Optimal. Leaf size=57 \[ \frac{(d+e x)^3 \left (a e^2+c d^2\right )}{3 e^3}+\frac{c (d+e x)^5}{5 e^3}-\frac{c d (d+e x)^4}{2 e^3} \]
[Out]
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Rubi [A] time = 0.0868488, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ \frac{(d+e x)^3 \left (a e^2+c d^2\right )}{3 e^3}+\frac{c (d+e x)^5}{5 e^3}-\frac{c d (d+e x)^4}{2 e^3} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)^2*(a + c*x^2),x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ 2 a d e \int x\, dx + \frac{c d e x^{4}}{2} + \frac{c e^{2} x^{5}}{5} + d^{2} \int a\, dx + x^{3} \left (\frac{a e^{2}}{3} + \frac{c d^{2}}{3}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)**2*(c*x**2+a),x)
[Out]
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Mathematica [A] time = 0.0165297, size = 53, normalized size = 0.93 \[ \frac{1}{3} x^3 \left (a e^2+c d^2\right )+a d^2 x+a d e x^2+\frac{1}{2} c d e x^4+\frac{1}{5} c e^2 x^5 \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)^2*(a + c*x^2),x]
[Out]
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Maple [A] time = 0.001, size = 48, normalized size = 0.8 \[{\frac{c{e}^{2}{x}^{5}}{5}}+{\frac{cde{x}^{4}}{2}}+{\frac{ \left ( a{e}^{2}+c{d}^{2} \right ){x}^{3}}{3}}+ade{x}^{2}+a{d}^{2}x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^2*(c*x^2+a),x)
[Out]
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Maxima [A] time = 0.71196, size = 63, normalized size = 1.11 \[ \frac{1}{5} \, c e^{2} x^{5} + \frac{1}{2} \, c d e x^{4} + a d e x^{2} + a d^{2} x + \frac{1}{3} \,{\left (c d^{2} + a e^{2}\right )} x^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)*(e*x + d)^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.192051, size = 1, normalized size = 0.02 \[ \frac{1}{5} x^{5} e^{2} c + \frac{1}{2} x^{4} e d c + \frac{1}{3} x^{3} d^{2} c + \frac{1}{3} x^{3} e^{2} a + x^{2} e d a + x d^{2} a \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)*(e*x + d)^2,x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.104374, size = 51, normalized size = 0.89 \[ a d^{2} x + a d e x^{2} + \frac{c d e x^{4}}{2} + \frac{c e^{2} x^{5}}{5} + x^{3} \left (\frac{a e^{2}}{3} + \frac{c d^{2}}{3}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)**2*(c*x**2+a),x)
[Out]
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GIAC/XCAS [A] time = 0.203901, size = 66, normalized size = 1.16 \[ \frac{1}{5} \, c x^{5} e^{2} + \frac{1}{2} \, c d x^{4} e + \frac{1}{3} \, c d^{2} x^{3} + \frac{1}{3} \, a x^{3} e^{2} + a d x^{2} e + a d^{2} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)*(e*x + d)^2,x, algorithm="giac")
[Out]