3.1.67 \(\int \frac {x^3 (a+b \log (c x^n))}{(d+e x)^7} \, dx\) [67]

Optimal. Leaf size=226 \[ -\frac {b d^2 n}{30 e^4 (d+e x)^5}+\frac {13 b d n}{120 e^4 (d+e x)^4}-\frac {19 b n}{180 e^4 (d+e x)^3}+\frac {b n}{120 d e^4 (d+e x)^2}+\frac {b n}{60 d^2 e^4 (d+e x)}+\frac {b n \log (x)}{60 d^3 e^4}+\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{6 e^4 (d+e x)^6}-\frac {3 d^2 \left (a+b \log \left (c x^n\right )\right )}{5 e^4 (d+e x)^5}+\frac {3 d \left (a+b \log \left (c x^n\right )\right )}{4 e^4 (d+e x)^4}-\frac {a+b \log \left (c x^n\right )}{3 e^4 (d+e x)^3}-\frac {b n \log (d+e x)}{60 d^3 e^4} \]

[Out]

-1/30*b*d^2*n/e^4/(e*x+d)^5+13/120*b*d*n/e^4/(e*x+d)^4-19/180*b*n/e^4/(e*x+d)^3+1/120*b*n/d/e^4/(e*x+d)^2+1/60
*b*n/d^2/e^4/(e*x+d)+1/60*b*n*ln(x)/d^3/e^4+1/6*d^3*(a+b*ln(c*x^n))/e^4/(e*x+d)^6-3/5*d^2*(a+b*ln(c*x^n))/e^4/
(e*x+d)^5+3/4*d*(a+b*ln(c*x^n))/e^4/(e*x+d)^4+1/3*(-a-b*ln(c*x^n))/e^4/(e*x+d)^3-1/60*b*n*ln(e*x+d)/d^3/e^4

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Rubi [A]
time = 0.14, antiderivative size = 226, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {45, 2382, 12, 1634} \begin {gather*} \frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{6 e^4 (d+e x)^6}-\frac {3 d^2 \left (a+b \log \left (c x^n\right )\right )}{5 e^4 (d+e x)^5}+\frac {3 d \left (a+b \log \left (c x^n\right )\right )}{4 e^4 (d+e x)^4}-\frac {a+b \log \left (c x^n\right )}{3 e^4 (d+e x)^3}+\frac {b n \log (x)}{60 d^3 e^4}-\frac {b n \log (d+e x)}{60 d^3 e^4}-\frac {b d^2 n}{30 e^4 (d+e x)^5}+\frac {b n}{60 d^2 e^4 (d+e x)}+\frac {13 b d n}{120 e^4 (d+e x)^4}-\frac {19 b n}{180 e^4 (d+e x)^3}+\frac {b n}{120 d e^4 (d+e x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x^3*(a + b*Log[c*x^n]))/(d + e*x)^7,x]

[Out]

-1/30*(b*d^2*n)/(e^4*(d + e*x)^5) + (13*b*d*n)/(120*e^4*(d + e*x)^4) - (19*b*n)/(180*e^4*(d + e*x)^3) + (b*n)/
(120*d*e^4*(d + e*x)^2) + (b*n)/(60*d^2*e^4*(d + e*x)) + (b*n*Log[x])/(60*d^3*e^4) + (d^3*(a + b*Log[c*x^n]))/
(6*e^4*(d + e*x)^6) - (3*d^2*(a + b*Log[c*x^n]))/(5*e^4*(d + e*x)^5) + (3*d*(a + b*Log[c*x^n]))/(4*e^4*(d + e*
x)^4) - (a + b*Log[c*x^n])/(3*e^4*(d + e*x)^3) - (b*n*Log[d + e*x])/(60*d^3*e^4)

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 1634

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[Px*(a + b*x)
^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && PolyQ[Px, x] && (IntegersQ[m, n] || IGtQ[m, -2]) &&
GtQ[Expon[Px, x], 2]

