3.69 \(\int \frac{x (a+b \log (c x^n))}{(d+e x)^7} \, dx\)
Optimal. Leaf size=174 \[ -\frac{a+b \log \left (c x^n\right )}{5 e^2 (d+e x)^5}+\frac{d \left (a+b \log \left (c x^n\right )\right )}{6 e^2 (d+e x)^6}+\frac{b n}{30 d^4 e^2 (d+e x)}+\frac{b n}{60 d^3 e^2 (d+e x)^2}+\frac{b n}{90 d^2 e^2 (d+e x)^3}+\frac{b n \log (x)}{30 d^5 e^2}-\frac{b n \log (d+e x)}{30 d^5 e^2}+\frac{b n}{120 d e^2 (d+e x)^4}-\frac{b n}{30 e^2 (d+e x)^5} \]
[Out]
-(b*n)/(30*e^2*(d + e*x)^5) + (b*n)/(120*d*e^2*(d + e*x)^4) + (b*n)/(90*d^2*e^2*(d + e*x)^3) + (b*n)/(60*d^3*e
^2*(d + e*x)^2) + (b*n)/(30*d^4*e^2*(d + e*x)) + (b*n*Log[x])/(30*d^5*e^2) + (d*(a + b*Log[c*x^n]))/(6*e^2*(d
+ e*x)^6) - (a + b*Log[c*x^n])/(5*e^2*(d + e*x)^5) - (b*n*Log[d + e*x])/(30*d^5*e^2)
________________________________________________________________________________________
Rubi [A] time = 0.117887, antiderivative size = 174, normalized size of antiderivative = 1.,
number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used =
{43, 2350, 12, 77} \[ -\frac{a+b \log \left (c x^n\right )}{5 e^2 (d+e x)^5}+\frac{d \left (a+b \log \left (c x^n\right )\right )}{6 e^2 (d+e x)^6}+\frac{b n}{30 d^4 e^2 (d+e x)}+\frac{b n}{60 d^3 e^2 (d+e x)^2}+\frac{b n}{90 d^2 e^2 (d+e x)^3}+\frac{b n \log (x)}{30 d^5 e^2}-\frac{b n \log (d+e x)}{30 d^5 e^2}+\frac{b n}{120 d e^2 (d+e x)^4}-\frac{b n}{30 e^2 (d+e x)^5} \]
Antiderivative was successfully verified.
[In]
Int[(x*(a + b*Log[c*x^n]))/(d + e*x)^7,x]
[Out]
-(b*n)/(30*e^2*(d + e*x)^5) + (b*n)/(120*d*e^2*(d + e*x)^4) + (b*n)/(90*d^2*e^2*(d + e*x)^3) + (b*n)/(60*d^3*e
^2*(d + e*x)^2) + (b*n)/(30*d^4*e^2*(d + e*x)) + (b*n*Log[x])/(30*d^5*e^2) + (d*(a + b*Log[c*x^n]))/(6*e^2*(d
+ e*x)^6) - (a + b*Log[c*x^n])/(5*e^2*(d + e*x)^5) - (b*n*Log[d + e*x])/(30*d^5*e^2)
Rule 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 77
Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))
Rubi steps
\begin{align*} \int \frac{x \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^7} \, dx &=\frac{d \left (a+b \log \left (c x^n\right )\right )}{6 e^2 (d+e x)^6}-\frac{a+b \log \left (c x^n\right )}{5 e^2 (d+e x)^5}-(b n) \int \frac{-d-6 e x}{30 e^2 x (d+e x)^6} \, dx\\ &=\frac{d \left (a+b \log \left (c x^n\right )\right )}{6 e^2 (d+e x)^6}-\frac{a+b \log \left (c x^n\right )}{5 e^2 (d+e x)^5}-\frac{(b