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

Optimal result

Integrand size = 21, antiderivative size = 226 \[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^7} \, dx=-\frac {b d^2 n}{30 e^4 (d+e x)^5}+\frac {13 b d n}{120 e^4 (d+e x)^4}-\frac {19 b n}{180 e^4 (d+e x)^3}+\frac {b n}{120 d e^4 (d+e x)^2}+\frac {b n}{60 d^2 e^4 (d+e x)}+\frac {b n \log (x)}{60 d^3 e^4}+\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{6 e^4 (d+e x)^6}-\frac {3 d^2 \left (a+b \log \left (c x^n\right )\right )}{5 e^4 (d+e x)^5}+\frac {3 d \left (a+b \log \left (c x^n\right )\right )}{4 e^4 (d+e x)^4}-\frac {a+b \log \left (c x^n\right )}{3 e^4 (d+e x)^3}-\frac {b n \log (d+e x)}{60 d^3 e^4} \]

[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

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {45, 2382, 12, 1634} \[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^7} \, dx=\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{6 e^4 (d+e x)^6}-\frac {3 d^2 \left (a+b \log \left (c x^n\right )\right )}{5 e^4 (d+e x)^5}+\frac {3 d \left (a+b \log \left (c x^n\right )\right )}{4 e^4 (d+e x)^4}-\frac {a+b \log \left (c x^n\right )}{3 e^4 (d+e x)^3}+\frac {b n \log (x)}{60 d^3 e^4}-\frac {b n \log (d+e x)}{60 d^3 e^4}-\frac {b d^2 n}{30 e^4 (d+e x)^5}+\frac {b n}{60 d^2 e^4 (d+e x)}+\frac {13 b d n}{120 e^4 (d+e x)^4}-\frac {19 b n}{180 e^4 (d+e x)^3}+\frac {b n}{120 d e^4 (d+e x)^2} \]

[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*} \text {integral}& = \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*}

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 281, normalized size of antiderivative = 1.24 \[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^7} \, dx=\frac {a d^3}{6 e^4 (d+e x)^6}-\frac {3 a d^2}{5 e^4 (d+e x)^5}-\frac {b d^2 n}{30 e^4 (d+e x)^5}+\frac {3 a d}{4 e^4 (d+e x)^4}+\frac {13 b d n}{120 e^4 (d+e x)^4}-\frac {a}{3 e^4 (d+e x)^3}-\frac {19 b n}{180 e^4 (d+e x)^3}+\frac {b n}{120 d e^4 (d+e x)^2}+\frac {b n}{60 d^2 e^4 (d+e x)}+\frac {b n \log (x)}{60 d^3 e^4}+\frac {b d^3 \log \left (c x^n\right )}{6 e^4 (d+e x)^6}-\frac {3 b d^2 \log \left (c x^n\right )}{5 e^4 (d+e x)^5}+\frac {3 b d \log \left (c x^n\right )}{4 e^4 (d+e x)^4}-\frac {b \log \left (c x^n\right )}{3 e^4 (d+e x)^3}-\frac {b n \log (d+e x)}{60 d^3 e^4} \]

[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)

Maple [A] (verified)

Time = 1.34 (sec) , antiderivative size = 413, normalized size of antiderivative = 1.83

