Integrand size = 21, antiderivative size = 105 \[ \int \frac {a+b x+c x^2}{x^4 (d+e x)} \, dx=-\frac {a}{3 d x^3}-\frac {b d-a e}{2 d^2 x^2}-\frac {c d^2-b d e+a e^2}{d^3 x}-\frac {e \left (c d^2-b d e+a e^2\right ) \log (x)}{d^4}+\frac {e \left (c d^2-b d e+a e^2\right ) \log (d+e x)}{d^4} \] Output:
-1/3*a/d/x^3-1/2*(-a*e+b*d)/d^2/x^2-(a*e^2-b*d*e+c*d^2)/d^3/x-e*(a*e^2-b*d *e+c*d^2)*ln(x)/d^4+e*(a*e^2-b*d*e+c*d^2)*ln(e*x+d)/d^4
Time = 0.08 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.94 \[ \int \frac {a+b x+c x^2}{x^4 (d+e x)} \, dx=-\frac {\frac {d \left (3 d x (b d+2 c d x-2 b e x)+a \left (2 d^2-3 d e x+6 e^2 x^2\right )\right )}{x^3}+6 e \left (c d^2+e (-b d+a e)\right ) \log (x)-6 e \left (c d^2+e (-b d+a e)\right ) \log (d+e x)}{6 d^4} \] Input:
Integrate[(a + b*x + c*x^2)/(x^4*(d + e*x)),x]
Output:
-1/6*((d*(3*d*x*(b*d + 2*c*d*x - 2*b*e*x) + a*(2*d^2 - 3*d*e*x + 6*e^2*x^2 )))/x^3 + 6*e*(c*d^2 + e*(-(b*d) + a*e))*Log[x] - 6*e*(c*d^2 + e*(-(b*d) + a*e))*Log[d + e*x])/d^4
Time = 0.27 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {1195, 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.
\(\displaystyle \int \frac {a+b x+c x^2}{x^4 (d+e x)} \, dx\) |
\(\Big \downarrow \) 1195 |
\(\displaystyle \int \left (\frac {e^2 \left (a e^2-b d e+c d^2\right )}{d^4 (d+e x)}-\frac {e \left (a e^2-b d e+c d^2\right )}{d^4 x}+\frac {a e^2-b d e+c d^2}{d^3 x^2}+\frac {b d-a e}{d^2 x^3}+\frac {a}{d x^4}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {e \log (x) \left (a e^2-b d e+c d^2\right )}{d^4}+\frac {e \log (d+e x) \left (a e^2-b d e+c d^2\right )}{d^4}-\frac {a e^2-b d e+c d^2}{d^3 x}-\frac {b d-a e}{2 d^2 x^2}-\frac {a}{3 d x^3}\) |
Input:
Int[(a + b*x + c*x^2)/(x^4*(d + e*x)),x]
Output:
-1/3*a/(d*x^3) - (b*d - a*e)/(2*d^2*x^2) - (c*d^2 - b*d*e + a*e^2)/(d^3*x) - (e*(c*d^2 - b*d*e + a*e^2)*Log[x])/d^4 + (e*(c*d^2 - b*d*e + a*e^2)*Log [d + e*x])/d^4
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x _) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x ] && IGtQ[p, 0]
Time = 0.76 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.97
method | result | size |
default | \(-\frac {a}{3 d \,x^{3}}-\frac {-a e +b d}{2 d^{2} x^{2}}-\frac {a \,e^{2}-b d e +c \,d^{2}}{d^{3} x}-\frac {e \left (a \,e^{2}-b d e +c \,d^{2}\right ) \ln \left (x \right )}{d^{4}}+\frac {e \left (a \,e^{2}-b d e +c \,d^{2}\right ) \ln \left (e x +d \right )}{d^{4}}\) | \(102\) |
norman | \(\frac {-\frac {a}{3 d}+\frac {\left (a e -b d \right ) x}{2 d^{2}}-\frac {\left (a \,e^{2}-b d e +c \,d^{2}\right ) x^{2}}{d^{3}}}{x^{3}}+\frac {e \left (a \,e^{2}-b d e +c \,d^{2}\right ) \ln \left (e x +d \right )}{d^{4}}-\frac {e \left (a \,e^{2}-b d e +c \,d^{2}\right ) \ln \left (x \right )}{d^{4}}\) | \(102\) |
