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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]