Integrand size = 20, antiderivative size = 113 \[ \int \frac {d+e x}{x^2 \left (b x+c x^2\right )^2} \, dx=-\frac {d}{3 b^2 x^3}+\frac {2 c d-b e}{2 b^3 x^2}-\frac {c (3 c d-2 b e)}{b^4 x}-\frac {c^2 (c d-b e)}{b^4 (b+c x)}-\frac {c^2 (4 c d-3 b e) \log (x)}{b^5}+\frac {c^2 (4 c d-3 b e) \log (b+c x)}{b^5} \]
-1/3*d/b^2/x^3+1/2*(-b*e+2*c*d)/b^3/x^2-c*(-2*b*e+3*c*d)/b^4/x-c^2*(-b*e+c *d)/b^4/(c*x+b)-c^2*(-3*b*e+4*c*d)*ln(x)/b^5+c^2*(-3*b*e+4*c*d)*ln(c*x+b)/ b^5
Time = 0.06 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.94 \[ \int \frac {d+e x}{x^2 \left (b x+c x^2\right )^2} \, dx=\frac {-\frac {2 b^3 d}{x^3}-\frac {3 b^2 (-2 c d+b e)}{x^2}+\frac {6 b c (-3 c d+2 b e)}{x}+\frac {6 b c^2 (-c d+b e)}{b+c x}+6 c^2 (-4 c d+3 b e) \log (x)+6 c^2 (4 c d-3 b e) \log (b+c x)}{6 b^5} \]
((-2*b^3*d)/x^3 - (3*b^2*(-2*c*d + b*e))/x^2 + (6*b*c*(-3*c*d + 2*b*e))/x + (6*b*c^2*(-(c*d) + b*e))/(b + c*x) + 6*c^2*(-4*c*d + 3*b*e)*Log[x] + 6*c ^2*(4*c*d - 3*b*e)*Log[b + c*x])/(6*b^5)
Time = 0.28 (sec) , antiderivative size = 113, 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.
\(\displaystyle \int \frac {d+e x}{x^2 \left (b x+c x^2\right )^2} \, dx\) |
\(\Big \downarrow \) 9 |
\(\displaystyle \int \frac {d+e x}{x^4 (b+c x)^2}dx\) |
\(\Big \downarrow \) 86 |
\(\displaystyle \int \left (-\frac {c^3 (3 b e-4 c d)}{b^5 (b+c x)}+\frac {c^2 (3 b e-4 c d)}{b^5 x}-\frac {c^3 (b e-c d)}{b^4 (b+c x)^2}-\frac {c (2 b e-3 c d)}{b^4 x^2}+\frac {b e-2 c d}{b^3 x^3}+\frac {d}{b^2 x^4}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {c^2 \log (x) (4 c d-3 b e)}{b^5}+\frac {c^2 (4 c d-3 b e) \log (b+c x)}{b^5}-\frac {c^2 (c d-b e)}{b^4 (b+c x)}-\frac {c (3 c d-2 b e)}{b^4 x}+\frac {2 c d-b e}{2 b^3 x^2}-\frac {d}{3 b^2 x^3}\) |
-1/3*d/(b^2*x^3) + (2*c*d - b*e)/(2*b^3*x^2) - (c*(3*c*d - 2*b*e))/(b^4*x) - (c^2*(c*d - b*e))/(b^4*(b + c*x)) - (c^2*(4*c*d - 3*b*e)*Log[x])/b^5 + (c^2*(4*c*d - 3*b*e)*Log[b + c*x])/b^5
3.1.58.3.1 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 = 0.05 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.95
method | result | size |
default | \(-\frac {d}{3 b^{2} x^{3}}-\frac {b e -2 c d}{2 b^{3} x^{2}}+\frac {c^{2} \left (3 b e -4 c d \right ) \ln \left (x \right )}{b^{5}}+\frac {c \left (2 b e -3 c d \right )}{b^{4} x}-\frac {c^{2} \left (3 b e -4 c d \right ) \ln \left (c x +b \right )}{b^{5}}+\frac {\left (b e -c d \right ) c^{2}}{b^{4} \left (c x +b \right )}\) | \(107\) |
