Integrand size = 21, antiderivative size = 149 \[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^3} \, dx=-\frac {3 b n x}{e^3}+\frac {(6 a+5 b n) x}{2 e^3}+\frac {3 b x \log \left (c x^n\right )}{e^3}-\frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{2 e (d+e x)^2}-\frac {x^2 \left (3 a+b n+3 b \log \left (c x^n\right )\right )}{2 e^2 (d+e x)}-\frac {d \left (6 a+5 b n+6 b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{2 e^4}-\frac {3 b d n \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{e^4} \] Output:
-3*b*n*x/e^3+1/2*(5*b*n+6*a)*x/e^3+3*b*x*ln(c*x^n)/e^3-1/2*x^3*(a+b*ln(c*x ^n))/e/(e*x+d)^2-1/2*x^2*(3*a+b*n+3*b*ln(c*x^n))/e^2/(e*x+d)-1/2*d*(6*a+5* b*n+6*b*ln(c*x^n))*ln(1+e*x/d)/e^4-3*b*d*n*polylog(2,-e*x/d)/e^4
Time = 0.16 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.01 \[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^3} \, dx=\frac {2 a e x-2 b e n x+2 b e x \log \left (c x^n\right )+\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^2}-\frac {6 d^2 \left (a+b \log \left (c x^n\right )\right )}{d+e x}+6 b d n (\log (x)-\log (d+e x))-b d n \left (\frac {d}{d+e x}+\log (x)-\log (d+e x)\right )-6 d \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )-6 b d n \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{2 e^4} \] Input:
Integrate[(x^3*(a + b*Log[c*x^n]))/(d + e*x)^3,x]
Output:
(2*a*e*x - 2*b*e*n*x + 2*b*e*x*Log[c*x^n] + (d^3*(a + b*Log[c*x^n]))/(d + e*x)^2 - (6*d^2*(a + b*Log[c*x^n]))/(d + e*x) + 6*b*d*n*(Log[x] - Log[d + e*x]) - b*d*n*(d/(d + e*x) + Log[x] - Log[d + e*x]) - 6*d*(a + b*Log[c*x^n ])*Log[1 + (e*x)/d] - 6*b*d*n*PolyLog[2, -((e*x)/d)])/(2*e^4)
Time = 0.54 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.04, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {2784, 2784, 2793, 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.
\(\displaystyle \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^3} \, dx\) |
\(\Big \downarrow \) 2784 |
\(\displaystyle \frac {\int \frac {x^2 \left (3 a+b n+3 b \log \left (c x^n\right )\right )}{(d+e x)^2}dx}{2 e}-\frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{2 e (d+e x)^2}\) |
\(\Big \downarrow \) 2784 |
\(\displaystyle \frac {\frac {\int \frac {x \left (6 a+5 b n+6 b \log \left (c x^n\right )\right )}{d+e x}dx}{e}-\frac {x^2 \left (3 a+3 b \log \left (c x^n\right )+b n\right )}{e (d+e x)}}{2 e}-\frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{2 e (d+e x)^2}\) |
\(\Big \downarrow \) 2793 |
