Integrand size = 21, antiderivative size = 145 \[ \int \frac {d+e x}{x^3 \left (a+b x+c x^2\right )} \, dx=-\frac {d}{2 a x^2}+\frac {b d-a e}{a^2 x}+\frac {\left (b^3 d-3 a b c d-a b^2 e+2 a^2 c e\right ) \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{a^3 \sqrt {b^2-4 a c}}+\frac {\left (b^2 d-a c d-a b e\right ) \log (x)}{a^3}-\frac {\left (b^2 d-a c d-a b e\right ) \log \left (a+b x+c x^2\right )}{2 a^3} \] Output:
-1/2*d/a/x^2+(-a*e+b*d)/a^2/x+(2*a^2*c*e-a*b^2*e-3*a*b*c*d+b^3*d)*arctanh( (2*c*x+b)/(-4*a*c+b^2)^(1/2))/a^3/(-4*a*c+b^2)^(1/2)+(-a*b*e-a*c*d+b^2*d)* ln(x)/a^3-1/2*(-a*b*e-a*c*d+b^2*d)*ln(c*x^2+b*x+a)/a^3
Time = 0.13 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.97 \[ \int \frac {d+e x}{x^3 \left (a+b x+c x^2\right )} \, dx=\frac {-\frac {a^2 d}{x^2}+\frac {2 a (b d-a e)}{x}+\frac {2 \left (-b^3 d+3 a b c d+a b^2 e-2 a^2 c e\right ) \arctan \left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )}{\sqrt {-b^2+4 a c}}+2 \left (b^2 d-a c d-a b e\right ) \log (x)+\left (-b^2 d+a c d+a b e\right ) \log (a+x (b+c x))}{2 a^3} \] Input:
Integrate[(d + e*x)/(x^3*(a + b*x + c*x^2)),x]
Output:
(-((a^2*d)/x^2) + (2*a*(b*d - a*e))/x + (2*(-(b^3*d) + 3*a*b*c*d + a*b^2*e - 2*a^2*c*e)*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c] + 2*(b^2*d - a*c*d - a*b*e)*Log[x] + (-(b^2*d) + a*c*d + a*b*e)*Log[a + x*( b + c*x)])/(2*a^3)
Time = 0.40 (sec) , antiderivative size = 145, 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.
| \(\displaystyle \int \frac {d+e x}{x^3 \left (a+b x+c x^2\right )} \, dx\) |
| \(\Big \downarrow \) 1200 |
| \(\displaystyle \int \left (\frac {-a b e-a c d+b^2 d}{a^3 x}+\frac {a e-b d}{a^2 x^2}+\frac {-a^2 c e-c x \left (-a b e-a c d+b^2 d\right )+a b^2 e+2 a b c d+b^3 (-d)}{a^3 \left (a+b x+c x^2\right )}+\frac {d}{a x^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\left (-a b e-a c d+b^2 d\right ) \log \left (a+b x+c x^2\right )}{2 a^3}+\frac {\log (x) \left (-a b e-a c d+b^2 d\right )}{a^3}+\frac {b d-a e}{a^2 x}+\frac {\text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right ) \left (2 a^2 c e-a b^2 e-3 a b c d+b^3 d\right )}{a^3 \sqrt {b^2-4 a c}}-\frac {d}{2 a x^2}\) |
Input:
Int[(d + e*x)/(x^3*(a + b*x + c*x^2)),x]
Output:
