Integrand size = 15, antiderivative size = 116 \[ \int \frac {1}{x^{20} \sqrt [4]{a+b x^4}} \, dx=-\frac {\left (a+b x^4\right )^{3/4}}{19 a x^{19}}+\frac {16 b \left (a+b x^4\right )^{3/4}}{285 a^2 x^{15}}-\frac {64 b^2 \left (a+b x^4\right )^{3/4}}{1045 a^3 x^{11}}+\frac {512 b^3 \left (a+b x^4\right )^{3/4}}{7315 a^4 x^7}-\frac {2048 b^4 \left (a+b x^4\right )^{3/4}}{21945 a^5 x^3} \]
-1/19*(b*x^4+a)^(3/4)/a/x^19+16/285*b*(b*x^4+a)^(3/4)/a^2/x^15-64/1045*b^2 *(b*x^4+a)^(3/4)/a^3/x^11+512/7315*b^3*(b*x^4+a)^(3/4)/a^4/x^7-2048/21945* b^4*(b*x^4+a)^(3/4)/a^5/x^3
Time = 0.34 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.55 \[ \int \frac {1}{x^{20} \sqrt [4]{a+b x^4}} \, dx=\frac {\left (a+b x^4\right )^{3/4} \left (-1155 a^4+1232 a^3 b x^4-1344 a^2 b^2 x^8+1536 a b^3 x^{12}-2048 b^4 x^{16}\right )}{21945 a^5 x^{19}} \]
((a + b*x^4)^(3/4)*(-1155*a^4 + 1232*a^3*b*x^4 - 1344*a^2*b^2*x^8 + 1536*a *b^3*x^12 - 2048*b^4*x^16))/(21945*a^5*x^19)
Time = 0.25 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.16, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {803, 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.
\(\displaystyle \int \frac {1}{x^{20} \sqrt [4]{a+b x^4}} \, dx\) |
\(\Big \downarrow \) 803 |
\(\displaystyle -\frac {16 b \int \frac {1}{x^{16} \sqrt [4]{b x^4+a}}dx}{19 a}-\frac {\left (a+b x^4\right )^{3/4}}{19 a x^{19}}\) |
\(\Big \downarrow \) 803 |
\(\displaystyle -\frac {16 b \left (-\frac {4 b \int \frac {1}{x^{12} \sqrt [4]{b x^4+a}}dx}{5 a}-\frac {\left (a+b x^4\right )^{3/4}}{15 a x^{15}}\right )}{19 a}-\frac {\left (a+b x^4\right )^{3/4}}{19 a x^{19}}\) |
\(\Big \downarrow \) 803 |
\(\displaystyle -\frac {16 b \left (-\frac {4 b \left (-\frac {8 b \int \frac {1}{x^8 \sqrt [4]{b x^4+a}}dx}{11 a}-\frac {\left (a+b x^4\right )^{3/4}}{11 a x^{11}}\right )}{5 a}-\frac {\left (a+b x^4\right )^{3/4}}{15 a x^{15}}\right )}{19 a}-\frac {\left (a+b x^4\right )^{3/4}}{19 a x^{19}}\) |
\(\Big \downarrow \) 803 |
\(\displaystyle -\frac {16 b \left (-\frac {4 b \left (-\frac {8 b \left (-\frac {4 b \int \frac {1}{x^4 \sqrt [4]{b x^4+a}}dx}{7 a}-\frac {\left (a+b x^4\right )^{3/4}}{7 a x^7}\right )}{11 a}-\frac {\left (a+b x^4\right )^{3/4}}{11 a x^{11}}\right )}{5 a}-\frac {\left (a+b x^4\right )^{3/4}}{15 a x^{15}}\right )}{19 a}-\frac {\left (a+b x^4\right )^{3/4}}{19 a x^{19}}\) |
