Optimal. Leaf size=96 \[ a^2 d x+\frac{1}{4} x^4 \left (2 a c e+b^2 e+2 b c d\right )+\frac{1}{3} x^3 \left (2 a b e+2 a c d+b^2 d\right )+\frac{1}{2} a x^2 (a e+2 b d)+\frac{1}{5} c x^5 (2 b e+c d)+\frac{1}{6} c^2 e x^6 \]
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Rubi [A] time = 0.210046, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056 \[ a^2 d x+\frac{1}{4} x^4 \left (2 a c e+b^2 e+2 b c d\right )+\frac{1}{3} x^3 \left (2 a b e+2 a c d+b^2 d\right )+\frac{1}{2} a x^2 (a e+2 b d)+\frac{1}{5} c x^5 (2 b e+c d)+\frac{1}{6} c^2 e x^6 \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)*(a + b*x + c*x^2)^2,x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ a^{2} \int d\, dx + a \left (a e + 2 b d\right ) \int x\, dx + \frac{c^{2} e x^{6}}{6} + \frac{c x^{5} \left (2 b e + c d\right )}{5} + x^{4} \left (\frac{a c e}{2} + \frac{b^{2} e}{4} + \frac{b c d}{2}\right ) + x^{3} \left (\frac{2 a b e}{3} + \frac{2 a c d}{3} + \frac{b^{2} d}{3}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)*(c*x**2+b*x+a)**2,x)
[Out]
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Mathematica [A] time = 0.039563, size = 96, normalized size = 1. \[ a^2 d x+\frac{1}{4} x^4 \left (2 a c e+b^2 e+2 b c d\right )+\frac{1}{3} x^3 \left (2 a b e+2 a c d+b^2 d\right )+\frac{1}{2} a x^2 (a e+2 b d)+\frac{1}{5} c x^5 (2 b e+c d)+\frac{1}{6} c^2 e x^6 \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)*(a + b*x + c*x^2)^2,x]
[Out]
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Maple [A] time = 0.001, size = 91, normalized size = 1. \[{\frac{{c}^{2}e{x}^{6}}{6}}+{\frac{ \left ( 2\,bce+{c}^{2}d \right ){x}^{5}}{5}}+{\frac{ \left ( 2\,bcd+e \left ( 2\,ac+{b}^{2} \right ) \right ){x}^{4}}{4}}+{\frac{ \left ( d \left ( 2\,ac+{b}^{2} \right ) +2\,bea \right ){x}^{3}}{3}}+{\frac{ \left ( e{a}^{2}+2\,bda \right ){x}^{2}}{2}}+{a}^{2}dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)*(c*x^2+b*x+a)^2,x)
[Out]
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Maxima [A] time = 0.810993, size = 122, normalized size = 1.27 \[ \frac{1}{6} \, c^{2} e x^{6} + \frac{1}{5} \,{\left (c^{2} d + 2 \, b c e\right )} x^{5} + \frac{1}{4} \,{\left (2 \, b c d +{\left (b^{2} + 2 \, a c\right )} e\right )} x^{4} + a^{2} d x + \frac{1}{3} \,{\left (2 \, a b e +{\left (b^{2} + 2 \, a c\right )} d\right )} x^{3} + \frac{1}{2} \,{\left (2 \, a b d + a^{2} e\right )} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^2*(e*x + d),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.182866, size = 1, normalized size = 0.01 \[ \frac{1}{6} x^{6} e c^{2} + \frac{1}{5} x^{5} d c^{2} + \frac{2}{5} x^{5} e c b + \frac{1}{2} x^{4} d c b + \frac{1}{4} x^{4} e b^{2} + \frac{1}{2} x^{4} e c a + \frac{1}{3} x^{3} d b^{2} + \frac{2}{3} x^{3} d c a + \frac{2}{3} x^{3} e b a + x^{2} d b a + \frac{1}{2} x^{2} e a^{2} + x d a^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^2*(e*x + d),x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.15036, size = 100, normalized size = 1.04 \[ a^{2} d x + \frac{c^{2} e x^{6}}{6} + x^{5} \left (\frac{2 b c e}{5} + \frac{c^{2} d}{5}\right ) + x^{4} \left (\frac{a c e}{2} + \frac{b^{2} e}{4} + \frac{b c d}{2}\right ) + x^{3} \left (\frac{2 a b e}{3} + \frac{2 a c d}{3} + \frac{b^{2} d}{3}\right ) + x^{2} \left (\frac{a^{2} e}{2} + a b d\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)*(c*x**2+b*x+a)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.202601, size = 142, normalized size = 1.48 \[ \frac{1}{6} \, c^{2} x^{6} e + \frac{1}{5} \, c^{2} d x^{5} + \frac{2}{5} \, b c x^{5} e + \frac{1}{2} \, b c d x^{4} + \frac{1}{4} \, b^{2} x^{4} e + \frac{1}{2} \, a c x^{4} e + \frac{1}{3} \, b^{2} d x^{3} + \frac{2}{3} \, a c d x^{3} + \frac{2}{3} \, a b x^{3} e + a b d x^{2} + \frac{1}{2} \, a^{2} x^{2} e + a^{2} d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^2*(e*x + d),x, algorithm="giac")
[Out]