Integrand size = 18, antiderivative size = 87 \[ \int \frac {d+e x}{\left (a+b x+c x^2\right )^2} \, dx=-\frac {b d-2 a e+(2 c d-b e) x}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {2 (2 c d-b e) \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}} \] Output:
-(b*d-2*a*e+(-b*e+2*c*d)*x)/(-4*a*c+b^2)/(c*x^2+b*x+a)+2*(-b*e+2*c*d)*arct anh((2*c*x+b)/(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(3/2)
Time = 0.08 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.01 \[ \int \frac {d+e x}{\left (a+b x+c x^2\right )^2} \, dx=\frac {\frac {-b d+2 a e-2 c d x+b e x}{a+x (b+c x)}+\frac {2 (-2 c d+b e) \arctan \left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )}{\sqrt {-b^2+4 a c}}}{b^2-4 a c} \] Input:
Integrate[(d + e*x)/(a + b*x + c*x^2)^2,x]
Output:
((-(b*d) + 2*a*e - 2*c*d*x + b*e*x)/(a + x*(b + c*x)) + (2*(-2*c*d + b*e)* ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c])/(b^2 - 4*a*c)
Time = 0.23 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {1159, 1083, 219}
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.
\(\displaystyle \int \frac {d+e x}{\left (a+b x+c x^2\right )^2} \, dx\) |
\(\Big \downarrow \) 1159 |
\(\displaystyle -\frac {(2 c d-b e) \int \frac {1}{c x^2+b x+a}dx}{b^2-4 a c}-\frac {-2 a e+x (2 c d-b e)+b d}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\) |
\(\Big \downarrow \) 1083 |
\(\displaystyle \frac {2 (2 c d-b e) \int \frac {1}{b^2-(b+2 c x)^2-4 a c}d(b+2 c x)}{b^2-4 a c}-\frac {-2 a e+x (2 c d-b e)+b d}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {2 (2 c d-b e) \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}}-\frac {-2 a e+x (2 c d-b e)+b d}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\) |
Input:
Int[(d + e*x)/(a + b*x + c*x^2)^2,x]
Output:
-((b*d - 2*a*e + (2*c*d - b*e)*x)/((b^2 - 4*a*c)*(a + b*x + c*x^2))) + (2* (2*c*d - b*e)*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(b^2 - 4*a*c)^(3/2)
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)^(p_), x_Symbol ] :> Simp[((b*d - 2*a*e + (2*c*d - b*e)*x)/((p + 1)*(b^2 - 4*a*c)))*(a + b* x + c*x^2)^(p + 1), x] - Simp[(2*p + 3)*((2*c*d - b*e)/((p + 1)*(b^2 - 4*a* c))) Int[(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] & & LtQ[p, -1] && NeQ[p, -3/2]
Time = 1.20 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.02
method | result | size |
default | \(\frac {b d -2 a e +\left (-b e +2 c d \right ) x}{\left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )}+\frac {2 \left (-b e +2 c d \right ) \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\left (4 a c -b^{2}\right )^{\frac {3}{2}}}\) | \(89\) |
risch | \(\frac {-\frac {\left (b e -2 c d \right ) x}{4 a c -b^{2}}-\frac {2 a e -b d}{4 a c -b^{2}}}{c \,x^{2}+b x +a}+\frac {\ln \left (\left (-8 a \,c^{2}+2 b^{2} c \right ) x -\left (-4 a c +b^{2}\right )^{\frac {3}{2}}-4 a b c +b^{3}\right ) b e}{\left (-4 a c +b^{2}\right )^{\frac {3}{2}}}-\frac {2 \ln \left (\left (-8 a \,c^{2}+2 b^{2} c \right ) x -\left (-4 a c +b^{2}\right )^{\frac {3}{2}}-4 a b c +b^{3}\right ) c d}{\left (-4 a c +b^{2}\right )^{\frac {3}{2}}}-\frac {\ln \left (\left (8 a \,c^{2}-2 b^{2} c \right ) x -\left (-4 a c +b^{2}\right )^{\frac {3}{2}}+4 a b c -b^{3}\right ) b e}{\left (-4 a c +b^{2}\right )^{\frac {3}{2}}}+\frac {2 \ln \left (\left (8 a \,c^{2}-2 b^{2} c \right ) x -\left (-4 a c +b^{2}\right )^{\frac {3}{2}}+4 a b c -b^{3}\right ) c d}{\left (-4 a c +b^{2}\right )^{\frac {3}{2}}}\) | \(269\) |
Input:
int((e*x+d)/(c*x^2+b*x+a)^2,x,method=_RETURNVERBOSE)
Output:
(b*d-2*a*e+(-b*e+2*c*d)*x)/(4*a*c-b^2)/(c*x^2+b*x+a)+2*(-b*e+2*c*d)/(4*a*c -b^2)^(3/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))
Leaf count of result is larger than twice the leaf count of optimal. 220 vs. \(2 (83) = 166\).
