Integrand size = 20, antiderivative size = 85 \[ \int \frac {d+e x}{x \left (b x+c x^2\right )^2} \, dx=-\frac {d}{2 b^2 x^2}+\frac {2 c d-b e}{b^3 x}+\frac {c (c d-b e)}{b^3 (b+c x)}+\frac {c (3 c d-2 b e) \log (x)}{b^4}-\frac {c (3 c d-2 b e) \log (b+c x)}{b^4} \] Output:
-1/2*d/b^2/x^2+(-b*e+2*c*d)/b^3/x+c*(-b*e+c*d)/b^3/(c*x+b)+c*(-2*b*e+3*c*d )*ln(x)/b^4-c*(-2*b*e+3*c*d)*ln(c*x+b)/b^4
Time = 0.05 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.00 \[ \int \frac {d+e x}{x \left (b x+c x^2\right )^2} \, dx=\frac {-\frac {b \left (-6 c^2 d x^2+b^2 (d+2 e x)+b c x (-3 d+4 e x)\right )}{x^2 (b+c x)}+2 c (3 c d-2 b e) \log (x)+2 c (-3 c d+2 b e) \log (b+c x)}{2 b^4} \] Input:
Integrate[(d + e*x)/(x*(b*x + c*x^2)^2),x]
Output:
(-((b*(-6*c^2*d*x^2 + b^2*(d + 2*e*x) + b*c*x*(-3*d + 4*e*x)))/(x^2*(b + c *x))) + 2*c*(3*c*d - 2*b*e)*Log[x] + 2*c*(-3*c*d + 2*b*e)*Log[b + c*x])/(2 *b^4)
Time = 0.41 (sec) , antiderivative size = 85, 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.
\(\displaystyle \int \frac {d+e x}{x \left (b x+c x^2\right )^2} \, dx\) |
\(\Big \downarrow \) 9 |
\(\displaystyle \int \frac {d+e x}{x^3 (b+c x)^2}dx\) |
\(\Big \downarrow \) 86 |
\(\displaystyle \int \left (\frac {c^2 (2 b e-3 c d)}{b^4 (b+c x)}-\frac {c (2 b e-3 c d)}{b^4 x}+\frac {c^2 (b e-c d)}{b^3 (b+c x)^2}+\frac {b e-2 c d}{b^3 x^2}+\frac {d}{b^2 x^3}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {c \log (x) (3 c d-2 b e)}{b^4}-\frac {c (3 c d-2 b e) \log (b+c x)}{b^4}+\frac {2 c d-b e}{b^3 x}+\frac {c (c d-b e)}{b^3 (b+c x)}-\frac {d}{2 b^2 x^2}\) |
Input:
Int[(d + e*x)/(x*(b*x + c*x^2)^2),x]
Output:
-1/2*d/(b^2*x^2) + (2*c*d - b*e)/(b^3*x) + (c*(c*d - b*e))/(b^3*(b + c*x)) + (c*(3*c*d - 2*b*e)*Log[x])/b^4 - (c*(3*c*d - 2*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 = 0.71 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.00
method | result | size |
default | \(\frac {c \left (2 b e -3 c d \right ) \ln \left (c x +b \right )}{b^{4}}-\frac {\left (b e -c d \right ) c}{b^{3} \left (c x +b \right )}-\frac {d}{2 b^{2} x^{2}}-\frac {b e -2 c d}{b^{3} x}-\frac {c \left (2 b e -3 c d \right ) \ln \left (x \right )}{b^{4}}\) | \(85\) |
norman | \(\frac {\frac {c \left (2 b c e -3 c^{2} d \right ) x^{3}}{b^{4}}-\frac {d}{2 b}-\frac {\left (2 b e -3 c d \right ) x}{2 b^{2}}}{\left (c x +b \right ) x^{2}}+\frac {c \left (2 b e -3 c d \right ) \ln \left (c x +b \right )}{b^{4}}-\frac {c \left (2 b e -3 c d \right ) \ln \left (x \right )}{b^{4}}\) | \(92\) |
risch | \(\frac {-\frac {c \left (2 b e -3 c d \right ) x^{2}}{b^{3}}-\frac {\left (2 b e -3 c d \right ) x}{2 b^{2}}-\frac {d}{2 b}}{\left (c x +b \right ) x^{2}}-\frac {2 c \ln \left (x \right ) e}{b^{3}}+\frac {3 c^{2} \ln \left (x \right ) d}{b^{4}}+\frac {2 c \ln \left (-c x -b \right ) e}{b^{3}}-\frac {3 c^{2} \ln \left (-c x -b \right ) d}{b^{4}}\) | \(107\) |
