3.1769 \(\int \left (a+\frac{b}{x}\right )^{5/2} x^{5/2} \, dx\)

Optimal. Leaf size=23 \[ \frac{2 x^{7/2} \left (a+\frac{b}{x}\right )^{7/2}}{7 a} \]

[Out]

(2*(a + b/x)^(7/2)*x^(7/2))/(7*a)

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Rubi [A]  time = 0.0262264, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059 \[ \frac{2 x^{7/2} \left (a+\frac{b}{x}\right )^{7/2}}{7 a} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x)^(5/2)*x^(5/2),x]

[Out]

(2*(a + b/x)^(7/2)*x^(7/2))/(7*a)

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Rubi in Sympy [A]  time = 2.65608, size = 17, normalized size = 0.74 \[ \frac{2 x^{\frac{7}{2}} \left (a + \frac{b}{x}\right )^{\frac{7}{2}}}{7 a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b/x)**(5/2)*x**(5/2),x)

[Out]

2*x**(7/2)*(a + b/x)**(7/2)/(7*a)

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Mathematica [A]  time = 0.0457233, size = 30, normalized size = 1.3 \[ \frac{2 \sqrt{x} \sqrt{a+\frac{b}{x}} (a x+b)^3}{7 a} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x)^(5/2)*x^(5/2),x]

[Out]

(2*Sqrt[a + b/x]*Sqrt[x]*(b + a*x)^3)/(7*a)

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Maple [A]  time = 0.004, size = 25, normalized size = 1.1 \[{\frac{2\,ax+2\,b}{7\,a} \left ({\frac{ax+b}{x}} \right ) ^{{\frac{5}{2}}}{x}^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x)^(5/2)*x^(5/2),x)

[Out]

2/7*(a*x+b)*((a*x+b)/x)^(5/2)*x^(5/2)/a

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Maxima [A]  time = 1.43819, size = 23, normalized size = 1. \[ \frac{2 \,{\left (a + \frac{b}{x}\right )}^{\frac{7}{2}} x^{\frac{7}{2}}}{7 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x)^(5/2)*x^(5/2),x, algorithm="maxima")

[Out]

2/7*(a + b/x)^(7/2)*x^(7/2)/a

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Fricas [A]  time = 0.232119, size = 62, normalized size = 2.7 \[ \frac{2 \,{\left (a^{3} x^{3} + 3 \, a^{2} b x^{2} + 3 \, a b^{2} x + b^{3}\right )} \sqrt{x} \sqrt{\frac{a x + b}{x}}}{7 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x)^(5/2)*x^(5/2),x, algorithm="fricas")

[Out]

2/7*(a^3*x^3 + 3*a^2*b*x^2 + 3*a*b^2*x + b^3)*sqrt(x)*sqrt((a*x + b)/x)/a

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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((a+b/x)**(5/2)*x**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.23479, size = 81, normalized size = 3.52 \[ \frac{2 \,{\left (15 \,{\left (a x + b\right )}^{\frac{7}{2}} - 42 \,{\left (a x + b\right )}^{\frac{5}{2}} b + 70 \,{\left (a x + b\right )}^{\frac{3}{2}} b^{2} + 14 \,{\left (3 \,{\left (a x + b\right )}^{\frac{5}{2}} - 5 \,{\left (a x + b\right )}^{\frac{3}{2}} b\right )} b\right )}}{105 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x)^(5/2)*x^(5/2),x, algorithm="giac")

[Out]

2/105*(15*(a*x + b)^(7/2) - 42*(a*x + b)^(5/2)*b + 70*(a*x + b)^(3/2)*b^2 + 14*(
3*(a*x + b)^(5/2) - 5*(a*x + b)^(3/2)*b)*b)/a