Time = 0.08 (sec) , antiderivative size = 459, normalized size of antiderivative = 5.28 \[ \int \frac {d+e x}{\left (a+b x+c x^2\right )^2} \, dx=\left [\frac {{\left (2 \, a c d - a b e + {\left (2 \, c^{2} d - b c e\right )} x^{2} + {\left (2 \, b c d - b^{2} e\right )} x\right )} \sqrt {b^{2} - 4 \, a c} \log \left (\frac {2 \, c^{2} x^{2} + 2 \, b c x + b^{2} - 2 \, a c + \sqrt {b^{2} - 4 \, a c} {\left (2 \, c x + b\right )}}{c x^{2} + b x + a}\right ) - {\left (b^{3} - 4 \, a b c\right )} d + 2 \, {\left (a b^{2} - 4 \, a^{2} c\right )} e - {\left (2 \, {\left (b^{2} c - 4 \, a c^{2}\right )} d - {\left (b^{3} - 4 \, a b c\right )} e\right )} x}{a b^{4} - 8 \, a^{2} b^{2} c + 16 \, a^{3} c^{2} + {\left (b^{4} c - 8 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} x^{2} + {\left (b^{5} - 8 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} x}, \frac {2 \, {\left (2 \, a c d - a b e + {\left (2 \, c^{2} d - b c e\right )} x^{2} + {\left (2 \, b c d - b^{2} e\right )} x\right )} \sqrt {-b^{2} + 4 \, a c} \arctan \left (-\frac {\sqrt {-b^{2} + 4 \, a c} {\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) - {\left (b^{3} - 4 \, a b c\right )} d + 2 \, {\left (a b^{2} - 4 \, a^{2} c\right )} e - {\left (2 \, {\left (b^{2} c - 4 \, a c^{2}\right )} d - {\left (b^{3} - 4 \, a b c\right )} e\right )} x}{a b^{4} - 8 \, a^{2} b^{2} c + 16 \, a^{3} c^{2} + {\left (b^{4} c - 8 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} x^{2} + {\left (b^{5} - 8 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} x}\right ] \] Input:
integrate((e*x+d)/(c*x^2+b*x+a)^2,x, algorithm="fricas")
Output:
[((2*a*c*d - a*b*e + (2*c^2*d - b*c*e)*x^2 + (2*b*c*d - b^2*e)*x)*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)) - (b^3 - 4*a*b*c)*d + 2*(a*b^2 - 4*a^2*c)*e - ( 2*(b^2*c - 4*a*c^2)*d - (b^3 - 4*a*b*c)*e)*x)/(a*b^4 - 8*a^2*b^2*c + 16*a^ 3*c^2 + (b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)*x^2 + (b^5 - 8*a*b^3*c + 16*a^2 *b*c^2)*x), (2*(2*a*c*d - a*b*e + (2*c^2*d - b*c*e)*x^2 + (2*b*c*d - b^2*e )*x)*sqrt(-b^2 + 4*a*c)*arctan(-sqrt(-b^2 + 4*a*c)*(2*c*x + b)/(b^2 - 4*a* c)) - (b^3 - 4*a*b*c)*d + 2*(a*b^2 - 4*a^2*c)*e - (2*(b^2*c - 4*a*c^2)*d - (b^3 - 4*a*b*c)*e)*x)/(a*b^4 - 8*a^2*b^2*c + 16*a^3*c^2 + (b^4*c - 8*a*b^ 2*c^2 + 16*a^2*c^3)*x^2 + (b^5 - 8*a*b^3*c + 16*a^2*b*c^2)*x)]
Leaf count of result is larger than twice the leaf count of optimal. 359 vs. \(2 (78) = 156\).
