3.70 \(\int \frac{a+b \log (c x^n)}{(d+e x)^7} \, dx\)
Optimal. Leaf size=152 \[ -\frac{a+b \log \left (c x^n\right )}{6 e (d+e x)^6}+\frac{b n}{6 d^5 e (d+e x)}+\frac{b n}{12 d^4 e (d+e x)^2}+\frac{b n}{18 d^3 e (d+e x)^3}+\frac{b n}{24 d^2 e (d+e x)^4}+\frac{b n \log (x)}{6 d^6 e}-\frac{b n \log (d+e x)}{6 d^6 e}+\frac{b n}{30 d e (d+e x)^5} \]
[Out]
(b*n)/(30*d*e*(d + e*x)^5) + (b*n)/(24*d^2*e*(d + e*x)^4) + (b*n)/(18*d^3*e*(d + e*x)^3) + (b*n)/(12*d^4*e*(d
+ e*x)^2) + (b*n)/(6*d^5*e*(d + e*x)) + (b*n*Log[x])/(6*d^6*e) - (a + b*Log[c*x^n])/(6*e*(d + e*x)^6) - (b*n*L
og[d + e*x])/(6*d^6*e)
________________________________________________________________________________________
Rubi [A] time = 0.0654523, antiderivative size = 152, normalized size of antiderivative = 1.,
number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used =
{2319, 44} \[ -\frac{a+b \log \left (c x^n\right )}{6 e (d+e x)^6}+\frac{b n}{6 d^5 e (d+e x)}+\frac{b n}{12 d^4 e (d+e x)^2}+\frac{b n}{18 d^3 e (d+e x)^3}+\frac{b n}{24 d^2 e (d+e x)^4}+\frac{b n \log (x)}{6 d^6 e}-\frac{b n \log (d+e x)}{6 d^6 e}+\frac{b n}{30 d e (d+e x)^5} \]
Antiderivative was successfully verified.
[In]
Int[(a + b*Log[c*x^n])/(d + e*x)^7,x]
[Out]
(b*n)/(30*d*e*(d + e*x)^5) + (b*n)/(24*d^2*e*(d + e*x)^4) + (b*n)/(18*d^3*e*(d + e*x)^3) + (b*n)/(12*d^4*e*(d
+ e*x)^2) + (b*n)/(6*d^5*e*(d + e*x)) + (b*n*Log[x])/(6*d^6*e) - (a + b*Log[c*x^n])/(6*e*(d + e*x)^6) - (b*n*L
og[d + e*x])/(6*d^6*e)
Rule 2319
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_.), x_Symbol] :> Simp[((d + e*x)^(q + 1
)*(a + b*Log[c*x^n])^p)/(e*(q + 1)), x] - Dist[(b*n*p)/(e*(q + 1)), Int[((d + e*x)^(q + 1)*(a + b*Log[c*x^n])^
(p - 1))/x, x], x] /; FreeQ[{a, b, c, d, e, n, p, q}, x] && GtQ[p, 0] && NeQ[q, -1] && (EqQ[p, 1] || (Integers
Q[2*p, 2*q] && !IGtQ[q, 0]) || (EqQ[p, 2] && NeQ[q, 1]))
Rule 44
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}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] && !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])
Rubi steps
\begin{align*} \int \frac{a+b \log \left (c x^n\right )}{(d+e x)^7} \, dx &=-\frac{a+b \log \left (c x^n\right )}{6 e (d+e x)^6}+\frac{(b n) \int \frac{1}{x (d+e x)^6} \, dx}{6 e}\\ &=-\frac{a+b \log \left (c x^n\right )}{6 e (d+e x)^6}+\frac{(b n) \int \left (\frac{1}{d^6 x}-\frac{e}{d (d+e x)^6}-\frac{e}{d^2 (d+e x)^5}-\frac{e}{d^3 (d+e x)^4}-\frac{e}{d^4 (d+e x)^3}-\frac{e}{d^5 (d+e x)^2}-\frac{e}{d^6 (d+e x)}\right ) \, dx}{6 e}\\ &=\frac{b n}{30 d e (d+e x)^5}+\frac{b n}{24 d^2 e (d+e x)^4}+\frac{b n}{18 d^3 e (d+e x)^3}+\frac{b n}{12 d^4 e (d+e x)^2}+\frac{b n}{6 d^5 e (d+e x)}+\frac{b n \log (x)}{6 d^6 e}-\frac{a+b \log \left (c x^n\right )}{6 e (d+e x)^6}-\frac{b n \log (d+e x)}{6 d^6 e}\\ \end{align*}
Mathematica [A] time = 0.130409, size = 99, normalized size = 0.65 \[ \frac{\frac{b n \left (\frac{d \left (470 d^2 e^2 x^2+385 d^3 e x+137 d^4+270 d e^3 x^3+60 e^4 x^4\right )}{(d+e x)^5}-60 \log (d+e x)+60 \log (x)\right )}{60 d^6}-\frac{a+b \log \left (c x^n\right )}{(d+e x)^6}}{6 e} \]
Antiderivative was successfully verified.