risch | \(\frac {-\frac {a}{3 d}+\frac {\left (a e -b d \right ) x}{2 d^{2}}-\frac {\left (a \,e^{2}-b d e +c \,d^{2}\right ) x^{2}}{d^{3}}}{x^{3}}+\frac {e^{3} \ln \left (-e x -d \right ) a}{d^{4}}-\frac {e^{2} \ln \left (-e x -d \right ) b}{d^{3}}+\frac {e \ln \left (-e x -d \right ) c}{d^{2}}-\frac {e^{3} \ln \left (x \right ) a}{d^{4}}+\frac {e^{2} \ln \left (x \right ) b}{d^{3}}-\frac {e \ln \left (x \right ) c}{d^{2}}\) | \(131\) |
parallelrisch | \(-\frac {6 \ln \left (x \right ) x^{3} a \,e^{3}-6 \ln \left (x \right ) x^{3} b d \,e^{2}+6 \ln \left (x \right ) x^{3} c \,d^{2} e -6 \ln \left (e x +d \right ) x^{3} a \,e^{3}+6 \ln \left (e x +d \right ) x^{3} b d \,e^{2}-6 \ln \left (e x +d \right ) x^{3} c \,d^{2} e +6 a d \,e^{2} x^{2}-6 b \,d^{2} e \,x^{2}+6 d^{3} c \,x^{2}-3 x a \,d^{2} e +3 b \,d^{3} x +2 a \,d^{3}}{6 d^{4} x^{3}}\) | \(142\) |
Input:
int((c*x^2+b*x+a)/x^4/(e*x+d),x,method=_RETURNVERBOSE)
Output:
-1/3*a/d/x^3-1/2*(-a*e+b*d)/d^2/x^2-(a*e^2-b*d*e+c*d^2)/d^3/x-e*(a*e^2-b*d *e+c*d^2)*ln(x)/d^4+e*(a*e^2-b*d*e+c*d^2)*ln(e*x+d)/d^4
Time = 0.08 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.06 \[ \int \frac {a+b x+c x^2}{x^4 (d+e x)} \, dx=\frac {6 \, {\left (c d^{2} e - b d e^{2} + a e^{3}\right )} x^{3} \log \left (e x + d\right ) - 6 \, {\left (c d^{2} e - b d e^{2} + a e^{3}\right )} x^{3} \log \left (x\right ) - 2 \, a d^{3} - 6 \, {\left (c d^{3} - b d^{2} e + a d e^{2}\right )} x^{2} - 3 \, {\left (b d^{3} - a d^{2} e\right )} x}{6 \, d^{4} x^{3}} \] Input:
integrate((c*x^2+b*x+a)/x^4/(e*x+d),x, algorithm="fricas")
Output:
1/6*(6*(c*d^2*e - b*d*e^2 + a*e^3)*x^3*log(e*x + d) - 6*(c*d^2*e - b*d*e^2 + a*e^3)*x^3*log(x) - 2*a*d^3 - 6*(c*d^3 - b*d^2*e + a*d*e^2)*x^2 - 3*(b* d^3 - a*d^2*e)*x)/(d^4*x^3)
Leaf count of result is larger than twice the leaf count of optimal. 233 vs. \(2 (95) = 190\).
Time = 0.41 (sec) , antiderivative size = 233, normalized size of antiderivative = 2.22 \[ \int \frac {a+b x+c x^2}{x^4 (d+e x)} \, dx=\frac {- 2 a d^{2} + x^{2} \left (- 6 a e^{2} + 6 b d e - 6 c d^{2}\right ) + x \left (3 a d e - 3 b d^{2}\right )}{6 d^{3} x^{3}} - \frac {e \left (a e^{2} - b d e + c d^{2}\right ) \log {\left (x + \frac {a d e^{3} - b d^{2} e^{2} + c d^{3} e - d e \left (a e^{2} - b d e + c d^{2}\right )}{2 a e^{4} - 2 b d e^{3} + 2 c d^{2} e^{2}} \right )}}{d^{4}} + \frac {e \left (a e^{2} - b d e + c d^{2}\right ) \log {\left (x + \frac {a d e^{3} - b d^{2} e^{2} + c d^{3} e + d e \left (a e^{2} - b d e + c d^{2}\right )}{2 a e^{4} - 2 b d e^{3} + 2 c d^{2} e^{2}} \right )}}{d^{4}} \] Input:
integrate((c*x**2+b*x+a)/x**4/(e*x+d),x)
Output:
(-2*a*d**2 + x**2*(-6*a*e**2 + 6*b*d*e - 6*c*d**2) + x*(3*a*d*e - 3*b*d**2 ))/(6*d**3*x**3) - e*(a*e**2 - b*d*e + c*d**2)*log(x + (a*d*e**3 - b*d**2* e**2 + c*d**3*e - d*e*(a*e**2 - b*d*e + c*d**2))/(2*a*e**4 - 2*b*d*e**3 + 2*c*d**2*e**2))/d**4 + e*(a*e**2 - b*d*e + c*d**2)*log(x + (a*d*e**3 - b*d **2*e**2 + c*d**3*e + d*e*(a*e**2 - b*d*e + c*d**2))/(2*a*e**4 - 2*b*d*e** 3 + 2*c*d**2*e**2))/d**4
