Integrand size = 16, antiderivative size = 121 \[ \int \frac {1}{x^{20} \sqrt [4]{a-b x^4}} \, dx=-\frac {\left (a-b x^4\right )^{3/4}}{19 a x^{19}}-\frac {16 b \left (a-b x^4\right )^{3/4}}{285 a^2 x^{15}}-\frac {64 b^2 \left (a-b x^4\right )^{3/4}}{1045 a^3 x^{11}}-\frac {512 b^3 \left (a-b x^4\right )^{3/4}}{7315 a^4 x^7}-\frac {2048 b^4 \left (a-b x^4\right )^{3/4}}{21945 a^5 x^3} \]
-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^7-2048/21 945*b^4*(-b*x^4+a)^(3/4)/a^5/x^3
Time = 0.42 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.54 \[ \int \frac {1}{x^{20} \sqrt [4]{a-b x^4}} \, dx=\frac {\left (a-b x^4\right )^{3/4} \left (-1155 a^4-1232 a^3 b x^4-1344 a^2 b^2 x^8-1536 a b^3 x^{12}-2048 b^4 x^{16}\right )}{21945 a^5 x^{19}} \]
((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 = 139, normalized size of antiderivative = 1.15, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, 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 |
\(\displaystyle \frac {16 b \int \frac {1}{x^{16} \sqrt [4]{a-b x^4}}dx}{19 a}-\frac {\left (a-b x^4\right )^{3/4}}{19 a x^{19}}\) |
\(\Big \downarrow \) 803 |
\(\displaystyle \frac {16 b \left (\frac {4 b \int \frac {1}{x^{12} \sqrt [4]{a-b x^4}}dx}{5 a}-\frac {\left (a-b x^4\right )^{3/4}}{15 a x^{15}}\right )}{19 a}-\frac {\left (a-b x^4\right )^{3/4}}{19 a x^{19}}\) |
\(\Big \downarrow \) 803 |
\(\displaystyle \frac {16 b \left (\frac {4 b \left (\frac {8 b \int \frac {1}{x^8 \sqrt [4]{a-b x^4}}dx}{11 a}-\frac {\left (a-b x^4\right )^{3/4}}{11 a x^{11}}\right )}{5 a}-\frac {\left (a-b x^4\right )^{3/4}}{15 a x^{15}}\right )}{19 a}-\frac {\left (a-b x^4\right )^{3/4}}{19 a x^{19}}\) |
\(\Big \downarrow \) 803 |
\(\displaystyle \frac {16 b \left (\frac {4 b \left (\frac {8 b \left (\frac {4 b \int \frac {1}{x^4 \sqrt [4]{a-b x^4}}dx}{7 a}-\frac {\left (a-b x^4\right )^{3/4}}{7 a x^7}\right )}{11 a}-\frac {\left (a-b x^4\right )^{3/4}}{11 a x^{11}}\right )}{5 a}-\frac {\left (a-b x^4\right )^{3/4}}{15 a x^{15}}\right )}{19 a}-\frac {\left (a-b x^4\right )^{3/4}}{19 a x^{19}}\) |
\(\Big \downarrow \) 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)
3.13.26.3.1 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 = 4.70 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.51
method | result | size |
gosper | \(-\frac {\left (-b \,x^{4}+a \right )^{\frac {3}{4}} \left (2048 x^{16} b^{4}+1536 a \,b^{3} x^{12}+1344 a^{2} b^{2} x^{8}+1232 a^{3} b \,x^{4}+1155 a^{4}\right )}{21945 x^{19} a^{5}}\) | \(62\) |
trager | \(-\frac {\left (-b \,x^{4}+a \right )^{\frac {3}{4}} \left (2048 x^{16} b^{4}+1536 a \,b^{3} x^{12}+1344 a^{2} b^{2} x^{8}+1232 a^{3} b \,x^{4}+1155 a^{4}\right )}{21945 x^{19} a^{5}}\) | \(62\) |
risch | \(-\frac {\left (-b \,x^{4}+a \right )^{\frac {3}{4}} \left (2048 x^{16} b^{4}+1536 a \,b^{3} x^{12}+1344 a^{2} b^{2} x^{8}+1232 a^{3} b \,x^{4}+1155 a^{4}\right )}{21945 x^{19} a^{5}}\) | \(62\) |
pseudoelliptic | \(-\frac {\left (-b \,x^{4}+a \right )^{\frac {3}{4}} \left (2048 x^{16} b^{4}+1536 a \,b^{3} x^{12}+1344 a^{2} b^{2} x^{8}+1232 a^{3} b \,x^{4}+1155 a^{4}\right )}{21945 x^{19} a^{5}}\) | \(62\) |
-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+ 1232*a^3*b*x^4+1155*a^4)/x^19/a^5
Time = 0.25 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.50 \[ \int \frac {1}{x^{20} \sqrt [4]{a-b x^4}} \, dx=-\frac {{\left (2048 \, b^{4} x^{16} + 1536 \, a b^{3} x^{12} + 1344 \, a^{2} b^{2} x^{8} + 1232 \, a^{3} b x^{4} + 1155 \, a^{4}\right )} {\left (-b x^{4} + a\right )}^{\frac {3}{4}}}{21945 \, a^{5} x^{19}} \]
-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)
Result contains complex when optimal does not.
