3.1.66 \(\int \frac {x^4 (a+b \log (c x^n))}{(d+e x)^7} \, dx\) [66]

Optimal. Leaf size=163 \[ -\frac {b n x^5}{30 d^2 (d+e x)^5}+\frac {b d^2 n}{120 e^5 (d+e x)^4}-\frac {2 b d n}{45 e^5 (d+e x)^3}+\frac {b n}{10 e^5 (d+e x)^2}-\frac {2 b n}{15 d e^5 (d+e x)}+\frac {x^5 \left (a+b \log \left (c x^n\right )\right )}{6 d (d+e x)^6}+\frac {x^5 \left (a+b \log \left (c x^n\right )\right )}{30 d^2 (d+e x)^5}-\frac {b n \log (d+e x)}{30 d^2 e^5} \]

[Out]

-1/30*b*n*x^5/d^2/(e*x+d)^5+1/120*b*d^2*n/e^5/(e*x+d)^4-2/45*b*d*n/e^5/(e*x+d)^3+1/10*b*n/e^5/(e*x+d)^2-2/15*b
*n/d/e^5/(e*x+d)+1/6*x^5*(a+b*ln(c*x^n))/d/(e*x+d)^6+1/30*x^5*(a+b*ln(c*x^n))/d^2/(e*x+d)^5-1/30*b*n*ln(e*x+d)
/d^2/e^5

________________________________________________________________________________________

Rubi [A]
time = 0.09, antiderivative size = 163, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {47, 37, 2382, 12, 79, 45} \begin {gather*} \frac {x^5 \left (a+b \log \left (c x^n\right )\right )}{30 d^2 (d+e x)^5}+\frac {x^5 \left (a+b \log \left (c x^n\right )\right )}{6 d (d+e x)^6}+\frac {b d^2 n}{120 e^5 (d+e x)^4}-\frac {b n \log (d+e x)}{30 d^2 e^5}-\frac {b n x^5}{30 d^2 (d+e x)^5}-\frac {2 b n}{15 d e^5 (d+e x)}+\frac {b n}{10 e^5 (d+e x)^2}-\frac {2 b d n}{45 e^5 (d+e x)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x^4*(a + b*Log[c*x^n]))/(d + e*x)^7,x]

[Out]

-1/30*(b*n*x^5)/(d^2*(d + e*x)^5) + (b*d^2*n)/(120*e^5*(d + e*x)^4) - (2*b*d*n)/(45*e^5*(d + e*x)^3) + (b*n)/(
10*e^5*(d + e*x)^2) - (2*b*n)/(15*d*e^5*(d + e*x)) + (x^5*(a + b*Log[c*x^n]))/(6*d*(d + e*x)^6) + (x^5*(a + b*
Log[c*x^n]))/(30*d^2*(d + e*x)^5) - (b*n*Log[d + e*x])/(30*d^2*e^5)

Rule 12

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

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n +
1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

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 47

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n + 1
)/((b*c - a*d)*(m + 1))), x] - Dist[d*(Simplify[m + n + 2]/((b*c - a*d)*(m + 1))), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

Rule 79

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(-(b*e - a*f
))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1
) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e,
f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || L
tQ[p, n]))))

