*
*5*x**5*log(c*x**n)/(360*d**11*e**2 + 2160*d**10*e**3*x + 5400*d**9*e**4*x**2 + 7200*d**8*e**5*x**3 + 5400*d**
7*e**6*x**4 + 2160*d**6*e**7*x**5 + 360*d**5*e**8*x**6) - 12*b*e**6*n*x**6*log(d/e + x)/(360*d**11*e**2 + 2160
*d**10*e**3*x + 5400*d**9*e**4*x**2 + 7200*d**8*e**5*x**3 + 5400*d**7*e**6*x**4 + 2160*d**6*e**7*x**5 + 360*d*
*5*e**8*x**6) + 12*b*e**6*x**6*log(c*x**n)/(360*d**11*e**2 + 2160*d**10*e**3*x + 5400*d**9*e**4*x**2 + 7200*d*
*8*e**5*x**3 + 5400*d**7*e**6*x**4 + 2160*d**6*e**7*x**5 + 360*d**5*e**8*x**6), True))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 352 vs. \(2 (155) = 310\).
time = 4.31, size = 352, normalized size = 2.02 \begin {gather*} -\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 \left (x\right ) - 72 \, b d n x^{5} e^{5} \log \left (x\right ) - 180 \, b d^{2} n x^{4} e^{4} \log \left (x\right ) - 240 \, b d^{3} n x^{3} e^{3} \log \left (x\right ) - 180 \, b d^{4} n x^{2} e^{2} \log \left (x\right ) - 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 \left (c\right ) - 13 \, b d^{6} n + 72 \, a d^{5} x e + 12 \, b d^{6} \log \left (c\right ) + 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 )}} \end {gather*}

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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Mupad [B]
time = 4.04, size = 251, normalized size = 1.44 \begin {gather*} \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} \end {gather*}

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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