Integrand size = 15, antiderivative size = 78 \[ \int \frac {1}{\left (a+\frac {b}{x^3}\right )^{3/2} x^{13}} \, dx=-\frac {2 a^3}{3 b^4 \sqrt {a+\frac {b}{x^3}}}-\frac {2 a^2 \sqrt {a+\frac {b}{x^3}}}{b^4}+\frac {2 a \left (a+\frac {b}{x^3}\right )^{3/2}}{3 b^4}-\frac {2 \left (a+\frac {b}{x^3}\right )^{5/2}}{15 b^4} \] Output:
-2/3*a^3/b^4/(a+b/x^3)^(1/2)-2*a^2*(a+b/x^3)^(1/2)/b^4+2/3*a*(a+b/x^3)^(3/ 2)/b^4-2/15*(a+b/x^3)^(5/2)/b^4
Time = 2.08 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.65 \[ \int \frac {1}{\left (a+\frac {b}{x^3}\right )^{3/2} x^{13}} \, dx=-\frac {2 \left (b^3-2 a b^2 x^3+8 a^2 b x^6+16 a^3 x^9\right )}{15 b^4 \sqrt {a+\frac {b}{x^3}} x^9} \] Input:
Integrate[1/((a + b/x^3)^(3/2)*x^13),x]
Output:
(-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)
Time = 0.34 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {798, 53, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {1}{x^{13} \left (a+\frac {b}{x^3}\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle -\frac {1}{3} \int \frac {1}{\left (a+\frac {b}{x^3}\right )^{3/2} x^9}d\frac {1}{x^3}\) |
\(\Big \downarrow \) 53 |
\(\displaystyle -\frac {1}{3} \int \left (-\frac {a^3}{b^3 \left (a+\frac {b}{x^3}\right )^{3/2}}+\frac {3 a^2}{b^3 \sqrt {a+\frac {b}{x^3}}}-\frac {3 \sqrt {a+\frac {b}{x^3}} a}{b^3}+\frac {\left (a+\frac {b}{x^3}\right )^{3/2}}{b^3}\right )d\frac {1}{x^3}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{3} \left (-\frac {2 a^3}{b^4 \sqrt {a+\frac {b}{x^3}}}-\frac {6 a^2 \sqrt {a+\frac {b}{x^3}}}{b^4}+\frac {2 a \left (a+\frac {b}{x^3}\right )^{3/2}}{b^4}-\frac {2 \left (a+\frac {b}{x^3}\right )^{5/2}}{5 b^4}\right )\) |
Input:
Int[1/((a + b/x^3)^(3/2)*x^13),x]
Output:
((-2*a^3)/(b^4*Sqrt[a + b/x^3]) - (6*a^2*Sqrt[a + b/x^3])/b^4 + (2*a*(a + b/x^3)^(3/2))/b^4 - (2*(a + b/x^3)^(5/2))/(5*b^4))/3
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] && 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])
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[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]]
Time = 0.51 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.71
method | result | size |
orering | \(-\frac {2 \left (16 a^{3} x^{9}+8 a^{2} b \,x^{6}-2 a \,b^{2} x^{3}+b^{3}\right ) \left (a \,x^{3}+b \right )}{15 b^{4} x^{12} \left (a +\frac {b}{x^{3}}\right )^{\frac {3}{2}}}\) | \(55\) |
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 | \(\frac {2 \left (a \,x^{3}+b \right ) \left (-20 a^{3} x^{10}+4 \sqrt {a \,x^{4}+b x}\, \sqrt {x \left (a \,x^{3}+b \right )}\, a^{2} x^{6}-15 a^{2} b \,x^{7}+3 \sqrt {a \,x^{4}+b x}\, \sqrt {x \left (a \,x^{3}+b \right )}\, a b \,x^{3}-\sqrt {a \,x^{4}+b x}\, \sqrt {x \left (a \,x^{3}+b \right )}\, b^{2}\right )}{15 \left (\frac {a \,x^{3}+b}{x^{3}}\right )^{\frac {3}{2}} x^{13} b^{4}}\) | \(133\) |
Input:
int(1/(a+b/x^3)^(3/2)/x^13,x,method=_RETURNVERBOSE)
Output:
-2/15*(16*a^3*x^9+8*a^2*b*x^6-2*a*b^2*x^3+b^3)/b^4/x^12*(a*x^3+b)/(a+b/x^3 )^(3/2)
Time = 0.08 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.81 \[ \int \frac {1}{\left (a+\frac {b}{x^3}\right )^{3/2} x^{13}} \, dx=-\frac {2 \, {\left (16 \, a^{3} x^{9} + 8 \, a^{2} b x^{6} - 2 \, a b^{2} x^{3} + b^{3}\right )} \sqrt {\frac {a x^{3} + b}{x^{3}}}}{15 \, {\left (a b^{4} x^{9} + b^{5} x^{6}\right )}} \] Input:
integrate(1/(a+b/x^3)^(3/2)/x^13,x, algorithm="fricas")
Output:
-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)
Leaf count of result is larger than twice the leaf count of optimal. 2048 vs. \(2 (73) = 146\).