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^4 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^7} \, dx &=\frac {x^5 \left (a+b \log \left (c x^n\right )\right )}{6 d (d+e x)^6}+\frac {x^5 \left (a+b \log \left (c x^n\right )\right )}{30 d^2 (d+e x)^5}-(b n) \int \frac {x^4 (6 d+e x)}{30 d^2 (d+e x)^6} \, dx\\ &=\frac {x^5 \left (a+b \log \left (c x^n\right )\right )}{6 d (d+e x)^6}+\frac {x^5 \left (a+b \log \left (c x^n\right )\right )}{30 d^2 (d+e x)^5}-\frac {(b n) \int \frac {x^4 (6 d+e x)}{(d+e x)^6} \, dx}{30 d^2}\\ &=-\frac {b n x^5}{30 d^2 (d+e x)^5}+\frac {x^5 \left (a+b \log \left (c x^n\right )\right )}{6 d (d+e x)^6}+\frac {x^5 \left (a+b \log \left (c x^n\right )\right )}{30 d^2 (d+e x)^5}-\frac {(b n) \int \frac {x^4}{(d+e x)^5} \, dx}{30 d^2}\\ &=-\frac {b n x^5}{30 d^2 (d+e x)^5}+\frac {x^5 \left (a+b \log \left (c x^n\right )\right )}{6 d (d+e x)^6}+\frac {x^5 \left (a+b \log \left (c x^n\right )\right )}{30 d^2 (d+e x)^5}-\frac {(b n) \int \left (\frac {d^4}{e^4 (d+e x)^5}-\frac {4 d^3}{e^4 (d+e x)^4}+\frac {6 d^2}{e^4 (d+e x)^3}-\frac {4 d}{e^4 (d+e x)^2}+\frac {1}{e^4 (d+e x)}\right ) \, dx}{30 d^2}\\ &=-\frac {b n x^5}{30 d^2 (d+e x)^5}+\frac {b d^2 n}{120 e^5 (d+e x)^4}-\frac {2 b d n}{45 e^5 (d+e x)^3}+\frac {b n}{10 e^5 (d+e x)^2}-\frac {2 b n}{15 d e^5 (d+e x)}+\frac {x^5 \left (a+b \log \left (c x^n\right )\right )}{6 d (d+e x)^6}+\frac {x^5 \left (a+b \log \left (c x^n\right )\right )}{30 d^2 (d+e x)^5}-\frac {b n \log (d+e x)}{30 d^2 e^5}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.19, size = 316, normalized size = 1.94 \begin {gather*} -\frac {12 a d^6+13 b d^6 n+72 a d^5 e x+66 b d^5 e n x+180 a d^4 e^2 x^2+129 b d^4 e^2 n x^2+240 a d^3 e^3 x^3+112 b d^3 e^3 n x^3+180 a d^2 e^4 x^4+24 b d^2 e^4 n x^4-12 b d e^5 n x^5-12 b n (d+e x)^6 \log (x)+12 b d^2 \left (d^4+6 d^3 e x+15 d^2 e^2 x^2+20 d e^3 x^3+15 e^4 x^4\right ) \log \left (c x^n\right )+12 b d^6 n \log (d+e x)+72 b d^5 e n x \log (d+e x)+180 b d^4 e^2 n x^2 \log (d+e x)+240 b d^3 e^3 n x^3 \log (d+e x)+180 b d^2 e^4 n x^4 \log (d+e x)+72 b d e^5 n x^5 \log (d+e x)+12 b e^6 n x^6 \log (d+e x)}{360 d^2 e^5 (d+e x)^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x^4*(a + b*Log[c*x^n]))/(d + e*x)^7,x]

[Out]

-1/360*(12*a*d^6 + 13*b*d^6*n + 72*a*d^5*e*x + 66*b*d^5*e*n*x + 180*a*d^4*e^2*x^2 + 129*b*d^4*e^2*n*x^2 + 240*
a*d^3*e^3*x^3 + 112*b*d^3*e^3*n*x^3 + 180*a*d^2*e^4*x^4 + 24*b*d^2*e^4*n*x^4 - 12*b*d*e^5*n*x^5 - 12*b*n*(d +
e*x)^6*Log[x] + 12*b*d^2*(d^4 + 6*d^3*e*x + 15*d^2*e^2*x^2 + 20*d*e^3*x^3 + 15*e^4*x^4)*Log[c*x^n] + 12*b*d^6*
n*Log[d + e*x] + 72*b*d^5*e*n*x*Log[d + e*x] + 180*b*d^4*e^2*n*x^2*Log[d + e*x] + 240*b*d^3*e^3*n*x^3*Log[d +
e*x] + 180*b*d^2*e^4*n*x^4*Log[d + e*x] + 72*b*d*e^5*n*x^5*Log[d + e*x] + 12*b*e^6*n*x^6*Log[d + e*x])/(d^2*e^
5*(d + e*x)^6)

________________________________________________________________________________________

Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.15, size = 1022, normalized size = 6.27

method	result	size
risch	\(-\frac {b \left (15 x^{4} e^{4}+20 d \,x^{3} e^{3}+15 d^{2} x^{2} e^{2}+6 d^{3} x e +d^{4}\right ) \ln \left (x^{n}\right )}{30 \left (e x +d \right )^{6} e^{5}}+\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}-180 a \,d^{2} e^{4} x^{4}-240 a \,d^{3} e^{3} x^{3}-180 a \,d^{4} e^{2} x^{2}-72 a \,d^{5} e x -13 b \,d^{6} n -120 i \pi b \,d^{3} e^{3} x^{3} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-90 i \pi b \,d^{4} e^{2} x^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-90 i \pi b \,d^{4} e^{2} x^{2} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-120 i \pi b \,d^{3} e^{3} x^{3} \mathrm {csgn}\left (i x^{n}\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}+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 -129 b \,d^{4} e^{2} n \,x^{2}-66 b \,d^{5} e n x +12 b d \,e^{5} n \,x^{5}-24 b \,d^{2} e^{4} n \,x^{4}-112 b \,d^{3} e^{3} n \,x^{3}-90 i \pi b \,d^{2} e^{4} x^{4} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-90 i \pi b \,d^{2} e^{4} x^{4} \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 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}-180 \ln \left (c \right ) b \,d^{2} e^{4} x^{4}-240 \ln \left (c \right ) b \,d^{3} e^{3} x^{3}-180 \ln \left (c \right ) b \,d^{4} e^{2} x^{2}-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}+90 i \pi b \,d^{2} e^{4} x^{4} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+120 i \pi b \,d^{3} e^{3} x^{3} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+90 i \pi b \,d^{4} e^{2} x^{2} \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 )+120 i \pi b \,d^{3} e^{3} x^{3} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )+90 i \pi b \,d^{4} e^{2} x^{2} \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 )+90 i \pi b \,d^{2} e^{4} x^{4} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{360 d^{2} e^{5} \left (e x +d \right )^{6}}\)	\(1022\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(a+b*ln(c*x^n))/(e*x+d)^7,x,method=_RETURNVERBOSE)