[In]
Integrate[(a + b*Log[c*x^n])/(d + e*x)^7,x]
[Out]
(-((a + b*Log[c*x^n])/(d + e*x)^6) + (b*n*((d*(137*d^4 + 385*d^3*e*x + 470*d^2*e^2*x^2 + 270*d*e^3*x^3 + 60*e^
4*x^4))/(d + e*x)^5 + 60*Log[x] - 60*Log[d + e*x]))/(60*d^6))/(6*e)
________________________________________________________________________________________
Maple [C] time = 0.109, size = 431, normalized size = 2.8 \begin{align*} -{\frac{b\ln \left ({x}^{n} \right ) }{6\, \left ( ex+d \right ) ^{6}e}}-{\frac{60\,a{d}^{6}-30\,i\pi \,b{d}^{6} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}-60\,bd{e}^{5}n{x}^{5}-330\,b{d}^{2}{e}^{4}n{x}^{4}-740\,b{d}^{3}{e}^{3}n{x}^{3}-855\,b{d}^{4}{e}^{2}n{x}^{2}-522\,b{d}^{5}enx+30\,i\pi \,b{d}^{6} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) +30\,i\pi \,b{d}^{6}{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}-30\,i\pi \,b{d}^{6}{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ) -137\,b{d}^{6}n+360\,\ln \left ( ex+d \right ) bd{e}^{5}n{x}^{5}+900\,\ln \left ( ex+d \right ) b{d}^{2}{e}^{4}n{x}^{4}+1200\,\ln \left ( ex+d \right ) b{d}^{3}{e}^{3}n{x}^{3}+900\,\ln \left ( ex+d \right ) b{d}^{4}{e}^{2}n{x}^{2}+360\,\ln \left ( ex+d \right ) b{d}^{5}enx-360\,\ln \left ( -x \right ) bd{e}^{5}n{x}^{5}-900\,\ln \left ( -x \right ) b{d}^{2}{e}^{4}n{x}^{4}-1200\,\ln \left ( -x \right ) b{d}^{3}{e}^{3}n{x}^{3}-900\,\ln \left ( -x \right ) b{d}^{4}{e}^{2}n{x}^{2}-360\,\ln \left ( -x \right ) b{d}^{5}enx+60\,\ln \left ( c \right ) b{d}^{6}+60\,\ln \left ( ex+d \right ) b{e}^{6}n{x}^{6}-60\,\ln \left ( -x \right ) b{e}^{6}n{x}^{6}+60\,\ln \left ( ex+d \right ) b{d}^{6}n-60\,\ln \left ( -x \right ) b{d}^{6}n}{360\,{d}^{6}e \left ( ex+d \right ) ^{6}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a+b*ln(c*x^n))/(e*x+d)^7,x)
[Out]
-1/6*b/e/(e*x+d)^6*ln(x^n)-1/360*(60*a*d^6-30*I*Pi*b*d^6*csgn(I*c*x^n)^3-60*b*d*e^5*n*x^5-330*b*d^2*e^4*n*x^4-
740*b*d^3*e^3*n*x^3-855*b*d^4*e^2*n*x^2-522*b*d^5*e*n*x+30*I*Pi*b*d^6*csgn(I*c*x^n)^2*csgn(I*c)+30*I*Pi*b*d^6*
csgn(I*x^n)*csgn(I*c*x^n)^2-30*I*Pi*b*d^6*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-137*b*d^6*n+360*ln(e*x+d)*b*d*e^
5*n*x^5+900*ln(e*x+d)*b*d^2*e^4*n*x^4+1200*ln(e*x+d)*b*d^3*e^3*n*x^3+900*ln(e*x+d)*b*d^4*e^2*n*x^2+360*ln(e*x+
d)*b*d^5*e*n*x-360*ln(-x)*b*d*e^5*n*x^5-900*ln(-x)*b*d^2*e^4*n*x^4-1200*ln(-x)*b*d^3*e^3*n*x^3-900*ln(-x)*b*d^
4*e^2*n*x^2-360*ln(-x)*b*d^5*e*n*x+60*ln(c)*b*d^6+60*ln(e*x+d)*b*e^6*n*x^6-60*ln(-x)*b*e^6*n*x^6+60*ln(e*x+d)*