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^3 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^7} \, dx &=\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{6 e^4 (d+e x)^6}-\frac {3 d^2 \left (a+b \log \left (c x^n\right )\right )}{5 e^4 (d+e x)^5}+\frac {3 d \left (a+b \log \left (c x^n\right )\right )}{4 e^4 (d+e x)^4}-\frac {a+b \log \left (c x^n\right )}{3 e^4 (d+e x)^3}-(b n) \int \frac {-d^3-6 d^2 e x-15 d e^2 x^2-20 e^3 x^3}{60 e^4 x (d+e x)^6} \, dx\\ &=\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{6 e^4 (d+e x)^6}-\frac {3 d^2 \left (a+b \log \left (c x^n\right )\right )}{5 e^4 (d+e x)^5}+\frac {3 d \left (a+b \log \left (c x^n\right )\right )}{4 e^4 (d+e x)^4}-\frac {a+b \log \left (c x^n\right )}{3 e^4 (d+e x)^3}-\frac {(b n) \int \frac {-d^3-6 d^2 e x-15 d e^2 x^2-20 e^3 x^3}{x (d+e x)^6} \, dx}{60 e^4}\\ &=\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{6 e^4 (d+e x)^6}-\frac {3 d^2 \left (a+b \log \left (c x^n\right )\right )}{5 e^4 (d+e x)^5}+\frac {3 d \left (a+b \log \left (c x^n\right )\right )}{4 e^4 (d+e x)^4}-\frac {a+b \log \left (c x^n\right )}{3 e^4 (d+e x)^3}-\frac {(b n) \int \left (-\frac {1}{d^3 x}-\frac {10 d^2 e}{(d+e x)^6}+\frac {26 d e}{(d+e x)^5}-\frac {19 e}{(d+e x)^4}+\frac {e}{d (d+e x)^3}+\frac {e}{d^2 (d+e x)^2}+\frac {e}{d^3 (d+e x)}\right ) \, dx}{60 e^4}\\ &=-\frac {b d^2 n}{30 e^4 (d+e x)^5}+\frac {13 b d n}{120 e^4 (d+e x)^4}-\frac {19 b n}{180 e^4 (d+e x)^3}+\frac {b n}{120 d e^4 (d+e x)^2}+\frac {b n}{60 d^2 e^4 (d+e x)}+\frac {b n \log (x)}{60 d^3 e^4}+\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{6 e^4 (d+e x)^6}-\frac {3 d^2 \left (a+b \log \left (c x^n\right )\right )}{5 e^4 (d+e x)^5}+\frac {3 d \left (a+b \log \left (c x^n\right )\right )}{4 e^4 (d+e x)^4}-\frac {a+b \log \left (c x^n\right )}{3 e^4 (d+e x)^3}-\frac {b n \log (d+e x)}{60 d^3 e^4}\\ \end {align*}

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Mathematica [A]
time = 0.16, size = 281, normalized size = 1.24 \begin {gather*} \frac {a d^3}{6 e^4 (d+e x)^6}-\frac {3 a d^2}{5 e^4 (d+e x)^5}-\frac {b d^2 n}{30 e^4 (d+e x)^5}+\frac {3 a d}{4 e^4 (d+e x)^4}+\frac {13 b d n}{120 e^4 (d+e x)^4}-\frac {a}{3 e^4 (d+e x)^3}-\frac {19 b n}{180 e^4 (d+e x)^3}+\frac {b n}{120 d e^4 (d+e x)^2}+\frac {b n}{60 d^2 e^4 (d+e x)}+\frac {b n \log (x)}{60 d^3 e^4}+\frac {b d^3 \log \left (c x^n\right )}{6 e^4 (d+e x)^6}-\frac {3 b d^2 \log \left (c x^n\right )}{5 e^4 (d+e x)^5}+\frac {3 b d \log \left (c x^n\right )}{4 e^4 (d+e x)^4}-\frac {b \log \left (c x^n\right )}{3 e^4 (d+e x)^3}-\frac {b n \log (d+e x)}{60 d^3 e^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x^3*(a + b*Log[c*x^n]))/(d + e*x)^7,x]

[Out]

(a*d^3)/(6*e^4*(d + e*x)^6) - (3*a*d^2)/(5*e^4*(d + e*x)^5) - (b*d^2*n)/(30*e^4*(d + e*x)^5) + (3*a*d)/(4*e^4*
(d + e*x)^4) + (13*b*d*n)/(120*e^4*(d + e*x)^4) - a/(3*e^4*(d + e*x)^3) - (19*b*n)/(180*e^4*(d + e*x)^3) + (b*
n)/(120*d*e^4*(d + e*x)^2) + (b*n)/(60*d^2*e^4*(d + e*x)) + (b*n*Log[x])/(60*d^3*e^4) + (b*d^3*Log[c*x^n])/(6*
e^4*(d + e*x)^6) - (3*b*d^2*Log[c*x^n])/(5*e^4*(d + e*x)^5) + (3*b*d*Log[c*x^n])/(4*e^4*(d + e*x)^4) - (b*Log[
c*x^n])/(3*e^4*(d + e*x)^3) - (b*n*Log[d + e*x])/(60*d^3*e^4)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.16, size = 867, normalized size = 3.84