[Out]

-1/30*b*(15*e^4*x^4+20*d*e^3*x^3+15*d^2*e^2*x^2+6*d^3*e*x+d^4)/(e*x+d)^6/e^5*ln(x^n)+1/360*(-12*ln(c)*b*d^6-12
0*I*Pi*b*d^3*e^3*x^3*csgn(I*c)*csgn(I*c*x^n)^2-36*I*Pi*b*d^5*e*x*csgn(I*x^n)*csgn(I*c*x^n)^2-120*I*Pi*b*d^3*e^
3*x^3*csgn(I*x^n)*csgn(I*c*x^n)^2-6*I*Pi*b*d^6*csgn(I*x^n)*csgn(I*c*x^n)^2-72*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+90*I*Pi*b*d^2*e^4*x^4*csgn(I*c*x^n)
^3+120*I*Pi*b*d^3*e^3*x^3*csgn(I*c*x^n)^3+90*I*Pi*b*d^4*e^2*x^2*csgn(I*c*x^n)^3-90*I*Pi*b*d^2*e^4*x^4*csgn(I*x
^n)*csgn(I*c*x^n)^2-180*a*d^2*e^4*x^4-240*a*d^3*e^3*x^3-180*a*d^4*e^2*x^2-72*a*d^5*e*x-13*b*d^6*n-6*I*Pi*b*d^6
*csgn(I*c)*csgn(I*c*x^n)^2+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)+
180*ln(-x)*b*d^4*e^2*n*x^2+72*ln(-x)*b*d^5*e*n*x-36*I*Pi*b*d^5*e*x*csgn(I*c)*csgn(I*c*x^n)^2-90*I*Pi*b*d^4*e^2
*x^2*csgn(I*c)*csgn(I*c*x^n)^2-90*I*Pi*b*d^4*e^2*x^2*csgn(I*x^n)*csgn(I*c*x^n)^2-90*I*Pi*b*d^2*e^4*x^4*csgn(I*
c)*csgn(I*c*x^n)^2-12*a*d^6-12*ln(e*x+d)*b*d^6*n+12*ln(-x)*b*d^6*n-129*b*d^4*e^2*n*x^2-66*b*d^5*e*n*x+12*b*d*e
^5*n*x^5-24*b*d^2*e^4*n*x^4-112*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+90*I*Pi*b*d^2*e
^4*x^4*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)-180*ln(c)*b*d^2*e^4*x^4-240*ln(c)*b*d^3*e^3*x^3-180*ln(c)*b*d^4*e^2
*x^2-72*ln(c)*b*d^5*e*x+6*I*Pi*b*d^6*csgn(I*c*x^n)^3+36*I*Pi*b*d^5*e*x*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+120
*I*Pi*b*d^3*e^3*x^3*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+90*I*Pi*b*d^4*e^2*x^2*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x
^n))/d^2/e^5/(e*x+d)^6

________________________________________________________________________________________

Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 331 vs. \(2 (150) = 300\).
time = 0.31, size = 331, normalized size = 2.03 \begin {gather*} \frac {1}{360} \, b n {\left (\frac {12 \, x^{4} e^{4} - 36 \, d x^{3} e^{3} - 76 \, d^{2} x^{2} e^{2} - 53 \, d^{3} x e - 13 \, d^{4}}{d x^{5} e^{10} + 5 \, d^{2} x^{4} e^{9} + 10 \, d^{3} x^{3} e^{8} + 10 \, d^{4} x^{2} e^{7} + 5 \, d^{5} x e^{6} + d^{6} e^{5}} - \frac {12 \, e^{\left (-5\right )} \log \left (x e + d\right )}{d^{2}} + \frac {12 \, e^{\left (-5\right )} \log \left (x\right )}{d^{2}}\right )} - \frac {{\left (15 \, x^{4} e^{4} + 20 \, d x^{3} e^{3} + 15 \, d^{2} x^{2} e^{2} + 6 \, d^{3} x e + d^{4}\right )} b \log \left (c x^{n}\right )}{30 \, {\left (x^{6} e^{11} + 6 \, d x^{5} e^{10} + 15 \, d^{2} x^{4} e^{9} + 20 \, d^{3} x^{3} e^{8} + 15 \, d^{4} x^{2} e^{7} + 6 \, d^{5} x e^{6} + d^{6} e^{5}\right )}} - \frac {{\left (15 \, x^{4} e^{4} + 20 \, d x^{3} e^{3} + 15 \, d^{2} x^{2} e^{2} + 6 \, d^{3} x e + d^{4}\right )} a}{30 \, {\left (x^{6} e^{11} + 6 \, d x^{5} e^{10} + 15 \, d^{2} x^{4} e^{9} + 20 \, d^{3} x^{3} e^{8} + 15 \, d^{4} x^{2} e^{7} + 6 \, d^{5} x e^{6} + d^{6} e^{5}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(a+b*log(c*x^n))/(e*x+d)^7,x, algorithm="maxima")

