3.2014 \(\int \frac{\left (a+\frac{b}{x^3}\right )^{3/2}}{x^{13}} \, dx\)

Optimal. Leaf size=80 \[ \frac{2 a^3 \left (a+\frac{b}{x^3}\right )^{5/2}}{15 b^4}-\frac{2 a^2 \left (a+\frac{b}{x^3}\right )^{7/2}}{7 b^4}-\frac{2 \left (a+\frac{b}{x^3}\right )^{11/2}}{33 b^4}+\frac{2 a \left (a+\frac{b}{x^3}\right )^{9/2}}{9 b^4} \]

[Out]

(2*a^3*(a + b/x^3)^(5/2))/(15*b^4) - (2*a^2*(a + b/x^3)^(7/2))/(7*b^4) + (2*a*(a
+ b/x^3)^(9/2))/(9*b^4) - (2*(a + b/x^3)^(11/2))/(33*b^4)

_______________________________________________________________________________________

Rubi [A]  time = 0.119652, 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 \[ \frac{2 a^3 \left (a+\frac{b}{x^3}\right )^{5/2}}{15 b^4}-\frac{2 a^2 \left (a+\frac{b}{x^3}\right )^{7/2}}{7 b^4}-\frac{2 \left (a+\frac{b}{x^3}\right )^{11/2}}{33 b^4}+\frac{2 a \left (a+\frac{b}{x^3}\right )^{9/2}}{9 b^4} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x^3)^(3/2)/x^13,x]

[Out]

(2*a^3*(a + b/x^3)^(5/2))/(15*b^4) - (2*a^2*(a + b/x^3)^(7/2))/(7*b^4) + (2*a*(a
+ b/x^3)^(9/2))/(9*b^4) - (2*(a + b/x^3)^(11/2))/(33*b^4)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 14.2659, size = 75, normalized size = 0.94 \[ \frac{2 a^{3} \left (a + \frac{b}{x^{3}}\right )^{\frac{5}{2}}}{15 b^{4}} - \frac{2 a^{2} \left (a + \frac{b}{x^{3}}\right )^{\frac{7}{2}}}{7 b^{4}} + \frac{2 a \left (a + \frac{b}{x^{3}}\right )^{\frac{9}{2}}}{9 b^{4}} - \frac{2 \left (a + \frac{b}{x^{3}}\right )^{\frac{11}{2}}}{33 b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*a**3*(a + b/x**3)**(5/2)/(15*b**4) - 2*a**2*(a + b/x**3)**(7/2)/(7*b**4) + 2*a
*(a + b/x**3)**(9/2)/(9*b**4) - 2*(a + b/x**3)**(11/2)/(33*b**4)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0543504, size = 62, normalized size = 0.78 \[ \frac{2 \sqrt{a+\frac{b}{x^3}} \left (a x^3+b\right )^2 \left (16 a^3 x^9-40 a^2 b x^6+70 a b^2 x^3-105 b^3\right )}{3465 b^4 x^{15}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x^3)^(3/2)/x^13,x]

[Out]

(2*Sqrt[a + b/x^3]*(b + a*x^3)^2*(-105*b^3 + 70*a*b^2*x^3 - 40*a^2*b*x^6 + 16*a^
3*x^9))/(3465*b^4*x^15)

_______________________________________________________________________________________

Maple [A]  time = 0.01, size = 61, normalized size = 0.8 \[{\frac{ \left ( 2\,a{x}^{3}+2\,b \right ) \left ( 16\,{a}^{3}{x}^{9}-40\,{a}^{2}b{x}^{6}+70\,a{b}^{2}{x}^{3}-105\,{b}^{3} \right ) }{3465\,{x}^{12}{b}^{4}} \left ({\frac{a{x}^{3}+b}{{x}^{3}}} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x^3)^(3/2)/x^13,x)

[Out]

2/3465*(a*x^3+b)*(16*a^3*x^9-40*a^2*b*x^6+70*a*b^2*x^3-105*b^3)*((a*x^3+b)/x^3)^
(3/2)/x^12/b^4

