Optimal. Leaf size=69 \[ \frac{(d+e x)^4 \left (a e^2-b d e+c d^2\right )}{4 e^3}-\frac{(d+e x)^5 (2 c d-b e)}{5 e^3}+\frac{c (d+e x)^6}{6 e^3} \]
[Out]
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Rubi [A] time = 0.172506, antiderivative size = 69, 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 \[ \frac{(d+e x)^4 \left (a e^2-b d e+c d^2\right )}{4 e^3}-\frac{(d+e x)^5 (2 c d-b e)}{5 e^3}+\frac{c (d+e x)^6}{6 e^3} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)^3*(a + b*x + c*x^2),x]
[Out]
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Rubi in Sympy [A] time = 19.8388, size = 60, normalized size = 0.87 \[ \frac{c \left (d + e x\right )^{6}}{6 e^{3}} + \frac{\left (d + e x\right )^{5} \left (b e - 2 c d\right )}{5 e^{3}} + \frac{\left (d + e x\right )^{4} \left (a e^{2} - b d e + c d^{2}\right )}{4 e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)**3*(c*x**2+b*x+a),x)
[Out]
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Mathematica [A] time = 0.0470263, size = 104, normalized size = 1.51 \[ \frac{1}{4} e x^4 \left (a e^2+3 b d e+3 c d^2\right )+\frac{1}{3} d x^3 \left (3 a e^2+3 b d e+c d^2\right )+\frac{1}{2} d^2 x^2 (3 a e+b d)+a d^3 x+\frac{1}{5} e^2 x^5 (b e+3 c d)+\frac{1}{6} c e^3 x^6 \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)^3*(a + b*x + c*x^2),x]
[Out]
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Maple [A] time = 0.001, size = 103, normalized size = 1.5 \[{\frac{{e}^{3}c{x}^{6}}{6}}+{\frac{ \left ({e}^{3}b+3\,d{e}^{2}c \right ){x}^{5}}{5}}+{\frac{ \left ( a{e}^{3}+3\,d{e}^{2}b+3\,{d}^{2}ec \right ){x}^{4}}{4}}+{\frac{ \left ( 3\,ad{e}^{2}+3\,{d}^{2}eb+c{d}^{3} \right ){x}^{3}}{3}}+{\frac{ \left ( 3\,{d}^{2}ea+{d}^{3}b \right ){x}^{2}}{2}}+{d}^{3}ax \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^3*(c*x^2+b*x+a),x)
[Out]
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Maxima [A] time = 0.789762, size = 138, normalized size = 2. \[ \frac{1}{6} \, c e^{3} x^{6} + \frac{1}{5} \,{\left (3 \, c d e^{2} + b e^{3}\right )} x^{5} + a d^{3} x + \frac{1}{4} \,{\left (3 \, c d^{2} e + 3 \, b d e^{2} + a e^{3}\right )} x^{4} + \frac{1}{3} \,{\left (c d^{3} + 3 \, b d^{2} e + 3 \, a d e^{2}\right )} x^{3} + \frac{1}{2} \,{\left (b d^{3} + 3 \, a d^{2} e\right )} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)*(e*x + d)^3,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.18132, size = 1, normalized size = 0.01 \[ \frac{1}{6} x^{6} e^{3} c + \frac{3}{5} x^{5} e^{2} d c + \frac{1}{5} x^{5} e^{3} b + \frac{3}{4} x^{4} e d^{2} c + \frac{3}{4} x^{4} e^{2} d b + \frac{1}{4} x^{4} e^{3} a + \frac{1}{3} x^{3} d^{3} c + x^{3} e d^{2} b + x^{3} e^{2} d a + \frac{1}{2} x^{2} d^{3} b + \frac{3}{2} x^{2} e d^{2} a + x d^{3} a \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)*(e*x + d)^3,x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.14489, size = 110, normalized size = 1.59 \[ a d^{3} x + \frac{c e^{3} x^{6}}{6} + x^{5} \left (\frac{b e^{3}}{5} + \frac{3 c d e^{2}}{5}\right ) + x^{4} \left (\frac{a e^{3}}{4} + \frac{3 b d e^{2}}{4} + \frac{3 c d^{2} e}{4}\right ) + x^{3} \left (a d e^{2} + b d^{2} e + \frac{c d^{3}}{3}\right ) + x^{2} \left (\frac{3 a d^{2} e}{2} + \frac{b d^{3}}{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)**3*(c*x**2+b*x+a),x)
[Out]
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GIAC/XCAS [A] time = 0.201858, size = 144, normalized size = 2.09 \[ \frac{1}{6} \, c x^{6} e^{3} + \frac{3}{5} \, c d x^{5} e^{2} + \frac{3}{4} \, c d^{2} x^{4} e + \frac{1}{3} \, c d^{3} x^{3} + \frac{1}{5} \, b x^{5} e^{3} + \frac{3}{4} \, b d x^{4} e^{2} + b d^{2} x^{3} e + \frac{1}{2} \, b d^{3} x^{2} + \frac{1}{4} \, a x^{4} e^{3} + a d x^{3} e^{2} + \frac{3}{2} \, a d^{2} x^{2} e + a d^{3} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)*(e*x + d)^3,x, algorithm="giac")
[Out]