Integrand size = 20, antiderivative size = 140 \[ \int \frac {d+e x}{x \left (b x+c x^2\right )^3} \, dx=-\frac {d}{3 b^3 x^3}+\frac {3 c d-b e}{2 b^4 x^2}-\frac {3 c (2 c d-b e)}{b^5 x}-\frac {c^2 (c d-b e)}{2 b^4 (b+c x)^2}-\frac {c^2 (4 c d-3 b e)}{b^5 (b+c x)}-\frac {2 c^2 (5 c d-3 b e) \log (x)}{b^6}+\frac {2 c^2 (5 c d-3 b e) \log (b+c x)}{b^6} \]
-1/3*d/b^3/x^3+1/2*(-b*e+3*c*d)/b^4/x^2-3*c*(-b*e+2*c*d)/b^5/x-1/2*c^2*(-b *e+c*d)/b^4/(c*x+b)^2-c^2*(-3*b*e+4*c*d)/b^5/(c*x+b)-2*c^2*(-3*b*e+5*c*d)* ln(x)/b^6+2*c^2*(-3*b*e+5*c*d)*ln(c*x+b)/b^6
Time = 0.08 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.92 \[ \int \frac {d+e x}{x \left (b x+c x^2\right )^3} \, dx=\frac {\frac {b \left (-60 c^4 d x^4+18 b c^3 x^3 (-5 d+2 e x)-b^4 (2 d+3 e x)+b^3 c x (5 d+12 e x)+2 b^2 c^2 x^2 (-10 d+27 e x)\right )}{x^3 (b+c x)^2}+12 c^2 (-5 c d+3 b e) \log (x)+12 c^2 (5 c d-3 b e) \log (b+c x)}{6 b^6} \]
((b*(-60*c^4*d*x^4 + 18*b*c^3*x^3*(-5*d + 2*e*x) - b^4*(2*d + 3*e*x) + b^3 *c*x*(5*d + 12*e*x) + 2*b^2*c^2*x^2*(-10*d + 27*e*x)))/(x^3*(b + c*x)^2) + 12*c^2*(-5*c*d + 3*b*e)*Log[x] + 12*c^2*(5*c*d - 3*b*e)*Log[b + c*x])/(6* b^6)
Time = 0.33 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {9, 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.
| \(\displaystyle \int \frac {d+e x}{x \left (b x+c x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 9 |
| \(\displaystyle \int \frac {d+e x}{x^4 (b+c x)^3}dx\) |
| \(\Big \downarrow \) 86 |
| \(\displaystyle \int \left (-\frac {2 c^3 (3 b e-5 c d)}{b^6 (b+c x)}+\frac {2 c^2 (3 b e-5 c d)}{b^6 x}-\frac {c^3 (3 b e-4 c d)}{b^5 (b+c x)^2}-\frac {3 c (b e-2 c d)}{b^5 x^2}-\frac {c^3 (b e-c d)}{b^4 (b+c x)^3}+\frac {b e-3 c d}{b^4 x^3}+\frac {d}{b^3 x^4}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 c^2 \log (x) (5 c d-3 b e)}{b^6}+\frac {2 c^2 (5 c d-3 b e) \log (b+c x)}{b^6}-\frac {c^2 (4 c d-3 b e)}{b^5 (b+c x)}-\frac {3 c (2 c d-b e)}{b^5 x}-\frac {c^2 (c d-b e)}{2 b^4 (b+c x)^2}+\frac {3 c d-b e}{2 b^4 x^2}-\frac {d}{3 b^3 x^3}\) |
-1/3*d/(b^3*x^3) + (3*c*d - b*e)/(2*b^4*x^2) - (3*c*(2*c*d - b*e))/(b^5*x) - (c^2*(c*d - b*e))/(2*b^4*(b + c*x)^2) - (c^2*(4*c*d - 3*b*e))/(b^5*(b + c*x)) - (2*c^2*(5*c*d - 3*b*e)*Log[x])/b^6 + (2*c^2*(5*c*d - 3*b*e)*Log[b + c*x])/b^6
3.1.66.3.1 Defintions of rubi rules used
Int[(u_.)*(Px_)^(p_.)*((e_.)*(x_))^(m_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Simp[1/e^(p*r) Int[u*(e*x)^(m + p*r)*ExpandToSum[Px/x^r, x]^p, x], x] /; IGtQ[r, 0]] /; FreeQ[{e, m}, x] && PolyQ[Px, x] && IntegerQ[p] && !MonomialQ[Px, 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]))))
