Integrand size = 15, antiderivative size = 59 \[ \int \frac {1}{\left (a+\frac {b}{x^3}\right )^{3/2} x^{10}} \, dx=\frac {2 a^2}{3 b^3 \sqrt {a+\frac {b}{x^3}}}+\frac {4 a \sqrt {a+\frac {b}{x^3}}}{3 b^3}-\frac {2 \left (a+\frac {b}{x^3}\right )^{3/2}}{9 b^3} \] Output:
2/3*a^2/b^3/(a+b/x^3)^(1/2)+4/3*a*(a+b/x^3)^(1/2)/b^3-2/9*(a+b/x^3)^(3/2)/ b^3
Time = 0.38 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.71 \[ \int \frac {1}{\left (a+\frac {b}{x^3}\right )^{3/2} x^{10}} \, dx=\frac {2 \left (-b^2+4 a b x^3+8 a^2 x^6\right )}{9 b^3 \sqrt {a+\frac {b}{x^3}} x^6} \] Input:
Integrate[1/((a + b/x^3)^(3/2)*x^10),x]
Output:
(2*(-b^2 + 4*a*b*x^3 + 8*a^2*x^6))/(9*b^3*Sqrt[a + b/x^3]*x^6)
Time = 0.31 (sec) , antiderivative size = 59, 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.
\(\displaystyle \int \frac {1}{x^{10} \left (a+\frac {b}{x^3}\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle -\frac {1}{3} \int \frac {1}{\left (a+\frac {b}{x^3}\right )^{3/2} x^6}d\frac {1}{x^3}\) |
\(\Big \downarrow \) 53 |
\(\displaystyle -\frac {1}{3} \int \left (\frac {a^2}{b^2 \left (a+\frac {b}{x^3}\right )^{3/2}}-\frac {2 a}{b^2 \sqrt {a+\frac {b}{x^3}}}+\frac {\sqrt {a+\frac {b}{x^3}}}{b^2}\right )d\frac {1}{x^3}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{3} \left (\frac {2 a^2}{b^3 \sqrt {a+\frac {b}{x^3}}}+\frac {4 a \sqrt {a+\frac {b}{x^3}}}{b^3}-\frac {2 \left (a+\frac {b}{x^3}\right )^{3/2}}{3 b^3}\right )\) |
Input:
Int[1/((a + b/x^3)^(3/2)*x^10),x]
Output:
((2*a^2)/(b^3*Sqrt[a + b/x^3]) + (4*a*Sqrt[a + b/x^3])/b^3 - (2*(a + b/x^3 )^(3/2))/(3*b^3))/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 = 0.52 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.78
method | result | size |
orering | \(\frac {2 \left (8 a^{2} x^{6}+4 a b \,x^{3}-b^{2}\right ) \left (a \,x^{3}+b \right )}{9 b^{3} x^{9} \left (a +\frac {b}{x^{3}}\right )^{\frac {3}{2}}}\) | \(46\) |
gosper | \(\frac {2 \left (a \,x^{3}+b \right ) \left (8 a^{2} x^{6}+4 a b \,x^{3}-b^{2}\right )}{9 x^{9} b^{3} \left (\frac {a \,x^{3}+b}{x^{3}}\right )^{\frac {3}{2}}}\) | \(50\) |
trager | \(\frac {2 \left (8 a^{2} x^{6}+4 a b \,x^{3}-b^{2}\right ) \sqrt {-\frac {-a \,x^{3}-b}{x^{3}}}}{9 x^{3} b^{3} \left (a \,x^{3}+b \right )}\) | \(56\) |
risch | \(\frac {2 \left (a \,x^{3}+b \right ) \left (5 a \,x^{3}-b \right )}{9 b^{3} x^{6} \sqrt {\frac {a \,x^{3}+b}{x^{3}}}}+\frac {2 a^{2}}{3 b^{3} \sqrt {\frac {a \,x^{3}+b}{x^{3}}}}\) | \(61\) |
default | \(-\frac {2 \left (a \,x^{3}+b \right ) \left (-9 a^{2} x^{7}+\sqrt {x \left (a \,x^{3}+b \right )}\, \sqrt {a \,x^{4}+b x}\, a \,x^{3}-6 a b \,x^{4}+\sqrt {x \left (a \,x^{3}+b \right )}\, \sqrt {a \,x^{4}+b x}\, b \right )}{9 \left (\frac {a \,x^{3}+b}{x^{3}}\right )^{\frac {3}{2}} x^{10} b^{3}}\) | \(96\) |
Input:
int(1/(a+b/x^3)^(3/2)/x^10,x,method=_RETURNVERBOSE)
Output:
2/9*(8*a^2*x^6+4*a*b*x^3-b^2)/b^3/x^9*(a*x^3+b)/(a+b/x^3)^(3/2)
Time = 0.08 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.92 \[ \int \frac {1}{\left (a+\frac {b}{x^3}\right )^{3/2} x^{10}} \, dx=\frac {2 \, {\left (8 \, a^{2} x^{6} + 4 \, a b x^{3} - b^{2}\right )} \sqrt {\frac {a x^{3} + b}{x^{3}}}}{9 \, {\left (a b^{3} x^{6} + b^{4} x^{3}\right )}} \] Input:
integrate(1/(a+b/x^3)^(3/2)/x^10,x, algorithm="fricas")
Output:
2/9*(8*a^2*x^6 + 4*a*b*x^3 - b^2)*sqrt((a*x^3 + b)/x^3)/(a*b^3*x^6 + b^4*x ^3)
Leaf count of result is larger than twice the leaf count of optimal. 466 vs. \(2 (54) = 108\).
