3.1995 \(\int \frac{\sqrt{a+\frac{b}{x^3}}}{x^{13}} \, dx\)

Optimal. Leaf size=80 \[ \frac{2 a^3 \left (a+\frac{b}{x^3}\right )^{3/2}}{9 b^4}-\frac{2 a^2 \left (a+\frac{b}{x^3}\right )^{5/2}}{5 b^4}-\frac{2 \left (a+\frac{b}{x^3}\right )^{9/2}}{27 b^4}+\frac{2 a \left (a+\frac{b}{x^3}\right )^{7/2}}{7 b^4} \]

[Out]

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

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Rubi [A]  time = 0.113603, antiderivative size = 80, 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 \[ \frac{2 a^3 \left (a+\frac{b}{x^3}\right )^{3/2}}{9 b^4}-\frac{2 a^2 \left (a+\frac{b}{x^3}\right )^{5/2}}{5 b^4}-\frac{2 \left (a+\frac{b}{x^3}\right )^{9/2}}{27 b^4}+\frac{2 a \left (a+\frac{b}{x^3}\right )^{7/2}}{7 b^4} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a + b/x^3]/x^13,x]

[Out]

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

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Rubi in Sympy [A]  time = 14.1343, size = 75, normalized size = 0.94 \[ \frac{2 a^{3} \left (a + \frac{b}{x^{3}}\right )^{\frac{3}{2}}}{9 b^{4}} - \frac{2 a^{2} \left (a + \frac{b}{x^{3}}\right )^{\frac{5}{2}}}{5 b^{4}} + \frac{2 a \left (a + \frac{b}{x^{3}}\right )^{\frac{7}{2}}}{7 b^{4}} - \frac{2 \left (a + \frac{b}{x^{3}}\right )^{\frac{9}{2}}}{27 b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b/x**3)**(1/2)/x**13,x)

[Out]

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

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Mathematica [A]  time = 0.0460779, size = 64, normalized size = 0.8 \[ \frac{2 \sqrt{a+\frac{b}{x^3}} \left (16 a^4 x^{12}-8 a^3 b x^9+6 a^2 b^2 x^6-5 a b^3 x^3-35 b^4\right )}{945 b^4 x^{12}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a + b/x^3]/x^13,x]

[Out]

(2*Sqrt[a + b/x^3]*(-35*b^4 - 5*a*b^3*x^3 + 6*a^2*b^2*x^6 - 8*a^3*b*x^9 + 16*a^4
*x^12))/(945*b^4*x^12)

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Maple [A]  time = 0.011, size = 61, normalized size = 0.8 \[{\frac{ \left ( 2\,a{x}^{3}+2\,b \right ) \left ( 16\,{a}^{3}{x}^{9}-24\,{a}^{2}b{x}^{6}+30\,a{b}^{2}{x}^{3}-35\,{b}^{3} \right ) }{945\,{x}^{12}{b}^{4}}\sqrt{{\frac{a{x}^{3}+b}{{x}^{3}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x^3)^(1/2)/x^13,x)

[Out]

2/945*(a*x^3+b)*(16*a^3*x^9-24*a^2*b*x^6+30*a*b^2*x^3-35*b^3)*((a*x^3+b)/x^3)^(1
/2)/x^12/b^4

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Maxima [A]  time = 1.43603, size = 86, normalized size = 1.08 \[ -\frac{2 \,{\left (a + \frac{b}{x^{3}}\right )}^{\frac{9}{2}}}{27 \, b^{4}} + \frac{2 \,{\left (a + \frac{b}{x^{3}}\right )}^{\frac{7}{2}} a}{7 \, b^{4}} - \frac{2 \,{\left (a + \frac{b}{x^{3}}\right )}^{\frac{5}{2}} a^{2}}{5 \, b^{4}} + \frac{2 \,{\left (a + \frac{b}{x^{3}}\right )}^{\frac{3}{2}} a^{3}}{9 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(a + b/x^3)/x^13,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.230959, size = 86, normalized size = 1.08 \[ \frac{2 \,{\left (16 \, a^{4} x^{12} - 8 \, a^{3} b x^{9} + 6 \, a^{2} b^{2} x^{6} - 5 \, a b^{3} x^{3} - 35 \, b^{4}\right )} \sqrt{\frac{a x^{3} + b}{x^{3}}}}{945 \, b^{4} x^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(a + b/x^3)/x^13,x, algorithm="fricas")

[Out]

2/945*(16*a^4*x^12 - 8*a^3*b*x^9 + 6*a^2*b^2*x^6 - 5*a*b^3*x^3 - 35*b^4)*sqrt((a
*x^3 + b)/x^3)/(b^4*x^12)

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Sympy [A]  time = 25.0441, size = 2317, normalized size = 28.96 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x**3)**(1/2)/x**13,x)

[Out]

