Optimal. Leaf size=19 \[ \frac{2}{3} d \left (a+b x+c x^2\right )^{3/2} \]
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Rubi [A] time = 0.0136825, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042 \[ \frac{2}{3} d \left (a+b x+c x^2\right )^{3/2} \]
Antiderivative was successfully verified.
[In] Int[(b*d + 2*c*d*x)*Sqrt[a + b*x + c*x^2],x]
[Out]
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Rubi in Sympy [A] time = 4.83825, size = 17, normalized size = 0.89 \[ \frac{2 d \left (a + b x + c x^{2}\right )^{\frac{3}{2}}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((2*c*d*x+b*d)*(c*x**2+b*x+a)**(1/2),x)
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Mathematica [A] time = 0.024273, size = 18, normalized size = 0.95 \[ \frac{2}{3} d (a+x (b+c x))^{3/2} \]
Antiderivative was successfully verified.
[In] Integrate[(b*d + 2*c*d*x)*Sqrt[a + b*x + c*x^2],x]
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Maple [A] time = 0.005, size = 16, normalized size = 0.8 \[{\frac{2\,d}{3} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((2*c*d*x+b*d)*(c*x^2+b*x+a)^(1/2),x)
[Out]
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Maxima [A] time = 0.672085, size = 20, normalized size = 1.05 \[ \frac{2}{3} \,{\left (c x^{2} + b x + a\right )}^{\frac{3}{2}} d \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*c*d*x + b*d)*sqrt(c*x^2 + b*x + a),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.224001, size = 38, normalized size = 2. \[ \frac{2}{3} \,{\left (c d x^{2} + b d x + a d\right )} \sqrt{c x^{2} + b x + a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*c*d*x + b*d)*sqrt(c*x^2 + b*x + a),x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.403914, size = 65, normalized size = 3.42 \[ \frac{2 a d \sqrt{a + b x + c x^{2}}}{3} + \frac{2 b d x \sqrt{a + b x + c x^{2}}}{3} + \frac{2 c d x^{2} \sqrt{a + b x + c x^{2}}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*c*d*x+b*d)*(c*x**2+b*x+a)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.212344, size = 20, normalized size = 1.05 \[ \frac{2}{3} \,{\left (c x^{2} + b x + a\right )}^{\frac{3}{2}} d \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*c*d*x + b*d)*sqrt(c*x^2 + b*x + a),x, algorithm="giac")
[Out]