3.622 \(\int \frac{1}{x^{7/2} (2+b x)^{3/2}} \, dx\)

Optimal. Leaf size=74 \[ -\frac{2 b^2 \sqrt{b x+2}}{5 \sqrt{x}}+\frac{2 b \sqrt{b x+2}}{5 x^{3/2}}-\frac{3 \sqrt{b x+2}}{5 x^{5/2}}+\frac{1}{x^{5/2} \sqrt{b x+2}} \]

[Out]

1/(x^(5/2)*Sqrt[2 + b*x]) - (3*Sqrt[2 + b*x])/(5*x^(5/2)) + (2*b*Sqrt[2 + b*x])/
(5*x^(3/2)) - (2*b^2*Sqrt[2 + b*x])/(5*Sqrt[x])

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Rubi [A]  time = 0.0532417, antiderivative size = 74, 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 \[ -\frac{2 b^2 \sqrt{b x+2}}{5 \sqrt{x}}+\frac{2 b \sqrt{b x+2}}{5 x^{3/2}}-\frac{3 \sqrt{b x+2}}{5 x^{5/2}}+\frac{1}{x^{5/2} \sqrt{b x+2}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^(7/2)*(2 + b*x)^(3/2)),x]

[Out]

1/(x^(5/2)*Sqrt[2 + b*x]) - (3*Sqrt[2 + b*x])/(5*x^(5/2)) + (2*b*Sqrt[2 + b*x])/
(5*x^(3/2)) - (2*b^2*Sqrt[2 + b*x])/(5*Sqrt[x])

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Rubi in Sympy [A]  time = 5.93663, size = 70, normalized size = 0.95 \[ - \frac{2 b^{2} \sqrt{b x + 2}}{5 \sqrt{x}} + \frac{2 b \sqrt{b x + 2}}{5 x^{\frac{3}{2}}} - \frac{3 \sqrt{b x + 2}}{5 x^{\frac{5}{2}}} + \frac{1}{x^{\frac{5}{2}} \sqrt{b x + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**(7/2)/(b*x+2)**(3/2),x)

[Out]

-2*b**2*sqrt(b*x + 2)/(5*sqrt(x)) + 2*b*sqrt(b*x + 2)/(5*x**(3/2)) - 3*sqrt(b*x
+ 2)/(5*x**(5/2)) + 1/(x**(5/2)*sqrt(b*x + 2))

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Mathematica [A]  time = 0.0247523, size = 39, normalized size = 0.53 \[ \frac{-2 b^3 x^3-2 b^2 x^2+b x-1}{5 x^{5/2} \sqrt{b x+2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^(7/2)*(2 + b*x)^(3/2)),x]

[Out]

(-1 + b*x - 2*b^2*x^2 - 2*b^3*x^3)/(5*x^(5/2)*Sqrt[2 + b*x])

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Maple [A]  time = 0.007, size = 35, normalized size = 0.5 \[ -{\frac{2\,{b}^{3}{x}^{3}+2\,{b}^{2}{x}^{2}-bx+1}{5}{x}^{-{\frac{5}{2}}}{\frac{1}{\sqrt{bx+2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^(7/2)/(b*x+2)^(3/2),x)

[Out]

-1/5*(2*b^3*x^3+2*b^2*x^2-b*x+1)/x^(5/2)/(b*x+2)^(1/2)

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Maxima [A]  time = 1.34621, size = 76, normalized size = 1.03 \[ -\frac{b^{3} \sqrt{x}}{8 \, \sqrt{b x + 2}} - \frac{3 \, \sqrt{b x + 2} b^{2}}{8 \, \sqrt{x}} + \frac{{\left (b x + 2\right )}^{\frac{3}{2}} b}{8 \, x^{\frac{3}{2}}} - \frac{{\left (b x + 2\right )}^{\frac{5}{2}}}{40 \, x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + 2)^(3/2)*x^(7/2)),x, algorithm="maxima")

[Out]

-1/8*b^3*sqrt(x)/sqrt(b*x + 2) - 3/8*sqrt(b*x + 2)*b^2/sqrt(x) + 1/8*(b*x + 2)^(
3/2)*b/x^(3/2) - 1/40*(b*x + 2)^(5/2)/x^(5/2)

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Fricas [A]  time = 0.214832, size = 46, normalized size = 0.62 \[ -\frac{2 \, b^{3} x^{3} + 2 \, b^{2} x^{2} - b x + 1}{5 \, \sqrt{b x + 2} x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + 2)^(3/2)*x^(7/2)),x, algorithm="fricas")

[Out]

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

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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**(7/2)/(b*x+2)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.220174, size = 144, normalized size = 1.95 \[ -\frac{b^{\frac{9}{2}}}{2 \,{\left ({\left (\sqrt{b x + 2} \sqrt{b} - \sqrt{{\left (b x + 2\right )} b - 2 \, b}\right )}^{2} + 2 \, b\right )}{\left | b \right |}} - \frac{{\left (\frac{60 \, b^{6}}{{\left | b \right |}} +{\left (\frac{11 \,{\left (b x + 2\right )} b^{6}}{{\left | b \right |}} - \frac{50 \, b^{6}}{{\left | b \right |}}\right )}{\left (b x + 2\right )}\right )} \sqrt{b x + 2}}{40 \,{\left ({\left (b x + 2\right )} b - 2 \, b\right )}^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + 2)^(3/2)*x^(7/2)),x, algorithm="giac")

[Out]

-1/2*b^(9/2)/(((sqrt(b*x + 2)*sqrt(b) - sqrt((b*x + 2)*b - 2*b))^2 + 2*b)*abs(b)
) - 1/40*(60*b^6/abs(b) + (11*(b*x + 2)*b^6/abs(b) - 50*b^6/abs(b))*(b*x + 2))*s
qrt(b*x + 2)/((b*x + 2)*b - 2*b)^(5/2)