3.1585 \(\int \frac{b+2 c x}{(a+b x+c x^2)^{3/2}} \, dx\)

Optimal. Leaf size=16 \[ -\frac{2}{\sqrt{a+b x+c x^2}} \]

[Out]

-2/Sqrt[a + b*x + c*x^2]

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Rubi [A]  time = 0.0064971, antiderivative size = 16, 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} \[ -\frac{2}{\sqrt{a+b x+c x^2}} \]

Antiderivative was successfully verified.

[In]

Int[(b + 2*c*x)/(a + b*x + c*x^2)^(3/2),x]

[Out]

-2/Sqrt[a + b*x + c*x^2]

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

Mathematica [A]  time = 0.0069801, size = 15, normalized size = 0.94 \[ -\frac{2}{\sqrt{a+x (b+c x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[(b + 2*c*x)/(a + b*x + c*x^2)^(3/2),x]

[Out]

-2/Sqrt[a + x*(b + c*x)]

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Maple [A]  time = 0.003, size = 15, normalized size = 0.9 \begin{align*} -2\,{\frac{1}{\sqrt{c{x}^{2}+bx+a}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.973827, size = 19, normalized size = 1.19 \begin{align*} -\frac{2}{\sqrt{c x^{2} + b x + a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)/(c*x^2+b*x+a)^(3/2),x, algorithm="maxima")

[Out]

-2/sqrt(c*x^2 + b*x + a)

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Fricas [A]  time = 1.82914, size = 35, normalized size = 2.19 \begin{align*} -\frac{2}{\sqrt{c x^{2} + b x + a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)/(c*x^2+b*x+a)^(3/2),x, algorithm="fricas")

[Out]

-2/sqrt(c*x^2 + b*x + a)

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Sympy [A]  time = 0.753052, size = 15, normalized size = 0.94 \begin{align*} - \frac{2}{\sqrt{a + b x + c x^{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)**(3/2),x)

[Out]

-2/sqrt(a + b*x + c*x**2)

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Giac [A]  time = 1.37347, size = 19, normalized size = 1.19 \begin{align*} -\frac{2}{\sqrt{c x^{2} + b x + a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)/(c*x^2+b*x+a)^(3/2),x, algorithm="giac")

[Out]

-2/sqrt(c*x^2 + b*x + a)