3.67 \(\int \frac{x^3 (a+b \log (c x^n))}{(d+e x)^7} \, dx\)
Optimal. Leaf size=226 \[ \frac{d^3 \left (a+b \log \left (c x^n\right )\right )}{6 e^4 (d+e x)^6}-\frac{3 d^2 \left (a+b \log \left (c x^n\right )\right )}{5 e^4 (d+e x)^5}+\frac{3 d \left (a+b \log \left (c x^n\right )\right )}{4 e^4 (d+e x)^4}-\frac{a+b \log \left (c x^n\right )}{3 e^4 (d+e x)^3}-\frac{b d^2 n}{30 e^4 (d+e x)^5}+\frac{b n}{60 d^2 e^4 (d+e x)}+\frac{b n \log (x)}{60 d^3 e^4}-\frac{b n \log (d+e x)}{60 d^3 e^4}+\frac{13 b d n}{120 e^4 (d+e x)^4}-\frac{19 b n}{180 e^4 (d+e x)^3}+\frac{b n}{120 d e^4 (d+e x)^2} \]
[Out]
-(b*d^2*n)/(30*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)/(1
20*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)
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Rubi [A] time = 0.203683, antiderivative size = 226, normalized size of antiderivative = 1.,
number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used =
{43, 2350, 12, 1620} \[ \frac{d^3 \left (a+b \log \left (c x^n\right )\right )}{6 e^4 (d+e x)^6}-\frac{3 d^2 \left (a+b \log \left (c x^n\right )\right )}{5 e^4 (d+e x)^5}+\frac{3 d \left (a+b \log \left (c x^n\right )\right )}{4 e^4 (d+e x)^4}-\frac{a+b \log \left (c x^n\right )}{3 e^4 (d+e x)^3}-\frac{b d^2 n}{30 e^4 (d+e x)^5}+\frac{b n}{60 d^2 e^4 (d+e x)}+\frac{b n \log (x)}{60 d^3 e^4}-\frac{b n \log (d+e x)}{60 d^3 e^4}+\frac{13 b d n}{120 e^4 (d+e x)^4}-\frac{19 b n}{180 e^4 (d+e x)^3}+\frac{b n}{120 d e^4 (d+e x)^2} \]
Antiderivative was successfully verified.
[In]
Int[(x^3*(a + b*Log[c*x^n]))/(d + e*x)^7,x]
[Out]
-(b*d^2*n)/(30*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)/(1
20*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 43
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 2350
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> Wit
h[{u = IntHide[(f*x)^m*(d + e*x^r)^q, x]}, Dist[a + b*Log[c*x^n], u, x] - Dist[b*n, Int[SimplifyIntegrand[u/x,
x], x], x] /; ((EqQ[r, 1] || EqQ[r, 2]) && IntegerQ[m] && IntegerQ[q - 1/2]) || InverseFunctionFreeQ[u, x]] /
; FreeQ[{a, b, c, d, e, f, m, n, q, r}, x] && IntegerQ[2*q] && ((IntegerQ[m] && IntegerQ[r]) || IGtQ[q, 0])
Rule 12
Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] && !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]
Rule 1620
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]
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*}
Mathematica [A] time = 0.222935, size = 281, normalized size = 1.24 \[ \frac{a d^3}{6 e^4 (d+e x)^6}-\frac{3 a d^2}{5 e^4 (d+e x)^5}+\frac{3 a d}{4 e^4 (d+e x)^4}-\frac{a}{3 e^4 (d+e x)^3}+\frac{b d^3 \log \left (c x^n\right )}{6 e^4 (d+e x)^6}-\frac{3 b d^2 \log \left (c x^n\right )}{5 e^4 (d+e x)^5}+\frac{3 b d \log \left (c x^n\right )}{4 e^4 (d+e x)^4}-\frac{b \log \left (c x^n\right )}{3 e^4 (d+e x)^3}-\frac{b d^2 n}{30 e^4 (d+e x)^5}+\frac{b n}{60 d^2 e^4 (d+e x)}+\frac{b n \log (x)}{60 d^3 e^4}-\frac{b n \log (d+e x)}{60 d^3 e^4}+\frac{13 b d n}{120 e^4 (d+e x)^4}-\frac{19 b n}{180 e^4 (d+e x)^3}+\frac{b n}{120 d e^4 (d+e x)^2} \]
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] time = 0.151, size = 867, normalized size = 3.8 \begin{align*} \text{result too large to display} \end{align*}
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)
[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*(18*I*Pi*b*d^5*e*x*csgn(I*x^n)*csg
