3.1535 \(\int \frac{(b+2 c x) (d+e x)}{\left (a+b x+c x^2\right )^2} \, dx\)

Optimal. Leaf size=55 \[ -\frac{2 e \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\sqrt{b^2-4 a c}}-\frac{d+e x}{a+b x+c x^2} \]

[Out]

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

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Rubi [A]  time = 0.084856, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ -\frac{2 e \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\sqrt{b^2-4 a c}}-\frac{d+e x}{a+b x+c x^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 16.1854, size = 51, normalized size = 0.93 \[ - \frac{2 e \operatorname{atanh}{\left (\frac{b + 2 c x}{\sqrt{- 4 a c + b^{2}}} \right )}}{\sqrt{- 4 a c + b^{2}}} - \frac{d + e x}{a + b x + c x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.053114, size = 58, normalized size = 1.05 \[ \frac{2 e \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\sqrt{4 a c-b^2}}-\frac{d+e x}{a+x (b+c x)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.01, size = 58, normalized size = 1.1 \[{\frac{-ex-d}{c{x}^{2}+bx+a}}+2\,{\frac{e}{\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.319274, size = 1, normalized size = 0.02 \[ \left [\frac{{\left (c e x^{2} + b e x + a e\right )} \log \left (-\frac{b^{3} - 4 \, a b c + 2 \,{\left (b^{2} c - 4 \, a c^{2}\right )} x -{\left (2 \, c^{2} x^{2} + 2 \, b c x + b^{2} - 2 \, a c\right )} \sqrt{b^{2} - 4 \, a c}}{c x^{2} + b x + a}\right ) - \sqrt{b^{2} - 4 \, a c}{\left (e x + d\right )}}{{\left (c x^{2} + b x + a\right )} \sqrt{b^{2} - 4 \, a c}}, \frac{2 \,{\left (c e x^{2} + b e x + a e\right )} \arctan \left (-\frac{\sqrt{-b^{2} + 4 \, a c}{\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) - \sqrt{-b^{2} + 4 \, a c}{\left (e x + d\right )}}{{\left (c x^{2} + b x + a\right )} \sqrt{-b^{2} + 4 \, a c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 5.28545, size = 156, normalized size = 2.84 \[ - e \sqrt{- \frac{1}{4 a c - b^{2}}} \log{\left (x + \frac{- 4 a c e \sqrt{- \frac{1}{4 a c - b^{2}}} + b^{2} e \sqrt{- \frac{1}{4 a c - b^{2}}} + b e}{2 c e} \right )} + e \sqrt{- \frac{1}{4 a c - b^{2}}} \log{\left (x + \frac{4 a c e \sqrt{- \frac{1}{4 a c - b^{2}}} - b^{2} e \sqrt{- \frac{1}{4 a c - b^{2}}} + b e}{2 c e} \right )} - \frac{d + e x}{a + b x + c x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-e*sqrt(-1/(4*a*c - b**2))*log(x + (-4*a*c*e*sqrt(-1/(4*a*c - b**2)) + b**2*e*sq
rt(-1/(4*a*c - b**2)) + b*e)/(2*c*e)) + e*sqrt(-1/(4*a*c - b**2))*log(x + (4*a*c
*e*sqrt(-1/(4*a*c - b**2)) - b**2*e*sqrt(-1/(4*a*c - b**2)) + b*e)/(2*c*e)) - (d
+ e*x)/(a + b*x + c*x**2)

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GIAC/XCAS [A]  time = 0.270548, size = 77, normalized size = 1.4 \[ \frac{2 \, \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right ) e}{\sqrt{-b^{2} + 4 \, a c}} - \frac{x e + d}{c x^{2} + b x + a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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