Integrand size = 14, antiderivative size = 70 \[ \int \frac {1}{c+\frac {a}{x^2}+\frac {b}{x}} \, dx=\frac {x}{c}-\frac {\left (b^2-2 a c\right ) \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c^2 \sqrt {b^2-4 a c}}-\frac {b \log \left (a+b x+c x^2\right )}{2 c^2} \] Output:
x/c-(-2*a*c+b^2)*arctanh((2*c*x+b)/(-4*a*c+b^2)^(1/2))/c^2/(-4*a*c+b^2)^(1 /2)-1/2*b*ln(c*x^2+b*x+a)/c^2
Time = 0.06 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.04 \[ \int \frac {1}{c+\frac {a}{x^2}+\frac {b}{x}} \, dx=\frac {x}{c}+\frac {\left (b^2-2 a c\right ) \arctan \left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )}{c^2 \sqrt {-b^2+4 a c}}-\frac {b \log \left (a+b x+c x^2\right )}{2 c^2} \] Input:
Integrate[(c + a/x^2 + b/x)^(-1),x]
Output:
x/c + ((b^2 - 2*a*c)*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(c^2*Sqrt[-b^ 2 + 4*a*c]) - (b*Log[a + b*x + c*x^2])/(2*c^2)
Time = 0.24 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {1679, 1143, 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.
\(\displaystyle \int \frac {1}{\frac {a}{x^2}+\frac {b}{x}+c} \, dx\) |
\(\Big \downarrow \) 1679 |
\(\displaystyle \int \frac {x^2}{a+b x+c x^2}dx\) |
\(\Big \downarrow \) 1143 |
\(\displaystyle \int \left (\frac {1}{c}-\frac {a+b x}{c \left (a+b x+c x^2\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {\left (b^2-2 a c\right ) \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c^2 \sqrt {b^2-4 a c}}-\frac {b \log \left (a+b x+c x^2\right )}{2 c^2}+\frac {x}{c}\) |
Input:
Int[(c + a/x^2 + b/x)^(-1),x]
Output:
x/c - ((b^2 - 2*a*c)*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(c^2*Sqrt[b^2 - 4*a*c]) - (b*Log[a + b*x + c*x^2])/(2*c^2)
Int[((d_.) + (e_.)*(x_))^(m_)/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[m, 1]
Int[((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[x^( 2*n*p)*(c + b/x^n + a/x^(2*n))^p, x] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && LtQ[n, 0] && IntegerQ[p]
Time = 0.04 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.07
method | result | size |
default | \(\frac {x}{c}+\frac {-\frac {b \ln \left (c \,x^{2}+b x +a \right )}{2 c}+\frac {2 \left (-a +\frac {b^{2}}{2 c}\right ) \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{c}\) | \(75\) |
risch | \(\frac {x}{c}-\frac {2 \ln \left (-8 a^{2} c^{2}+6 a \,b^{2} c -b^{4}-2 \sqrt {-\left (4 a c -b^{2}\right ) \left (2 a c -b^{2}\right )^{2}}\, c x -\sqrt {-\left (4 a c -b^{2}\right ) \left (2 a c -b^{2}\right )^{2}}\, b \right ) a b}{c \left (4 a c -b^{2}\right )}+\frac {\ln \left (-8 a^{2} c^{2}+6 a \,b^{2} c -b^{4}-2 \sqrt {-\left (4 a c -b^{2}\right ) \left (2 a c -b^{2}\right )^{2}}\, c x -\sqrt {-\left (4 a c -b^{2}\right ) \left (2 a c -b^{2}\right )^{2}}\, b \right ) b^{3}}{2 c^{2} \left (4 a c -b^{2}\right )}+\frac {\ln \left (-8 a^{2} c^{2}+6 a \,b^{2} c -b^{4}-2 \sqrt {-\left (4 a c -b^{2}\right ) \left (2 a c -b^{2}\right )^{2}}\, c x -\sqrt {-\left (4 a c -b^{2}\right ) \left (2 a c -b^{2}\right )^{2}}\, b \right ) \sqrt {-\left (4 a c -b^{2}\right ) \left (2 a c -b^{2}\right )^{2}}}{2 c^{2} \left (4 a c -b^{2}\right )}-\frac {2 \ln \left (-8 a^{2} c^{2}+6 a \,b^{2} c -b^{4}+2 \sqrt {-\left (4 a c -b^{2}\right ) \left (2 a c -b^{2}\right )^{2}}\, c x +\sqrt {-\left (4 a c -b^{2}\right ) \left (2 a c -b^{2}\right )^{2}}\, b \right ) a b}{c \left (4 a c -b^{2}\right )}+\frac {\ln \left (-8 a^{2} c^{2}+6 a \,b^{2} c -b^{4}+2 \sqrt {-\left (4 a c -b^{2}\right ) \left (2 a c -b^{2}\right )^{2}}\, c x +\sqrt {-\left (4 a c -b^{2}\right ) \left (2 a c -b^{2}\right )^{2}}\, b \right ) b^{3}}{2 c^{2} \left (4 a c -b^{2}\right )}-\frac {\ln \left (-8 a^{2} c^{2}+6 a \,b^{2} c -b^{4}+2 \sqrt {-\left (4 a c -b^{2}\right ) \left (2 a c -b^{2}\right )^{2}}\, c x +\sqrt {-\left (4 a c -b^{2}\right ) \left (2 a c -b^{2}\right )^{2}}\, b \right ) \sqrt {-\left (4 a c -b^{2}\right ) \left (2 a c -b^{2}\right )^{2}}}{2 c^{2} \left (4 a c -b^{2}\right )}\) | \(654\) |
Input:
int(1/(c+a/x^2+b/x),x,method=_RETURNVERBOSE)
Output:
x/c+1/c*(-1/2*b/c*ln(c*x^2+b*x+a)+2*(-a+1/2*b^2/c)/(4*a*c-b^2)^(1/2)*arcta n((2*c*x+b)/(4*a*c-b^2)^(1/2)))
Time = 0.08 (sec) , antiderivative size = 235, normalized size of antiderivative = 3.36 \[ \int \frac {1}{c+\frac {a}{x^2}+\frac {b}{x}} \, dx=\left [-\frac {{\left (b^{2} - 2 \, a c\right )} \sqrt {b^{2} - 4 \, a c} \log \left (\frac {2 \, c^{2} x^{2} + 2 \, b c x + b^{2} - 2 \, a c + \sqrt {b^{2} - 4 \, a c} {\left (2 \, c x + b\right )}}{c x^{2} + b x + a}\right ) - 2 \, {\left (b^{2} c - 4 \, a c^{2}\right )} x + {\left (b^{3} - 4 \, a b c\right )} \log \left (c x^{2} + b x + a\right )}{2 \, {\left (b^{2} c^{2} - 4 \, a c^{3}\right )}}, -\frac {2 \, {\left (b^{2} - 2 \, a c\right )} \sqrt {-b^{2} + 4 \, a c} \arctan \left (-\frac {\sqrt {-b^{2} + 4 \, a c} {\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) - 2 \, {\left (b^{2} c - 4 \, a c^{2}\right )} x + {\left (b^{3} - 4 \, a b c\right )} \log \left (c x^{2} + b x + a\right )}{2 \, {\left (b^{2} c^{2} - 4 \, a c^{3}\right )}}\right ] \] Input:
integrate(1/(c+a/x^2+b/x),x, algorithm="fricas")
Output:
[-1/2*((b^2 - 2*a*c)*sqrt(b^2 - 4*a*c)*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)) - 2*(b^2*c - 4*a*c ^2)*x + (b^3 - 4*a*b*c)*log(c*x^2 + b*x + a))/(b^2*c^2 - 4*a*c^3), -1/2*(2 *(b^2 - 2*a*c)*sqrt(-b^2 + 4*a*c)*arctan(-sqrt(-b^2 + 4*a*c)*(2*c*x + b)/( b^2 - 4*a*c)) - 2*(b^2*c - 4*a*c^2)*x + (b^3 - 4*a*b*c)*log(c*x^2 + b*x + a))/(b^2*c^2 - 4*a*c^3)]
Leaf count of result is larger than twice the leaf count of optimal. 306 vs. \(2 (65) = 130\).
Time = 0.35 (sec) , antiderivative size = 306, normalized size of antiderivative = 4.37 \[ \int \frac {1}{c+\frac {a}{x^2}+\frac {b}{x}} \, dx=\left (- \frac {b}{2 c^{2}} - \frac {\sqrt {- 4 a c + b^{2}} \cdot \left (2 a c - b^{2}\right )}{2 c^{2} \cdot \left (4 a c - b^{2}\right )}\right ) \log {\left (x + \frac {- a b - 4 a c^{2} \left (- \frac {b}{2 c^{2}} - \frac {\sqrt {- 4 a c + b^{2}} \cdot \left (2 a c - b^{2}\right )}{2 c^{2} \cdot \left (4 a c - b^{2}\right )}\right ) + b^{2} c \left (- \frac {b}{2 c^{2}} - \frac {\sqrt {- 4 a c + b^{2}} \cdot \left (2 a c - b^{2}\right )}{2 c^{2} \cdot \left (4 a c - b^{2}\right )}\right )}{2 a c - b^{2}} \right )} + \left (- \frac {b}{2 c^{2}} + \frac {\sqrt {- 4 a c + b^{2}} \cdot \left (2 a c - b^{2}\right )}{2 c^{2} \cdot \left (4 a c - b^{2}\right )}\right ) \log {\left (x + \frac {- a b - 4 a c^{2} \left (- \frac {b}{2 c^{2}} + \frac {\sqrt {- 4 a c + b^{2}} \cdot \left (2 a c - b^{2}\right )}{2 c^{2} \cdot \left (4 a c - b^{2}\right )}\right ) + b^{2} c \left (- \frac {b}{2 c^{2}} + \frac {\sqrt {- 4 a c + b^{2}} \cdot \left (2 a c - b^{2}\right )}{2 c^{2} \cdot \left (4 a c - b^{2}\right )}\right )}{2 a c - b^{2}} \right )} + \frac {x}{c} \] Input:
