3.1159 \(\int \frac{(b d+2 c d x)^4}{\left (a+b x+c x^2\right )^2} \, dx\)

Optimal. Leaf size=76 \[ -12 c d^4 \sqrt{b^2-4 a c} \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )-\frac{d^4 (b+2 c x)^3}{a+b x+c x^2}+12 c d^4 (b+2 c x) \]

[Out]

12*c*d^4*(b + 2*c*x) - (d^4*(b + 2*c*x)^3)/(a + b*x + c*x^2) - 12*c*Sqrt[b^2 - 4
*a*c]*d^4*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]]

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Rubi [A]  time = 0.134061, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -12 c d^4 \sqrt{b^2-4 a c} \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )-\frac{d^4 (b+2 c x)^3}{a+b x+c x^2}+12 c d^4 (b+2 c x) \]

Antiderivative was successfully verified.

[In]  Int[(b*d + 2*c*d*x)^4/(a + b*x + c*x^2)^2,x]

[Out]

12*c*d^4*(b + 2*c*x) - (d^4*(b + 2*c*x)^3)/(a + b*x + c*x^2) - 12*c*Sqrt[b^2 - 4
*a*c]*d^4*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]]

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Rubi in Sympy [A]  time = 35.289, size = 78, normalized size = 1.03 \[ 12 b c d^{4} + 24 c^{2} d^{4} x - 12 c d^{4} \sqrt{- 4 a c + b^{2}} \operatorname{atanh}{\left (\frac{b + 2 c x}{\sqrt{- 4 a c + b^{2}}} \right )} - \frac{d^{4} \left (b + 2 c x\right )^{3}}{a + b x + c x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((2*c*d*x+b*d)**4/(c*x**2+b*x+a)**2,x)

[Out]

12*b*c*d**4 + 24*c**2*d**4*x - 12*c*d**4*sqrt(-4*a*c + b**2)*atanh((b + 2*c*x)/s
qrt(-4*a*c + b**2)) - d**4*(b + 2*c*x)**3/(a + b*x + c*x**2)

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Mathematica [A]  time = 0.102854, size = 77, normalized size = 1.01 \[ d^4 \left (-\frac{\left (b^2-4 a c\right ) (b+2 c x)}{a+x (b+c x)}-12 c \sqrt{4 a c-b^2} \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )+16 c^2 x\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(b*d + 2*c*d*x)^4/(a + b*x + c*x^2)^2,x]

[Out]

d^4*(16*c^2*x - ((b^2 - 4*a*c)*(b + 2*c*x))/(a + x*(b + c*x)) - 12*c*Sqrt[-b^2 +
4*a*c]*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])

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Maple [A]  time = 0.01, size = 133, normalized size = 1.8 \[ 16\,{c}^{2}{d}^{4}x+8\,{\frac{{d}^{4}a{c}^{2}x}{c{x}^{2}+bx+a}}-2\,{\frac{{d}^{4}x{b}^{2}c}{c{x}^{2}+bx+a}}+4\,{\frac{{d}^{4}abc}{c{x}^{2}+bx+a}}-{\frac{{d}^{4}{b}^{3}}{c{x}^{2}+bx+a}}-12\,{d}^{4}c\sqrt{4\,ac-{b}^{2}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2*c*d*x+b*d)^4/(c*x^2+b*x+a)^2,x)

[Out]

16*c^2*d^4*x+8*d^4/(c*x^2+b*x+a)*a*c^2*x-2*d^4/(c*x^2+b*x+a)*x*b^2*c+4*d^4/(c*x^
2+b*x+a)*a*b*c-d^4/(c*x^2+b*x+a)*b^3-12*d^4*c*(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*d*x + b*d)^4/(c*x^2 + b*x + a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.219527, size = 1, normalized size = 0.01 \[ \left [\frac{16 \, c^{3} d^{4} x^{3} + 16 \, b c^{2} d^{4} x^{2} - 2 \,{\left (b^{2} c - 12 \, a c^{2}\right )} d^{4} x -{\left (b^{3} - 4 \, a b c\right )} d^{4} + 6 \,{\left (c^{2} d^{4} x^{2} + b c d^{4} x + a c d^{4}\right )} \sqrt{b^{2} - 4 \, a c} \log \left (\frac{2 \, c^{2} x^{2} + 2 \, b c x + b^{2} - 2 \, a c - \sqrt{b^{2} - 4 \, a c}{\left (2 \, c x + b\right )}}{c x^{2} + b x + a}\right )}{c x^{2} + b x + a}, \frac{16 \, c^{3} d^{4} x^{3} + 16 \, b c^{2} d^{4} x^{2} - 2 \,{\left (b^{2} c - 12 \, a c^{2}\right )} d^{4} x -{\left (b^{3} - 4 \, a b c\right )} d^{4} - 12 \,{\left (c^{2} d^{4} x^{2} + b c d^{4} x + a c d^{4}\right )} \sqrt{-b^{2} + 4 \, a c} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{c x^{2} + b x + a}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*d*x + b*d)^4/(c*x^2 + b*x + a)^2,x, algorithm="fricas")

