3.1645 \(\int \left (a+\frac{b}{x}\right ) x^{5/2} \, dx\)

Optimal. Leaf size=21 \[ \frac{2}{7} a x^{7/2}+\frac{2}{5} b x^{5/2} \]

[Out]

(2*b*x^(5/2))/5 + (2*a*x^(7/2))/7

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Rubi [A]  time = 0.0146677, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \frac{2}{7} a x^{7/2}+\frac{2}{5} b x^{5/2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x)*x^(5/2),x]

[Out]

(2*b*x^(5/2))/5 + (2*a*x^(7/2))/7

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Rubi in Sympy [A]  time = 2.83753, size = 19, normalized size = 0.9 \[ \frac{2 a x^{\frac{7}{2}}}{7} + \frac{2 b x^{\frac{5}{2}}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*a*x**(7/2)/7 + 2*b*x**(5/2)/5

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Mathematica [A]  time = 0.00662717, size = 17, normalized size = 0.81 \[ \frac{2}{35} x^{5/2} (5 a x+7 b) \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x)*x^(5/2),x]

[Out]

(2*x^(5/2)*(7*b + 5*a*x))/35

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Maple [A]  time = 0.005, size = 14, normalized size = 0.7 \[{\frac{10\,ax+14\,b}{35}{x}^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x)*x^(5/2),x)

[Out]

2/35*(5*a*x+7*b)*x^(5/2)

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Maxima [A]  time = 1.4446, size = 20, normalized size = 0.95 \[ \frac{2}{35} \,{\left (5 \, a + \frac{7 \, b}{x}\right )} x^{\frac{7}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/35*(5*a + 7*b/x)*x^(7/2)

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Fricas [A]  time = 0.227039, size = 24, normalized size = 1.14 \[ \frac{2}{35} \,{\left (5 \, a x^{3} + 7 \, b x^{2}\right )} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/35*(5*a*x^3 + 7*b*x^2)*sqrt(x)

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Sympy [A]  time = 5.92101, size = 19, normalized size = 0.9 \[ \frac{2 a x^{\frac{7}{2}}}{7} + \frac{2 b x^{\frac{5}{2}}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x)*x**(5/2),x)

[Out]

2*a*x**(7/2)/7 + 2*b*x**(5/2)/5

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GIAC/XCAS [A]  time = 0.22279, size = 18, normalized size = 0.86 \[ \frac{2}{7} \, a x^{\frac{7}{2}} + \frac{2}{5} \, b x^{\frac{5}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/7*a*x^(7/2) + 2/5*b*x^(5/2)