method	result	size
risch	\(-\frac {b \left (20 e^{3} x^{3}+15 d \,e^{2} x^{2}+6 d^{2} e x +d^{3}\right ) \ln \left (x^{n}\right )}{60 \left (e x +d \right )^{6} e^{4}}+\frac {-6 \ln \left (c \right ) b \,d^{6}-36 \ln \left (e x +d \right ) b d \,e^{5} n \,x^{5}-90 \ln \left (e x +d \right ) b \,d^{2} e^{4} n \,x^{4}-120 \ln \left (e x +d \right ) b \,d^{3} e^{3} n \,x^{3}-90 \ln \left (e x +d \right ) b \,d^{4} e^{2} n \,x^{2}-36 \ln \left (e x +d \right ) b \,d^{5} e n x +36 \ln \left (-x \right ) b d \,e^{5} n \,x^{5}+90 \ln \left (-x \right ) b \,d^{2} e^{4} n \,x^{4}+120 \ln \left (-x \right ) b \,d^{3} e^{3} n \,x^{3}-3 i \pi b \,d^{6} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-120 a \,d^{3} e^{3} x^{3}-90 a \,d^{4} e^{2} x^{2}-36 a \,d^{5} e x -2 b \,d^{6} n -60 i \pi b \,d^{3} e^{3} x^{3} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-45 i \pi b \,d^{4} e^{2} x^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-45 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}-60 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}-3 i \pi b \,d^{6} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+90 \ln \left (-x \right ) b \,d^{4} e^{2} n \,x^{2}+36 \ln \left (-x \right ) b \,d^{5} e n x -6 a \,d^{6}-6 \ln \left (e x +d \right ) b \,d^{6} n +6 \ln \left (-x \right ) b \,d^{6} n +3 b \,d^{4} e^{2} n \,x^{2}-6 b \,d^{5} e n x +6 b d \,e^{5} n \,x^{5}+33 b \,d^{2} e^{4} n \,x^{4}+34 b \,d^{3} e^{3} n \,x^{3}-18 i \pi b \,d^{5} e x \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-18 i \pi b \,d^{5} e x \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-6 \ln \left (e x +d \right ) b \,e^{6} n \,x^{6}+6 \ln \left (-x \right ) b \,e^{6} n \,x^{6}-120 \ln \left (c \right ) b \,d^{3} e^{3} x^{3}-90 \ln \left (c \right ) b \,d^{4} e^{2} x^{2}-36 \ln \left (c \right ) b \,d^{5} e x +3 i \pi b \,d^{6} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+60 i \pi b \,d^{3} e^{3} x^{3} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+45 i \pi b \,d^{4} e^{2} x^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+18 i \pi b \,d^{5} e x \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+3 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 )+60 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 )+45 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 )+18 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^{4} d^{3} \left (e x +d \right )^{6}}\)	\(867\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(a+b*ln(c*x^n))/(e*x+d)^7,x,method=_RETURNVERBOSE)

[Out]

-1/60*b*(20*e^3*x^3+15*d*e^2*x^2+6*d^2*e*x+d^3)/(e*x+d)^6/e^4*ln(x^n)+1/360*(3*I*Pi*b*d^6*csgn(I*c*x^n)^3-6*ln
(c)*b*d^6-45*I*Pi*b*d^4*e^2*x^2*csgn(I*x^n)*csgn(I*c*x^n)^2-36*ln(e*x+d)*b*d*e^5*n*x^5-90*ln(e*x+d)*b*d^2*e^4*
n*x^4-120*ln(e*x+d)*b*d^3*e^3*n*x^3-90*ln(e*x+d)*b*d^4*e^2*n*x^2-36*ln(e*x+d)*b*d^5*e*n*x+36*ln(-x)*b*d*e^5*n*
x^5+90*ln(-x)*b*d^2*e^4*n*x^4+120*ln(-x)*b*d^3*e^3*n*x^3-60*I*Pi*b*d^3*e^3*x^3*csgn(I*x^n)*csgn(I*c*x^n)^2-120
*a*d^3*e^3*x^3-90*a*d^4*e^2*x^2-36*a*d^5*e*x-2*b*d^6*n-60*I*Pi*b*d^3*e^3*x^3*csgn(I*c)*csgn(I*c*x^n)^2-18*I*Pi
*b*d^5*e*x*csgn(I*c)*csgn(I*c*x^n)^2-3*I*Pi*b*d^6*csgn(I*c)*csgn(I*c*x^n)^2-3*I*Pi*b*d^6*csgn(I*x^n)*csgn(I*c*
x^n)^2-18*I*Pi*b*d^5*e*x*csgn(I*x^n)*csgn(I*c*x^n)^2+45*I*Pi*b*d^4*e^2*x^2*csgn(I*c*x^n)^3-45*I*Pi*b*d^4*e^2*x
^2*csgn(I*c)*csgn(I*c*x^n)^2+90*ln(-x)*b*d^4*e^2*n*x^2+36*ln(-x)*b*d^5*e*n*x-6*a*d^6-6*ln(e*x+d)*b*d^6*n+6*ln(
-x)*b*d^6*n+3*b*d^4*e^2*n*x^2-6*b*d^5*e*n*x+6*b*d*e^5*n*x^5+33*b*d^2*e^4*n*x^4+34*b*d^3*e^3*n*x^3-6*ln(e*x+d)*
b*e^6*n*x^6+6*ln(-x)*b*e^6*n*x^6-120*ln(c)*b*d^3*e^3*x^3-90*ln(c)*b*d^4*e^2*x^2-36*ln(c)*b*d^5*e*x+45*I*Pi*b*d
^4*e^2*x^2*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+60*I*Pi*b*d^3*e^3*x^3*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+60*I*
Pi*b*d^3*e^3*x^3*csgn(I*c*x^n)^3+18*I*Pi*b*d^5*e*x*csgn(I*c*x^n)^3+3*I*Pi*b*d^6*csgn(I*c)*csgn(I*x^n)*csgn(I*c
*x^n)+18*I*Pi*b*d^5*e*x*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n))/e^4/d^3/(e*x+d)^6