b*d^6*n-60*ln(-x)*b*d^6*n)/d^6/e/(e*x+d)^6
________________________________________________________________________________________
Maxima [B] time = 1.23373, size = 373, normalized size = 2.45 \begin{align*} \frac{1}{360} \, b n{\left (\frac{60 \, e^{4} x^{4} + 270 \, d e^{3} x^{3} + 470 \, d^{2} e^{2} x^{2} + 385 \, d^{3} e x + 137 \, d^{4}}{d^{5} e^{6} x^{5} + 5 \, d^{6} e^{5} x^{4} + 10 \, d^{7} e^{4} x^{3} + 10 \, d^{8} e^{3} x^{2} + 5 \, d^{9} e^{2} x + d^{10} e} - \frac{60 \, \log \left (e x + d\right )}{d^{6} e} + \frac{60 \, \log \left (x\right )}{d^{6} e}\right )} - \frac{b \log \left (c x^{n}\right )}{6 \,{\left (e^{7} x^{6} + 6 \, d e^{6} x^{5} + 15 \, d^{2} e^{5} x^{4} + 20 \, d^{3} e^{4} x^{3} + 15 \, d^{4} e^{3} x^{2} + 6 \, d^{5} e^{2} x + d^{6} e\right )}} - \frac{a}{6 \,{\left (e^{7} x^{6} + 6 \, d e^{6} x^{5} + 15 \, d^{2} e^{5} x^{4} + 20 \, d^{3} e^{4} x^{3} + 15 \, d^{4} e^{3} x^{2} + 6 \, d^{5} e^{2} x + d^{6} e\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*log(c*x^n))/(e*x+d)^7,x, algorithm="maxima")
[Out]
1/360*b*n*((60*e^4*x^4 + 270*d*e^3*x^3 + 470*d^2*e^2*x^2 + 385*d^3*e*x + 137*d^4)/(d^5*e^6*x^5 + 5*d^6*e^5*x^4
+ 10*d^7*e^4*x^3 + 10*d^8*e^3*x^2 + 5*d^9*e^2*x + d^10*e) - 60*log(e*x + d)/(d^6*e) + 60*log(x)/(d^6*e)) - 1/
6*b*log(c*x^n)/(e^7*x^6 + 6*d*e^6*x^5 + 15*d^2*e^5*x^4 + 20*d^3*e^4*x^3 + 15*d^4*e^3*x^2 + 6*d^5*e^2*x + d^6*e
) - 1/6*a/(e^7*x^6 + 6*d*e^6*x^5 + 15*d^2*e^5*x^4 + 20*d^3*e^4*x^3 + 15*d^4*e^3*x^2 + 6*d^5*e^2*x + d^6*e)
________________________________________________________________________________________
Fricas [B] time = 1.45231, size = 703, normalized size = 4.62 \begin{align*} \frac{60 \, b d e^{5} n x^{5} + 330 \, b d^{2} e^{4} n x^{4} + 740 \, b d^{3} e^{3} n x^{3} + 855 \, b d^{4} e^{2} n x^{2} + 522 \, b d^{5} e n x + 137 \, b d^{6} n - 60 \, b d^{6} \log \left (c\right ) - 60 \, a d^{6} - 60 \,{\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 ) + 60 \,{\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\right )} \log \left (x\right )}{360 \,{\left (d^{6} e^{7} x^{6} + 6 \, d^{7} e^{6} x^{5} + 15 \, d^{8} e^{5} x^{4} + 20 \, d^{9} e^{4} x^{3} + 15 \, d^{10} e^{3} x^{2} + 6 \, d^{11} e^{2} x + d^{12} e\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*log(c*x^n))/(e*x+d)^7,x, algorithm="fricas")
[Out]
1/360*(60*b*d*e^5*n*x^5 + 330*b*d^2*e^4*n*x^4 + 740*b*d^3*e^3*n*x^3 + 855*b*d^4*e^2*n*x^2 + 522*b*d^5*e*n*x +
