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]

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

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Rubi [A]  time = 0.01, antiderivative size = 19, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.01, 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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fricas [A]  time = 0.94, size = 28, normalized size = 1.47 \[ \frac {2}{3} \, {\left (c d x^{2} + b d x + a d\right )} \sqrt {c x^{2} + b x + a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*d*x+b*d)*(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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giac [A]  time = 0.15, size = 15, normalized size = 0.79 \[ \frac {2}{3} \, {\left (c x^{2} + b x + a\right )}^{\frac {3}{2}} d \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(c*x^2 + b*x + a)^(3/2)*d

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maple [A]  time = 0.05, size = 16, normalized size = 0.84 \[ \frac {2 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} d}{3} \]

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 = 1.30, size = 15, normalized size = 0.79 \[ \frac {2}{3} \, {\left (c x^{2} + b x + a\right )}^{\frac {3}{2}} d \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*d*x+b*d)*(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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mupad [B]  time = 0.48, size = 15, normalized size = 0.79 \[ \frac {2\,d\,{\left (c\,x^2+b\,x+a\right )}^{3/2}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*d*(a + b*x + c*x^2)^(3/2))/3

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sympy [B]  time = 0.19, 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*sqrt(a + b*x + c*x**2)/3

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