Time = 0.03 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.01 \[ \int \frac {a+b x+c x^2}{x^4 (d+e x)} \, dx=\frac {{\left (c d^{2} e - b d e^{2} + a e^{3}\right )} \log \left (e x + d\right )}{d^{4}} - \frac {{\left (c d^{2} e - b d e^{2} + a e^{3}\right )} \log \left (x\right )}{d^{4}} - \frac {2 \, a d^{2} + 6 \, {\left (c d^{2} - b d e + a e^{2}\right )} x^{2} + 3 \, {\left (b d^{2} - a d e\right )} x}{6 \, d^{3} x^{3}} \] Input:
integrate((c*x^2+b*x+a)/x^4/(e*x+d),x, algorithm="maxima")
Output:
(c*d^2*e - b*d*e^2 + a*e^3)*log(e*x + d)/d^4 - (c*d^2*e - b*d*e^2 + a*e^3) *log(x)/d^4 - 1/6*(2*a*d^2 + 6*(c*d^2 - b*d*e + a*e^2)*x^2 + 3*(b*d^2 - a* d*e)*x)/(d^3*x^3)
Time = 0.23 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.12 \[ \int \frac {a+b x+c x^2}{x^4 (d+e x)} \, dx=-\frac {{\left (c d^{2} e - b d e^{2} + a e^{3}\right )} \log \left ({\left | x \right |}\right )}{d^{4}} + \frac {{\left (c d^{2} e^{2} - b d e^{3} + a e^{4}\right )} \log \left ({\left | e x + d \right |}\right )}{d^{4} e} - \frac {2 \, a d^{3} + 6 \, {\left (c d^{3} - b d^{2} e + a d e^{2}\right )} x^{2} + 3 \, {\left (b d^{3} - a d^{2} e\right )} x}{6 \, d^{4} x^{3}} \] Input:
integrate((c*x^2+b*x+a)/x^4/(e*x+d),x, algorithm="giac")
Output:
-(c*d^2*e - b*d*e^2 + a*e^3)*log(abs(x))/d^4 + (c*d^2*e^2 - b*d*e^3 + a*e^ 4)*log(abs(e*x + d))/(d^4*e) - 1/6*(2*a*d^3 + 6*(c*d^3 - b*d^2*e + a*d*e^2 )*x^2 + 3*(b*d^3 - a*d^2*e)*x)/(d^4*x^3)
Time = 10.69 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.15 \[ \int \frac {a+b x+c x^2}{x^4 (d+e x)} \, dx=\frac {2\,e\,\mathrm {atanh}\left (\frac {e\,\left (d+2\,e\,x\right )\,\left (c\,d^2-b\,d\,e+a\,e^2\right )}{d\,\left (c\,d^2\,e-b\,d\,e^2+a\,e^3\right )}\right )\,\left (c\,d^2-b\,d\,e+a\,e^2\right )}{d^4}-\frac {\frac {a}{3\,d}-\frac {x\,\left (a\,e-b\,d\right )}{2\,d^2}+\frac {x^2\,\left (c\,d^2-b\,d\,e+a\,e^2\right )}{d^3}}{x^3} \] Input:
int((a + b*x + c*x^2)/(x^4*(d + e*x)),x)
Output:
(2*e*atanh((e*(d + 2*e*x)*(a*e^2 + c*d^2 - b*d*e))/(d*(a*e^3 - b*d*e^2 + c *d^2*e)))*(a*e^2 + c*d^2 - b*d*e))/d^4 - (a/(3*d) - (x*(a*e - b*d))/(2*d^2 ) + (x^2*(a*e^2 + c*d^2 - b*d*e))/d^3)/x^3
Time = 0.21 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.34 \[ \int \frac {a+b x+c x^2}{x^4 (d+e x)} \, dx=\frac {6 \,\mathrm {log}\left (e x +d \right ) a \,e^{3} x^{3}-6 \,\mathrm {log}\left (e x +d \right ) b d \,e^{2} x^{3}+6 \,\mathrm {log}\left (e x +d \right ) c \,d^{2} e \,x^{3}-6 \,\mathrm {log}\left (x \right ) a \,e^{3} x^{3}+6 \,\mathrm {log}\left (x \right ) b d \,e^{2} x^{3}-6 \,\mathrm {log}\left (x \right ) c \,d^{2} e \,x^{3}-2 a \,d^{3}+3 a \,d^{2} e x -6 a d \,e^{2} x^{2}-3 b \,d^{3} x +6 b \,d^{2} e \,x^{2}-6 c \,d^{3} x^{2}}{6 d^{4} x^{3}} \] Input:
int((c*x^2+b*x+a)/x^4/(e*x+d),x)
Output:
(6*log(d + e*x)*a*e**3*x**3 - 6*log(d + e*x)*b*d*e**2*x**3 + 6*log(d + e*x )*c*d**2*e*x**3 - 6*log(x)*a*e**3*x**3 + 6*log(x)*b*d*e**2*x**3 - 6*log(x) *c*d**2*e*x**3 - 2*a*d**3 + 3*a*d**2*e*x - 6*a*d*e**2*x**2 - 3*b*d**3*x + 6*b*d**2*e*x**2 - 6*c*d**3*x**2)/(6*d**4*x**3)