Integrand size = 15, antiderivative size = 92 \[ \int \frac {1}{x^{16} \sqrt [4]{a+b x^4}} \, dx=-\frac {\left (a+b x^4\right )^{3/4}}{15 a x^{15}}+\frac {4 b \left (a+b x^4\right )^{3/4}}{55 a^2 x^{11}}-\frac {32 b^2 \left (a+b x^4\right )^{3/4}}{385 a^3 x^7}+\frac {128 b^3 \left (a+b x^4\right )^{3/4}}{1155 a^4 x^3} \]
-1/15*(b*x^4+a)^(3/4)/a/x^15+4/55*b*(b*x^4+a)^(3/4)/a^2/x^11-32/385*b^2*(b *x^4+a)^(3/4)/a^3/x^7+128/1155*b^3*(b*x^4+a)^(3/4)/a^4/x^3
Time = 0.29 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.58 \[ \int \frac {1}{x^{16} \sqrt [4]{a+b x^4}} \, dx=\frac {\left (a+b x^4\right )^{3/4} \left (-77 a^3+84 a^2 b x^4-96 a b^2 x^8+128 b^3 x^{12}\right )}{1155 a^4 x^{15}} \]
Time = 0.22 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.13, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, 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.
\(\displaystyle \int \frac {1}{x^{16} \sqrt [4]{a+b x^4}} \, dx\) |
\(\Big \downarrow \) 803 |
\(\displaystyle -\frac {4 b \int \frac {1}{x^{12} \sqrt [4]{b x^4+a}}dx}{5 a}-\frac {\left (a+b x^4\right )^{3/4}}{15 a x^{15}}\) |
\(\Big \downarrow \) 803 |
\(\displaystyle -\frac {4 b \left (-\frac {8 b \int \frac {1}{x^8 \sqrt [4]{b x^4+a}}dx}{11 a}-\frac {\left (a+b x^4\right )^{3/4}}{11 a x^{11}}\right )}{5 a}-\frac {\left (a+b x^4\right )^{3/4}}{15 a x^{15}}\) |
\(\Big \downarrow \) 803 |
\(\displaystyle -\frac {4 b \left (-\frac {8 b \left (-\frac {4 b \int \frac {1}{x^4 \sqrt [4]{b x^4+a}}dx}{7 a}-\frac {\left (a+b x^4\right )^{3/4}}{7 a x^7}\right )}{11 a}-\frac {\left (a+b x^4\right )^{3/4}}{11 a x^{11}}\right )}{5 a}-\frac {\left (a+b x^4\right )^{3/4}}{15 a x^{15}}\) |
\(\Big \downarrow \) 796 |
\(\displaystyle -\frac {4 b \left (-\frac {8 b \left (\frac {4 b \left (a+b x^4\right )^{3/4}}{21 a^2 x^3}-\frac {\left (a+b x^4\right )^{3/4}}{7 a x^7}\right )}{11 a}-\frac {\left (a+b x^4\right )^{3/4}}{11 a x^{11}}\right )}{5 a}-\frac {\left (a+b x^4\right )^{3/4}}{15 a x^{15}}\) |
-1/15*(a + b*x^4)^(3/4)/(a*x^15) - (4*b*(-1/11*(a + b*x^4)^(3/4)/(a*x^11) - (8*b*(-1/7*(a + b*x^4)^(3/4)/(a*x^7) + (4*b*(a + b*x^4)^(3/4))/(21*a^2*x ^3)))/(11*a)))/(5*a)
3.12.1.3.1 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 = 4.38 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.54
method | result | size |
gosper | \(-\frac {\left (b \,x^{4}+a \right )^{\frac {3}{4}} \left (-128 b^{3} x^{12}+96 a \,b^{2} x^{8}-84 a^{2} b \,x^{4}+77 a^{3}\right )}{1155 x^{15} a^{4}}\) | \(50\) |
trager | \(-\frac {\left (b \,x^{4}+a \right )^{\frac {3}{4}} \left (-128 b^{3} x^{12}+96 a \,b^{2} x^{8}-84 a^{2} b \,x^{4}+77 a^{3}\right )}{1155 x^{15} a^{4}}\) | \(50\) |
risch | \(-\frac {\left (b \,x^{4}+a \right )^{\frac {3}{4}} \left (-128 b^{3} x^{12}+96 a \,b^{2} x^{8}-84 a^{2} b \,x^{4}+77 a^{3}\right )}{1155 x^{15} a^{4}}\) | \(50\) |
pseudoelliptic | \(-\frac {\left (b \,x^{4}+a \right )^{\frac {3}{4}} \left (-128 b^{3} x^{12}+96 a \,b^{2} x^{8}-84 a^{2} b \,x^{4}+77 a^{3}\right )}{1155 x^{15} a^{4}}\) | \(50\) |
Time = 0.25 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.53 \[ \int \frac {1}{x^{16} \sqrt [4]{a+b x^4}} \, dx=\frac {{\left (128 \, b^{3} x^{12} - 96 \, a b^{2} x^{8} + 84 \, a^{2} b x^{4} - 77 \, a^{3}\right )} {\left (b x^{4} + a\right )}^{\frac {3}{4}}}{1155 \, a^{4} x^{15}} \]
Leaf count of result is larger than twice the leaf count of optimal. 692 vs. \(2 (85) = 170\).