method	result	size
parallelrisch	\(\frac {-450 x^{2} \ln \left (c \,x^{n}\right ) b \,d^{4} e^{4}-600 x^{3} \ln \left (c \,x^{n}\right ) b \,d^{3} e^{5}-36 x^{5} b d \,e^{7} n -96 x b \,d^{5} e^{3} n -150 x^{2} b \,d^{4} e^{4} n -50 x^{3} b \,d^{3} e^{5} n +30 \ln \left (x \right ) x^{6} b \,e^{8} n -30 \ln \left (e x +d \right ) x^{6} b \,e^{8} n +30 \ln \left (x \right ) b \,d^{6} e^{2} n -30 \ln \left (e x +d \right ) b \,d^{6} e^{2} n -180 x \ln \left (c \,x^{n}\right ) b \,d^{5} e^{3}-30 a \,d^{6} e^{2}+180 \ln \left (x \right ) x^{5} b d \,e^{7} n -180 \ln \left (e x +d \right ) x^{5} b d \,e^{7} n +450 \ln \left (x \right ) x^{4} b \,d^{2} e^{6} n -450 \ln \left (e x +d \right ) x^{4} b \,d^{2} e^{6} n +600 \ln \left (x \right ) x^{3} b \,d^{3} e^{5} n -600 \ln \left (e x +d \right ) x^{3} b \,d^{3} e^{5} n +450 \ln \left (x \right ) x^{2} b \,d^{4} e^{4} n -450 \ln \left (e x +d \right ) x^{2} b \,d^{4} e^{4} n +180 \ln \left (x \right ) x b \,d^{5} e^{3} n -180 \ln \left (e x +d \right ) x b \,d^{5} e^{3} n -21 b \,d^{6} e^{2} n -30 \ln \left (c \,x^{n}\right ) b \,d^{6} e^{2}-11 x^{6} b \,e^{8} n -180 x a \,d^{5} e^{3}-450 x^{2} a \,d^{4} e^{4}-600 x^{3} a \,d^{3} e^{5}}{1800 d^{3} e^{6} \left (e x +d \right )^{6}}\)	\(413\)
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 {-90 \ln \left (e x +d \right ) b \,d^{2} e^{4} n \,x^{4}-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 +18 i \pi b \,d^{5} e x \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )+60 i \pi b \,d^{3} e^{3} x^{3} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )+45 i \pi b \,d^{4} e^{2} x^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )+36 \ln \left (-x \right ) b d \,e^{5} n \,x^{5}+3 i \pi b \,d^{6} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+18 i \pi b \,d^{5} e x \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+60 i \pi b \,d^{3} e^{3} x^{3} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}-60 i \pi b \,d^{3} e^{3} x^{3} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-60 i \pi b \,d^{3} e^{3} x^{3} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-45 i \pi b \,d^{4} e^{2} x^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-45 i \pi b \,d^{4} e^{2} x^{2} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-18 i \pi b \,d^{5} e x \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-90 \ln \left (e x +d \right ) b \,d^{4} e^{2} n \,x^{2}-2 b \,d^{6} n +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}+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 \ln \left (-x \right ) b \,e^{6} n \,x^{6}-120 \ln \left (e x +d \right ) b \,d^{3} e^{3} n \,x^{3}-18 i \pi b \,d^{5} e x \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-36 \ln \left (e x +d \right ) b d \,e^{5} n \,x^{5}-120 a \,d^{3} e^{3} x^{3}+3 b \,d^{4} e^{2} n \,x^{2}+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 b \,d^{5} e n x -36 \ln \left (e x +d \right ) b \,d^{5} e n x +6 \ln \left (-x \right ) b \,d^{6} n -6 \ln \left (e x +d \right ) b \,e^{6} n \,x^{6}+45 i \pi b \,d^{4} e^{2} x^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+3 i \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) b \,d^{6}-3 i \pi b \,d^{6} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-90 a \,d^{4} e^{2} x^{2}-36 a \,d^{5} e x -6 \ln \left (e x +d \right ) b \,d^{6} n -6 a \,d^{6}-3 i \pi b \,d^{6} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-6 \ln \left (c \right ) b \,d^{6}}{360 e^{4} d^{3} \left (e x +d \right )^{6}}\)	\(867\)

[In]

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

[Out]

1/1800*(-450*x^2*ln(c*x^n)*b*d^4*e^4-600*x^3*ln(c*x^n)*b*d^3*e^5-36*x^5*b*d*e^7*n-96*x*b*d^5*e^3*n-150*x^2*b*d
^4*e^4*n-50*x^3*b*d^3*e^5*n+30*ln(x)*x^6*b*e^8*n-30*ln(e*x+d)*x^6*b*e^8*n+30*ln(x)*b*d^6*e^2*n-30*ln(e*x+d)*b*
d^6*e^2*n-180*x*ln(c*x^n)*b*d^5*e^3-30*a*d^6*e^2+180*ln(x)*x^5*b*d*e^7*n-180*ln(e*x+d)*x^5*b*d*e^7*n+450*ln(x)
*x^4*b*d^2*e^6*n-450*ln(e*x+d)*x^4*b*d^2*e^6*n+600*ln(x)*x^3*b*d^3*e^5*n-600*ln(e*x+d)*x^3*b*d^3*e^5*n+450*ln(
x)*x^2*b*d^4*e^4*n-450*ln(e*x+d)*x^2*b*d^4*e^4*n+180*ln(x)*x*b*d^5*e^3*n-180*ln(e*x+d)*x*b*d^5*e^3*n-21*b*d^6*
e^2*n-30*ln(c*x^n)*b*d^6*e^2-11*x^6*b*e^8*n-180*x*a*d^5*e^3-450*x^2*a*d^4*e^4-600*x^3*a*d^3*e^5)/d^3/e^6/(e*x+
d)^6

