3.1149 \(\int \frac{(b d+2 c d x)^2}{a+b x+c x^2} \, dx\)

Optimal. Leaf size=49 \[ 2 d^2 (b+2 c x)-2 d^2 \sqrt{b^2-4 a c} \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0804815, antiderivative size = 49, 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 \[ 2 d^2 (b+2 c x)-2 d^2 \sqrt{b^2-4 a c} \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 19.4544, size = 51, normalized size = 1.04 \[ 2 b d^{2} + 4 c d^{2} x - 2 d^{2} \sqrt{- 4 a c + b^{2}} \operatorname{atanh}{\left (\frac{b + 2 c x}{\sqrt{- 4 a c + b^{2}}} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0492054, size = 47, normalized size = 0.96 \[ d^2 \left (4 c x-2 \sqrt{4 a c-b^2} \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.004, size = 88, normalized size = 1.8 \[ 4\,c{d}^{2}x-8\,{\frac{a{d}^{2}c}{\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }+2\,{\frac{{b}^{2}{d}^{2}}{\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4*c*d^2*x-8*d^2/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a*c+2*d^2/
(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b^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*d*x + b*d)^2/(c*x^2 + b*x + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.221089, size = 1, normalized size = 0.02 \[ \left [4 \, c d^{2} x + \sqrt{b^{2} - 4 \, a c} d^{2} \log \left (\frac{2 \, c^{2} x^{2} + 2 \, b c x + b^{2} - 2 \, a c - \sqrt{b^{2} - 4 \, a c}{\left (2 \, c x + b\right )}}{c x^{2} + b x + a}\right ), 4 \, c d^{2} x - 2 \, \sqrt{-b^{2} + 4 \, a c} d^{2} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.78211, size = 99, normalized size = 2.02 \[ 4 c d^{2} x + d^{2} \sqrt{- 4 a c + b^{2}} \log{\left (x + \frac{b d^{2} - d^{2} \sqrt{- 4 a c + b^{2}}}{2 c d^{2}} \right )} - d^{2} \sqrt{- 4 a c + b^{2}} \log{\left (x + \frac{b d^{2} + d^{2} \sqrt{- 4 a c + b^{2}}}{2 c d^{2}} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.215538, size = 77, normalized size = 1.57 \[ 4 \, c d^{2} x + \frac{2 \,{\left (b^{2} d^{2} - 4 \, a c d^{2}\right )} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{\sqrt{-b^{2} + 4 \, a c}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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