3.893 \(\int \frac{x (d+e x)}{\left (a+b x+c x^2\right )^2} \, dx\)

Optimal. Leaf size=99 \[ \frac{x \left (2 a c e+b^2 (-e)+b c d\right )+a (2 c d-b e)}{c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac{2 (b d-2 a e) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}} \]

[Out]

(a*(2*c*d - b*e) + (b*c*d - b^2*e + 2*a*c*e)*x)/(c*(b^2 - 4*a*c)*(a + b*x + c*x^
2)) - (2*(b*d - 2*a*e)*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(b^2 - 4*a*c)^(3/
2)

_______________________________________________________________________________________

Rubi [A]  time = 0.117059, antiderivative size = 99, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ \frac{x \left (2 a c e+b^2 (-e)+b c d\right )+a (2 c d-b e)}{c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac{2 (b d-2 a e) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[(x*(d + e*x))/(a + b*x + c*x^2)^2,x]

[Out]

(a*(2*c*d - b*e) + (b*c*d - b^2*e + 2*a*c*e)*x)/(c*(b^2 - 4*a*c)*(a + b*x + c*x^
2)) - (2*(b*d - 2*a*e)*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(b^2 - 4*a*c)^(3/
2)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 13.9659, size = 71, normalized size = 0.72 \[ \frac{\left (2 a + b x\right ) \left (d + e x\right )}{\left (- 4 a c + b^{2}\right ) \left (a + b x + c x^{2}\right )} + \frac{2 \left (2 a e - b d\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{\sqrt{- 4 a c + b^{2}}} \right )}}{\left (- 4 a c + b^{2}\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x*(e*x+d)/(c*x**2+b*x+a)**2,x)

[Out]

(2*a + b*x)*(d + e*x)/((-4*a*c + b**2)*(a + b*x + c*x**2)) + 2*(2*a*e - b*d)*ata
nh((b + 2*c*x)/sqrt(-4*a*c + b**2))/(-4*a*c + b**2)**(3/2)

_______________________________________________________________________________________

Mathematica [A]  time = 0.158278, size = 99, normalized size = 1. \[ \frac{a b e-2 a c (d+e x)+b x (b e-c d)}{c \left (4 a c-b^2\right ) (a+x (b+c x))}-\frac{2 (b d-2 a e) \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\left (4 a c-b^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(x*(d + e*x))/(a + b*x + c*x^2)^2,x]

[Out]

(a*b*e + b*(-(c*d) + b*e)*x - 2*a*c*(d + e*x))/(c*(-b^2 + 4*a*c)*(a + x*(b + c*x
))) - (2*(b*d - 2*a*e)*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(3
/2)

_______________________________________________________________________________________

Maple [A]  time = 0.009, size = 147, normalized size = 1.5 \[{\frac{1}{c{x}^{2}+bx+a} \left ( -{\frac{ \left ( 2\,ace-{b}^{2}e+bcd \right ) x}{c \left ( 4\,ac-{b}^{2} \right ) }}+{\frac{a \left ( be-2\,cd \right ) }{c \left ( 4\,ac-{b}^{2} \right ) }} \right ) }+4\,{\frac{ae}{ \left ( 4\,ac-{b}^{2} \right ) ^{3/2}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }-2\,{\frac{bd}{ \left ( 4\,ac-{b}^{2} \right ) ^{3/2}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(e*x+d)/(c*x^2+b*x+a)^2,x)

[Out]

(-(2*a*c*e-b^2*e+b*c*d)/c/(4*a*c-b^2)*x+a*(b*e-2*c*d)/c/(4*a*c-b^2))/(c*x^2+b*x+
a)+4/(4*a*c-b^2)^(3/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a*e-2/(4*a*c-b^2)^(3/
2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b*d

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)*x/(c*x^2 + b*x + a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.286065, size = 1, normalized size = 0.01 \[ \left [\frac{{\left (a b c d - 2 \, a^{2} c e +{\left (b c^{2} d - 2 \, a c^{2} e\right )} x^{2} +{\left (b^{2} c d - 2 \, a b c e\right )} x\right )} \log \left (-\frac{b^{3} - 4 \, a b c + 2 \,{\left (b^{2} c - 4 \, a c^{2}\right )} x -{\left (2 \, c^{2} x^{2} + 2 \, b c x + b^{2} - 2 \, a c\right )} \sqrt{b^{2} - 4 \, a c}}{c x^{2} + b x + a}\right ) +{\left (2 \, a c d - a b e +{\left (b c d -{\left (b^{2} - 2 \, a c\right )} e\right )} x\right )} \sqrt{b^{2} - 4 \, a c}}{{\left (a b^{2} c - 4 \, a^{2} c^{2} +{\left (b^{2} c^{2} - 4 \, a c^{3}\right )} x^{2} +{\left (b^{3} c - 4 \, a b c^{2}\right )} x\right )} \sqrt{b^{2} - 4 \, a c}}, \frac{2 \,{\left (a b c d - 2 \, a^{2} c e +{\left (b c^{2} d - 2 \, a c^{2} e\right )} x^{2} +{\left (b^{2} c d - 2 \, a b c e\right )} x\right )} \arctan \left (-\frac{\sqrt{-b^{2} + 4 \, a c}{\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) +{\left (2 \, a c d - a b e +{\left (b c d -{\left (b^{2} - 2 \, a c\right )} e\right )} x\right )} \sqrt{-b^{2} + 4 \, a c}}{{\left (a b^{2} c - 4 \, a^{2} c^{2} +{\left (b^{2} c^{2} - 4 \, a c^{3}\right )} x^{2} +{\left (b^{3} c - 4 \, a b c^{2}\right )} x\right )} \sqrt{-b^{2} + 4 \, a c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)*x/(c*x^2 + b*x + a)^2,x, algorithm="fricas")

