3.1175 \(\int \frac{b d+2 c d x}{\left (a+b x+c x^2\right )^3} \, dx\)

Optimal. Leaf size=17 \[ -\frac{d}{2 \left (a+b x+c x^2\right )^2} \]

[Out]

-d/(2*(a + b*x + c*x^2)^2)

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Rubi [A]  time = 0.0128742, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045 \[ -\frac{d}{2 \left (a+b x+c x^2\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

-d/(2*(a + b*x + c*x^2)^2)

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Rubi in Sympy [A]  time = 4.82198, size = 15, normalized size = 0.88 \[ - \frac{d}{2 \left (a + b x + c x^{2}\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-d/(2*(a + b*x + c*x**2)**2)

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Mathematica [A]  time = 0.0116794, size = 16, normalized size = 0.94 \[ -\frac{d}{2 (a+x (b+c x))^2} \]

Antiderivative was successfully verified.

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

[Out]

-d/(2*(a + x*(b + c*x))^2)

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Maple [A]  time = 0.001, size = 16, normalized size = 0.9 \[ -{\frac{d}{2\, \left ( c{x}^{2}+bx+a \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.676904, size = 20, normalized size = 1.18 \[ -\frac{d}{2 \,{\left (c x^{2} + b x + a\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*d*x + b*d)/(c*x^2 + b*x + a)^3,x, algorithm="maxima")

[Out]

-1/2*d/(c*x^2 + b*x + a)^2

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Fricas [A]  time = 0.201001, size = 54, normalized size = 3.18 \[ -\frac{d}{2 \,{\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x +{\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*d*x + b*d)/(c*x^2 + b*x + a)^3,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 3.99537, size = 44, normalized size = 2.59 \[ - \frac{d}{2 a^{2} + 4 a b x + 4 b c x^{3} + 2 c^{2} x^{4} + x^{2} \left (4 a c + 2 b^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.218094, size = 20, normalized size = 1.18 \[ -\frac{d}{2 \,{\left (c x^{2} + b x + a\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*d*x + b*d)/(c*x^2 + b*x + a)^3,x, algorithm="giac")

[Out]

-1/2*d/(c*x^2 + b*x + a)^2