Integrand size = 19, antiderivative size = 117 \[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^4} \, dx=-\frac {b n}{6 e^2 (d+e x)^2}+\frac {b n}{6 d e^2 (d+e x)}+\frac {b n \log (x)}{6 d^2 e^2}+\frac {d \left (a+b \log \left (c x^n\right )\right )}{3 e^2 (d+e x)^3}-\frac {a+b \log \left (c x^n\right )}{2 e^2 (d+e x)^2}-\frac {b n \log (d+e x)}{6 d^2 e^2} \] Output:
-1/6*b*n/e^2/(e*x+d)^2+1/6*b*n/d/e^2/(e*x+d)+1/6*b*n*ln(x)/d^2/e^2+1/3*d*( a+b*ln(c*x^n))/e^2/(e*x+d)^3-1/2*(a+b*ln(c*x^n))/e^2/(e*x+d)^2-1/6*b*n*ln( e*x+d)/d^2/e^2
Time = 0.12 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.15 \[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^4} \, dx=\frac {d \left (a+b \log \left (c x^n\right )\right )}{3 e^2 (d+e x)^3}-\frac {a+b \log \left (c x^n\right )}{2 e^2 (d+e x)^2}-\frac {b n \left (\frac {1}{(d+e x)^2}+\frac {2}{d (d+e x)}+\frac {2 \log (x)}{d^2}-\frac {2 \log (d+e x)}{d^2}\right )}{6 e^2}+\frac {b n \left (\frac {1}{d (d+e x)}+\frac {\log (x)}{d^2}-\frac {\log (d+e x)}{d^2}\right )}{2 e^2} \] Input:
Integrate[(x*(a + b*Log[c*x^n]))/(d + e*x)^4,x]
Output:
(d*(a + b*Log[c*x^n]))/(3*e^2*(d + e*x)^3) - (a + b*Log[c*x^n])/(2*e^2*(d + e*x)^2) - (b*n*((d + e*x)^(-2) + 2/(d*(d + e*x)) + (2*Log[x])/d^2 - (2*L og[d + e*x])/d^2))/(6*e^2) + (b*n*(1/(d*(d + e*x)) + Log[x]/d^2 - Log[d + e*x]/d^2))/(2*e^2)
Time = 0.31 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.83, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {2782, 27, 86, 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.
| \(\displaystyle \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^4} \, dx\) |
| \(\Big \downarrow \) 2782 |
| \(\displaystyle -b n \int -\frac {d+3 e x}{6 e^2 x (d+e x)^3}dx-\frac {a+b \log \left (c x^n\right )}{2 e^2 (d+e x)^2}+\frac {d \left (a+b \log \left (c x^n\right )\right )}{3 e^2 (d+e x)^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b n \int \frac {d+3 e x}{x (d+e x)^3}dx}{6 e^2}-\frac {a+b \log \left (c x^n\right )}{2 e^2 (d+e x)^2}+\frac {d \left (a+b \log \left (c x^n\right )\right )}{3 e^2 (d+e x)^3}\) |
| \(\Big \downarrow \) 86 |
| \(\displaystyle \frac {b n \int \left (-\frac {e}{d^2 (d+e x)}-\frac {e}{d (d+e x)^2}+\frac {2 e}{(d+e x)^3}+\frac {1}{d^2 x}\right )dx}{6 e^2}-\frac {a+b \log \left (c x^n\right )}{2 e^2 (d+e x)^2}+\frac {d \left (a+b \log \left (c x^n\right )\right )}{3 e^2 (d+e x)^3}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {a+b \log \left (c x^n\right )}{2 e^2 (d+e x)^2}+\frac {d \left (a+b \log \left (c x^n\right )\right )}{3 e^2 (d+e x)^3}+\frac {b n \left (-\frac {\log (d+e x)}{d^2}+\frac {\log (x)}{d^2}+\frac {1}{d (d+e x)}-\frac {1}{(d+e x)^2}\right )}{6 e^2}\) |
Input:
Int[(x*(a + b*Log[c*x^n]))/(d + e*x)^4,x]
Output:
(d*(a + b*Log[c*x^n]))/(3*e^2*(d + e*x)^3) - (a + b*Log[c*x^n])/(2*e^2*(d + e*x)^2) + (b*n*(-(d + e*x)^(-2) + 1/(d*(d + e*x)) + Log[x]/d^2 - Log[d + e*x]/d^2))/(6*e^2)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_ .), x_] :> Int[ExpandIntegrand[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && ((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]))))
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(x_)^(m_.)*((d_) + (e_.)*(x_))^(q _), x_Symbol] :> With[{u = IntHide[x^m*(d + e*x)^q, x]}, Simp[(a + b*Log[c* x^n]) u, x] - Simp[b*n Int[SimplifyIntegrand[u/x, x], x], x]] /; FreeQ[ {a, b, c, d, e, n}, x] && ILtQ[m + q + 2, 0] && IGtQ[m, 0]