Time = 1.38 (sec) , antiderivative size = 692, normalized size of antiderivative = 7.52 \[ \int \frac {1}{x^{16} \sqrt [4]{a+b x^4}} \, dx=- \frac {231 a^{6} b^{\frac {39}{4}} \left (\frac {a}{b x^{4}} + 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {15}{4}\right )}{256 a^{7} b^{9} x^{12} \Gamma \left (\frac {1}{4}\right ) + 768 a^{6} b^{10} x^{16} \Gamma \left (\frac {1}{4}\right ) + 768 a^{5} b^{11} x^{20} \Gamma \left (\frac {1}{4}\right ) + 256 a^{4} b^{12} x^{24} \Gamma \left (\frac {1}{4}\right )} - \frac {441 a^{5} b^{\frac {43}{4}} x^{4} \left (\frac {a}{b x^{4}} + 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {15}{4}\right )}{256 a^{7} b^{9} x^{12} \Gamma \left (\frac {1}{4}\right ) + 768 a^{6} b^{10} x^{16} \Gamma \left (\frac {1}{4}\right ) + 768 a^{5} b^{11} x^{20} \Gamma \left (\frac {1}{4}\right ) + 256 a^{4} b^{12} x^{24} \Gamma \left (\frac {1}{4}\right )} - \frac {225 a^{4} b^{\frac {47}{4}} x^{8} \left (\frac {a}{b x^{4}} + 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {15}{4}\right )}{256 a^{7} b^{9} x^{12} \Gamma \left (\frac {1}{4}\right ) + 768 a^{6} b^{10} x^{16} \Gamma \left (\frac {1}{4}\right ) + 768 a^{5} b^{11} x^{20} \Gamma \left (\frac {1}{4}\right ) + 256 a^{4} b^{12} x^{24} \Gamma \left (\frac {1}{4}\right )} + \frac {45 a^{3} b^{\frac {51}{4}} x^{12} \left (\frac {a}{b x^{4}} + 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {15}{4}\right )}{256 a^{7} b^{9} x^{12} \Gamma \left (\frac {1}{4}\right ) + 768 a^{6} b^{10} x^{16} \Gamma \left (\frac {1}{4}\right ) + 768 a^{5} b^{11} x^{20} \Gamma \left (\frac {1}{4}\right ) + 256 a^{4} b^{12} x^{24} \Gamma \left (\frac {1}{4}\right )} + \frac {540 a^{2} b^{\frac {55}{4}} x^{16} \left (\frac {a}{b x^{4}} + 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {15}{4}\right )}{256 a^{7} b^{9} x^{12} \Gamma \left (\frac {1}{4}\right ) + 768 a^{6} b^{10} x^{16} \Gamma \left (\frac {1}{4}\right ) + 768 a^{5} b^{11} x^{20} \Gamma \left (\frac {1}{4}\right ) + 256 a^{4} b^{12} x^{24} \Gamma \left (\frac {1}{4}\right )} + \frac {864 a b^{\frac {59}{4}} x^{20} \left (\frac {a}{b x^{4}} + 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {15}{4}\right )}{256 a^{7} b^{9} x^{12} \Gamma \left (\frac {1}{4}\right ) + 768 a^{6} b^{10} x^{16} \Gamma \left (\frac {1}{4}\right ) + 768 a^{5} b^{11} x^{20} \Gamma \left (\frac {1}{4}\right ) + 256 a^{4} b^{12} x^{24} \Gamma \left (\frac {1}{4}\right )} + \frac {384 b^{\frac {63}{4}} x^{24} \left (\frac {a}{b x^{4}} + 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {15}{4}\right )}{256 a^{7} b^{9} x^{12} \Gamma \left (\frac {1}{4}\right ) + 768 a^{6} b^{10} x^{16} \Gamma \left (\frac {1}{4}\right ) + 768 a^{5} b^{11} x^{20} \Gamma \left (\frac {1}{4}\right ) + 256 a^{4} b^{12} x^{24} \Gamma \left (\frac {1}{4}\right )} \]
