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

Optimal. Leaf size=59 \[ -\frac{2 b^2 \sqrt{b x+2}}{15 \sqrt{x}}+\frac{2 b \sqrt{b x+2}}{15 x^{3/2}}-\frac{\sqrt{b x+2}}{5 x^{5/2}} \]

[Out]

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

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Rubi [A]  time = 0.0363981, 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 \[ -\frac{2 b^2 \sqrt{b x+2}}{15 \sqrt{x}}+\frac{2 b \sqrt{b x+2}}{15 x^{3/2}}-\frac{\sqrt{b x+2}}{5 x^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 4.43052, size = 53, normalized size = 0.9 \[ - \frac{2 b^{2} \sqrt{b x + 2}}{15 \sqrt{x}} + \frac{2 b \sqrt{b x + 2}}{15 x^{\frac{3}{2}}} - \frac{\sqrt{b x + 2}}{5 x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0194562, size = 32, normalized size = 0.54 \[ -\frac{\sqrt{b x+2} \left (2 b^2 x^2-2 b x+3\right )}{15 x^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.007, size = 27, normalized size = 0.5 \[ -{\frac{2\,{b}^{2}{x}^{2}-2\,bx+3}{15}\sqrt{bx+2}{x}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^(7/2)/(b*x+2)^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.34078, size = 55, normalized size = 0.93 \[ -\frac{\sqrt{b x + 2} b^{2}}{4 \, \sqrt{x}} + \frac{{\left (b x + 2\right )}^{\frac{3}{2}} b}{6 \, x^{\frac{3}{2}}} - \frac{{\left (b x + 2\right )}^{\frac{5}{2}}}{20 \, x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*sqrt(b*x + 2)*b^2/sqrt(x) + 1/6*(b*x + 2)^(3/2)*b/x^(3/2) - 1/20*(b*x + 2)^
(5/2)/x^(5/2)

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Fricas [A]  time = 0.208186, size = 35, normalized size = 0.59 \[ -\frac{{\left (2 \, b^{2} x^{2} - 2 \, b x + 3\right )} \sqrt{b x + 2}}{15 \, x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.213216, size = 74, normalized size = 1.25 \[ -\frac{{\left (15 \, b^{5} + 2 \,{\left ({\left (b x + 2\right )} b^{5} - 5 \, b^{5}\right )}{\left (b x + 2\right )}\right )} \sqrt{b x + 2} b}{15 \,{\left ({\left (b x + 2\right )} b - 2 \, b\right )}^{\frac{5}{2}}{\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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