Integrand size = 18, antiderivative size = 66 \[ \int \frac {d+e x}{a+b x+c x^2} \, dx=-\frac {(2 c d-b e) \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c \sqrt {b^2-4 a c}}+\frac {e \log \left (a+b x+c x^2\right )}{2 c} \] Output:
-(-b*e+2*c*d)*arctanh((2*c*x+b)/(-4*a*c+b^2)^(1/2))/c/(-4*a*c+b^2)^(1/2)+1 /2*e*ln(c*x^2+b*x+a)/c
Time = 0.06 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.00 \[ \int \frac {d+e x}{a+b x+c x^2} \, dx=\frac {-\frac {2 (-2 c d+b e) \arctan \left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )}{\sqrt {-b^2+4 a c}}+e \log (a+x (b+c x))}{2 c} \] Input:
Integrate[(d + e*x)/(a + b*x + c*x^2),x]
Output:
((-2*(-2*c*d + b*e)*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4* a*c] + e*Log[a + x*(b + c*x)])/(2*c)
Time = 0.23 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {1142, 1083, 219, 1103}
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.
\(\displaystyle \int \frac {d+e x}{a+b x+c x^2} \, dx\) |
\(\Big \downarrow \) 1142 |
\(\displaystyle \frac {(2 c d-b e) \int \frac {1}{c x^2+b x+a}dx}{2 c}+\frac {e \int \frac {b+2 c x}{c x^2+b x+a}dx}{2 c}\) |
\(\Big \downarrow \) 1083 |
\(\displaystyle \frac {e \int \frac {b+2 c x}{c x^2+b x+a}dx}{2 c}-\frac {(2 c d-b e) \int \frac {1}{b^2-(b+2 c x)^2-4 a c}d(b+2 c x)}{c}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {e \int \frac {b+2 c x}{c x^2+b x+a}dx}{2 c}-\frac {(2 c d-b e) \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c \sqrt {b^2-4 a c}}\) |
\(\Big \downarrow \) 1103 |
\(\displaystyle \frac {e \log \left (a+b x+c x^2\right )}{2 c}-\frac {(2 c d-b e) \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c \sqrt {b^2-4 a c}}\) |
Input:
Int[(d + e*x)/(a + b*x + c*x^2),x]
Output:
-(((2*c*d - b*e)*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(c*Sqrt[b^2 - 4*a *c])) + (e*Log[a + b*x + c*x^2])/(2*c)
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[(2*c*d - b*e)/(2*c) Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
Time = 1.23 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.94
method | result | size |
default | \(\frac {e \ln \left (c \,x^{2}+b x +a \right )}{2 c}+\frac {2 \left (d -\frac {b e}{2 c}\right ) \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}\) | \(62\) |
risch | \(\frac {2 \ln \left (-4 a b c e +8 a \,c^{2} d +e \,b^{3}-2 c d \,b^{2}-2 \sqrt {-\left (b e -2 c d \right )^{2} \left (4 a c -b^{2}\right )}\, c x -\sqrt {-\left (b e -2 c d \right )^{2} \left (4 a c -b^{2}\right )}\, b \right ) a e}{4 a c -b^{2}}-\frac {\ln \left (-4 a b c e +8 a \,c^{2} d +e \,b^{3}-2 c d \,b^{2}-2 \sqrt {-\left (b e -2 c d \right )^{2} \left (4 a c -b^{2}\right )}\, c x -\sqrt {-\left (b e -2 c d \right )^{2} \left (4 a c -b^{2}\right )}\, b \right ) e \,b^{2}}{2 \left (4 a c -b^{2}\right ) c}+\frac {\ln \left (-4 a b c e +8 a \,c^{2} d +e \,b^{3}-2 c d \,b^{2}-2 \sqrt {-\left (b e -2 c d \right )^{2} \left (4 a c -b^{2}\right )}\, c x -\sqrt {-\left (b e -2 c d \right )^{2} \left (4 a c -b^{2}\right )}\, b \right ) \sqrt {-\left (b e -2 c d \right )^{2} \left (4 a c -b^{2}\right )}}{2 \left (4 a c -b^{2}\right ) c}+\frac {2 \ln \left (-4 a b c e +8 a \,c^{2} d +e \,b^{3}-2 c d \,b^{2}+2 \sqrt {-\left (b e -2 c d \right )^{2} \left (4 a c -b^{2}\right )}\, c x +\sqrt {-\left (b e -2 c d \right )^{2} \left (4 a c -b^{2}\right )}\, b \right ) a e}{4 a c -b^{2}}-\frac {\ln \left (-4 a b c e +8 a \,c^{2} d +e \,b^{3}-2 c d \,b^{2}+2 \sqrt {-\left (b e -2 c d \right )^{2} \left (4 a c -b^{2}\right )}\, c x +\sqrt {-\left (b e -2 c d \right )^{2} \left (4 a c -b^{2}\right )}\, b \right ) e \,b^{2}}{2 \left (4 a c -b^{2}\right ) c}-\frac {\ln \left (-4 a b c e +8 a \,c^{2} d +e \,b^{3}-2 c d \,b^{2}+2 \sqrt {-\left (b e -2 c d \right )^{2} \left (4 a c -b^{2}\right )}\, c x +\sqrt {-\left (b e -2 c d \right )^{2} \left (4 a c -b^{2}\right )}\, b \right ) \sqrt {-\left (b e -2 c d \right )^{2} \left (4 a c -b^{2}\right )}}{2 \left (4 a c -b^{2}\right ) c}\) | \(647\) |
Input:
int((e*x+d)/(c*x^2+b*x+a),x,method=_RETURNVERBOSE)
Output:
1/2*e*ln(c*x^2+b*x+a)/c+2*(d-1/2*b/c*e)/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b) /(4*a*c-b^2)^(1/2))
Time = 0.08 (sec) , antiderivative size = 204, normalized size of antiderivative = 3.09 \[ \int \frac {d+e x}{a+b x+c x^2} \, dx=\left [\frac {{\left (b^{2} - 4 \, a c\right )} e \log \left (c x^{2} + b x + a\right ) - \sqrt {b^{2} - 4 \, a c} {\left (2 \, c d - b e\right )} \log \left (\frac {2 \, c^{2} x^{2} + 2 \, b c x + b^{2} - 2 \, a c + \sqrt {b^{2} - 4 \, a c} {\left (2 \, c x + b\right )}}{c x^{2} + b x + a}\right )}{2 \, {\left (b^{2} c - 4 \, a c^{2}\right )}}, \frac {{\left (b^{2} - 4 \, a c\right )} e \log \left (c x^{2} + b x + a\right ) - 2 \, \sqrt {-b^{2} + 4 \, a c} {\left (2 \, c d - b e\right )} \arctan \left (-\frac {\sqrt {-b^{2} + 4 \, a c} {\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right )}{2 \, {\left (b^{2} c - 4 \, a c^{2}\right )}}\right ] \] Input:
integrate((e*x+d)/(c*x^2+b*x+a),x, algorithm="fricas")
Output:
[1/2*((b^2 - 4*a*c)*e*log(c*x^2 + b*x + a) - sqrt(b^2 - 4*a*c)*(2*c*d - b* e)*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^2*c - 4*a*c^2), 1/2*((b^2 - 4*a*c)*e*log(c*x^2 + b *x + a) - 2*sqrt(-b^2 + 4*a*c)*(2*c*d - b*e)*arctan(-sqrt(-b^2 + 4*a*c)*(2 *c*x + b)/(b^2 - 4*a*c)))/(b^2*c - 4*a*c^2)]
Leaf count of result is larger than twice the leaf count of optimal. 280 vs. \(2 (58) = 116\).