-231*a**6*b**(39/4)*(a/(b*x**4) + 1)**(3/4)*gamma(-15/4)/(256*a**7*b**9*x* *12*gamma(1/4) + 768*a**6*b**10*x**16*gamma(1/4) + 768*a**5*b**11*x**20*ga mma(1/4) + 256*a**4*b**12*x**24*gamma(1/4)) - 441*a**5*b**(43/4)*x**4*(a/( b*x**4) + 1)**(3/4)*gamma(-15/4)/(256*a**7*b**9*x**12*gamma(1/4) + 768*a** 6*b**10*x**16*gamma(1/4) + 768*a**5*b**11*x**20*gamma(1/4) + 256*a**4*b**1 2*x**24*gamma(1/4)) - 225*a**4*b**(47/4)*x**8*(a/(b*x**4) + 1)**(3/4)*gamm a(-15/4)/(256*a**7*b**9*x**12*gamma(1/4) + 768*a**6*b**10*x**16*gamma(1/4) + 768*a**5*b**11*x**20*gamma(1/4) + 256*a**4*b**12*x**24*gamma(1/4)) + 45 *a**3*b**(51/4)*x**12*(a/(b*x**4) + 1)**(3/4)*gamma(-15/4)/(256*a**7*b**9* x**12*gamma(1/4) + 768*a**6*b**10*x**16*gamma(1/4) + 768*a**5*b**11*x**20* gamma(1/4) + 256*a**4*b**12*x**24*gamma(1/4)) + 540*a**2*b**(55/4)*x**16*( a/(b*x**4) + 1)**(3/4)*gamma(-15/4)/(256*a**7*b**9*x**12*gamma(1/4) + 768* a**6*b**10*x**16*gamma(1/4) + 768*a**5*b**11*x**20*gamma(1/4) + 256*a**4*b **12*x**24*gamma(1/4)) + 864*a*b**(59/4)*x**20*(a/(b*x**4) + 1)**(3/4)*gam ma(-15/4)/(256*a**7*b**9*x**12*gamma(1/4) + 768*a**6*b**10*x**16*gamma(1/4 ) + 768*a**5*b**11*x**20*gamma(1/4) + 256*a**4*b**12*x**24*gamma(1/4)) + 3 84*b**(63/4)*x**24*(a/(b*x**4) + 1)**(3/4)*gamma(-15/4)/(256*a**7*b**9*x** 12*gamma(1/4) + 768*a**6*b**10*x**16*gamma(1/4) + 768*a**5*b**11*x**20*gam ma(1/4) + 256*a**4*b**12*x**24*gamma(1/4))
Time = 0.21 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.75 \[ \int \frac {1}{x^{16} \sqrt [4]{a+b x^4}} \, dx=\frac {\frac {385 \, {\left (b x^{4} + a\right )}^{\frac {3}{4}} b^{3}}{x^{3}} - \frac {495 \, {\left (b x^{4} + a\right )}^{\frac {7}{4}} b^{2}}{x^{7}} + \frac {315 \, {\left (b x^{4} + a\right )}^{\frac {11}{4}} b}{x^{11}} - \frac {77 \, {\left (b x^{4} + a\right )}^{\frac {15}{4}}}{x^{15}}}{1155 \, a^{4}} \]
1/1155*(385*(b*x^4 + a)^(3/4)*b^3/x^3 - 495*(b*x^4 + a)^(7/4)*b^2/x^7 + 31 5*(b*x^4 + a)^(11/4)*b/x^11 - 77*(b*x^4 + a)^(15/4)/x^15)/a^4
\[ \int \frac {1}{x^{16} \sqrt [4]{a+b x^4}} \, dx=\int { \frac {1}{{\left (b x^{4} + a\right )}^{\frac {1}{4}} x^{16}} \,d x } \]
Time = 5.92 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.83 \[ \int \frac {1}{x^{16} \sqrt [4]{a+b x^4}} \, dx=\frac {4\,b\,{\left (b\,x^4+a\right )}^{3/4}}{55\,a^2\,x^{11}}-\frac {{\left (b\,x^4+a\right )}^{3/4}}{15\,a\,x^{15}}+\frac {128\,b^3\,{\left (b\,x^4+a\right )}^{3/4}}{1155\,a^4\,x^3}-\frac {32\,b^2\,{\left (b\,x^4+a\right )}^{3/4}}{385\,a^3\,x^7} \]