32*a**(29/2)*b**(23/2)*x**30*sqrt(a*x**3/b + 1)/(945*a**(21/2)*b**15*x**(63/2) +
5670*a**(19/2)*b**16*x**(57/2) + 14175*a**(17/2)*b**17*x**(51/2) + 18900*a**(15
/2)*b**18*x**(45/2) + 14175*a**(13/2)*b**19*x**(39/2) + 5670*a**(11/2)*b**20*x**
(33/2) + 945*a**(9/2)*b**21*x**(27/2)) + 176*a**(27/2)*b**(25/2)*x**27*sqrt(a*x*
*3/b + 1)/(945*a**(21/2)*b**15*x**(63/2) + 5670*a**(19/2)*b**16*x**(57/2) + 1417
5*a**(17/2)*b**17*x**(51/2) + 18900*a**(15/2)*b**18*x**(45/2) + 14175*a**(13/2)*
b**19*x**(39/2) + 5670*a**(11/2)*b**20*x**(33/2) + 945*a**(9/2)*b**21*x**(27/2))
+ 396*a**(25/2)*b**(27/2)*x**24*sqrt(a*x**3/b + 1)/(945*a**(21/2)*b**15*x**(63/
2) + 5670*a**(19/2)*b**16*x**(57/2) + 14175*a**(17/2)*b**17*x**(51/2) + 18900*a*
*(15/2)*b**18*x**(45/2) + 14175*a**(13/2)*b**19*x**(39/2) + 5670*a**(11/2)*b**20
*x**(33/2) + 945*a**(9/2)*b**21*x**(27/2)) + 462*a**(23/2)*b**(29/2)*x**21*sqrt(
a*x**3/b + 1)/(945*a**(21/2)*b**15*x**(63/2) + 5670*a**(19/2)*b**16*x**(57/2) +
14175*a**(17/2)*b**17*x**(51/2) + 18900*a**(15/2)*b**18*x**(45/2) + 14175*a**(13
/2)*b**19*x**(39/2) + 5670*a**(11/2)*b**20*x**(33/2) + 945*a**(9/2)*b**21*x**(27
/2)) + 210*a**(21/2)*b**(31/2)*x**18*sqrt(a*x**3/b + 1)/(945*a**(21/2)*b**15*x**
(63/2) + 5670*a**(19/2)*b**16*x**(57/2) + 14175*a**(17/2)*b**17*x**(51/2) + 1890
0*a**(15/2)*b**18*x**(45/2) + 14175*a**(13/2)*b**19*x**(39/2) + 5670*a**(11/2)*b
**20*x**(33/2) + 945*a**(9/2)*b**21*x**(27/2)) - 378*a**(19/2)*b**(33/2)*x**15*s
qrt(a*x**3/b + 1)/(945*a**(21/2)*b**15*x**(63/2) + 5670*a**(19/2)*b**16*x**(57/2
) + 14175*a**(17/2)*b**17*x**(51/2) + 18900*a**(15/2)*b**18*x**(45/2) + 14175*a*
*(13/2)*b**19*x**(39/2) + 5670*a**(11/2)*b**20*x**(33/2) + 945*a**(9/2)*b**21*x*
*(27/2)) - 1134*a**(17/2)*b**(35/2)*x**12*sqrt(a*x**3/b + 1)/(945*a**(21/2)*b**1
5*x**(63/2) + 5670*a**(19/2)*b**16*x**(57/2) + 14175*a**(17/2)*b**17*x**(51/2) +
18900*a**(15/2)*b**18*x**(45/2) + 14175*a**(13/2)*b**19*x**(39/2) + 5670*a**(11
/2)*b**20*x**(33/2) + 945*a**(9/2)*b**21*x**(27/2)) - 1494*a**(15/2)*b**(37/2)*x
**9*sqrt(a*x**3/b + 1)/(945*a**(21/2)*b**15*x**(63/2) + 5670*a**(19/2)*b**16*x**
(57/2) + 14175*a**(17/2)*b**17*x**(51/2) + 18900*a**(15/2)*b**18*x**(45/2) + 141
75*a**(13/2)*b**19*x**(39/2) + 5670*a**(11/2)*b**20*x**(33/2) + 945*a**(9/2)*b**
21*x**(27/2)) - 1098*a**(13/2)*b**(39/2)*x**6*sqrt(a*x**3/b + 1)/(945*a**(21/2)*
b**15*x**(63/2) + 5670*a**(19/2)*b**16*x**(57/2) + 14175*a**(17/2)*b**17*x**(51/
2) + 18900*a**(15/2)*b**18*x**(45/2) + 14175*a**(13/2)*b**19*x**(39/2) + 5670*a*
*(11/2)*b**20*x**(33/2) + 945*a**(9/2)*b**21*x**(27/2)) - 430*a**(11/2)*b**(41/2
)*x**3*sqrt(a*x**3/b + 1)/(945*a**(21/2)*b**15*x**(63/2) + 5670*a**(19/2)*b**16*
x**(57/2) + 14175*a**(17/2)*b**17*x**(51/2) + 18900*a**(15/2)*b**18*x**(45/2) +