Time = 0.41 (sec) , antiderivative size = 280, normalized size of antiderivative = 4.24 \[ \int \frac {d+e x}{a+b x+c x^2} \, dx=\left (\frac {e}{2 c} - \frac {\sqrt {- 4 a c + b^{2}} \left (b e - 2 c d\right )}{2 c \left (4 a c - b^{2}\right )}\right ) \log {\left (x + \frac {- 4 a c \left (\frac {e}{2 c} - \frac {\sqrt {- 4 a c + b^{2}} \left (b e - 2 c d\right )}{2 c \left (4 a c - b^{2}\right )}\right ) + 2 a e + b^{2} \left (\frac {e}{2 c} - \frac {\sqrt {- 4 a c + b^{2}} \left (b e - 2 c d\right )}{2 c \left (4 a c - b^{2}\right )}\right ) - b d}{b e - 2 c d} \right )} + \left (\frac {e}{2 c} + \frac {\sqrt {- 4 a c + b^{2}} \left (b e - 2 c d\right )}{2 c \left (4 a c - b^{2}\right )}\right ) \log {\left (x + \frac {- 4 a c \left (\frac {e}{2 c} + \frac {\sqrt {- 4 a c + b^{2}} \left (b e - 2 c d\right )}{2 c \left (4 a c - b^{2}\right )}\right ) + 2 a e + b^{2} \left (\frac {e}{2 c} + \frac {\sqrt {- 4 a c + b^{2}} \left (b e - 2 c d\right )}{2 c \left (4 a c - b^{2}\right )}\right ) - b d}{b e - 2 c d} \right )} \] Input:
integrate((e*x+d)/(c*x**2+b*x+a),x)
Output:
(e/(2*c) - sqrt(-4*a*c + b**2)*(b*e - 2*c*d)/(2*c*(4*a*c - b**2)))*log(x + (-4*a*c*(e/(2*c) - sqrt(-4*a*c + b**2)*(b*e - 2*c*d)/(2*c*(4*a*c - b**2)) ) + 2*a*e + b**2*(e/(2*c) - sqrt(-4*a*c + b**2)*(b*e - 2*c*d)/(2*c*(4*a*c - b**2))) - b*d)/(b*e - 2*c*d)) + (e/(2*c) + sqrt(-4*a*c + b**2)*(b*e - 2* c*d)/(2*c*(4*a*c - b**2)))*log(x + (-4*a*c*(e/(2*c) + sqrt(-4*a*c + b**2)* (b*e - 2*c*d)/(2*c*(4*a*c - b**2))) + 2*a*e + b**2*(e/(2*c) + sqrt(-4*a*c + b**2)*(b*e - 2*c*d)/(2*c*(4*a*c - b**2))) - b*d)/(b*e - 2*c*d))
Exception generated. \[ \int \frac {d+e x}{a+b x+c x^2} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((e*x+d)/(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 = 0.23 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.95 \[ \int \frac {d+e x}{a+b x+c x^2} \, dx=\frac {e \log \left (c x^{2} + b x + a\right )}{2 \, c} + \frac {{\left (2 \, c d - b e\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{\sqrt {-b^{2} + 4 \, a c} c} \] Input:
integrate((e*x+d)/(c*x^2+b*x+a),x, algorithm="giac")
Output:
1/2*e*log(c*x^2 + b*x + a)/c + (2*c*d - b*e)*arctan((2*c*x + b)/sqrt(-b^2 + 4*a*c))/(sqrt(-b^2 + 4*a*c)*c)
Time = 0.08 (sec) , antiderivative size = 162, normalized size of antiderivative = 2.45 \[ \int \frac {d+e x}{a+b x+c x^2} \, dx=\frac {2\,d\,\mathrm {atan}\left (\frac {b}{\sqrt {4\,a\,c-b^2}}+\frac {2\,c\,x}{\sqrt {4\,a\,c-b^2}}\right )}{\sqrt {4\,a\,c-b^2}}-\frac {b^2\,e\,\ln \left (c\,x^2+b\,x+a\right )}{2\,\left (4\,a\,c^2-b^2\,c\right )}+\frac {2\,a\,c\,e\,\ln \left (c\,x^2+b\,x+a\right )}{4\,a\,c^2-b^2\,c}-\frac {b\,e\,\mathrm {atan}\left (\frac {b}{\sqrt {4\,a\,c-b^2}}+\frac {2\,c\,x}{\sqrt {4\,a\,c-b^2}}\right )}{c\,\sqrt {4\,a\,c-b^2}} \] Input:
int((d + e*x)/(a + b*x + c*x^2),x)
Output:
(2*d*atan(b/(4*a*c - b^2)^(1/2) + (2*c*x)/(4*a*c - b^2)^(1/2)))/(4*a*c - b ^2)^(1/2) - (b^2*e*log(a + b*x + c*x^2))/(2*(4*a*c^2 - b^2*c)) + (2*a*c*e* log(a + b*x + c*x^2))/(4*a*c^2 - b^2*c) - (b*e*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))
Time = 0.23 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.86 \[ \int \frac {d+e x}{a+b x+c x^2} \, dx=\frac {-2 \sqrt {4 a c -b^{2}}\, \mathit {atan} \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right ) b e +4 \sqrt {4 a c -b^{2}}\, \mathit {atan} \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right ) c d +4 \,\mathrm {log}\left (c \,x^{2}+b x +a \right ) a c e -\mathrm {log}\left (c \,x^{2}+b x +a \right ) b^{2} e}{2 c \left (4 a c -b^{2}\right )} \] Input:
int((e*x+d)/(c*x^2+b*x+a),x)
Output:
( - 2*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*b*e + 4*sqrt (4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*c*d + 4*log(a + b*x + c*x**2)*a*c*e - log(a + b*x + c*x**2)*b**2*e)/(2*c*(4*a*c - b**2))