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 = 0.0247017, 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 \[ \frac{2 a}{b^2 \sqrt{a+b x}}+\frac{2 \sqrt{a+b x}}{b^2} \]

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

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Rubi in Sympy [A]  time = 4.91127, size = 27, normalized size = 0.9 \[ \frac{2 a}{b^{2} \sqrt{a + b x}} + \frac{2 \sqrt{a + b x}}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x/(b*x+a)**(3/2),x)

[Out]

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

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Mathematica [A]  time = 0.0146373, 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 = 0.006, size = 20, normalized size = 0.7 \[ 2\,{\frac{bx+2\,a}{{b}^{2}\sqrt{bx+a}}} \]

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 = 1.33926, size = 35, normalized size = 1.17 \[ \frac{2 \, \sqrt{b x + a}}{b^{2}} + \frac{2 \, a}{\sqrt{b x + a} b^{2}} \]

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 = 0.222025, size = 26, normalized size = 0.87 \[ \frac{2 \,{\left (b x + 2 \, a\right )}}{\sqrt{b x + a} b^{2}} \]

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^2)

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Sympy [A]  time = 2.01795, size = 37, normalized size = 1.23 \[ \begin{cases} \frac{4 a}{b^{2} \sqrt{a + b x}} + \frac{2 x}{b \sqrt{a + b x}} & \text{for}\: b \neq 0 \\\frac{x^{2}}{2 a^{\frac{3}{2}}} & \text{otherwise} \end{cases} \]

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/XCAS [A]  time = 0.204764, size = 39, normalized size = 1.3 \[ \frac{2 \,{\left (\frac{\sqrt{b x + a}}{b} + \frac{a}{\sqrt{b x + a} b}\right )}}{b} \]

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