parallelrisch | \(-\frac {4 \ln \left (x \right ) x^{3} b \,c^{2} e -6 \ln \left (x \right ) x^{3} c^{3} d -4 \ln \left (c x +b \right ) x^{3} b \,c^{2} e +6 \ln \left (c x +b \right ) x^{3} c^{3} d +4 \ln \left (x \right ) x^{2} b^{2} c e -6 \ln \left (x \right ) x^{2} b \,c^{2} d -4 \ln \left (c x +b \right ) x^{2} b^{2} c e +6 \ln \left (c x +b \right ) x^{2} b \,c^{2} d -4 b \,c^{2} x^{3} e +6 c^{3} d \,x^{3}+2 b^{3} e x -3 b^{2} c x d +b^{3} d}{2 b^{4} x^{2} \left (c x +b \right )}\) | \(166\) |
Input:
int((e*x+d)/x/(c*x^2+b*x)^2,x,method=_RETURNVERBOSE)
Output:
c*(2*b*e-3*c*d)/b^4*ln(c*x+b)-(b*e-c*d)/b^3*c/(c*x+b)-1/2*d/b^2/x^2-(b*e-2 *c*d)/b^3/x-c*(2*b*e-3*c*d)/b^4*ln(x)
Time = 0.08 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.78 \[ \int \frac {d+e x}{x \left (b x+c x^2\right )^2} \, dx=-\frac {b^{3} d - 2 \, {\left (3 \, b c^{2} d - 2 \, b^{2} c e\right )} x^{2} - {\left (3 \, b^{2} c d - 2 \, b^{3} e\right )} x + 2 \, {\left ({\left (3 \, c^{3} d - 2 \, b c^{2} e\right )} x^{3} + {\left (3 \, b c^{2} d - 2 \, b^{2} c e\right )} x^{2}\right )} \log \left (c x + b\right ) - 2 \, {\left ({\left (3 \, c^{3} d - 2 \, b c^{2} e\right )} x^{3} + {\left (3 \, b c^{2} d - 2 \, b^{2} c e\right )} x^{2}\right )} \log \left (x\right )}{2 \, {\left (b^{4} c x^{3} + b^{5} x^{2}\right )}} \] Input:
integrate((e*x+d)/x/(c*x^2+b*x)^2,x, algorithm="fricas")
Output:
-1/2*(b^3*d - 2*(3*b*c^2*d - 2*b^2*c*e)*x^2 - (3*b^2*c*d - 2*b^3*e)*x + 2* ((3*c^3*d - 2*b*c^2*e)*x^3 + (3*b*c^2*d - 2*b^2*c*e)*x^2)*log(c*x + b) - 2 *((3*c^3*d - 2*b*c^2*e)*x^3 + (3*b*c^2*d - 2*b^2*c*e)*x^2)*log(x))/(b^4*c* x^3 + b^5*x^2)
Leaf count of result is larger than twice the leaf count of optimal. 184 vs. \(2 (80) = 160\).
Time = 0.29 (sec) , antiderivative size = 184, normalized size of antiderivative = 2.16 \[ \int \frac {d+e x}{x \left (b x+c x^2\right )^2} \, dx=\frac {- b^{2} d + x^{2} \left (- 4 b c e + 6 c^{2} d\right ) + x \left (- 2 b^{2} e + 3 b c d\right )}{2 b^{4} x^{2} + 2 b^{3} c x^{3}} - \frac {c \left (2 b e - 3 c d\right ) \log {\left (x + \frac {2 b^{2} c e - 3 b c^{2} d - b c \left (2 b e - 3 c d\right )}{4 b c^{2} e - 6 c^{3} d} \right )}}{b^{4}} + \frac {c \left (2 b e - 3 c d\right ) \log {\left (x + \frac {2 b^{2} c e - 3 b c^{2} d + b c \left (2 b e - 3 c d\right )}{4 b c^{2} e - 6 c^{3} d} \right )}}{b^{4}} \] Input:
integrate((e*x+d)/x/(c*x**2+b*x)**2,x)
Output:
(-b**2*d + x**2*(-4*b*c*e + 6*c**2*d) + x*(-2*b**2*e + 3*b*c*d))/(2*b**4*x **2 + 2*b**3*c*x**3) - c*(2*b*e - 3*c*d)*log(x + (2*b**2*c*e - 3*b*c**2*d - b*c*(2*b*e - 3*c*d))/(4*b*c**2*e - 6*c**3*d))/b**4 + c*(2*b*e - 3*c*d)*l og(x + (2*b**2*c*e - 3*b*c**2*d + b*c*(2*b*e - 3*c*d))/(4*b*c**2*e - 6*c** 3*d))/b**4
Time = 0.03 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.18 \[ \int \frac {d+e x}{x \left (b x+c x^2\right )^2} \, dx=-\frac {b^{2} d - 2 \, {\left (3 \, c^{2} d - 2 \, b c e\right )} x^{2} - {\left (3 \, b c d - 2 \, b^{2} e\right )} x}{2 \, {\left (b^{3} c x^{3} + b^{4} x^{2}\right )}} - \frac {{\left (3 \, c^{2} d - 2 \, b c e\right )} \log \left (c x + b\right )}{b^{4}} + \frac {{\left (3 \, c^{2} d - 2 \, b c e\right )} \log \left (x\right )}{b^{4}} \] Input:
integrate((e*x+d)/x/(c*x^2+b*x)^2,x, algorithm="maxima")
Output:
-1/2*(b^2*d - 2*(3*c^2*d - 2*b*c*e)*x^2 - (3*b*c*d - 2*b^2*e)*x)/(b^3*c*x^ 3 + b^4*x^2) - (3*c^2*d - 2*b*c*e)*log(c*x + b)/b^4 + (3*c^2*d - 2*b*c*e)* log(x)/b^4
Time = 0.12 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.26 \[ \int \frac {d+e x}{x \left (b x+c x^2\right )^2} \, dx=\frac {{\left (3 \, c^{2} d - 2 \, b c e\right )} \log \left ({\left | x \right |}\right )}{b^{4}} - \frac {{\left (3 \, c^{3} d - 2 \, b c^{2} e\right )} \log \left ({\left | c x + b \right |}\right )}{b^{4} c} - \frac {b^{3} d - 2 \, {\left (3 \, b c^{2} d - 2 \, b^{2} c e\right )} x^{2} - {\left (3 \, b^{2} c d - 2 \, b^{3} e\right )} x}{2 \, {\left (c x + b\right )} b^{4} x^{2}} \] Input:
integrate((e*x+d)/x/(c*x^2+b*x)^2,x, algorithm="giac")
Output:
(3*c^2*d - 2*b*c*e)*log(abs(x))/b^4 - (3*c^3*d - 2*b*c^2*e)*log(abs(c*x + b))/(b^4*c) - 1/2*(b^3*d - 2*(3*b*c^2*d - 2*b^2*c*e)*x^2 - (3*b^2*c*d - 2* b^3*e)*x)/((c*x + b)*b^4*x^2)
Time = 5.17 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.24 \[ \int \frac {d+e x}{x \left (b x+c x^2\right )^2} \, dx=-\frac {\frac {d}{2\,b}+\frac {x\,\left (2\,b\,e-3\,c\,d\right )}{2\,b^2}+\frac {c\,x^2\,\left (2\,b\,e-3\,c\,d\right )}{b^3}}{c\,x^3+b\,x^2}-\frac {2\,c\,\mathrm {atanh}\left (\frac {c\,\left (2\,b\,e-3\,c\,d\right )\,\left (b+2\,c\,x\right )}{b\,\left (3\,c^2\,d-2\,b\,c\,e\right )}\right )\,\left (2\,b\,e-3\,c\,d\right )}{b^4} \] Input:
int((d + e*x)/(x*(b*x + c*x^2)^2),x)
Output:
- (d/(2*b) + (x*(2*b*e - 3*c*d))/(2*b^2) + (c*x^2*(2*b*e - 3*c*d))/b^3)/(b *x^2 + c*x^3) - (2*c*atanh((c*(2*b*e - 3*c*d)*(b + 2*c*x))/(b*(3*c^2*d - 2 *b*c*e)))*(2*b*e - 3*c*d))/b^4
Time = 0.24 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.95 \[ \int \frac {d+e x}{x \left (b x+c x^2\right )^2} \, dx=\frac {4 \,\mathrm {log}\left (c x +b \right ) b^{2} c e \,x^{2}-6 \,\mathrm {log}\left (c x +b \right ) b \,c^{2} d \,x^{2}+4 \,\mathrm {log}\left (c x +b \right ) b \,c^{2} e \,x^{3}-6 \,\mathrm {log}\left (c x +b \right ) c^{3} d \,x^{3}-4 \,\mathrm {log}\left (x \right ) b^{2} c e \,x^{2}+6 \,\mathrm {log}\left (x \right ) b \,c^{2} d \,x^{2}-4 \,\mathrm {log}\left (x \right ) b \,c^{2} e \,x^{3}+6 \,\mathrm {log}\left (x \right ) c^{3} d \,x^{3}-b^{3} d -2 b^{3} e x +3 b^{2} c d x +4 b \,c^{2} e \,x^{3}-6 c^{3} d \,x^{3}}{2 b^{4} x^{2} \left (c x +b \right )} \] Input:
int((e*x+d)/x/(c*x^2+b*x)^2,x)
Output:
(4*log(b + c*x)*b**2*c*e*x**2 - 6*log(b + c*x)*b*c**2*d*x**2 + 4*log(b + c *x)*b*c**2*e*x**3 - 6*log(b + c*x)*c**3*d*x**3 - 4*log(x)*b**2*c*e*x**2 + 6*log(x)*b*c**2*d*x**2 - 4*log(x)*b*c**2*e*x**3 + 6*log(x)*c**3*d*x**3 - b **3*d - 2*b**3*e*x + 3*b**2*c*d*x + 4*b*c**2*e*x**3 - 6*c**3*d*x**3)/(2*b* *4*x**2*(b + c*x))