3.511 \(\int \frac{\sqrt{2+b x}}{x^{7/2}} \, dx\)

Optimal. Leaf size=38 \[ \frac{b (b x+2)^{3/2}}{15 x^{3/2}}-\frac{(b x+2)^{3/2}}{5 x^{5/2}} \]

[Out]

-(2 + b*x)^(3/2)/(5*x^(5/2)) + (b*(2 + b*x)^(3/2))/(15*x^(3/2))

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Rubi [A]  time = 0.0223227, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{b (b x+2)^{3/2}}{15 x^{3/2}}-\frac{(b x+2)^{3/2}}{5 x^{5/2}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[2 + b*x]/x^(7/2),x]

[Out]

-(2 + b*x)^(3/2)/(5*x^(5/2)) + (b*(2 + b*x)^(3/2))/(15*x^(3/2))

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Rubi in Sympy [A]  time = 2.86492, size = 31, normalized size = 0.82 \[ \frac{b \left (b x + 2\right )^{\frac{3}{2}}}{15 x^{\frac{3}{2}}} - \frac{\left (b x + 2\right )^{\frac{3}{2}}}{5 x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b*(b*x + 2)**(3/2)/(15*x**(3/2)) - (b*x + 2)**(3/2)/(5*x**(5/2))

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Mathematica [A]  time = 0.0151269, size = 31, normalized size = 0.82 \[ \frac{\sqrt{b x+2} \left (b^2 x^2-b x-6\right )}{15 x^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[2 + b*x]*(-6 - b*x + b^2*x^2))/(15*x^(5/2))

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Maple [A]  time = 0.005, size = 18, normalized size = 0.5 \[{\frac{bx-3}{15} \left ( bx+2 \right ) ^{{\frac{3}{2}}}{x}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+2)^(1/2)/x^(7/2),x)

[Out]

1/15*(b*x+2)^(3/2)*(b*x-3)/x^(5/2)

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Maxima [A]  time = 1.34219, size = 35, normalized size = 0.92 \[ \frac{{\left (b x + 2\right )}^{\frac{3}{2}} b}{6 \, x^{\frac{3}{2}}} - \frac{{\left (b x + 2\right )}^{\frac{5}{2}}}{10 \, x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*(b*x + 2)^(3/2)*b/x^(3/2) - 1/10*(b*x + 2)^(5/2)/x^(5/2)

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Fricas [A]  time = 0.209882, size = 34, normalized size = 0.89 \[ \frac{{\left (b^{2} x^{2} - b x - 6\right )} \sqrt{b x + 2}}{15 \, x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/15*(b^2*x^2 - b*x - 6)*sqrt(b*x + 2)/x^(5/2)

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Sympy [A]  time = 140.378, size = 56, normalized size = 1.47 \[ \frac{b^{\frac{5}{2}} \sqrt{1 + \frac{2}{b x}}}{15} - \frac{b^{\frac{3}{2}} \sqrt{1 + \frac{2}{b x}}}{15 x} - \frac{2 \sqrt{b} \sqrt{1 + \frac{2}{b x}}}{5 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b**(5/2)*sqrt(1 + 2/(b*x))/15 - b**(3/2)*sqrt(1 + 2/(b*x))/(15*x) - 2*sqrt(b)*sq
rt(1 + 2/(b*x))/(5*x**2)

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GIAC/XCAS [A]  time = 0.208415, size = 57, normalized size = 1.5 \[ \frac{{\left ({\left (b x + 2\right )} b^{5} - 5 \, b^{5}\right )}{\left (b x + 2\right )}^{\frac{3}{2}} b}{15 \,{\left ({\left (b x + 2\right )} b - 2 \, b\right )}^{\frac{5}{2}}{\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/15*((b*x + 2)*b^5 - 5*b^5)*(b*x + 2)^(3/2)*b/(((b*x + 2)*b - 2*b)^(5/2)*abs(b)
)