norman | \(\frac {\frac {c \left (-3 e \,c^{2} b +4 d \,c^{3}\right ) x^{4}}{b^{5}}-\frac {d}{3 b}-\frac {\left (3 b e -4 c d \right ) x}{6 b^{2}}+\frac {c \left (3 b e -4 c d \right ) x^{2}}{2 b^{3}}}{x^{3} \left (c x +b \right )}+\frac {c^{2} \left (3 b e -4 c d \right ) \ln \left (x \right )}{b^{5}}-\frac {c^{2} \left (3 b e -4 c d \right ) \ln \left (c x +b \right )}{b^{5}}\) | \(116\) |
risch | \(\frac {\frac {c^{2} \left (3 b e -4 c d \right ) x^{3}}{b^{4}}+\frac {c \left (3 b e -4 c d \right ) x^{2}}{2 b^{3}}-\frac {\left (3 b e -4 c d \right ) x}{6 b^{2}}-\frac {d}{3 b}}{x^{3} \left (c x +b \right )}+\frac {3 c^{2} \ln \left (-x \right ) e}{b^{4}}-\frac {4 c^{3} \ln \left (-x \right ) d}{b^{5}}-\frac {3 c^{2} \ln \left (c x +b \right ) e}{b^{4}}+\frac {4 c^{3} \ln \left (c x +b \right ) d}{b^{5}}\) | \(128\) |
parallelrisch | \(\frac {18 \ln \left (x \right ) x^{4} b \,c^{3} e -24 \ln \left (x \right ) x^{4} c^{4} d -18 \ln \left (c x +b \right ) x^{4} b \,c^{3} e +24 \ln \left (c x +b \right ) x^{4} c^{4} d +18 \ln \left (x \right ) x^{3} b^{2} c^{2} e -24 \ln \left (x \right ) x^{3} b \,c^{3} d -18 \ln \left (c x +b \right ) x^{3} b^{2} c^{2} e +24 \ln \left (c x +b \right ) x^{3} b \,c^{3} d -18 x^{4} b \,c^{3} e +24 x^{4} c^{4} d +9 x^{2} b^{3} c e -12 x^{2} b^{2} c^{2} d -3 x \,b^{4} e +4 x \,b^{3} c d -2 d \,b^{4}}{6 b^{5} x^{3} \left (c x +b \right )}\) | \(193\) |
-1/3*d/b^2/x^3-1/2*(b*e-2*c*d)/b^3/x^2+c^2*(3*b*e-4*c*d)/b^5*ln(x)+c*(2*b* e-3*c*d)/b^4/x-c^2*(3*b*e-4*c*d)/b^5*ln(c*x+b)+(b*e-c*d)/b^4*c^2/(c*x+b)
Time = 0.30 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.59 \[ \int \frac {d+e x}{x^2 \left (b x+c x^2\right )^2} \, dx=-\frac {2 \, b^{4} d + 6 \, {\left (4 \, b c^{3} d - 3 \, b^{2} c^{2} e\right )} x^{3} + 3 \, {\left (4 \, b^{2} c^{2} d - 3 \, b^{3} c e\right )} x^{2} - {\left (4 \, b^{3} c d - 3 \, b^{4} e\right )} x - 6 \, {\left ({\left (4 \, c^{4} d - 3 \, b c^{3} e\right )} x^{4} + {\left (4 \, b c^{3} d - 3 \, b^{2} c^{2} e\right )} x^{3}\right )} \log \left (c x + b\right ) + 6 \, {\left ({\left (4 \, c^{4} d - 3 \, b c^{3} e\right )} x^{4} + {\left (4 \, b c^{3} d - 3 \, b^{2} c^{2} e\right )} x^{3}\right )} \log \left (x\right )}{6 \, {\left (b^{5} c x^{4} + b^{6} x^{3}\right )}} \]
-1/6*(2*b^4*d + 6*(4*b*c^3*d - 3*b^2*c^2*e)*x^3 + 3*(4*b^2*c^2*d - 3*b^3*c *e)*x^2 - (4*b^3*c*d - 3*b^4*e)*x - 6*((4*c^4*d - 3*b*c^3*e)*x^4 + (4*b*c^ 3*d - 3*b^2*c^2*e)*x^3)*log(c*x + b) + 6*((4*c^4*d - 3*b*c^3*e)*x^4 + (4*b *c^3*d - 3*b^2*c^2*e)*x^3)*log(x))/(b^5*c*x^4 + b^6*x^3)
Leaf count of result is larger than twice the leaf count of optimal. 219 vs. \(2 (104) = 208\).
