3.1194 \(\int (b d+2 c d x) \sqrt{a+b x+c x^2} \, dx\)

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 = 0.0063804, 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, Rules used = {629} \[ \frac{2}{3} d \left (a+b x+c x^2\right )^{3/2} \]

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

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

Mathematica [A]  time = 0.0090106, 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 = 0.043, size = 16, normalized size = 0.8 \begin{align*}{\frac{2\,d}{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*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 = 1.17951, size = 20, normalized size = 1.05 \begin{align*} \frac{2}{3} \,{\left (c x^{2} + b x + a\right )}^{\frac{3}{2}} d \end{align*}

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, algorithm="maxima")

[Out]

2/3*(c*x^2 + b*x + a)^(3/2)*d

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Fricas [A]  time = 2.18912, size = 69, normalized size = 3.63 \begin{align*} \frac{2}{3} \,{\left (c d x^{2} + b d x + a d\right )} \sqrt{c x^{2} + b x + a} \end{align*}

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, 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 [B]  time = 0.255468, size = 65, normalized size = 3.42 \begin{align*} \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} \end{align*}

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*sqrt(a + b*x + c*x**2)/3

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Giac [A]  time = 1.1255, size = 20, normalized size = 1.05 \begin{align*} \frac{2}{3} \,{\left (c x^{2} + b x + a\right )}^{\frac{3}{2}} d \end{align*}

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, algorithm="giac")

[Out]

2/3*(c*x^2 + b*x + a)^(3/2)*d