3.613 \(\int \frac{1}{x^{5/2} \sqrt{2+b x}} \, dx\)

Optimal. Leaf size=38 \[ \frac{b \sqrt{b x+2}}{3 \sqrt{x}}-\frac{\sqrt{b x+2}}{3 x^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.0230381, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{b \sqrt{b x+2}}{3 \sqrt{x}}-\frac{\sqrt{b x+2}}{3 x^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^(5/2)*Sqrt[2 + b*x]),x]

[Out]

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

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Rubi in Sympy [A]  time = 3.06112, size = 31, normalized size = 0.82 \[ \frac{b \sqrt{b x + 2}}{3 \sqrt{x}} - \frac{\sqrt{b x + 2}}{3 x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.015859, size = 23, normalized size = 0.61 \[ \frac{(b x-1) \sqrt{b x+2}}{3 x^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^(5/2)*Sqrt[2 + b*x]),x]

[Out]

((-1 + b*x)*Sqrt[2 + b*x])/(3*x^(3/2))

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Maple [A]  time = 0.006, size = 18, normalized size = 0.5 \[{\frac{bx-1}{3}\sqrt{bx+2}{x}^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^(5/2)/(b*x+2)^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.3405, size = 35, normalized size = 0.92 \[ \frac{\sqrt{b x + 2} b}{2 \, \sqrt{x}} - \frac{{\left (b x + 2\right )}^{\frac{3}{2}}}{6 \, x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b*x + 2)*x^(5/2)),x, algorithm="maxima")

[Out]

1/2*sqrt(b*x + 2)*b/sqrt(x) - 1/6*(b*x + 2)^(3/2)/x^(3/2)

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Fricas [A]  time = 0.209167, size = 23, normalized size = 0.61 \[ \frac{\sqrt{b x + 2}{\left (b x - 1\right )}}{3 \, x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b*x + 2)*x^(5/2)),x, algorithm="fricas")

[Out]

1/3*sqrt(b*x + 2)*(b*x - 1)/x^(3/2)

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Sympy [A]  time = 23.9987, size = 34, normalized size = 0.89 \[ \frac{b^{\frac{3}{2}} \sqrt{1 + \frac{2}{b x}}}{3} - \frac{\sqrt{b} \sqrt{1 + \frac{2}{b x}}}{3 x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b**(3/2)*sqrt(1 + 2/(b*x))/3 - sqrt(b)*sqrt(1 + 2/(b*x))/(3*x)

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GIAC/XCAS [A]  time = 0.216265, size = 57, normalized size = 1.5 \[ \frac{{\left ({\left (b x + 2\right )} b^{3} - 3 \, b^{3}\right )} \sqrt{b x + 2} b}{3 \,{\left ({\left (b x + 2\right )} b - 2 \, b\right )}^{\frac{3}{2}}{\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b*x + 2)*x^(5/2)),x, algorithm="giac")

[Out]

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