Integrand size = 17, antiderivative size = 104 \[ \int \frac {1}{x^6 \left (a+b x^{3/2}\right )^{2/3}} \, dx=-\frac {\sqrt [3]{a+b x^{3/2}}}{5 a x^5}+\frac {9 b \sqrt [3]{a+b x^{3/2}}}{35 a^2 x^{7/2}}-\frac {27 b^2 \sqrt [3]{a+b x^{3/2}}}{70 a^3 x^2}+\frac {81 b^3 \sqrt [3]{a+b x^{3/2}}}{70 a^4 \sqrt {x}} \]
-1/5*(a+b*x^(3/2))^(1/3)/a/x^5+9/35*b*(a+b*x^(3/2))^(1/3)/a^2/x^(7/2)-27/7 0*b^2*(a+b*x^(3/2))^(1/3)/a^3/x^2+81/70*b^3*(a+b*x^(3/2))^(1/3)/a^4/x^(1/2 )
Time = 0.29 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.57 \[ \int \frac {1}{x^6 \left (a+b x^{3/2}\right )^{2/3}} \, dx=\frac {\sqrt [3]{a+b x^{3/2}} \left (-14 a^3+18 a^2 b x^{3/2}-27 a b^2 x^3+81 b^3 x^{9/2}\right )}{70 a^4 x^5} \]
((a + b*x^(3/2))^(1/3)*(-14*a^3 + 18*a^2*b*x^(3/2) - 27*a*b^2*x^3 + 81*b^3 *x^(9/2)))/(70*a^4*x^5)
Time = 0.24 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.12, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {803, 803, 803, 796}
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.
\(\displaystyle \int \frac {1}{x^6 \left (a+b x^{3/2}\right )^{2/3}} \, dx\) |
\(\Big \downarrow \) 803 |
\(\displaystyle -\frac {9 b \int \frac {1}{x^{9/2} \left (b x^{3/2}+a\right )^{2/3}}dx}{10 a}-\frac {\sqrt [3]{a+b x^{3/2}}}{5 a x^5}\) |
\(\Big \downarrow \) 803 |
\(\displaystyle -\frac {9 b \left (-\frac {6 b \int \frac {1}{x^3 \left (b x^{3/2}+a\right )^{2/3}}dx}{7 a}-\frac {2 \sqrt [3]{a+b x^{3/2}}}{7 a x^{7/2}}\right )}{10 a}-\frac {\sqrt [3]{a+b x^{3/2}}}{5 a x^5}\) |
\(\Big \downarrow \) 803 |
\(\displaystyle -\frac {9 b \left (-\frac {6 b \left (-\frac {3 b \int \frac {1}{x^{3/2} \left (b x^{3/2}+a\right )^{2/3}}dx}{4 a}-\frac {\sqrt [3]{a+b x^{3/2}}}{2 a x^2}\right )}{7 a}-\frac {2 \sqrt [3]{a+b x^{3/2}}}{7 a x^{7/2}}\right )}{10 a}-\frac {\sqrt [3]{a+b x^{3/2}}}{5 a x^5}\) |
\(\Big \downarrow \) 796 |
\(\displaystyle -\frac {9 b \left (-\frac {6 b \left (\frac {3 b \sqrt [3]{a+b x^{3/2}}}{2 a^2 \sqrt {x}}-\frac {\sqrt [3]{a+b x^{3/2}}}{2 a x^2}\right )}{7 a}-\frac {2 \sqrt [3]{a+b x^{3/2}}}{7 a x^{7/2}}\right )}{10 a}-\frac {\sqrt [3]{a+b x^{3/2}}}{5 a x^5}\) |
-1/5*(a + b*x^(3/2))^(1/3)/(a*x^5) - (9*b*((-2*(a + b*x^(3/2))^(1/3))/(7*a *x^(7/2)) - (6*b*(-1/2*(a + b*x^(3/2))^(1/3)/(a*x^2) + (3*b*(a + b*x^(3/2) )^(1/3))/(2*a^2*Sqrt[x])))/(7*a)))/(10*a)
3.23.74.3.1 Defintions of rubi rules used
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c* x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*c*(m + 1))), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]
Int[(x_)^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x^(m + 1)*(( a + b*x^n)^(p + 1)/(a*(m + 1))), x] - Simp[b*((m + n*(p + 1) + 1)/(a*(m + 1 ))) Int[x^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, m, n, p}, x] && I LtQ[Simplify[(m + 1)/n + p + 1], 0] && NeQ[m, -1]
\[\int \frac {1}{x^{6} \left (a +b \,x^{\frac {3}{2}}\right )^{\frac {2}{3}}}d x\]
Time = 0.53 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.51 \[ \int \frac {1}{x^6 \left (a+b x^{3/2}\right )^{2/3}} \, dx=-\frac {{\left (27 \, a b^{2} x^{3} + 14 \, a^{3} - 9 \, {\left (9 \, b^{3} x^{4} + 2 \, a^{2} b x\right )} \sqrt {x}\right )} {\left (b x^{\frac {3}{2}} + a\right )}^{\frac {1}{3}}}{70 \, a^{4} x^{5}} \]
-1/70*(27*a*b^2*x^3 + 14*a^3 - 9*(9*b^3*x^4 + 2*a^2*b*x)*sqrt(x))*(b*x^(3/ 2) + a)^(1/3)/(a^4*x^5)
Leaf count of result is larger than twice the leaf count of optimal. 736 vs. \(2 (95) = 190\).
