3.1152 \(\int \frac{1}{(b d+2 c d x)^2 \left (a+b x+c x^2\right )} \, dx\)

Optimal. Leaf size=61 \[ \frac{2}{d^2 \left (b^2-4 a c\right ) (b+2 c x)}-\frac{2 \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{d^2 \left (b^2-4 a c\right )^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.108927, antiderivative size = 61, 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 \[ \frac{2}{d^2 \left (b^2-4 a c\right ) (b+2 c x)}-\frac{2 \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{d^2 \left (b^2-4 a c\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/((b*d + 2*c*d*x)^2*(a + b*x + c*x^2)),x]

[Out]

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

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Rubi in Sympy [A]  time = 20.4485, size = 56, normalized size = 0.92 \[ - \frac{2 \operatorname{atanh}{\left (\frac{b + 2 c x}{\sqrt{- 4 a c + b^{2}}} \right )}}{d^{2} \left (- 4 a c + b^{2}\right )^{\frac{3}{2}}} + \frac{2}{d^{2} \left (b + 2 c x\right ) \left (- 4 a c + b^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(2*c*d*x+b*d)**2/(c*x**2+b*x+a),x)

[Out]

-2*atanh((b + 2*c*x)/sqrt(-4*a*c + b**2))/(d**2*(-4*a*c + b**2)**(3/2)) + 2/(d**
2*(b + 2*c*x)*(-4*a*c + b**2))

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Mathematica [A]  time = 0.0938424, size = 63, normalized size = 1.03 \[ \frac{\frac{2}{\left (b^2-4 a c\right ) (b+2 c x)}-\frac{2 \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\left (4 a c-b^2\right )^{3/2}}}{d^2} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((b*d + 2*c*d*x)^2*(a + b*x + c*x^2)),x]

[Out]

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

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Maple [A]  time = 0.007, size = 64, normalized size = 1.1 \[ -2\,{\frac{1}{{d}^{2} \left ( 4\,ac-{b}^{2} \right ) \left ( 2\,cx+b \right ) }}-2\,{\frac{1}{{d}^{2} \left ( 4\,ac-{b}^{2} \right ) ^{3/2}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(2*c*d*x+b*d)^2/(c*x^2+b*x+a),x)

[Out]

-2/d^2/(4*a*c-b^2)/(2*c*x+b)-2/d^2/(4*a*c-b^2)^(3/2)*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(1/((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 = 0.219403, size = 1, normalized size = 0.02 \[ \left [-\frac{{\left (2 \, c x + b\right )} \log \left (\frac{b^{3} - 4 \, a b c + 2 \,{\left (b^{2} c - 4 \, a c^{2}\right )} x +{\left (2 \, c^{2} x^{2} + 2 \, b c x + b^{2} - 2 \, a c\right )} \sqrt{b^{2} - 4 \, a c}}{c x^{2} + b x + a}\right ) - 2 \, \sqrt{b^{2} - 4 \, a c}}{{\left (2 \,{\left (b^{2} c - 4 \, a c^{2}\right )} d^{2} x +{\left (b^{3} - 4 \, a b c\right )} d^{2}\right )} \sqrt{b^{2} - 4 \, a c}}, \frac{2 \,{\left ({\left (2 \, c x + b\right )} \arctan \left (-\frac{\sqrt{-b^{2} + 4 \, a c}{\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) + \sqrt{-b^{2} + 4 \, a c}\right )}}{{\left (2 \,{\left (b^{2} c - 4 \, a c^{2}\right )} d^{2} x +{\left (b^{3} - 4 \, a b c\right )} d^{2}\right )} \sqrt{-b^{2} + 4 \, a c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((2*c*d*x + b*d)^2*(c*x^2 + b*x + a)),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 3.82077, size = 240, normalized size = 3.93 \[ - \frac{2}{4 a b c d^{2} - b^{3} d^{2} + x \left (8 a c^{2} d^{2} - 2 b^{2} c d^{2}\right )} + \frac{\sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} \log{\left (x + \frac{- 16 a^{2} c^{2} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} + 8 a b^{2} c \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} - b^{4} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} + b}{2 c} \right )}}{d^{2}} - \frac{\sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} \log{\left (x + \frac{16 a^{2} c^{2} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} - 8 a b^{2} c \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} + b^{4} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} + b}{2 c} \right )}}{d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(2*c*d*x+b*d)**2/(c*x**2+b*x+a),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.216075, size = 157, normalized size = 2.57 \[ \frac{2 \, c^{2} d^{3}}{{\left (b^{2} c^{2} d^{4} - 4 \, a c^{3} d^{4}\right )}{\left (2 \, c d x + b d\right )}} + \frac{2 \, \arctan \left (\frac{\frac{b^{2} d}{2 \, c d x + b d} - \frac{4 \, a c d}{2 \, c d x + b d}}{\sqrt{-b^{2} + 4 \, a c}}\right )}{{\left (b^{2} - 4 \, a c\right )} \sqrt{-b^{2} + 4 \, a c} d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((2*c*d*x + b*d)^2*(c*x^2 + b*x + a)),x, algorithm="giac")

[Out]

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