Time = 0.06 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.94
| method | result | size |
| default | \(-\frac {d}{3 b^{3} x^{3}}-\frac {b e -3 c d}{2 b^{4} x^{2}}+\frac {3 c \left (b e -2 c d \right )}{b^{5} x}+\frac {2 c^{2} \left (3 b e -5 c d \right ) \ln \left (x \right )}{b^{6}}-\frac {2 c^{2} \left (3 b e -5 c d \right ) \ln \left (c x +b \right )}{b^{6}}+\frac {c^{2} \left (3 b e -4 c d \right )}{b^{5} \left (c x +b \right )}+\frac {\left (b e -c d \right ) c^{2}}{2 b^{4} \left (c x +b \right )^{2}}\) | \(132\) |
| norman | \(\frac {\frac {\left (9 b \,c^{4} e -15 c^{5} d \right ) x^{3}}{b^{4} c^{2}}-\frac {d}{3 b}-\frac {\left (3 b e -5 c d \right ) x}{6 b^{2}}+\frac {2 c \left (3 b e -5 c d \right ) x^{2}}{3 b^{3}}+\frac {2 \left (3 b \,c^{4} e -5 c^{5} d \right ) x^{4}}{c \,b^{5}}}{x^{3} \left (c x +b \right )^{2}}+\frac {2 c^{2} \left (3 b e -5 c d \right ) \ln \left (x \right )}{b^{6}}-\frac {2 c^{2} \left (3 b e -5 c d \right ) \ln \left (c x +b \right )}{b^{6}}\) | \(144\) |
| risch | \(\frac {\frac {2 c^{3} \left (3 b e -5 c d \right ) x^{4}}{b^{5}}+\frac {3 c^{2} \left (3 b e -5 c d \right ) x^{3}}{b^{4}}+\frac {2 c \left (3 b e -5 c d \right ) x^{2}}{3 b^{3}}-\frac {\left (3 b e -5 c d \right ) x}{6 b^{2}}-\frac {d}{3 b}}{x^{3} \left (c x +b \right )^{2}}-\frac {6 c^{2} \ln \left (c x +b \right ) e}{b^{5}}+\frac {10 c^{3} \ln \left (c x +b \right ) d}{b^{6}}+\frac {6 c^{2} \ln \left (-x \right ) e}{b^{5}}-\frac {10 c^{3} \ln \left (-x \right ) d}{b^{6}}\) | \(149\) |
| parallelrisch | \(\frac {36 \ln \left (x \right ) x^{5} b \,c^{6} e -60 \ln \left (x \right ) x^{5} c^{7} d -36 \ln \left (c x +b \right ) x^{5} b \,c^{6} e +60 \ln \left (c x +b \right ) x^{5} c^{7} d +72 \ln \left (x \right ) x^{4} b^{2} c^{5} e -120 \ln \left (x \right ) x^{4} b \,c^{6} d -72 \ln \left (c x +b \right ) x^{4} b^{2} c^{5} e +120 \ln \left (c x +b \right ) x^{4} b \,c^{6} d +36 \ln \left (x \right ) x^{3} b^{3} c^{4} e -60 \ln \left (x \right ) x^{3} b^{2} c^{5} d -36 \ln \left (c x +b \right ) x^{3} b^{3} c^{4} e +60 \ln \left (c x +b \right ) x^{3} b^{2} c^{5} d +36 x^{4} b^{2} c^{5} e -60 x^{4} b \,c^{6} d +54 x^{3} b^{3} c^{4} e -90 x^{3} b^{2} c^{5} d +12 x^{2} b^{4} c^{3} e -20 x^{2} b^{3} c^{4} d -3 x \,b^{5} c^{2} e +5 x \,b^{4} c^{3} d -2 b^{5} c^{2} d}{6 c^{2} b^{6} x^{3} \left (c x +b \right )^{2}}\) | \(297\) |
-1/3*d/b^3/x^3-1/2*(b*e-3*c*d)/b^4/x^2+3*c*(b*e-2*c*d)/b^5/x+2*c^2*(3*b*e- 5*c*d)/b^6*ln(x)-2*c^2*(3*b*e-5*c*d)/b^6*ln(c*x+b)+c^2*(3*b*e-4*c*d)/b^5/( c*x+b)+1/2*(b*e-c*d)/b^4*c^2/(c*x+b)^2