n) \int \frac{-d-6 e x}{x (d+e x)^6} \, dx}{30 e^2}\\ &=\frac{d \left (a+b \log \left (c x^n\right )\right )}{6 e^2 (d+e x)^6}-\frac{a+b \log \left (c x^n\right )}{5 e^2 (d+e x)^5}-\frac{(b n) \int \left (-\frac{1}{d^5 x}-\frac{5 e}{(d+e x)^6}+\frac{e}{d (d+e x)^5}+\frac{e}{d^2 (d+e x)^4}+\frac{e}{d^3 (d+e x)^3}+\frac{e}{d^4 (d+e x)^2}+\frac{e}{d^5 (d+e x)}\right ) \, dx}{30 e^2}\\ &=-\frac{b n}{30 e^2 (d+e x)^5}+\frac{b n}{120 d e^2 (d+e x)^4}+\frac{b n}{90 d^2 e^2 (d+e x)^3}+\frac{b n}{60 d^3 e^2 (d+e x)^2}+\frac{b n}{30 d^4 e^2 (d+e x)}+\frac{b n \log (x)}{30 d^5 e^2}+\frac{d \left (a+b \log \left (c x^n\right )\right )}{6 e^2 (d+e x)^6}-\frac{a+b \log \left (c x^n\right )}{5 e^2 (d+e x)^5}-\frac{b n \log (d+e x)}{30 d^5 e^2}\\ \end{align*}
Mathematica [A] time = 0.14792, size = 160, normalized size = 0.92 \[ \frac{-72 a d^5 (d+e x)+60 a d^6-72 b d^5 (d+e x) \log \left (c x^n\right )+60 b d^6 \log \left (c x^n\right )-12 b d^5 n (d+e x)+3 b d^4 n (d+e x)^2+4 b d^3 n (d+e x)^3+6 b d^2 n (d+e x)^4+12 b d n (d+e x)^5+12 b n \log (x) (d+e x)^6-12 b n (d+e x)^6 \log (d+e x)}{360 d^5 e^2 (d+e x)^6} \]
Antiderivative was successfully verified.
[In]
Integrate[(x*(a + b*Log[c*x^n]))/(d + e*x)^7,x]
[Out]
(60*a*d^6 - 72*a*d^5*(d + e*x) - 12*b*d^5*n*(d + e*x) + 3*b*d^4*n*(d + e*x)^2 + 4*b*d^3*n*(d + e*x)^3 + 6*b*d^
2*n*(d + e*x)^4 + 12*b*d*n*(d + e*x)^5 + 12*b*n*(d + e*x)^6*Log[x] + 60*b*d^6*Log[c*x^n] - 72*b*d^5*(d + e*x)*
Log[c*x^n] - 12*b*n*(d + e*x)^6*Log[d + e*x])/(360*d^5*e^2*(d + e*x)^6)
________________________________________________________________________________________
Maple [C] time = 0.128, size = 557, normalized size = 3.2 \begin{align*} -{\frac{b \left ( 6\,ex+d \right ) \ln \left ({x}^{n} \right ) }{30\, \left ( ex+d \right ) ^{6}{e}^{2}}}-{\frac{12\,a{d}^{6}-12\,bd{e}^{5}n{x}^{5}-66\,b{d}^{2}{e}^{4}n{x}^{4}-148\,b{d}^{3}{e}^{3}n{x}^{3}-171\,b{d}^{4}{e}^{2}n{x}^{2}-90\,b{d}^{5}enx-36\,i\pi \,b{d}^{5}ex \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}-6\,i\pi \,b{d}^{6}{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ) +72\,a{d}^{5}ex+36\,i\pi \,b{d}^{5}ex \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) -36\,i\pi \,b{d}^{5}ex{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ) +36\,i\pi \,b{d}^{5}ex{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}+6\,i\pi \,b{d}^{6}{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}+6\,i\pi \,b{d}^{6} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) -13\,b{d}^{6}n+72\,\ln \left ( ex+d \right ) bd{e}^{5}n{x}^{5}+180\,\ln \left ( ex+d \right ) b{d}^{2}{e}^{4}n{x}^{4}+240\,\ln \left ( ex+d \right ) b{d}^{3}{e}^{3}n{x}^{3}+180\,\ln \left ( ex+d \right ) b{d}^{4}{e}^{2}n{x}^{2}+72\,\ln \left ( ex+d \right ) b{d}^{5}enx-72\,\ln \left ( -x \right ) bd{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}-180\,\ln \left ( -x \right ) b{d}^{4}{e}^{2}n{x}^{2}-72\,\ln \left ( -x \right ) b{d}^{5}enx+12\,\ln \left ( c \right ) b{d}^{6}-6\,i\pi \,b{d}^{6} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}+12\,\ln \left ( ex+d \right ) b{e}^{6}n{x}^{6}-12\,\ln \left ( -x \right ) b{e}^{6}n{x}^{6}+72\,\ln \left ( c \right ) b{d}^{5}ex+12\,\ln \left ( ex+d \right ) b{d}^{6}n-12\,\ln \left ( -x \right ) b{d}^{6}n}{360\,{e}^{2}{d}^{5} \left ( ex+d \right ) ^{6}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x*(a+b*ln(c*x^n))/(e*x+d)^7,x)
[Out]
-1/30*b*(6*e*x+d)/(e*x+d)^6/e^2*ln(x^n)-1/360*(12*a*d^6-12*b*d*e^5*n*x^5-66*b*d^2*e^4*n*x^4-148*b*d^3*e^3*n*x^
3-171*b*d^4*e^2*n*x^2-90*b*d^5*e*n*x-36*I*Pi*b*d^5*e*x*csgn(I*c*x^n)^3-6*I*Pi*b*d^6*csgn(I*x^n)*csgn(I*c*x^n)*
csgn(I*c)+72*a*d^5*e*x+36*I*Pi*b*d^5*e*x*csgn(I*c*x^n)^2*csgn(I*c)-36*I*Pi*b*d^5*e*x*csgn(I*x^n)*csgn(I*c*x^n)
*csgn(I*c)+36*I*Pi*b*d^5*e*x*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+6*I*Pi*b*d^6
*csgn(I*c*x^n)^2*csgn(I*c)-13*b*d^6*n+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^3-180*ln(-x)*b*d^4*e^2*n*x^2-72*ln(-x)*b*d^5*e*n*x+12*ln(c)*b*d^6-6*I*Pi*b
*d^6*csgn(I*c*x^n)^3+12*ln(e*x+d)*b*e^6*n*x^6-12*ln(-x)*b*e^6*n*x^6+72*ln(c)*b*d^5*e*x+12*ln(e*x+d)*b*d^6*n-12
*ln(-x)*b*d^6*n)/e^2/d^5/(e*x+d)^6
________________________________________________________________________________________
Maxima [A] time = 1.20077, size = 397, normalized size = 2.28 \begin{align*} \frac{1}{360} \, b n{\left (\frac{12 \, e^{4} x^{4} + 54 \, d e^{3} x^{3} + 94 \, d^{2} e^{2} x^{2} + 77 \, d^{3} e x + 13 \, d^{4}}{d^{4} e^{7} x^{5} + 5 \, d^{5} e^{6} x^{4} + 10 \, d^{6} e^{5} x^{3} + 10 \, d^{7} e^{4} x^{2} + 5 \, d^{8} e^{3} x + d^{9} e^{2}} - \frac{12 \, \log \left (e x + d\right )}{d^{5} e^{2}} + \frac{12 \, \log \left (x\right )}{d^{5} e^{2}}\right )} - \frac{{\left (6 \, e x + d\right )} b \log \left (c x^{n}\right )}{30 \,{\left (e^{8} x^{6} + 6 \, d e^{7} x^{5} + 15 \, d^{2} e^{6} x^{4} + 20 \, d^{3} e^{5} x^{3} + 15 \, d^{4} e^{4} x^{2} + 6 \, d^{5} e^{3} x + d^{6} e^{2}\right )}} - \frac{{\left (6 \, e x + d\right )} a}{30 \,{\left (e^{8} x^{6} + 6 \, d e^{7} x^{5} + 15 \, d^{2} e^{6} x^{4} + 20 \, d^{3} e^{5} x^{3} + 15 \, d^{4} e^{4} x^{2} + 6 \, d^{5} e^{3} x + d^{6} e^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x*(a+b*log(c*x^n))/(e*x+d)^7,x, algorithm="maxima")
[Out]
1/360*b*n*((12*e^4*x^4 + 54*d*e^3*x^3 + 94*d^2*e^2*x^2 + 77*d^3*e*x + 13*d^4)/(d^4*e^7*x^5 + 5*d^5*e^6*x^4 + 1
0*d^6*e^5*x^3 + 10*d^7*e^4*x^2 + 5*d^8*e^3*x + d^9*e^2) - 12*log(e*x + d)/(d^5*e^2) + 12*log(x)/(d^5*e^2)) - 1
/30*(6*e*x + d)*b*log(c*x^n)/(e^8*x^6 + 6*d*e^7*x^5 + 15*d^2*e^6*x^4 + 20*d^3*e^5*x^3 + 15*d^4*e^4*x^2 + 6*d^5
*e^3*x + d^6*e^2) - 1/30*(6*e*x + d)*a/(e^8*x^6 + 6*d*e^7*x^5 + 15*d^2*e^6*x^4 + 20*d^3*e^5*x^3 + 15*d^4*e^4*x
^2 + 6*d^5*e^3*x + d^6*e^2)
________________________________________________________________________________________
Fricas [B] time = 1.49607, size = 722, normalized size = 4.15 \begin{align*} \frac{12 \, b d e^{5} n x^{5} + 66 \, b d^{2} e^{4} n x^{4} + 148 \, b d^{3} e^{3} n x^{3} + 171 \, b d^{4} e^{2} n x^{2} + 13 \, b d^{6} n - 12 \, a d^{6} + 18 \,{\left (5 \, b d^{5} e n - 4 \, a d^{5} e\right )} x - 12 \,{\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 ) - 12 \,{\left (6 \, b d^{5} e x + b d^{6}\right )} \log \left (c\right ) + 12 \,{\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}\right )} \log \left (x\right )}{360 \,{\left (d^{5} e^{8} x^{6} + 6 \, d^{6} e^{7} x^{5} + 15 \, d^{7} e^{6} x^{4} + 20 \, d^{8} e^{5} x^{3} + 15 \, d^{9} e^{4} x^{2} + 6 \, d^{10} e^{3} x + d^{11} e^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x*(a+b*log(c*x^n))/(e*x+d)^7,x, algorithm="fricas")
[Out]
1/360*(12*b*d*e^5*n*x^5 + 66*b*d^2*e^4*n*x^4 + 148*b*d^3*e^3*n*x^3 + 171*b*d^4*e^2*n*x^2 + 13*b*d^6*n - 12*a*d
^6 + 18*(5*b*d^5*e*n - 4*a*d^5*e)*x - 12*(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) - 12*(6*b*d^5*e*x + b*d^6)*log(c) + 12*(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)*log(x))/(d^5*e^8*x^6
+ 6*d^6*e^7*x^5 + 15*d^7*e^6*x^4 + 20*d^8*e^5*x^3 + 15*d^9*e^4*x^2 + 6*d^10*e^3*x + d^11*e^2)