137*b*d^6*n - 60*b*d^6*log(c) - 60*a*d^6 - 60*(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) + 60*(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)*log(x))/(d^6*e^7*x^6 + 6*d^7*e^6*x
^5 + 15*d^8*e^5*x^4 + 20*d^9*e^4*x^3 + 15*d^10*e^3*x^2 + 6*d^11*e^2*x + d^12*e)
________________________________________________________________________________________
Sympy [A] time = 154.069, size = 2519, normalized size = 16.57 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*ln(c*x**n))/(e*x+d)**7,x)
[Out]
Piecewise((zoo*(-a/(6*x**6) - b*n*log(x)/(6*x**6) - b*n/(36*x**6) - b*log(c)/(6*x**6)), Eq(d, 0) & Eq(e, 0)),
((-a/(6*x**6) - b*n*log(x)/(6*x**6) - b*n/(36*x**6) - b*log(c)/(6*x**6))/e**7, Eq(d, 0)), ((a*x + b*n*x*log(x)
- b*n*x + b*x*log(c))/d**7, Eq(e, 0)), (-60*a*d**6/(360*d**12*e + 2160*d**11*e**2*x + 5400*d**10*e**3*x**2 +
7200*d**9*e**4*x**3 + 5400*d**8*e**5*x**4 + 2160*d**7*e**6*x**5 + 360*d**6*e**7*x**6) - 60*b*d**6*n*log(d/e +
x)/(360*d**12*e + 2160*d**11*e**2*x + 5400*d**10*e**3*x**2 + 7200*d**9*e**4*x**3 + 5400*d**8*e**5*x**4 + 2160*
d**7*e**6*x**5 + 360*d**6*e**7*x**6) + 137*b*d**6*n/(360*d**12*e + 2160*d**11*e**2*x + 5400*d**10*e**3*x**2 +
7200*d**9*e**4*x**3 + 5400*d**8*e**5*x**4 + 2160*d**7*e**6*x**5 + 360*d**6*e**7*x**6) + 360*b*d**5*e*n*x*log(x
)/(360*d**12*e + 2160*d**11*e**2*x + 5400*d**10*e**3*x**2 + 7200*d**9*e**4*x**3 + 5400*d**8*e**5*x**4 + 2160*d
**7*e**6*x**5 + 360*d**6*e**7*x**6) - 360*b*d**5*e*n*x*log(d/e + x)/(360*d**12*e + 2160*d**11*e**2*x + 5400*d*
*10*e**3*x**2 + 7200*d**9*e**4*x**3 + 5400*d**8*e**5*x**4 + 2160*d**7*e**6*x**5 + 360*d**6*e**7*x**6) + 522*b*
d**5*e*n*x/(360*d**12*e + 2160*d**11*e**2*x + 5400*d**10*e**3*x**2 + 7200*d**9*e**4*x**3 + 5400*d**8*e**5*x**4
+ 2160*d**7*e**6*x**5 + 360*d**6*e**7*x**6) + 360*b*d**5*e*x*log(c)/(360*d**12*e + 2160*d**11*e**2*x + 5400*d
**10*e**3*x**2 + 7200*d**9*e**4*x**3 + 5400*d**8*e**5*x**4 + 2160*d**7*e**6*x**5 + 360*d**6*e**7*x**6) + 900*b
*d**4*e**2*n*x**2*log(x)/(360*d**12*e + 2160*d**11*e**2*x + 5400*d**10*e**3*x**2 + 7200*d**9*e**4*x**3 + 5400*
d**8*e**5*x**4 + 2160*d**7*e**6*x**5 + 360*d**6*e**7*x**6) - 900*b*d**4*e**2*n*x**2*log(d/e + x)/(360*d**12*e
+ 2160*d**11*e**2*x + 5400*d**10*e**3*x**2 + 7200*d**9*e**4*x**3 + 5400*d**8*e**5*x**4 + 2160*d**7*e**6*x**5 +
360*d**6*e**7*x**6) + 855*b*d**4*e**2*n*x**2/(360*d**12*e + 2160*d**11*e**2*x + 5400*d**10*e**3*x**2 + 7200*d
