3.1174 \(\int \frac{(b d+2 c d x)^2}{\left (a+b x+c x^2\right )^3} \, dx\)

Optimal. Leaf size=100 \[ \frac{4 c^2 d^2 \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}}-\frac{c d^2 (b+2 c x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac{d^2 (b+2 c x)}{2 \left (a+b x+c x^2\right )^2} \]

[Out]

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

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Rubi [A]  time = 0.133189, antiderivative size = 100, 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 \[ \frac{4 c^2 d^2 \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}}-\frac{c d^2 (b+2 c x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac{d^2 (b+2 c x)}{2 \left (a+b x+c x^2\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[(b*d + 2*c*d*x)^2/(a + b*x + c*x^2)^3,x]

[Out]

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

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Rubi in Sympy [A]  time = 24.3231, size = 94, normalized size = 0.94 \[ \frac{4 c^{2} d^{2} \operatorname{atanh}{\left (\frac{b + 2 c x}{\sqrt{- 4 a c + b^{2}}} \right )}}{\left (- 4 a c + b^{2}\right )^{\frac{3}{2}}} - \frac{c d^{2} \left (b + 2 c x\right )}{\left (- 4 a c + b^{2}\right ) \left (a + b x + c x^{2}\right )} - \frac{d^{2} \left (b + 2 c x\right )}{2 \left (a + b x + c x^{2}\right )^{2}} \]

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)**3,x)

[Out]

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

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Mathematica [A]  time = 0.175237, size = 98, normalized size = 0.98 \[ d^2 \left (\frac{4 c^2 \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\left (4 a c-b^2\right )^{3/2}}-\frac{(b+2 c x) \left (2 c \left (c x^2-a\right )+b^2+2 b c x\right )}{2 \left (b^2-4 a c\right ) (a+x (b+c x))^2}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(b*d + 2*c*d*x)^2/(a + b*x + c*x^2)^3,x]

[Out]

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

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Maple [B]  time = 0.014, size = 245, normalized size = 2.5 \[ 2\,{\frac{{c}^{3}{d}^{2}{x}^{3}}{ \left ( c{x}^{2}+bx+a \right ) ^{2} \left ( 4\,ac-{b}^{2} \right ) }}+3\,{\frac{b{c}^{2}{d}^{2}{x}^{2}}{ \left ( c{x}^{2}+bx+a \right ) ^{2} \left ( 4\,ac-{b}^{2} \right ) }}-2\,{\frac{{c}^{2}{d}^{2}xa}{ \left ( c{x}^{2}+bx+a \right ) ^{2} \left ( 4\,ac-{b}^{2} \right ) }}+2\,{\frac{c{d}^{2}x{b}^{2}}{ \left ( c{x}^{2}+bx+a \right ) ^{2} \left ( 4\,ac-{b}^{2} \right ) }}-{\frac{abc{d}^{2}}{ \left ( c{x}^{2}+bx+a \right ) ^{2} \left ( 4\,ac-{b}^{2} \right ) }}+{\frac{{b}^{3}{d}^{2}}{2\, \left ( c{x}^{2}+bx+a \right ) ^{2} \left ( 4\,ac-{b}^{2} \right ) }}+4\,{\frac{{c}^{2}{d}^{2}}{ \left ( 4\,ac-{b}^{2} \right ) ^{3/2}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2*c*d*x+b*d)^2/(c*x^2+b*x+a)^3,x)

[Out]

