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

Optimal. Leaf size=71 \[ -\frac{2 \sqrt{b x+2}}{3 x^{3/2}}+\frac{1}{x^{3/2} \sqrt{b x+2}}+\frac{1}{3 x^{3/2} (b x+2)^{3/2}}+\frac{2 b \sqrt{b x+2}}{3 \sqrt{x}} \]

[Out]

1/(3*x^(3/2)*(2 + b*x)^(3/2)) + 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 = 0.0449746, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ -\frac{2 \sqrt{b x+2}}{3 x^{3/2}}+\frac{1}{x^{3/2} \sqrt{b x+2}}+\frac{1}{3 x^{3/2} (b x+2)^{3/2}}+\frac{2 b \sqrt{b x+2}}{3 \sqrt{x}} \]

Antiderivative was successfully verified.

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

[Out]

1/(3*x^(3/2)*(2 + b*x)^(3/2)) + 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 = 5.42585, size = 66, normalized size = 0.93 \[ \frac{2 b \sqrt{b x + 2}}{3 \sqrt{x}} - \frac{2 \sqrt{b x + 2}}{3 x^{\frac{3}{2}}} + \frac{1}{x^{\frac{3}{2}} \sqrt{b x + 2}} + \frac{1}{3 x^{\frac{3}{2}} \left (b x + 2\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

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Mathematica [A]  time = 0.0260629, size = 40, normalized size = 0.56 \[ \frac{2 b^3 x^3+6 b^2 x^2+3 b x-1}{3 x^{3/2} (b x+2)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.006, size = 35, normalized size = 0.5 \[{\frac{2\,{b}^{3}{x}^{3}+6\,{b}^{2}{x}^{2}+3\,bx-1}{3}{x}^{-{\frac{3}{2}}} \left ( bx+2 \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^(5/2)/(b*x+2)^(5/2),x)

[Out]

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

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Maxima [A]  time = 1.34373, size = 74, normalized size = 1.04 \[ \frac{3 \, \sqrt{b x + 2} b}{8 \, \sqrt{x}} - \frac{{\left (b^{3} - \frac{9 \,{\left (b x + 2\right )} b^{2}}{x}\right )} x^{\frac{3}{2}}}{24 \,{\left (b x + 2\right )}^{\frac{3}{2}}} - \frac{{\left (b x + 2\right )}^{\frac{3}{2}}}{24 \, x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/8*sqrt(b*x + 2)*b/sqrt(x) - 1/24*(b^3 - 9*(b*x + 2)*b^2/x)*x^(3/2)/(b*x + 2)^(
3/2) - 1/24*(b*x + 2)^(3/2)/x^(3/2)

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Fricas [A]  time = 0.212732, size = 61, normalized size = 0.86 \[ \frac{2 \, b^{3} x^{3} + 6 \, b^{2} x^{2} + 3 \, b x - 1}{3 \,{\left (b x^{2} + 2 \, x\right )} \sqrt{b x + 2} \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.231582, size = 213, normalized size = 3. \[ \frac{{\left (4 \,{\left (b x + 2\right )} b^{2}{\left | b \right |} - 9 \, b^{2}{\left | b \right |}\right )} \sqrt{b x + 2}}{12 \,{\left ({\left (b x + 2\right )} b - 2 \, b\right )}^{\frac{3}{2}}} + \frac{3 \,{\left (\sqrt{b x + 2} \sqrt{b} - \sqrt{{\left (b x + 2\right )} b - 2 \, b}\right )}^{4} b^{\frac{7}{2}} + 18 \,{\left (\sqrt{b x + 2} \sqrt{b} - \sqrt{{\left (b x + 2\right )} b - 2 \, b}\right )}^{2} b^{\frac{9}{2}} + 16 \, b^{\frac{11}{2}}}{3 \,{\left ({\left (\sqrt{b x + 2} \sqrt{b} - \sqrt{{\left (b x + 2\right )} b - 2 \, b}\right )}^{2} + 2 \, b\right )}^{3}{\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/12*(4*(b*x + 2)*b^2*abs(b) - 9*b^2*abs(b))*sqrt(b*x + 2)/((b*x + 2)*b - 2*b)^(
3/2) + 1/3*(3*(sqrt(b*x + 2)*sqrt(b) - sqrt((b*x + 2)*b - 2*b))^4*b^(7/2) + 18*(
sqrt(b*x + 2)*sqrt(b) - sqrt((b*x + 2)*b - 2*b))^2*b^(9/2) + 16*b^(11/2))/(((sqr
t(b*x + 2)*sqrt(b) - sqrt((b*x + 2)*b - 2*b))^2 + 2*b)^3*abs(b))