Optimal. Leaf size=14 \[ \frac{(c+d x)^3}{3 d} \]
[Out]
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Rubi [A] time = 0.0203599, antiderivative size = 14, 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 \[ \frac{(c+d x)^3}{3 d} \]
Antiderivative was successfully verified.
[In] Int[(a*c + (b*c + a*d)*x + b*d*x^2)^2/(a + b*x)^2,x]
[Out]
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Rubi in Sympy [A] time = 7.64738, size = 8, normalized size = 0.57 \[ \frac{\left (c + d x\right )^{3}}{3 d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((a*c+(a*d+b*c)*x+b*d*x**2)**2/(b*x+a)**2,x)
[Out]
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Mathematica [A] time = 0.00259954, size = 14, normalized size = 1. \[ \frac{(c+d x)^3}{3 d} \]
Antiderivative was successfully verified.
[In] Integrate[(a*c + (b*c + a*d)*x + b*d*x^2)^2/(a + b*x)^2,x]
[Out]
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Maple [A] time = 0.002, size = 13, normalized size = 0.9 \[{\frac{ \left ( dx+c \right ) ^{3}}{3\,d}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((a*c+(a*d+b*c)*x+x^2*b*d)^2/(b*x+a)^2,x)
[Out]
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Maxima [A] time = 0.724111, size = 27, normalized size = 1.93 \[ \frac{1}{3} \, d^{2} x^{3} + c d x^{2} + c^{2} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*d*x^2 + a*c + (b*c + a*d)*x)^2/(b*x + a)^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.199503, size = 27, normalized size = 1.93 \[ \frac{1}{3} \, d^{2} x^{3} + c d x^{2} + c^{2} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*d*x^2 + a*c + (b*c + a*d)*x)^2/(b*x + a)^2,x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.21576, size = 19, normalized size = 1.36 \[ c^{2} x + c d x^{2} + \frac{d^{2} x^{3}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a*c+(a*d+b*c)*x+b*d*x**2)**2/(b*x+a)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.210549, size = 113, normalized size = 8.07 \[ \frac{{\left (\frac{3 \, b^{2} c^{2}}{{\left (b x + a\right )}^{2}} + \frac{3 \, b c d}{b x + a} - \frac{6 \, a b c d}{{\left (b x + a\right )}^{2}} - \frac{3 \, a d^{2}}{b x + a} + \frac{3 \, a^{2} d^{2}}{{\left (b x + a\right )}^{2}} + d^{2}\right )}{\left (b x + a\right )}^{3}}{3 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*d*x^2 + a*c + (b*c + a*d)*x)^2/(b*x + a)^2,x, algorithm="giac")
[Out]