[Out]

[((a*b*c*d - 2*a^2*c*e + (b*c^2*d - 2*a*c^2*e)*x^2 + (b^2*c*d - 2*a*b*c*e)*x)*lo
g(-(b^3 - 4*a*b*c + 2*(b^2*c - 4*a*c^2)*x - (2*c^2*x^2 + 2*b*c*x + b^2 - 2*a*c)*
sqrt(b^2 - 4*a*c))/(c*x^2 + b*x + a)) + (2*a*c*d - a*b*e + (b*c*d - (b^2 - 2*a*c
)*e)*x)*sqrt(b^2 - 4*a*c))/((a*b^2*c - 4*a^2*c^2 + (b^2*c^2 - 4*a*c^3)*x^2 + (b^
3*c - 4*a*b*c^2)*x)*sqrt(b^2 - 4*a*c)), (2*(a*b*c*d - 2*a^2*c*e + (b*c^2*d - 2*a
*c^2*e)*x^2 + (b^2*c*d - 2*a*b*c*e)*x)*arctan(-sqrt(-b^2 + 4*a*c)*(2*c*x + b)/(b
^2 - 4*a*c)) + (2*a*c*d - a*b*e + (b*c*d - (b^2 - 2*a*c)*e)*x)*sqrt(-b^2 + 4*a*c
))/((a*b^2*c - 4*a^2*c^2 + (b^2*c^2 - 4*a*c^3)*x^2 + (b^3*c - 4*a*b*c^2)*x)*sqrt
(-b^2 + 4*a*c))]

_______________________________________________________________________________________

Sympy [A]  time = 4.52514, size = 379, normalized size = 3.83 \[ - \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} \left (2 a e - b d\right ) \log{\left (x + \frac{- 16 a^{2} c^{2} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} \left (2 a e - b d\right ) + 8 a b^{2} c \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} \left (2 a e - b d\right ) + 2 a b e - b^{4} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} \left (2 a e - b d\right ) - b^{2} d}{4 a c e - 2 b c d} \right )} + \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} \left (2 a e - b d\right ) \log{\left (x + \frac{16 a^{2} c^{2} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} \left (2 a e - b d\right ) - 8 a b^{2} c \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} \left (2 a e - b d\right ) + 2 a b e + b^{4} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} \left (2 a e - b d\right ) - b^{2} d}{4 a c e - 2 b c d} \right )} - \frac{- a b e + 2 a c d + x \left (2 a c e - b^{2} e + b c d\right )}{4 a^{2} c^{2} - a b^{2} c + x^{2} \left (4 a c^{3} - b^{2} c^{2}\right ) + x \left (4 a b c^{2} - b^{3} c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*(e*x+d)/(c*x**2+b*x+a)**2,x)

[Out]

-sqrt(-1/(4*a*c - b**2)**3)*(2*a*e - b*d)*log(x + (-16*a**2*c**2*sqrt(-1/(4*a*c
- b**2)**3)*(2*a*e - b*d) + 8*a*b**2*c*sqrt(-1/(4*a*c - b**2)**3)*(2*a*e - b*d)
+ 2*a*b*e - b**4*sqrt(-1/(4*a*c - b**2)**3)*(2*a*e - b*d) - b**2*d)/(4*a*c*e - 2
*b*c*d)) + sqrt(-1/(4*a*c - b**2)**3)*(2*a*e - b*d)*log(x + (16*a**2*c**2*sqrt(-
1/(4*a*c - b**2)**3)*(2*a*e - b*d) - 8*a*b**2*c*sqrt(-1/(4*a*c - b**2)**3)*(2*a*
e - b*d) + 2*a*b*e + b**4*sqrt(-1/(4*a*c - b**2)**3)*(2*a*e - b*d) - b**2*d)/(4*
a*c*e - 2*b*c*d)) - (-a*b*e + 2*a*c*d + x*(2*a*c*e - b**2*e + b*c*d))/(4*a**2*c*
*2 - a*b**2*c + x**2*(4*a*c**3 - b**2*c**2) + x*(4*a*b*c**2 - b**3*c))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.273135, size = 153, normalized size = 1.55 \[ \frac{2 \,{\left (b d - 2 \, a e\right )} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{{\left (b^{2} - 4 \, a c\right )} \sqrt{-b^{2} + 4 \, a c}} + \frac{b c d x - b^{2} x e + 2 \, a c x e + 2 \, a c d - a b e}{{\left (b^{2} c - 4 \, a c^{2}\right )}{\left (c x^{2} + b x + a\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)*x/(c*x^2 + b*x + a)^2,x, algorithm="giac")

[Out]

2*(b*d - 2*a*e)*arctan((2*c*x + b)/sqrt(-b^2 + 4*a*c))/((b^2 - 4*a*c)*sqrt(-b^2
+ 4*a*c)) + (b*c*d*x - b^2*x*e + 2*a*c*x*e + 2*a*c*d - a*b*e)/((b^2*c - 4*a*c^2)
*(c*x^2 + b*x + a))