3.2013 \(\int \frac{\left (a+\frac{b}{x^3}\right )^{3/2}}{x^{10}} \, dx\)

Optimal. Leaf size=59 \[ -\frac{2 a^2 \left (a+\frac{b}{x^3}\right )^{5/2}}{15 b^3}-\frac{2 \left (a+\frac{b}{x^3}\right )^{9/2}}{27 b^3}+\frac{4 a \left (a+\frac{b}{x^3}\right )^{7/2}}{21 b^3} \]

[Out]

(-2*a^2*(a + b/x^3)^(5/2))/(15*b^3) + (4*a*(a + b/x^3)^(7/2))/(21*b^3) - (2*(a +
b/x^3)^(9/2))/(27*b^3)

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Rubi [A]  time = 0.0954765, 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 \[ -\frac{2 a^2 \left (a+\frac{b}{x^3}\right )^{5/2}}{15 b^3}-\frac{2 \left (a+\frac{b}{x^3}\right )^{9/2}}{27 b^3}+\frac{4 a \left (a+\frac{b}{x^3}\right )^{7/2}}{21 b^3} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x^3)^(3/2)/x^10,x]

[Out]

(-2*a^2*(a + b/x^3)^(5/2))/(15*b^3) + (4*a*(a + b/x^3)^(7/2))/(21*b^3) - (2*(a +
b/x^3)^(9/2))/(27*b^3)

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Rubi in Sympy [A]  time = 10.574, size = 54, normalized size = 0.92 \[ - \frac{2 a^{2} \left (a + \frac{b}{x^{3}}\right )^{\frac{5}{2}}}{15 b^{3}} + \frac{4 a \left (a + \frac{b}{x^{3}}\right )^{\frac{7}{2}}}{21 b^{3}} - \frac{2 \left (a + \frac{b}{x^{3}}\right )^{\frac{9}{2}}}{27 b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b/x**3)**(3/2)/x**10,x)

[Out]

-2*a**2*(a + b/x**3)**(5/2)/(15*b**3) + 4*a*(a + b/x**3)**(7/2)/(21*b**3) - 2*(a
+ b/x**3)**(9/2)/(27*b**3)

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Mathematica [A]  time = 0.0462347, size = 51, normalized size = 0.86 \[ -\frac{2 \sqrt{a+\frac{b}{x^3}} \left (a x^3+b\right )^2 \left (8 a^2 x^6-20 a b x^3+35 b^2\right )}{945 b^3 x^{12}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x^3)^(3/2)/x^10,x]

[Out]

(-2*Sqrt[a + b/x^3]*(b + a*x^3)^2*(35*b^2 - 20*a*b*x^3 + 8*a^2*x^6))/(945*b^3*x^
12)

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Maple [A]  time = 0.009, size = 50, normalized size = 0.9 \[ -{\frac{ \left ( 2\,a{x}^{3}+2\,b \right ) \left ( 8\,{a}^{2}{x}^{6}-20\,ab{x}^{3}+35\,{b}^{2} \right ) }{945\,{b}^{3}{x}^{9}} \left ({\frac{a{x}^{3}+b}{{x}^{3}}} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x^3)^(3/2)/x^10,x)

[Out]

-2/945*(a*x^3+b)*(8*a^2*x^6-20*a*b*x^3+35*b^2)*((a*x^3+b)/x^3)^(3/2)/b^3/x^9

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Maxima [A]  time = 1.42616, size = 63, normalized size = 1.07 \[ -\frac{2 \,{\left (a + \frac{b}{x^{3}}\right )}^{\frac{9}{2}}}{27 \, b^{3}} + \frac{4 \,{\left (a + \frac{b}{x^{3}}\right )}^{\frac{7}{2}} a}{21 \, b^{3}} - \frac{2 \,{\left (a + \frac{b}{x^{3}}\right )}^{\frac{5}{2}} a^{2}}{15 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x^3)^(3/2)/x^10,x, algorithm="maxima")

[Out]

-2/27*(a + b/x^3)^(9/2)/b^3 + 4/21*(a + b/x^3)^(7/2)*a/b^3 - 2/15*(a + b/x^3)^(5
/2)*a^2/b^3

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Fricas [A]  time = 0.239493, size = 86, normalized size = 1.46 \[ -\frac{2 \,{\left (8 \, a^{4} x^{12} - 4 \, a^{3} b x^{9} + 3 \, a^{2} b^{2} x^{6} + 50 \, a b^{3} x^{3} + 35 \, b^{4}\right )} \sqrt{\frac{a x^{3} + b}{x^{3}}}}{945 \, b^{3} x^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x^3)^(3/2)/x^10,x, algorithm="fricas")

[Out]

-2/945*(8*a^4*x^12 - 4*a^3*b*x^9 + 3*a^2*b^2*x^6 + 50*a*b^3*x^3 + 35*b^4)*sqrt((
a*x^3 + b)/x^3)/(b^3*x^12)

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Sympy [A]  time = 19.3299, size = 1001, normalized size = 16.97 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x**3)**(3/2)/x**10,x)

[Out]