2*d^2/(c*x^2+b*x+a)^2*c^3/(4*a*c-b^2)*x^3+3*d^2/(c*x^2+b*x+a)^2*b*c^2/(4*a*c-b^2
)*x^2-2*d^2/(c*x^2+b*x+a)^2*c^2/(4*a*c-b^2)*x*a+2*d^2/(c*x^2+b*x+a)^2*c/(4*a*c-b
^2)*x*b^2-d^2/(c*x^2+b*x+a)^2*b/(4*a*c-b^2)*a*c+1/2*d^2/(c*x^2+b*x+a)^2*b^3/(4*a
*c-b^2)+4*d^2*c^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((2*c*d*x + b*d)^2/(c*x^2 + b*x + a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.219888, size = 1, normalized size = 0.01 \[ \left [-\frac{4 \,{\left (c^{4} d^{2} x^{4} + 2 \, b c^{3} d^{2} x^{3} + 2 \, a b c^{2} d^{2} x + a^{2} c^{2} d^{2} +{\left (b^{2} c^{2} + 2 \, a c^{3}\right )} d^{2} x^{2}\right )} \log \left (-\frac{b^{3} - 4 \, a b c + 2 \,{\left (b^{2} c - 4 \, a c^{2}\right )} x -{\left (2 \, c^{2} x^{2} + 2 \, b c x + b^{2} - 2 \, a c\right )} \sqrt{b^{2} - 4 \, a c}}{c x^{2} + b x + a}\right ) +{\left (4 \, c^{3} d^{2} x^{3} + 6 \, b c^{2} d^{2} x^{2} + 4 \,{\left (b^{2} c - a c^{2}\right )} d^{2} x +{\left (b^{3} - 2 \, a b c\right )} d^{2}\right )} \sqrt{b^{2} - 4 \, a c}}{2 \,{\left ({\left (b^{2} c^{2} - 4 \, a c^{3}\right )} x^{4} + a^{2} b^{2} - 4 \, a^{3} c + 2 \,{\left (b^{3} c - 4 \, a b c^{2}\right )} x^{3} +{\left (b^{4} - 2 \, a b^{2} c - 8 \, a^{2} c^{2}\right )} x^{2} + 2 \,{\left (a b^{3} - 4 \, a^{2} b c\right )} x\right )} \sqrt{b^{2} - 4 \, a c}}, -\frac{8 \,{\left (c^{4} d^{2} x^{4} + 2 \, b c^{3} d^{2} x^{3} + 2 \, a b c^{2} d^{2} x + a^{2} c^{2} d^{2} +{\left (b^{2} c^{2} + 2 \, a c^{3}\right )} d^{2} x^{2}\right )} \arctan \left (-\frac{\sqrt{-b^{2} + 4 \, a c}{\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) +{\left (4 \, c^{3} d^{2} x^{3} + 6 \, b c^{2} d^{2} x^{2} + 4 \,{\left (b^{2} c - a c^{2}\right )} d^{2} x +{\left (b^{3} - 2 \, a b c\right )} d^{2}\right )} \sqrt{-b^{2} + 4 \, a c}}{2 \,{\left ({\left (b^{2} c^{2} - 4 \, a c^{3}\right )} x^{4} + a^{2} b^{2} - 4 \, a^{3} c + 2 \,{\left (b^{3} c - 4 \, a b c^{2}\right )} x^{3} +{\left (b^{4} - 2 \, a b^{2} c - 8 \, a^{2} c^{2}\right )} x^{2} + 2 \,{\left (a b^{3} - 4 \, a^{2} b c\right )} x\right )} \sqrt{-b^{2} + 4 \, a c}}\right ] \]

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)^3,x, algorithm="fricas")

[Out]

[-1/2*(4*(c^4*d^2*x^4 + 2*b*c^3*d^2*x^3 + 2*a*b*c^2*d^2*x + a^2*c^2*d^2 + (b^2*c
^2 + 2*a*c^3)*d^2*x^2)*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)) + (4*c^3*d^2*x^3
+ 6*b*c^2*d^2*x^2 + 4*(b^2*c - a*c^2)*d^2*x + (b^3 - 2*a*b*c)*d^2)*sqrt(b^2 - 4*
a*c))/(((b^2*c^2 - 4*a*c^3)*x^4 + a^2*b^2 - 4*a^3*c + 2*(b^3*c - 4*a*b*c^2)*x^3
+ (b^4 - 2*a*b^2*c - 8*a^2*c^2)*x^2 + 2*(a*b^3 - 4*a^2*b*c)*x)*sqrt(b^2 - 4*a*c)
), -1/2*(8*(c^4*d^2*x^4 + 2*b*c^3*d^2*x^3 + 2*a*b*c^2*d^2*x + a^2*c^2*d^2 + (b^2
*c^2 + 2*a*c^3)*d^2*x^2)*arctan(-sqrt(-b^2 + 4*a*c)*(2*c*x + b)/(b^2 - 4*a*c)) +
(4*c^3*d^2*x^3 + 6*b*c^2*d^2*x^2 + 4*(b^2*c - a*c^2)*d^2*x + (b^3 - 2*a*b*c)*d^
2)*sqrt(-b^2 + 4*a*c))/(((b^2*c^2 - 4*a*c^3)*x^4 + a^2*b^2 - 4*a^3*c + 2*(b^3*c
- 4*a*b*c^2)*x^3 + (b^4 - 2*a*b^2*c - 8*a^2*c^2)*x^2 + 2*(a*b^3 - 4*a^2*b*c)*x)*
sqrt(-b^2 + 4*a*c))]

