3.1173 \(\int \frac{(b d+2 c d x)^3}{\left (a+b x+c x^2\right )^3} \, dx\)

Optimal. Leaf size=37 \[ -\frac{d^3 (b+2 c x)^4}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2} \]

[Out]

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

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Rubi [A]  time = 0.0506991, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042 \[ -\frac{d^3 (b+2 c x)^4}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 14.5795, size = 34, normalized size = 0.92 \[ - \frac{d^{3} \left (b + 2 c x\right )^{4}}{2 \left (- 4 a c + b^{2}\right ) \left (a + b x + c x^{2}\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0507944, size = 38, normalized size = 1.03 \[ -\frac{d^3 \left (4 c \left (a+2 c x^2\right )+b^2+8 b c x\right )}{2 (a+x (b+c x))^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.013, size = 40, normalized size = 1.1 \[{\frac{{d}^{3}}{ \left ( c{x}^{2}+bx+a \right ) ^{2}} \left ( -4\,{c}^{2}{x}^{2}-4\,bxc-2\,ac-{\frac{{b}^{2}}{2}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.700991, size = 96, normalized size = 2.59 \[ -\frac{8 \, c^{2} d^{3} x^{2} + 8 \, b c d^{3} x +{\left (b^{2} + 4 \, a c\right )} d^{3}}{2 \,{\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x +{\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(8*c^2*d^3*x^2 + 8*b*c*d^3*x + (b^2 + 4*a*c)*d^3)/(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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Fricas [A]  time = 0.204792, size = 96, normalized size = 2.59 \[ -\frac{8 \, c^{2} d^{3} x^{2} + 8 \, b c d^{3} x +{\left (b^{2} + 4 \, a c\right )} d^{3}}{2 \,{\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x +{\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(8*c^2*d^3*x^2 + 8*b*c*d^3*x + (b^2 + 4*a*c)*d^3)/(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 = 7.36427, size = 80, normalized size = 2.16 \[ - \frac{4 a c d^{3} + b^{2} d^{3} + 8 b c d^{3} x + 8 c^{2} d^{3} x^{2}}{2 a^{2} + 4 a b x + 4 b c x^{3} + 2 c^{2} x^{4} + x^{2} \left (4 a c + 2 b^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(4*a*c*d**3 + b**2*d**3 + 8*b*c*d**3*x + 8*c**2*d**3*x**2)/(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 = 0.218612, size = 65, normalized size = 1.76 \[ -\frac{8 \, c^{2} d^{3} x^{2} + 8 \, b c d^{3} x + b^{2} d^{3} + 4 \, a c d^{3}}{2 \,{\left (c x^{2} + b x + a\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(8*c^2*d^3*x^2 + 8*b*c*d^3*x + b^2*d^3 + 4*a*c*d^3)/(c*x^2 + b*x + a)^2