3.337 \(\int \frac{x}{\sqrt{a+b x}} \, dx\)

Optimal. Leaf size=32 \[ \frac{2 (a+b x)^{3/2}}{3 b^2}-\frac{2 a \sqrt{a+b x}}{b^2} \]

[Out]

(-2*a*Sqrt[a + b*x])/b^2 + (2*(a + b*x)^(3/2))/(3*b^2)

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Rubi [A]  time = 0.0252409, antiderivative size = 32, 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 \[ \frac{2 (a+b x)^{3/2}}{3 b^2}-\frac{2 a \sqrt{a+b x}}{b^2} \]

Antiderivative was successfully verified.

[In]  Int[x/Sqrt[a + b*x],x]

[Out]

(-2*a*Sqrt[a + b*x])/b^2 + (2*(a + b*x)^(3/2))/(3*b^2)

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Rubi in Sympy [A]  time = 4.87291, size = 29, normalized size = 0.91 \[ - \frac{2 a \sqrt{a + b x}}{b^{2}} + \frac{2 \left (a + b x\right )^{\frac{3}{2}}}{3 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x/(b*x+a)**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.0123686, size = 23, normalized size = 0.72 \[ \frac{2 (b x-2 a) \sqrt{a+b x}}{3 b^2} \]

Antiderivative was successfully verified.

[In]  Integrate[x/Sqrt[a + b*x],x]

[Out]

(2*(-2*a + b*x)*Sqrt[a + b*x])/(3*b^2)

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Maple [A]  time = 0.003, size = 21, normalized size = 0.7 \[ -{\frac{-2\,bx+4\,a}{3\,{b}^{2}}\sqrt{bx+a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(b*x+a)^(1/2),x)

[Out]

-2/3*(b*x+a)^(1/2)*(-b*x+2*a)/b^2

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Maxima [A]  time = 1.34679, size = 35, normalized size = 1.09 \[ \frac{2 \,{\left (b x + a\right )}^{\frac{3}{2}}}{3 \, b^{2}} - \frac{2 \, \sqrt{b x + a} a}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/sqrt(b*x + a),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.230085, size = 26, normalized size = 0.81 \[ \frac{2 \, \sqrt{b x + a}{\left (b x - 2 \, a\right )}}{3 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/sqrt(b*x + a),x, algorithm="fricas")

[Out]

2/3*sqrt(b*x + a)*(b*x - 2*a)/b^2

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Sympy [A]  time = 1.79893, size = 162, normalized size = 5.06 \[ - \frac{4 a^{\frac{7}{2}} \sqrt{1 + \frac{b x}{a}}}{3 a^{2} b^{2} + 3 a b^{3} x} + \frac{4 a^{\frac{7}{2}}}{3 a^{2} b^{2} + 3 a b^{3} x} - \frac{2 a^{\frac{5}{2}} b x \sqrt{1 + \frac{b x}{a}}}{3 a^{2} b^{2} + 3 a b^{3} x} + \frac{4 a^{\frac{5}{2}} b x}{3 a^{2} b^{2} + 3 a b^{3} x} + \frac{2 a^{\frac{3}{2}} b^{2} x^{2} \sqrt{1 + \frac{b x}{a}}}{3 a^{2} b^{2} + 3 a b^{3} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(b*x+a)**(1/2),x)

[Out]

-4*a**(7/2)*sqrt(1 + b*x/a)/(3*a**2*b**2 + 3*a*b**3*x) + 4*a**(7/2)/(3*a**2*b**2
+ 3*a*b**3*x) - 2*a**(5/2)*b*x*sqrt(1 + b*x/a)/(3*a**2*b**2 + 3*a*b**3*x) + 4*a
**(5/2)*b*x/(3*a**2*b**2 + 3*a*b**3*x) + 2*a**(3/2)*b**2*x**2*sqrt(1 + b*x/a)/(3
*a**2*b**2 + 3*a*b**3*x)

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GIAC/XCAS [A]  time = 0.211079, size = 31, normalized size = 0.97 \[ \frac{2 \,{\left ({\left (b x + a\right )}^{\frac{3}{2}} - 3 \, \sqrt{b x + a} a\right )}}{3 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/sqrt(b*x + a),x, algorithm="giac")

[Out]

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