Integrand size = 21, antiderivative size = 104 \[ \int \frac {d+e x}{x^2 \left (a+b x+c x^2\right )} \, dx=-\frac {d}{a x}-\frac {\left (b^2 d-2 a c d-a b e\right ) \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{a^2 \sqrt {b^2-4 a c}}-\frac {(b d-a e) \log (x)}{a^2}+\frac {(b d-a e) \log \left (a+b x+c x^2\right )}{2 a^2} \]
-d/a/x-(-a*e+b*d)*ln(x)/a^2+1/2*(-a*e+b*d)*ln(c*x^2+b*x+a)/a^2-(-a*b*e-2*a *c*d+b^2*d)*arctanh((2*c*x+b)/(-4*a*c+b^2)^(1/2))/a^2/(-4*a*c+b^2)^(1/2)
Time = 0.06 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.96 \[ \int \frac {d+e x}{x^2 \left (a+b x+c x^2\right )} \, dx=\frac {-\frac {2 a d}{x}+\frac {2 \left (b^2 d-2 a c d-a b e\right ) \arctan \left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )}{\sqrt {-b^2+4 a c}}+2 (-b d+a e) \log (x)+(b d-a e) \log (a+x (b+c x))}{2 a^2} \]
((-2*a*d)/x + (2*(b^2*d - 2*a*c*d - a*b*e)*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c] + 2*(-(b*d) + a*e)*Log[x] + (b*d - a*e)*Log[a + x*(b + c*x)])/(2*a^2)
Time = 0.32 (sec) , antiderivative size = 104, 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 = {1200, 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.
\(\displaystyle \int \frac {d+e x}{x^2 \left (a+b x+c x^2\right )} \, dx\) |
\(\Big \downarrow \) 1200 |
\(\displaystyle \int \left (\frac {c x (b d-a e)-a b e-a c d+b^2 d}{a^2 \left (a+b x+c x^2\right )}+\frac {a e-b d}{a^2 x}+\frac {d}{a x^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {\text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right ) \left (-a b e-2 a c d+b^2 d\right )}{a^2 \sqrt {b^2-4 a c}}+\frac {(b d-a e) \log \left (a+b x+c x^2\right )}{2 a^2}-\frac {\log (x) (b d-a e)}{a^2}-\frac {d}{a x}\) |
-(d/(a*x)) - ((b^2*d - 2*a*c*d - a*b*e)*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a *c]])/(a^2*Sqrt[b^2 - 4*a*c]) - ((b*d - a*e)*Log[x])/a^2 + ((b*d - a*e)*Lo g[a + b*x + c*x^2])/(2*a^2)
3.9.87.3.1 Defintions of rubi rules used
Int[(((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.))/((a_.) + (b_.)* (x_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*((f + g* x)^n/(a + b*x + c*x^2)), x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && In tegersQ[n]
Time = 0.46 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.17
method | result | size |