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Maxima [A]
time = 0.29, size = 313, normalized size = 1.38 \begin {gather*} \frac {1}{360} \, b n {\left (\frac {6 \, x^{4} e^{4} + 27 \, d x^{3} e^{3} + 7 \, d^{2} x^{2} e^{2} - 4 \, d^{3} x e - 2 \, d^{4}}{d^{2} x^{5} e^{9} + 5 \, d^{3} x^{4} e^{8} + 10 \, d^{4} x^{3} e^{7} + 10 \, d^{5} x^{2} e^{6} + 5 \, d^{6} x e^{5} + d^{7} e^{4}} - \frac {6 \, e^{\left (-4\right )} \log \left (x e + d\right )}{d^{3}} + \frac {6 \, e^{\left (-4\right )} \log \left (x\right )}{d^{3}}\right )} - \frac {{\left (20 \, x^{3} e^{3} + 15 \, d x^{2} e^{2} + 6 \, d^{2} x e + d^{3}\right )} b \log \left (c x^{n}\right )}{60 \, {\left (x^{6} e^{10} + 6 \, d x^{5} e^{9} + 15 \, d^{2} x^{4} e^{8} + 20 \, d^{3} x^{3} e^{7} + 15 \, d^{4} x^{2} e^{6} + 6 \, d^{5} x e^{5} + d^{6} e^{4}\right )}} - \frac {{\left (20 \, x^{3} e^{3} + 15 \, d x^{2} e^{2} + 6 \, d^{2} x e + d^{3}\right )} a}{60 \, {\left (x^{6} e^{10} + 6 \, d x^{5} e^{9} + 15 \, d^{2} x^{4} e^{8} + 20 \, d^{3} x^{3} e^{7} + 15 \, d^{4} x^{2} e^{6} + 6 \, d^{5} x e^{5} + d^{6} e^{4}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(a+b*log(c*x^n))/(e*x+d)^7,x, algorithm="maxima")

[Out]

1/360*b*n*((6*x^4*e^4 + 27*d*x^3*e^3 + 7*d^2*x^2*e^2 - 4*d^3*x*e - 2*d^4)/(d^2*x^5*e^9 + 5*d^3*x^4*e^8 + 10*d^
4*x^3*e^7 + 10*d^5*x^2*e^6 + 5*d^6*x*e^5 + d^7*e^4) - 6*e^(-4)*log(x*e + d)/d^3 + 6*e^(-4)*log(x)/d^3) - 1/60*
(20*x^3*e^3 + 15*d*x^2*e^2 + 6*d^2*x*e + d^3)*b*log(c*x^n)/(x^6*e^10 + 6*d*x^5*e^9 + 15*d^2*x^4*e^8 + 20*d^3*x
^3*e^7 + 15*d^4*x^2*e^6 + 6*d^5*x*e^5 + d^6*e^4) - 1/60*(20*x^3*e^3 + 15*d*x^2*e^2 + 6*d^2*x*e + d^3)*a/(x^6*e
^10 + 6*d*x^5*e^9 + 15*d^2*x^4*e^8 + 20*d^3*x^3*e^7 + 15*d^4*x^2*e^6 + 6*d^5*x*e^5 + d^6*e^4)

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Fricas [A]
time = 0.36, size = 319, normalized size = 1.41 \begin {gather*} \frac {6 \, b d n x^{5} e^{5} + 33 \, b d^{2} n x^{4} e^{4} - 2 \, b d^{6} n - 6 \, a d^{6} + 2 \, {\left (17 \, b d^{3} n - 60 \, a d^{3}\right )} x^{3} e^{3} + 3 \, {\left (b d^{4} n - 30 \, a d^{4}\right )} x^{2} e^{2} - 6 \, {\left (b d^{5} n + 6 \, a d^{5}\right )} x e - 6 \, {\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 ) - 6 \, {\left (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 ) + 6 \, {\left (b n x^{6} e^{6} + 6 \, b d n x^{5} e^{5} + 15 \, b d^{2} n x^{4} e^{4}\right )} \log \left (x\right )}{360 \, {\left (d^{3} x^{6} e^{10} + 6 \, d^{4} x^{5} e^{9} + 15 \, d^{5} x^{4} e^{8} + 20 \, d^{6} x^{3} e^{7} + 15 \, d^{7} x^{2} e^{6} + 6 \, d^{8} x e^{5} + d^{9} e^{4}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(a+b*log(c*x^n))/(e*x+d)^7,x, algorithm="fricas")

[Out]

