Integrand size = 15, antiderivative size = 68 \[ \int \frac {\left (a+b x^4\right )^{5/4}}{x^{18}} \, dx=-\frac {\left (a+b x^4\right )^{9/4}}{17 a x^{17}}+\frac {8 b \left (a+b x^4\right )^{9/4}}{221 a^2 x^{13}}-\frac {32 b^2 \left (a+b x^4\right )^{9/4}}{1989 a^3 x^9} \]
-1/17*(b*x^4+a)^(9/4)/a/x^17+8/221*b*(b*x^4+a)^(9/4)/a^2/x^13-32/1989*b^2* (b*x^4+a)^(9/4)/a^3/x^9
Time = 0.33 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.62 \[ \int \frac {\left (a+b x^4\right )^{5/4}}{x^{18}} \, dx=\frac {\left (a+b x^4\right )^{9/4} \left (-117 a^2+72 a b x^4-32 b^2 x^8\right )}{1989 a^3 x^{17}} \]
Time = 0.21 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.09, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {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.
\(\displaystyle \int \frac {\left (a+b x^4\right )^{5/4}}{x^{18}} \, dx\) |
\(\Big \downarrow \) 803 |
\(\displaystyle -\frac {8 b \int \frac {\left (b x^4+a\right )^{5/4}}{x^{14}}dx}{17 a}-\frac {\left (a+b x^4\right )^{9/4}}{17 a x^{17}}\) |
\(\Big \downarrow \) 803 |
\(\displaystyle -\frac {8 b \left (-\frac {4 b \int \frac {\left (b x^4+a\right )^{5/4}}{x^{10}}dx}{13 a}-\frac {\left (a+b x^4\right )^{9/4}}{13 a x^{13}}\right )}{17 a}-\frac {\left (a+b x^4\right )^{9/4}}{17 a x^{17}}\) |
\(\Big \downarrow \) 796 |
\(\displaystyle -\frac {8 b \left (\frac {4 b \left (a+b x^4\right )^{9/4}}{117 a^2 x^9}-\frac {\left (a+b x^4\right )^{9/4}}{13 a x^{13}}\right )}{17 a}-\frac {\left (a+b x^4\right )^{9/4}}{17 a x^{17}}\) |
-1/17*(a + b*x^4)^(9/4)/(a*x^17) - (8*b*(-1/13*(a + b*x^4)^(9/4)/(a*x^13) + (4*b*(a + b*x^4)^(9/4))/(117*a^2*x^9)))/(17*a)
3.11.69.3.1 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]
Time = 4.46 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.57
method | result | size |
gosper | \(-\frac {\left (b \,x^{4}+a \right )^{\frac {9}{4}} \left (32 b^{2} x^{8}-72 a b \,x^{4}+117 a^{2}\right )}{1989 x^{17} a^{3}}\) | \(39\) |
pseudoelliptic | \(-\frac {\left (b \,x^{4}+a \right )^{\frac {9}{4}} \left (32 b^{2} x^{8}-72 a b \,x^{4}+117 a^{2}\right )}{1989 x^{17} a^{3}}\) | \(39\) |
trager | \(-\frac {\left (32 x^{16} b^{4}-8 a \,b^{3} x^{12}+5 a^{2} b^{2} x^{8}+162 a^{3} b \,x^{4}+117 a^{4}\right ) \left (b \,x^{4}+a \right )^{\frac {1}{4}}}{1989 x^{17} a^{3}}\) | \(61\) |
risch | \(-\frac {\left (32 x^{16} b^{4}-8 a \,b^{3} x^{12}+5 a^{2} b^{2} x^{8}+162 a^{3} b \,x^{4}+117 a^{4}\right ) \left (b \,x^{4}+a \right )^{\frac {1}{4}}}{1989 x^{17} a^{3}}\) | \(61\) |
Time = 0.31 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.88 \[ \int \frac {\left (a+b x^4\right )^{5/4}}{x^{18}} \, dx=-\frac {{\left (32 \, b^{4} x^{16} - 8 \, a b^{3} x^{12} + 5 \, a^{2} b^{2} x^{8} + 162 \, a^{3} b x^{4} + 117 \, a^{4}\right )} {\left (b x^{4} + a\right )}^{\frac {1}{4}}}{1989 \, a^{3} x^{17}} \]
-1/1989*(32*b^4*x^16 - 8*a*b^3*x^12 + 5*a^2*b^2*x^8 + 162*a^3*b*x^4 + 117* a^4)*(b*x^4 + a)^(1/4)/(a^3*x^17)
Leaf count of result is larger than twice the leaf count of optimal. 609 vs. \(2 (61) = 122\).