**9*e**4*x**3 + 5400*d**8*e**5*x**4 + 2160*d**7*e**6*x**5 + 360*d**6*e**7*x**6) + 900*b*d**4*e**2*x**2*log(c)/
(360*d**12*e + 2160*d**11*e**2*x + 5400*d**10*e**3*x**2 + 7200*d**9*e**4*x**3 + 5400*d**8*e**5*x**4 + 2160*d**
7*e**6*x**5 + 360*d**6*e**7*x**6) + 1200*b*d**3*e**3*n*x**3*log(x)/(360*d**12*e + 2160*d**11*e**2*x + 5400*d**
10*e**3*x**2 + 7200*d**9*e**4*x**3 + 5400*d**8*e**5*x**4 + 2160*d**7*e**6*x**5 + 360*d**6*e**7*x**6) - 1200*b*
d**3*e**3*n*x**3*log(d/e + x)/(360*d**12*e + 2160*d**11*e**2*x + 5400*d**10*e**3*x**2 + 7200*d**9*e**4*x**3 +
5400*d**8*e**5*x**4 + 2160*d**7*e**6*x**5 + 360*d**6*e**7*x**6) + 740*b*d**3*e**3*n*x**3/(360*d**12*e + 2160*d
**11*e**2*x + 5400*d**10*e**3*x**2 + 7200*d**9*e**4*x**3 + 5400*d**8*e**5*x**4 + 2160*d**7*e**6*x**5 + 360*d**
6*e**7*x**6) + 1200*b*d**3*e**3*x**3*log(c)/(360*d**12*e + 2160*d**11*e**2*x + 5400*d**10*e**3*x**2 + 7200*d**
9*e**4*x**3 + 5400*d**8*e**5*x**4 + 2160*d**7*e**6*x**5 + 360*d**6*e**7*x**6) + 900*b*d**2*e**4*n*x**4*log(x)/
(360*d**12*e + 2160*d**11*e**2*x + 5400*d**10*e**3*x**2 + 7200*d**9*e**4*x**3 + 5400*d**8*e**5*x**4 + 2160*d**
7*e**6*x**5 + 360*d**6*e**7*x**6) - 900*b*d**2*e**4*n*x**4*log(d/e + x)/(360*d**12*e + 2160*d**11*e**2*x + 540
0*d**10*e**3*x**2 + 7200*d**9*e**4*x**3 + 5400*d**8*e**5*x**4 + 2160*d**7*e**6*x**5 + 360*d**6*e**7*x**6) + 33
0*b*d**2*e**4*n*x**4/(360*d**12*e + 2160*d**11*e**2*x + 5400*d**10*e**3*x**2 + 7200*d**9*e**4*x**3 + 5400*d**8
*e**5*x**4 + 2160*d**7*e**6*x**5 + 360*d**6*e**7*x**6) + 900*b*d**2*e**4*x**4*log(c)/(360*d**12*e + 2160*d**11
*e**2*x + 5400*d**10*e**3*x**2 + 7200*d**9*e**4*x**3 + 5400*d**8*e**5*x**4 + 2160*d**7*e**6*x**5 + 360*d**6*e*
*7*x**6) + 360*b*d*e**5*n*x**5*log(x)/(360*d**12*e + 2160*d**11*e**2*x + 5400*d**10*e**3*x**2 + 7200*d**9*e**4
*x**3 + 5400*d**8*e**5*x**4 + 2160*d**7*e**6*x**5 + 360*d**6*e**7*x**6) - 360*b*d*e**5*n*x**5*log(d/e + x)/(36
0*d**12*e + 2160*d**11*e**2*x + 5400*d**10*e**3*x**2 + 7200*d**9*e**4*x**3 + 5400*d**8*e**5*x**4 + 2160*d**7*e
**6*x**5 + 360*d**6*e**7*x**6) + 60*b*d*e**5*n*x**5/(360*d**12*e + 2160*d**11*e**2*x + 5400*d**10*e**3*x**2 +
7200*d**9*e**4*x**3 + 5400*d**8*e**5*x**4 + 2160*d**7*e**6*x**5 + 360*d**6*e**7*x**6) + 360*b*d*e**5*x**5*log(
c)/(360*d**12*e + 2160*d**11*e**2*x + 5400*d**10*e**3*x**2 + 7200*d**9*e**4*x**3 + 5400*d**8*e**5*x**4 + 2160*
d**7*e**6*x**5 + 360*d**6*e**7*x**6) + 60*b*e**6*n*x**6*log(x)/(360*d**12*e + 2160*d**11*e**2*x + 5400*d**10*e
