Integrand size = 18, antiderivative size = 88 \[ \int \frac {x (d+e x)}{\left (b x+c x^2\right )^3} \, dx=-\frac {d}{b^3 x}-\frac {c d-b e}{2 b^2 (b+c x)^2}-\frac {2 c d-b e}{b^3 (b+c x)}-\frac {(3 c d-b e) \log (x)}{b^4}+\frac {(3 c d-b e) \log (b+c x)}{b^4} \] Output:
-d/b^3/x-1/2*(-b*e+c*d)/b^2/(c*x+b)^2-(-b*e+2*c*d)/b^3/(c*x+b)-(-b*e+3*c*d )*ln(x)/b^4+(-b*e+3*c*d)*ln(c*x+b)/b^4
Time = 0.03 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.92 \[ \int \frac {x (d+e x)}{\left (b x+c x^2\right )^3} \, dx=\frac {-\frac {2 b d}{x}+\frac {b^2 (-c d+b e)}{(b+c x)^2}+\frac {2 b (-2 c d+b e)}{b+c x}+2 (-3 c d+b e) \log (x)+2 (3 c d-b e) \log (b+c x)}{2 b^4} \] Input:
Integrate[(x*(d + e*x))/(b*x + c*x^2)^3,x]
Output:
((-2*b*d)/x + (b^2*(-(c*d) + b*e))/(b + c*x)^2 + (2*b*(-2*c*d + b*e))/(b + c*x) + 2*(-3*c*d + b*e)*Log[x] + 2*(3*c*d - b*e)*Log[b + c*x])/(2*b^4)
Time = 0.41 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, 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.
\(\displaystyle \int \frac {x (d+e x)}{\left (b x+c x^2\right )^3} \, dx\) |
\(\Big \downarrow \) 9 |
\(\displaystyle \int \frac {d+e x}{x^2 (b+c x)^3}dx\) |
\(\Big \downarrow \) 86 |
\(\displaystyle \int \left (\frac {b e-3 c d}{b^4 x}-\frac {c (b e-3 c d)}{b^4 (b+c x)}-\frac {c (b e-2 c d)}{b^3 (b+c x)^2}+\frac {d}{b^3 x^2}-\frac {c (b e-c d)}{b^2 (b+c x)^3}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {\log (x) (3 c d-b e)}{b^4}+\frac {(3 c d-b e) \log (b+c x)}{b^4}-\frac {2 c d-b e}{b^3 (b+c x)}-\frac {d}{b^3 x}-\frac {c d-b e}{2 b^2 (b+c x)^2}\) |
Input:
Int[(x*(d + e*x))/(b*x + c*x^2)^3,x]
Output:
-(d/(b^3*x)) - (c*d - b*e)/(2*b^2*(b + c*x)^2) - (2*c*d - b*e)/(b^3*(b + c *x)) - ((3*c*d - b*e)*Log[x])/b^4 + ((3*c*d - b*e)*Log[b + c*x])/b^4
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 = 0.79 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.94
method | result | size |
default | \(-\frac {\left (b e -3 c d \right ) \ln \left (c x +b \right )}{b^{4}}+\frac {b e -2 c d}{b^{3} \left (c x +b \right )}+\frac {b e -c d}{2 b^{2} \left (c x +b \right )^{2}}-\frac {d}{b^{3} x}+\frac {\left (b e -3 c d \right ) \ln \left (x \right )}{b^{4}}\) | \(83\) |
norman | \(\frac {-\frac {d x}{b}+\frac {2 c \left (-b e +3 c d \right ) x^{3}}{b^{3}}+\frac {c^{2} \left (-3 b e +9 c d \right ) x^{4}}{2 b^{4}}}{x^{2} \left (c x +b \right )^{2}}+\frac {\left (b e -3 c d \right ) \ln \left (x \right )}{b^{4}}-\frac {\left (b e -3 c d \right ) \ln \left (c x +b \right )}{b^{4}}\) | \(92\) |
