3.512 \(\int \frac{\sqrt{2+b x}}{x^{9/2}} \, dx\)

Optimal. Leaf size=59 \[ -\frac{2 b^2 (b x+2)^{3/2}}{105 x^{3/2}}+\frac{2 b (b x+2)^{3/2}}{35 x^{5/2}}-\frac{(b x+2)^{3/2}}{7 x^{7/2}} \]

[Out]

-(2 + b*x)^(3/2)/(7*x^(7/2)) + (2*b*(2 + b*x)^(3/2))/(35*x^(5/2)) - (2*b^2*(2 +
b*x)^(3/2))/(105*x^(3/2))

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Rubi [A]  time = 0.0359808, antiderivative size = 59, 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 \[ -\frac{2 b^2 (b x+2)^{3/2}}{105 x^{3/2}}+\frac{2 b (b x+2)^{3/2}}{35 x^{5/2}}-\frac{(b x+2)^{3/2}}{7 x^{7/2}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[2 + b*x]/x^(9/2),x]

[Out]

-(2 + b*x)^(3/2)/(7*x^(7/2)) + (2*b*(2 + b*x)^(3/2))/(35*x^(5/2)) - (2*b^2*(2 +
b*x)^(3/2))/(105*x^(3/2))

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Rubi in Sympy [A]  time = 4.11006, size = 53, normalized size = 0.9 \[ - \frac{2 b^{2} \left (b x + 2\right )^{\frac{3}{2}}}{105 x^{\frac{3}{2}}} + \frac{2 b \left (b x + 2\right )^{\frac{3}{2}}}{35 x^{\frac{5}{2}}} - \frac{\left (b x + 2\right )^{\frac{3}{2}}}{7 x^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+2)**(1/2)/x**(9/2),x)

[Out]

-2*b**2*(b*x + 2)**(3/2)/(105*x**(3/2)) + 2*b*(b*x + 2)**(3/2)/(35*x**(5/2)) - (
b*x + 2)**(3/2)/(7*x**(7/2))

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Mathematica [A]  time = 0.017793, size = 40, normalized size = 0.68 \[ -\frac{\sqrt{b x+2} \left (2 b^3 x^3-2 b^2 x^2+3 b x+30\right )}{105 x^{7/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[2 + b*x]/x^(9/2),x]

[Out]

-(Sqrt[2 + b*x]*(30 + 3*b*x - 2*b^2*x^2 + 2*b^3*x^3))/(105*x^(7/2))

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Maple [A]  time = 0.006, size = 27, normalized size = 0.5 \[ -{\frac{2\,{b}^{2}{x}^{2}-6\,bx+15}{105} \left ( bx+2 \right ) ^{{\frac{3}{2}}}{x}^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+2)^(1/2)/x^(9/2),x)

[Out]

-1/105*(b*x+2)^(3/2)*(2*b^2*x^2-6*b*x+15)/x^(7/2)

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Maxima [A]  time = 1.3399, size = 55, normalized size = 0.93 \[ -\frac{{\left (b x + 2\right )}^{\frac{3}{2}} b^{2}}{12 \, x^{\frac{3}{2}}} + \frac{{\left (b x + 2\right )}^{\frac{5}{2}} b}{10 \, x^{\frac{5}{2}}} - \frac{{\left (b x + 2\right )}^{\frac{7}{2}}}{28 \, x^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b*x + 2)/x^(9/2),x, algorithm="maxima")

[Out]

-1/12*(b*x + 2)^(3/2)*b^2/x^(3/2) + 1/10*(b*x + 2)^(5/2)*b/x^(5/2) - 1/28*(b*x +
2)^(7/2)/x^(7/2)

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Fricas [A]  time = 0.209734, size = 46, normalized size = 0.78 \[ -\frac{{\left (2 \, b^{3} x^{3} - 2 \, b^{2} x^{2} + 3 \, b x + 30\right )} \sqrt{b x + 2}}{105 \, x^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b*x + 2)/x^(9/2),x, algorithm="fricas")

[Out]

-1/105*(2*b^3*x^3 - 2*b^2*x^2 + 3*b*x + 30)*sqrt(b*x + 2)/x^(7/2)

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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((b*x+2)**(1/2)/x**(9/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.208748, size = 74, normalized size = 1.25 \[ -\frac{{\left (35 \, b^{7} + 2 \,{\left ({\left (b x + 2\right )} b^{7} - 7 \, b^{7}\right )}{\left (b x + 2\right )}\right )}{\left (b x + 2\right )}^{\frac{3}{2}} b}{105 \,{\left ({\left (b x + 2\right )} b - 2 \, b\right )}^{\frac{7}{2}}{\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b*x + 2)/x^(9/2),x, algorithm="giac")

[Out]

-1/105*(35*b^7 + 2*((b*x + 2)*b^7 - 7*b^7)*(b*x + 2))*(b*x + 2)^(3/2)*b/(((b*x +
2)*b - 2*b)^(7/2)*abs(b))