[Out]

1/360*b*n*((12*x^4*e^4 - 36*d*x^3*e^3 - 76*d^2*x^2*e^2 - 53*d^3*x*e - 13*d^4)/(d*x^5*e^10 + 5*d^2*x^4*e^9 + 10
*d^3*x^3*e^8 + 10*d^4*x^2*e^7 + 5*d^5*x*e^6 + d^6*e^5) - 12*e^(-5)*log(x*e + d)/d^2 + 12*e^(-5)*log(x)/d^2) -
1/30*(15*x^4*e^4 + 20*d*x^3*e^3 + 15*d^2*x^2*e^2 + 6*d^3*x*e + d^4)*b*log(c*x^n)/(x^6*e^11 + 6*d*x^5*e^10 + 15
*d^2*x^4*e^9 + 20*d^3*x^3*e^8 + 15*d^4*x^2*e^7 + 6*d^5*x*e^6 + d^6*e^5) - 1/30*(15*x^4*e^4 + 20*d*x^3*e^3 + 15
*d^2*x^2*e^2 + 6*d^3*x*e + d^4)*a/(x^6*e^11 + 6*d*x^5*e^10 + 15*d^2*x^4*e^9 + 20*d^3*x^3*e^8 + 15*d^4*x^2*e^7
+ 6*d^5*x*e^6 + d^6*e^5)

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 329 vs. \(2 (150) = 300\).
time = 0.36, size = 329, normalized size = 2.02 \begin {gather*} \frac {12 \, b d n x^{5} e^{5} - 13 \, b d^{6} n - 12 \, a d^{6} - 12 \, {\left (2 \, b d^{2} n + 15 \, a d^{2}\right )} x^{4} e^{4} - 16 \, {\left (7 \, b d^{3} n + 15 \, a d^{3}\right )} x^{3} e^{3} - 3 \, {\left (43 \, b d^{4} n + 60 \, a d^{4}\right )} x^{2} e^{2} - 6 \, {\left (11 \, b d^{5} n + 12 \, 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 (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 ) + 12 \, {\left (b n x^{6} e^{6} + 6 \, b d n x^{5} e^{5}\right )} \log \left (x\right )}{360 \, {\left (d^{2} x^{6} e^{11} + 6 \, d^{3} x^{5} e^{10} + 15 \, d^{4} x^{4} e^{9} + 20 \, d^{5} x^{3} e^{8} + 15 \, d^{6} x^{2} e^{7} + 6 \, d^{7} x e^{6} + d^{8} e^{5}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(a+b*log(c*x^n))/(e*x+d)^7,x, algorithm="fricas")

[Out]

1/360*(12*b*d*n*x^5*e^5 - 13*b*d^6*n - 12*a*d^6 - 12*(2*b*d^2*n + 15*a*d^2)*x^4*e^4 - 16*(7*b*d^3*n + 15*a*d^3
)*x^3*e^3 - 3*(43*b*d^4*n + 60*a*d^4)*x^2*e^2 - 6*(11*b*d^5*n + 12*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) - 1
2*(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) + 12*(b*n*x^6*e^6 + 6*
b*d*n*x^5*e^5)*log(x))/(d^2*x^6*e^11 + 6*d^3*x^5*e^10 + 15*d^4*x^4*e^9 + 20*d^5*x^3*e^8 + 15*d^6*x^2*e^7 + 6*d
^7*x*e^6 + d^8*e^5)