Time = 0.81 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.75
| method | result | size |
| parallelrisch | \(\frac {-\ln \left (e x +d \right ) b \,d^{4} e n +x b \,d^{3} e^{2} n +x^{2} b \,d^{2} e^{3} n -3 x \ln \left (c \,x^{n}\right ) b \,d^{3} e^{2}+\ln \left (x \right ) b \,d^{4} e n -\ln \left (e x +d \right ) x^{3} b d \,e^{4} n +3 \ln \left (x \right ) x^{2} b \,d^{2} e^{3} n -3 \ln \left (e x +d \right ) x^{2} b \,d^{2} e^{3} n +3 \ln \left (x \right ) x b \,d^{3} e^{2} n -3 \ln \left (e x +d \right ) x b \,d^{3} e^{2} n +\ln \left (x \right ) x^{3} b d \,e^{4} n +3 x^{2} a \,d^{2} e^{3}+x^{3} a d \,e^{4}-\ln \left (c \,x^{n}\right ) b \,d^{4} e}{6 d^{3} e^{3} \left (e x +d \right )^{3}}\) | \(205\) |
| risch | \(-\frac {b \left (3 e x +d \right ) \ln \left (x^{n}\right )}{6 \left (e x +d \right )^{3} e^{2}}-\frac {-3 i \pi b \,d^{2} e x \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+3 i \pi b \,d^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right ) e x -i \pi b \,d^{3} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )-i \pi b \,d^{3} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+i \pi b \,d^{3} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-3 i \pi b \,d^{2} e x \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )+i \pi b \,d^{3} \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )+3 i \pi b \,d^{2} e x \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+2 \ln \left (e x +d \right ) b \,e^{3} n \,x^{3}-2 \ln \left (-x \right ) b \,e^{3} n \,x^{3}+6 \ln \left (e x +d \right ) b d \,e^{2} n \,x^{2}-6 \ln \left (-x \right ) b d \,e^{2} n \,x^{2}+6 \ln \left (e x +d \right ) b \,d^{2} e n x -6 \ln \left (-x \right ) b \,d^{2} e n x -2 b d \,e^{2} n \,x^{2}+6 \ln \left (c \right ) b \,d^{2} e x +2 \ln \left (e x +d \right ) b \,d^{3} n -2 \ln \left (-x \right ) b \,d^{3} n -2 b \,d^{2} e n x +2 d^{3} b \ln \left (c \right )+6 a \,d^{2} e x +2 a \,d^{3}}{12 d^{2} e^{2} \left (e x +d \right )^{3}}\) | \(403\) |
Input:
int(x*(a+b*ln(c*x^n))/(e*x+d)^4,x,method=_RETURNVERBOSE)
Output:
1/6*(-ln(e*x+d)*b*d^4*e*n+x*b*d^3*e^2*n+x^2*b*d^2*e^3*n-3*x*ln(c*x^n)*b*d^ 3*e^2+ln(x)*b*d^4*e*n-ln(e*x+d)*x^3*b*d*e^4*n+3*ln(x)*x^2*b*d^2*e^3*n-3*ln (e*x+d)*x^2*b*d^2*e^3*n+3*ln(x)*x*b*d^3*e^2*n-3*ln(e*x+d)*x*b*d^3*e^2*n+ln (x)*x^3*b*d*e^4*n+3*x^2*a*d^2*e^3+x^3*a*d*e^4-ln(c*x^n)*b*d^4*e)/d^3/e^3/( e*x+d)^3
Time = 0.08 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.38 \[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^4} \, dx=\frac {b d e^{2} n x^{2} - a d^{3} + {\left (b d^{2} e n - 3 \, a d^{2} e\right )} x - {\left (b e^{3} n x^{3} + 3 \, b d e^{2} n x^{2} + 3 \, b d^{2} e n x + b d^{3} n\right )} \log \left (e x + d\right ) - {\left (3 \, b d^{2} e x + b d^{3}\right )} \log \left (c\right ) + {\left (b e^{3} n x^{3} + 3 \, b d e^{2} n x^{2}\right )} \log \left (x\right )}{6 \, {\left (d^{2} e^{5} x^{3} + 3 \, d^{3} e^{4} x^{2} + 3 \, d^{4} e^{3} x + d^{5} e^{2}\right )}} \] Input:
integrate(x*(a+b*log(c*x^n))/(e*x+d)^4,x, algorithm="fricas")
Output:
1/6*(b*d*e^2*n*x^2 - a*d^3 + (b*d^2*e*n - 3*a*d^2*e)*x - (b*e^3*n*x^3 + 3* b*d*e^2*n*x^2 + 3*b*d^2*e*n*x + b*d^3*n)*log(e*x + d) - (3*b*d^2*e*x + b*d ^3)*log(c) + (b*e^3*n*x^3 + 3*b*d*e^2*n*x^2)*log(x))/(d^2*e^5*x^3 + 3*d^3* e^4*x^2 + 3*d^4*e^3*x + d^5*e^2)
Leaf count of result is larger than twice the leaf count of optimal. 661 vs. \(2 (110) = 220\).
