3.2476 \(\int \frac{A+B x}{(a+b x+c x^2)^{3/2}} \, dx\)

Optimal. Leaf size=45 \[ -\frac{2 (-2 a B-x (b B-2 A c)+A b)}{\left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}} \]

[Out]

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

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Rubi [A]  time = 0.0111191, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {636} \[ -\frac{2 (-2 a B-x (b B-2 A c)+A b)}{\left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 636

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

Rubi steps

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

Mathematica [A]  time = 0.17701, size = 44, normalized size = 0.98 \[ \frac{2 B (2 a+b x)-2 A (b+2 c x)}{\left (b^2-4 a c\right ) \sqrt{a+x (b+c x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0., size = 45, normalized size = 1. \begin{align*} 2\,{\frac{2\,Acx-Bbx+Ab-2\,aB}{\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((B*x+A)/(c*x^2+b*x+a)^(3/2),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.20212, size = 161, normalized size = 3.58 \begin{align*} \frac{2 \, \sqrt{c x^{2} + b x + a}{\left (2 \, B a - A b +{\left (B b - 2 \, A c\right )} x\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((B*x+A)/(c*x^2+b*x+a)^(3/2),x, algorithm="fricas")

[Out]

2*sqrt(c*x^2 + b*x + a)*(2*B*a - A*b + (B*b - 2*A*c)*x)/(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{A + B x}{\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((B*x+A)/(c*x**2+b*x+a)**(3/2),x)

[Out]

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

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Giac [A]  time = 1.15008, size = 74, normalized size = 1.64 \begin{align*} \frac{2 \,{\left (\frac{{\left (B b - 2 \, A c\right )} x}{b^{2} - 4 \, a c} + \frac{2 \, B a - A 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((B*x+A)/(c*x^2+b*x+a)^(3/2),x, algorithm="giac")

[Out]

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