-1/2*d/(a*x^2) + (b*d - a*e)/(a^2*x) + ((b^3*d - 3*a*b*c*d - a*b^2*e + 2*a ^2*c*e)*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(a^3*Sqrt[b^2 - 4*a*c]) + ((b^2*d - a*c*d - a*b*e)*Log[x])/a^3 - ((b^2*d - a*c*d - a*b*e)*Log[a + b* x + c*x^2])/(2*a^3)
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 = 1.33 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.20
| method | result | size |
| default | \(-\frac {d}{2 a \,x^{2}}-\frac {a e -b d}{a^{2} x}+\frac {\left (-a b e -a c d +b^{2} d \right ) \ln \left (x \right )}{a^{3}}+\frac {\frac {\left (a b c e +a \,c^{2} d -c d \,b^{2}\right ) \ln \left (c \,x^{2}+b x +a \right )}{2 c}+\frac {2 \left (-a^{2} c e +e a \,b^{2}+2 a b c d -b^{3} d -\frac {\left (a b c e +a \,c^{2} d -c d \,b^{2}\right ) b}{2 c}\right ) \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{a^{3}}\) | \(174\) |
| risch | \(\frac {-\frac {\left (a e -b d \right ) x}{a^{2}}-\frac {d}{2 a}}{x^{2}}-\frac {\ln \left (x \right ) b e}{a^{2}}-\frac {c d \ln \left (x \right )}{a^{2}}+\frac {\ln \left (x \right ) b^{2} d}{a^{3}}+\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (4 c \,a^{4}-a^{3} b^{2}\right ) \textit {\_Z}^{2}+\left (-4 a^{2} b c e -4 a^{2} c^{2} d +a \,b^{3} e +5 c d a \,b^{2}-b^{4} d \right ) \textit {\_Z} +c^{2} a \,e^{2}-c^{2} d e b +c^{3} d^{2}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (6 c \,a^{5}-2 a^{4} b^{2}\right ) \textit {\_R}^{2}+\left (-2 a^{3} b c e -3 a^{3} c^{2} d +2 a^{2} b^{2} c d \right ) \textit {\_R} +a^{2} c^{2} e^{2}-2 a b \,c^{2} d e +b^{2} c^{2} d^{2}\right ) x -a^{5} b \,\textit {\_R}^{2}+\left (a^{4} c e -a^{3} b^{2} e -2 a^{3} d c b +a^{2} b^{3} d \right ) \textit {\_R} +a^{2} b c \,e^{2}+a^{2} c^{2} d e -2 a \,b^{2} c d e -b \,c^{2} d^{2} a +b^{3} c \,d^{2}\right )\right )\) | \(315\) |
Input:
int((e*x+d)/x^3/(c*x^2+b*x+a),x,method=_RETURNVERBOSE)
Output:
-1/2*d/a/x^2-(a*e-b*d)/a^2/x+(-a*b*e-a*c*d+b^2*d)*ln(x)/a^3+1/a^3*(1/2*(a* b*c*e+a*c^2*d-b^2*c*d)/c*ln(c*x^2+b*x+a)+2*(-a^2*c*e+e*a*b^2+2*a*b*c*d-b^3 *d-1/2*(a*b*c*e+a*c^2*d-b^2*c*d)*b/c)/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/( 4*a*c-b^2)^(1/2)))