\(\Big \downarrow \) 796 |
\(\displaystyle -\frac {16 b \left (-\frac {4 b \left (-\frac {8 b \left (\frac {4 b \left (a+b x^4\right )^{3/4}}{21 a^2 x^3}-\frac {\left (a+b x^4\right )^{3/4}}{7 a x^7}\right )}{11 a}-\frac {\left (a+b x^4\right )^{3/4}}{11 a x^{11}}\right )}{5 a}-\frac {\left (a+b x^4\right )^{3/4}}{15 a x^{15}}\right )}{19 a}-\frac {\left (a+b x^4\right )^{3/4}}{19 a x^{19}}\) |
-1/19*(a + b*x^4)^(3/4)/(a*x^19) - (16*b*(-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)))/(19*a)
3.12.2.3.1 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 = 4.48 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.53
method | result | size |
gosper | \(-\frac {\left (b \,x^{4}+a \right )^{\frac {3}{4}} \left (2048 x^{16} b^{4}-1536 a \,b^{3} x^{12}+1344 a^{2} b^{2} x^{8}-1232 a^{3} b \,x^{4}+1155 a^{4}\right )}{21945 x^{19} a^{5}}\) | \(61\) |
trager | \(-\frac {\left (b \,x^{4}+a \right )^{\frac {3}{4}} \left (2048 x^{16} b^{4}-1536 a \,b^{3} x^{12}+1344 a^{2} b^{2} x^{8}-1232 a^{3} b \,x^{4}+1155 a^{4}\right )}{21945 x^{19} a^{5}}\) | \(61\) |
risch | \(-\frac {\left (b \,x^{4}+a \right )^{\frac {3}{4}} \left (2048 x^{16} b^{4}-1536 a \,b^{3} x^{12}+1344 a^{2} b^{2} x^{8}-1232 a^{3} b \,x^{4}+1155 a^{4}\right )}{21945 x^{19} a^{5}}\) | \(61\) |
pseudoelliptic | \(-\frac {\left (b \,x^{4}+a \right )^{\frac {3}{4}} \left (2048 x^{16} b^{4}-1536 a \,b^{3} x^{12}+1344 a^{2} b^{2} x^{8}-1232 a^{3} b \,x^{4}+1155 a^{4}\right )}{21945 x^{19} a^{5}}\) | \(61\) |
-1/21945*(b*x^4+a)^(3/4)*(2048*b^4*x^16-1536*a*b^3*x^12+1344*a^2*b^2*x^8-1 232*a^3*b*x^4+1155*a^4)/x^19/a^5
Time = 0.25 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.52 \[ \int \frac {1}{x^{20} \sqrt [4]{a+b x^4}} \, dx=-\frac {{\left (2048 \, b^{4} x^{16} - 1536 \, a b^{3} x^{12} + 1344 \, a^{2} b^{2} x^{8} - 1232 \, a^{3} b x^{4} + 1155 \, a^{4}\right )} {\left (b x^{4} + a\right )}^{\frac {3}{4}}}{21945 \, a^{5} x^{19}} \]
-1/21945*(2048*b^4*x^16 - 1536*a*b^3*x^12 + 1344*a^2*b^2*x^8 - 1232*a^3*b* x^4 + 1155*a^4)*(b*x^4 + a)^(3/4)/(a^5*x^19)
Leaf count of result is larger than twice the leaf count of optimal. 1046 vs. \(2 (109) = 218\).
