3.615 \(\int \frac{1}{x^{9/2} \sqrt{2+b x}} \, dx\)

Optimal. Leaf size=80 \[ \frac{2 b^3 \sqrt{b x+2}}{35 \sqrt{x}}-\frac{2 b^2 \sqrt{b x+2}}{35 x^{3/2}}+\frac{3 b \sqrt{b x+2}}{35 x^{5/2}}-\frac{\sqrt{b x+2}}{7 x^{7/2}} \]

[Out]

-Sqrt[2 + b*x]/(7*x^(7/2)) + (3*b*Sqrt[2 + b*x])/(35*x^(5/2)) - (2*b^2*Sqrt[2 +
b*x])/(35*x^(3/2)) + (2*b^3*Sqrt[2 + b*x])/(35*Sqrt[x])

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Rubi [A]  time = 0.0522062, antiderivative size = 80, 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 \[ \frac{2 b^3 \sqrt{b x+2}}{35 \sqrt{x}}-\frac{2 b^2 \sqrt{b x+2}}{35 x^{3/2}}+\frac{3 b \sqrt{b x+2}}{35 x^{5/2}}-\frac{\sqrt{b x+2}}{7 x^{7/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^(9/2)*Sqrt[2 + b*x]),x]

[Out]

-Sqrt[2 + b*x]/(7*x^(7/2)) + (3*b*Sqrt[2 + b*x])/(35*x^(5/2)) - (2*b^2*Sqrt[2 +
b*x])/(35*x^(3/2)) + (2*b^3*Sqrt[2 + b*x])/(35*Sqrt[x])

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Rubi in Sympy [A]  time = 5.96022, size = 73, normalized size = 0.91 \[ \frac{2 b^{3} \sqrt{b x + 2}}{35 \sqrt{x}} - \frac{2 b^{2} \sqrt{b x + 2}}{35 x^{\frac{3}{2}}} + \frac{3 b \sqrt{b x + 2}}{35 x^{\frac{5}{2}}} - \frac{\sqrt{b x + 2}}{7 x^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*b**3*sqrt(b*x + 2)/(35*sqrt(x)) - 2*b**2*sqrt(b*x + 2)/(35*x**(3/2)) + 3*b*sqr
t(b*x + 2)/(35*x**(5/2)) - sqrt(b*x + 2)/(7*x**(7/2))

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Mathematica [A]  time = 0.0205807, size = 40, normalized size = 0.5 \[ \frac{\sqrt{b x+2} \left (2 b^3 x^3-2 b^2 x^2+3 b x-5\right )}{35 x^{7/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^(9/2)*Sqrt[2 + b*x]),x]

[Out]

(Sqrt[2 + b*x]*(-5 + 3*b*x - 2*b^2*x^2 + 2*b^3*x^3))/(35*x^(7/2))

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Maple [A]  time = 0.006, size = 35, normalized size = 0.4 \[{\frac{2\,{b}^{3}{x}^{3}-2\,{b}^{2}{x}^{2}+3\,bx-5}{35}\sqrt{bx+2}{x}^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^(9/2)/(b*x+2)^(1/2),x)

[Out]

1/35*(b*x+2)^(1/2)*(2*b^3*x^3-2*b^2*x^2+3*b*x-5)/x^(7/2)

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Maxima [A]  time = 1.34723, size = 76, normalized size = 0.95 \[ \frac{\sqrt{b x + 2} b^{3}}{8 \, \sqrt{x}} - \frac{{\left (b x + 2\right )}^{\frac{3}{2}} b^{2}}{8 \, x^{\frac{3}{2}}} + \frac{3 \,{\left (b x + 2\right )}^{\frac{5}{2}} b}{40 \, x^{\frac{5}{2}}} - \frac{{\left (b x + 2\right )}^{\frac{7}{2}}}{56 \, x^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b*x + 2)*x^(9/2)),x, algorithm="maxima")

[Out]

1/8*sqrt(b*x + 2)*b^3/sqrt(x) - 1/8*(b*x + 2)^(3/2)*b^2/x^(3/2) + 3/40*(b*x + 2)
^(5/2)*b/x^(5/2) - 1/56*(b*x + 2)^(7/2)/x^(7/2)

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Fricas [A]  time = 0.246375, size = 46, normalized size = 0.57 \[ \frac{{\left (2 \, b^{3} x^{3} - 2 \, b^{2} x^{2} + 3 \, b x - 5\right )} \sqrt{b x + 2}}{35 \, x^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b*x + 2)*x^(9/2)),x, algorithm="fricas")

[Out]

1/35*(2*b^3*x^3 - 2*b^2*x^2 + 3*b*x - 5)*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(1/x**(9/2)/(b*x+2)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.213796, size = 92, normalized size = 1.15 \[ -\frac{{\left (35 \, b^{7} -{\left (35 \, b^{7} + 2 \,{\left ({\left (b x + 2\right )} b^{7} - 7 \, b^{7}\right )}{\left (b x + 2\right )}\right )}{\left (b x + 2\right )}\right )} \sqrt{b x + 2} b}{35 \,{\left ({\left (b x + 2\right )} b - 2 \, b\right )}^{\frac{7}{2}}{\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b*x + 2)*x^(9/2)),x, algorithm="giac")

[Out]

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