________________________________________________________________________________________
Sympy [A] time = 135.978, size = 2480, normalized size = 14.25 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x*(a+b*ln(c*x**n))/(e*x+d)**7,x)
[Out]
Piecewise((zoo*(-a/(5*x**5) - b*n*log(x)/(5*x**5) - b*n/(25*x**5) - b*log(c)/(5*x**5)), Eq(d, 0) & Eq(e, 0)),
((a*x**2/2 + b*n*x**2*log(x)/2 - b*n*x**2/4 + b*x**2*log(c)/2)/d**7, Eq(e, 0)), ((-a/(5*x**5) - b*n*log(x)/(5*
x**5) - b*n/(25*x**5) - b*log(c)/(5*x**5))/e**7, Eq(d, 0)), (-12*a*d**6/(360*d**11*e**2 + 2160*d**10*e**3*x +
5400*d**9*e**4*x**2 + 7200*d**8*e**5*x**3 + 5400*d**7*e**6*x**4 + 2160*d**6*e**7*x**5 + 360*d**5*e**8*x**6) -
72*a*d**5*e*x/(360*d**11*e**2 + 2160*d**10*e**3*x + 5400*d**9*e**4*x**2 + 7200*d**8*e**5*x**3 + 5400*d**7*e**6
*x**4 + 2160*d**6*e**7*x**5 + 360*d**5*e**8*x**6) - 12*b*d**6*n*log(d/e + x)/(360*d**11*e**2 + 2160*d**10*e**3
*x + 5400*d**9*e**4*x**2 + 7200*d**8*e**5*x**3 + 5400*d**7*e**6*x**4 + 2160*d**6*e**7*x**5 + 360*d**5*e**8*x**
6) + 13*b*d**6*n/(360*d**11*e**2 + 2160*d**10*e**3*x + 5400*d**9*e**4*x**2 + 7200*d**8*e**5*x**3 + 5400*d**7*e
**6*x**4 + 2160*d**6*e**7*x**5 + 360*d**5*e**8*x**6) - 72*b*d**5*e*n*x*log(d/e + x)/(360*d**11*e**2 + 2160*d**
10*e**3*x + 5400*d**9*e**4*x**2 + 7200*d**8*e**5*x**3 + 5400*d**7*e**6*x**4 + 2160*d**6*e**7*x**5 + 360*d**5*e
**8*x**6) + 90*b*d**5*e*n*x/(360*d**11*e**2 + 2160*d**10*e**3*x + 5400*d**9*e**4*x**2 + 7200*d**8*e**5*x**3 +
5400*d**7*e**6*x**4 + 2160*d**6*e**7*x**5 + 360*d**5*e**8*x**6) + 180*b*d**4*e**2*n*x**2*log(x)/(360*d**11*e**
2 + 2160*d**10*e**3*x + 5400*d**9*e**4*x**2 + 7200*d**8*e**5*x**3 + 5400*d**7*e**6*x**4 + 2160*d**6*e**7*x**5
+ 360*d**5*e**8*x**6) - 180*b*d**4*e**2*n*x**2*log(d/e + x)/(360*d**11*e**2 + 2160*d**10*e**3*x + 5400*d**9*e*
*4*x**2 + 7200*d**8*e**5*x**3 + 5400*d**7*e**6*x**4 + 2160*d**6*e**7*x**5 + 360*d**5*e**8*x**6) + 171*b*d**4*e
**2*n*x**2/(360*d**11*e**2 + 2160*d**10*e**3*x + 5400*d**9*e**4*x**2 + 7200*d**8*e**5*x**3 + 5400*d**7*e**6*x*
*4 + 2160*d**6*e**7*x**5 + 360*d**5*e**8*x**6) + 180*b*d**4*e**2*x**2*log(c)/(360*d**11*e**2 + 2160*d**10*e**3
*x + 5400*d**9*e**4*x**2 + 7200*d**8*e**5*x**3 + 5400*d**7*e**6*x**4 + 2160*d**6*e**7*x**5 + 360*d**5*e**8*x**