n(I*c*x^n)*csgn(I*c)+60*I*Pi*b*d^3*e^3*x^3*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+45*I*Pi*b*d^4*e^2*x^2*csgn(I*x^
n)*csgn(I*c*x^n)*csgn(I*c)-6*a*d^6+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+3*b*d^4*e^2*n*x^2-6*b
*d^5*e*n*x-18*I*Pi*b*d^5*e*x*csgn(I*x^n)*csgn(I*c*x^n)^2-18*I*Pi*b*d^5*e*x*csgn(I*c*x^n)^2*csgn(I*c)-60*I*Pi*b
*d^3*e^3*x^3*csgn(I*c*x^n)^2*csgn(I*c)-45*I*Pi*b*d^4*e^2*x^2*csgn(I*c*x^n)^2*csgn(I*c)-120*a*d^3*e^3*x^3-90*a*
d^4*e^2*x^2-36*a*d^5*e*x-45*I*Pi*b*d^4*e^2*x^2*csgn(I*x^n)*csgn(I*c*x^n)^2-60*I*Pi*b*d^3*e^3*x^3*csgn(I*x^n)*c
sgn(I*c*x^n)^2-3*I*Pi*b*d^6*csgn(I*x^n)*csgn(I*c*x^n)^2-3*I*Pi*b*d^6*csgn(I*c*x^n)^2*csgn(I*c)-2*b*d^6*n+3*I*P
i*b*d^6*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-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+90*ln(-x)*b*d^4*e^2*n*x^2+36*ln(-x)*b*d^5*e*n*x+60*I*Pi*b*d^3*e^3*x^3
*csgn(I*c*x^n)^3+45*I*Pi*b*d^4*e^2*x^2*csgn(I*c*x^n)^3+18*I*Pi*b*d^5*e*x*csgn(I*c*x^n)^3-6*ln(c)*b*d^6+3*I*Pi*
b*d^6*csgn(I*c*x^n)^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-6*ln(e*x+d)*b*d^6*n+6*ln(-x)*b*d^6*n)/e^4/d^3/(e*x+d)^6
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Maxima [A] time = 1.23264, size = 456, normalized size = 2.02 \begin{align*} \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 )}} \end{align*}
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*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)
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Fricas [A] time = 1.55594, size = 748, normalized size = 3.31 \begin{align*} \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 )}} \end{align*}
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*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)
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Sympy [A] time = 103.044, size = 2280, normalized size = 10.09 \begin{align*} \text{result too large to display} \end{align*}
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*log(x)/(3*x**3) - b*n/(9*x**3) - b*log(c)/(3*x**3)), Eq(d, 0) & Eq(e, 0)), (
(a*x**4/4 + b*n*x**4*log(x)/4 - b*n*x**4/16 + b*x**4*log(c)/4)/d**7, Eq(e, 0)), ((-a/(3*x**3) - b*n*log(x)/(3*
x**3) - b*n/(9*x**3) - b*log(c)/(3*x**3))/e**7, Eq(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 + 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*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 + 5400*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 + 3
60*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 + 21
60*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(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**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)/(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(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) - 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 + 5
400*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)/(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(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*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)/(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), True))
________________________________________________________________________________________
Giac [A] time = 1.22734, size = 502, normalized size = 2.22 \begin{align*} -\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{align*}
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)