3.1716 \(\int \frac{\left (a+\frac{b}{x}\right )^{5/2}}{x^3} \, dx\)

Optimal. Leaf size=38 \[ \frac{2 a \left (a+\frac{b}{x}\right )^{7/2}}{7 b^2}-\frac{2 \left (a+\frac{b}{x}\right )^{9/2}}{9 b^2} \]

[Out]

(2*a*(a + b/x)^(7/2))/(7*b^2) - (2*(a + b/x)^(9/2))/(9*b^2)

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Rubi [A]  time = 0.0556546, antiderivative size = 38, 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 \[ \frac{2 a \left (a+\frac{b}{x}\right )^{7/2}}{7 b^2}-\frac{2 \left (a+\frac{b}{x}\right )^{9/2}}{9 b^2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x)^(5/2)/x^3,x]

[Out]

(2*a*(a + b/x)^(7/2))/(7*b^2) - (2*(a + b/x)^(9/2))/(9*b^2)

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Rubi in Sympy [A]  time = 6.93288, size = 31, normalized size = 0.82 \[ \frac{2 a \left (a + \frac{b}{x}\right )^{\frac{7}{2}}}{7 b^{2}} - \frac{2 \left (a + \frac{b}{x}\right )^{\frac{9}{2}}}{9 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*a*(a + b/x)**(7/2)/(7*b**2) - 2*(a + b/x)**(9/2)/(9*b**2)

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Mathematica [A]  time = 0.0434502, size = 36, normalized size = 0.95 \[ \frac{2 \sqrt{a+\frac{b}{x}} (a x+b)^3 (2 a x-7 b)}{63 b^2 x^4} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[a + b/x]*(b + a*x)^3*(-7*b + 2*a*x))/(63*b^2*x^4)

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Maple [A]  time = 0.007, size = 33, normalized size = 0.9 \[{\frac{ \left ( 2\,ax+2\,b \right ) \left ( 2\,ax-7\,b \right ) }{63\,{b}^{2}{x}^{2}} \left ({\frac{ax+b}{x}} \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x)^(5/2)/x^3,x)

[Out]

2/63*(a*x+b)*(2*a*x-7*b)*((a*x+b)/x)^(5/2)/b^2/x^2

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Maxima [A]  time = 1.41531, size = 41, normalized size = 1.08 \[ -\frac{2 \,{\left (a + \frac{b}{x}\right )}^{\frac{9}{2}}}{9 \, b^{2}} + \frac{2 \,{\left (a + \frac{b}{x}\right )}^{\frac{7}{2}} a}{7 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/9*(a + b/x)^(9/2)/b^2 + 2/7*(a + b/x)^(7/2)*a/b^2

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Fricas [A]  time = 0.227142, size = 81, normalized size = 2.13 \[ \frac{2 \,{\left (2 \, a^{4} x^{4} - a^{3} b x^{3} - 15 \, a^{2} b^{2} x^{2} - 19 \, a b^{3} x - 7 \, b^{4}\right )} \sqrt{\frac{a x + b}{x}}}{63 \, b^{2} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/63*(2*a^4*x^4 - a^3*b*x^3 - 15*a^2*b^2*x^2 - 19*a*b^3*x - 7*b^4)*sqrt((a*x + b
)/x)/(b^2*x^4)