6) + 240*b*d**3*e**3*n*x**3*log(x)/(360*d**11*e**2 + 2160*d**10*e**3*x + 5400*d**9*e**4*x**2 + 7200*d**8*e**5*
x**3 + 5400*d**7*e**6*x**4 + 2160*d**6*e**7*x**5 + 360*d**5*e**8*x**6) - 240*b*d**3*e**3*n*x**3*log(d/e + x)/(
360*d**11*e**2 + 2160*d**10*e**3*x + 5400*d**9*e**4*x**2 + 7200*d**8*e**5*x**3 + 5400*d**7*e**6*x**4 + 2160*d*
*6*e**7*x**5 + 360*d**5*e**8*x**6) + 148*b*d**3*e**3*n*x**3/(360*d**11*e**2 + 2160*d**10*e**3*x + 5400*d**9*e*
*4*x**2 + 7200*d**8*e**5*x**3 + 5400*d**7*e**6*x**4 + 2160*d**6*e**7*x**5 + 360*d**5*e**8*x**6) + 240*b*d**3*e
**3*x**3*log(c)/(360*d**11*e**2 + 2160*d**10*e**3*x + 5400*d**9*e**4*x**2 + 7200*d**8*e**5*x**3 + 5400*d**7*e*
*6*x**4 + 2160*d**6*e**7*x**5 + 360*d**5*e**8*x**6) + 180*b*d**2*e**4*n*x**4*log(x)/(360*d**11*e**2 + 2160*d**
10*e**3*x + 5400*d**9*e**4*x**2 + 7200*d**8*e**5*x**3 + 5400*d**7*e**6*x**4 + 2160*d**6*e**7*x**5 + 360*d**5*e
**8*x**6) - 180*b*d**2*e**4*n*x**4*log(d/e + x)/(360*d**11*e**2 + 2160*d**10*e**3*x + 5400*d**9*e**4*x**2 + 72
00*d**8*e**5*x**3 + 5400*d**7*e**6*x**4 + 2160*d**6*e**7*x**5 + 360*d**5*e**8*x**6) + 66*b*d**2*e**4*n*x**4/(3
60*d**11*e**2 + 2160*d**10*e**3*x + 5400*d**9*e**4*x**2 + 7200*d**8*e**5*x**3 + 5400*d**7*e**6*x**4 + 2160*d**
6*e**7*x**5 + 360*d**5*e**8*x**6) + 180*b*d**2*e**4*x**4*log(c)/(360*d**11*e**2 + 2160*d**10*e**3*x + 5400*d**
9*e**4*x**2 + 7200*d**8*e**5*x**3 + 5400*d**7*e**6*x**4 + 2160*d**6*e**7*x**5 + 360*d**5*e**8*x**6) + 72*b*d*e
**5*n*x**5*log(x)/(360*d**11*e**2 + 2160*d**10*e**3*x + 5400*d**9*e**4*x**2 + 7200*d**8*e**5*x**3 + 5400*d**7*
e**6*x**4 + 2160*d**6*e**7*x**5 + 360*d**5*e**8*x**6) - 72*b*d*e**5*n*x**5*log(d/e + x)/(360*d**11*e**2 + 2160
*d**10*e**3*x + 5400*d**9*e**4*x**2 + 7200*d**8*e**5*x**3 + 5400*d**7*e**6*x**4 + 2160*d**6*e**7*x**5 + 360*d*
*5*e**8*x**6) + 12*b*d*e**5*n*x**5/(360*d**11*e**2 + 2160*d**10*e**3*x + 5400*d**9*e**4*x**2 + 7200*d**8*e**5*
x**3 + 5400*d**7*e**6*x**4 + 2160*d**6*e**7*x**5 + 360*d**5*e**8*x**6) + 72*b*d*e**5*x**5*log(c)/(360*d**11*e*
*2 + 2160*d**10*e**3*x + 5400*d**9*e**4*x**2 + 7200*d**8*e**5*x**3 + 5400*d**7*e**6*x**4 + 2160*d**6*e**7*x**5
+ 360*d**5*e**8*x**6) + 12*b*e**6*n*x**6*log(x)/(360*d**11*e**2 + 2160*d**10*e**3*x + 5400*d**9*e**4*x**2 + 7
200*d**8*e**5*x**3 + 5400*d**7*e**6*x**4 + 2160*d**6*e**7*x**5 + 360*d**5*e**8*x**6) - 12*b*e**6*n*x**6*log(d/