Time = 0.51 (sec) , antiderivative size = 359, normalized size of antiderivative = 4.13 \[ \int \frac {d+e x}{\left (a+b x+c x^2\right )^2} \, dx=\sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} \left (b e - 2 c d\right ) \log {\left (x + \frac {- 16 a^{2} c^{2} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} \left (b e - 2 c d\right ) + 8 a b^{2} c \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} \left (b e - 2 c d\right ) - b^{4} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} \left (b e - 2 c d\right ) + b^{2} e - 2 b c d}{2 b c e - 4 c^{2} d} \right )} - \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} \left (b e - 2 c d\right ) \log {\left (x + \frac {16 a^{2} c^{2} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} \left (b e - 2 c d\right ) - 8 a b^{2} c \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} \left (b e - 2 c d\right ) + b^{4} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} \left (b e - 2 c d\right ) + b^{2} e - 2 b c d}{2 b c e - 4 c^{2} d} \right )} + \frac {- 2 a e + b d + x \left (- b e + 2 c d\right )}{4 a^{2} c - a b^{2} + x^{2} \cdot \left (4 a c^{2} - b^{2} c\right ) + x \left (4 a b c - b^{3}\right )} \] Input:
integrate((e*x+d)/(c*x**2+b*x+a)**2,x)
Output:
sqrt(-1/(4*a*c - b**2)**3)*(b*e - 2*c*d)*log(x + (-16*a**2*c**2*sqrt(-1/(4 *a*c - b**2)**3)*(b*e - 2*c*d) + 8*a*b**2*c*sqrt(-1/(4*a*c - b**2)**3)*(b* e - 2*c*d) - b**4*sqrt(-1/(4*a*c - b**2)**3)*(b*e - 2*c*d) + b**2*e - 2*b* c*d)/(2*b*c*e - 4*c**2*d)) - sqrt(-1/(4*a*c - b**2)**3)*(b*e - 2*c*d)*log( x + (16*a**2*c**2*sqrt(-1/(4*a*c - b**2)**3)*(b*e - 2*c*d) - 8*a*b**2*c*sq rt(-1/(4*a*c - b**2)**3)*(b*e - 2*c*d) + b**4*sqrt(-1/(4*a*c - b**2)**3)*( b*e - 2*c*d) + b**2*e - 2*b*c*d)/(2*b*c*e - 4*c**2*d)) + (-2*a*e + b*d + x *(-b*e + 2*c*d))/(4*a**2*c - a*b**2 + x**2*(4*a*c**2 - b**2*c) + x*(4*a*b* c - b**3))
Exception generated. \[ \int \frac {d+e x}{\left (a+b x+c x^2\right )^2} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((e*x+d)/(c*x^2+b*x+a)^2,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 = 96, normalized size of antiderivative = 1.10 \[ \int \frac {d+e x}{\left (a+b x+c x^2\right )^2} \, dx=-\frac {2 \, {\left (2 \, c d - b e\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (b^{2} - 4 \, a c\right )} \sqrt {-b^{2} + 4 \, a c}} - \frac {2 \, c d x - b e x + b d - 2 \, a e}{{\left (c x^{2} + b x + a\right )} {\left (b^{2} - 4 \, a c\right )}} \] Input:
integrate((e*x+d)/(c*x^2+b*x+a)^2,x, algorithm="giac")
Output:
-2*(2*c*d - b*e)*arctan((2*c*x + b)/sqrt(-b^2 + 4*a*c))/((b^2 - 4*a*c)*sqr t(-b^2 + 4*a*c)) - (2*c*d*x - b*e*x + b*d - 2*a*e)/((c*x^2 + b*x + a)*(b^2 - 4*a*c))
