3.1706 \(\int \frac{\left (a+\frac{b}{x}\right )^{3/2}}{x^4} \, dx\)

Optimal. Leaf size=59 \[ -\frac{2 a^2 \left (a+\frac{b}{x}\right )^{5/2}}{5 b^3}-\frac{2 \left (a+\frac{b}{x}\right )^{9/2}}{9 b^3}+\frac{4 a \left (a+\frac{b}{x}\right )^{7/2}}{7 b^3} \]

[Out]

(-2*a^2*(a + b/x)^(5/2))/(5*b^3) + (4*a*(a + b/x)^(7/2))/(7*b^3) - (2*(a + b/x)^
(9/2))/(9*b^3)

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Rubi [A]  time = 0.0781626, antiderivative size = 59, 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 \[ -\frac{2 a^2 \left (a+\frac{b}{x}\right )^{5/2}}{5 b^3}-\frac{2 \left (a+\frac{b}{x}\right )^{9/2}}{9 b^3}+\frac{4 a \left (a+\frac{b}{x}\right )^{7/2}}{7 b^3} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x)^(3/2)/x^4,x]

[Out]

(-2*a^2*(a + b/x)^(5/2))/(5*b^3) + (4*a*(a + b/x)^(7/2))/(7*b^3) - (2*(a + b/x)^
(9/2))/(9*b^3)

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Rubi in Sympy [A]  time = 10.0348, size = 49, normalized size = 0.83 \[ - \frac{2 a^{2} \left (a + \frac{b}{x}\right )^{\frac{5}{2}}}{5 b^{3}} + \frac{4 a \left (a + \frac{b}{x}\right )^{\frac{7}{2}}}{7 b^{3}} - \frac{2 \left (a + \frac{b}{x}\right )^{\frac{9}{2}}}{9 b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*a**2*(a + b/x)**(5/2)/(5*b**3) + 4*a*(a + b/x)**(7/2)/(7*b**3) - 2*(a + b/x)*
*(9/2)/(9*b**3)

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Mathematica [A]  time = 0.038733, size = 47, normalized size = 0.8 \[ -\frac{2 \sqrt{a+\frac{b}{x}} (a x+b)^2 \left (8 a^2 x^2-20 a b x+35 b^2\right )}{315 b^3 x^4} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[a + b/x]*(b + a*x)^2*(35*b^2 - 20*a*b*x + 8*a^2*x^2))/(315*b^3*x^4)

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Maple [A]  time = 0.008, size = 44, normalized size = 0.8 \[ -{\frac{ \left ( 2\,ax+2\,b \right ) \left ( 8\,{a}^{2}{x}^{2}-20\,abx+35\,{b}^{2} \right ) }{315\,{b}^{3}{x}^{3}} \left ({\frac{ax+b}{x}} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x)^(3/2)/x^4,x)

[Out]

-2/315*(a*x+b)*(8*a^2*x^2-20*a*b*x+35*b^2)*((a*x+b)/x)^(3/2)/b^3/x^3

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Maxima [A]  time = 1.44267, size = 63, normalized size = 1.07 \[ -\frac{2 \,{\left (a + \frac{b}{x}\right )}^{\frac{9}{2}}}{9 \, b^{3}} + \frac{4 \,{\left (a + \frac{b}{x}\right )}^{\frac{7}{2}} a}{7 \, b^{3}} - \frac{2 \,{\left (a + \frac{b}{x}\right )}^{\frac{5}{2}} a^{2}}{5 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/9*(a + b/x)^(9/2)/b^3 + 4/7*(a + b/x)^(7/2)*a/b^3 - 2/5*(a + b/x)^(5/2)*a^2/b
^3

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Fricas [A]  time = 0.221235, size = 81, normalized size = 1.37 \[ -\frac{2 \,{\left (8 \, a^{4} x^{4} - 4 \, a^{3} b x^{3} + 3 \, a^{2} b^{2} x^{2} + 50 \, a b^{3} x + 35 \, b^{4}\right )} \sqrt{\frac{a x + b}{x}}}{315 \, b^{3} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/315*(8*a^4*x^4 - 4*a^3*b*x^3 + 3*a^2*b^2*x^2 + 50*a*b^3*x + 35*b^4)*sqrt((a*x
+ b)/x)/(b^3*x^4)