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 343, normalized size of antiderivative = 1.52 \[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^7} \, dx=\frac {6 \, b d e^{5} n x^{5} + 33 \, b d^{2} e^{4} n x^{4} - 2 \, b d^{6} n - 6 \, a d^{6} + 2 \, {\left (17 \, b d^{3} e^{3} n - 60 \, a d^{3} e^{3}\right )} x^{3} + 3 \, {\left (b d^{4} e^{2} n - 30 \, a d^{4} e^{2}\right )} x^{2} - 6 \, {\left (b d^{5} e n + 6 \, a d^{5} e\right )} x - 6 \, {\left (b e^{6} n x^{6} + 6 \, b d e^{5} n x^{5} + 15 \, b d^{2} e^{4} n x^{4} + 20 \, b d^{3} e^{3} n x^{3} + 15 \, b d^{4} e^{2} n x^{2} + 6 \, b d^{5} e n x + b d^{6} n\right )} \log \left (e x + d\right ) - 6 \, {\left (20 \, b d^{3} e^{3} x^{3} + 15 \, b d^{4} e^{2} x^{2} + 6 \, b d^{5} e x + b d^{6}\right )} \log \left (c\right ) + 6 \, {\left (b e^{6} n x^{6} + 6 \, b d e^{5} n x^{5} + 15 \, b d^{2} e^{4} n x^{4}\right )} \log \left (x\right )}{360 \, {\left (d^{3} e^{10} x^{6} + 6 \, d^{4} e^{9} x^{5} + 15 \, d^{5} e^{8} x^{4} + 20 \, d^{6} e^{7} x^{3} + 15 \, d^{7} e^{6} x^{2} + 6 \, d^{8} e^{5} x + d^{9} e^{4}\right )}} \]

[In]

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

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1979 vs. \(2 (224) = 448\).

Time = 75.70 (sec) , antiderivative size = 1979, normalized size of antiderivative = 8.76 \[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^7} \, dx=\text {Too large to display} \]

[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))

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 338, normalized size of antiderivative = 1.50 \[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^7} \, dx=\frac {1}{360} \, b n {\left (\frac {6 \, e^{4} x^{4} + 27 \, d e^{3} x^{3} + 7 \, d^{2} e^{2} x^{2} - 4 \, d^{3} e x - 2 \, d^{4}}{d^{2} e^{9} x^{5} + 5 \, d^{3} e^{8} x^{4} + 10 \, d^{4} e^{7} x^{3} + 10 \, d^{5} e^{6} x^{2} + 5 \, d^{6} e^{5} x + d^{7} e^{4}} - \frac {6 \, \log \left (e x + d\right )}{d^{3} e^{4}} + \frac {6 \, \log \left (x\right )}{d^{3} e^{4}}\right )} - \frac {{\left (20 \, e^{3} x^{3} + 15 \, d e^{2} x^{2} + 6 \, d^{2} e x + d^{3}\right )} b \log \left (c x^{n}\right )}{60 \, {\left (e^{10} x^{6} + 6 \, d e^{9} x^{5} + 15 \, d^{2} e^{8} x^{4} + 20 \, d^{3} e^{7} x^{3} + 15 \, d^{4} e^{6} x^{2} + 6 \, d^{5} e^{5} x + d^{6} e^{4}\right )}} - \frac {{\left (20 \, e^{3} x^{3} + 15 \, d e^{2} x^{2} + 6 \, d^{2} e x + d^{3}\right )} a}{60 \, {\left (e^{10} x^{6} + 6 \, d e^{9} x^{5} + 15 \, d^{2} e^{8} x^{4} + 20 \, d^{3} e^{7} x^{3} + 15 \, d^{4} e^{6} x^{2} + 6 \, d^{5} e^{5} x + d^{6} e^{4}\right )}} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.40 (sec) , antiderivative size = 361, normalized size of antiderivative = 1.60 \[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^7} \, dx=-\frac {{\left (20 \, b e^{3} n x^{3} + 15 \, b d e^{2} n x^{2} + 6 \, b d^{2} e n x + b d^{3} n\right )} \log \left (x\right )}{60 \, {\left (e^{10} x^{6} + 6 \, d e^{9} x^{5} + 15 \, d^{2} e^{8} x^{4} + 20 \, d^{3} e^{7} x^{3} + 15 \, d^{4} e^{6} x^{2} + 6 \, d^{5} e^{5} x + d^{6} e^{4}\right )}} + \frac {6 \, b e^{5} n x^{5} + 33 \, b d e^{4} n x^{4} + 34 \, b d^{2} e^{3} n x^{3} - 120 \, b d^{2} e^{3} x^{3} \log \left (c\right ) + 3 \, b d^{3} e^{2} n x^{2} - 120 \, a d^{2} e^{3} x^{3} - 90 \, b d^{3} e^{2} x^{2} \log \left (c\right ) - 6 \, b d^{4} e n x - 90 \, a d^{3} e^{2} x^{2} - 36 \, b d^{4} e x \log \left (c\right ) - 2 \, b d^{5} n - 36 \, a d^{4} e x - 6 \, b d^{5} \log \left (c\right ) - 6 \, a d^{5}}{360 \, {\left (d^{2} e^{10} x^{6} + 6 \, d^{3} e^{9} x^{5} + 15 \, d^{4} e^{8} x^{4} + 20 \, d^{5} e^{7} x^{3} + 15 \, d^{6} e^{6} x^{2} + 6 \, d^{7} e^{5} x + d^{8} e^{4}\right )}} - \frac {b n \log \left (e x + d\right )}{60 \, d^{3} e^{4}} + \frac {b n \log \left (x\right )}{60 \, d^{3} e^{4}} \]