default | \(-\frac {d}{a x}+\frac {\left (a e -b d \right ) \ln \left (x \right )}{a^{2}}+\frac {\frac {\left (-a c e +b c d \right ) \ln \left (c \,x^{2}+b x +a \right )}{2 c}+\frac {2 \left (-a b e -a c d +b^{2} d -\frac {\left (-a c e +b c d \right ) b}{2 c}\right ) \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{a^{2}}\) | \(122\) |
risch | \(-\frac {d}{a x}+\frac {e \ln \left (x \right )}{a}-\frac {\ln \left (x \right ) b d}{a^{2}}+\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (4 a^{3} c -b^{2} a^{2}\right ) \textit {\_Z}^{2}+\left (4 a^{2} c e -a \,b^{2} e -4 a b c d +d \,b^{3}\right ) \textit {\_Z} +a c \,e^{2}-b c d e +c^{2} d^{2}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (6 a^{3} c -2 b^{2} a^{2}\right ) \textit {\_R}^{2}+\left (3 a^{2} c e -2 a b c d \right ) \textit {\_R} +c^{2} d^{2}\right ) x -a^{3} b \,\textit {\_R}^{2}+\left (b e \,a^{2}+a^{2} c d -b^{2} d a \right ) \textit {\_R} -a c d e +b c \,d^{2}\right )\right )\) | \(190\) |
-d/a/x+(a*e-b*d)/a^2*ln(x)+1/a^2*(1/2*(-a*c*e+b*c*d)/c*ln(c*x^2+b*x+a)+2*( -a*b*e-a*c*d+b^2*d-1/2*(-a*c*e+b*c*d)*b/c)/(4*a*c-b^2)^(1/2)*arctan((2*c*x +b)/(4*a*c-b^2)^(1/2)))
Time = 0.35 (sec) , antiderivative size = 361, normalized size of antiderivative = 3.47 \[ \int \frac {d+e x}{x^2 \left (a+b x+c x^2\right )} \, dx=\left [\frac {{\left (a b e - {\left (b^{2} - 2 \, a c\right )} d\right )} \sqrt {b^{2} - 4 \, a c} x \log \left (\frac {2 \, c^{2} x^{2} + 2 \, b c x + b^{2} - 2 \, a c + \sqrt {b^{2} - 4 \, a c} {\left (2 \, c x + b\right )}}{c x^{2} + b x + a}\right ) + {\left ({\left (b^{3} - 4 \, a b c\right )} d - {\left (a b^{2} - 4 \, a^{2} c\right )} e\right )} x \log \left (c x^{2} + b x + a\right ) - 2 \, {\left ({\left (b^{3} - 4 \, a b c\right )} d - {\left (a b^{2} - 4 \, a^{2} c\right )} e\right )} x \log \left (x\right ) - 2 \, {\left (a b^{2} - 4 \, a^{2} c\right )} d}{2 \, {\left (a^{2} b^{2} - 4 \, a^{3} c\right )} x}, \frac {2 \, {\left (a b e - {\left (b^{2} - 2 \, a c\right )} d\right )} \sqrt {-b^{2} + 4 \, a c} x \arctan \left (-\frac {\sqrt {-b^{2} + 4 \, a c} {\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) + {\left ({\left (b^{3} - 4 \, a b c\right )} d - {\left (a b^{2} - 4 \, a^{2} c\right )} e\right )} x \log \left (c x^{2} + b x + a\right ) - 2 \, {\left ({\left (b^{3} - 4 \, a b c\right )} d - {\left (a b^{2} - 4 \, a^{2} c\right )} e\right )} x \log \left (x\right ) - 2 \, {\left (a b^{2} - 4 \, a^{2} c\right )} d}{2 \, {\left (a^{2} b^{2} - 4 \, a^{3} c\right )} x}\right ] \]