Time = 0.32 (sec) , antiderivative size = 517, normalized size of antiderivative = 3.57 \[ \int \frac {d+e x}{x^3 \left (a+b x+c x^2\right )} \, dx=\left [\frac {\sqrt {b^{2} - 4 \, a c} {\left ({\left (b^{3} - 3 \, a b c\right )} d - {\left (a b^{2} - 2 \, a^{2} c\right )} e\right )} x^{2} \log \left (\frac {2 \, c^{2} x^{2} + 2 \, b c x + b^{2} - 2 \, a c + \sqrt {b^{2} - 4 \, a c} {\left (2 \, c x + b\right )}}{c x^{2} + b x + a}\right ) - {\left ({\left (b^{4} - 5 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} d - {\left (a b^{3} - 4 \, a^{2} b c\right )} e\right )} x^{2} \log \left (c x^{2} + b x + a\right ) + 2 \, {\left ({\left (b^{4} - 5 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} d - {\left (a b^{3} - 4 \, a^{2} b c\right )} e\right )} x^{2} \log \left (x\right ) - {\left (a^{2} b^{2} - 4 \, a^{3} c\right )} d + 2 \, {\left ({\left (a b^{3} - 4 \, a^{2} b c\right )} d - {\left (a^{2} b^{2} - 4 \, a^{3} c\right )} e\right )} x}{2 \, {\left (a^{3} b^{2} - 4 \, a^{4} c\right )} x^{2}}, \frac {2 \, \sqrt {-b^{2} + 4 \, a c} {\left ({\left (b^{3} - 3 \, a b c\right )} d - {\left (a b^{2} - 2 \, a^{2} c\right )} e\right )} x^{2} \arctan \left (-\frac {\sqrt {-b^{2} + 4 \, a c} {\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) - {\left ({\left (b^{4} - 5 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} d - {\left (a b^{3} - 4 \, a^{2} b c\right )} e\right )} x^{2} \log \left (c x^{2} + b x + a\right ) + 2 \, {\left ({\left (b^{4} - 5 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} d - {\left (a b^{3} - 4 \, a^{2} b c\right )} e\right )} x^{2} \log \left (x\right ) - {\left (a^{2} b^{2} - 4 \, a^{3} c\right )} d + 2 \, {\left ({\left (a b^{3} - 4 \, a^{2} b c\right )} d - {\left (a^{2} b^{2} - 4 \, a^{3} c\right )} e\right )} x}{2 \, {\left (a^{3} b^{2} - 4 \, a^{4} c\right )} x^{2}}\right ] \] Input:
integrate((e*x+d)/x^3/(c*x^2+b*x+a),x, algorithm="fricas")
Output:
[1/2*(sqrt(b^2 - 4*a*c)*((b^3 - 3*a*b*c)*d - (a*b^2 - 2*a^2*c)*e)*x^2*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^4 - 5*a*b^2*c + 4*a^2*c^2)*d - (a*b^3 - 4*a^2*b*c)*e)*x ^2*log(c*x^2 + b*x + a) + 2*((b^4 - 5*a*b^2*c + 4*a^2*c^2)*d - (a*b^3 - 4* a^2*b*c)*e)*x^2*log(x) - (a^2*b^2 - 4*a^3*c)*d + 2*((a*b^3 - 4*a^2*b*c)*d - (a^2*b^2 - 4*a^3*c)*e)*x)/((a^3*b^2 - 4*a^4*c)*x^2), 1/2*(2*sqrt(-b^2 + 4*a*c)*((b^3 - 3*a*b*c)*d - (a*b^2 - 2*a^2*c)*e)*x^2*arctan(-sqrt(-b^2 + 4 *a*c)*(2*c*x + b)/(b^2 - 4*a*c)) - ((b^4 - 5*a*b^2*c + 4*a^2*c^2)*d - (a*b ^3 - 4*a^2*b*c)*e)*x^2*log(c*x^2 + b*x + a) + 2*((b^4 - 5*a*b^2*c + 4*a^2* c^2)*d - (a*b^3 - 4*a^2*b*c)*e)*x^2*log(x) - (a^2*b^2 - 4*a^3*c)*d + 2*((a *b^3 - 4*a^2*b*c)*d - (a^2*b^2 - 4*a^3*c)*e)*x)/((a^3*b^2 - 4*a^4*c)*x^2)]
