\(\int \frac {\sqrt {a+\frac {b}{x^3}}}{x^{13}} \, dx\) [1997]

Optimal result

Integrand size = 15, antiderivative size = 80 \[ \int \frac {\sqrt {a+\frac {b}{x^3}}}{x^{13}} \, dx=\frac {2 a^3 \left (a+\frac {b}{x^3}\right )^{3/2}}{9 b^4}-\frac {2 a^2 \left (a+\frac {b}{x^3}\right )^{5/2}}{5 b^4}+\frac {2 a \left (a+\frac {b}{x^3}\right )^{7/2}}{7 b^4}-\frac {2 \left (a+\frac {b}{x^3}\right )^{9/2}}{27 b^4} \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {272, 45} \[ \int \frac {\sqrt {a+\frac {b}{x^3}}}{x^{13}} \, dx=\frac {2 a^3 \left (a+\frac {b}{x^3}\right )^{3/2}}{9 b^4}-\frac {2 a^2 \left (a+\frac {b}{x^3}\right )^{5/2}}{5 b^4}-\frac {2 \left (a+\frac {b}{x^3}\right )^{9/2}}{27 b^4}+\frac {2 a \left (a+\frac {b}{x^3}\right )^{7/2}}{7 b^4} \]

[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)

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps \begin{align*} \text {integral}& = -\left (\frac {1}{3} \text {Subst}\left (\int x^3 \sqrt {a+b x} \, dx,x,\frac {1}{x^3}\right )\right ) \\ & = -\left (\frac {1}{3} \text {Subst}\left (\int \left (-\frac {a^3 \sqrt {a+b x}}{b^3}+\frac {3 a^2 (a+b x)^{3/2}}{b^3}-\frac {3 a (a+b x)^{5/2}}{b^3}+\frac {(a+b x)^{7/2}}{b^3}\right ) \, dx,x,\frac {1}{x^3}\right )\right ) \\ & = \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 a \left (a+\frac {b}{x^3}\right )^{7/2}}{7 b^4}-\frac {2 \left (a+\frac {b}{x^3}\right )^{9/2}}{27 b^4} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.97 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.80 \[ \int \frac {\sqrt {a+\frac {b}{x^3}}}{x^{13}} \, dx=-\frac {2 \sqrt {a+\frac {b}{x^3}} \left (35 b^4+5 a b^3 x^3-6 a^2 b^2 x^6+8 a^3 b x^9-16 a^4 x^{12}\right )}{945 b^4 x^{12}} \]

[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)

Maple [A] (verified)

Time = 0.08 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.76

method	result	size
gosper	\(\frac {2 \left (a \,x^{3}+b \right ) \left (16 a^{3} x^{9}-24 a^{2} b \,x^{6}+30 a \,b^{2} x^{3}-35 b^{3}\right ) \sqrt {\frac {a \,x^{3}+b}{x^{3}}}}{945 x^{12} b^{4}}\)	\(61\)
risch	\(\frac {2 \sqrt {\frac {a \,x^{3}+b}{x^{3}}}\, \left (16 a^{4} x^{12}-8 a^{3} b \,x^{9}+6 a^{2} b^{2} x^{6}-5 a \,b^{3} x^{3}-35 b^{4}\right )}{945 x^{12} b^{4}}\)	\(65\)
trager	\(\frac {2 \left (16 a^{4} x^{12}-8 a^{3} b \,x^{9}+6 a^{2} b^{2} x^{6}-5 a \,b^{3} x^{3}-35 b^{4}\right ) \sqrt {-\frac {-a \,x^{3}-b}{x^{3}}}}{945 x^{12} b^{4}}\)	\(69\)
default	\(\frac {2 \sqrt {\frac {a \,x^{3}+b}{x^{3}}}\, \sqrt {a \,x^{4}+b x}\, \left (16 a^{4} x^{12}-8 a^{3} b \,x^{9}+6 a^{2} b^{2} x^{6}-5 a \,b^{3} x^{3}-35 b^{4}\right )}{945 x^{12} \sqrt {x \left (a \,x^{3}+b \right )}\, b^{4}}\)	\(87\)

[In]

int((a+b/x^3)^(1/2)/x^13,x,method=_RETURNVERBOSE)

[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

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.80 \[ \int \frac {\sqrt {a+\frac {b}{x^3}}}{x^{13}} \, dx=\frac {2 \, {\left (16 \, a^{4} x^{12} - 8 \, a^{3} b x^{9} + 6 \, a^{2} b^{2} x^{6} - 5 \, a b^{3} x^{3} - 35 \, b^{4}\right )} \sqrt {\frac {a x^{3} + b}{x^{3}}}}{945 \, b^{4} x^{12}} \]

[In]

integrate((a+b/x^3)^(1/2)/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)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2317 vs. \(2 (75) = 150\).

Time = 1.97 (sec) , antiderivative size = 2317, normalized size of antiderivative = 28.96 \[ \int \frac {\sqrt {a+\frac {b}{x^3}}}{x^{13}} \, dx=\text {Too large to display} \]

[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) + 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)) + 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) + 5
670*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)) - 378*a**(19/2)*b**(33/
2)*x**15*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)) - 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) + 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)) - 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**(2
7/2)) - 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)/(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)) - 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) + 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)) - 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**(6
3/2) + 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) + 14
175*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) + 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**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) + 56
70*a**(11/2)*b**20*x**(33/2) + 945*a**(9/2)*b**21*x**(27/2))

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.80 \[ \int \frac {\sqrt {a+\frac {b}{x^3}}}{x^{13}} \, dx=-\frac {2 \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {9}{2}}}{27 \, b^{4}} + \frac {2 \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {7}{2}} a}{7 \, b^{4}} - \frac {2 \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {5}{2}} a^{2}}{5 \, b^{4}} + \frac {2 \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {3}{2}} a^{3}}{9 \, b^{4}} \]

[In]

integrate((a+b/x^3)^(1/2)/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

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.71 \[ \int \frac {\sqrt {a+\frac {b}{x^3}}}{x^{13}} \, dx=-\frac {2 \, {\left (35 \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {9}{2}} - 135 \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {7}{2}} a + 189 \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {5}{2}} a^{2} - 105 \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {3}{2}} a^{3}\right )}}{945 \, b^{4}} \]

[In]

integrate((a+b/x^3)^(1/2)/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

Mupad [B] (verification not implemented)

Time = 7.03 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.12 \[ \int \frac {\sqrt {a+\frac {b}{x^3}}}{x^{13}} \, dx=\frac {32\,a^4\,\sqrt {a+\frac {b}{x^3}}}{945\,b^4}-\frac {2\,\sqrt {a+\frac {b}{x^3}}}{27\,x^{12}}-\frac {2\,a\,\sqrt {a+\frac {b}{x^3}}}{189\,b\,x^9}-\frac {16\,a^3\,\sqrt {a+\frac {b}{x^3}}}{945\,b^3\,x^3}+\frac {4\,a^2\,\sqrt {a+\frac {b}{x^3}}}{315\,b^2\,x^6} \]

[In]

int((a + b/x^3)^(1/2)/x^13,x)

[Out]

(32*a^4*(a + b/x^3)^(1/2))/(945*b^4) - (2*(a + b/x^3)^(1/2))/(27*x^12) - (2*a*(a + b/x^3)^(1/2))/(189*b*x^9) -
(16*a^3*(a + b/x^3)^(1/2))/(945*b^3*x^3) + (4*a^2*(a + b/x^3)^(1/2))/(315*b^2*x^6)