[Out]

[(16*c^3*d^4*x^3 + 16*b*c^2*d^4*x^2 - 2*(b^2*c - 12*a*c^2)*d^4*x - (b^3 - 4*a*b*
c)*d^4 + 6*(c^2*d^4*x^2 + b*c*d^4*x + a*c*d^4)*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)))/(c*
x^2 + b*x + a), (16*c^3*d^4*x^3 + 16*b*c^2*d^4*x^2 - 2*(b^2*c - 12*a*c^2)*d^4*x
- (b^3 - 4*a*b*c)*d^4 - 12*(c^2*d^4*x^2 + b*c*d^4*x + a*c*d^4)*sqrt(-b^2 + 4*a*c
)*arctan((2*c*x + b)/sqrt(-b^2 + 4*a*c)))/(c*x^2 + b*x + a)]

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Sympy [A]  time = 4.94295, size = 173, normalized size = 2.28 \[ 16 c^{2} d^{4} x + c d^{4} \sqrt{- 144 a c + 36 b^{2}} \log{\left (x + \frac{6 b c d^{4} - c d^{4} \sqrt{- 144 a c + 36 b^{2}}}{12 c^{2} d^{4}} \right )} - c d^{4} \sqrt{- 144 a c + 36 b^{2}} \log{\left (x + \frac{6 b c d^{4} + c d^{4} \sqrt{- 144 a c + 36 b^{2}}}{12 c^{2} d^{4}} \right )} + \frac{4 a b c d^{4} - b^{3} d^{4} + x \left (8 a c^{2} d^{4} - 2 b^{2} c d^{4}\right )}{a + b x + c x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*d*x+b*d)**4/(c*x**2+b*x+a)**2,x)

[Out]

16*c**2*d**4*x + c*d**4*sqrt(-144*a*c + 36*b**2)*log(x + (6*b*c*d**4 - c*d**4*sq
rt(-144*a*c + 36*b**2))/(12*c**2*d**4)) - c*d**4*sqrt(-144*a*c + 36*b**2)*log(x
+ (6*b*c*d**4 + c*d**4*sqrt(-144*a*c + 36*b**2))/(12*c**2*d**4)) + (4*a*b*c*d**4
- b**3*d**4 + x*(8*a*c**2*d**4 - 2*b**2*c*d**4))/(a + b*x + c*x**2)

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GIAC/XCAS [A]  time = 0.214502, size = 151, normalized size = 1.99 \[ 16 \, c^{2} d^{4} x + \frac{12 \,{\left (b^{2} c d^{4} - 4 \, a c^{2} d^{4}\right )} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{\sqrt{-b^{2} + 4 \, a c}} - \frac{2 \, b^{2} c d^{4} x - 8 \, a c^{2} d^{4} x + b^{3} d^{4} - 4 \, a b c d^{4}}{c x^{2} + b x + a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*d*x + b*d)^4/(c*x^2 + b*x + a)^2,x, algorithm="giac")

[Out]

16*c^2*d^4*x + 12*(b^2*c*d^4 - 4*a*c^2*d^4)*arctan((2*c*x + b)/sqrt(-b^2 + 4*a*c
))/sqrt(-b^2 + 4*a*c) - (2*b^2*c*d^4*x - 8*a*c^2*d^4*x + b^3*d^4 - 4*a*b*c*d^4)/
(c*x^2 + b*x + a)