________________________________________________________________________________________

Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1972 vs. \(2 (155) = 310\).
time = 84.83, size = 1972, normalized size = 12.10 \begin {gather*} \begin {cases} \tilde {\infty } \left (- \frac {a}{2 x^{2}} - \frac {b n}{4 x^{2}} - \frac {b \log {\left (c x^{n} \right )}}{2 x^{2}}\right ) & \text {for}\: d = 0 \wedge e = 0 \\\frac {- \frac {a}{2 x^{2}} - \frac {b n}{4 x^{2}} - \frac {b \log {\left (c x^{n} \right )}}{2 x^{2}}}{e^{7}} & \text {for}\: d = 0 \\\frac {\frac {a x^{5}}{5} - \frac {b n x^{5}}{25} + \frac {b x^{5} \log {\left (c x^{n} \right )}}{5}}{d^{7}} & \text {for}\: e = 0 \\- \frac {12 a d^{6}}{360 d^{8} e^{5} + 2160 d^{7} e^{6} x + 5400 d^{6} e^{7} x^{2} + 7200 d^{5} e^{8} x^{3} + 5400 d^{4} e^{9} x^{4} + 2160 d^{3} e^{10} x^{5} + 360 d^{2} e^{11} x^{6}} - \frac {72 a d^{5} e x}{360 d^{8} e^{5} + 2160 d^{7} e^{6} x + 5400 d^{6} e^{7} x^{2} + 7200 d^{5} e^{8} x^{3} + 5400 d^{4} e^{9} x^{4} + 2160 d^{3} e^{10} x^{5} + 360 d^{2} e^{11} x^{6}} - \frac {180 a d^{4} e^{2} x^{2}}{360 d^{8} e^{5} + 2160 d^{7} e^{6} x + 5400 d^{6} e^{7} x^{2} + 7200 d^{5} e^{8} x^{3} + 5400 d^{4} e^{9} x^{4} + 2160 d^{3} e^{10} x^{5} + 360 d^{2} e^{11} x^{6}} - \frac {240 a d^{3} e^{3} x^{3}}{360 d^{8} e^{5} + 2160 d^{7} e^{6} x + 5400 d^{6} e^{7} x^{2} + 7200 d^{5} e^{8} x^{3} + 5400 d^{4} e^{9} x^{4} + 2160 d^{3} e^{10} x^{5} + 360 d^{2} e^{11} x^{6}} - \frac {180 a d^{2} e^{4} x^{4}}{360 d^{8} e^{5} + 2160 d^{7} e^{6} x + 5400 d^{6} e^{7} x^{2} + 7200 d^{5} e^{8} x^{3} + 5400 d^{4} e^{9} x^{4} + 2160 d^{3} e^{10} x^{5} + 360 d^{2} e^{11} x^{6}} - \frac {12 b d^{6} n \log {\left (\frac {d}{e} + x \right )}}{360 d^{8} e^{5} + 2160 d^{7} e^{6} x + 5400 d^{6} e^{7} x^{2} + 7200 d^{5} e^{8} x^{3} + 5400 d^{4} e^{9} x^{4} + 2160 d^{3} e^{10} x^{5} + 360 d^{2} e^{11} x^{6}} - \frac {13 b d^{6} n}{360 d^{8} e^{5} + 2160 d^{7} e^{6} x + 5400 d^{6} e^{7} x^{2} + 7200 d^{5} e^{8} x^{3} + 5400 d^{4} e^{9} x^{4} + 2160 d^{3} e^{10} x^{5} + 360 d^{2} e^{11} x^{6}} - \frac {72 b d^{5} e n x \log {\left (\frac {d}{e} + x \right )}}{360 d^{8} e^{5} + 2160 d^{7} e^{6} x + 5400 d^{6} e^{7} x^{2} + 7200 d^{5} e^{8} x^{3} + 5400 d^{4} e^{9} x^{4} + 2160 d^{3} e^{10} x^{5} + 360 d^{2} e^{11} x^{6}} - \frac {66 b d^{5} e n x}{360 d^{8} e^{5} + 2160 d^{7} e^{6} x + 5400 d^{6} e^{7} x^{2} + 7200 d^{5} e^{8} x^{3} + 5400 d^{4} e^{9} x^{4} + 2160 d^{3} e^{10} x^{5} + 360 d^{2} e^{11} x^{6}} - \frac {180 b d^{4} e^{2} n x^{2} \log {\left (\frac {d}{e} + x \right )}}{360 d^{8} e^{5} + 2160 d^{7} e^{6} x + 5400 d^{6} e^{7} x^{2} + 7200 d^{5} e^{8} x^{3} + 5400 d^{4} e^{9} x^{4} + 2160 d^{3} e^{10} x^{5} + 360 d^{2} e^{11} x^{6}} - \frac {129 b d^{4} e^{2} n x^{2}}{360 d^{8} e^{5} + 2160 d^{7} e^{6} x + 5400 d^{6} e^{7} x^{2} + 7200 d^{5} e^{8} x^{3} + 5400 d^{4} e^{9} x^{4} + 2160 d^{3} e^{10} x^{5} + 360 d^{2} e^{11} x^{6}} - \frac {240 b d^{3} e^{3} n x^{3} \log {\left (\frac {d}{e} + x \right )}}{360 d^{8} e^{5} + 2160 d^{7} e^{6} x + 5400 d^{6} e^{7} x^{2} + 7200 d^{5} e^{8} x^{3} + 5400 d^{4} e^{9} x^{4} + 2160 d^{3} e^{10} x^{5} + 360 d^{2} e^{11} x^{6}} - \frac {112 b d^{3} e^{3} n x^{3}}{360 d^{8} e^{5} + 2160 d^{7} e^{6} x + 5400 d^{6} e^{7} x^{2} + 7200 d^{5} e^{8} x^{3} + 5400 d^{4} e^{9} x^{4} + 2160 d^{3} e^{10} x^{5} + 360 d^{2} e^{11} x^{6}} - \frac {180 b d^{2} e^{4} n x^{4} \log {\left (\frac {d}{e} + x \right )}}{360 d^{8} e^{5} + 2160 d^{7} e^{6} x + 5400 d^{6} e^{7} x^{2} + 7200 d^{5} e^{8} x^{3} + 5400 d^{4} e^{9} x^{4} + 2160 d^{3} e^{10} x^{5} + 360 d^{2} e^{11} x^{6}} - \frac {24 b d^{2} e^{4} n x^{4}}{360 d^{8} e^{5} + 2160 d^{7} e^{6} x + 5400 d^{6} e^{7} x^{2} + 7200 d^{5} e^{8} x^{3} + 5400 d^{4} e^{9} x^{4} + 2160 d^{3} e^{10} x^{5} + 360 d^{2} e^{11} x^{6}} - \frac {72 b d e^{5} n x^{5} \log {\left (\frac {d}{e} + x \right )}}{360 d^{8} e^{5} + 2160 d^{7} e^{6} x + 5400 d^{6} e^{7} x^{2} + 7200 d^{5} e^{8} x^{3} + 5400 d^{4} e^{9} x^{4} + 2160 d^{3} e^{10} x^{5} + 360 d^{2} e^{11} x^{6}} + \frac {12 b d e^{5} n x^{5}}{360 d^{8} e^{5} + 2160 d^{7} e^{6} x + 5400 d^{6} e^{7} x^{2} + 7200 d^{5} e^{8} x^{3} + 5400 d^{4} e^{9} x^{4} + 2160 d^{3} e^{10} x^{5} + 360 d^{2} e^{11} x^{6}} + \frac {72 b d e^{5} x^{5} \log {\left (c x^{n} \right )}}{360 d^{8} e^{5} + 2160 d^{7} e^{6} x + 5400 d^{6} e^{7} x^{2} + 7200 d^{5} e^{8} x^{3} + 5400 d^{4} e^{9} x^{4} + 2160 d^{3} e^{10} x^{5} + 360 d^{2} e^{11} x^{6}} - \frac {12 b e^{6} n x^{6} \log {\left (\frac {d}{e} + x \right )}}{360 d^{8} e^{5} + 2160 d^{7} e^{6} x + 5400 d^{6} e^{7} x^{2} + 7200 d^{5} e^{8} x^{3} + 5400 d^{4} e^{9} x^{4} + 2160 d^{3} e^{10} x^{5} + 360 d^{2} e^{11} x^{6}} + \frac {12 b e^{6} x^{6} \log {\left (c x^{n} \right )}}{360 d^{8} e^{5} + 2160 d^{7} e^{6} x + 5400 d^{6} e^{7} x^{2} + 7200 d^{5} e^{8} x^{3} + 5400 d^{4} e^{9} x^{4} + 2160 d^{3} e^{10} x^{5} + 360 d^{2} e^{11} x^{6}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*(a+b*ln(c*x**n))/(e*x+d)**7,x)

