Optimal. Leaf size=38 \[ \frac{(a+b x)^4 (b d-a e)}{4 b^2}+\frac{e (a+b x)^5}{5 b^2} \]
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Rubi [A] time = 0.0453112, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074 \[ \frac{(a+b x)^4 (b d-a e)}{4 b^2}+\frac{e (a+b x)^5}{5 b^2} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x)*(d + e*x)*(a^2 + 2*a*b*x + b^2*x^2),x]
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Rubi in Sympy [A] time = 20.6913, size = 31, normalized size = 0.82 \[ \frac{e \left (a + b x\right )^{5}}{5 b^{2}} - \frac{\left (a + b x\right )^{4} \left (a e - b d\right )}{4 b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)*(e*x+d)*(b**2*x**2+2*a*b*x+a**2),x)
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Mathematica [A] time = 0.0203039, size = 67, normalized size = 1.76 \[ a^3 d x+\frac{1}{2} a^2 x^2 (a e+3 b d)+\frac{1}{4} b^2 x^4 (3 a e+b d)+a b x^3 (a e+b d)+\frac{1}{5} b^3 e x^5 \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x)*(d + e*x)*(a^2 + 2*a*b*x + b^2*x^2),x]
[Out]
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Maple [B] time = 0.002, size = 94, normalized size = 2.5 \[{\frac{{b}^{3}e{x}^{5}}{5}}+{\frac{ \left ( \left ( ae+bd \right ){b}^{2}+2\,ae{b}^{2} \right ){x}^{4}}{4}}+{\frac{ \left ( a{b}^{2}d+2\, \left ( ae+bd \right ) ab+{a}^{2}be \right ){x}^{3}}{3}}+{\frac{ \left ( 2\,{a}^{2}bd+ \left ( ae+bd \right ){a}^{2} \right ){x}^{2}}{2}}+{a}^{3}dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)*(e*x+d)*(b^2*x^2+2*a*b*x+a^2),x)
[Out]
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Maxima [A] time = 0.70932, size = 93, normalized size = 2.45 \[ \frac{1}{5} \, b^{3} e x^{5} + a^{3} d x + \frac{1}{4} \,{\left (b^{3} d + 3 \, a b^{2} e\right )} x^{4} +{\left (a b^{2} d + a^{2} b e\right )} x^{3} + \frac{1}{2} \,{\left (3 \, a^{2} b d + a^{3} e\right )} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)*(b*x + a)*(e*x + d),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.25916, size = 1, normalized size = 0.03 \[ \frac{1}{5} x^{5} e b^{3} + \frac{1}{4} x^{4} d b^{3} + \frac{3}{4} x^{4} e b^{2} a + x^{3} d b^{2} a + x^{3} e b a^{2} + \frac{3}{2} x^{2} d b a^{2} + \frac{1}{2} x^{2} e a^{3} + x d a^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)*(b*x + a)*(e*x + d),x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.127732, size = 73, normalized size = 1.92 \[ a^{3} d x + \frac{b^{3} e x^{5}}{5} + x^{4} \left (\frac{3 a b^{2} e}{4} + \frac{b^{3} d}{4}\right ) + x^{3} \left (a^{2} b e + a b^{2} d\right ) + x^{2} \left (\frac{a^{3} e}{2} + \frac{3 a^{2} b d}{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x+a)*(e*x+d)*(b**2*x**2+2*a*b*x+a**2),x)
[Out]
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GIAC/XCAS [A] time = 0.306851, size = 103, normalized size = 2.71 \[ \frac{1}{5} \, b^{3} x^{5} e + \frac{1}{4} \, b^{3} d x^{4} + \frac{3}{4} \, a b^{2} x^{4} e + a b^{2} d x^{3} + a^{2} b x^{3} e + \frac{3}{2} \, a^{2} b d x^{2} + \frac{1}{2} \, a^{3} x^{2} e + a^{3} d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)*(b*x + a)*(e*x + d),x, algorithm="giac")
[Out]