[1/2*((a*b*e - (b^2 - 2*a*c)*d)*sqrt(b^2 - 4*a*c)*x*log((2*c^2*x^2 + 2*b*c *x + b^2 - 2*a*c + sqrt(b^2 - 4*a*c)*(2*c*x + b))/(c*x^2 + b*x + a)) + ((b ^3 - 4*a*b*c)*d - (a*b^2 - 4*a^2*c)*e)*x*log(c*x^2 + b*x + a) - 2*((b^3 - 4*a*b*c)*d - (a*b^2 - 4*a^2*c)*e)*x*log(x) - 2*(a*b^2 - 4*a^2*c)*d)/((a^2* b^2 - 4*a^3*c)*x), 1/2*(2*(a*b*e - (b^2 - 2*a*c)*d)*sqrt(-b^2 + 4*a*c)*x*a rctan(-sqrt(-b^2 + 4*a*c)*(2*c*x + b)/(b^2 - 4*a*c)) + ((b^3 - 4*a*b*c)*d - (a*b^2 - 4*a^2*c)*e)*x*log(c*x^2 + b*x + a) - 2*((b^3 - 4*a*b*c)*d - (a* b^2 - 4*a^2*c)*e)*x*log(x) - 2*(a*b^2 - 4*a^2*c)*d)/((a^2*b^2 - 4*a^3*c)*x )]
Timed out. \[ \int \frac {d+e x}{x^2 \left (a+b x+c x^2\right )} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {d+e x}{x^2 \left (a+b x+c x^2\right )} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` for more deta
Time = 0.26 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.98 \[ \int \frac {d+e x}{x^2 \left (a+b x+c x^2\right )} \, dx=\frac {{\left (b d - a e\right )} \log \left (c x^{2} + b x + a\right )}{2 \, a^{2}} - \frac {{\left (b d - a e\right )} \log \left ({\left | x \right |}\right )}{a^{2}} + \frac {{\left (b^{2} d - 2 \, a c d - a b e\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{\sqrt {-b^{2} + 4 \, a c} a^{2}} - \frac {d}{a x} \]
1/2*(b*d - a*e)*log(c*x^2 + b*x + a)/a^2 - (b*d - a*e)*log(abs(x))/a^2 + ( b^2*d - 2*a*c*d - a*b*e)*arctan((2*c*x + b)/sqrt(-b^2 + 4*a*c))/(sqrt(-b^2 + 4*a*c)*a^2) - d/(a*x)
Time = 11.63 (sec) , antiderivative size = 791, normalized size of antiderivative = 7.61 \[ \int \frac {d+e x}{x^2 \left (a+b x+c x^2\right )} \, dx=\frac {\ln \left (x\right )\,\left (a\,e-b\,d\right )}{a^2}-\frac {d}{a\,x}+\frac {\ln \left (\frac {b\,c^2\,d^2-a\,c^2\,d\,e}{a^2}+\frac {\left (\frac {e\,a^2\,b\,c+d\,a^2\,c^2-d\,a\,b^2\,c}{a^2}+\frac {\left (\frac {x\,\left (6\,a^3\,c^2-2\,a^2\,b^2\,c\right )}{a^2}-a\,b\,c\right )\,\left (d\,{\left (b^2-4\,a\,c\right )}^{3/2}+b^2\,d\,\sqrt {b^2-4\,a\,c}-2\,a\,e\,\left (4\,a\,c-b^2\right )+2\,b\,d\,\left (4\,a\,c-b^2\right )-2\,a\,b\,e\,\sqrt {b^2-4\,a\,c}\right )}{16\,a^3\,c-4\,a^2\,b^2}+\frac {x\,\left (3\,a^2\,c^2\,e-2\,a\,b\,c^2\,d\right )}{a^2}\right )\,\left (d\,{\left (b^2-4\,a\,c\right )}^{3/2}+b^2\,d\,\sqrt {b^2-4\,a\,c}-2\,a\,e\,\left (4\,a\,c-b^2\right )+2\,b\,d\,\left (4\,a\,c-b^2\right )-2\,a\,b\,e\,\sqrt {b^2-4\,a\,c}\right )}{16\,a^3\,c-4\,a^2\,b^2}+\frac {c^3\,d^2\,x}{a^2}\right )\,\left (d\,{\left (b^2-4\,a\,c\right )}^{3/2}+b^2\,d\,\sqrt {b^2-4\,a\,c}-2\,a\,e\,\left (4\,a\,c-b^2\right )+2\,b\,d\,\left (4\,a\,c-b^2\right )-2\,a\,b\,e\,\sqrt {b^2-4\,a\,c}\right )}{16\,a^3\,c-4\,a^2\,b^2}-\frac {\ln \left (\frac {b\,c^2\,d^2-a\,c^2\,d\,e}{a^2}-\frac {\left (\frac {e\,a^2\,b\,c+d\,a^2\,c^2-d\,a\,b^2\,c}{a^2}-\frac {\left (\frac {x\,\left (6\,a^3\,c^2-2\,a^2\,b^2\,c\right )}{a^2}-a\,b\,c\right )\,\left (d\,{\left (b^2-4\,a\,c\right )}^{3/2}+b^2\,d\,\sqrt {b^2-4\,a\,c}+2\,a\,e\,\left (4\,a\,c-b^2\right )-2\,b\,d\,\left (4\,a\,c-b^2\right )-2\,a\,b\,e\,\sqrt {b^2-4\,a\,c}\right )}{16\,a^3\,c-4\,a^2\,b^2}+\frac {x\,\left (3\,a^2\,c^2\,e-2\,a\,b\,c^2\,d\right )}{a^2}\right )\,\left (d\,{\left (b^2-4\,a\,c\right )}^{3/2}+b^2\,d\,\sqrt {b^2-4\,a\,c}+2\,a\,e\,\left (4\,a\,c-b^2\right )-2\,b\,d\,\left (4\,a\,c-b^2\right )-2\,a\,b\,e\,\sqrt {b^2-4\,a\,c}\right )}{16\,a^3\,c-4\,a^2\,b^2}+\frac {c^3\,d^2\,x}{a^2}\right )\,\left (d\,{\left (b^2-4\,a\,c\right )}^{3/2}+b^2\,d\,\sqrt {b^2-4\,a\,c}+2\,a\,e\,\left (4\,a\,c-b^2\right )-2\,b\,d\,\left (4\,a\,c-b^2\right )-2\,a\,b\,e\,\sqrt {b^2-4\,a\,c}\right )}{16\,a^3\,c-4\,a^2\,b^2} \]
(log(x)*(a*e - b*d))/a^2 - d/(a*x) + (log((b*c^2*d^2 - a*c^2*d*e)/a^2 + (( (a^2*c^2*d - a*b^2*c*d + a^2*b*c*e)/a^2 + (((x*(6*a^3*c^2 - 2*a^2*b^2*c))/ a^2 - a*b*c)*(d*(b^2 - 4*a*c)^(3/2) + b^2*d*(b^2 - 4*a*c)^(1/2) - 2*a*e*(4 *a*c - b^2) + 2*b*d*(4*a*c - b^2) - 2*a*b*e*(b^2 - 4*a*c)^(1/2)))/(16*a^3* c - 4*a^2*b^2) + (x*(3*a^2*c^2*e - 2*a*b*c^2*d))/a^2)*(d*(b^2 - 4*a*c)^(3/ 2) + b^2*d*(b^2 - 4*a*c)^(1/2) - 2*a*e*(4*a*c - b^2) + 2*b*d*(4*a*c - b^2) - 2*a*b*e*(b^2 - 4*a*c)^(1/2)))/(16*a^3*c - 4*a^2*b^2) + (c^3*d^2*x)/a^2) *(d*(b^2 - 4*a*c)^(3/2) + b^2*d*(b^2 - 4*a*c)^(1/2) - 2*a*e*(4*a*c - b^2) + 2*b*d*(4*a*c - b^2) - 2*a*b*e*(b^2 - 4*a*c)^(1/2)))/(16*a^3*c - 4*a^2*b^ 2) - (log((b*c^2*d^2 - a*c^2*d*e)/a^2 - (((a^2*c^2*d - a*b^2*c*d + a^2*b*c *e)/a^2 - (((x*(6*a^3*c^2 - 2*a^2*b^2*c))/a^2 - a*b*c)*(d*(b^2 - 4*a*c)^(3 /2) + b^2*d*(b^2 - 4*a*c)^(1/2) + 2*a*e*(4*a*c - b^2) - 2*b*d*(4*a*c - b^2 ) - 2*a*b*e*(b^2 - 4*a*c)^(1/2)))/(16*a^3*c - 4*a^2*b^2) + (x*(3*a^2*c^2*e - 2*a*b*c^2*d))/a^2)*(d*(b^2 - 4*a*c)^(3/2) + b^2*d*(b^2 - 4*a*c)^(1/2) + 2*a*e*(4*a*c - b^2) - 2*b*d*(4*a*c - b^2) - 2*a*b*e*(b^2 - 4*a*c)^(1/2))) /(16*a^3*c - 4*a^2*b^2) + (c^3*d^2*x)/a^2)*(d*(b^2 - 4*a*c)^(3/2) + b^2*d* (b^2 - 4*a*c)^(1/2) + 2*a*e*(4*a*c - b^2) - 2*b*d*(4*a*c - b^2) - 2*a*b*e* (b^2 - 4*a*c)^(1/2)))/(16*a^3*c - 4*a^2*b^2)