[Out]

Piecewise((zoo*(-a/(2*x**2) - b*n/(4*x**2) - b*log(c*x**n)/(2*x**2)), Eq(d, 0) & Eq(e, 0)), ((-a/(2*x**2) - b*
n/(4*x**2) - b*log(c*x**n)/(2*x**2))/e**7, Eq(d, 0)), ((a*x**5/5 - b*n*x**5/25 + b*x**5*log(c*x**n)/5)/d**7, E
q(e, 0)), (-12*a*d**6/(360*d**8*e**5 + 2160*d**7*e**6*x + 5400*d**6*e**7*x**2 + 7200*d**5*e**8*x**3 + 5400*d**
4*e**9*x**4 + 2160*d**3*e**10*x**5 + 360*d**2*e**11*x**6) - 72*a*d**5*e*x/(360*d**8*e**5 + 2160*d**7*e**6*x +
5400*d**6*e**7*x**2 + 7200*d**5*e**8*x**3 + 5400*d**4*e**9*x**4 + 2160*d**3*e**10*x**5 + 360*d**2*e**11*x**6)
- 180*a*d**4*e**2*x**2/(360*d**8*e**5 + 2160*d**7*e**6*x + 5400*d**6*e**7*x**2 + 7200*d**5*e**8*x**3 + 5400*d*
*4*e**9*x**4 + 2160*d**3*e**10*x**5 + 360*d**2*e**11*x**6) - 240*a*d**3*e**3*x**3/(360*d**8*e**5 + 2160*d**7*e
**6*x + 5400*d**6*e**7*x**2 + 7200*d**5*e**8*x**3 + 5400*d**4*e**9*x**4 + 2160*d**3*e**10*x**5 + 360*d**2*e**1
1*x**6) - 180*a*d**2*e**4*x**4/(360*d**8*e**5 + 2160*d**7*e**6*x + 5400*d**6*e**7*x**2 + 7200*d**5*e**8*x**3 +
5400*d**4*e**9*x**4 + 2160*d**3*e**10*x**5 + 360*d**2*e**11*x**6) - 12*b*d**6*n*log(d/e + x)/(360*d**8*e**5 +
2160*d**7*e**6*x + 5400*d**6*e**7*x**2 + 7200*d**5*e**8*x**3 + 5400*d**4*e**9*x**4 + 2160*d**3*e**10*x**5 + 3
60*d**2*e**11*x**6) - 13*b*d**6*n/(360*d**8*e**5 + 2160*d**7*e**6*x + 5400*d**6*e**7*x**2 + 7200*d**5*e**8*x**
3 + 5400*d**4*e**9*x**4 + 2160*d**3*e**10*x**5 + 360*d**2*e**11*x**6) - 72*b*d**5*e*n*x*log(d/e + x)/(360*d**8
*e**5 + 2160*d**7*e**6*x + 5400*d**6*e**7*x**2 + 7200*d**5*e**8*x**3 + 5400*d**4*e**9*x**4 + 2160*d**3*e**10*x
**5 + 360*d**2*e**11*x**6) - 66*b*d**5*e*n*x/(360*d**8*e**5 + 2160*d**7*e**6*x + 5400*d**6*e**7*x**2 + 7200*d*
*5*e**8*x**3 + 5400*d**4*e**9*x**4 + 2160*d**3*e**10*x**5 + 360*d**2*e**11*x**6) - 180*b*d**4*e**2*n*x**2*log(
d/e + x)/(360*d**8*e**5 + 2160*d**7*e**6*x + 5400*d**6*e**7*x**2 + 7200*d**5*e**8*x**3 + 5400*d**4*e**9*x**4 +
2160*d**3*e**10*x**5 + 360*d**2*e**11*x**6) - 129*b*d**4*e**2*n*x**2/(360*d**8*e**5 + 2160*d**7*e**6*x + 5400
*d**6*e**7*x**2 + 7200*d**5*e**8*x**3 + 5400*d**4*e**9*x**4 + 2160*d**3*e**10*x**5 + 360*d**2*e**11*x**6) - 24
0*b*d**3*e**3*n*x**3*log(d/e + x)/(360*d**8*e**5 + 2160*d**7*e**6*x + 5400*d**6*e**7*x**2 + 7200*d**5*e**8*x**
3 + 5400*d**4*e**9*x**4 + 2160*d**3*e**10*x**5 + 360*d**2*e**11*x**6) - 112*b*d**3*e**3*n*x**3/(360*d**8*e**5
+ 2160*d**7*e**6*x + 5400*d**6*e**7*x**2 + 7200*d**5*e**8*x**3 + 5400*d**4*e**9*x**4 + 2160*d**3*e**10*x**5 +
360*d**2*e**11*x**6) - 180*b*d**2*e**4*n*x**4*log(d/e + x)/(360*d**8*e**5 + 2160*d**7*e**6*x + 5400*d**6*e**7*
x**2 + 7200*d**5*e**8*x**3 + 5400*d**4*e**9*x**4 + 2160*d**3*e**10*x**5 + 360*d**2*e**11*x**6) - 24*b*d**2*e**
4*n*x**4/(360*d**8*e**5 + 2160*d**7*e**6*x + 5400*d**6*e**7*x**2 + 7200*d**5*e**8*x**3 + 5400*d**4*e**9*x**4 +
2160*d**3*e**10*x**5 + 360*d**2*e**11*x**6) - 72*b*d*e**5*n*x**5*log(d/e + x)/(360*d**8*e**5 + 2160*d**7*e**6
*x + 5400*d**6*e**7*x**2 + 7200*d**5*e**8*x**3 + 5400*d**4*e**9*x**4 + 2160*d**3*e**10*x**5 + 360*d**2*e**11*x
**6) + 12*b*d*e**5*n*x**5/(360*d**8*e**5 + 2160*d**7*e**6*x + 5400*d**6*e**7*x**2 + 7200*d**5*e**8*x**3 + 5400
*d**4*e**9*x**4 + 2160*d**3*e**10*x**5 + 360*d**2*e**11*x**6) + 72*b*d*e**5*x**5*log(c*x**n)/(360*d**8*e**5 +
2160*d**7*e**6*x + 5400*d**6*e**7*x**2 + 7200*d**5*e**8*x**3 + 5400*d**4*e**9*x**4 + 2160*d**3*e**10*x**5 + 36
0*d**2*e**11*x**6) - 12*b*e**6*n*x**6*log(d/e + x)/(360*d**8*e**5 + 2160*d**7*e**6*x + 5400*d**6*e**7*x**2 + 7
200*d**5*e**8*x**3 + 5400*d**4*e**9*x**4 + 2160*d**3*e**10*x**5 + 360*d**2*e**11*x**6) + 12*b*e**6*x**6*log(c*
x**n)/(360*d**8*e**5 + 2160*d**7*e**6*x + 5400*d**6*e**7*x**2 + 7200*d**5*e**8*x**3 + 5400*d**4*e**9*x**4 + 21
60*d**3*e**10*x**5 + 360*d**2*e**11*x**6), True))

