\(\int \frac {1}{(a+\frac {b}{x^3})^{3/2} x^{13}} \, dx\) [2043]

Optimal result

Integrand size = 15, antiderivative size = 78 \[ \int \frac {1}{\left (a+\frac {b}{x^3}\right )^{3/2} x^{13}} \, dx=-\frac {2 a^3}{3 b^4 \sqrt {a+\frac {b}{x^3}}}-\frac {2 a^2 \sqrt {a+\frac {b}{x^3}}}{b^4}+\frac {2 a \left (a+\frac {b}{x^3}\right )^{3/2}}{3 b^4}-\frac {2 \left (a+\frac {b}{x^3}\right )^{5/2}}{15 b^4} \]

[Out]

2/3*a*(a+b/x^3)^(3/2)/b^4-2/15*(a+b/x^3)^(5/2)/b^4-2/3*a^3/b^4/(a+b/x^3)^(1/2)-2*a^2*(a+b/x^3)^(1/2)/b^4

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 78, 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} \[ \int \frac {1}{\left (a+\frac {b}{x^3}\right )^{3/2} x^{13}} \, dx=-\frac {2 a^3}{3 b^4 \sqrt {a+\frac {b}{x^3}}}-\frac {2 a^2 \sqrt {a+\frac {b}{x^3}}}{b^4}+\frac {2 a \left (a+\frac {b}{x^3}\right )^{3/2}}{3 b^4}-\frac {2 \left (a+\frac {b}{x^3}\right )^{5/2}}{15 b^4} \]

[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)

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 \frac {x^3}{(a+b x)^{3/2}} \, dx,x,\frac {1}{x^3}\right )\right ) \\ & = -\left (\frac {1}{3} \text {Subst}\left (\int \left (-\frac {a^3}{b^3 (a+b x)^{3/2}}+\frac {3 a^2}{b^3 \sqrt {a+b x}}-\frac {3 a \sqrt {a+b x}}{b^3}+\frac {(a+b x)^{3/2}}{b^3}\right ) \, dx,x,\frac {1}{x^3}\right )\right ) \\ & = -\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} \\ \end{align*}

Mathematica [A] (verified)

Time = 2.66 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.65 \[ \int \frac {1}{\left (a+\frac {b}{x^3}\right )^{3/2} x^{13}} \, dx=-\frac {2 \left (b^3-2 a b^2 x^3+8 a^2 b x^6+16 a^3 x^9\right )}{15 b^4 \sqrt {a+\frac {b}{x^3}} x^9} \]

[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] (verified)

Time = 0.40 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.76

method	result	size
gosper	\(-\frac {2 \left (a \,x^{3}+b \right ) \left (16 a^{3} x^{9}+8 a^{2} b \,x^{6}-2 a \,b^{2} x^{3}+b^{3}\right )}{15 x^{12} b^{4} \left (\frac {a \,x^{3}+b}{x^{3}}\right )^{\frac {3}{2}}}\)	\(59\)
trager	\(-\frac {2 \left (16 a^{3} x^{9}+8 a^{2} b \,x^{6}-2 a \,b^{2} x^{3}+b^{3}\right ) \sqrt {-\frac {-a \,x^{3}-b}{x^{3}}}}{15 x^{6} b^{4} \left (a \,x^{3}+b \right )}\)	\(65\)
risch	\(-\frac {2 \left (a \,x^{3}+b \right ) \left (11 a^{2} x^{6}-3 a b \,x^{3}+b^{2}\right )}{15 b^{4} x^{9} \sqrt {\frac {a \,x^{3}+b}{x^{3}}}}-\frac {2 a^{3}}{3 b^{4} \sqrt {\frac {a \,x^{3}+b}{x^{3}}}}\)	\(70\)
default	\(\frac {2 \left (a \,x^{3}+b \right ) \left (-20 x^{10} a^{3}+4 \sqrt {x \left (a \,x^{3}+b \right )}\, \sqrt {a \,x^{4}+b x}\, a^{2} x^{6}-15 a^{2} b \,x^{7}+3 \sqrt {x \left (a \,x^{3}+b \right )}\, \sqrt {a \,x^{4}+b x}\, a b \,x^{3}-\sqrt {x \left (a \,x^{3}+b \right )}\, \sqrt {a \,x^{4}+b x}\, b^{2}\right )}{15 \left (\frac {a \,x^{3}+b}{x^{3}}\right )^{\frac {3}{2}} x^{13} b^{4}}\)	\(133\)

[In]

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

[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)

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.81 \[ \int \frac {1}{\left (a+\frac {b}{x^3}\right )^{3/2} x^{13}} \, dx=-\frac {2 \, {\left (16 \, a^{3} x^{9} + 8 \, a^{2} b x^{6} - 2 \, a b^{2} x^{3} + b^{3}\right )} \sqrt {\frac {a x^{3} + b}{x^{3}}}}{15 \, {\left (a b^{4} x^{9} + b^{5} x^{6}\right )}} \]

[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 [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2048 vs. \(2 (73) = 146\).

Time = 2.30 (sec) , antiderivative size = 2048, normalized size of antiderivative = 26.26 \[ \int \frac {1}{\left (a+\frac {b}{x^3}\right )^{3/2} x^{13}} \, dx=\text {Too large to display} \]

[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**1
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) + 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)) - 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) + 2
25*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**2
0*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**(5
1/2) + 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)) + 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**(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)) + 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) + 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**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**21*x**(15/2)) + 192*a**6*b**16*
x**(21/2)/(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)) + 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))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.82 \[ \int \frac {1}{\left (a+\frac {b}{x^3}\right )^{3/2} x^{13}} \, dx=-\frac {2 \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {5}{2}}}{15 \, b^{4}} + \frac {2 \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {3}{2}} a}{3 \, b^{4}} - \frac {2 \, \sqrt {a + \frac {b}{x^{3}}} a^{2}}{b^{4}} - \frac {2 \, a^{3}}{3 \, \sqrt {a + \frac {b}{x^{3}}} b^{4}} \]

[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)

Giac [F]

\[ \int \frac {1}{\left (a+\frac {b}{x^3}\right )^{3/2} x^{13}} \, dx=\int { \frac {1}{{\left (a + \frac {b}{x^{3}}\right )}^{\frac {3}{2}} x^{13}} \,d x } \]

[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)

Mupad [B] (verification not implemented)

Time = 6.17 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.05 \[ \int \frac {1}{\left (a+\frac {b}{x^3}\right )^{3/2} x^{13}} \, dx=\frac {2\,a\,\sqrt {a+\frac {b}{x^3}}}{5\,b^3\,x^3}-\frac {2\,\sqrt {a+\frac {b}{x^3}}}{15\,b^2\,x^6}-\frac {22\,a^2\,\sqrt {a+\frac {b}{x^3}}}{15\,b^4}-\frac {2\,a^3\,x^3\,\sqrt {a+\frac {b}{x^3}}}{3\,b^4\,\left (a\,x^3+b\right )} \]

[In]

int(1/(x^13*(a + b/x^3)^(3/2)),x)

[Out]

(2*a*(a + b/x^3)^(1/2))/(5*b^3*x^3) - (2*(a + b/x^3)^(1/2))/(15*b^2*x^6) - (22*a^2*(a + b/x^3)^(1/2))/(15*b^4)
- (2*a^3*x^3*(a + b/x^3)^(1/2))/(3*b^4*(b + a*x^3))