3.2041 \(\int \frac{1}{\left (a+\frac{b}{x^3}\right )^{3/2} x^{13}} \, dx\)
Optimal. Leaf size=78 \[ -\frac{2 a^3}{3 b^4 \sqrt{a+\frac{b}{x^3}}}-\frac{2 a^2 \sqrt{a+\frac{b}{x^3}}}{b^4}+\frac{2 a \left (a+\frac{b}{x^3}\right )^{3/2}}{3 b^4}-\frac{2 \left (a+\frac{b}{x^3}\right )^{5/2}}{15 b^4} \]
[Out]
(-2*a^3)/(3*b^4*Sqrt[a + b/x^3]) - (2*a^2*Sqrt[a + b/x^3])/b^4 + (2*a*(a + b/x^3
)^(3/2))/(3*b^4) - (2*(a + b/x^3)^(5/2))/(15*b^4)
_______________________________________________________________________________________
Rubi [A] time = 0.118436, antiderivative size = 78, 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
\[ -\frac{2 a^3}{3 b^4 \sqrt{a+\frac{b}{x^3}}}-\frac{2 a^2 \sqrt{a+\frac{b}{x^3}}}{b^4}+\frac{2 a \left (a+\frac{b}{x^3}\right )^{3/2}}{3 b^4}-\frac{2 \left (a+\frac{b}{x^3}\right )^{5/2}}{15 b^4} \]
Antiderivative was successfully verified.
[In] Int[1/((a + b/x^3)^(3/2)*x^13),x]
[Out]
(-2*a^3)/(3*b^4*Sqrt[a + b/x^3]) - (2*a^2*Sqrt[a + b/x^3])/b^4 + (2*a*(a + b/x^3
)^(3/2))/(3*b^4) - (2*(a + b/x^3)^(5/2))/(15*b^4)
_______________________________________________________________________________________
Rubi in Sympy [A] time = 14.0165, size = 73, normalized size = 0.94 \[ - \frac{2 a^{3}}{3 b^{4} \sqrt{a + \frac{b}{x^{3}}}} - \frac{2 a^{2} \sqrt{a + \frac{b}{x^{3}}}}{b^{4}} + \frac{2 a \left (a + \frac{b}{x^{3}}\right )^{\frac{3}{2}}}{3 b^{4}} - \frac{2 \left (a + \frac{b}{x^{3}}\right )^{\frac{5}{2}}}{15 b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(a+b/x**3)**(3/2)/x**13,x)
[Out]
-2*a**3/(3*b**4*sqrt(a + b/x**3)) - 2*a**2*sqrt(a + b/x**3)/b**4 + 2*a*(a + b/x*
*3)**(3/2)/(3*b**4) - 2*(a + b/x**3)**(5/2)/(15*b**4)
_______________________________________________________________________________________
Mathematica [A] time = 0.0542749, size = 51, normalized size = 0.65 \[ -\frac{2 \left (16 a^3 x^9+8 a^2 b x^6-2 a b^2 x^3+b^3\right )}{15 b^4 x^9 \sqrt{a+\frac{b}{x^3}}} \]
Antiderivative was successfully verified.
[In] Integrate[1/((a + b/x^3)^(3/2)*x^13),x]
[Out]
(-2*(b^3 - 2*a*b^2*x^3 + 8*a^2*b*x^6 + 16*a^3*x^9))/(15*b^4*Sqrt[a + b/x^3]*x^9)
_______________________________________________________________________________________
Maple [A] time = 0.01, size = 59, normalized size = 0.8 \[ -{\frac{ \left ( 2\,a{x}^{3}+2\,b \right ) \left ( 16\,{a}^{3}{x}^{9}+8\,{a}^{2}b{x}^{6}-2\,a{b}^{2}{x}^{3}+{b}^{3} \right ) }{15\,{x}^{12}{b}^{4}} \left ({\frac{a{x}^{3}+b}{{x}^{3}}} \right ) ^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(a+b/x^3)^(3/2)/x^13,x)
[Out]
-2/15*(a*x^3+b)*(16*a^3*x^9+8*a^2*b*x^6-2*a*b^2*x^3+b^3)/x^12/b^4/((a*x^3+b)/x^3
)^(3/2)
