3.1760 \(\int \left (a+\frac{b}{x}\right )^{3/2} x^{5/2} \, dx\)

Optimal. Leaf size=48 \[ \frac{2 x^{7/2} \left (a+\frac{b}{x}\right )^{5/2}}{7 a}-\frac{4 b x^{5/2} \left (a+\frac{b}{x}\right )^{5/2}}{35 a^2} \]

[Out]

(-4*b*(a + b/x)^(5/2)*x^(5/2))/(35*a^2) + (2*(a + b/x)^(5/2)*x^(7/2))/(7*a)

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Rubi [A]  time = 0.0541184, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ \frac{2 x^{7/2} \left (a+\frac{b}{x}\right )^{5/2}}{7 a}-\frac{4 b x^{5/2} \left (a+\frac{b}{x}\right )^{5/2}}{35 a^2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x)^(3/2)*x^(5/2),x]

[Out]

(-4*b*(a + b/x)^(5/2)*x^(5/2))/(35*a^2) + (2*(a + b/x)^(5/2)*x^(7/2))/(7*a)

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Rubi in Sympy [A]  time = 4.19298, size = 39, normalized size = 0.81 \[ \frac{2 x^{\frac{7}{2}} \left (a + \frac{b}{x}\right )^{\frac{5}{2}}}{7 a} - \frac{4 b x^{\frac{5}{2}} \left (a + \frac{b}{x}\right )^{\frac{5}{2}}}{35 a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b/x)**(3/2)*x**(5/2),x)

[Out]

2*x**(7/2)*(a + b/x)**(5/2)/(7*a) - 4*b*x**(5/2)*(a + b/x)**(5/2)/(35*a**2)

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Mathematica [A]  time = 0.0447067, size = 38, normalized size = 0.79 \[ \frac{2 \sqrt{x} \sqrt{a+\frac{b}{x}} (a x+b)^2 (5 a x-2 b)}{35 a^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x)^(3/2)*x^(5/2),x]

[Out]

(2*Sqrt[a + b/x]*Sqrt[x]*(b + a*x)^2*(-2*b + 5*a*x))/(35*a^2)

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Maple [A]  time = 0.006, size = 33, normalized size = 0.7 \[{\frac{ \left ( 2\,ax+2\,b \right ) \left ( 5\,ax-2\,b \right ) }{35\,{a}^{2}} \left ({\frac{ax+b}{x}} \right ) ^{{\frac{3}{2}}}{x}^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x)^(3/2)*x^(5/2),x)

[Out]

2/35*(a*x+b)*(5*a*x-2*b)*x^(3/2)*((a*x+b)/x)^(3/2)/a^2

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Maxima [A]  time = 1.42424, size = 47, normalized size = 0.98 \[ \frac{2 \,{\left (5 \,{\left (a + \frac{b}{x}\right )}^{\frac{7}{2}} x^{\frac{7}{2}} - 7 \,{\left (a + \frac{b}{x}\right )}^{\frac{5}{2}} b x^{\frac{5}{2}}\right )}}{35 \, a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x)^(3/2)*x^(5/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.231821, size = 65, normalized size = 1.35 \[ \frac{2 \,{\left (5 \, a^{3} x^{3} + 8 \, a^{2} b x^{2} + a b^{2} x - 2 \, b^{3}\right )} \sqrt{x} \sqrt{\frac{a x + b}{x}}}{35 \, a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x)^(3/2)*x^(5/2),x, algorithm="fricas")

[Out]

2/35*(5*a^3*x^3 + 8*a^2*b*x^2 + a*b^2*x - 2*b^3)*sqrt(x)*sqrt((a*x + b)/x)/a^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((a+b/x)**(3/2)*x**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.240434, size = 122, normalized size = 2.54 \[ \frac{2}{15} \, b{\left (\frac{2 \, b^{\frac{5}{2}}}{a^{2}} + \frac{3 \,{\left (a x + b\right )}^{\frac{5}{2}} - 5 \,{\left (a x + b\right )}^{\frac{3}{2}} b}{a^{2}}\right )}{\rm sign}\left (x\right ) - \frac{2}{105} \, a{\left (\frac{8 \, b^{\frac{7}{2}}}{a^{3}} - \frac{15 \,{\left (a x + b\right )}^{\frac{7}{2}} - 42 \,{\left (a x + b\right )}^{\frac{5}{2}} b + 35 \,{\left (a x + b\right )}^{\frac{3}{2}} b^{2}}{a^{3}}\right )}{\rm sign}\left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x)^(3/2)*x^(5/2),x, algorithm="giac")

[Out]

2/15*b*(2*b^(5/2)/a^2 + (3*(a*x + b)^(5/2) - 5*(a*x + b)^(3/2)*b)/a^2)*sign(x) -
2/105*a*(8*b^(7/2)/a^3 - (15*(a*x + b)^(7/2) - 42*(a*x + b)^(5/2)*b + 35*(a*x +
b)^(3/2)*b^2)/a^3)*sign(x)