-16*a**(23/2)*b**(9/2)*x**21*sqrt(a*x**3/b + 1)/(945*a**(15/2)*b**7*x**(45/2) +
2835*a**(13/2)*b**8*x**(39/2) + 2835*a**(11/2)*b**9*x**(33/2) + 945*a**(9/2)*b**
10*x**(27/2)) - 40*a**(21/2)*b**(11/2)*x**18*sqrt(a*x**3/b + 1)/(945*a**(15/2)*b
**7*x**(45/2) + 2835*a**(13/2)*b**8*x**(39/2) + 2835*a**(11/2)*b**9*x**(33/2) +
945*a**(9/2)*b**10*x**(27/2)) - 30*a**(19/2)*b**(13/2)*x**15*sqrt(a*x**3/b + 1)/
(945*a**(15/2)*b**7*x**(45/2) + 2835*a**(13/2)*b**8*x**(39/2) + 2835*a**(11/2)*b
**9*x**(33/2) + 945*a**(9/2)*b**10*x**(27/2)) - 110*a**(17/2)*b**(15/2)*x**12*sq
rt(a*x**3/b + 1)/(945*a**(15/2)*b**7*x**(45/2) + 2835*a**(13/2)*b**8*x**(39/2) +
2835*a**(11/2)*b**9*x**(33/2) + 945*a**(9/2)*b**10*x**(27/2)) - 380*a**(15/2)*b
**(17/2)*x**9*sqrt(a*x**3/b + 1)/(945*a**(15/2)*b**7*x**(45/2) + 2835*a**(13/2)*
b**8*x**(39/2) + 2835*a**(11/2)*b**9*x**(33/2) + 945*a**(9/2)*b**10*x**(27/2)) -
516*a**(13/2)*b**(19/2)*x**6*sqrt(a*x**3/b + 1)/(945*a**(15/2)*b**7*x**(45/2) +
2835*a**(13/2)*b**8*x**(39/2) + 2835*a**(11/2)*b**9*x**(33/2) + 945*a**(9/2)*b*
*10*x**(27/2)) - 310*a**(11/2)*b**(21/2)*x**3*sqrt(a*x**3/b + 1)/(945*a**(15/2)*
b**7*x**(45/2) + 2835*a**(13/2)*b**8*x**(39/2) + 2835*a**(11/2)*b**9*x**(33/2) +
945*a**(9/2)*b**10*x**(27/2)) - 70*a**(9/2)*b**(23/2)*sqrt(a*x**3/b + 1)/(945*a
**(15/2)*b**7*x**(45/2) + 2835*a**(13/2)*b**8*x**(39/2) + 2835*a**(11/2)*b**9*x*
*(33/2) + 945*a**(9/2)*b**10*x**(27/2)) + 16*a**12*b**4*x**(45/2)/(945*a**(15/2)
*b**7*x**(45/2) + 2835*a**(13/2)*b**8*x**(39/2) + 2835*a**(11/2)*b**9*x**(33/2)
+ 945*a**(9/2)*b**10*x**(27/2)) + 48*a**11*b**5*x**(39/2)/(945*a**(15/2)*b**7*x*
*(45/2) + 2835*a**(13/2)*b**8*x**(39/2) + 2835*a**(11/2)*b**9*x**(33/2) + 945*a*
*(9/2)*b**10*x**(27/2)) + 48*a**10*b**6*x**(33/2)/(945*a**(15/2)*b**7*x**(45/2)
+ 2835*a**(13/2)*b**8*x**(39/2) + 2835*a**(11/2)*b**9*x**(33/2) + 945*a**(9/2)*b
**10*x**(27/2)) + 16*a**9*b**7*x**(27/2)/(945*a**(15/2)*b**7*x**(45/2) + 2835*a*
*(13/2)*b**8*x**(39/2) + 2835*a**(11/2)*b**9*x**(33/2) + 945*a**(9/2)*b**10*x**(
27/2))

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GIAC/XCAS [A]  time = 0.246158, size = 143, normalized size = 2.42 \[ -\frac{2 \,{\left (\frac{3 \,{\left (15 \,{\left (a + \frac{b}{x^{3}}\right )}^{\frac{7}{2}} - 42 \,{\left (a + \frac{b}{x^{3}}\right )}^{\frac{5}{2}} a + 35 \,{\left (a + \frac{b}{x^{3}}\right )}^{\frac{3}{2}} a^{2}\right )} a}{b^{2}} + \frac{35 \,{\left (a + \frac{b}{x^{3}}\right )}^{\frac{9}{2}} - 135 \,{\left (a + \frac{b}{x^{3}}\right )}^{\frac{7}{2}} a + 189 \,{\left (a + \frac{b}{x^{3}}\right )}^{\frac{5}{2}} a^{2} - 105 \,{\left (a + \frac{b}{x^{3}}\right )}^{\frac{3}{2}} a^{3}}{b^{2}}\right )}}{945 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x^3)^(3/2)/x^10,x, algorithm="giac")

[Out]

-2/945*(3*(15*(a + b/x^3)^(7/2) - 42*(a + b/x^3)^(5/2)*a + 35*(a + b/x^3)^(3/2)*
a^2)*a/b^2 + (35*(a + b/x^3)^(9/2) - 135*(a + b/x^3)^(7/2)*a + 189*(a + b/x^3)^(
5/2)*a^2 - 105*(a + b/x^3)^(3/2)*a^3)/b^2)/b