_______________________________________________________________________________________

Maxima [A]  time = 1.4179, size = 86, normalized size = 1.08 \[ -\frac{2 \,{\left (a + \frac{b}{x^{3}}\right )}^{\frac{11}{2}}}{33 \, b^{4}} + \frac{2 \,{\left (a + \frac{b}{x^{3}}\right )}^{\frac{9}{2}} a}{9 \, b^{4}} - \frac{2 \,{\left (a + \frac{b}{x^{3}}\right )}^{\frac{7}{2}} a^{2}}{7 \, b^{4}} + \frac{2 \,{\left (a + \frac{b}{x^{3}}\right )}^{\frac{5}{2}} a^{3}}{15 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x^3)^(3/2)/x^13,x, algorithm="maxima")

[Out]

-2/33*(a + b/x^3)^(11/2)/b^4 + 2/9*(a + b/x^3)^(9/2)*a/b^4 - 2/7*(a + b/x^3)^(7/
2)*a^2/b^4 + 2/15*(a + b/x^3)^(5/2)*a^3/b^4

_______________________________________________________________________________________

Fricas [A]  time = 0.240963, size = 101, normalized size = 1.26 \[ \frac{2 \,{\left (16 \, a^{5} x^{15} - 8 \, a^{4} b x^{12} + 6 \, a^{3} b^{2} x^{9} - 5 \, a^{2} b^{3} x^{6} - 140 \, a b^{4} x^{3} - 105 \, b^{5}\right )} \sqrt{\frac{a x^{3} + b}{x^{3}}}}{3465 \, b^{4} x^{15}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x^3)^(3/2)/x^13,x, algorithm="fricas")

[Out]

2/3465*(16*a^5*x^15 - 8*a^4*b*x^12 + 6*a^3*b^2*x^9 - 5*a^2*b^3*x^6 - 140*a*b^4*x
^3 - 105*b^5)*sqrt((a*x^3 + b)/x^3)/(b^4*x^15)

_______________________________________________________________________________________

Sympy [A]  time = 33.0716, 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)**(3/2)/x**13,x)

[Out]