risch | \(\frac {\frac {c \left (b e -3 c d \right ) x^{2}}{b^{3}}+\frac {3 \left (b e -3 c d \right ) x}{2 b^{2}}-\frac {d}{b}}{\left (c x +b \right )^{2} x}-\frac {\ln \left (c x +b \right ) e}{b^{3}}+\frac {3 \ln \left (c x +b \right ) c d}{b^{4}}+\frac {\ln \left (-x \right ) e}{b^{3}}-\frac {3 \ln \left (-x \right ) c d}{b^{4}}\) | \(95\) |
parallelrisch | \(\frac {2 \ln \left (x \right ) x^{3} b \,c^{2} e -6 \ln \left (x \right ) x^{3} c^{3} d -2 \ln \left (c x +b \right ) x^{3} b \,c^{2} e +6 \ln \left (c x +b \right ) x^{3} c^{3} d +4 \ln \left (x \right ) x^{2} b^{2} c e -12 \ln \left (x \right ) x^{2} b \,c^{2} d -4 \ln \left (c x +b \right ) x^{2} b^{2} c e +12 \ln \left (c x +b \right ) x^{2} b \,c^{2} d -3 b \,c^{2} x^{3} e +9 c^{3} d \,x^{3}+2 \ln \left (x \right ) x \,b^{3} e -6 \ln \left (x \right ) x \,b^{2} c d -2 \ln \left (c x +b \right ) x \,b^{3} e +6 \ln \left (c x +b \right ) x \,b^{2} c d -4 b^{2} c e \,x^{2}+12 b \,c^{2} d \,x^{2}-2 b^{3} d}{2 b^{4} x \left (c x +b \right )^{2}}\) | \(218\) |
Input:
int(x*(e*x+d)/(c*x^2+b*x)^3,x,method=_RETURNVERBOSE)
Output:
-(b*e-3*c*d)/b^4*ln(c*x+b)+(b*e-2*c*d)/b^3/(c*x+b)+1/2*(b*e-c*d)/b^2/(c*x+ b)^2-d/b^3/x+(b*e-3*c*d)/b^4*ln(x)
Leaf count of result is larger than twice the leaf count of optimal. 195 vs. \(2 (86) = 172\).
Time = 0.09 (sec) , antiderivative size = 195, normalized size of antiderivative = 2.22 \[ \int \frac {x (d+e x)}{\left (b x+c x^2\right )^3} \, dx=-\frac {2 \, b^{3} d + 2 \, {\left (3 \, b c^{2} d - b^{2} c e\right )} x^{2} + 3 \, {\left (3 \, b^{2} c d - b^{3} e\right )} x - 2 \, {\left ({\left (3 \, c^{3} d - b c^{2} e\right )} x^{3} + 2 \, {\left (3 \, b c^{2} d - b^{2} c e\right )} x^{2} + {\left (3 \, b^{2} c d - b^{3} e\right )} x\right )} \log \left (c x + b\right ) + 2 \, {\left ({\left (3 \, c^{3} d - b c^{2} e\right )} x^{3} + 2 \, {\left (3 \, b c^{2} d - b^{2} c e\right )} x^{2} + {\left (3 \, b^{2} c d - b^{3} e\right )} x\right )} \log \left (x\right )}{2 \, {\left (b^{4} c^{2} x^{3} + 2 \, b^{5} c x^{2} + b^{6} x\right )}} \] Input:
integrate(x*(e*x+d)/(c*x^2+b*x)^3,x, algorithm="fricas")
Output:
-1/2*(2*b^3*d + 2*(3*b*c^2*d - b^2*c*e)*x^2 + 3*(3*b^2*c*d - b^3*e)*x - 2* ((3*c^3*d - b*c^2*e)*x^3 + 2*(3*b*c^2*d - b^2*c*e)*x^2 + (3*b^2*c*d - b^3* e)*x)*log(c*x + b) + 2*((3*c^3*d - b*c^2*e)*x^3 + 2*(3*b*c^2*d - b^2*c*e)* x^2 + (3*b^2*c*d - b^3*e)*x)*log(x))/(b^4*c^2*x^3 + 2*b^5*c*x^2 + b^6*x)
Leaf count of result is larger than twice the leaf count of optimal. 168 vs. \(2 (76) = 152\).