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Sympy [A]  time = 7.54556, size = 986, normalized size = 16.71 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-16*a**(23/2)*b**(9/2)*x**7*sqrt(a*x/b + 1)/(315*a**(15/2)*b**7*x**(15/2) + 945*
a**(13/2)*b**8*x**(13/2) + 945*a**(11/2)*b**9*x**(11/2) + 315*a**(9/2)*b**10*x**
(9/2)) - 40*a**(21/2)*b**(11/2)*x**6*sqrt(a*x/b + 1)/(315*a**(15/2)*b**7*x**(15/
2) + 945*a**(13/2)*b**8*x**(13/2) + 945*a**(11/2)*b**9*x**(11/2) + 315*a**(9/2)*
b**10*x**(9/2)) - 30*a**(19/2)*b**(13/2)*x**5*sqrt(a*x/b + 1)/(315*a**(15/2)*b**
7*x**(15/2) + 945*a**(13/2)*b**8*x**(13/2) + 945*a**(11/2)*b**9*x**(11/2) + 315*
a**(9/2)*b**10*x**(9/2)) - 110*a**(17/2)*b**(15/2)*x**4*sqrt(a*x/b + 1)/(315*a**
(15/2)*b**7*x**(15/2) + 945*a**(13/2)*b**8*x**(13/2) + 945*a**(11/2)*b**9*x**(11
/2) + 315*a**(9/2)*b**10*x**(9/2)) - 380*a**(15/2)*b**(17/2)*x**3*sqrt(a*x/b + 1
)/(315*a**(15/2)*b**7*x**(15/2) + 945*a**(13/2)*b**8*x**(13/2) + 945*a**(11/2)*b
**9*x**(11/2) + 315*a**(9/2)*b**10*x**(9/2)) - 516*a**(13/2)*b**(19/2)*x**2*sqrt
(a*x/b + 1)/(315*a**(15/2)*b**7*x**(15/2) + 945*a**(13/2)*b**8*x**(13/2) + 945*a
**(11/2)*b**9*x**(11/2) + 315*a**(9/2)*b**10*x**(9/2)) - 310*a**(11/2)*b**(21/2)
*x*sqrt(a*x/b + 1)/(315*a**(15/2)*b**7*x**(15/2) + 945*a**(13/2)*b**8*x**(13/2)
+ 945*a**(11/2)*b**9*x**(11/2) + 315*a**(9/2)*b**10*x**(9/2)) - 70*a**(9/2)*b**(
23/2)*sqrt(a*x/b + 1)/(315*a**(15/2)*b**7*x**(15/2) + 945*a**(13/2)*b**8*x**(13/
2) + 945*a**(11/2)*b**9*x**(11/2) + 315*a**(9/2)*b**10*x**(9/2)) + 16*a**12*b**4
*x**(15/2)/(315*a**(15/2)*b**7*x**(15/2) + 945*a**(13/2)*b**8*x**(13/2) + 945*a*
*(11/2)*b**9*x**(11/2) + 315*a**(9/2)*b**10*x**(9/2)) + 48*a**11*b**5*x**(13/2)/
(315*a**(15/2)*b**7*x**(15/2) + 945*a**(13/2)*b**8*x**(13/2) + 945*a**(11/2)*b**
9*x**(11/2) + 315*a**(9/2)*b**10*x**(9/2)) + 48*a**10*b**6*x**(11/2)/(315*a**(15
/2)*b**7*x**(15/2) + 945*a**(13/2)*b**8*x**(13/2) + 945*a**(11/2)*b**9*x**(11/2)
+ 315*a**(9/2)*b**10*x**(9/2)) + 16*a**9*b**7*x**(9/2)/(315*a**(15/2)*b**7*x**(
15/2) + 945*a**(13/2)*b**8*x**(13/2) + 945*a**(11/2)*b**9*x**(11/2) + 315*a**(9/
2)*b**10*x**(9/2))

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GIAC/XCAS [A]  time = 0.261963, size = 281, normalized size = 4.76 \[ \frac{2 \,{\left (420 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{6} a^{3}{\rm sign}\left (x\right ) + 1575 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{5} a^{\frac{5}{2}} b{\rm sign}\left (x\right ) + 2583 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{4} a^{2} b^{2}{\rm sign}\left (x\right ) + 2310 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{3} a^{\frac{3}{2}} b^{3}{\rm sign}\left (x\right ) + 1170 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{2} a b^{4}{\rm sign}\left (x\right ) + 315 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )} \sqrt{a} b^{5}{\rm sign}\left (x\right ) + 35 \, b^{6}{\rm sign}\left (x\right )\right )}}{315 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/315*(420*(sqrt(a)*x - sqrt(a*x^2 + b*x))^6*a^3*sign(x) + 1575*(sqrt(a)*x - sqr
t(a*x^2 + b*x))^5*a^(5/2)*b*sign(x) + 2583*(sqrt(a)*x - sqrt(a*x^2 + b*x))^4*a^2
*b^2*sign(x) + 2310*(sqrt(a)*x - sqrt(a*x^2 + b*x))^3*a^(3/2)*b^3*sign(x) + 1170
*(sqrt(a)*x - sqrt(a*x^2 + b*x))^2*a*b^4*sign(x) + 315*(sqrt(a)*x - sqrt(a*x^2 +
b*x))*sqrt(a)*b^5*sign(x) + 35*b^6*sign(x))/(sqrt(a)*x - sqrt(a*x^2 + b*x))^9