Time = 1.77 (sec) , antiderivative size = 609, normalized size of antiderivative = 8.96 \[ \int \frac {\left (a+b x^4\right )^{5/4}}{x^{18}} \, dx=\frac {117 a^{6} b^{\frac {17}{4}} \sqrt [4]{\frac {a}{b x^{4}} + 1} \Gamma \left (- \frac {17}{4}\right )}{64 a^{5} b^{4} x^{16} \Gamma \left (- \frac {5}{4}\right ) + 128 a^{4} b^{5} x^{20} \Gamma \left (- \frac {5}{4}\right ) + 64 a^{3} b^{6} x^{24} \Gamma \left (- \frac {5}{4}\right )} + \frac {396 a^{5} b^{\frac {21}{4}} x^{4} \sqrt [4]{\frac {a}{b x^{4}} + 1} \Gamma \left (- \frac {17}{4}\right )}{64 a^{5} b^{4} x^{16} \Gamma \left (- \frac {5}{4}\right ) + 128 a^{4} b^{5} x^{20} \Gamma \left (- \frac {5}{4}\right ) + 64 a^{3} b^{6} x^{24} \Gamma \left (- \frac {5}{4}\right )} + \frac {446 a^{4} b^{\frac {25}{4}} x^{8} \sqrt [4]{\frac {a}{b x^{4}} + 1} \Gamma \left (- \frac {17}{4}\right )}{64 a^{5} b^{4} x^{16} \Gamma \left (- \frac {5}{4}\right ) + 128 a^{4} b^{5} x^{20} \Gamma \left (- \frac {5}{4}\right ) + 64 a^{3} b^{6} x^{24} \Gamma \left (- \frac {5}{4}\right )} + \frac {164 a^{3} b^{\frac {29}{4}} x^{12} \sqrt [4]{\frac {a}{b x^{4}} + 1} \Gamma \left (- \frac {17}{4}\right )}{64 a^{5} b^{4} x^{16} \Gamma \left (- \frac {5}{4}\right ) + 128 a^{4} b^{5} x^{20} \Gamma \left (- \frac {5}{4}\right ) + 64 a^{3} b^{6} x^{24} \Gamma \left (- \frac {5}{4}\right )} + \frac {21 a^{2} b^{\frac {33}{4}} x^{16} \sqrt [4]{\frac {a}{b x^{4}} + 1} \Gamma \left (- \frac {17}{4}\right )}{64 a^{5} b^{4} x^{16} \Gamma \left (- \frac {5}{4}\right ) + 128 a^{4} b^{5} x^{20} \Gamma \left (- \frac {5}{4}\right ) + 64 a^{3} b^{6} x^{24} \Gamma \left (- \frac {5}{4}\right )} + \frac {56 a b^{\frac {37}{4}} x^{20} \sqrt [4]{\frac {a}{b x^{4}} + 1} \Gamma \left (- \frac {17}{4}\right )}{64 a^{5} b^{4} x^{16} \Gamma \left (- \frac {5}{4}\right ) + 128 a^{4} b^{5} x^{20} \Gamma \left (- \frac {5}{4}\right ) + 64 a^{3} b^{6} x^{24} \Gamma \left (- \frac {5}{4}\right )} + \frac {32 b^{\frac {41}{4}} x^{24} \sqrt [4]{\frac {a}{b x^{4}} + 1} \Gamma \left (- \frac {17}{4}\right )}{64 a^{5} b^{4} x^{16} \Gamma \left (- \frac {5}{4}\right ) + 128 a^{4} b^{5} x^{20} \Gamma \left (- \frac {5}{4}\right ) + 64 a^{3} b^{6} x^{24} \Gamma \left (- \frac {5}{4}\right )} \]
