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

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

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Rubi [A]  time = 0.01, antiderivative size = 18, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {2}{3} \left (a+b x+c x^2\right )^{3/2} \end {gather*}

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*}

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Mathematica [A]  time = 0.01, size = 17, normalized size = 0.94 \begin {gather*} \frac {2}{3} (a+x (b+c x))^{3/2} \end {gather*}

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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IntegrateAlgebraic [A]  time = 0.01, size = 18, normalized size = 1.00 \begin {gather*} \frac {2}{3} \left (a+b x+c x^2\right )^{3/2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(a + b*x + c*x^2)^(3/2))/3

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fricas [A]  time = 0.43, size = 14, normalized size = 0.78 \begin {gather*} \frac {2}{3} \, {\left (c x^{2} + b x + a\right )}^{\frac {3}{2}} \end {gather*}

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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giac [A]  time = 0.17, size = 14, normalized size = 0.78 \begin {gather*} \frac {2}{3} \, {\left (c x^{2} + b x + a\right )}^{\frac {3}{2}} \end {gather*}

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)

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maple [A]  time = 0.05, size = 15, normalized size = 0.83 \begin {gather*} \frac {2 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{3} \end {gather*}

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 = 0.62, size = 14, normalized size = 0.78 \begin {gather*} \frac {2}{3} \, {\left (c x^{2} + b x + a\right )}^{\frac {3}{2}} \end {gather*}

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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mupad [B]  time = 1.83, size = 14, normalized size = 0.78 \begin {gather*} \frac {2\,{\left (c\,x^2+b\,x+a\right )}^{3/2}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*(a + b*x + c*x^2)^(3/2))/3

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sympy [B]  time = 0.17, size = 60, normalized size = 3.33 \begin {gather*} \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 {gather*}

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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