14175*a**(13/2)*b**19*x**(39/2) + 5670*a**(11/2)*b**20*x**(33/2) + 945*a**(9/2)*
b**21*x**(27/2)) - 70*a**(9/2)*b**(43/2)*sqrt(a*x**3/b + 1)/(945*a**(21/2)*b**15
*x**(63/2) + 5670*a**(19/2)*b**16*x**(57/2) + 14175*a**(17/2)*b**17*x**(51/2) +
18900*a**(15/2)*b**18*x**(45/2) + 14175*a**(13/2)*b**19*x**(39/2) + 5670*a**(11/
2)*b**20*x**(33/2) + 945*a**(9/2)*b**21*x**(27/2)) - 32*a**15*b**11*x**(63/2)/(9
45*a**(21/2)*b**15*x**(63/2) + 5670*a**(19/2)*b**16*x**(57/2) + 14175*a**(17/2)*
b**17*x**(51/2) + 18900*a**(15/2)*b**18*x**(45/2) + 14175*a**(13/2)*b**19*x**(39
/2) + 5670*a**(11/2)*b**20*x**(33/2) + 945*a**(9/2)*b**21*x**(27/2)) - 192*a**14
*b**12*x**(57/2)/(945*a**(21/2)*b**15*x**(63/2) + 5670*a**(19/2)*b**16*x**(57/2)
+ 14175*a**(17/2)*b**17*x**(51/2) + 18900*a**(15/2)*b**18*x**(45/2) + 14175*a**
(13/2)*b**19*x**(39/2) + 5670*a**(11/2)*b**20*x**(33/2) + 945*a**(9/2)*b**21*x**
(27/2)) - 480*a**13*b**13*x**(51/2)/(945*a**(21/2)*b**15*x**(63/2) + 5670*a**(19
/2)*b**16*x**(57/2) + 14175*a**(17/2)*b**17*x**(51/2) + 18900*a**(15/2)*b**18*x*
*(45/2) + 14175*a**(13/2)*b**19*x**(39/2) + 5670*a**(11/2)*b**20*x**(33/2) + 945
*a**(9/2)*b**21*x**(27/2)) - 640*a**12*b**14*x**(45/2)/(945*a**(21/2)*b**15*x**(
63/2) + 5670*a**(19/2)*b**16*x**(57/2) + 14175*a**(17/2)*b**17*x**(51/2) + 18900
*a**(15/2)*b**18*x**(45/2) + 14175*a**(13/2)*b**19*x**(39/2) + 5670*a**(11/2)*b*
*20*x**(33/2) + 945*a**(9/2)*b**21*x**(27/2)) - 480*a**11*b**15*x**(39/2)/(945*a
**(21/2)*b**15*x**(63/2) + 5670*a**(19/2)*b**16*x**(57/2) + 14175*a**(17/2)*b**1
7*x**(51/2) + 18900*a**(15/2)*b**18*x**(45/2) + 14175*a**(13/2)*b**19*x**(39/2)
+ 5670*a**(11/2)*b**20*x**(33/2) + 945*a**(9/2)*b**21*x**(27/2)) - 192*a**10*b**
16*x**(33/2)/(945*a**(21/2)*b**15*x**(63/2) + 5670*a**(19/2)*b**16*x**(57/2) + 1
4175*a**(17/2)*b**17*x**(51/2) + 18900*a**(15/2)*b**18*x**(45/2) + 14175*a**(13/
2)*b**19*x**(39/2) + 5670*a**(11/2)*b**20*x**(33/2) + 945*a**(9/2)*b**21*x**(27/
2)) - 32*a**9*b**17*x**(27/2)/(945*a**(21/2)*b**15*x**(63/2) + 5670*a**(19/2)*b*
*16*x**(57/2) + 14175*a**(17/2)*b**17*x**(51/2) + 18900*a**(15/2)*b**18*x**(45/2
) + 14175*a**(13/2)*b**19*x**(39/2) + 5670*a**(11/2)*b**20*x**(33/2) + 945*a**(9
/2)*b**21*x**(27/2))

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GIAC/XCAS [A]  time = 0.232382, size = 77, normalized size = 0.96 \[ -\frac{2 \,{\left (35 \,{\left (a + \frac{b}{x^{3}}\right )}^{\frac{9}{2}} - 135 \,{\left (a + \frac{b}{x^{3}}\right )}^{\frac{7}{2}} a + 189 \,{\left (a + \frac{b}{x^{3}}\right )}^{\frac{5}{2}} a^{2} - 105 \,{\left (a + \frac{b}{x^{3}}\right )}^{\frac{3}{2}} a^{3}\right )}}{945 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(a + b/x^3)/x^13,x, algorithm="giac")

[Out]

-2/945*(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^4