3.1550 \(\int (b+2 c x) \sqrt{a+b x+c x^2} \, dx\)

Optimal. Leaf size=18 \[ \frac{2}{3} \left (a+b x+c x^2\right )^{3/2} \]

[Out]

(2*(a + b*x + c*x^2)^(3/2))/3

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Rubi [A]  time = 0.0058565, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048, Rules used = {629} \[ \frac{2}{3} \left (a+b x+c x^2\right )^{3/2} \]

Antiderivative was successfully verified.

[In]

Int[(b + 2*c*x)*Sqrt[a + b*x + c*x^2],x]

[Out]

(2*(a + b*x + c*x^2)^(3/2))/3

Rule 629

Int[((d_) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d*(a + b*x + c*x^2)^(p +
1))/(b*(p + 1)), x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[2*c*d - b*e, 0] && NeQ[p, -1]

Rubi steps

\begin{align*} \int (b+2 c x) \sqrt{a+b x+c x^2} \, dx &=\frac{2}{3} \left (a+b x+c x^2\right )^{3/2}\\ \end{align*}

Mathematica [A]  time = 0.0087581, size = 17, normalized size = 0.94 \[ \frac{2}{3} (a+x (b+c x))^{3/2} \]

Antiderivative was successfully verified.

[In]

Integrate[(b + 2*c*x)*Sqrt[a + b*x + c*x^2],x]

[Out]

(2*(a + x*(b + c*x))^(3/2))/3

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Maple [A]  time = 0.003, size = 15, normalized size = 0.8 \begin{align*}{\frac{2}{3} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*c*x+b)*(c*x^2+b*x+a)^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.00861, size = 19, normalized size = 1.06 \begin{align*} \frac{2}{3} \,{\left (c x^{2} + b x + a\right )}^{\frac{3}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)*(c*x^2+b*x+a)^(1/2),x, algorithm="maxima")

[Out]

2/3*(c*x^2 + b*x + a)^(3/2)

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Fricas [A]  time = 1.33353, size = 39, normalized size = 2.17 \begin{align*} \frac{2}{3} \,{\left (c x^{2} + b x + a\right )}^{\frac{3}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)*(c*x^2+b*x+a)^(1/2),x, algorithm="fricas")

[Out]

2/3*(c*x^2 + b*x + a)^(3/2)

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Sympy [B]  time = 0.178724, size = 60, normalized size = 3.33 \begin{align*} \frac{2 a \sqrt{a + b x + c x^{2}}}{3} + \frac{2 b x \sqrt{a + b x + c x^{2}}}{3} + \frac{2 c x^{2} \sqrt{a + b x + c x^{2}}}{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)*(c*x**2+b*x+a)**(1/2),x)

[Out]

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

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Giac [A]  time = 1.16113, size = 19, normalized size = 1.06 \begin{align*} \frac{2}{3} \,{\left (c x^{2} + b x + a\right )}^{\frac{3}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)*(c*x^2+b*x+a)^(1/2),x, algorithm="giac")

[Out]

2/3*(c*x^2 + b*x + a)^(3/2)