Time = 2.53 (sec) , antiderivative size = 2048, normalized size of antiderivative = 26.26 \[ \int \frac {1}{\left (a+\frac {b}{x^3}\right )^{3/2} x^{13}} \, dx=\text {Too large to display} \] Input:
integrate(1/(a+b/x**3)**(3/2)/x**13,x)
Output:
-32*a**(21/2)*b**(23/2)*x**24*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)) - 176*a**(19/2)*b**(25/2)*x**2 1*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**1 7*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) + 90*a**(7/2)*b**20*x**(21/2) + 15*a**(5/2)*b**21*x**(1 5/2)) - 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/...
Time = 0.03 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.82 \[ \int \frac {1}{\left (a+\frac {b}{x^3}\right )^{3/2} x^{13}} \, dx=-\frac {2 \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {5}{2}}}{15 \, b^{4}} + \frac {2 \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {3}{2}} a}{3 \, b^{4}} - \frac {2 \, \sqrt {a + \frac {b}{x^{3}}} a^{2}}{b^{4}} - \frac {2 \, a^{3}}{3 \, \sqrt {a + \frac {b}{x^{3}}} b^{4}} \] Input:
integrate(1/(a+b/x^3)^(3/2)/x^13,x, algorithm="maxima")
Output:
-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)
\[ \int \frac {1}{\left (a+\frac {b}{x^3}\right )^{3/2} x^{13}} \, dx=\int { \frac {1}{{\left (a + \frac {b}{x^{3}}\right )}^{\frac {3}{2}} x^{13}} \,d x } \] Input:
integrate(1/(a+b/x^3)^(3/2)/x^13,x, algorithm="giac")
Output:
integrate(1/((a + b/x^3)^(3/2)*x^13), x)
Time = 0.77 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.05 \[ \int \frac {1}{\left (a+\frac {b}{x^3}\right )^{3/2} x^{13}} \, dx=\frac {2\,a\,\sqrt {a+\frac {b}{x^3}}}{5\,b^3\,x^3}-\frac {2\,\sqrt {a+\frac {b}{x^3}}}{15\,b^2\,x^6}-\frac {22\,a^2\,\sqrt {a+\frac {b}{x^3}}}{15\,b^4}-\frac {2\,a^3\,x^3\,\sqrt {a+\frac {b}{x^3}}}{3\,b^4\,\left (a\,x^3+b\right )} \] Input:
int(1/(x^13*(a + b/x^3)^(3/2)),x)
Output:
(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))
Time = 0.36 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.78 \[ \int \frac {1}{\left (a+\frac {b}{x^3}\right )^{3/2} x^{13}} \, dx=\frac {2 \sqrt {a \,x^{3}+b}\, \left (-16 a^{3} x^{9}-8 a^{2} b \,x^{6}+2 a \,b^{2} x^{3}-b^{3}\right )}{15 \sqrt {x}\, b^{4} x^{7} \left (a \,x^{3}+b \right )} \] Input:
int(1/(a+b/x^3)^(3/2)/x^13,x)
Output:
(2*sqrt(a*x**3 + b)*( - 16*a**3*x**9 - 8*a**2*b*x**6 + 2*a*b**2*x**3 - b** 3))/(15*sqrt(x)*b**4*x**7*(a*x**3 + b))