Integrand size = 18, antiderivative size = 152 \[ \int \frac {a+b \log \left (c x^n\right )}{(d+e x)^7} \, dx=\frac {b n}{30 d e (d+e x)^5}+\frac {b n}{24 d^2 e (d+e x)^4}+\frac {b n}{18 d^3 e (d+e x)^3}+\frac {b n}{12 d^4 e (d+e x)^2}+\frac {b n}{6 d^5 e (d+e x)}+\frac {b n \log (x)}{6 d^6 e}-\frac {a+b \log \left (c x^n\right )}{6 e (d+e x)^6}-\frac {b n \log (d+e x)}{6 d^6 e} \]
1/30*b*n/d/e/(e*x+d)^5+1/24*b*n/d^2/e/(e*x+d)^4+1/18*b*n/d^3/e/(e*x+d)^3+1 /12*b*n/d^4/e/(e*x+d)^2+1/6*b*n/d^5/e/(e*x+d)+1/6*b*n*ln(x)/d^6/e+1/6*(-a- b*ln(c*x^n))/e/(e*x+d)^6-1/6*b*n*ln(e*x+d)/d^6/e
Time = 0.10 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.65 \[ \int \frac {a+b \log \left (c x^n\right )}{(d+e x)^7} \, dx=\frac {-\frac {a+b \log \left (c x^n\right )}{(d+e x)^6}+\frac {b n \left (\frac {d \left (137 d^4+385 d^3 e x+470 d^2 e^2 x^2+270 d e^3 x^3+60 e^4 x^4\right )}{(d+e x)^5}+60 \log (x)-60 \log (d+e x)\right )}{60 d^6}}{6 e} \]
(-((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*L og[d + e*x]))/(60*d^6))/(6*e)
Time = 0.27 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.78, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2756, 54, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {a+b \log \left (c x^n\right )}{(d+e x)^7} \, dx\) |
\(\Big \downarrow \) 2756 |
\(\displaystyle \frac {b n \int \frac {1}{x (d+e x)^6}dx}{6 e}-\frac {a+b \log \left (c x^n\right )}{6 e (d+e x)^6}\) |
\(\Big \downarrow \) 54 |
\(\displaystyle \frac {b n \int \left (-\frac {e}{d^6 (d+e x)}-\frac {e}{d^5 (d+e x)^2}-\frac {e}{d^4 (d+e x)^3}-\frac {e}{d^3 (d+e x)^4}-\frac {e}{d^2 (d+e x)^5}-\frac {e}{d (d+e x)^6}+\frac {1}{d^6 x}\right )dx}{6 e}-\frac {a+b \log \left (c x^n\right )}{6 e (d+e x)^6}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {b n \left (-\frac {\log (d+e x)}{d^6}+\frac {\log (x)}{d^6}+\frac {1}{d^5 (d+e x)}+\frac {1}{2 d^4 (d+e x)^2}+\frac {1}{3 d^3 (d+e x)^3}+\frac {1}{4 d^2 (d+e x)^4}+\frac {1}{5 d (d+e x)^5}\right )}{6 e}-\frac {a+b \log \left (c x^n\right )}{6 e (d+e x)^6}\) |
-1/6*(a + b*Log[c*x^n])/(e*(d + e*x)^6) + (b*n*(1/(5*d*(d + e*x)^5) + 1/(4 *d^2*(d + e*x)^4) + 1/(3*d^3*(d + e*x)^3) + 1/(2*d^4*(d + e*x)^2) + 1/(d^5 *(d + e*x)) + Log[x]/d^6 - Log[d + e*x]/d^6))/(6*e)
3.1.70.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[E xpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[m, 0] && IntegerQ[n] && !(IGtQ[n, 0] && LtQ[m + n + 2, 0])
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] - Simp[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] || (IntegersQ[2*p, 2*q] && !IGtQ[q, 0]) || (EqQ[p, 2] & & NeQ[q, 1]))
Leaf count of result is larger than twice the leaf count of optimal. \(385\) vs. \(2(139)=278\).