Timed out. \[ \int \frac {d+e x}{x^3 \left (a+b x+c x^2\right )} \, dx=\text {Timed out} \] Input:
integrate((e*x+d)/x**3/(c*x**2+b*x+a),x)
Output:
Timed out
Exception generated. \[ \int \frac {d+e x}{x^3 \left (a+b x+c x^2\right )} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((e*x+d)/x^3/(c*x^2+b*x+a),x, algorithm="maxima")
Output:
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 = 0.21 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.01 \[ \int \frac {d+e x}{x^3 \left (a+b x+c x^2\right )} \, dx=-\frac {{\left (b^{2} d - a c d - a b e\right )} \log \left (c x^{2} + b x + a\right )}{2 \, a^{3}} + \frac {{\left (b^{2} d - a c d - a b e\right )} \log \left ({\left | x \right |}\right )}{a^{3}} - \frac {{\left (b^{3} d - 3 \, a b c d - a b^{2} e + 2 \, a^{2} c e\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{\sqrt {-b^{2} + 4 \, a c} a^{3}} - \frac {a^{2} d - 2 \, {\left (a b d - a^{2} e\right )} x}{2 \, a^{3} x^{2}} \] Input:
integrate((e*x+d)/x^3/(c*x^2+b*x+a),x, algorithm="giac")
Output:
-1/2*(b^2*d - a*c*d - a*b*e)*log(c*x^2 + b*x + a)/a^3 + (b^2*d - a*c*d - a *b*e)*log(abs(x))/a^3 - (b^3*d - 3*a*b*c*d - a*b^2*e + 2*a^2*c*e)*arctan(( 2*c*x + b)/sqrt(-b^2 + 4*a*c))/(sqrt(-b^2 + 4*a*c)*a^3) - 1/2*(a^2*d - 2*( a*b*d - a^2*e)*x)/(a^3*x^2)
Time = 11.67 (sec) , antiderivative size = 814, normalized size of antiderivative = 5.61 \[ \int \frac {d+e x}{x^3 \left (a+b x+c x^2\right )} \, dx=\frac {\ln \left (6\,a^3\,c^2\,d-2\,a^2\,b^3\,e+2\,a\,b^4\,d+2\,b^5\,d\,x+7\,a^3\,b\,c\,e-2\,a\,b^4\,e\,x+2\,a\,b^3\,d\,\sqrt {b^2-4\,a\,c}+a^3\,c\,e\,\sqrt {b^2-4\,a\,c}+2\,b^4\,d\,x\,\sqrt {b^2-4\,a\,c}-9\,a^2\,b^2\,c\,d-2\,a^3\,c^2\,e\,x-2\,a^2\,b^2\,e\,\sqrt {b^2-4\,a\,c}-2\,a\,b^3\,e\,x\,\sqrt {b^2-4\,a\,c}+9\,a^2\,b\,c^2\,d\,x+8\,a^2\,b^2\,c\,e\,x+3\,a^2\,c^2\,d\,x\,\sqrt {b^2-4\,a\,c}-10\,a\,b^3\,c\,d\,x-3\,a^2\,b\,c\,d\,\sqrt {b^2-4\,a\,c}-6\,a\,b^2\,c\,d\,x\,\sqrt {b^2-4\,a\,c}+4\,a^2\,b\,c\,e\,x\,\sqrt {b^2-4\,a\,c}\right )\,\left (a^2\,\left (2\,c^2\,d+2\,b\,c\,e+c\,e\,\sqrt {b^2-4\,a\,c}\right )+\frac {b^4\,d}{2}-a\,\left (\frac {b^3\,e}{2}+\frac {b^2\,e\,\sqrt {b^2-4\,a\,c}}{2}+\frac {5\,b^2\,c\,d}{2}+\frac {3\,b\,c\,d\,\sqrt {b^2-4\,a\,c}}{2}\right )+\frac {b^3\,d\,\sqrt {b^2-4\,a\,c}}{2}\right )}{4\,a^4\,c-a^3\,b^2}-\frac {\ln \left (x\right )\,\left (a\,\left (b\,e+c\,d\right )-b^2\,d\right )}{a^3}-\frac {\frac {d}{2\,a}+\frac {x\,\left (a\,e-b\,d\right )}{a^2}}{x^2}+\frac {\ln \left (2\,a^2\,b^3\,e-6\,a^3\,c^2\,d-2\,a\,b^4\,d-2\,b^5\,d\,x-7\,a^3\,b\,c\,e+2\,a\,b^4\,e\,x+2\,a\,b^3\,d\,\sqrt {b^2-4\,a\,c}+a^3\,c\,e\,\sqrt {b^2-4\,a\,c}+2\,b^4\,d\,x\,\sqrt {b^2-4\,a\,c}+9\,a^2\,b^2\,c\,d+2\,a^3\,c^2\,e\,x-2\,a^2\,b^2\,e\,\sqrt {b^2-4\,a\,c}-2\,a\,b^3\,e\,x\,\sqrt {b^2-4\,a\,c}-9\,a^2\,b\,c^2\,d\,x-8\,a^2\,b^2\,c\,e\,x+3\,a^2\,c^2\,d\,x\,\sqrt {b^2-4\,a\,c}+10\,a\,b^3\,c\,d\,x-3\,a^2\,b\,c\,d\,\sqrt {b^2-4\,a\,c}-6\,a\,b^2\,c\,d\,x\,\sqrt {b^2-4\,a\,c}+4\,a^2\,b\,c\,e\,x\,\sqrt {b^2-4\,a\,c}\right )\,\left (a^2\,\left (2\,c^2\,d+2\,b\,c\,e-c\,e\,\sqrt {b^2-4\,a\,c}\right )+\frac {b^4\,d}{2}-a\,\left (\frac {b^3\,e}{2}-\frac {b^2\,e\,\sqrt {b^2-4\,a\,c}}{2}+\frac {5\,b^2\,c\,d}{2}-\frac {3\,b\,c\,d\,\sqrt {b^2-4\,a\,c}}{2}\right )-\frac {b^3\,d\,\sqrt {b^2-4\,a\,c}}{2}\right )}{4\,a^4\,c-a^3\,b^2} \] Input:
int((d + e*x)/(x^3*(a + b*x + c*x^2)),x)
Output:
(log(6*a^3*c^2*d - 2*a^2*b^3*e + 2*a*b^4*d + 2*b^5*d*x + 7*a^3*b*c*e - 2*a *b^4*e*x + 2*a*b^3*d*(b^2 - 4*a*c)^(1/2) + a^3*c*e*(b^2 - 4*a*c)^(1/2) + 2 *b^4*d*x*(b^2 - 4*a*c)^(1/2) - 9*a^2*b^2*c*d - 2*a^3*c^2*e*x - 2*a^2*b^2*e *(b^2 - 4*a*c)^(1/2) - 2*a*b^3*e*x*(b^2 - 4*a*c)^(1/2) + 9*a^2*b*c^2*d*x + 8*a^2*b^2*c*e*x + 3*a^2*c^2*d*x*(b^2 - 4*a*c)^(1/2) - 10*a*b^3*c*d*x - 3* a^2*b*c*d*(b^2 - 4*a*c)^(1/2) - 6*a*b^2*c*d*x*(b^2 - 4*a*c)^(1/2) + 4*a^2* b*c*e*x*(b^2 - 4*a*c)^(1/2))*(a^2*(2*c^2*d + 2*b*c*e + c*e*(b^2 - 4*a*c)^( 1/2)) + (b^4*d)/2 - a*((b^3*e)/2 + (b^2*e*(b^2 - 4*a*c)^(1/2))/2 + (5*b^2* c*d)/2 + (3*b*c*d*(b^2 - 4*a*c)^(1/2))/2) + (b^3*d*(b^2 - 4*a*c)^(1/2))/2) )/(4*a^4*c - a^3*b^2) - (log(x)*(a*(b*e + c*d) - b^2*d))/a^3 - (d/(2*a) + (x*(a*e - b*d))/a^2)/x^2 + (log(2*a^2*b^3*e - 6*a^3*c^2*d - 2*a*b^4*d - 2* b^5*d*x - 7*a^3*b*c*e + 2*a*b^4*e*x + 2*a*b^3*d*(b^2 - 4*a*c)^(1/2) + a^3* c*e*(b^2 - 4*a*c)^(1/2) + 2*b^4*d*x*(b^2 - 4*a*c)^(1/2) + 9*a^2*b^2*c*d + 2*a^3*c^2*e*x - 2*a^2*b^2*e*(b^2 - 4*a*c)^(1/2) - 2*a*b^3*e*x*(b^2 - 4*a*c )^(1/2) - 9*a^2*b*c^2*d*x - 8*a^2*b^2*c*e*x + 3*a^2*c^2*d*x*(b^2 - 4*a*c)^ (1/2) + 10*a*b^3*c*d*x - 3*a^2*b*c*d*(b^2 - 4*a*c)^(1/2) - 6*a*b^2*c*d*x*( b^2 - 4*a*c)^(1/2) + 4*a^2*b*c*e*x*(b^2 - 4*a*c)^(1/2))*(a^2*(2*c^2*d + 2* b*c*e - c*e*(b^2 - 4*a*c)^(1/2)) + (b^4*d)/2 - a*((b^3*e)/2 - (b^2*e*(b^2 - 4*a*c)^(1/2))/2 + (5*b^2*c*d)/2 - (3*b*c*d*(b^2 - 4*a*c)^(1/2))/2) - (b^ 3*d*(b^2 - 4*a*c)^(1/2))/2))/(4*a^4*c - a^3*b^2)