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Sympy [A]  time = 6.2294, size = 430, normalized size = 4.3 \[ - 2 c^{2} d^{2} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} \log{\left (x + \frac{- 32 a^{2} c^{4} d^{2} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} + 16 a b^{2} c^{3} d^{2} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} - 2 b^{4} c^{2} d^{2} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} + 2 b c^{2} d^{2}}{4 c^{3} d^{2}} \right )} + 2 c^{2} d^{2} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} \log{\left (x + \frac{32 a^{2} c^{4} d^{2} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} - 16 a b^{2} c^{3} d^{2} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} + 2 b^{4} c^{2} d^{2} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} + 2 b c^{2} d^{2}}{4 c^{3} d^{2}} \right )} + \frac{- 2 a b c d^{2} + b^{3} d^{2} + 6 b c^{2} d^{2} x^{2} + 4 c^{3} d^{2} x^{3} + x \left (- 4 a c^{2} d^{2} + 4 b^{2} c d^{2}\right )}{8 a^{3} c - 2 a^{2} b^{2} + x^{4} \left (8 a c^{3} - 2 b^{2} c^{2}\right ) + x^{3} \left (16 a b c^{2} - 4 b^{3} c\right ) + x^{2} \left (16 a^{2} c^{2} + 4 a b^{2} c - 2 b^{4}\right ) + x \left (16 a^{2} b c - 4 a b^{3}\right )} \]

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)**3,x)

[Out]

-2*c**2*d**2*sqrt(-1/(4*a*c - b**2)**3)*log(x + (-32*a**2*c**4*d**2*sqrt(-1/(4*a
*c - b**2)**3) + 16*a*b**2*c**3*d**2*sqrt(-1/(4*a*c - b**2)**3) - 2*b**4*c**2*d*
*2*sqrt(-1/(4*a*c - b**2)**3) + 2*b*c**2*d**2)/(4*c**3*d**2)) + 2*c**2*d**2*sqrt
(-1/(4*a*c - b**2)**3)*log(x + (32*a**2*c**4*d**2*sqrt(-1/(4*a*c - b**2)**3) - 1
6*a*b**2*c**3*d**2*sqrt(-1/(4*a*c - b**2)**3) + 2*b**4*c**2*d**2*sqrt(-1/(4*a*c
- b**2)**3) + 2*b*c**2*d**2)/(4*c**3*d**2)) + (-2*a*b*c*d**2 + b**3*d**2 + 6*b*c
**2*d**2*x**2 + 4*c**3*d**2*x**3 + x*(-4*a*c**2*d**2 + 4*b**2*c*d**2))/(8*a**3*c
- 2*a**2*b**2 + x**4*(8*a*c**3 - 2*b**2*c**2) + x**3*(16*a*b*c**2 - 4*b**3*c) +
x**2*(16*a**2*c**2 + 4*a*b**2*c - 2*b**4) + x*(16*a**2*b*c - 4*a*b**3))

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GIAC/XCAS [A]  time = 0.217478, size = 181, normalized size = 1.81 \[ -\frac{4 \, c^{2} d^{2} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{{\left (b^{2} - 4 \, a c\right )} \sqrt{-b^{2} + 4 \, a c}} - \frac{4 \, c^{3} d^{2} x^{3} + 6 \, b c^{2} d^{2} x^{2} + 4 \, b^{2} c d^{2} x - 4 \, a c^{2} d^{2} x + b^{3} d^{2} - 2 \, a b c d^{2}}{2 \,{\left (c x^{2} + b x + a\right )}^{2}{\left (b^{2} - 4 \, a c\right )}} \]

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)^3,x, algorithm="giac")

[Out]

-4*c^2*d^2*arctan((2*c*x + b)/sqrt(-b^2 + 4*a*c))/((b^2 - 4*a*c)*sqrt(-b^2 + 4*a
*c)) - 1/2*(4*c^3*d^2*x^3 + 6*b*c^2*d^2*x^2 + 4*b^2*c*d^2*x - 4*a*c^2*d^2*x + b^
3*d^2 - 2*a*b*c*d^2)/((c*x^2 + b*x + a)^2*(b^2 - 4*a*c))