1/360*(6*b*d*n*x^5*e^5 + 33*b*d^2*n*x^4*e^4 - 2*b*d^6*n - 6*a*d^6 + 2*(17*b*d^3*n - 60*a*d^3)*x^3*e^3 + 3*(b*d
^4*n - 30*a*d^4)*x^2*e^2 - 6*(b*d^5*n + 6*a*d^5)*x*e - 6*(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) - 6*(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) + 6*(b*n*x^6*e^6 + 6*b*d*n*x^5*e^5 + 15*b*d^2*n*x^4*e^4)*log(x))/(d^
3*x^6*e^10 + 6*d^4*x^5*e^9 + 15*d^5*x^4*e^8 + 20*d^6*x^3*e^7 + 15*d^7*x^2*e^6 + 6*d^8*x*e^5 + d^9*e^4)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1979 vs. \(2 (224) = 448\).
time = 84.07, size = 1979, normalized size = 8.76 \begin {gather*} \begin {cases} \tilde {\infty } \left (- \frac {a}{3 x^{3}} - \frac {b n}{9 x^{3}} - \frac {b \log {\left (c x^{n} \right )}}{3 x^{3}}\right ) & \text {for}\: d = 0 \wedge e = 0 \\\frac {\frac {a x^{4}}{4} - \frac {b n x^{4}}{16} + \frac {b x^{4} \log {\left (c x^{n} \right )}}{4}}{d^{7}} & \text {for}\: e = 0 \\\frac {- \frac {a}{3 x^{3}} - \frac {b n}{9 x^{3}} - \frac {b \log {\left (c x^{n} \right )}}{3 x^{3}}}{e^{7}} & \text {for}\: d = 0 \\- \frac {6 a d^{6}}{360 d^{9} e^{4} + 2160 d^{8} e^{5} x + 5400 d^{7} e^{6} x^{2} + 7200 d^{6} e^{7} x^{3} + 5400 d^{5} e^{8} x^{4} + 2160 d^{4} e^{9} x^{5} + 360 d^{3} e^{10} x^{6}} - \frac {36 a d^{5} e x}{360 d^{9} e^{4} + 2160 d^{8} e^{5} x + 5400 d^{7} e^{6} x^{2} + 7200 d^{6} e^{7} x^{3} + 5400 d^{5} e^{8} x^{4} + 2160 d^{4} e^{9} x^{5} + 360 d^{3} e^{10} x^{6}} - \frac {90 a d^{4} e^{2} x^{2}}{360 d^{9} e^{4} + 2160 d^{8} e^{5} x + 5400 d^{7} e^{6} x^{2} + 7200 d^{6} e^{7} x^{3} + 5400 d^{5} e^{8} x^{4} + 2160 d^{4} e^{9} x^{5} + 360 d^{3} e^{10} x^{6}} - \frac {120 a d^{3} e^{3} x^{3}}{360 d^{9} e^{4} + 2160 d^{8} e^{5} x + 5400 d^{7} e^{6} x^{2} + 7200 d^{6} e^{7} x^{3} + 5400 d^{5} e^{8} x^{4} + 2160 d^{4} e^{9} x^{5} + 360 d^{3} e^{10} x^{6}} - \frac {6 b d^{6} n \log {\left (\frac {d}{e} + x \right )}}{360 d^{9} e^{4} + 2160 d^{8} e^{5} x + 5400 d^{7} e^{6} x^{2} + 7200 d^{6} e^{7} x^{3} + 5400 d^{5} e^{8} x^{4} + 2160 d^{4} e^{9} x^{5} + 360 d^{3} e^{10} x^{6}} - \frac {2 b d^{6} n}{360 d^{9} e^{4} + 2160 d^{8} e^{5} x + 5400 d^{7} e^{6} x^{2} + 7200 d^{6} e^{7} x^{3} + 5400 d^{5} e^{8} x^{4} + 2160 d^{4} e^{9} x^{5} + 360 d^{3} e^{10} x^{6}} - \frac {36 b d^{5} e n x \log {\left (\frac {d}{e} + x \right )}}{360 d^{9} e^{4} + 2160 d^{8} e^{5} x + 5400 d^{7} e^{6} x^{2} + 7200 d^{6} e^{7} x^{3} + 5400 d^{5} e^{8} x^{4} + 2160 d^{4} e^{9} x^{5} + 360 d^{3} e^{10} x^{6}} - \frac {6 b d^{5} e n x}{360 d^{9} e^{4} + 2160 d^{8} e^{5} x + 5400 d^{7} e^{6} x^{2} + 7200 d^{6} e^{7} x^{3} + 5400 d^{5} e^{8} x^{4} + 2160 d^{4} e^{9} x^{5} + 360 d^{3} e^{10} x^{6}} - \frac {90 b d^{4} e^{2} n x^{2} \log {\left (\frac {d}{e} + x \right )}}{360 d^{9} e^{4} + 2160 d^{8} e^{5} x + 5400 d^{7} e^{6} x^{2} + 7200 d^{6} e^{7} x^{3} + 5400 d^{5} e^{8} x^{4} + 2160 d^{4} e^{9} x^{5} + 360 d^{3} e^{10} x^{6}} + \frac {3 b d^{4} e^{2} n x^{2}}{360 d^{9} e^{4} + 2160 d^{8} e^{5} x + 5400 d^{7} e^{6} x^{2} + 7200 d^{6} e^{7} x^{3} + 5400 d^{5} e^{8} x^{4} + 2160 d^{4} e^{9} x^{5} + 360 d^{3} e^{10} x^{6}} - \frac {120 b d^{3} e^{3} n x^{3} \log {\left (\frac {d}{e} + x \right )}}{360 d^{9} e^{4} + 2160 d^{8} e^{5} x + 5400 d^{7} e^{6} x^{2} + 7200 d^{6} e^{7} x^{3} + 5400 d^{5} e^{8} x^{4} + 2160 d^{4} e^{9} x^{5} + 360 d^{3} e^{10} x^{6}} + \frac {34 b d^{3} e^{3} n x^{3}}{360 d^{9} e^{4} + 2160 d^{8} e^{5} x + 5400 d^{7} e^{6} x^{2} + 7200 d^{6} e^{7} x^{3} + 5400 d^{5} e^{8} x^{4} + 2160 d^{4} e^{9} x^{5} + 360 d^{3} e^{10} x^{6}} - \frac {90 b d^{2} e^{4} n x^{4} \log {\left (\frac {d}{e} + x \right )}}{360 d^{9} e^{4} + 2160 d^{8} e^{5} x + 5400 d^{7} e^{6} x^{2} + 7200 d^{6} e^{7} x^{3} + 5400 d^{5} e^{8} x^{4} + 2160 d^{4} e^{9} x^{5} + 360 d^{3} e^{10} x^{6}} + \frac {33 b d^{2} e^{4} n x^{4}}{360 d^{9} e^{4} + 2160 d^{8} e^{5} x + 5400 d^{7} e^{6} x^{2} + 7200 d^{6} e^{7} x^{3} + 5400 d^{5} e^{8} x^{4} + 2160 d^{4} e^{9} x^{5} + 360 d^{3} e^{10} x^{6}} + \frac {90 b d^{2} e^{4} x^{4} \log {\left (c x^{n} \right )}}{360 d^{9} e^{4} + 2160 d^{8} e^{5} x + 5400 d^{7} e^{6} x^{2} + 7200 d^{6} e^{7} x^{3} + 5400 d^{5} e^{8} x^{4} + 2160 d^{4} e^{9} x^{5} + 360 d^{3} e^{10} x^{6}} - \frac {36 b d e^{5} n x^{5} \log {\left (\frac {d}{e} + x \right )}}{360 d^{9} e^{4} + 2160 d^{8} e^{5} x + 5400 d^{7} e^{6} x^{2} + 7200 d^{6} e^{7} x^{3} + 5400 d^{5} e^{8} x^{4} + 2160 d^{4} e^{9} x^{5} + 360 d^{3} e^{10} x^{6}} + \frac {6 b d e^{5} n x^{5}}{360 d^{9} e^{4} + 2160 d^{8} e^{5} x + 5400 d^{7} e^{6} x^{2} + 7200 d^{6} e^{7} x^{3} + 5400 d^{5} e^{8} x^{4} + 2160 d^{4} e^{9} x^{5} + 360 d^{3} e^{10} x^{6}} + \frac {36 b d e^{5} x^{5} \log {\left (c x^{n} \right )}}{360 d^{9} e^{4} + 2160 d^{8} e^{5} x + 5400 d^{7} e^{6} x^{2} + 7200 d^{6} e^{7} x^{3} + 5400 d^{5} e^{8} x^{4} + 2160 d^{4} e^{9} x^{5} + 360 d^{3} e^{10} x^{6}} - \frac {6 b e^{6} n x^{6} \log {\left (\frac {d}{e} + x \right )}}{360 d^{9} e^{4} + 2160 d^{8} e^{5} x + 5400 d^{7} e^{6} x^{2} + 7200 d^{6} e^{7} x^{3} + 5400 d^{5} e^{8} x^{4} + 2160 d^{4} e^{9} x^{5} + 360 d^{3} e^{10} x^{6}} + \frac {6 b e^{6} x^{6} \log {\left (c x^{n} \right )}}{360 d^{9} e^{4} + 2160 d^{8} e^{5} x + 5400 d^{7} e^{6} x^{2} + 7200 d^{6} e^{7} x^{3} + 5400 d^{5} e^{8} x^{4} + 2160 d^{4} e^{9} x^{5} + 360 d^{3} e^{10} x^{6}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(a+b*ln(c*x**n))/(e*x+d)**7,x)