Time = 0.32 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.94 \[ \int \frac {d+e x}{x^2 \left (b x+c x^2\right )^2} \, dx=\frac {- 2 b^{3} d + x^{3} \cdot \left (18 b c^{2} e - 24 c^{3} d\right ) + x^{2} \cdot \left (9 b^{2} c e - 12 b c^{2} d\right ) + x \left (- 3 b^{3} e + 4 b^{2} c d\right )}{6 b^{5} x^{3} + 6 b^{4} c x^{4}} + \frac {c^{2} \cdot \left (3 b e - 4 c d\right ) \log {\left (x + \frac {3 b^{2} c^{2} e - 4 b c^{3} d - b c^{2} \cdot \left (3 b e - 4 c d\right )}{6 b c^{3} e - 8 c^{4} d} \right )}}{b^{5}} - \frac {c^{2} \cdot \left (3 b e - 4 c d\right ) \log {\left (x + \frac {3 b^{2} c^{2} e - 4 b c^{3} d + b c^{2} \cdot \left (3 b e - 4 c d\right )}{6 b c^{3} e - 8 c^{4} d} \right )}}{b^{5}} \]
(-2*b**3*d + x**3*(18*b*c**2*e - 24*c**3*d) + x**2*(9*b**2*c*e - 12*b*c**2 *d) + x*(-3*b**3*e + 4*b**2*c*d))/(6*b**5*x**3 + 6*b**4*c*x**4) + c**2*(3* b*e - 4*c*d)*log(x + (3*b**2*c**2*e - 4*b*c**3*d - b*c**2*(3*b*e - 4*c*d)) /(6*b*c**3*e - 8*c**4*d))/b**5 - c**2*(3*b*e - 4*c*d)*log(x + (3*b**2*c**2 *e - 4*b*c**3*d + b*c**2*(3*b*e - 4*c*d))/(6*b*c**3*e - 8*c**4*d))/b**5
Time = 0.18 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.14 \[ \int \frac {d+e x}{x^2 \left (b x+c x^2\right )^2} \, dx=-\frac {2 \, b^{3} d + 6 \, {\left (4 \, c^{3} d - 3 \, b c^{2} e\right )} x^{3} + 3 \, {\left (4 \, b c^{2} d - 3 \, b^{2} c e\right )} x^{2} - {\left (4 \, b^{2} c d - 3 \, b^{3} e\right )} x}{6 \, {\left (b^{4} c x^{4} + b^{5} x^{3}\right )}} + \frac {{\left (4 \, c^{3} d - 3 \, b c^{2} e\right )} \log \left (c x + b\right )}{b^{5}} - \frac {{\left (4 \, c^{3} d - 3 \, b c^{2} e\right )} \log \left (x\right )}{b^{5}} \]
-1/6*(2*b^3*d + 6*(4*c^3*d - 3*b*c^2*e)*x^3 + 3*(4*b*c^2*d - 3*b^2*c*e)*x^ 2 - (4*b^2*c*d - 3*b^3*e)*x)/(b^4*c*x^4 + b^5*x^3) + (4*c^3*d - 3*b*c^2*e) *log(c*x + b)/b^5 - (4*c^3*d - 3*b*c^2*e)*log(x)/b^5
Time = 0.25 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.19 \[ \int \frac {d+e x}{x^2 \left (b x+c x^2\right )^2} \, dx=-\frac {{\left (4 \, c^{3} d - 3 \, b c^{2} e\right )} \log \left ({\left | x \right |}\right )}{b^{5}} + \frac {{\left (4 \, c^{4} d - 3 \, b c^{3} e\right )} \log \left ({\left | c x + b \right |}\right )}{b^{5} c} - \frac {2 \, b^{4} d + 6 \, {\left (4 \, b c^{3} d - 3 \, b^{2} c^{2} e\right )} x^{3} + 3 \, {\left (4 \, b^{2} c^{2} d - 3 \, b^{3} c e\right )} x^{2} - {\left (4 \, b^{3} c d - 3 \, b^{4} e\right )} x}{6 \, {\left (c x + b\right )} b^{5} x^{3}} \]
-(4*c^3*d - 3*b*c^2*e)*log(abs(x))/b^5 + (4*c^4*d - 3*b*c^3*e)*log(abs(c*x + b))/(b^5*c) - 1/6*(2*b^4*d + 6*(4*b*c^3*d - 3*b^2*c^2*e)*x^3 + 3*(4*b^2 *c^2*d - 3*b^3*c*e)*x^2 - (4*b^3*c*d - 3*b^4*e)*x)/((c*x + b)*b^5*x^3)
Time = 10.05 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.17 \[ \int \frac {d+e x}{x^2 \left (b x+c x^2\right )^2} \, dx=\frac {2\,c^2\,\mathrm {atanh}\left (\frac {c^2\,\left (3\,b\,e-4\,c\,d\right )\,\left (b+2\,c\,x\right )}{b\,\left (4\,c^3\,d-3\,b\,c^2\,e\right )}\right )\,\left (3\,b\,e-4\,c\,d\right )}{b^5}-\frac {\frac {d}{3\,b}+\frac {x\,\left (3\,b\,e-4\,c\,d\right )}{6\,b^2}-\frac {c\,x^2\,\left (3\,b\,e-4\,c\,d\right )}{2\,b^3}-\frac {c^2\,x^3\,\left (3\,b\,e-4\,c\,d\right )}{b^4}}{c\,x^4+b\,x^3} \]