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

Optimal. Leaf size=28 \[ -\frac{a B-A b x}{a b \sqrt{a+b x^2}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.006717, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {637} \[ -\frac{a B-A b x}{a b \sqrt{a+b x^2}} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*x)/(a + b*x^2)^(3/2),x]

[Out]

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

Rule 637

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

Rubi steps

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

Mathematica [A]  time = 0.014167, size = 27, normalized size = 0.96 \[ \frac{A b x-a B}{a b \sqrt{a+b x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.001, size = 26, normalized size = 0.9 \begin{align*}{\frac{Abx-Ba}{ab}{\frac{1}{\sqrt{b{x}^{2}+a}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 0.992055, size = 42, normalized size = 1.5 \begin{align*} \frac{A x}{\sqrt{b x^{2} + a} a} - \frac{B}{\sqrt{b x^{2} + a} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.60548, size = 69, normalized size = 2.46 \begin{align*} \frac{{\left (A b x - B a\right )} \sqrt{b x^{2} + a}}{a b^{2} x^{2} + a^{2} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 3.63549, size = 46, normalized size = 1.64 \begin{align*} \frac{A x}{a^{\frac{3}{2}} \sqrt{1 + \frac{b x^{2}}{a}}} + B \left (\begin{cases} - \frac{1}{b \sqrt{a + b x^{2}}} & \text{for}\: b \neq 0 \\\frac{x^{2}}{2 a^{\frac{3}{2}}} & \text{otherwise} \end{cases}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

A*x/(a**(3/2)*sqrt(1 + b*x**2/a)) + B*Piecewise((-1/(b*sqrt(a + b*x**2)), Ne(b, 0)), (x**2/(2*a**(3/2)), True)
)

________________________________________________________________________________________

Giac [A]  time = 1.20417, size = 31, normalized size = 1.11 \begin{align*} \frac{\frac{A x}{a} - \frac{B}{b}}{\sqrt{b x^{2} + a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(A*x/a - B/b)/sqrt(b*x^2 + a)