3.1736 \(\int \frac{1}{\left (a+\frac{b}{x}\right )^{3/2} x^4} \, dx\)

Optimal. Leaf size=55 \[ \frac{2 a^2}{b^3 \sqrt{a+\frac{b}{x}}}+\frac{4 a \sqrt{a+\frac{b}{x}}}{b^3}-\frac{2 \left (a+\frac{b}{x}\right )^{3/2}}{3 b^3} \]

[Out]

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

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Rubi [A]  time = 0.0781181, antiderivative size = 55, 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 \[ \frac{2 a^2}{b^3 \sqrt{a+\frac{b}{x}}}+\frac{4 a \sqrt{a+\frac{b}{x}}}{b^3}-\frac{2 \left (a+\frac{b}{x}\right )^{3/2}}{3 b^3} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b/x)^(3/2)*x^4),x]

[Out]

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

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Rubi in Sympy [A]  time = 9.74647, size = 46, normalized size = 0.84 \[ \frac{2 a^{2}}{b^{3} \sqrt{a + \frac{b}{x}}} + \frac{4 a \sqrt{a + \frac{b}{x}}}{b^{3}} - \frac{2 \left (a + \frac{b}{x}\right )^{\frac{3}{2}}}{3 b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a+b/x)**(3/2)/x**4,x)

[Out]

2*a**2/(b**3*sqrt(a + b/x)) + 4*a*sqrt(a + b/x)/b**3 - 2*(a + b/x)**(3/2)/(3*b**
3)

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Mathematica [A]  time = 0.0411181, size = 46, normalized size = 0.84 \[ \sqrt{\frac{a x+b}{x}} \left (\frac{16 a}{3 b^3}-\frac{2 a}{b^2 (a x+b)}-\frac{2}{3 b^2 x}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b/x)^(3/2)*x^4),x]

[Out]

Sqrt[(b + a*x)/x]*((16*a)/(3*b^3) - 2/(3*b^2*x) - (2*a)/(b^2*(b + a*x)))

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Maple [A]  time = 0.009, size = 44, normalized size = 0.8 \[{\frac{ \left ( 2\,ax+2\,b \right ) \left ( 8\,{a}^{2}{x}^{2}+4\,abx-{b}^{2} \right ) }{3\,{b}^{3}{x}^{3}} \left ({\frac{ax+b}{x}} \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x)^(3/2)/x^4,x)

[Out]

2/3*(a*x+b)*(8*a^2*x^2+4*a*b*x-b^2)/x^3/b^3/((a*x+b)/x)^(3/2)

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Maxima [A]  time = 1.43559, size = 63, normalized size = 1.15 \[ -\frac{2 \,{\left (a + \frac{b}{x}\right )}^{\frac{3}{2}}}{3 \, b^{3}} + \frac{4 \, \sqrt{a + \frac{b}{x}} a}{b^{3}} + \frac{2 \, a^{2}}{\sqrt{a + \frac{b}{x}} b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x)^(3/2)*x^4),x, algorithm="maxima")

[Out]

-2/3*(a + b/x)^(3/2)/b^3 + 4*sqrt(a + b/x)*a/b^3 + 2*a^2/(sqrt(a + b/x)*b^3)

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Fricas [A]  time = 0.236646, size = 51, normalized size = 0.93 \[ \frac{2 \,{\left (8 \, a^{2} x^{2} + 4 \, a b x - b^{2}\right )}}{3 \, b^{3} x^{2} \sqrt{\frac{a x + b}{x}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x)^(3/2)*x^4),x, algorithm="fricas")

[Out]

2/3*(8*a^2*x^2 + 4*a*b*x - b^2)/(b^3*x^2*sqrt((a*x + b)/x))