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Sympy [A]  time = 7.77481, size = 416, normalized size = 10.95 \[ \frac{4 a^{\frac{19}{2}} b^{\frac{3}{2}} x^{5} \sqrt{\frac{a x}{b} + 1}}{63 a^{\frac{11}{2}} b^{3} x^{\frac{11}{2}} + 63 a^{\frac{9}{2}} b^{4} x^{\frac{9}{2}}} + \frac{2 a^{\frac{17}{2}} b^{\frac{5}{2}} x^{4} \sqrt{\frac{a x}{b} + 1}}{63 a^{\frac{11}{2}} b^{3} x^{\frac{11}{2}} + 63 a^{\frac{9}{2}} b^{4} x^{\frac{9}{2}}} - \frac{32 a^{\frac{15}{2}} b^{\frac{7}{2}} x^{3} \sqrt{\frac{a x}{b} + 1}}{63 a^{\frac{11}{2}} b^{3} x^{\frac{11}{2}} + 63 a^{\frac{9}{2}} b^{4} x^{\frac{9}{2}}} - \frac{68 a^{\frac{13}{2}} b^{\frac{9}{2}} x^{2} \sqrt{\frac{a x}{b} + 1}}{63 a^{\frac{11}{2}} b^{3} x^{\frac{11}{2}} + 63 a^{\frac{9}{2}} b^{4} x^{\frac{9}{2}}} - \frac{52 a^{\frac{11}{2}} b^{\frac{11}{2}} x \sqrt{\frac{a x}{b} + 1}}{63 a^{\frac{11}{2}} b^{3} x^{\frac{11}{2}} + 63 a^{\frac{9}{2}} b^{4} x^{\frac{9}{2}}} - \frac{14 a^{\frac{9}{2}} b^{\frac{13}{2}} \sqrt{\frac{a x}{b} + 1}}{63 a^{\frac{11}{2}} b^{3} x^{\frac{11}{2}} + 63 a^{\frac{9}{2}} b^{4} x^{\frac{9}{2}}} - \frac{4 a^{10} b x^{\frac{11}{2}}}{63 a^{\frac{11}{2}} b^{3} x^{\frac{11}{2}} + 63 a^{\frac{9}{2}} b^{4} x^{\frac{9}{2}}} - \frac{4 a^{9} b^{2} x^{\frac{9}{2}}}{63 a^{\frac{11}{2}} b^{3} x^{\frac{11}{2}} + 63 a^{\frac{9}{2}} b^{4} x^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4*a**(19/2)*b**(3/2)*x**5*sqrt(a*x/b + 1)/(63*a**(11/2)*b**3*x**(11/2) + 63*a**(
9/2)*b**4*x**(9/2)) + 2*a**(17/2)*b**(5/2)*x**4*sqrt(a*x/b + 1)/(63*a**(11/2)*b*
*3*x**(11/2) + 63*a**(9/2)*b**4*x**(9/2)) - 32*a**(15/2)*b**(7/2)*x**3*sqrt(a*x/
b + 1)/(63*a**(11/2)*b**3*x**(11/2) + 63*a**(9/2)*b**4*x**(9/2)) - 68*a**(13/2)*
b**(9/2)*x**2*sqrt(a*x/b + 1)/(63*a**(11/2)*b**3*x**(11/2) + 63*a**(9/2)*b**4*x*
*(9/2)) - 52*a**(11/2)*b**(11/2)*x*sqrt(a*x/b + 1)/(63*a**(11/2)*b**3*x**(11/2)
+ 63*a**(9/2)*b**4*x**(9/2)) - 14*a**(9/2)*b**(13/2)*sqrt(a*x/b + 1)/(63*a**(11/
2)*b**3*x**(11/2) + 63*a**(9/2)*b**4*x**(9/2)) - 4*a**10*b*x**(11/2)/(63*a**(11/
2)*b**3*x**(11/2) + 63*a**(9/2)*b**4*x**(9/2)) - 4*a**9*b**2*x**(9/2)/(63*a**(11
/2)*b**3*x**(11/2) + 63*a**(9/2)*b**4*x**(9/2))

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GIAC/XCAS [A]  time = 0.269879, size = 323, normalized size = 8.5 \[ \frac{2 \,{\left (63 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{7} a^{\frac{7}{2}}{\rm sign}\left (x\right ) + 273 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{6} a^{3} b{\rm sign}\left (x\right ) + 567 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{5} a^{\frac{5}{2}} b^{2}{\rm sign}\left (x\right ) + 693 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{4} a^{2} b^{3}{\rm sign}\left (x\right ) + 525 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{3} a^{\frac{3}{2}} b^{4}{\rm sign}\left (x\right ) + 243 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{2} a b^{5}{\rm sign}\left (x\right ) + 63 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )} \sqrt{a} b^{6}{\rm sign}\left (x\right ) + 7 \, b^{7}{\rm sign}\left (x\right )\right )}}{63 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/63*(63*(sqrt(a)*x - sqrt(a*x^2 + b*x))^7*a^(7/2)*sign(x) + 273*(sqrt(a)*x - sq
rt(a*x^2 + b*x))^6*a^3*b*sign(x) + 567*(sqrt(a)*x - sqrt(a*x^2 + b*x))^5*a^(5/2)
*b^2*sign(x) + 693*(sqrt(a)*x - sqrt(a*x^2 + b*x))^4*a^2*b^3*sign(x) + 525*(sqrt
(a)*x - sqrt(a*x^2 + b*x))^3*a^(3/2)*b^4*sign(x) + 243*(sqrt(a)*x - sqrt(a*x^2 +
b*x))^2*a*b^5*sign(x) + 63*(sqrt(a)*x - sqrt(a*x^2 + b*x))*sqrt(a)*b^6*sign(x)
+ 7*b^7*sign(x))/(sqrt(a)*x - sqrt(a*x^2 + b*x))^9