Time = 1.66 (sec) , antiderivative size = 466, normalized size of antiderivative = 7.90 \[ \int \frac {1}{\left (a+\frac {b}{x^3}\right )^{3/2} x^{10}} \, dx=\frac {16 a^{\frac {9}{2}} b^{\frac {7}{2}} x^{9} \sqrt {\frac {a x^{3}}{b} + 1}}{9 a^{\frac {7}{2}} b^{6} x^{\frac {21}{2}} + 18 a^{\frac {5}{2}} b^{7} x^{\frac {15}{2}} + 9 a^{\frac {3}{2}} b^{8} x^{\frac {9}{2}}} + \frac {24 a^{\frac {7}{2}} b^{\frac {9}{2}} x^{6} \sqrt {\frac {a x^{3}}{b} + 1}}{9 a^{\frac {7}{2}} b^{6} x^{\frac {21}{2}} + 18 a^{\frac {5}{2}} b^{7} x^{\frac {15}{2}} + 9 a^{\frac {3}{2}} b^{8} x^{\frac {9}{2}}} + \frac {6 a^{\frac {5}{2}} b^{\frac {11}{2}} x^{3} \sqrt {\frac {a x^{3}}{b} + 1}}{9 a^{\frac {7}{2}} b^{6} x^{\frac {21}{2}} + 18 a^{\frac {5}{2}} b^{7} x^{\frac {15}{2}} + 9 a^{\frac {3}{2}} b^{8} x^{\frac {9}{2}}} - \frac {2 a^{\frac {3}{2}} b^{\frac {13}{2}} \sqrt {\frac {a x^{3}}{b} + 1}}{9 a^{\frac {7}{2}} b^{6} x^{\frac {21}{2}} + 18 a^{\frac {5}{2}} b^{7} x^{\frac {15}{2}} + 9 a^{\frac {3}{2}} b^{8} x^{\frac {9}{2}}} - \frac {16 a^{5} b^{3} x^{\frac {21}{2}}}{9 a^{\frac {7}{2}} b^{6} x^{\frac {21}{2}} + 18 a^{\frac {5}{2}} b^{7} x^{\frac {15}{2}} + 9 a^{\frac {3}{2}} b^{8} x^{\frac {9}{2}}} - \frac {32 a^{4} b^{4} x^{\frac {15}{2}}}{9 a^{\frac {7}{2}} b^{6} x^{\frac {21}{2}} + 18 a^{\frac {5}{2}} b^{7} x^{\frac {15}{2}} + 9 a^{\frac {3}{2}} b^{8} x^{\frac {9}{2}}} - \frac {16 a^{3} b^{5} x^{\frac {9}{2}}}{9 a^{\frac {7}{2}} b^{6} x^{\frac {21}{2}} + 18 a^{\frac {5}{2}} b^{7} x^{\frac {15}{2}} + 9 a^{\frac {3}{2}} b^{8} x^{\frac {9}{2}}} \] Input:
integrate(1/(a+b/x**3)**(3/2)/x**10,x)
Output:
16*a**(9/2)*b**(7/2)*x**9*sqrt(a*x**3/b + 1)/(9*a**(7/2)*b**6*x**(21/2) + 18*a**(5/2)*b**7*x**(15/2) + 9*a**(3/2)*b**8*x**(9/2)) + 24*a**(7/2)*b**(9 /2)*x**6*sqrt(a*x**3/b + 1)/(9*a**(7/2)*b**6*x**(21/2) + 18*a**(5/2)*b**7* x**(15/2) + 9*a**(3/2)*b**8*x**(9/2)) + 6*a**(5/2)*b**(11/2)*x**3*sqrt(a*x **3/b + 1)/(9*a**(7/2)*b**6*x**(21/2) + 18*a**(5/2)*b**7*x**(15/2) + 9*a** (3/2)*b**8*x**(9/2)) - 2*a**(3/2)*b**(13/2)*sqrt(a*x**3/b + 1)/(9*a**(7/2) *b**6*x**(21/2) + 