117*a**6*b**(17/4)*(a/(b*x**4) + 1)**(1/4)*gamma(-17/4)/(64*a**5*b**4*x**1 6*gamma(-5/4) + 128*a**4*b**5*x**20*gamma(-5/4) + 64*a**3*b**6*x**24*gamma (-5/4)) + 396*a**5*b**(21/4)*x**4*(a/(b*x**4) + 1)**(1/4)*gamma(-17/4)/(64 *a**5*b**4*x**16*gamma(-5/4) + 128*a**4*b**5*x**20*gamma(-5/4) + 64*a**3*b **6*x**24*gamma(-5/4)) + 446*a**4*b**(25/4)*x**8*(a/(b*x**4) + 1)**(1/4)*g amma(-17/4)/(64*a**5*b**4*x**16*gamma(-5/4) + 128*a**4*b**5*x**20*gamma(-5 /4) + 64*a**3*b**6*x**24*gamma(-5/4)) + 164*a**3*b**(29/4)*x**12*(a/(b*x** 4) + 1)**(1/4)*gamma(-17/4)/(64*a**5*b**4*x**16*gamma(-5/4) + 128*a**4*b** 5*x**20*gamma(-5/4) + 64*a**3*b**6*x**24*gamma(-5/4)) + 21*a**2*b**(33/4)* x**16*(a/(b*x**4) + 1)**(1/4)*gamma(-17/4)/(64*a**5*b**4*x**16*gamma(-5/4) + 128*a**4*b**5*x**20*gamma(-5/4) + 64*a**3*b**6*x**24*gamma(-5/4)) + 56* a*b**(37/4)*x**20*(a/(b*x**4) + 1)**(1/4)*gamma(-17/4)/(64*a**5*b**4*x**16 *gamma(-5/4) + 128*a**4*b**5*x**20*gamma(-5/4) + 64*a**3*b**6*x**24*gamma( -5/4)) + 32*b**(41/4)*x**24*(a/(b*x**4) + 1)**(1/4)*gamma(-17/4)/(64*a**5* b**4*x**16*gamma(-5/4) + 128*a**4*b**5*x**20*gamma(-5/4) + 64*a**3*b**6*x* *24*gamma(-5/4))
Time = 0.20 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.76 \[ \int \frac {\left (a+b x^4\right )^{5/4}}{x^{18}} \, dx=-\frac {\frac {221 \, {\left (b x^{4} + a\right )}^{\frac {9}{4}} b^{2}}{x^{9}} - \frac {306 \, {\left (b x^{4} + a\right )}^{\frac {13}{4}} b}{x^{13}} + \frac {117 \, {\left (b x^{4} + a\right )}^{\frac {17}{4}}}{x^{17}}}{1989 \, a^{3}} \]
-1/1989*(221*(b*x^4 + a)^(9/4)*b^2/x^9 - 306*(b*x^4 + a)^(13/4)*b/x^13 + 1 17*(b*x^4 + a)^(17/4)/x^17)/a^3
\[ \int \frac {\left (a+b x^4\right )^{5/4}}{x^{18}} \, dx=\int { \frac {{\left (b x^{4} + a\right )}^{\frac {5}{4}}}{x^{18}} \,d x } \]
Time = 7.00 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.34 \[ \int \frac {\left (a+b x^4\right )^{5/4}}{x^{18}} \, dx=\frac {8\,b^3\,{\left (b\,x^4+a\right )}^{1/4}}{1989\,a^2\,x^5}-\frac {18\,b\,{\left (b\,x^4+a\right )}^{1/4}}{221\,x^{13}}-\frac {32\,b^4\,{\left (b\,x^4+a\right )}^{1/4}}{1989\,a^3\,x}-\frac {a\,{\left (b\,x^4+a\right )}^{1/4}}{17\,x^{17}}-\frac {5\,b^2\,{\left (b\,x^4+a\right )}^{1/4}}{1989\,a\,x^9} \]