[In]

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

[Out]

-1/60*(20*b*e^3*n*x^3 + 15*b*d*e^2*n*x^2 + 6*b*d^2*e*n*x + b*d^3*n)*log(x)/(e^10*x^6 + 6*d*e^9*x^5 + 15*d^2*e^
8*x^4 + 20*d^3*e^7*x^3 + 15*d^4*e^6*x^2 + 6*d^5*e^5*x + d^6*e^4) + 1/360*(6*b*e^5*n*x^5 + 33*b*d*e^4*n*x^4 + 3
4*b*d^2*e^3*n*x^3 - 120*b*d^2*e^3*x^3*log(c) + 3*b*d^3*e^2*n*x^2 - 120*a*d^2*e^3*x^3 - 90*b*d^3*e^2*x^2*log(c)
- 6*b*d^4*e*n*x - 90*a*d^3*e^2*x^2 - 36*b*d^4*e*x*log(c) - 2*b*d^5*n - 36*a*d^4*e*x - 6*b*d^5*log(c) - 6*a*d^
5)/(d^2*e^10*x^6 + 6*d^3*e^9*x^5 + 15*d^4*e^8*x^4 + 20*d^5*e^7*x^3 + 15*d^6*e^6*x^2 + 6*d^7*e^5*x + d^8*e^4) -
1/60*b*n*log(e*x + d)/(d^3*e^4) + 1/60*b*n*log(x)/(d^3*e^4)

Mupad [B] (verification not implemented)

Time = 1.01 (sec) , antiderivative size = 296, normalized size of antiderivative = 1.31 \[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^7} \, dx=-\frac {x^3\,\left (20\,a\,e^3-\frac {17\,b\,e^3\,n}{3}\right )+x\,\left (6\,a\,d^2\,e+b\,d^2\,e\,n\right )+a\,d^3+x^2\,\left (15\,a\,d\,e^2-\frac {b\,d\,e^2\,n}{2}\right )+\frac {b\,d^3\,n}{3}-\frac {11\,b\,e^4\,n\,x^4}{2\,d}-\frac {b\,e^5\,n\,x^5}{d^2}}{60\,d^6\,e^4+360\,d^5\,e^5\,x+900\,d^4\,e^6\,x^2+1200\,d^3\,e^7\,x^3+900\,d^2\,e^8\,x^4+360\,d\,e^9\,x^5+60\,e^{10}\,x^6}-\frac {\ln \left (c\,x^n\right )\,\left (\frac {b\,d^3}{60\,e^4}+\frac {b\,x^3}{3\,e}+\frac {b\,d\,x^2}{4\,e^2}+\frac {b\,d^2\,x}{10\,e^3}\right )}{d^6+6\,d^5\,e\,x+15\,d^4\,e^2\,x^2+20\,d^3\,e^3\,x^3+15\,d^2\,e^4\,x^4+6\,d\,e^5\,x^5+e^6\,x^6}-\frac {b\,n\,\mathrm {atanh}\left (\frac {2\,e\,x}{d}+1\right )}{30\,d^3\,e^4} \]

[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)