Time = 1.32 (sec) , antiderivative size = 386, normalized size of antiderivative = 2.54
method | result | size |
parallelrisch | \(\frac {-360 \ln \left (e x +d \right ) x^{5} b d \,e^{10} n +900 \ln \left (x \right ) x^{4} b \,d^{2} e^{9} n -900 \ln \left (e x +d \right ) x^{4} b \,d^{2} e^{9} n +1200 \ln \left (x \right ) x^{3} b \,d^{3} e^{8} n -1200 \ln \left (e x +d \right ) x^{3} b \,d^{3} e^{8} n +360 \ln \left (x \right ) x^{5} b d \,e^{10} n +60 \ln \left (x \right ) x^{6} b \,e^{11} n -60 \ln \left (e x +d \right ) x^{6} b \,e^{11} n +900 x^{4} a \,d^{2} e^{9}+360 x^{5} a d \,e^{10}+360 x a \,d^{5} e^{6}+900 x^{2} a \,d^{4} e^{7}+1200 x^{3} a \,d^{3} e^{8}-2000 x^{3} b \,d^{3} e^{8} n -1200 x^{2} b \,d^{4} e^{7} n -300 x b \,d^{5} e^{6} n +60 \ln \left (x \right ) b \,d^{6} e^{5} n -60 \ln \left (e x +d \right ) b \,d^{6} e^{5} n +900 \ln \left (x \right ) x^{2} b \,d^{4} e^{7} n -900 \ln \left (e x +d \right ) x^{2} b \,d^{4} e^{7} n +360 \ln \left (x \right ) x b \,d^{5} e^{6} n -360 \ln \left (e x +d \right ) x b \,d^{5} e^{6} n -762 x^{5} b d \,e^{10} n -1725 x^{4} b \,d^{2} e^{9} n -137 x^{6} b \,e^{11} n -60 \ln \left (c \,x^{n}\right ) b \,d^{6} e^{5}+60 x^{6} a \,e^{11}}{360 e^{6} d^{6} \left (e x +d \right )^{6}}\) | \(386\) |
risch | \(-\frac {b \ln \left (x^{n}\right )}{6 e \left (e x +d \right )^{6}}-\frac {900 \ln \left (e x +d \right ) b \,d^{2} e^{4} n \,x^{4}-360 \ln \left (-x \right ) b d \,e^{5} n \,x^{5}-30 i \pi b \,d^{6} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+900 \ln \left (e x +d \right ) b \,d^{4} e^{2} n \,x^{2}-137 b \,d^{6} n -900 \ln \left (-x \right ) b \,d^{2} e^{4} n \,x^{4}-1200 \ln \left (-x \right ) b \,d^{3} e^{3} n \,x^{3}-900 \ln \left (-x \right ) b \,d^{4} e^{2} n \,x^{2}-360 \ln \left (-x \right ) b \,d^{5} e n x -60 \ln \left (-x \right ) b \,e^{6} n \,x^{6}+1200 \ln \left (e x +d \right ) b \,d^{3} e^{3} n \,x^{3}+360 \ln \left (e x +d \right ) b d \,e^{5} n \,x^{5}-855 b \,d^{4} e^{2} n \,x^{2}-60 b d \,e^{5} n \,x^{5}-330 b \,d^{2} e^{4} n \,x^{4}-740 b \,d^{3} e^{3} n \,x^{3}-522 b \,d^{5} e n x +360 \ln \left (e x +d \right ) b \,d^{5} e n x -60 \ln \left (-x \right ) b \,d^{6} n +60 \ln \left (e x +d \right ) b \,e^{6} n \,x^{6}-30 i \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) b \,d^{6}+30 i \pi b \,d^{6} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+60 \ln \left (e x +d \right ) b \,d^{6} n +60 a \,d^{6}+30 i \pi b \,d^{6} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+60 \ln \left (c \right ) b \,d^{6}}{360 d^{6} e \left (e x +d \right )^{6}}\) | \(431\) |
1/360*(-360*ln(e*x+d)*x^5*b*d*e^10*n+900*ln(x)*x^4*b*d^2*e^9*n-900*ln(e*x+ d)*x^4*b*d^2*e^9*n+1200*ln(x)*x^3*b*d^3*e^8*n-1200*ln(e*x+d)*x^3*b*d^3*e^8 *n+360*ln(x)*x^5*b*d*e^10*n+60*ln(x)*x^6*b*e^11*n-60*ln(e*x+d)*x^6*b*e^11* n+900*x^4*a*d^2*e^9+360*x^5*a*d*e^10+360*x*a*d^5*e^6+900*x^2*a*d^4*e^7+120 0*x^3*a*d^3*e^8-2000*x^3*b*d^3*e^8*n-1200*x^2*b*d^4*e^7*n-300*x*b*d^5*e^6* n+60*ln(x)*b*d^6*e^5*n-60*ln(e*x+d)*b*d^6*e^5*n+900*ln(x)*x^2*b*d^4*e^7*n- 900*ln(e*x+d)*x^2*b*d^4*e^7*n+360*ln(x)*x*b*d^5*e^6*n-360*ln(e*x+d)*x*b*d^ 5*e^6*n-762*x^5*b*d*e^10*n-1725*x^4*b*d^2*e^9*n-137*x^6*b*e^11*n-60*ln(c*x ^n)*b*d^6*e^5+60*x^6*a*e^11)/e^6/d^6/(e*x+d)^6
Leaf count of result is larger than twice the leaf count of optimal. 310 vs. \(2 (136) = 272\).