[Out]

Piecewise((zoo*(-a/(3*x**3) - b*n/(9*x**3) - b*log(c*x**n)/(3*x**3)), Eq(d, 0) & Eq(e, 0)), ((a*x**4/4 - b*n*x
**4/16 + b*x**4*log(c*x**n)/4)/d**7, Eq(e, 0)), ((-a/(3*x**3) - b*n/(9*x**3) - b*log(c*x**n)/(3*x**3))/e**7, E
q(d, 0)), (-6*a*d**6/(360*d**9*e**4 + 2160*d**8*e**5*x + 5400*d**7*e**6*x**2 + 7200*d**6*e**7*x**3 + 5400*d**5
*e**8*x**4 + 2160*d**4*e**9*x**5 + 360*d**3*e**10*x**6) - 36*a*d**5*e*x/(360*d**9*e**4 + 2160*d**8*e**5*x + 54
00*d**7*e**6*x**2 + 7200*d**6*e**7*x**3 + 5400*d**5*e**8*x**4 + 2160*d**4*e**9*x**5 + 360*d**3*e**10*x**6) - 9
0*a*d**4*e**2*x**2/(360*d**9*e**4 + 2160*d**8*e**5*x + 5400*d**7*e**6*x**2 + 7200*d**6*e**7*x**3 + 5400*d**5*e
**8*x**4 + 2160*d**4*e**9*x**5 + 360*d**3*e**10*x**6) - 120*a*d**3*e**3*x**3/(360*d**9*e**4 + 2160*d**8*e**5*x
+ 5400*d**7*e**6*x**2 + 7200*d**6*e**7*x**3 + 5400*d**5*e**8*x**4 + 2160*d**4*e**9*x**5 + 360*d**3*e**10*x**6
) - 6*b*d**6*n*log(d/e + x)/(360*d**9*e**4 + 2160*d**8*e**5*x + 5400*d**7*e**6*x**2 + 7200*d**6*e**7*x**3 + 54
00*d**5*e**8*x**4 + 2160*d**4*e**9*x**5 + 360*d**3*e**10*x**6) - 2*b*d**6*n/(360*d**9*e**4 + 2160*d**8*e**5*x
+ 5400*d**7*e**6*x**2 + 7200*d**6*e**7*x**3 + 5400*d**5*e**8*x**4 + 2160*d**4*e**9*x**5 + 360*d**3*e**10*x**6)
- 36*b*d**5*e*n*x*log(d/e + x)/(360*d**9*e**4 + 2160*d**8*e**5*x + 5400*d**7*e**6*x**2 + 7200*d**6*e**7*x**3
+ 5400*d**5*e**8*x**4 + 2160*d**4*e**9*x**5 + 360*d**3*e**10*x**6) - 6*b*d**5*e*n*x/(360*d**9*e**4 + 2160*d**8
*e**5*x + 5400*d**7*e**6*x**2 + 7200*d**6*e**7*x**3 + 5400*d**5*e**8*x**4 + 2160*d**4*e**9*x**5 + 360*d**3*e**
10*x**6) - 90*b*d**4*e**2*n*x**2*log(d/e + x)/(360*d**9*e**4 + 2160*d**8*e**5*x + 5400*d**7*e**6*x**2 + 7200*d
**6*e**7*x**3 + 5400*d**5*e**8*x**4 + 2160*d**4*e**9*x**5 + 360*d**3*e**10*x**6) + 3*b*d**4*e**2*n*x**2/(360*d
**9*e**4 + 2160*d**8*e**5*x + 5400*d**7*e**6*x**2 + 7200*d**6*e**7*x**3 + 5400*d**5*e**8*x**4 + 2160*d**4*e**9
*x**5 + 360*d**3*e**10*x**6) - 120*b*d**3*e**3*n*x**3*log(d/e + x)/(360*d**9*e**4 + 2160*d**8*e**5*x + 5400*d*
*7*e**6*x**2 + 7200*d**6*e**7*x**3 + 5400*d**5*e**8*x**4 + 2160*d**4*e**9*x**5 + 360*d**3*e**10*x**6) + 34*b*d
**3*e**3*n*x**3/(360*d**9*e**4 + 2160*d**8*e**5*x + 5400*d**7*e**6*x**2 + 7200*d**6*e**7*x**3 + 5400*d**5*e**8
*x**4 + 2160*d**4*e**9*x**5 + 360*d**3*e**10*x**6) - 90*b*d**2*e**4*n*x**4*log(d/e + x)/(360*d**9*e**4 + 2160*
d**8*e**5*x + 5400*d**7*e**6*x**2 + 7200*d**6*e**7*x**3 + 5400*d**5*e**8*x**4 + 2160*d**4*e**9*x**5 + 360*d**3
*e**10*x**6) + 33*b*d**2*e**4*n*x**4/(360*d**9*e**4 + 2160*d**8*e**5*x + 5400*d**7*e**6*x**2 + 7200*d**6*e**7*
x**3 + 5400*d**5*e**8*x**4 + 2160*d**4*e**9*x**5 + 360*d**3*e**10*x**6) + 90*b*d**2*e**4*x**4*log(c*x**n)/(360
*d**9*e**4 + 2160*d**8*e**5*x + 5400*d**7*e**6*x**2 + 7200*d**6*e**7*x**3 + 5400*d**5*e**8*x**4 + 2160*d**4*e*
*9*x**5 + 360*d**3*e**10*x**6) - 36*b*d*e**5*n*x**5*log(d/e + x)/(360*d**9*e**4 + 2160*d**8*e**5*x + 5400*d**7
*e**6*x**2 + 7200*d**6*e**7*x**3 + 5400*d**5*e**8*x**4 + 2160*d**4*e**9*x**5 + 360*d**3*e**10*x**6) + 6*b*d*e*
*5*n*x**5/(360*d**9*e**4 + 2160*d**8*e**5*x + 5400*d**7*e**6*x**2 + 7200*d**6*e**7*x**3 + 5400*d**5*e**8*x**4
+ 2160*d**4*e**9*x**5 + 360*d**3*e**10*x**6) + 36*b*d*e**5*x**5*log(c*x**n)/(360*d**9*e**4 + 2160*d**8*e**5*x
+ 5400*d**7*e**6*x**2 + 7200*d**6*e**7*x**3 + 5400*d**5*e**8*x**4 + 2160*d**4*e**9*x**5 + 360*d**3*e**10*x**6)
- 6*b*e**6*n*x**6*log(d/e + x)/(360*d**9*e**4 + 2160*d**8*e**5*x + 5400*d**7*e**6*x**2 + 7200*d**6*e**7*x**3
+ 5400*d**5*e**8*x**4 + 2160*d**4*e**9*x**5 + 360*d**3*e**10*x**6) + 6*b*e**6*x**6*log(c*x**n)/(360*d**9*e**4
+ 2160*d**8*e**5*x + 5400*d**7*e**6*x**2 + 7200*d**6*e**7*x**3 + 5400*d**5*e**8*x**4 + 2160*d**4*e**9*x**5 + 3
60*d**3*e**10*x**6), True))

