Optimal. Leaf size=65 \[ \frac{e (a+b x)^4 (b d-a e)}{2 b^3}+\frac{(a+b x)^3 (b d-a e)^2}{3 b^3}+\frac{e^2 (a+b x)^5}{5 b^3} \]
[Out]
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Rubi [A] time = 0.125051, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ \frac{e (a+b x)^4 (b d-a e)}{2 b^3}+\frac{(a+b x)^3 (b d-a e)^2}{3 b^3}+\frac{e^2 (a+b x)^5}{5 b^3} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)^2*(a^2 + 2*a*b*x + b^2*x^2),x]
[Out]
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Rubi in Sympy [A] time = 33.9954, size = 54, normalized size = 0.83 \[ \frac{b^{2} \left (d + e x\right )^{5}}{5 e^{3}} + \frac{b \left (d + e x\right )^{4} \left (a e - b d\right )}{2 e^{3}} + \frac{\left (d + e x\right )^{3} \left (a e - b d\right )^{2}}{3 e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)**2*(b**2*x**2+2*a*b*x+a**2),x)
[Out]
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Mathematica [A] time = 0.0268671, size = 79, normalized size = 1.22 \[ \frac{1}{3} x^3 \left (a^2 e^2+4 a b d e+b^2 d^2\right )+a^2 d^2 x+\frac{1}{2} b e x^4 (a e+b d)+a d x^2 (a e+b d)+\frac{1}{5} b^2 e^2 x^5 \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)^2*(a^2 + 2*a*b*x + b^2*x^2),x]
[Out]
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Maple [A] time = 0.001, size = 87, normalized size = 1.3 \[{\frac{{b}^{2}{e}^{2}{x}^{5}}{5}}+{\frac{ \left ( 2\,{e}^{2}ab+2\,{b}^{2}de \right ){x}^{4}}{4}}+{\frac{ \left ({a}^{2}{e}^{2}+4\,deab+{b}^{2}{d}^{2} \right ){x}^{3}}{3}}+{\frac{ \left ( 2\,de{a}^{2}+2\,{d}^{2}ab \right ){x}^{2}}{2}}+{a}^{2}{d}^{2}x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^2*(b^2*x^2+2*a*b*x+a^2),x)
[Out]
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Maxima [A] time = 0.690422, size = 109, normalized size = 1.68 \[ \frac{1}{5} \, b^{2} e^{2} x^{5} + a^{2} d^{2} x + \frac{1}{2} \,{\left (b^{2} d e + a b e^{2}\right )} x^{4} + \frac{1}{3} \,{\left (b^{2} d^{2} + 4 \, a b d e + a^{2} e^{2}\right )} x^{3} +{\left (a b d^{2} + a^{2} d e\right )} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)*(e*x + d)^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.179596, size = 1, normalized size = 0.02 \[ \frac{1}{5} x^{5} e^{2} b^{2} + \frac{1}{2} x^{4} e d b^{2} + \frac{1}{2} x^{4} e^{2} b a + \frac{1}{3} x^{3} d^{2} b^{2} + \frac{4}{3} x^{3} e d b a + \frac{1}{3} x^{3} e^{2} a^{2} + x^{2} d^{2} b a + x^{2} e d a^{2} + x d^{2} a^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)*(e*x + d)^2,x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.138325, size = 87, normalized size = 1.34 \[ a^{2} d^{2} x + \frac{b^{2} e^{2} x^{5}}{5} + x^{4} \left (\frac{a b e^{2}}{2} + \frac{b^{2} d e}{2}\right ) + x^{3} \left (\frac{a^{2} e^{2}}{3} + \frac{4 a b d e}{3} + \frac{b^{2} d^{2}}{3}\right ) + x^{2} \left (a^{2} d e + a b d^{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)**2*(b**2*x**2+2*a*b*x+a**2),x)
[Out]
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GIAC/XCAS [A] time = 0.211514, size = 120, normalized size = 1.85 \[ \frac{1}{5} \, b^{2} x^{5} e^{2} + \frac{1}{2} \, b^{2} d x^{4} e + \frac{1}{3} \, b^{2} d^{2} x^{3} + \frac{1}{2} \, a b x^{4} e^{2} + \frac{4}{3} \, a b d x^{3} e + a b d^{2} x^{2} + \frac{1}{3} \, a^{2} x^{3} e^{2} + a^{2} d x^{2} e + a^{2} d^{2} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)*(e*x + d)^2,x, algorithm="giac")
[Out]