Time = 0.28 (sec) , antiderivative size = 263, normalized size of antiderivative = 1.88 \[ \int \frac {d+e x}{x \left (b x+c x^2\right )^3} \, dx=-\frac {2 \, b^{5} d + 12 \, {\left (5 \, b c^{4} d - 3 \, b^{2} c^{3} e\right )} x^{4} + 18 \, {\left (5 \, b^{2} c^{3} d - 3 \, b^{3} c^{2} e\right )} x^{3} + 4 \, {\left (5 \, b^{3} c^{2} d - 3 \, b^{4} c e\right )} x^{2} - {\left (5 \, b^{4} c d - 3 \, b^{5} e\right )} x - 12 \, {\left ({\left (5 \, c^{5} d - 3 \, b c^{4} e\right )} x^{5} + 2 \, {\left (5 \, b c^{4} d - 3 \, b^{2} c^{3} e\right )} x^{4} + {\left (5 \, b^{2} c^{3} d - 3 \, b^{3} c^{2} e\right )} x^{3}\right )} \log \left (c x + b\right ) + 12 \, {\left ({\left (5 \, c^{5} d - 3 \, b c^{4} e\right )} x^{5} + 2 \, {\left (5 \, b c^{4} d - 3 \, b^{2} c^{3} e\right )} x^{4} + {\left (5 \, b^{2} c^{3} d - 3 \, b^{3} c^{2} e\right )} x^{3}\right )} \log \left (x\right )}{6 \, {\left (b^{6} c^{2} x^{5} + 2 \, b^{7} c x^{4} + b^{8} x^{3}\right )}} \]
-1/6*(2*b^5*d + 12*(5*b*c^4*d - 3*b^2*c^3*e)*x^4 + 18*(5*b^2*c^3*d - 3*b^3 *c^2*e)*x^3 + 4*(5*b^3*c^2*d - 3*b^4*c*e)*x^2 - (5*b^4*c*d - 3*b^5*e)*x - 12*((5*c^5*d - 3*b*c^4*e)*x^5 + 2*(5*b*c^4*d - 3*b^2*c^3*e)*x^4 + (5*b^2*c ^3*d - 3*b^3*c^2*e)*x^3)*log(c*x + b) + 12*((5*c^5*d - 3*b*c^4*e)*x^5 + 2* (5*b*c^4*d - 3*b^2*c^3*e)*x^4 + (5*b^2*c^3*d - 3*b^3*c^2*e)*x^3)*log(x))/( b^6*c^2*x^5 + 2*b^7*c*x^4 + b^8*x^3)
Time = 0.42 (sec) , antiderivative size = 262, normalized size of antiderivative = 1.87 \[ \int \frac {d+e x}{x \left (b x+c x^2\right )^3} \, dx=\frac {- 2 b^{4} d + x^{4} \cdot \left (36 b c^{3} e - 60 c^{4} d\right ) + x^{3} \cdot \left (54 b^{2} c^{2} e - 90 b c^{3} d\right ) + x^{2} \cdot \left (12 b^{3} c e - 20 b^{2} c^{2} d\right ) + x \left (- 3 b^{4} e + 5 b^{3} c d\right )}{6 b^{7} x^{3} + 12 b^{6} c x^{4} + 6 b^{5} c^{2} x^{5}} + \frac {2 c^{2} \cdot \left (3 b e - 5 c d\right ) \log {\left (x + \frac {6 b^{2} c^{2} e - 10 b c^{3} d - 2 b c^{2} \cdot \left (3 b e - 5 c d\right )}{12 b c^{3} e - 20 c^{4} d} \right )}}{b^{6}} - \frac {2 c^{2} \cdot \left (3 b e - 5 c d\right ) \log {\left (x + \frac {6 b^{2} c^{2} e - 10 b c^{3} d + 2 b c^{2} \cdot \left (3 b e - 5 c d\right )}{12 b c^{3} e - 20 c^{4} d} \right )}}{b^{6}} \]
(-2*b**4*d + x**4*(36*b*c**3*e - 60*c**4*d) + x**3*(54*b**2*c**2*e - 90*b* c**3*d) + x**2*(12*b**3*c*e - 20*b**2*c**2*d) + x*(-3*b**4*e + 5*b**3*c*d) )/(6*b**7*x**3 + 12*b**6*c*x**4 + 6*b**5*c**2*x**5) + 2*c**2*(3*b*e - 5*c* d)*log(x + (6*b**2*c**2*e - 10*b*c**3*d - 2*b*c**2*(3*b*e - 5*c*d))/(12*b* c**3*e - 20*c**4*d))/b**6 - 2*c**2*(3*b*e - 5*c*d)*log(x + (6*b**2*c**2*e - 10*b*c**3*d + 2*b*c**2*(3*b*e - 5*c*d))/(12*b*c**3*e - 20*c**4*d))/b**6
