3.1651 \(\int \left (a+\frac{b}{x}\right )^2 x^{5/2} \, dx\)

Optimal. Leaf size=36 \[ \frac{2}{7} a^2 x^{7/2}+\frac{4}{5} a b x^{5/2}+\frac{2}{3} b^2 x^{3/2} \]

[Out]

(2*b^2*x^(3/2))/3 + (4*a*b*x^(5/2))/5 + (2*a^2*x^(7/2))/7

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Rubi [A]  time = 0.0351325, antiderivative size = 36, 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 \[ \frac{2}{7} a^2 x^{7/2}+\frac{4}{5} a b x^{5/2}+\frac{2}{3} b^2 x^{3/2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x)^2*x^(5/2),x]

[Out]

(2*b^2*x^(3/2))/3 + (4*a*b*x^(5/2))/5 + (2*a^2*x^(7/2))/7

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Rubi in Sympy [A]  time = 5.48002, size = 34, normalized size = 0.94 \[ \frac{2 a^{2} x^{\frac{7}{2}}}{7} + \frac{4 a b x^{\frac{5}{2}}}{5} + \frac{2 b^{2} x^{\frac{3}{2}}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*a**2*x**(7/2)/7 + 4*a*b*x**(5/2)/5 + 2*b**2*x**(3/2)/3

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Mathematica [A]  time = 0.0118147, size = 28, normalized size = 0.78 \[ \frac{2}{105} x^{3/2} \left (15 a^2 x^2+42 a b x+35 b^2\right ) \]

Antiderivative was successfully verified.

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

[Out]

(2*x^(3/2)*(35*b^2 + 42*a*b*x + 15*a^2*x^2))/105

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Maple [A]  time = 0.007, size = 25, normalized size = 0.7 \[{\frac{30\,{a}^{2}{x}^{2}+84\,abx+70\,{b}^{2}}{105}{x}^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x)^2*x^(5/2),x)

[Out]

2/105*(15*a^2*x^2+42*a*b*x+35*b^2)*x^(3/2)

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Maxima [A]  time = 1.43676, size = 35, normalized size = 0.97 \[ \frac{2}{105} \,{\left (15 \, a^{2} + \frac{42 \, a b}{x} + \frac{35 \, b^{2}}{x^{2}}\right )} x^{\frac{7}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/105*(15*a^2 + 42*a*b/x + 35*b^2/x^2)*x^(7/2)

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Fricas [A]  time = 0.230686, size = 36, normalized size = 1. \[ \frac{2}{105} \,{\left (15 \, a^{2} x^{3} + 42 \, a b x^{2} + 35 \, b^{2} x\right )} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/105*(15*a^2*x^3 + 42*a*b*x^2 + 35*b^2*x)*sqrt(x)

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Sympy [A]  time = 9.34088, size = 34, normalized size = 0.94 \[ \frac{2 a^{2} x^{\frac{7}{2}}}{7} + \frac{4 a b x^{\frac{5}{2}}}{5} + \frac{2 b^{2} x^{\frac{3}{2}}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*a**2*x**(7/2)/7 + 4*a*b*x**(5/2)/5 + 2*b**2*x**(3/2)/3

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GIAC/XCAS [A]  time = 0.226156, size = 32, normalized size = 0.89 \[ \frac{2}{7} \, a^{2} x^{\frac{7}{2}} + \frac{4}{5} \, a b x^{\frac{5}{2}} + \frac{2}{3} \, b^{2} x^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/7*a^2*x^(7/2) + 4/5*a*b*x^(5/2) + 2/3*b^2*x^(3/2)