Time = 0.12 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.83 \[ \int \frac {d+e x}{\left (a+b x+c x^2\right )^2} \, dx=\frac {2\,\mathrm {atan}\left (\frac {\left (4\,a\,c-b^2\right )\,\left (\frac {\left (b^3-4\,a\,b\,c\right )\,\left (b\,e-2\,c\,d\right )}{{\left (4\,a\,c-b^2\right )}^{5/2}}-\frac {2\,c\,x\,\left (b\,e-2\,c\,d\right )}{{\left (4\,a\,c-b^2\right )}^{3/2}}\right )}{b\,e-2\,c\,d}\right )\,\left (b\,e-2\,c\,d\right )}{{\left (4\,a\,c-b^2\right )}^{3/2}}-\frac {\frac {2\,a\,e-b\,d}{4\,a\,c-b^2}+\frac {x\,\left (b\,e-2\,c\,d\right )}{4\,a\,c-b^2}}{c\,x^2+b\,x+a} \] Input:
int((d + e*x)/(a + b*x + c*x^2)^2,x)
Output:
(2*atan(((4*a*c - b^2)*(((b^3 - 4*a*b*c)*(b*e - 2*c*d))/(4*a*c - b^2)^(5/2 ) - (2*c*x*(b*e - 2*c*d))/(4*a*c - b^2)^(3/2)))/(b*e - 2*c*d))*(b*e - 2*c* d))/(4*a*c - b^2)^(3/2) - ((2*a*e - b*d)/(4*a*c - b^2) + (x*(b*e - 2*c*d)) /(4*a*c - b^2))/(a + b*x + c*x^2)
Time = 0.21 (sec) , antiderivative size = 404, normalized size of antiderivative = 4.64 \[ \int \frac {d+e x}{\left (a+b x+c x^2\right )^2} \, dx=\frac {-2 \sqrt {4 a c -b^{2}}\, \mathit {atan} \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right ) a \,b^{2} e +4 \sqrt {4 a c -b^{2}}\, \mathit {atan} \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right ) a b c d -2 \sqrt {4 a c -b^{2}}\, \mathit {atan} \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right ) b^{3} e x +4 \sqrt {4 a c -b^{2}}\, \mathit {atan} \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right ) b^{2} c d x -2 \sqrt {4 a c -b^{2}}\, \mathit {atan} \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right ) b^{2} c e \,x^{2}+4 \sqrt {4 a c -b^{2}}\, \mathit {atan} \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right ) b \,c^{2} d \,x^{2}-4 a^{2} b c e -8 a^{2} c^{2} d +a \,b^{3} e +6 a \,b^{2} c d +4 a b \,c^{2} e \,x^{2}-8 a \,c^{3} d \,x^{2}-b^{4} d -b^{3} c e \,x^{2}+2 b^{2} c^{2} d \,x^{2}}{b \left (16 a^{2} c^{3} x^{2}-8 a \,b^{2} c^{2} x^{2}+b^{4} c \,x^{2}+16 a^{2} b \,c^{2} x -8 a \,b^{3} c x +b^{5} x +16 a^{3} c^{2}-8 a^{2} b^{2} c +a \,b^{4}\right )} \] Input:
int((e*x+d)/(c*x^2+b*x+a)^2,x)
Output:
( - 2*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a*b**2*e + 4 *sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a*b*c*d - 2*sqrt( 4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*b**3*e*x + 4*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*b**2*c*d*x - 2*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*b**2*c*e*x**2 + 4*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*b*c**2*d*x**2 - 4*a**2*b*c*e - 8*a**2*c**2*d + a*b**3*e + 6*a*b**2*c*d + 4*a*b*c**2*e*x**2 - 8*a*c**3*d*x **2 - b**4*d - b**3*c*e*x**2 + 2*b**2*c**2*d*x**2)/(b*(16*a**3*c**2 - 8*a* *2*b**2*c + 16*a**2*b*c**2*x + 16*a**2*c**3*x**2 + a*b**4 - 8*a*b**3*c*x - 8*a*b**2*c**2*x**2 + b**5*x + b**4*c*x**2))