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Sympy [A]  time = 7.9295, size = 457, normalized size = 8.31 \[ \frac{16 a^{\frac{9}{2}} b^{\frac{7}{2}} x^{3} \sqrt{\frac{a x}{b} + 1}}{3 a^{\frac{7}{2}} b^{6} x^{\frac{7}{2}} + 6 a^{\frac{5}{2}} b^{7} x^{\frac{5}{2}} + 3 a^{\frac{3}{2}} b^{8} x^{\frac{3}{2}}} + \frac{24 a^{\frac{7}{2}} b^{\frac{9}{2}} x^{2} \sqrt{\frac{a x}{b} + 1}}{3 a^{\frac{7}{2}} b^{6} x^{\frac{7}{2}} + 6 a^{\frac{5}{2}} b^{7} x^{\frac{5}{2}} + 3 a^{\frac{3}{2}} b^{8} x^{\frac{3}{2}}} + \frac{6 a^{\frac{5}{2}} b^{\frac{11}{2}} x \sqrt{\frac{a x}{b} + 1}}{3 a^{\frac{7}{2}} b^{6} x^{\frac{7}{2}} + 6 a^{\frac{5}{2}} b^{7} x^{\frac{5}{2}} + 3 a^{\frac{3}{2}} b^{8} x^{\frac{3}{2}}} - \frac{2 a^{\frac{3}{2}} b^{\frac{13}{2}} \sqrt{\frac{a x}{b} + 1}}{3 a^{\frac{7}{2}} b^{6} x^{\frac{7}{2}} + 6 a^{\frac{5}{2}} b^{7} x^{\frac{5}{2}} + 3 a^{\frac{3}{2}} b^{8} x^{\frac{3}{2}}} - \frac{16 a^{5} b^{3} x^{\frac{7}{2}}}{3 a^{\frac{7}{2}} b^{6} x^{\frac{7}{2}} + 6 a^{\frac{5}{2}} b^{7} x^{\frac{5}{2}} + 3 a^{\frac{3}{2}} b^{8} x^{\frac{3}{2}}} - \frac{32 a^{4} b^{4} x^{\frac{5}{2}}}{3 a^{\frac{7}{2}} b^{6} x^{\frac{7}{2}} + 6 a^{\frac{5}{2}} b^{7} x^{\frac{5}{2}} + 3 a^{\frac{3}{2}} b^{8} x^{\frac{3}{2}}} - \frac{16 a^{3} b^{5} x^{\frac{3}{2}}}{3 a^{\frac{7}{2}} b^{6} x^{\frac{7}{2}} + 6 a^{\frac{5}{2}} b^{7} x^{\frac{5}{2}} + 3 a^{\frac{3}{2}} b^{8} x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a+b/x)**(3/2)/x**4,x)

[Out]

16*a**(9/2)*b**(7/2)*x**3*sqrt(a*x/b + 1)/(3*a**(7/2)*b**6*x**(7/2) + 6*a**(5/2)
*b**7*x**(5/2) + 3*a**(3/2)*b**8*x**(3/2)) + 24*a**(7/2)*b**(9/2)*x**2*sqrt(a*x/
b + 1)/(3*a**(7/2)*b**6*x**(7/2) + 6*a**(5/2)*b**7*x**(5/2) + 3*a**(3/2)*b**8*x*
*(3/2)) + 6*a**(5/2)*b**(11/2)*x*sqrt(a*x/b + 1)/(3*a**(7/2)*b**6*x**(7/2) + 6*a
**(5/2)*b**7*x**(5/2) + 3*a**(3/2)*b**8*x**(3/2)) - 2*a**(3/2)*b**(13/2)*sqrt(a*
x/b + 1)/(3*a**(7/2)*b**6*x**(7/2) + 6*a**(5/2)*b**7*x**(5/2) + 3*a**(3/2)*b**8*
x**(3/2)) - 16*a**5*b**3*x**(7/2)/(3*a**(7/2)*b**6*x**(7/2) + 6*a**(5/2)*b**7*x*
*(5/2) + 3*a**(3/2)*b**8*x**(3/2)) - 32*a**4*b**4*x**(5/2)/(3*a**(7/2)*b**6*x**(
7/2) + 6*a**(5/2)*b**7*x**(5/2) + 3*a**(3/2)*b**8*x**(3/2)) - 16*a**3*b**5*x**(3
/2)/(3*a**(7/2)*b**6*x**(7/2) + 6*a**(5/2)*b**7*x**(5/2) + 3*a**(3/2)*b**8*x**(3
/2))

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GIAC/XCAS [A]  time = 0.256802, size = 93, normalized size = 1.69 \[ \frac{2}{3} \, b{\left (\frac{3 \, a^{2}}{b^{4} \sqrt{\frac{a x + b}{x}}} + \frac{6 \, a b^{8} \sqrt{\frac{a x + b}{x}} - \frac{{\left (a x + b\right )} b^{8} \sqrt{\frac{a x + b}{x}}}{x}}{b^{12}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x)^(3/2)*x^4),x, algorithm="giac")

[Out]

2/3*b*(3*a^2/(b^4*sqrt((a*x + b)/x)) + (6*a*b^8*sqrt((a*x + b)/x) - (a*x + b)*b^
8*sqrt((a*x + b)/x)/x)/b^12)