e + x)/(360*d**11*e**2 + 2160*d**10*e**3*x + 5400*d**9*e**4*x**2 + 7200*d**8*e**5*x**3 + 5400*d**7*e**6*x**4 +
2160*d**6*e**7*x**5 + 360*d**5*e**8*x**6) + 12*b*e**6*x**6*log(c)/(360*d**11*e**2 + 2160*d**10*e**3*x + 5400*
d**9*e**4*x**2 + 7200*d**8*e**5*x**3 + 5400*d**7*e**6*x**4 + 2160*d**6*e**7*x**5 + 360*d**5*e**8*x**6), True))
________________________________________________________________________________________
Giac [B] time = 1.26505, size = 475, normalized size = 2.73 \begin{align*} -\frac{12 \, b n x^{6} e^{6} \log \left (x e + d\right ) + 72 \, b d n x^{5} e^{5} \log \left (x e + d\right ) + 180 \, b d^{2} n x^{4} e^{4} \log \left (x e + d\right ) + 240 \, b d^{3} n x^{3} e^{3} \log \left (x e + d\right ) + 180 \, b d^{4} n x^{2} e^{2} \log \left (x e + d\right ) + 72 \, b d^{5} n x e \log \left (x e + d\right ) - 12 \, b n x^{6} e^{6} \log \left (x\right ) - 72 \, b d n x^{5} e^{5} \log \left (x\right ) - 180 \, b d^{2} n x^{4} e^{4} \log \left (x\right ) - 240 \, b d^{3} n x^{3} e^{3} \log \left (x\right ) - 180 \, b d^{4} n x^{2} e^{2} \log \left (x\right ) - 12 \, b d n x^{5} e^{5} - 66 \, b d^{2} n x^{4} e^{4} - 148 \, b d^{3} n x^{3} e^{3} - 171 \, b d^{4} n x^{2} e^{2} - 90 \, b d^{5} n x e + 12 \, b d^{6} n \log \left (x e + d\right ) + 72 \, b d^{5} x e \log \left (c\right ) - 13 \, b d^{6} n + 72 \, a d^{5} x e + 12 \, b d^{6} \log \left (c\right ) + 12 \, a d^{6}}{360 \,{\left (d^{5} x^{6} e^{8} + 6 \, d^{6} x^{5} e^{7} + 15 \, d^{7} x^{4} e^{6} + 20 \, d^{8} x^{3} e^{5} + 15 \, d^{9} x^{2} e^{4} + 6 \, d^{10} x e^{3} + d^{11} e^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x*(a+b*log(c*x^n))/(e*x+d)^7,x, algorithm="giac")
[Out]
-1/360*(12*b*n*x^6*e^6*log(x*e + d) + 72*b*d*n*x^5*e^5*log(x*e + d) + 180*b*d^2*n*x^4*e^4*log(x*e + d) + 240*b
*d^3*n*x^3*e^3*log(x*e + d) + 180*b*d^4*n*x^2*e^2*log(x*e + d) + 72*b*d^5*n*x*e*log(x*e + d) - 12*b*n*x^6*e^6*
log(x) - 72*b*d*n*x^5*e^5*log(x) - 180*b*d^2*n*x^4*e^4*log(x) - 240*b*d^3*n*x^3*e^3*log(x) - 180*b*d^4*n*x^2*e
^2*log(x) - 12*b*d*n*x^5*e^5 - 66*b*d^2*n*x^4*e^4 - 148*b*d^3*n*x^3*e^3 - 171*b*d^4*n*x^2*e^2 - 90*b*d^5*n*x*e
+ 12*b*d^6*n*log(x*e + d) + 72*b*d^5*x*e*log(c) - 13*b*d^6*n + 72*a*d^5*x*e + 12*b*d^6*log(c) + 12*a*d^6)/(d^
5*x^6*e^8 + 6*d^6*x^5*e^7 + 15*d^7*x^4*e^6 + 20*d^8*x^3*e^5 + 15*d^9*x^2*e^4 + 6*d^10*x*e^3 + d^11*e^2)