Time = 0.32 (sec) , antiderivative size = 310, normalized size of antiderivative = 2.04 \[ \int \frac {a+b \log \left (c x^n\right )}{(d+e x)^7} \, dx=\frac {60 \, b d e^{5} n x^{5} + 330 \, b d^{2} e^{4} n x^{4} + 740 \, b d^{3} e^{3} n x^{3} + 855 \, b d^{4} e^{2} n x^{2} + 522 \, b d^{5} e n x + 137 \, b d^{6} n - 60 \, b d^{6} \log \left (c\right ) - 60 \, a d^{6} - 60 \, {\left (b e^{6} n x^{6} + 6 \, b d e^{5} n x^{5} + 15 \, b d^{2} e^{4} n x^{4} + 20 \, b d^{3} e^{3} n x^{3} + 15 \, b d^{4} e^{2} n x^{2} + 6 \, b d^{5} e n x + b d^{6} n\right )} \log \left (e x + d\right ) + 60 \, {\left (b e^{6} n x^{6} + 6 \, b d e^{5} n x^{5} + 15 \, b d^{2} e^{4} n x^{4} + 20 \, b d^{3} e^{3} n x^{3} + 15 \, b d^{4} e^{2} n x^{2} + 6 \, b d^{5} e n x\right )} \log \left (x\right )}{360 \, {\left (d^{6} e^{7} x^{6} + 6 \, d^{7} e^{6} x^{5} + 15 \, d^{8} e^{5} x^{4} + 20 \, d^{9} e^{4} x^{3} + 15 \, d^{10} e^{3} x^{2} + 6 \, d^{11} e^{2} x + d^{12} e\right )}} \]
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 )
Leaf count of result is larger than twice the leaf count of optimal. 1955 vs. \(2 (134) = 268\).
Time = 76.21 (sec) , antiderivative size = 1955, normalized size of antiderivative = 12.86 \[ \int \frac {a+b \log \left (c x^n\right )}{(d+e x)^7} \, dx=\text {Too large to display} \]
Piecewise((zoo*(-a/(6*x**6) - b*n/(36*x**6) - b*log(c*x**n)/(6*x**6)), Eq( d, 0) & Eq(e, 0)), ((a*x - b*n*x + b*x*log(c*x**n))/d**7, Eq(e, 0)), ((-a/ (6*x**6) - b*n/(36*x**6) - b*log(c*x**n)/(6*x**6))/e**7, Eq(d, 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(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*x**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) - 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 ...
Leaf count of result is larger than twice the leaf count of optimal. 276 vs. \(2 (136) = 272\).