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 382 vs. \(2 (150) = 300\).
time = 4.39, size = 382, normalized size = 2.34 \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 ) - 12 \, b d n x^{5} e^{5} + 24 \, b d^{2} n x^{4} e^{4} + 112 \, b d^{3} n x^{3} e^{3} + 129 \, b d^{4} n x^{2} e^{2} + 66 \, b d^{5} n x e + 12 \, b d^{6} n \log \left (x e + d\right ) + 180 \, b d^{2} x^{4} e^{4} \log \left (c\right ) + 240 \, b d^{3} x^{3} e^{3} \log \left (c\right ) + 180 \, b d^{4} x^{2} e^{2} \log \left (c\right ) + 72 \, b d^{5} x e \log \left (c\right ) + 13 \, b d^{6} n + 180 \, a d^{2} x^{4} e^{4} + 240 \, a d^{3} x^{3} e^{3} + 180 \, a d^{4} x^{2} e^{2} + 72 \, a d^{5} x e + 12 \, b d^{6} \log \left (c\right ) + 12 \, a d^{6}}{360 \, {\left (d^{2} x^{6} e^{11} + 6 \, d^{3} x^{5} e^{10} + 15 \, d^{4} x^{4} e^{9} + 20 \, d^{5} x^{3} e^{8} + 15 \, d^{6} x^{2} e^{7} + 6 \, d^{7} x e^{6} + d^{8} e^{5}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(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) - 12*b*d*n*x^5*e^5 + 24*b*d^2*n*x^4*e^4 + 112*b*d^3*n*x^3*e^3 + 129*b*d^4*n*x
^2*e^2 + 66*b*d^5*n*x*e + 12*b*d^6*n*log(x*e + d) + 180*b*d^2*x^4*e^4*log(c) + 240*b*d^3*x^3*e^3*log(c) + 180*
b*d^4*x^2*e^2*log(c) + 72*b*d^5*x*e*log(c) + 13*b*d^6*n + 180*a*d^2*x^4*e^4 + 240*a*d^3*x^3*e^3 + 180*a*d^4*x^
2*e^2 + 72*a*d^5*x*e + 12*b*d^6*log(c) + 12*a*d^6)/(d^2*x^6*e^11 + 6*d^3*x^5*e^10 + 15*d^4*x^4*e^9 + 20*d^5*x^
3*e^8 + 15*d^6*x^2*e^7 + 6*d^7*x*e^6 + d^8*e^5)

________________________________________________________________________________________

Mupad [B]
time = 4.26, size = 320, normalized size = 1.96 \begin {gather*} -\frac {x^4\,\left (15\,a\,e^4+2\,b\,e^4\,n\right )+x\,\left (6\,a\,d^3\,e+\frac {11\,b\,d^3\,e\,n}{2}\right )+a\,d^4+x^2\,\left (15\,a\,d^2\,e^2+\frac {43\,b\,d^2\,e^2\,n}{4}\right )+x^3\,\left (20\,a\,d\,e^3+\frac {28\,b\,d\,e^3\,n}{3}\right )+\frac {13\,b\,d^4\,n}{12}-\frac {b\,e^5\,n\,x^5}{d}}{30\,d^6\,e^5+180\,d^5\,e^6\,x+450\,d^4\,e^7\,x^2+600\,d^3\,e^8\,x^3+450\,d^2\,e^9\,x^4+180\,d\,e^{10}\,x^5+30\,e^{11}\,x^6}-\frac {\ln \left (c\,x^n\right )\,\left (\frac {b\,d^4}{30\,e^5}+\frac {b\,x^4}{2\,e}+\frac {b\,d^2\,x^2}{2\,e^3}+\frac {2\,b\,d\,x^3}{3\,e^2}+\frac {b\,d^3\,x}{5\,e^4}\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^2\,e^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^4*(a + b*log(c*x^n)))/(d + e*x)^7,x)

[Out]

- (x^4*(15*a*e^4 + 2*b*e^4*n) + x*(6*a*d^3*e + (11*b*d^3*e*n)/2) + a*d^4 + x^2*(15*a*d^2*e^2 + (43*b*d^2*e^2*n
)/4) + x^3*(20*a*d*e^3 + (28*b*d*e^3*n)/3) + (13*b*d^4*n)/12 - (b*e^5*n*x^5)/d)/(30*d^6*e^5 + 30*e^11*x^6 + 18
0*d^5*e^6*x + 180*d*e^10*x^5 + 450*d^4*e^7*x^2 + 600*d^3*e^8*x^3 + 450*d^2*e^9*x^4) - (log(c*x^n)*((b*d^4)/(30
*e^5) + (b*x^4)/(2*e) + (b*d^2*x^2)/(2*e^3) + (2*b*d*x^3)/(3*e^2) + (b*d^3*x)/(5*e^4)))/(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^2*e^
5)

________________________________________________________________________________________