\(\displaystyle \frac {\frac {\int \left (\frac {6 a+5 b n+6 b \log \left (c x^n\right )}{e}-\frac {d \left (6 a+5 b n+6 b \log \left (c x^n\right )\right )}{e (d+e x)}\right )dx}{e}-\frac {x^2 \left (3 a+3 b \log \left (c x^n\right )+b n\right )}{e (d+e x)}}{2 e}-\frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{2 e (d+e x)^2}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\frac {-\frac {d \log \left (\frac {e x}{d}+1\right ) \left (6 a+6 b \log \left (c x^n\right )+5 b n\right )}{e^2}+\frac {x (6 a+5 b n)}{e}+\frac {6 b x \log \left (c x^n\right )}{e}-\frac {6 b d n \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{e^2}-\frac {6 b n x}{e}}{e}-\frac {x^2 \left (3 a+3 b \log \left (c x^n\right )+b n\right )}{e (d+e x)}}{2 e}-\frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{2 e (d+e x)^2}\) |
Input:
Int[(x^3*(a + b*Log[c*x^n]))/(d + e*x)^3,x]
Output:
-1/2*(x^3*(a + b*Log[c*x^n]))/(e*(d + e*x)^2) + (-((x^2*(3*a + b*n + 3*b*L og[c*x^n]))/(e*(d + e*x))) + ((-6*b*n*x)/e + ((6*a + 5*b*n)*x)/e + (6*b*x* Log[c*x^n])/e - (d*(6*a + 5*b*n + 6*b*Log[c*x^n])*Log[1 + (e*x)/d])/e^2 - (6*b*d*n*PolyLog[2, -((e*x)/d)])/e^2)/e)/(2*e)
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)* (x_))^(q_.), x_Symbol] :> Simp[(f*x)^m*(d + e*x)^(q + 1)*((a + b*Log[c*x^n] )/(e*(q + 1))), x] - Simp[f/(e*(q + 1)) Int[(f*x)^(m - 1)*(d + e*x)^(q + 1)*(a*m + b*n + b*m*Log[c*x^n]), x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && ILtQ[q, -1] && GtQ[m, 0]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)* (x_)^(r_.))^(q_.), x_Symbol] :> With[{u = ExpandIntegrand[a + b*Log[c*x^n], (f*x)^m*(d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, f, m, n, q, r}, x] && IntegerQ[q] && (GtQ[q, 0] || (IntegerQ[m] && Integer Q[r]))
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.60 (sec) , antiderivative size = 303, normalized size of antiderivative = 2.03
method | result | size |
risch | \(\frac {b \ln \left (x^{n}\right ) x}{e^{3}}-\frac {3 b \ln \left (x^{n}\right ) d^{2}}{e^{4} \left (e x +d \right )}-\frac {3 b \ln \left (x^{n}\right ) d \ln \left (e x +d \right )}{e^{4}}+\frac {b \ln \left (x^{n}\right ) d^{3}}{2 e^{4} \left (e x +d \right )^{2}}-\frac {b n x}{e^{3}}-\frac {b n d}{e^{4}}-\frac {5 b n d \ln \left (e x +d \right )}{2 e^{4}}-\frac {b n \,d^{2}}{2 e^{4} \left (e x +d \right )}+\frac {5 b n d \ln \left (e x \right )}{2 e^{4}}+\frac {3 b n d \ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right )}{e^{4}}+\frac {3 b n d \operatorname {dilog}\left (-\frac {e x}{d}\right )}{e^{4}}+\left (\frac {i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}-\frac {i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )}{2}-\frac {i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{2}+\frac {i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )}{2}+b \ln \left (c \right )+a \right ) \left (\frac {x}{e^{3}}-\frac {3 d^{2}}{e^{4} \left (e x +d \right )}-\frac {3 d \ln \left (e x +d \right )}{e^{4}}+\frac {d^{3}}{2 e^{4} \left (e x +d \right )^{2}}\right )\) | \(303\) |
Input:
int(x^3*(a+b*ln(c*x^n))/(e*x+d)^3,x,method=_RETURNVERBOSE)
Output:
b*ln(x^n)*x/e^3-3*b*ln(x^n)/e^4*d^2/(e*x+d)-3*b*ln(x^n)/e^4*d*ln(e*x+d)+1/ 2*b*ln(x^n)*d^3/e^4/(e*x+d)^2-b*n*x/e^3-b*n/e^4*d-5/2*b*n/e^4*d*ln(e*x+d)- 1/2*b*n/e^4*d^2/(e*x+d)+5/2*b*n/e^4*d*ln(e*x)+3*b*n/e^4*d*ln(e*x+d)*ln(-e* x/d)+3*b*n/e^4*d*dilog(-e*x/d)+(1/2*I*Pi*b*csgn(I*x^n)*csgn(I*c*x^n)^2-1/2 *I*Pi*b*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-1/2*I*Pi*b*csgn(I*c*x^n)^3+1/2 *I*Pi*b*csgn(I*c*x^n)^2*csgn(I*c)+b*ln(c)+a)*(x/e^3-3/e^4*d^2/(e*x+d)-3/e^ 4*d*ln(e*x+d)+1/2*d^3/e^4/(e*x+d)^2)
\[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^3} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} x^{3}}{{\left (e x + d\right )}^{3}} \,d x } \] Input:
integrate(x^3*(a+b*log(c*x^n))/(e*x+d)^3,x, algorithm="fricas")
Output:
integral((b*x^3*log(c*x^n) + a*x^3)/(e^3*x^3 + 3*d*e^2*x^2 + 3*d^2*e*x + d ^3), x)
Time = 32.09 (sec) , antiderivative size = 391, normalized size of antiderivative = 2.62 \[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^3} \, dx =\text {Too large to display} \] Input:
integrate(x**3*(a+b*ln(c*x**n))/(e*x+d)**3,x)
Output:
-a*d**3*Piecewise((x/d**3, Eq(e, 0)), (-1/(2*e*(d + e*x)**2), True))/e**3 + 3*a*d**2*Piecewise((x/d**2, Eq(e, 0)), (-1/(d*e + e**2*x), True))/e**3 - 3*a*d*Piecewise((x/d, Eq(e, 0)), (log(d + e*x)/e, True))/e**3 + a*x/e**3 + b*d**3*n*Piecewise((x/d**3, Eq(e, 0)), (-1/(2*d**2*e + 2*d*e**2*x) - log (x)/(2*d**2*e) + log(d/e + x)/(2*d**2*e), True))/e**3 - b*d**3*Piecewise(( x/d**3, Eq(e, 0)), (-1/(2*e*(d + e*x)**2), True))*log(c*x**n)/e**3 - 3*b*d **2*n*Piecewise((x/d**2, Eq(e, 0)), (-log(x)/(d*e) + log(d/e + x)/(d*e), T rue))/e**3 + 3*b*d**2*Piecewise((x/d**2, Eq(e, 0)), (-1/(d*e + e**2*x), Tr ue))*log(c*x**n)/e**3 + 3*b*d*n*Piecewise((x/d, Eq(e, 0)), (Piecewise((-po lylog(2, e*x*exp_polar(I*pi)/d), (Abs(x) < 1) & (1/Abs(x) < 1)), (log(d)*l og(x) - polylog(2, e*x*exp_polar(I*pi)/d), Abs(x) < 1), (-log(d)*log(1/x) - polylog(2, e*x*exp_polar(I*pi)/d), 1/Abs(x) < 1), (-meijerg(((), (1, 1)) , ((0, 0), ()), x)*log(d) + meijerg(((1, 1), ()), ((), (0, 0)), x)*log(d) - polylog(2, e*x*exp_polar(I*pi)/d), True))/e, True))/e**3 - 3*b*d*Piecewi se((x/d, Eq(e, 0)), (log(d + e*x)/e, True))*log(c*x**n)/e**3 - b*n*x/e**3 + b*x*log(c*x**n)/e**3
\[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^3} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} x^{3}}{{\left (e x + d\right )}^{3}} \,d x } \] Input:
integrate(x^3*(a+b*log(c*x^n))/(e*x+d)^3,x, algorithm="maxima")
Output:
-1/2*a*((6*d^2*e*x + 5*d^3)/(e^6*x^2 + 2*d*e^5*x + d^2*e^4) - 2*x/e^3 + 6* d*log(e*x + d)/e^4) + b*integrate((x^3*log(c) + x^3*log(x^n))/(e^3*x^3 + 3 *d*e^2*x^2 + 3*d^2*e*x + d^3), x)
\[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^3} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} x^{3}}{{\left (e x + d\right )}^{3}} \,d x } \] Input:
integrate(x^3*(a+b*log(c*x^n))/(e*x+d)^3,x, algorithm="giac")
Output:
integrate((b*log(c*x^n) + a)*x^3/(e*x + d)^3, x)
Timed out. \[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^3} \, dx=\int \frac {x^3\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{{\left (d+e\,x\right )}^3} \,d x \] Input:
int((x^3*(a + b*log(c*x^n)))/(d + e*x)^3,x)
Output:
int((x^3*(a + b*log(c*x^n)))/(d + e*x)^3, x)
\[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^3} \, dx=\frac {6 \left (\int \frac {\mathrm {log}\left (x^{n} c \right )}{e^{3} x^{4}+3 d \,e^{2} x^{3}+3 d^{2} e \,x^{2}+d^{3} x}d x \right ) b \,d^{6} n +12 \left (\int \frac {\mathrm {log}\left (x^{n} c \right )}{e^{3} x^{4}+3 d \,e^{2} x^{3}+3 d^{2} e \,x^{2}+d^{3} x}d x \right ) b \,d^{5} e n x +6 \left (\int \frac {\mathrm {log}\left (x^{n} c \right )}{e^{3} x^{4}+3 d \,e^{2} x^{3}+3 d^{2} e \,x^{2}+d^{3} x}d x \right ) b \,d^{4} e^{2} n \,x^{2}-6 \,\mathrm {log}\left (e x +d \right ) a \,d^{3} n -12 \,\mathrm {log}\left (e x +d \right ) a \,d^{2} e n x -6 \,\mathrm {log}\left (e x +d \right ) a d \,e^{2} n \,x^{2}-14 \,\mathrm {log}\left (e x +d \right ) b \,d^{3} n^{2}-28 \,\mathrm {log}\left (e x +d \right ) b \,d^{2} e \,n^{2} x -14 \,\mathrm {log}\left (e x +d \right ) b d \,e^{2} n^{2} x^{2}-3 \mathrm {log}\left (x^{n} c \right )^{2} b \,d^{3}-6 \mathrm {log}\left (x^{n} c \right )^{2} b \,d^{2} e x -3 \mathrm {log}\left (x^{n} c \right )^{2} b d \,e^{2} x^{2}-9 \,\mathrm {log}\left (x^{n} c \right ) b \,d^{3} n +9 \,\mathrm {log}\left (x^{n} c \right ) b d \,e^{2} n \,x^{2}+2 \,\mathrm {log}\left (x^{n} c \right ) b \,e^{3} n \,x^{3}+9 \,\mathrm {log}\left (x \right ) b \,d^{3} n^{2}+18 \,\mathrm {log}\left (x \right ) b \,d^{2} e \,n^{2} x +9 \,\mathrm {log}\left (x \right ) b d \,e^{2} n^{2} x^{2}-3 a \,d^{3} n +6 a d \,e^{2} n \,x^{2}+2 a \,e^{3} n \,x^{3}+2 b \,d^{3} n^{2}-4 b d \,e^{2} n^{2} x^{2}-2 b \,e^{3} n^{2} x^{3}}{2 e^{4} n \left (e^{2} x^{2}+2 d e x +d^{2}\right )} \] Input:
int(x^3*(a+b*log(c*x^n))/(e*x+d)^3,x)
Output:
(6*int(log(x**n*c)/(d**3*x + 3*d**2*e*x**2 + 3*d*e**2*x**3 + e**3*x**4),x) *b*d**6*n + 12*int(log(x**n*c)/(d**3*x + 3*d**2*e*x**2 + 3*d*e**2*x**3 + e **3*x**4),x)*b*d**5*e*n*x + 6*int(log(x**n*c)/(d**3*x + 3*d**2*e*x**2 + 3* d*e**2*x**3 + e**3*x**4),x)*b*d**4*e**2*n*x**2 - 6*log(d + e*x)*a*d**3*n - 12*log(d + e*x)*a*d**2*e*n*x - 6*log(d + e*x)*a*d*e**2*n*x**2 - 14*log(d + e*x)*b*d**3*n**2 - 28*log(d + e*x)*b*d**2*e*n**2*x - 14*log(d + e*x)*b*d *e**2*n**2*x**2 - 3*log(x**n*c)**2*b*d**3 - 6*log(x**n*c)**2*b*d**2*e*x - 3*log(x**n*c)**2*b*d*e**2*x**2 - 9*log(x**n*c)*b*d**3*n + 9*log(x**n*c)*b* d*e**2*n*x**2 + 2*log(x**n*c)*b*e**3*n*x**3 + 9*log(x)*b*d**3*n**2 + 18*lo g(x)*b*d**2*e*n**2*x + 9*log(x)*b*d*e**2*n**2*x**2 - 3*a*d**3*n + 6*a*d*e* *2*n*x**2 + 2*a*e**3*n*x**3 + 2*b*d**3*n**2 - 4*b*d*e**2*n**2*x**2 - 2*b*e **3*n**2*x**3)/(2*e**4*n*(d**2 + 2*d*e*x + e**2*x**2))