18*a**(5/2)*b**7*x**(15/2) + 9*a**(3/2)*b**8*x**(9/2)) - 16*a**5*b**3*x**(21/2)/(9*a**(7/2)*b**6*x**(21/2) + 18*a**(5/2)*b**7*x**( 15/2) + 9*a**(3/2)*b**8*x**(9/2)) - 32*a**4*b**4*x**(15/2)/(9*a**(7/2)*b** 6*x**(21/2) + 18*a**(5/2)*b**7*x**(15/2) + 9*a**(3/2)*b**8*x**(9/2)) - 16* a**3*b**5*x**(9/2)/(9*a**(7/2)*b**6*x**(21/2) + 18*a**(5/2)*b**7*x**(15/2) + 9*a**(3/2)*b**8*x**(9/2))
Time = 0.06 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.80 \[ \int \frac {1}{\left (a+\frac {b}{x^3}\right )^{3/2} x^{10}} \, dx=-\frac {2 \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {3}{2}}}{9 \, b^{3}} + \frac {4 \, \sqrt {a + \frac {b}{x^{3}}} a}{3 \, b^{3}} + \frac {2 \, a^{2}}{3 \, \sqrt {a + \frac {b}{x^{3}}} b^{3}} \] Input:
integrate(1/(a+b/x^3)^(3/2)/x^10,x, algorithm="maxima")
Output:
-2/9*(a + b/x^3)^(3/2)/b^3 + 4/3*sqrt(a + b/x^3)*a/b^3 + 2/3*a^2/(sqrt(a + b/x^3)*b^3)
\[ \int \frac {1}{\left (a+\frac {b}{x^3}\right )^{3/2} x^{10}} \, dx=\int { \frac {1}{{\left (a + \frac {b}{x^{3}}\right )}^{\frac {3}{2}} x^{10}} \,d x } \] Input:
integrate(1/(a+b/x^3)^(3/2)/x^10,x, algorithm="giac")
Output:
integrate(1/((a + b/x^3)^(3/2)*x^10), x)
Time = 0.61 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.05 \[ \int \frac {1}{\left (a+\frac {b}{x^3}\right )^{3/2} x^{10}} \, dx=\frac {10\,a\,\sqrt {a+\frac {b}{x^3}}}{9\,b^3}-\frac {2\,\sqrt {a+\frac {b}{x^3}}}{9\,b^2\,x^3}+\frac {2\,a^2\,x^3\,\sqrt {a+\frac {b}{x^3}}}{3\,b^3\,\left (a\,x^3+b\right )} \] Input:
int(1/(x^10*(a + b/x^3)^(3/2)),x)
Output:
(10*a*(a + b/x^3)^(1/2))/(9*b^3) - (2*(a + b/x^3)^(1/2))/(9*b^2*x^3) + (2* a^2*x^3*(a + b/x^3)^(1/2))/(3*b^3*(b + a*x^3))
Time = 0.30 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.85 \[ \int \frac {1}{\left (a+\frac {b}{x^3}\right )^{3/2} x^{10}} \, dx=\frac {2 \sqrt {a \,x^{3}+b}\, \left (8 a^{2} x^{6}+4 a b \,x^{3}-b^{2}\right )}{9 \sqrt {x}\, b^{3} x^{4} \left (a \,x^{3}+b \right )} \] Input:
int(1/(a+b/x^3)^(3/2)/x^10,x)
Output:
(2*sqrt(a*x**3 + b)*(8*a**2*x**6 + 4*a*b*x**3 - b**2))/(9*sqrt(x)*b**3*x** 4*(a*x**3 + b))