3.621 \(\int \frac{1}{x^{5/2} (2+b x)^{3/2}} \, dx\)

Optimal. Leaf size=53 \[ -\frac{2 \sqrt{b x+2}}{3 x^{3/2}}+\frac{1}{x^{3/2} \sqrt{b x+2}}+\frac{2 b \sqrt{b x+2}}{3 \sqrt{x}} \]

[Out]

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

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Rubi [A]  time = 0.0363974, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ -\frac{2 \sqrt{b x+2}}{3 x^{3/2}}+\frac{1}{x^{3/2} \sqrt{b x+2}}+\frac{2 b \sqrt{b x+2}}{3 \sqrt{x}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^(5/2)*(2 + b*x)^(3/2)),x]

[Out]

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

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Rubi in Sympy [A]  time = 4.38866, size = 49, normalized size = 0.92 \[ \frac{2 b \sqrt{b x + 2}}{3 \sqrt{x}} - \frac{2 \sqrt{b x + 2}}{3 x^{\frac{3}{2}}} + \frac{1}{x^{\frac{3}{2}} \sqrt{b x + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**(5/2)/(b*x+2)**(3/2),x)

[Out]

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

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Mathematica [A]  time = 0.0203605, size = 32, normalized size = 0.6 \[ \frac{2 b^2 x^2+2 b x-1}{3 x^{3/2} \sqrt{b x+2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^(5/2)*(2 + b*x)^(3/2)),x]

[Out]

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

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Maple [A]  time = 0.006, size = 27, normalized size = 0.5 \[{\frac{2\,{b}^{2}{x}^{2}+2\,bx-1}{3}{x}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{bx+2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^(5/2)/(b*x+2)^(3/2),x)

[Out]

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

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Maxima [A]  time = 1.35263, size = 55, normalized size = 1.04 \[ \frac{b^{2} \sqrt{x}}{4 \, \sqrt{b x + 2}} + \frac{\sqrt{b x + 2} b}{2 \, \sqrt{x}} - \frac{{\left (b x + 2\right )}^{\frac{3}{2}}}{12 \, x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + 2)^(3/2)*x^(5/2)),x, algorithm="maxima")

[Out]

1/4*b^2*sqrt(x)/sqrt(b*x + 2) + 1/2*sqrt(b*x + 2)*b/sqrt(x) - 1/12*(b*x + 2)^(3/
2)/x^(3/2)

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Fricas [A]  time = 0.212198, size = 35, normalized size = 0.66 \[ \frac{2 \, b^{2} x^{2} + 2 \, b x - 1}{3 \, \sqrt{b x + 2} x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + 2)^(3/2)*x^(5/2)),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 102.32, size = 170, normalized size = 3.21 \[ \frac{2 b^{\frac{15}{2}} x^{3} \sqrt{1 + \frac{2}{b x}}}{3 b^{6} x^{3} + 12 b^{5} x^{2} + 12 b^{4} x} + \frac{6 b^{\frac{13}{2}} x^{2} \sqrt{1 + \frac{2}{b x}}}{3 b^{6} x^{3} + 12 b^{5} x^{2} + 12 b^{4} x} + \frac{3 b^{\frac{11}{2}} x \sqrt{1 + \frac{2}{b x}}}{3 b^{6} x^{3} + 12 b^{5} x^{2} + 12 b^{4} x} - \frac{2 b^{\frac{9}{2}} \sqrt{1 + \frac{2}{b x}}}{3 b^{6} x^{3} + 12 b^{5} x^{2} + 12 b^{4} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**(5/2)/(b*x+2)**(3/2),x)

[Out]

2*b**(15/2)*x**3*sqrt(1 + 2/(b*x))/(3*b**6*x**3 + 12*b**5*x**2 + 12*b**4*x) + 6*
b**(13/2)*x**2*sqrt(1 + 2/(b*x))/(3*b**6*x**3 + 12*b**5*x**2 + 12*b**4*x) + 3*b*
*(11/2)*x*sqrt(1 + 2/(b*x))/(3*b**6*x**3 + 12*b**5*x**2 + 12*b**4*x) - 2*b**(9/2
)*sqrt(1 + 2/(b*x))/(3*b**6*x**3 + 12*b**5*x**2 + 12*b**4*x)

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GIAC/XCAS [A]  time = 0.218849, size = 116, normalized size = 2.19 \[ \frac{b^{\frac{7}{2}}}{{\left ({\left (\sqrt{b x + 2} \sqrt{b} - \sqrt{{\left (b x + 2\right )} b - 2 \, b}\right )}^{2} + 2 \, b\right )}{\left | b \right |}} + \frac{{\left (5 \,{\left (b x + 2\right )} b^{2}{\left | b \right |} - 12 \, b^{2}{\left | b \right |}\right )} \sqrt{b x + 2}}{12 \,{\left ({\left (b x + 2\right )} b - 2 \, b\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + 2)^(3/2)*x^(5/2)),x, algorithm="giac")

[Out]

b^(7/2)/(((sqrt(b*x + 2)*sqrt(b) - sqrt((b*x + 2)*b - 2*b))^2 + 2*b)*abs(b)) + 1
/12*(5*(b*x + 2)*b^2*abs(b) - 12*b^2*abs(b))*sqrt(b*x + 2)/((b*x + 2)*b - 2*b)^(
3/2)