### 3.2386 $$\int \frac{1}{(a+b x+c x^2)^{3/2}} \, dx$$

Optimal. Leaf size=32 $-\frac{2 (b+2 c x)}{\left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}$

[Out]

(-2*(b + 2*c*x))/((b^2 - 4*a*c)*Sqrt[a + b*x + c*x^2])

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Rubi [A]  time = 0.0042317, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 14, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.071, Rules used = {613} $-\frac{2 (b+2 c x)}{\left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}$

Antiderivative was successfully veriﬁed.

[In]

Int[(a + b*x + c*x^2)^(-3/2),x]

[Out]

(-2*(b + 2*c*x))/((b^2 - 4*a*c)*Sqrt[a + b*x + c*x^2])

Rule 613

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[(-2*(b + 2*c*x))/((b^2 - 4*a*c)*Sqrt[a + b*x
+ c*x^2]), x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rubi steps

\begin{align*} \int \frac{1}{\left (a+b x+c x^2\right )^{3/2}} \, dx &=-\frac{2 (b+2 c x)}{\left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0172564, size = 31, normalized size = 0.97 $-\frac{2 (b+2 c x)}{\left (b^2-4 a c\right ) \sqrt{a+x (b+c x)}}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(a + b*x + c*x^2)^(-3/2),x]

[Out]

(-2*(b + 2*c*x))/((b^2 - 4*a*c)*Sqrt[a + x*(b + c*x)])

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Maple [A]  time = 0.043, size = 33, normalized size = 1. \begin{align*} 2\,{\frac{2\,cx+b}{\sqrt{c{x}^{2}+bx+a} \left ( 4\,ac-{b}^{2} \right ) }} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(1/(c*x^2+b*x+a)^(3/2),x)

[Out]

2*(2*c*x+b)/(c*x^2+b*x+a)^(1/2)/(4*a*c-b^2)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*x^2+b*x+a)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.76706, size = 135, normalized size = 4.22 \begin{align*} -\frac{2 \, \sqrt{c x^{2} + b x + a}{\left (2 \, c x + b\right )}}{a b^{2} - 4 \, a^{2} c +{\left (b^{2} c - 4 \, a c^{2}\right )} x^{2} +{\left (b^{3} - 4 \, a b c\right )} x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*x^2+b*x+a)^(3/2),x, algorithm="fricas")

[Out]

-2*sqrt(c*x^2 + b*x + a)*(2*c*x + b)/(a*b^2 - 4*a^2*c + (b^2*c - 4*a*c^2)*x^2 + (b^3 - 4*a*b*c)*x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a + b x + c x^{2}\right )^{\frac{3}{2}}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*x**2+b*x+a)**(3/2),x)

[Out]

Integral((a + b*x + c*x**2)**(-3/2), x)

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Giac [A]  time = 1.13739, size = 55, normalized size = 1.72 \begin{align*} -\frac{2 \,{\left (\frac{2 \, c x}{b^{2} - 4 \, a c} + \frac{b}{b^{2} - 4 \, a c}\right )}}{\sqrt{c x^{2} + b x + a}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*x^2+b*x+a)^(3/2),x, algorithm="giac")

[Out]

-2*(2*c*x/(b^2 - 4*a*c) + b/(b^2 - 4*a*c))/sqrt(c*x^2 + b*x + a)