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Giac [A]
time = 2.63, size = 372, normalized size = 1.65 \begin {gather*} -\frac {6 \, b n x^{6} e^{6} \log \left (x e + d\right ) + 36 \, b d n x^{5} e^{5} \log \left (x e + d\right ) + 90 \, b d^{2} n x^{4} e^{4} \log \left (x e + d\right ) + 120 \, b d^{3} n x^{3} e^{3} \log \left (x e + d\right ) + 90 \, b d^{4} n x^{2} e^{2} \log \left (x e + d\right ) + 36 \, b d^{5} n x e \log \left (x e + d\right ) - 6 \, b n x^{6} e^{6} \log \left (x\right ) - 36 \, b d n x^{5} e^{5} \log \left (x\right ) - 90 \, b d^{2} n x^{4} e^{4} \log \left (x\right ) - 6 \, b d n x^{5} e^{5} - 33 \, b d^{2} n x^{4} e^{4} - 34 \, b d^{3} n x^{3} e^{3} - 3 \, b d^{4} n x^{2} e^{2} + 6 \, b d^{5} n x e + 6 \, b d^{6} n \log \left (x e + d\right ) + 120 \, b d^{3} x^{3} e^{3} \log \left (c\right ) + 90 \, b d^{4} x^{2} e^{2} \log \left (c\right ) + 36 \, b d^{5} x e \log \left (c\right ) + 2 \, b d^{6} n + 120 \, a d^{3} x^{3} e^{3} + 90 \, a d^{4} x^{2} e^{2} + 36 \, a d^{5} x e + 6 \, b d^{6} \log \left (c\right ) + 6 \, a d^{6}}{360 \, {\left (d^{3} x^{6} e^{10} + 6 \, d^{4} x^{5} e^{9} + 15 \, d^{5} x^{4} e^{8} + 20 \, d^{6} x^{3} e^{7} + 15 \, d^{7} x^{2} e^{6} + 6 \, d^{8} x e^{5} + d^{9} e^{4}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(a+b*log(c*x^n))/(e*x+d)^7,x, algorithm="giac")

[Out]

-1/360*(6*b*n*x^6*e^6*log(x*e + d) + 36*b*d*n*x^5*e^5*log(x*e + d) + 90*b*d^2*n*x^4*e^4*log(x*e + d) + 120*b*d
^3*n*x^3*e^3*log(x*e + d) + 90*b*d^4*n*x^2*e^2*log(x*e + d) + 36*b*d^5*n*x*e*log(x*e + d) - 6*b*n*x^6*e^6*log(
x) - 36*b*d*n*x^5*e^5*log(x) - 90*b*d^2*n*x^4*e^4*log(x) - 6*b*d*n*x^5*e^5 - 33*b*d^2*n*x^4*e^4 - 34*b*d^3*n*x
^3*e^3 - 3*b*d^4*n*x^2*e^2 + 6*b*d^5*n*x*e + 6*b*d^6*n*log(x*e + d) + 120*b*d^3*x^3*e^3*log(c) + 90*b*d^4*x^2*
e^2*log(c) + 36*b*d^5*x*e*log(c) + 2*b*d^6*n + 120*a*d^3*x^3*e^3 + 90*a*d^4*x^2*e^2 + 36*a*d^5*x*e + 6*b*d^6*l
og(c) + 6*a*d^6)/(d^3*x^6*e^10 + 6*d^4*x^5*e^9 + 15*d^5*x^4*e^8 + 20*d^6*x^3*e^7 + 15*d^7*x^2*e^6 + 6*d^8*x*e^
5 + d^9*e^4)

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Mupad [B]
time = 4.23, size = 296, normalized size = 1.31 \begin {gather*} -\frac {x^3\,\left (20\,a\,e^3-\frac {17\,b\,e^3\,n}{3}\right )+x\,\left (6\,a\,d^2\,e+b\,d^2\,e\,n\right )+a\,d^3+x^2\,\left (15\,a\,d\,e^2-\frac {b\,d\,e^2\,n}{2}\right )+\frac {b\,d^3\,n}{3}-\frac {11\,b\,e^4\,n\,x^4}{2\,d}-\frac {b\,e^5\,n\,x^5}{d^2}}{60\,d^6\,e^4+360\,d^5\,e^5\,x+900\,d^4\,e^6\,x^2+1200\,d^3\,e^7\,x^3+900\,d^2\,e^8\,x^4+360\,d\,e^9\,x^5+60\,e^{10}\,x^6}-\frac {\ln \left (c\,x^n\right )\,\left (\frac {b\,d^3}{60\,e^4}+\frac {b\,x^3}{3\,e}+\frac {b\,d\,x^2}{4\,e^2}+\frac {b\,d^2\,x}{10\,e^3}\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 )}{30\,d^3\,e^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^3*(a + b*log(c*x^n)))/(d + e*x)^7,x)

[Out]

- (x^3*(20*a*e^3 - (17*b*e^3*n)/3) + x*(6*a*d^2*e + b*d^2*e*n) + a*d^3 + x^2*(15*a*d*e^2 - (b*d*e^2*n)/2) + (b
*d^3*n)/3 - (11*b*e^4*n*x^4)/(2*d) - (b*e^5*n*x^5)/d^2)/(60*d^6*e^4 + 60*e^10*x^6 + 360*d^5*e^5*x + 360*d*e^9*
x^5 + 900*d^4*e^6*x^2 + 1200*d^3*e^7*x^3 + 900*d^2*e^8*x^4) - (log(c*x^n)*((b*d^3)/(60*e^4) + (b*x^3)/(3*e) +
(b*d*x^2)/(4*e^2) + (b*d^2*x)/(10*e^3)))/(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))/(30*d^3*e^4)

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