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

Optimal. Leaf size=30 \[ \frac{2 a}{b^2 \sqrt{a+b x}}+\frac{2 \sqrt{a+b x}}{b^2} \]

[Out]

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

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Rubi [A]  time = 0.0076305, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {43} \[ \frac{2 a}{b^2 \sqrt{a+b x}}+\frac{2 \sqrt{a+b x}}{b^2} \]

Antiderivative was successfully verified.

[In]

Int[x/(a + b*x)^(3/2),x]

[Out]

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

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

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

Mathematica [A]  time = 0.0379939, size = 21, normalized size = 0.7 \[ \frac{2 (2 a+b x)}{b^2 \sqrt{a+b x}} \]

Antiderivative was successfully verified.

[In]

Integrate[x/(a + b*x)^(3/2),x]

[Out]

(2*(2*a + b*x))/(b^2*Sqrt[a + b*x])

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Maple [A]  time = 0.001, size = 20, normalized size = 0.7 \begin{align*} 2\,{\frac{bx+2\,a}{{b}^{2}\sqrt{bx+a}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.02808, size = 35, normalized size = 1.17 \begin{align*} \frac{2 \, \sqrt{b x + a}}{b^{2}} + \frac{2 \, a}{\sqrt{b x + a} b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(b*x + a)/b^2 + 2*a/(sqrt(b*x + a)*b^2)

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Fricas [A]  time = 1.46682, size = 61, normalized size = 2.03 \begin{align*} \frac{2 \,{\left (b x + 2 \, a\right )} \sqrt{b x + a}}{b^{3} x + a b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.737788, size = 37, normalized size = 1.23 \begin{align*} \begin{cases} \frac{4 a}{b^{2} \sqrt{a + b x}} + \frac{2 x}{b \sqrt{a + b x}} & \text{for}\: b \neq 0 \\\frac{x^{2}}{2 a^{\frac{3}{2}}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((4*a/(b**2*sqrt(a + b*x)) + 2*x/(b*sqrt(a + b*x)), Ne(b, 0)), (x**2/(2*a**(3/2)), True))

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Giac [A]  time = 1.1926, size = 39, normalized size = 1.3 \begin{align*} \frac{2 \,{\left (\frac{\sqrt{b x + a}}{b} + \frac{a}{\sqrt{b x + a} b}\right )}}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(sqrt(b*x + a)/b + a/(sqrt(b*x + a)*b))/b