Time = 6.54 (sec) , antiderivative size = 661, normalized size of antiderivative = 5.65 \[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^4} \, dx=\begin {cases} \tilde {\infty } \left (- \frac {a}{2 x^{2}} - \frac {b n}{4 x^{2}} - \frac {b \log {\left (c x^{n} \right )}}{2 x^{2}}\right ) & \text {for}\: d = 0 \wedge e = 0 \\\frac {\frac {a x^{2}}{2} - \frac {b n x^{2}}{4} + \frac {b x^{2} \log {\left (c x^{n} \right )}}{2}}{d^{4}} & \text {for}\: e = 0 \\\frac {- \frac {a}{2 x^{2}} - \frac {b n}{4 x^{2}} - \frac {b \log {\left (c x^{n} \right )}}{2 x^{2}}}{e^{4}} & \text {for}\: d = 0 \\- \frac {a d^{3}}{6 d^{5} e^{2} + 18 d^{4} e^{3} x + 18 d^{3} e^{4} x^{2} + 6 d^{2} e^{5} x^{3}} - \frac {3 a d^{2} e x}{6 d^{5} e^{2} + 18 d^{4} e^{3} x + 18 d^{3} e^{4} x^{2} + 6 d^{2} e^{5} x^{3}} - \frac {b d^{3} n \log {\left (\frac {d}{e} + x \right )}}{6 d^{5} e^{2} + 18 d^{4} e^{3} x + 18 d^{3} e^{4} x^{2} + 6 d^{2} e^{5} x^{3}} - \frac {3 b d^{2} e n x \log {\left (\frac {d}{e} + x \right )}}{6 d^{5} e^{2} + 18 d^{4} e^{3} x + 18 d^{3} e^{4} x^{2} + 6 d^{2} e^{5} x^{3}} + \frac {b d^{2} e n x}{6 d^{5} e^{2} + 18 d^{4} e^{3} x + 18 d^{3} e^{4} x^{2} + 6 d^{2} e^{5} x^{3}} - \frac {3 b d e^{2} n x^{2} \log {\left (\frac {d}{e} + x \right )}}{6 d^{5} e^{2} + 18 d^{4} e^{3} x + 18 d^{3} e^{4} x^{2} + 6 d^{2} e^{5} x^{3}} + \frac {b d e^{2} n x^{2}}{6 d^{5} e^{2} + 18 d^{4} e^{3} x + 18 d^{3} e^{4} x^{2} + 6 d^{2} e^{5} x^{3}} + \frac {3 b d e^{2} x^{2} \log {\left (c x^{n} \right )}}{6 d^{5} e^{2} + 18 d^{4} e^{3} x + 18 d^{3} e^{4} x^{2} + 6 d^{2} e^{5} x^{3}} - \frac {b e^{3} n x^{3} \log {\left (\frac {d}{e} + x \right )}}{6 d^{5} e^{2} + 18 d^{4} e^{3} x + 18 d^{3} e^{4} x^{2} + 6 d^{2} e^{5} x^{3}} + \frac {b e^{3} x^{3} \log {\left (c x^{n} \right )}}{6 d^{5} e^{2} + 18 d^{4} e^{3} x + 18 d^{3} e^{4} x^{2} + 6 d^{2} e^{5} x^{3}} & \text {otherwise} \end {cases} \] Input:
integrate(x*(a+b*ln(c*x**n))/(e*x+d)**4,x)
Output:
Piecewise((zoo*(-a/(2*x**2) - b*n/(4*x**2) - b*log(c*x**n)/(2*x**2)), Eq(d , 0) & Eq(e, 0)), ((a*x**2/2 - b*n*x**2/4 + b*x**2*log(c*x**n)/2)/d**4, Eq (e, 0)), ((-a/(2*x**2) - b*n/(4*x**2) - b*log(c*x**n)/(2*x**2))/e**4, Eq(d , 0)), (-a*d**3/(6*d**5*e**2 + 18*d**4*e**3*x + 18*d**3*e**4*x**2 + 6*d**2 *e**5*x**3) - 3*a*d**2*e*x/(6*d**5*e**2 + 18*d**4*e**3*x + 18*d**3*e**4*x* *2 + 6*d**2*e**5*x**3) - b*d**3*n*log(d/e + x)/(6*d**5*e**2 + 18*d**4*e**3 *x + 18*d**3*e**4*x**2 + 6*d**2*e**5*x**3) - 3*b*d**2*e*n*x*log(d/e + x)/( 6*d**5*e**2 + 18*d**4*e**3*x + 18*d**3*e**4*x**2 + 6*d**2*e**5*x**3) + b*d **2*e*n*x/(6*d**5*e**2 + 18*d**4*e**3*x + 18*d**3*e**4*x**2 + 6*d**2*e**5* x**3) - 3*b*d*e**2*n*x**2*log(d/e + x)/(6*d**5*e**2 + 18*d**4*e**3*x + 18* d**3*e**4*x**2 + 6*d**2*e**5*x**3) + b*d*e**2*n*x**2/(6*d**5*e**2 + 18*d** 4*e**3*x + 18*d**3*e**4*x**2 + 6*d**2*e**5*x**3) + 3*b*d*e**2*x**2*log(c*x **n)/(6*d**5*e**2 + 18*d**4*e**3*x + 18*d**3*e**4*x**2 + 6*d**2*e**5*x**3) - b*e**3*n*x**3*log(d/e + x)/(6*d**5*e**2 + 18*d**4*e**3*x + 18*d**3*e**4 *x**2 + 6*d**2*e**5*x**3) + b*e**3*x**3*log(c*x**n)/(6*d**5*e**2 + 18*d**4 *e**3*x + 18*d**3*e**4*x**2 + 6*d**2*e**5*x**3), True))
Time = 0.04 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.28 \[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^4} \, dx=\frac {1}{6} \, b n {\left (\frac {x}{d e^{3} x^{2} + 2 \, d^{2} e^{2} x + d^{3} e} - \frac {\log \left (e x + d\right )}{d^{2} e^{2}} + \frac {\log \left (x\right )}{d^{2} e^{2}}\right )} - \frac {{\left (3 \, e x + d\right )} b \log \left (c x^{n}\right )}{6 \, {\left (e^{5} x^{3} + 3 \, d e^{4} x^{2} + 3 \, d^{2} e^{3} x + d^{3} e^{2}\right )}} - \frac {{\left (3 \, e x + d\right )} a}{6 \, {\left (e^{5} x^{3} + 3 \, d e^{4} x^{2} + 3 \, d^{2} e^{3} x + d^{3} e^{2}\right )}} \] Input:
integrate(x*(a+b*log(c*x^n))/(e*x+d)^4,x, algorithm="maxima")
Output:
1/6*b*n*(x/(d*e^3*x^2 + 2*d^2*e^2*x + d^3*e) - log(e*x + d)/(d^2*e^2) + lo g(x)/(d^2*e^2)) - 1/6*(3*e*x + d)*b*log(c*x^n)/(e^5*x^3 + 3*d*e^4*x^2 + 3* d^2*e^3*x + d^3*e^2) - 1/6*(3*e*x + d)*a/(e^5*x^3 + 3*d*e^4*x^2 + 3*d^2*e^ 3*x + d^3*e^2)
Time = 0.12 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.39 \[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^4} \, dx=-\frac {{\left (3 \, b e n x + b d n\right )} \log \left (x\right )}{6 \, {\left (e^{5} x^{3} + 3 \, d e^{4} x^{2} + 3 \, d^{2} e^{3} x + d^{3} e^{2}\right )}} + \frac {b e^{2} n x^{2} + b d e n x - 3 \, b d e x \log \left (c\right ) - 3 \, a d e x - b d^{2} \log \left (c\right ) - a d^{2}}{6 \, {\left (d e^{5} x^{3} + 3 \, d^{2} e^{4} x^{2} + 3 \, d^{3} e^{3} x + d^{4} e^{2}\right )}} - \frac {b n \log \left (e x + d\right )}{6 \, d^{2} e^{2}} + \frac {b n \log \left (x\right )}{6 \, d^{2} e^{2}} \] Input:
integrate(x*(a+b*log(c*x^n))/(e*x+d)^4,x, algorithm="giac")
Output:
-1/6*(3*b*e*n*x + b*d*n)*log(x)/(e^5*x^3 + 3*d*e^4*x^2 + 3*d^2*e^3*x + d^3 *e^2) + 1/6*(b*e^2*n*x^2 + b*d*e*n*x - 3*b*d*e*x*log(c) - 3*a*d*e*x - b*d^ 2*log(c) - a*d^2)/(d*e^5*x^3 + 3*d^2*e^4*x^2 + 3*d^3*e^3*x + d^4*e^2) - 1/ 6*b*n*log(e*x + d)/(d^2*e^2) + 1/6*b*n*log(x)/(d^2*e^2)
Time = 26.34 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.21 \[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^4} \, dx=-\frac {a\,d+x\,\left (3\,a\,e-b\,e\,n\right )-\frac {b\,e^2\,n\,x^2}{d}}{6\,d^3\,e^2+18\,d^2\,e^3\,x+18\,d\,e^4\,x^2+6\,e^5\,x^3}-\frac {\ln \left (c\,x^n\right )\,\left (\frac {b\,d}{6\,e^2}+\frac {b\,x}{2\,e}\right )}{d^3+3\,d^2\,e\,x+3\,d\,e^2\,x^2+e^3\,x^3}-\frac {b\,n\,\mathrm {atanh}\left (\frac {2\,e\,x}{d}+1\right )}{3\,d^2\,e^2} \] Input:
int((x*(a + b*log(c*x^n)))/(d + e*x)^4,x)
Output:
- (a*d + x*(3*a*e - b*e*n) - (b*e^2*n*x^2)/d)/(6*d^3*e^2 + 6*e^5*x^3 + 18* d^2*e^3*x + 18*d*e^4*x^2) - (log(c*x^n)*((b*d)/(6*e^2) + (b*x)/(2*e)))/(d^ 3 + e^3*x^3 + 3*d*e^2*x^2 + 3*d^2*e*x) - (b*n*atanh((2*e*x)/d + 1))/(3*d^2 *e^2)
Time = 0.17 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.38 \[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^4} \, dx=\frac {-3 \,\mathrm {log}\left (e x +d \right ) b \,d^{3} n -9 \,\mathrm {log}\left (e x +d \right ) b \,d^{2} e n x -9 \,\mathrm {log}\left (e x +d \right ) b d \,e^{2} n \,x^{2}-3 \,\mathrm {log}\left (e x +d \right ) b \,e^{3} n \,x^{3}+9 \,\mathrm {log}\left (x^{n} c \right ) b d \,e^{2} x^{2}+3 \,\mathrm {log}\left (x^{n} c \right ) b \,e^{3} x^{3}-3 a \,d^{3}-9 a \,d^{2} e x -b \,d^{3} n -b \,e^{3} n \,x^{3}}{18 d^{2} e^{2} \left (e^{3} x^{3}+3 d \,e^{2} x^{2}+3 d^{2} e x +d^{3}\right )} \] Input:
int(x*(a+b*log(c*x^n))/(e*x+d)^4,x)
Output:
( - 3*log(d + e*x)*b*d**3*n - 9*log(d + e*x)*b*d**2*e*n*x - 9*log(d + e*x) *b*d*e**2*n*x**2 - 3*log(d + e*x)*b*e**3*n*x**3 + 9*log(x**n*c)*b*d*e**2*x **2 + 3*log(x**n*c)*b*e**3*x**3 - 3*a*d**3 - 9*a*d**2*e*x - b*d**3*n - b*e **3*n*x**3)/(18*d**2*e**2*(d**3 + 3*d**2*e*x + 3*d*e**2*x**2 + e**3*x**3))