3.1202 \(\int \frac{A+B x}{(b x+c x^2)^{3/2}} \, dx\)

Optimal. Leaf size=33 \[ -\frac{2 (A b-x (b B-2 A c))}{b^2 \sqrt{b x+c x^2}} \]

[Out]

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

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Rubi [A]  time = 0.0099222, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {636} \[ -\frac{2 (A b-x (b B-2 A c))}{b^2 \sqrt{b x+c x^2}} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*x)/(b*x + c*x^2)^(3/2),x]

[Out]

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

Mathematica [A]  time = 0.0127149, size = 30, normalized size = 0.91 \[ \frac{2 b B x-2 A (b+2 c x)}{b^2 \sqrt{x (b+c x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.006, size = 37, normalized size = 1.1 \begin{align*} -2\,{\frac{x \left ( cx+b \right ) \left ( 2\,Acx-bBx+Ab \right ) }{{b}^{2} \left ( c{x}^{2}+bx \right ) ^{3/2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*x*(c*x+b)*(2*A*c*x-B*b*x+A*b)/b^2/(c*x^2+b*x)^(3/2)

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Maxima [A]  time = 1.01205, size = 74, normalized size = 2.24 \begin{align*} \frac{2 \, B x}{\sqrt{c x^{2} + b x} b} - \frac{4 \, A c x}{\sqrt{c x^{2} + b x} b^{2}} - \frac{2 \, A}{\sqrt{c x^{2} + b x} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.65545, size = 89, normalized size = 2.7 \begin{align*} -\frac{2 \, \sqrt{c x^{2} + b x}{\left (A b -{\left (B b - 2 \, A c\right )} x\right )}}{b^{2} c x^{2} + b^{3} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*sqrt(c*x^2 + b*x)*(A*b - (B*b - 2*A*c)*x)/(b^2*c*x^2 + b^3*x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{A + B x}{\left (x \left (b + c x\right )\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)**(3/2),x)

[Out]

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

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Giac [A]  time = 1.34254, size = 45, normalized size = 1.36 \begin{align*} -\frac{2 \,{\left (\frac{A}{b} - \frac{{\left (B b - 2 \, A c\right )} x}{b^{2}}\right )}}{\sqrt{c x^{2} + b x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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