Time = 6.88 (sec) , antiderivative size = 736, normalized size of antiderivative = 7.08 \[ \int \frac {1}{x^6 \left (a+b x^{3/2}\right )^{2/3}} \, dx=- \frac {56 a^{6} b^{\frac {28}{3}} x^{9} \sqrt [3]{\frac {a}{b x^{\frac {3}{2}}} + 1} \Gamma \left (- \frac {10}{3}\right )}{81 a^{7} b^{9} x^{\frac {27}{2}} \Gamma \left (\frac {2}{3}\right ) + 243 a^{6} b^{10} x^{15} \Gamma \left (\frac {2}{3}\right ) + 243 a^{5} b^{11} x^{\frac {33}{2}} \Gamma \left (\frac {2}{3}\right ) + 81 a^{4} b^{12} x^{18} \Gamma \left (\frac {2}{3}\right )} - \frac {96 a^{5} b^{\frac {31}{3}} x^{\frac {21}{2}} \sqrt [3]{\frac {a}{b x^{\frac {3}{2}}} + 1} \Gamma \left (- \frac {10}{3}\right )}{81 a^{7} b^{9} x^{\frac {27}{2}} \Gamma \left (\frac {2}{3}\right ) + 243 a^{6} b^{10} x^{15} \Gamma \left (\frac {2}{3}\right ) + 243 a^{5} b^{11} x^{\frac {33}{2}} \Gamma \left (\frac {2}{3}\right ) + 81 a^{4} b^{12} x^{18} \Gamma \left (\frac {2}{3}\right )} - \frac {60 a^{4} b^{\frac {34}{3}} x^{12} \sqrt [3]{\frac {a}{b x^{\frac {3}{2}}} + 1} \Gamma \left (- \frac {10}{3}\right )}{81 a^{7} b^{9} x^{\frac {27}{2}} \Gamma \left (\frac {2}{3}\right ) + 243 a^{6} b^{10} x^{15} \Gamma \left (\frac {2}{3}\right ) + 243 a^{5} b^{11} x^{\frac {33}{2}} \Gamma \left (\frac {2}{3}\right ) + 81 a^{4} b^{12} x^{18} \Gamma \left (\frac {2}{3}\right )} + \frac {160 a^{3} b^{\frac {37}{3}} x^{\frac {27}{2}} \sqrt [3]{\frac {a}{b x^{\frac {3}{2}}} + 1} \Gamma \left (- \frac {10}{3}\right )}{81 a^{7} b^{9} x^{\frac {27}{2}} \Gamma \left (\frac {2}{3}\right ) + 243 a^{6} b^{10} x^{15} \Gamma \left (\frac {2}{3}\right ) + 243 a^{5} b^{11} x^{\frac {33}{2}} \Gamma \left (\frac {2}{3}\right ) + 81 a^{4} b^{12} x^{18} \Gamma \left (\frac {2}{3}\right )} + \frac {720 a^{2} b^{\frac {40}{3}} x^{15} \sqrt [3]{\frac {a}{b x^{\frac {3}{2}}} + 1} \Gamma \left (- \frac {10}{3}\right )}{81 a^{7} b^{9} x^{\frac {27}{2}} \Gamma \left (\frac {2}{3}\right ) + 243 a^{6} b^{10} x^{15} \Gamma \left (\frac {2}{3}\right ) + 243 a^{5} b^{11} x^{\frac {33}{2}} \Gamma \left (\frac {2}{3}\right ) + 81 a^{4} b^{12} x^{18} \Gamma \left (\frac {2}{3}\right )} + \frac {864 a b^{\frac {43}{3}} x^{\frac {33}{2}} \sqrt [3]{\frac {a}{b x^{\frac {3}{2}}} + 1} \Gamma \left (- \frac {10}{3}\right )}{81 a^{7} b^{9} x^{\frac {27}{2}} \Gamma \left (\frac {2}{3}\right ) + 243 a^{6} b^{10} x^{15} \Gamma \left (\frac {2}{3}\right ) + 243 a^{5} b^{11} x^{\frac {33}{2}} \Gamma \left (\frac {2}{3}\right ) + 81 a^{4} b^{12} x^{18} \Gamma \left (\frac {2}{3}\right )} + \frac {324 b^{\frac {46}{3}} x^{18} \sqrt [3]{\frac {a}{b x^{\frac {3}{2}}} + 1} \Gamma \left (- \frac {10}{3}\right )}{81 a^{7} b^{9} x^{\frac {27}{2}} \Gamma \left (\frac {2}{3}\right ) + 243 a^{6} b^{10} x^{15} \Gamma \left (\frac {2}{3}\right ) + 243 a^{5} b^{11} x^{\frac {33}{2}} \Gamma \left (\frac {2}{3}\right ) + 81 a^{4} b^{12} x^{18} \Gamma \left (\frac {2}{3}\right )} \]