Time = 0.21 (sec) , antiderivative size = 276, normalized size of antiderivative = 1.82 \[ \int \frac {a+b \log \left (c x^n\right )}{(d+e x)^7} \, dx=\frac {1}{360} \, b n {\left (\frac {60 \, e^{4} x^{4} + 270 \, d e^{3} x^{3} + 470 \, d^{2} e^{2} x^{2} + 385 \, d^{3} e x + 137 \, d^{4}}{d^{5} e^{6} x^{5} + 5 \, d^{6} e^{5} x^{4} + 10 \, d^{7} e^{4} x^{3} + 10 \, d^{8} e^{3} x^{2} + 5 \, d^{9} e^{2} x + d^{10} e} - \frac {60 \, \log \left (e x + d\right )}{d^{6} e} + \frac {60 \, \log \left (x\right )}{d^{6} e}\right )} - \frac {b \log \left (c x^{n}\right )}{6 \, {\left (e^{7} x^{6} + 6 \, d e^{6} x^{5} + 15 \, d^{2} e^{5} x^{4} + 20 \, d^{3} e^{4} x^{3} + 15 \, d^{4} e^{3} x^{2} + 6 \, d^{5} e^{2} x + d^{6} e\right )}} - \frac {a}{6 \, {\left (e^{7} x^{6} + 6 \, d e^{6} x^{5} + 15 \, d^{2} e^{5} x^{4} + 20 \, d^{3} e^{4} x^{3} + 15 \, d^{4} e^{3} x^{2} + 6 \, d^{5} e^{2} x + d^{6} e\right )}} \]
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 + 1 37*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)
Time = 0.44 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.66 \[ \int \frac {a+b \log \left (c x^n\right )}{(d+e x)^7} \, dx=-\frac {b n \log \left (x\right )}{6 \, {\left (e^{7} x^{6} + 6 \, d e^{6} x^{5} + 15 \, d^{2} e^{5} x^{4} + 20 \, d^{3} e^{4} x^{3} + 15 \, d^{4} e^{3} x^{2} + 6 \, d^{5} e^{2} x + d^{6} e\right )}} + \frac {60 \, b e^{5} n x^{5} + 330 \, b d e^{4} n x^{4} + 740 \, b d^{2} e^{3} n x^{3} + 855 \, b d^{3} e^{2} n x^{2} + 522 \, b d^{4} e n x + 137 \, b d^{5} n - 60 \, b d^{5} \log \left (c\right ) - 60 \, a d^{5}}{360 \, {\left (d^{5} e^{7} x^{6} + 6 \, d^{6} e^{6} x^{5} + 15 \, d^{7} e^{5} x^{4} + 20 \, d^{8} e^{4} x^{3} + 15 \, d^{9} e^{3} x^{2} + 6 \, d^{10} e^{2} x + d^{11} e\right )}} - \frac {b n \log \left (e x + d\right )}{6 \, d^{6} e} + \frac {b n \log \left (x\right )}{6 \, d^{6} e} \]
-1/6*b*n*log(x)/(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) + 1/360*(60*b*e^5*n*x^5 + 330*b*d*e ^4*n*x^4 + 740*b*d^2*e^3*n*x^3 + 855*b*d^3*e^2*n*x^2 + 522*b*d^4*e*n*x + 1 37*b*d^5*n - 60*b*d^5*log(c) - 60*a*d^5)/(d^5*e^7*x^6 + 6*d^6*e^6*x^5 + 15 *d^7*e^5*x^4 + 20*d^8*e^4*x^3 + 15*d^9*e^3*x^2 + 6*d^10*e^2*x + d^11*e) - 1/6*b*n*log(e*x + d)/(d^6*e) + 1/6*b*n*log(x)/(d^6*e)
Time = 0.75 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.53 \[ \int \frac {a+b \log \left (c x^n\right )}{(d+e x)^7} \, dx=\frac {\frac {137\,b\,n}{60}-a+\frac {57\,b\,e^2\,n\,x^2}{4\,d^2}+\frac {37\,b\,e^3\,n\,x^3}{3\,d^3}+\frac {11\,b\,e^4\,n\,x^4}{2\,d^4}+\frac {b\,e^5\,n\,x^5}{d^5}+\frac {87\,b\,e\,n\,x}{10\,d}}{6\,d^6\,e+36\,d^5\,e^2\,x+90\,d^4\,e^3\,x^2+120\,d^3\,e^4\,x^3+90\,d^2\,e^5\,x^4+36\,d\,e^6\,x^5+6\,e^7\,x^6}-\frac {b\,\ln \left (c\,x^n\right )}{6\,e\,\left (d^6+6\,d^5\,e\,x+15\,d^4\,e^2\,x^2+20\,d^3\,e^3\,x^3+15\,d^2\,e^4\,x^4+6\,d\,e^5\,x^5+e^6\,x^6\right )}-\frac {b\,n\,\mathrm {atanh}\left (\frac {2\,e\,x}{d}+1\right )}{3\,d^6\,e} \]
((137*b*n)/60 - a + (57*b*e^2*n*x^2)/(4*d^2) + (37*b*e^3*n*x^3)/(3*d^3) + (11*b*e^4*n*x^4)/(2*d^4) + (b*e^5*n*x^5)/d^5 + (87*b*e*n*x)/(10*d))/(6*d^6 *e + 6*e^7*x^6 + 36*d^5*e^2*x + 36*d*e^6*x^5 + 90*d^4*e^3*x^2 + 120*d^3*e^ 4*x^3 + 90*d^2*e^5*x^4) - (b*log(c*x^n))/(6*e*(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*a tanh((2*e*x)/d + 1))/(3*d^6*e)