Time = 0.27 (sec) , antiderivative size = 407, normalized size of antiderivative = 2.81 \[ \int \frac {d+e x}{x^3 \left (a+b x+c x^2\right )} \, dx=\frac {-4 \sqrt {4 a c -b^{2}}\, \mathit {atan} \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right ) a^{2} c e \,x^{2}+2 \sqrt {4 a c -b^{2}}\, \mathit {atan} \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right ) a \,b^{2} e \,x^{2}+6 \sqrt {4 a c -b^{2}}\, \mathit {atan} \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right ) a b c d \,x^{2}-2 \sqrt {4 a c -b^{2}}\, \mathit {atan} \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right ) b^{3} d \,x^{2}+4 \,\mathrm {log}\left (c \,x^{2}+b x +a \right ) a^{2} b c e \,x^{2}+4 \,\mathrm {log}\left (c \,x^{2}+b x +a \right ) a^{2} c^{2} d \,x^{2}-\mathrm {log}\left (c \,x^{2}+b x +a \right ) a \,b^{3} e \,x^{2}-5 \,\mathrm {log}\left (c \,x^{2}+b x +a \right ) a \,b^{2} c d \,x^{2}+\mathrm {log}\left (c \,x^{2}+b x +a \right ) b^{4} d \,x^{2}-8 \,\mathrm {log}\left (x \right ) a^{2} b c e \,x^{2}-8 \,\mathrm {log}\left (x \right ) a^{2} c^{2} d \,x^{2}+2 \,\mathrm {log}\left (x \right ) a \,b^{3} e \,x^{2}+10 \,\mathrm {log}\left (x \right ) a \,b^{2} c d \,x^{2}-2 \,\mathrm {log}\left (x \right ) b^{4} d \,x^{2}-4 a^{3} c d -8 a^{3} c e x +a^{2} b^{2} d +2 a^{2} b^{2} e x +8 a^{2} b c d x -2 a \,b^{3} d x}{2 a^{3} x^{2} \left (4 a c -b^{2}\right )} \] Input:
int((e*x+d)/x^3/(c*x^2+b*x+a),x)
Output:
( - 4*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**2*c*e*x** 2 + 2*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a*b**2*e*x** 2 + 6*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a*b*c*d*x**2 - 2*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*b**3*d*x**2 + 4*log(a + b*x + c*x**2)*a**2*b*c*e*x**2 + 4*log(a + b*x + c*x**2)*a**2*c* *2*d*x**2 - log(a + b*x + c*x**2)*a*b**3*e*x**2 - 5*log(a + b*x + c*x**2)* a*b**2*c*d*x**2 + log(a + b*x + c*x**2)*b**4*d*x**2 - 8*log(x)*a**2*b*c*e* x**2 - 8*log(x)*a**2*c**2*d*x**2 + 2*log(x)*a*b**3*e*x**2 + 10*log(x)*a*b* *2*c*d*x**2 - 2*log(x)*b**4*d*x**2 - 4*a**3*c*d - 8*a**3*c*e*x + a**2*b**2 *d + 2*a**2*b**2*e*x + 8*a**2*b*c*d*x - 2*a*b**3*d*x)/(2*a**3*x**2*(4*a*c - b**2))