Time = 0.30 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.91 \[ \int \frac {x (d+e x)}{\left (b x+c x^2\right )^3} \, dx=\frac {- 2 b^{2} d + x^{2} \cdot \left (2 b c e - 6 c^{2} d\right ) + x \left (3 b^{2} e - 9 b c d\right )}{2 b^{5} x + 4 b^{4} c x^{2} + 2 b^{3} c^{2} x^{3}} + \frac {\left (b e - 3 c d\right ) \log {\left (x + \frac {b^{2} e - 3 b c d - b \left (b e - 3 c d\right )}{2 b c e - 6 c^{2} d} \right )}}{b^{4}} - \frac {\left (b e - 3 c d\right ) \log {\left (x + \frac {b^{2} e - 3 b c d + b \left (b e - 3 c d\right )}{2 b c e - 6 c^{2} d} \right )}}{b^{4}} \] Input:
integrate(x*(e*x+d)/(c*x**2+b*x)**3,x)
Output:
(-2*b**2*d + x**2*(2*b*c*e - 6*c**2*d) + x*(3*b**2*e - 9*b*c*d))/(2*b**5*x + 4*b**4*c*x**2 + 2*b**3*c**2*x**3) + (b*e - 3*c*d)*log(x + (b**2*e - 3*b *c*d - b*(b*e - 3*c*d))/(2*b*c*e - 6*c**2*d))/b**4 - (b*e - 3*c*d)*log(x + (b**2*e - 3*b*c*d + b*(b*e - 3*c*d))/(2*b*c*e - 6*c**2*d))/b**4
Time = 0.03 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.18 \[ \int \frac {x (d+e x)}{\left (b x+c x^2\right )^3} \, dx=-\frac {2 \, b^{2} d + 2 \, {\left (3 \, c^{2} d - b c e\right )} x^{2} + 3 \, {\left (3 \, b c d - b^{2} e\right )} x}{2 \, {\left (b^{3} c^{2} x^{3} + 2 \, b^{4} c x^{2} + b^{5} x\right )}} + \frac {{\left (3 \, c d - b e\right )} \log \left (c x + b\right )}{b^{4}} - \frac {{\left (3 \, c d - b e\right )} \log \left (x\right )}{b^{4}} \] Input:
integrate(x*(e*x+d)/(c*x^2+b*x)^3,x, algorithm="maxima")
Output:
-1/2*(2*b^2*d + 2*(3*c^2*d - b*c*e)*x^2 + 3*(3*b*c*d - b^2*e)*x)/(b^3*c^2* x^3 + 2*b^4*c*x^2 + b^5*x) + (3*c*d - b*e)*log(c*x + b)/b^4 - (3*c*d - b*e )*log(x)/b^4
Time = 0.12 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.17 \[ \int \frac {x (d+e x)}{\left (b x+c x^2\right )^3} \, dx=-\frac {{\left (3 \, c d - b e\right )} \log \left ({\left | x \right |}\right )}{b^{4}} + \frac {{\left (3 \, c^{2} d - b c e\right )} \log \left ({\left | c x + b \right |}\right )}{b^{4} c} - \frac {2 \, b^{3} d + 2 \, {\left (3 \, b c^{2} d - b^{2} c e\right )} x^{2} + 3 \, {\left (3 \, b^{2} c d - b^{3} e\right )} x}{2 \, {\left (c x + b\right )}^{2} b^{4} x} \] Input:
integrate(x*(e*x+d)/(c*x^2+b*x)^3,x, algorithm="giac")
