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

Optimal. Leaf size=55 \[ -\frac{2 \sqrt{b x+2}}{3 \sqrt{x}}+\frac{2}{3 \sqrt{x} \sqrt{b x+2}}+\frac{1}{3 \sqrt{x} (b x+2)^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.0324738, antiderivative size = 55, 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 \[ -\frac{2 \sqrt{b x+2}}{3 \sqrt{x}}+\frac{2}{3 \sqrt{x} \sqrt{b x+2}}+\frac{1}{3 \sqrt{x} (b x+2)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 4.53927, size = 49, normalized size = 0.89 \[ - \frac{2 \sqrt{b x + 2}}{3 \sqrt{x}} + \frac{2}{3 \sqrt{x} \sqrt{b x + 2}} + \frac{1}{3 \sqrt{x} \left (b x + 2\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0223111, size = 32, normalized size = 0.58 \[ -\frac{2 b^2 x^2+6 b x+3}{3 \sqrt{x} (b x+2)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

-(3 + 6*b*x + 2*b^2*x^2)/(3*Sqrt[x]*(2 + b*x)^(3/2))

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Maple [A]  time = 0.006, size = 27, normalized size = 0.5 \[ -{\frac{2\,{b}^{2}{x}^{2}+6\,bx+3}{3} \left ( bx+2 \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^(3/2)/(b*x+2)^(5/2),x)

[Out]

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

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Maxima [A]  time = 1.32382, size = 54, normalized size = 0.98 \[ \frac{{\left (b^{2} - \frac{6 \,{\left (b x + 2\right )} b}{x}\right )} x^{\frac{3}{2}}}{12 \,{\left (b x + 2\right )}^{\frac{3}{2}}} - \frac{\sqrt{b x + 2}}{4 \, \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/12*(b^2 - 6*(b*x + 2)*b/x)*x^(3/2)/(b*x + 2)^(3/2) - 1/4*sqrt(b*x + 2)/sqrt(x)

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Fricas [A]  time = 0.209134, size = 35, normalized size = 0.64 \[ -\frac{2 \, b^{2} x^{2} + 6 \, b x + 3}{3 \,{\left (b x + 2\right )}^{\frac{3}{2}} \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 102.687, size = 117, normalized size = 2.13 \[ - \frac{2 b^{\frac{13}{2}} x^{2} \sqrt{1 + \frac{2}{b x}}}{3 b^{6} x^{2} + 12 b^{5} x + 12 b^{4}} - \frac{6 b^{\frac{11}{2}} x \sqrt{1 + \frac{2}{b x}}}{3 b^{6} x^{2} + 12 b^{5} x + 12 b^{4}} - \frac{3 b^{\frac{9}{2}} \sqrt{1 + \frac{2}{b x}}}{3 b^{6} x^{2} + 12 b^{5} x + 12 b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.229389, size = 196, normalized size = 3.56 \[ -\frac{\sqrt{b x + 2} b^{2}}{4 \, \sqrt{{\left (b x + 2\right )} b - 2 \, b}{\left | b \right |}} - \frac{3 \,{\left (\sqrt{b x + 2} \sqrt{b} - \sqrt{{\left (b x + 2\right )} b - 2 \, b}\right )}^{4} b^{\frac{5}{2}} + 24 \,{\left (\sqrt{b x + 2} \sqrt{b} - \sqrt{{\left (b x + 2\right )} b - 2 \, b}\right )}^{2} b^{\frac{7}{2}} + 20 \, b^{\frac{9}{2}}}{3 \,{\left ({\left (\sqrt{b x + 2} \sqrt{b} - \sqrt{{\left (b x + 2\right )} b - 2 \, b}\right )}^{2} + 2 \, b\right )}^{3}{\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*sqrt(b*x + 2)*b^2/(sqrt((b*x + 2)*b - 2*b)*abs(b)) - 1/3*(3*(sqrt(b*x + 2)*
sqrt(b) - sqrt((b*x + 2)*b - 2*b))^4*b^(5/2) + 24*(sqrt(b*x + 2)*sqrt(b) - sqrt(
(b*x + 2)*b - 2*b))^2*b^(7/2) + 20*b^(9/2))/(((sqrt(b*x + 2)*sqrt(b) - sqrt((b*x
+ 2)*b - 2*b))^2 + 2*b)^3*abs(b))