32*a**(33/2)*b**(23/2)*x**33*sqrt(a*x**3/b + 1)/(3465*a**(23/2)*b**15*x**(69/2)
+ 20790*a**(21/2)*b**16*x**(63/2) + 51975*a**(19/2)*b**17*x**(57/2) + 69300*a**(
17/2)*b**18*x**(51/2) + 51975*a**(15/2)*b**19*x**(45/2) + 20790*a**(13/2)*b**20*
x**(39/2) + 3465*a**(11/2)*b**21*x**(33/2)) + 176*a**(31/2)*b**(25/2)*x**30*sqrt
(a*x**3/b + 1)/(3465*a**(23/2)*b**15*x**(69/2) + 20790*a**(21/2)*b**16*x**(63/2)
+ 51975*a**(19/2)*b**17*x**(57/2) + 69300*a**(17/2)*b**18*x**(51/2) + 51975*a**
(15/2)*b**19*x**(45/2) + 20790*a**(13/2)*b**20*x**(39/2) + 3465*a**(11/2)*b**21*
x**(33/2)) + 396*a**(29/2)*b**(27/2)*x**27*sqrt(a*x**3/b + 1)/(3465*a**(23/2)*b*
*15*x**(69/2) + 20790*a**(21/2)*b**16*x**(63/2) + 51975*a**(19/2)*b**17*x**(57/2
) + 69300*a**(17/2)*b**18*x**(51/2) + 51975*a**(15/2)*b**19*x**(45/2) + 20790*a*
*(13/2)*b**20*x**(39/2) + 3465*a**(11/2)*b**21*x**(33/2)) + 462*a**(27/2)*b**(29
/2)*x**24*sqrt(a*x**3/b + 1)/(3465*a**(23/2)*b**15*x**(69/2) + 20790*a**(21/2)*b
**16*x**(63/2) + 51975*a**(19/2)*b**17*x**(57/2) + 69300*a**(17/2)*b**18*x**(51/
2) + 51975*a**(15/2)*b**19*x**(45/2) + 20790*a**(13/2)*b**20*x**(39/2) + 3465*a*
*(11/2)*b**21*x**(33/2)) - 1848*a**(23/2)*b**(33/2)*x**18*sqrt(a*x**3/b + 1)/(34
65*a**(23/2)*b**15*x**(69/2) + 20790*a**(21/2)*b**16*x**(63/2) + 51975*a**(19/2)
*b**17*x**(57/2) + 69300*a**(17/2)*b**18*x**(51/2) + 51975*a**(15/2)*b**19*x**(4
5/2) + 20790*a**(13/2)*b**20*x**(39/2) + 3465*a**(11/2)*b**21*x**(33/2)) - 5544*
a**(21/2)*b**(35/2)*x**15*sqrt(a*x**3/b + 1)/(3465*a**(23/2)*b**15*x**(69/2) + 2
0790*a**(21/2)*b**16*x**(63/2) + 51975*a**(19/2)*b**17*x**(57/2) + 69300*a**(17/
2)*b**18*x**(51/2) + 51975*a**(15/2)*b**19*x**(45/2) + 20790*a**(13/2)*b**20*x**
(39/2) + 3465*a**(11/2)*b**21*x**(33/2)) - 8844*a**(19/2)*b**(37/2)*x**12*sqrt(a
*x**3/b + 1)/(3465*a**(23/2)*b**15*x**(69/2) + 20790*a**(21/2)*b**16*x**(63/2) +
51975*a**(19/2)*b**17*x**(57/2) + 69300*a**(17/2)*b**18*x**(51/2) + 51975*a**(1
5/2)*b**19*x**(45/2) + 20790*a**(13/2)*b**20*x**(39/2) + 3465*a**(11/2)*b**21*x*
*(33/2)) - 8448*a**(17/2)*b**(39/2)*x**9*sqrt(a*x**3/b + 1)/(3465*a**(23/2)*b**1
5*x**(69/2) + 20790*a**(21/2)*b**16*x**(63/2) + 51975*a**(19/2)*b**17*x**(57/2)
+ 69300*a**(17/2)*b**18*x**(51/2) + 51975*a**(15/2)*b**19*x**(45/2) + 20790*a**(
13/2)*b**20*x**(39/2) + 3465*a**(11/2)*b**21*x**(33/2)) - 4840*a**(15/2)*b**(41/
2)*x**6*sqrt(a*x**3/b + 1)/(3465*a**(23/2)*b**15*x**(69/2) + 20790*a**(21/2)*b**
16*x**(63/2) + 51975*a**(19/2)*b**17*x**(57/2) + 69300*a**(17/2)*b**18*x**(51/2)
+ 51975*a**(15/2)*b**19*x**(45/2) + 20790*a**(13/2)*b**20*x**(39/2) + 3465*a**(
11/2)*b**21*x**(33/2)) - 1540*a**(13/2)*b**(43/2)*x**3*sqrt(a*x**3/b + 1)/(3465*
a**(23/2)*b**15*x**(69/2) + 20790*a**(21/2)*b**16*x**(63/2) + 51975*a**(19/2)*b*
*17*x**(57/2) + 69300*a**(17/2)*b**18*x**(51/2) + 51975*a**(15/2)*b**19*x**(45/2
) + 20790*a**(13/2)*b**20*x**(39/2) + 3465*a**(11/2)*b**21*x**(33/2)) - 210*a**(
11/2)*b**(45/2)*sqrt(a*x**3/b + 1)/(3465*a**(23/2)*b**15*x**(69/2) + 20790*a**(2
1/2)*b**16*x**(63/2) + 51975*a**(19/2)*b**17*x**(57/2) + 69300*a**(17/2)*b**18*x
