Optimal. Leaf size=28 \[ \frac{(a+b x)^2}{2 (c+d x)^2 (b c-a d)} \]
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Rubi [A] time = 0.0313347, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069 \[ \frac{(a+b x)^2}{2 (c+d x)^2 (b c-a d)} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x)^4/(a*c + (b*c + a*d)*x + b*d*x^2)^3,x]
[Out]
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Rubi in Sympy [A] time = 9.47573, size = 22, normalized size = 0.79 \[ - \frac{\left (a + b x\right )^{2}}{2 \left (c + d x\right )^{2} \left (a d - b c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)**4/(a*c+(a*d+b*c)*x+b*d*x**2)**3,x)
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Mathematica [A] time = 0.0172292, size = 26, normalized size = 0.93 \[ -\frac{a d+b (c+2 d x)}{2 d^2 (c+d x)^2} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x)^4/(a*c + (b*c + a*d)*x + b*d*x^2)^3,x]
[Out]
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Maple [A] time = 0.007, size = 35, normalized size = 1.3 \[ -{\frac{b}{{d}^{2} \left ( dx+c \right ) }}-{\frac{ad-bc}{2\,{d}^{2} \left ( dx+c \right ) ^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)^4/(a*c+(a*d+b*c)*x+x^2*b*d)^3,x)
[Out]
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Maxima [A] time = 0.763088, size = 51, normalized size = 1.82 \[ -\frac{2 \, b d x + b c + a d}{2 \,{\left (d^{4} x^{2} + 2 \, c d^{3} x + c^{2} d^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^4/(b*d*x^2 + a*c + (b*c + a*d)*x)^3,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.200592, size = 51, normalized size = 1.82 \[ -\frac{2 \, b d x + b c + a d}{2 \,{\left (d^{4} x^{2} + 2 \, c d^{3} x + c^{2} d^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^4/(b*d*x^2 + a*c + (b*c + a*d)*x)^3,x, algorithm="fricas")
[Out]
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Sympy [A] time = 1.87418, size = 39, normalized size = 1.39 \[ - \frac{a d + b c + 2 b d x}{2 c^{2} d^{2} + 4 c d^{3} x + 2 d^{4} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x+a)**4/(a*c+(a*d+b*c)*x+b*d*x**2)**3,x)
[Out]
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GIAC/XCAS [A] time = 0.214342, size = 32, normalized size = 1.14 \[ -\frac{2 \, b d x + b c + a d}{2 \,{\left (d x + c\right )}^{2} d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^4/(b*d*x^2 + a*c + (b*c + a*d)*x)^3,x, algorithm="giac")
[Out]