Time = 0.18 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.18 \[ \int \frac {d+e x}{x \left (b x+c x^2\right )^3} \, dx=-\frac {2 \, b^{4} d + 12 \, {\left (5 \, c^{4} d - 3 \, b c^{3} e\right )} x^{4} + 18 \, {\left (5 \, b c^{3} d - 3 \, b^{2} c^{2} e\right )} x^{3} + 4 \, {\left (5 \, b^{2} c^{2} d - 3 \, b^{3} c e\right )} x^{2} - {\left (5 \, b^{3} c d - 3 \, b^{4} e\right )} x}{6 \, {\left (b^{5} c^{2} x^{5} + 2 \, b^{6} c x^{4} + b^{7} x^{3}\right )}} + \frac {2 \, {\left (5 \, c^{3} d - 3 \, b c^{2} e\right )} \log \left (c x + b\right )}{b^{6}} - \frac {2 \, {\left (5 \, c^{3} d - 3 \, b c^{2} e\right )} \log \left (x\right )}{b^{6}} \]
-1/6*(2*b^4*d + 12*(5*c^4*d - 3*b*c^3*e)*x^4 + 18*(5*b*c^3*d - 3*b^2*c^2*e )*x^3 + 4*(5*b^2*c^2*d - 3*b^3*c*e)*x^2 - (5*b^3*c*d - 3*b^4*e)*x)/(b^5*c^ 2*x^5 + 2*b^6*c*x^4 + b^7*x^3) + 2*(5*c^3*d - 3*b*c^2*e)*log(c*x + b)/b^6 - 2*(5*c^3*d - 3*b*c^2*e)*log(x)/b^6
Time = 0.26 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.14 \[ \int \frac {d+e x}{x \left (b x+c x^2\right )^3} \, dx=-\frac {2 \, {\left (5 \, c^{3} d - 3 \, b c^{2} e\right )} \log \left ({\left | x \right |}\right )}{b^{6}} + \frac {2 \, {\left (5 \, c^{4} d - 3 \, b c^{3} e\right )} \log \left ({\left | c x + b \right |}\right )}{b^{6} c} - \frac {2 \, b^{5} d + 12 \, {\left (5 \, b c^{4} d - 3 \, b^{2} c^{3} e\right )} x^{4} + 18 \, {\left (5 \, b^{2} c^{3} d - 3 \, b^{3} c^{2} e\right )} x^{3} + 4 \, {\left (5 \, b^{3} c^{2} d - 3 \, b^{4} c e\right )} x^{2} - {\left (5 \, b^{4} c d - 3 \, b^{5} e\right )} x}{6 \, {\left (c x + b\right )}^{2} b^{6} x^{3}} \]
-2*(5*c^3*d - 3*b*c^2*e)*log(abs(x))/b^6 + 2*(5*c^4*d - 3*b*c^3*e)*log(abs (c*x + b))/(b^6*c) - 1/6*(2*b^5*d + 12*(5*b*c^4*d - 3*b^2*c^3*e)*x^4 + 18* (5*b^2*c^3*d - 3*b^3*c^2*e)*x^3 + 4*(5*b^3*c^2*d - 3*b^4*c*e)*x^2 - (5*b^4 *c*d - 3*b^5*e)*x)/((c*x + b)^2*b^6*x^3)
Time = 9.80 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.16 \[ \int \frac {d+e x}{x \left (b x+c x^2\right )^3} \, dx=\frac {\frac {2\,c\,x^2\,\left (3\,b\,e-5\,c\,d\right )}{3\,b^3}-\frac {x\,\left (3\,b\,e-5\,c\,d\right )}{6\,b^2}-\frac {d}{3\,b}+\frac {3\,c^2\,x^3\,\left (3\,b\,e-5\,c\,d\right )}{b^4}+\frac {2\,c^3\,x^4\,\left (3\,b\,e-5\,c\,d\right )}{b^5}}{b^2\,x^3+2\,b\,c\,x^4+c^2\,x^5}+\frac {4\,c^2\,\mathrm {atanh}\left (\frac {2\,c^2\,\left (3\,b\,e-5\,c\,d\right )\,\left (b+2\,c\,x\right )}{b\,\left (10\,c^3\,d-6\,b\,c^2\,e\right )}\right )\,\left (3\,b\,e-5\,c\,d\right )}{b^6} \]