_______________________________________________________________________________________
Maxima [A] time = 1.43794, size = 86, normalized size = 1.1 \[ -\frac{2 \,{\left (a + \frac{b}{x^{3}}\right )}^{\frac{5}{2}}}{15 \, b^{4}} + \frac{2 \,{\left (a + \frac{b}{x^{3}}\right )}^{\frac{3}{2}} a}{3 \, b^{4}} - \frac{2 \, \sqrt{a + \frac{b}{x^{3}}} a^{2}}{b^{4}} - \frac{2 \, a^{3}}{3 \, \sqrt{a + \frac{b}{x^{3}}} b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((a + b/x^3)^(3/2)*x^13),x, algorithm="maxima")
[Out]
-2/15*(a + b/x^3)^(5/2)/b^4 + 2/3*(a + b/x^3)^(3/2)*a/b^4 - 2*sqrt(a + b/x^3)*a^
2/b^4 - 2/3*a^3/(sqrt(a + b/x^3)*b^4)
_______________________________________________________________________________________
Fricas [A] time = 0.24679, size = 85, normalized size = 1.09 \[ -\frac{2 \,{\left (16 \, a^{3} x^{9} + 8 \, a^{2} b x^{6} - 2 \, a b^{2} x^{3} + b^{3}\right )} \sqrt{\frac{a x^{3} + b}{x^{3}}}}{15 \,{\left (a b^{4} x^{9} + b^{5} x^{6}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((a + b/x^3)^(3/2)*x^13),x, algorithm="fricas")
[Out]
-2/15*(16*a^3*x^9 + 8*a^2*b*x^6 - 2*a*b^2*x^3 + b^3)*sqrt((a*x^3 + b)/x^3)/(a*b^
4*x^9 + b^5*x^6)
_______________________________________________________________________________________
Sympy [A] time = 57.3701, size = 2048, normalized size = 26.26 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(a+b/x**3)**(3/2)/x**13,x)
[Out]
-32*a**(21/2)*b**(23/2)*x**24*sqrt(a*x**3/b + 1)/(15*a**(17/2)*b**15*x**(51/2) +
90*a**(15/2)*b**16*x**(45/2) + 225*a**(13/2)*b**17*x**(39/2) + 300*a**(11/2)*b*
*18*x**(33/2) + 225*a**(9/2)*b**19*x**(27/2) + 90*a**(7/2)*b**20*x**(21/2) + 15*
a**(5/2)*b**21*x**(15/2)) - 176*a**(19/2)*b**(25/2)*x**21*sqrt(a*x**3/b + 1)/(15
*a**(17/2)*b**15*x**(51/2) + 90*a**(15/2)*b**16*x**(45/2) + 225*a**(13/2)*b**17*
x**(39/2) + 300*a**(11/2)*b**18*x**(33/2) + 225*a**(9/2)*b**19*x**(27/2) + 90*a*
*(7/2)*b**20*x**(21/2) + 15*a**(5/2)*b**21*x**(15/2)) - 396*a**(17/2)*b**(27/2)*
x**18*sqrt(a*x**3/b + 1)/(15*a**(17/2)*b**15*x**(51/2) + 90*a**(15/2)*b**16*x**(
45/2) + 225*a**(13/2)*b**17*x**(39/2) + 300*a**(11/2)*b**18*x**(33/2) + 225*a**(
9/2)*b**19*x**(27/2) + 90*a**(7/2)*b**20*x**(21/2) + 15*a**(5/2)*b**21*x**(15/2)
) - 462*a**(15/2)*b**(29/2)*x**15*sqrt(a*x**3/b + 1)/(15*a**(17/2)*b**15*x**(51/
2) + 90*a**(15/2)*b**16*x**(45/2) + 225*a**(13/2)*b**17*x**(39/2) + 300*a**(11/2
)*b**18*x**(33/2) + 225*a**(9/2)*b**19*x**(27/2) + 90*a**(7/2)*b**20*x**(21/2) +
15*a**(5/2)*b**21*x**(15/2)) - 290*a**(13/2)*b**(31/2)*x**12*sqrt(a*x**3/b + 1)
/(15*a**(17/2)*b**15*x**(51/2) + 90*a**(15/2)*b**16*x**(45/2) + 225*a**(13/2)*b*
*17*x**(39/2) + 300*a**(11/2)*b**18*x**(33/2) + 225*a**(9/2)*b**19*x**(27/2) + 9
0*a**(7/2)*b**20*x**(21/2) + 15*a**(5/2)*b**21*x**(15/2)) - 92*a**(11/2)*b**(33/
2)*x**9*sqrt(a*x**3/b + 1)/(15*a**(17/2)*b**15*x**(51/2) + 90*a**(15/2)*b**16*x*
*(45/2) + 225*a**(13/2)*b**17*x**(39/2) + 300*a**(11/2)*b**18*x**(33/2) + 225*a*
*(9/2)*b**19*x**(27/2) + 90*a**(7/2)*b**20*x**(21/2) + 15*a**(5/2)*b**21*x**(15/