**(51/2) + 51975*a**(15/2)*b**19*x**(45/2) + 20790*a**(13/2)*b**20*x**(39/2) + 3
465*a**(11/2)*b**21*x**(33/2)) - 32*a**17*b**11*x**(69/2)/(3465*a**(23/2)*b**15*
x**(69/2) + 20790*a**(21/2)*b**16*x**(63/2) + 51975*a**(19/2)*b**17*x**(57/2) +
69300*a**(17/2)*b**18*x**(51/2) + 51975*a**(15/2)*b**19*x**(45/2) + 20790*a**(13
/2)*b**20*x**(39/2) + 3465*a**(11/2)*b**21*x**(33/2)) - 192*a**16*b**12*x**(63/2
)/(3465*a**(23/2)*b**15*x**(69/2) + 20790*a**(21/2)*b**16*x**(63/2) + 51975*a**(
19/2)*b**17*x**(57/2) + 69300*a**(17/2)*b**18*x**(51/2) + 51975*a**(15/2)*b**19*
x**(45/2) + 20790*a**(13/2)*b**20*x**(39/2) + 3465*a**(11/2)*b**21*x**(33/2)) -
480*a**15*b**13*x**(57/2)/(3465*a**(23/2)*b**15*x**(69/2) + 20790*a**(21/2)*b**1
6*x**(63/2) + 51975*a**(19/2)*b**17*x**(57/2) + 69300*a**(17/2)*b**18*x**(51/2)
+ 51975*a**(15/2)*b**19*x**(45/2) + 20790*a**(13/2)*b**20*x**(39/2) + 3465*a**(1
1/2)*b**21*x**(33/2)) - 640*a**14*b**14*x**(51/2)/(3465*a**(23/2)*b**15*x**(69/2
) + 20790*a**(21/2)*b**16*x**(63/2) + 51975*a**(19/2)*b**17*x**(57/2) + 69300*a*
*(17/2)*b**18*x**(51/2) + 51975*a**(15/2)*b**19*x**(45/2) + 20790*a**(13/2)*b**2
0*x**(39/2) + 3465*a**(11/2)*b**21*x**(33/2)) - 480*a**13*b**15*x**(45/2)/(3465*
a**(23/2)*b**15*x**(69/2) + 20790*a**(21/2)*b**16*x**(63/2) + 51975*a**(19/2)*b*
*17*x**(57/2) + 69300*a**(17/2)*b**18*x**(51/2) + 51975*a**(15/2)*b**19*x**(45/2
) + 20790*a**(13/2)*b**20*x**(39/2) + 3465*a**(11/2)*b**21*x**(33/2)) - 192*a**1
2*b**16*x**(39/2)/(3465*a**(23/2)*b**15*x**(69/2) + 20790*a**(21/2)*b**16*x**(63
/2) + 51975*a**(19/2)*b**17*x**(57/2) + 69300*a**(17/2)*b**18*x**(51/2) + 51975*
a**(15/2)*b**19*x**(45/2) + 20790*a**(13/2)*b**20*x**(39/2) + 3465*a**(11/2)*b**
21*x**(33/2)) - 32*a**11*b**17*x**(33/2)/(3465*a**(23/2)*b**15*x**(69/2) + 20790
*a**(21/2)*b**16*x**(63/2) + 51975*a**(19/2)*b**17*x**(57/2) + 69300*a**(17/2)*b
**18*x**(51/2) + 51975*a**(15/2)*b**19*x**(45/2) + 20790*a**(13/2)*b**20*x**(39/
2) + 3465*a**(11/2)*b**21*x**(33/2))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.24161, size = 181, normalized size = 2.26 \[ -\frac{2 \,{\left (\frac{11 \,{\left (35 \,{\left (a + \frac{b}{x^{3}}\right )}^{\frac{9}{2}} - 135 \,{\left (a + \frac{b}{x^{3}}\right )}^{\frac{7}{2}} a + 189 \,{\left (a + \frac{b}{x^{3}}\right )}^{\frac{5}{2}} a^{2} - 105 \,{\left (a + \frac{b}{x^{3}}\right )}^{\frac{3}{2}} a^{3}\right )} a}{b^{3}} + \frac{315 \,{\left (a + \frac{b}{x^{3}}\right )}^{\frac{11}{2}} - 1540 \,{\left (a + \frac{b}{x^{3}}\right )}^{\frac{9}{2}} a + 2970 \,{\left (a + \frac{b}{x^{3}}\right )}^{\frac{7}{2}} a^{2} - 2772 \,{\left (a + \frac{b}{x^{3}}\right )}^{\frac{5}{2}} a^{3} + 1155 \,{\left (a + \frac{b}{x^{3}}\right )}^{\frac{3}{2}} a^{4}}{b^{3}}\right )}}{10395 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x^3)^(3/2)/x^13,x, algorithm="giac")

[Out]

-2/10395*(11*(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)*a/b^3 + (315*(a + b/x^3)^(11/2) - 1540*(a
+ b/x^3)^(9/2)*a + 2970*(a + b/x^3)^(7/2)*a^2 - 2772*(a + b/x^3)^(5/2)*a^3 + 115
5*(a + b/x^3)^(3/2)*a^4)/b^3)/b