3.625 \(\int \frac{\sqrt{x}}{(2+b x)^{5/2}} \, dx\)

Optimal. Leaf size=18 \[ \frac{x^{3/2}}{3 (b x+2)^{3/2}} \]

[Out]

x^(3/2)/(3*(2 + b*x)^(3/2))

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Rubi [A]  time = 0.0104337, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ \frac{x^{3/2}}{3 (b x+2)^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[x]/(2 + b*x)^(5/2),x]

[Out]

x^(3/2)/(3*(2 + b*x)^(3/2))

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Rubi in Sympy [A]  time = 2.13042, size = 14, normalized size = 0.78 \[ \frac{x^{\frac{3}{2}}}{3 \left (b x + 2\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(1/2)/(b*x+2)**(5/2),x)

[Out]

x**(3/2)/(3*(b*x + 2)**(3/2))

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Mathematica [A]  time = 0.0158017, size = 18, normalized size = 1. \[ \frac{x^{3/2}}{3 (b x+2)^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[x]/(2 + b*x)^(5/2),x]

[Out]

x^(3/2)/(3*(2 + b*x)^(3/2))

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Maple [A]  time = 0.006, size = 13, normalized size = 0.7 \[{\frac{1}{3}{x}^{{\frac{3}{2}}} \left ( bx+2 \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(1/2)/(b*x+2)^(5/2),x)

[Out]

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

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Maxima [A]  time = 1.34672, size = 16, normalized size = 0.89 \[ \frac{x^{\frac{3}{2}}}{3 \,{\left (b x + 2\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x)/(b*x + 2)^(5/2),x, algorithm="maxima")

[Out]

1/3*x^(3/2)/(b*x + 2)^(3/2)

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Fricas [A]  time = 0.208844, size = 16, normalized size = 0.89 \[ \frac{x^{\frac{3}{2}}}{3 \,{\left (b x + 2\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x)/(b*x + 2)^(5/2),x, algorithm="fricas")

[Out]

1/3*x^(3/2)/(b*x + 2)^(3/2)

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Sympy [A]  time = 13.0641, size = 27, normalized size = 1.5 \[ \frac{x^{\frac{3}{2}}}{3 b x \sqrt{b x + 2} + 6 \sqrt{b x + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**(1/2)/(b*x+2)**(5/2),x)

[Out]

x**(3/2)/(3*b*x*sqrt(b*x + 2) + 6*sqrt(b*x + 2))

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GIAC/XCAS [A]  time = 0.219529, size = 111, normalized size = 6.17 \[ \frac{4 \,{\left (3 \,{\left (\sqrt{b x + 2} \sqrt{b} - \sqrt{{\left (b x + 2\right )} b - 2 \, b}\right )}^{4} \sqrt{b} + 4 \, b^{\frac{5}{2}}\right )}{\left | b \right |}}{3 \,{\left ({\left (\sqrt{b x + 2} \sqrt{b} - \sqrt{{\left (b x + 2\right )} b - 2 \, b}\right )}^{2} + 2 \, b\right )}^{3} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x)/(b*x + 2)^(5/2),x, algorithm="giac")

[Out]

4/3*(3*(sqrt(b*x + 2)*sqrt(b) - sqrt((b*x + 2)*b - 2*b))^4*sqrt(b) + 4*b^(5/2))*
abs(b)/(((sqrt(b*x + 2)*sqrt(b) - sqrt((b*x + 2)*b - 2*b))^2 + 2*b)^3*b^2)