2)) - 16*a**(9/2)*b**(35/2)*x**6*sqrt(a*x**3/b + 1)/(15*a**(17/2)*b**15*x**(51/2
) + 90*a**(15/2)*b**16*x**(45/2) + 225*a**(13/2)*b**17*x**(39/2) + 300*a**(11/2)
*b**18*x**(33/2) + 225*a**(9/2)*b**19*x**(27/2) + 90*a**(7/2)*b**20*x**(21/2) +
15*a**(5/2)*b**21*x**(15/2)) - 6*a**(7/2)*b**(37/2)*x**3*sqrt(a*x**3/b + 1)/(15*
a**(17/2)*b**15*x**(51/2) + 90*a**(15/2)*b**16*x**(45/2) + 225*a**(13/2)*b**17*x
**(39/2) + 300*a**(11/2)*b**18*x**(33/2) + 225*a**(9/2)*b**19*x**(27/2) + 90*a**
(7/2)*b**20*x**(21/2) + 15*a**(5/2)*b**21*x**(15/2)) - 2*a**(5/2)*b**(39/2)*sqrt
(a*x**3/b + 1)/(15*a**(17/2)*b**15*x**(51/2) + 90*a**(15/2)*b**16*x**(45/2) + 22
5*a**(13/2)*b**17*x**(39/2) + 300*a**(11/2)*b**18*x**(33/2) + 225*a**(9/2)*b**19
*x**(27/2) + 90*a**(7/2)*b**20*x**(21/2) + 15*a**(5/2)*b**21*x**(15/2)) + 32*a**
11*b**11*x**(51/2)/(15*a**(17/2)*b**15*x**(51/2) + 90*a**(15/2)*b**16*x**(45/2)
+ 225*a**(13/2)*b**17*x**(39/2) + 300*a**(11/2)*b**18*x**(33/2) + 225*a**(9/2)*b
**19*x**(27/2) + 90*a**(7/2)*b**20*x**(21/2) + 15*a**(5/2)*b**21*x**(15/2)) + 19
2*a**10*b**12*x**(45/2)/(15*a**(17/2)*b**15*x**(51/2) + 90*a**(15/2)*b**16*x**(4
5/2) + 225*a**(13/2)*b**17*x**(39/2) + 300*a**(11/2)*b**18*x**(33/2) + 225*a**(9
/2)*b**19*x**(27/2) + 90*a**(7/2)*b**20*x**(21/2) + 15*a**(5/2)*b**21*x**(15/2))
+ 480*a**9*b**13*x**(39/2)/(15*a**(17/2)*b**15*x**(51/2) + 90*a**(15/2)*b**16*x
**(45/2) + 225*a**(13/2)*b**17*x**(39/2) + 300*a**(11/2)*b**18*x**(33/2) + 225*a
**(9/2)*b**19*x**(27/2) + 90*a**(7/2)*b**20*x**(21/2) + 15*a**(5/2)*b**21*x**(15
/2)) + 640*a**8*b**14*x**(33/2)/(15*a**(17/2)*b**15*x**(51/2) + 90*a**(15/2)*b**
16*x**(45/2) + 225*a**(13/2)*b**17*x**(39/2) + 300*a**(11/2)*b**18*x**(33/2) + 2
25*a**(9/2)*b**19*x**(27/2) + 90*a**(7/2)*b**20*x**(21/2) + 15*a**(5/2)*b**21*x*
*(15/2)) + 480*a**7*b**15*x**(27/2)/(15*a**(17/2)*b**15*x**(51/2) + 90*a**(15/2)
*b**16*x**(45/2) + 225*a**(13/2)*b**17*x**(39/2) + 300*a**(11/2)*b**18*x**(33/2)
+ 225*a**(9/2)*b**19*x**(27/2) + 90*a**(7/2)*b**20*x**(21/2) + 15*a**(5/2)*b**2
1*x**(15/2)) + 192*a**6*b**16*x**(21/2)/(15*a**(17/2)*b**15*x**(51/2) + 90*a**(1
5/2)*b**16*x**(45/2) + 225*a**(13/2)*b**17*x**(39/2) + 300*a**(11/2)*b**18*x**(3
3/2) + 225*a**(9/2)*b**19*x**(27/2) + 90*a**(7/2)*b**20*x**(21/2) + 15*a**(5/2)*
b**21*x**(15/2)) + 32*a**5*b**17*x**(15/2)/(15*a**(17/2)*b**15*x**(51/2) + 90*a*
*(15/2)*b**16*x**(45/2) + 225*a**(13/2)*b**17*x**(39/2) + 300*a**(11/2)*b**18*x*
*(33/2) + 225*a**(9/2)*b**19*x**(27/2) + 90*a**(7/2)*b**20*x**(21/2) + 15*a**(5/
2)*b**21*x**(15/2))
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (a + \frac{b}{x^{3}}\right )}^{\frac{3}{2}} x^{13}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((a + b/x^3)^(3/2)*x^13),x, algorithm="giac")
[Out]
integrate(1/((a + b/x^3)^(3/2)*x^13), x)