**3*x**2 + 7200*d**9*e**4*x**3 + 5400*d**8*e**5*x**4 + 2160*d**7*e**6*x**5 + 360*d**6*e**7*x**6) - 60*b*e**6*n
*x**6*log(d/e + x)/(360*d**12*e + 2160*d**11*e**2*x + 5400*d**10*e**3*x**2 + 7200*d**9*e**4*x**3 + 5400*d**8*e
**5*x**4 + 2160*d**7*e**6*x**5 + 360*d**6*e**7*x**6) + 60*b*e**6*x**6*log(c)/(360*d**12*e + 2160*d**11*e**2*x
+ 5400*d**10*e**3*x**2 + 7200*d**9*e**4*x**3 + 5400*d**8*e**5*x**4 + 2160*d**7*e**6*x**5 + 360*d**6*e**7*x**6)
, True))
________________________________________________________________________________________
Giac [B] time = 1.21837, size = 464, normalized size = 3.05 \begin{align*} -\frac{60 \, b n x^{6} e^{6} \log \left (x e + d\right ) + 360 \, b d n x^{5} e^{5} \log \left (x e + d\right ) + 900 \, b d^{2} n x^{4} e^{4} \log \left (x e + d\right ) + 1200 \, b d^{3} n x^{3} e^{3} \log \left (x e + d\right ) + 900 \, b d^{4} n x^{2} e^{2} \log \left (x e + d\right ) + 360 \, b d^{5} n x e \log \left (x e + d\right ) - 60 \, b n x^{6} e^{6} \log \left (x\right ) - 360 \, b d n x^{5} e^{5} \log \left (x\right ) - 900 \, b d^{2} n x^{4} e^{4} \log \left (x\right ) - 1200 \, b d^{3} n x^{3} e^{3} \log \left (x\right ) - 900 \, b d^{4} n x^{2} e^{2} \log \left (x\right ) - 360 \, b d^{5} n x e \log \left (x\right ) - 60 \, b d n x^{5} e^{5} - 330 \, b d^{2} n x^{4} e^{4} - 740 \, b d^{3} n x^{3} e^{3} - 855 \, b d^{4} n x^{2} e^{2} - 522 \, b d^{5} n x e + 60 \, b d^{6} n \log \left (x e + d\right ) - 137 \, b d^{6} n + 60 \, b d^{6} \log \left (c\right ) + 60 \, a d^{6}}{360 \,{\left (d^{6} x^{6} e^{7} + 6 \, d^{7} x^{5} e^{6} + 15 \, d^{8} x^{4} e^{5} + 20 \, d^{9} x^{3} e^{4} + 15 \, d^{10} x^{2} e^{3} + 6 \, d^{11} x e^{2} + d^{12} e\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*log(c*x^n))/(e*x+d)^7,x, algorithm="giac")
[Out]
-1/360*(60*b*n*x^6*e^6*log(x*e + d) + 360*b*d*n*x^5*e^5*log(x*e + d) + 900*b*d^2*n*x^4*e^4*log(x*e + d) + 1200
*b*d^3*n*x^3*e^3*log(x*e + d) + 900*b*d^4*n*x^2*e^2*log(x*e + d) + 360*b*d^5*n*x*e*log(x*e + d) - 60*b*n*x^6*e
^6*log(x) - 360*b*d*n*x^5*e^5*log(x) - 900*b*d^2*n*x^4*e^4*log(x) - 1200*b*d^3*n*x^3*e^3*log(x) - 900*b*d^4*n*
x^2*e^2*log(x) - 360*b*d^5*n*x*e*log(x) - 60*b*d*n*x^5*e^5 - 330*b*d^2*n*x^4*e^4 - 740*b*d^3*n*x^3*e^3 - 855*b
*d^4*n*x^2*e^2 - 522*b*d^5*n*x*e + 60*b*d^6*n*log(x*e + d) - 137*b*d^6*n + 60*b*d^6*log(c) + 60*a*d^6)/(d^6*x^
6*e^7 + 6*d^7*x^5*e^6 + 15*d^8*x^4*e^5 + 20*d^9*x^3*e^4 + 15*d^10*x^2*e^3 + 6*d^11*x*e^2 + d^12*e)