Time = 1.92 (sec) , antiderivative size = 1046, normalized size of antiderivative = 9.02 \[ \int \frac {1}{x^{20} \sqrt [4]{a+b x^4}} \, dx=\text {Too large to display} \]
3465*a**8*b**(67/4)*(a/(b*x**4) + 1)**(3/4)*gamma(-19/4)/(1024*a**9*b**16* x**16*gamma(1/4) + 4096*a**8*b**17*x**20*gamma(1/4) + 6144*a**7*b**18*x**2 4*gamma(1/4) + 4096*a**6*b**19*x**28*gamma(1/4) + 1024*a**5*b**20*x**32*ga mma(1/4)) + 10164*a**7*b**(71/4)*x**4*(a/(b*x**4) + 1)**(3/4)*gamma(-19/4) /(1024*a**9*b**16*x**16*gamma(1/4) + 4096*a**8*b**17*x**20*gamma(1/4) + 61 44*a**7*b**18*x**24*gamma(1/4) + 4096*a**6*b**19*x**28*gamma(1/4) + 1024*a **5*b**20*x**32*gamma(1/4)) + 10038*a**6*b**(75/4)*x**8*(a/(b*x**4) + 1)** (3/4)*gamma(-19/4)/(1024*a**9*b**16*x**16*gamma(1/4) + 4096*a**8*b**17*x** 20*gamma(1/4) + 6144*a**7*b**18*x**24*gamma(1/4) + 4096*a**6*b**19*x**28*g amma(1/4) + 1024*a**5*b**20*x**32*gamma(1/4)) + 3204*a**5*b**(79/4)*x**12* (a/(b*x**4) + 1)**(3/4)*gamma(-19/4)/(1024*a**9*b**16*x**16*gamma(1/4) + 4 096*a**8*b**17*x**20*gamma(1/4) + 6144*a**7*b**18*x**24*gamma(1/4) + 4096* a**6*b**19*x**28*gamma(1/4) + 1024*a**5*b**20*x**32*gamma(1/4)) + 585*a**4 *b**(83/4)*x**16*(a/(b*x**4) + 1)**(3/4)*gamma(-19/4)/(1024*a**9*b**16*x** 16*gamma(1/4) + 4096*a**8*b**17*x**20*gamma(1/4) + 6144*a**7*b**18*x**24*g amma(1/4) + 4096*a**6*b**19*x**28*gamma(1/4) + 1024*a**5*b**20*x**32*gamma (1/4)) + 9360*a**3*b**(87/4)*x**20*(a/(b*x**4) + 1)**(3/4)*gamma(-19/4)/(1 024*a**9*b**16*x**16*gamma(1/4) + 4096*a**8*b**17*x**20*gamma(1/4) + 6144* a**7*b**18*x**24*gamma(1/4) + 4096*a**6*b**19*x**28*gamma(1/4) + 1024*a**5 *b**20*x**32*gamma(1/4)) + 22464*a**2*b**(91/4)*x**24*(a/(b*x**4) + 1)*...
Time = 0.25 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.74 \[ \int \frac {1}{x^{20} \sqrt [4]{a+b x^4}} \, dx=-\frac {\frac {7315 \, {\left (b x^{4} + a\right )}^{\frac {3}{4}} b^{4}}{x^{3}} - \frac {12540 \, {\left (b x^{4} + a\right )}^{\frac {7}{4}} b^{3}}{x^{7}} + \frac {11970 \, {\left (b x^{4} + a\right )}^{\frac {11}{4}} b^{2}}{x^{11}} - \frac {5852 \, {\left (b x^{4} + a\right )}^{\frac {15}{4}} b}{x^{15}} + \frac {1155 \, {\left (b x^{4} + a\right )}^{\frac {19}{4}}}{x^{19}}}{21945 \, a^{5}} \]
-1/21945*(7315*(b*x^4 + a)^(3/4)*b^4/x^3 - 12540*(b*x^4 + a)^(7/4)*b^3/x^7 + 11970*(b*x^4 + a)^(11/4)*b^2/x^11 - 5852*(b*x^4 + a)^(15/4)*b/x^15 + 11 55*(b*x^4 + a)^(19/4)/x^19)/a^5
\[ \int \frac {1}{x^{20} \sqrt [4]{a+b x^4}} \, dx=\int { \frac {1}{{\left (b x^{4} + a\right )}^{\frac {1}{4}} x^{20}} \,d x } \]
Time = 6.00 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.83 \[ \int \frac {1}{x^{20} \sqrt [4]{a+b x^4}} \, dx=\frac {16\,b\,{\left (b\,x^4+a\right )}^{3/4}}{285\,a^2\,x^{15}}-\frac {{\left (b\,x^4+a\right )}^{3/4}}{19\,a\,x^{19}}-\frac {2048\,b^4\,{\left (b\,x^4+a\right )}^{3/4}}{21945\,a^5\,x^3}+\frac {512\,b^3\,{\left (b\,x^4+a\right )}^{3/4}}{7315\,a^4\,x^7}-\frac {64\,b^2\,{\left (b\,x^4+a\right )}^{3/4}}{1045\,a^3\,x^{11}} \]