3.1183 \(\int (b d+2 c d x) \sqrt{a+b x+c x^2} \, dx\)

Optimal. Leaf size=19 \[ \frac{2}{3} d \left (a+b x+c x^2\right )^{3/2} \]

[Out]

(2*d*(a + b*x + c*x^2)^(3/2))/3

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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]

(2*d*(a + b*x + c*x^2)^(3/2))/3

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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)

[Out]

2*d*(a + b*x + c*x**2)**(3/2)/3

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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]

[Out]

(2*d*(a + x*(b + c*x))^(3/2))/3

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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]

2/3*d*(c*x^2+b*x+a)^(3/2)

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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]

2/3*(c*x^2 + b*x + a)^(3/2)*d

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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]

2/3*(c*d*x^2 + b*d*x + a*d)*sqrt(c*x^2 + b*x + a)

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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]

2*a*d*sqrt(a + b*x + c*x**2)/3 + 2*b*d*x*sqrt(a + b*x + c*x**2)/3 + 2*c*d*x**2*s
qrt(a + b*x + c*x**2)/3

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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]

2/3*(c*x^2 + b*x + a)^(3/2)*d