integrate(1/(c+a/x**2+b/x),x)
Output:
(-b/(2*c**2) - sqrt(-4*a*c + b**2)*(2*a*c - b**2)/(2*c**2*(4*a*c - b**2))) *log(x + (-a*b - 4*a*c**2*(-b/(2*c**2) - sqrt(-4*a*c + b**2)*(2*a*c - b**2 )/(2*c**2*(4*a*c - b**2))) + b**2*c*(-b/(2*c**2) - sqrt(-4*a*c + b**2)*(2* a*c - b**2)/(2*c**2*(4*a*c - b**2))))/(2*a*c - b**2)) + (-b/(2*c**2) + sqr t(-4*a*c + b**2)*(2*a*c - b**2)/(2*c**2*(4*a*c - b**2)))*log(x + (-a*b - 4 *a*c**2*(-b/(2*c**2) + sqrt(-4*a*c + b**2)*(2*a*c - b**2)/(2*c**2*(4*a*c - b**2))) + b**2*c*(-b/(2*c**2) + sqrt(-4*a*c + b**2)*(2*a*c - b**2)/(2*c** 2*(4*a*c - b**2))))/(2*a*c - b**2)) + x/c
Exception generated. \[ \int \frac {1}{c+\frac {a}{x^2}+\frac {b}{x}} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(1/(c+a/x^2+b/x),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 = 0.13 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.96 \[ \int \frac {1}{c+\frac {a}{x^2}+\frac {b}{x}} \, dx=\frac {x}{c} - \frac {b \log \left (c x^{2} + b x + a\right )}{2 \, c^{2}} + \frac {{\left (b^{2} - 2 \, a c\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{\sqrt {-b^{2} + 4 \, a c} c^{2}} \] Input:
integrate(1/(c+a/x^2+b/x),x, algorithm="giac")
Output:
x/c - 1/2*b*log(c*x^2 + b*x + a)/c^2 + (b^2 - 2*a*c)*arctan((2*c*x + b)/sq rt(-b^2 + 4*a*c))/(sqrt(-b^2 + 4*a*c)*c^2)
Time = 20.56 (sec) , antiderivative size = 172, normalized size of antiderivative = 2.46 \[ \int \frac {1}{c+\frac {a}{x^2}+\frac {b}{x}} \, dx=\frac {x}{c}+\frac {b^3\,\ln \left (c\,x^2+b\,x+a\right )}{2\,\left (4\,a\,c^3-b^2\,c^2\right )}-\frac {2\,a\,\mathrm {atan}\left (\frac {b}{\sqrt {4\,a\,c-b^2}}+\frac {2\,c\,x}{\sqrt {4\,a\,c-b^2}}\right )}{c\,\sqrt {4\,a\,c-b^2}}+\frac {b^2\,\mathrm {atan}\left (\frac {b}{\sqrt {4\,a\,c-b^2}}+\frac {2\,c\,x}{\sqrt {4\,a\,c-b^2}}\right )}{c^2\,\sqrt {4\,a\,c-b^2}}-\frac {2\,a\,b\,c\,\ln \left (c\,x^2+b\,x+a\right )}{4\,a\,c^3-b^2\,c^2} \] Input:
int(1/(c + a/x^2 + b/x),x)
Output:
x/c + (b^3*log(a + b*x + c*x^2))/(2*(4*a*c^3 - b^2*c^2)) - (2*a*atan(b/(4* a*c - b^2)^(1/2) + (2*c*x)/(4*a*c - b^2)^(1/2)))/(c*(4*a*c - b^2)^(1/2)) + (b^2*atan(b/(4*a*c - b^2)^(1/2) + (2*c*x)/(4*a*c - b^2)^(1/2)))/(c^2*(4*a *c - b^2)^(1/2)) - (2*a*b*c*log(a + b*x + c*x^2))/(4*a*c^3 - b^2*c^2)
Time = 0.16 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.94 \[ \int \frac {1}{c+\frac {a}{x^2}+\frac {b}{x}} \, dx=\frac {-4 \sqrt {4 a c -b^{2}}\, \mathit {atan} \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right ) a c +2 \sqrt {4 a c -b^{2}}\, \mathit {atan} \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right ) b^{2}-4 \,\mathrm {log}\left (c \,x^{2}+b x +a \right ) a b c +\mathrm {log}\left (c \,x^{2}+b x +a \right ) b^{3}+8 a \,c^{2} x -2 b^{2} c x}{2 c^{2} \left (4 a c -b^{2}\right )} \] Input:
int(1/(c+a/x^2+b/x),x)
Output:
( - 4*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a*c + 2*sqrt (4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*b**2 - 4*log(a + b*x + c*x**2)*a*b*c + log(a + b*x + c*x**2)*b**3 + 8*a*c**2*x - 2*b**2*c*x)/(2* c**2*(4*a*c - b**2))