Time = 1.99 (sec) , antiderivative size = 2788, normalized size of antiderivative = 23.04 \[ \int \frac {1}{x^{20} \sqrt [4]{a-b x^4}} \, dx=\text {Too large to display} \]
Piecewise((-3465*a**8*b**(67/4)*(a/(b*x**4) - 1)**(3/4)*exp(-3*I*pi/4)*gam ma(-19/4)/(1024*a**9*b**16*x**16*exp(I*pi/4)*gamma(1/4) - 4096*a**8*b**17* x**20*exp(I*pi/4)*gamma(1/4) + 6144*a**7*b**18*x**24*exp(I*pi/4)*gamma(1/4 ) - 4096*a**6*b**19*x**28*exp(I*pi/4)*gamma(1/4) + 1024*a**5*b**20*x**32*e xp(I*pi/4)*gamma(1/4)) + 10164*a**7*b**(71/4)*x**4*(a/(b*x**4) - 1)**(3/4) *exp(-3*I*pi/4)*gamma(-19/4)/(1024*a**9*b**16*x**16*exp(I*pi/4)*gamma(1/4) - 4096*a**8*b**17*x**20*exp(I*pi/4)*gamma(1/4) + 6144*a**7*b**18*x**24*ex p(I*pi/4)*gamma(1/4) - 4096*a**6*b**19*x**28*exp(I*pi/4)*gamma(1/4) + 1024 *a**5*b**20*x**32*exp(I*pi/4)*gamma(1/4)) - 10038*a**6*b**(75/4)*x**8*(a/( b*x**4) - 1)**(3/4)*exp(-3*I*pi/4)*gamma(-19/4)/(1024*a**9*b**16*x**16*exp (I*pi/4)*gamma(1/4) - 4096*a**8*b**17*x**20*exp(I*pi/4)*gamma(1/4) + 6144* a**7*b**18*x**24*exp(I*pi/4)*gamma(1/4) - 4096*a**6*b**19*x**28*exp(I*pi/4 )*gamma(1/4) + 1024*a**5*b**20*x**32*exp(I*pi/4)*gamma(1/4)) + 3204*a**5*b **(79/4)*x**12*(a/(b*x**4) - 1)**(3/4)*exp(-3*I*pi/4)*gamma(-19/4)/(1024*a **9*b**16*x**16*exp(I*pi/4)*gamma(1/4) - 4096*a**8*b**17*x**20*exp(I*pi/4) *gamma(1/4) + 6144*a**7*b**18*x**24*exp(I*pi/4)*gamma(1/4) - 4096*a**6*b** 19*x**28*exp(I*pi/4)*gamma(1/4) + 1024*a**5*b**20*x**32*exp(I*pi/4)*gamma( 1/4)) - 585*a**4*b**(83/4)*x**16*(a/(b*x**4) - 1)**(3/4)*exp(-3*I*pi/4)*ga mma(-19/4)/(1024*a**9*b**16*x**16*exp(I*pi/4)*gamma(1/4) - 4096*a**8*b**17 *x**20*exp(I*pi/4)*gamma(1/4) + 6144*a**7*b**18*x**24*exp(I*pi/4)*gamma...
Time = 0.20 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.75 \[ \int \frac {1}{x^{20} \sqrt [4]{a-b x^4}} \, dx=-\frac {\frac {7315 \, {\left (-b x^{4} + a\right )}^{\frac {3}{4}} b^{4}}{x^{3}} + \frac {12540 \, {\left (-b x^{4} + a\right )}^{\frac {7}{4}} b^{3}}{x^{7}} + \frac {11970 \, {\left (-b x^{4} + a\right )}^{\frac {11}{4}} b^{2}}{x^{11}} + \frac {5852 \, {\left (-b x^{4} + a\right )}^{\frac {15}{4}} b}{x^{15}} + \frac {1155 \, {\left (-b x^{4} + a\right )}^{\frac {19}{4}}}{x^{19}}}{21945 \, a^{5}} \]
-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 + 1155*(-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 = 5.89 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.83 \[ \int \frac {1}{x^{20} \sqrt [4]{a-b x^4}} \, dx=-\frac {{\left (a-b\,x^4\right )}^{3/4}}{19\,a\,x^{19}}-\frac {16\,b\,{\left (a-b\,x^4\right )}^{3/4}}{285\,a^2\,x^{15}}-\frac {2048\,b^4\,{\left (a-b\,x^4\right )}^{3/4}}{21945\,a^5\,x^3}-\frac {512\,b^3\,{\left (a-b\,x^4\right )}^{3/4}}{7315\,a^4\,x^7}-\frac {64\,b^2\,{\left (a-b\,x^4\right )}^{3/4}}{1045\,a^3\,x^{11}} \]