-56*a**6*b**(28/3)*x**9*(a/(b*x**(3/2)) + 1)**(1/3)*gamma(-10/3)/(81*a**7* b**9*x**(27/2)*gamma(2/3) + 243*a**6*b**10*x**15*gamma(2/3) + 243*a**5*b** 11*x**(33/2)*gamma(2/3) + 81*a**4*b**12*x**18*gamma(2/3)) - 96*a**5*b**(31 /3)*x**(21/2)*(a/(b*x**(3/2)) + 1)**(1/3)*gamma(-10/3)/(81*a**7*b**9*x**(2 7/2)*gamma(2/3) + 243*a**6*b**10*x**15*gamma(2/3) + 243*a**5*b**11*x**(33/ 2)*gamma(2/3) + 81*a**4*b**12*x**18*gamma(2/3)) - 60*a**4*b**(34/3)*x**12* (a/(b*x**(3/2)) + 1)**(1/3)*gamma(-10/3)/(81*a**7*b**9*x**(27/2)*gamma(2/3 ) + 243*a**6*b**10*x**15*gamma(2/3) + 243*a**5*b**11*x**(33/2)*gamma(2/3) + 81*a**4*b**12*x**18*gamma(2/3)) + 160*a**3*b**(37/3)*x**(27/2)*(a/(b*x** (3/2)) + 1)**(1/3)*gamma(-10/3)/(81*a**7*b**9*x**(27/2)*gamma(2/3) + 243*a **6*b**10*x**15*gamma(2/3) + 243*a**5*b**11*x**(33/2)*gamma(2/3) + 81*a**4 *b**12*x**18*gamma(2/3)) + 720*a**2*b**(40/3)*x**15*(a/(b*x**(3/2)) + 1)** (1/3)*gamma(-10/3)/(81*a**7*b**9*x**(27/2)*gamma(2/3) + 243*a**6*b**10*x** 15*gamma(2/3) + 243*a**5*b**11*x**(33/2)*gamma(2/3) + 81*a**4*b**12*x**18* gamma(2/3)) + 864*a*b**(43/3)*x**(33/2)*(a/(b*x**(3/2)) + 1)**(1/3)*gamma( -10/3)/(81*a**7*b**9*x**(27/2)*gamma(2/3) + 243*a**6*b**10*x**15*gamma(2/3 ) + 243*a**5*b**11*x**(33/2)*gamma(2/3) + 81*a**4*b**12*x**18*gamma(2/3)) + 324*b**(46/3)*x**18*(a/(b*x**(3/2)) + 1)**(1/3)*gamma(-10/3)/(81*a**7*b* *9*x**(27/2)*gamma(2/3) + 243*a**6*b**10*x**15*gamma(2/3) + 243*a**5*b**11 *x**(33/2)*gamma(2/3) + 81*a**4*b**12*x**18*gamma(2/3))
Time = 0.20 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.66 \[ \int \frac {1}{x^6 \left (a+b x^{3/2}\right )^{2/3}} \, dx=\frac {\frac {140 \, {\left (b x^{\frac {3}{2}} + a\right )}^{\frac {1}{3}} b^{3}}{\sqrt {x}} - \frac {105 \, {\left (b x^{\frac {3}{2}} + a\right )}^{\frac {4}{3}} b^{2}}{x^{2}} + \frac {60 \, {\left (b x^{\frac {3}{2}} + a\right )}^{\frac {7}{3}} b}{x^{\frac {7}{2}}} - \frac {14 \, {\left (b x^{\frac {3}{2}} + a\right )}^{\frac {10}{3}}}{x^{5}}}{70 \, a^{4}} \]
1/70*(140*(b*x^(3/2) + a)^(1/3)*b^3/sqrt(x) - 105*(b*x^(3/2) + a)^(4/3)*b^ 2/x^2 + 60*(b*x^(3/2) + a)^(7/3)*b/x^(7/2) - 14*(b*x^(3/2) + a)^(10/3)/x^5 )/a^4
\[ \int \frac {1}{x^6 \left (a+b x^{3/2}\right )^{2/3}} \, dx=\int { \frac {1}{{\left (b x^{\frac {3}{2}} + a\right )}^{\frac {2}{3}} x^{6}} \,d x } \]
Timed out. \[ \int \frac {1}{x^6 \left (a+b x^{3/2}\right )^{2/3}} \, dx=\int \frac {1}{x^6\,{\left (a+b\,x^{3/2}\right )}^{2/3}} \,d x \]