Output:
-(3*c*d - b*e)*log(abs(x))/b^4 + (3*c^2*d - b*c*e)*log(abs(c*x + b))/(b^4* c) - 1/2*(2*b^3*d + 2*(3*b*c^2*d - b^2*c*e)*x^2 + 3*(3*b^2*c*d - b^3*e)*x) /((c*x + b)^2*b^4*x)
Time = 5.19 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.95 \[ \int \frac {x (d+e x)}{\left (b x+c x^2\right )^3} \, dx=\frac {\frac {3\,x\,\left (b\,e-3\,c\,d\right )}{2\,b^2}-\frac {d}{b}+\frac {c\,x^2\,\left (b\,e-3\,c\,d\right )}{b^3}}{b^2\,x+2\,b\,c\,x^2+c^2\,x^3}-\frac {2\,\mathrm {atanh}\left (\frac {2\,c\,x}{b}+1\right )\,\left (b\,e-3\,c\,d\right )}{b^4} \] Input:
int((x*(d + e*x))/(b*x + c*x^2)^3,x)
Output:
((3*x*(b*e - 3*c*d))/(2*b^2) - d/b + (c*x^2*(b*e - 3*c*d))/b^3)/(b^2*x + c ^2*x^3 + 2*b*c*x^2) - (2*atanh((2*c*x)/b + 1)*(b*e - 3*c*d))/b^4
Time = 0.19 (sec) , antiderivative size = 223, normalized size of antiderivative = 2.53 \[ \int \frac {x (d+e x)}{\left (b x+c x^2\right )^3} \, dx=\frac {-2 \,\mathrm {log}\left (c x +b \right ) b^{3} e x +6 \,\mathrm {log}\left (c x +b \right ) b^{2} c d x -4 \,\mathrm {log}\left (c x +b \right ) b^{2} c e \,x^{2}+12 \,\mathrm {log}\left (c x +b \right ) b \,c^{2} d \,x^{2}-2 \,\mathrm {log}\left (c x +b \right ) b \,c^{2} e \,x^{3}+6 \,\mathrm {log}\left (c x +b \right ) c^{3} d \,x^{3}+2 \,\mathrm {log}\left (x \right ) b^{3} e x -6 \,\mathrm {log}\left (x \right ) b^{2} c d x +4 \,\mathrm {log}\left (x \right ) b^{2} c e \,x^{2}-12 \,\mathrm {log}\left (x \right ) b \,c^{2} d \,x^{2}+2 \,\mathrm {log}\left (x \right ) b \,c^{2} e \,x^{3}-6 \,\mathrm {log}\left (x \right ) c^{3} d \,x^{3}-2 b^{3} d +2 b^{3} e x -6 b^{2} c d x -b \,c^{2} e \,x^{3}+3 c^{3} d \,x^{3}}{2 b^{4} x \left (c^{2} x^{2}+2 b c x +b^{2}\right )} \] Input:
int(x*(e*x+d)/(c*x^2+b*x)^3,x)
Output:
( - 2*log(b + c*x)*b**3*e*x + 6*log(b + c*x)*b**2*c*d*x - 4*log(b + c*x)*b **2*c*e*x**2 + 12*log(b + c*x)*b*c**2*d*x**2 - 2*log(b + c*x)*b*c**2*e*x** 3 + 6*log(b + c*x)*c**3*d*x**3 + 2*log(x)*b**3*e*x - 6*log(x)*b**2*c*d*x + 4*log(x)*b**2*c*e*x**2 - 12*log(x)*b*c**2*d*x**2 + 2*log(x)*b*c**2*e*x**3 - 6*log(x)*c**3*d*x**3 - 2*b**3*d + 2*b**3*e*x - 6*b**2*c*d*x - b*c**2*e* x**3 + 3*c**3*d*x**3)/(2*b**4*x*(b**2 + 2*b*c*x + c**2*x**2))