Integrand size = 16, antiderivative size = 96 \[ \int \frac {1}{x^{16} \sqrt [4]{a-b x^4}} \, dx=-\frac {\left (a-b x^4\right )^{3/4}}{15 a x^{15}}-\frac {4 b \left (a-b x^4\right )^{3/4}}{55 a^2 x^{11}}-\frac {32 b^2 \left (a-b x^4\right )^{3/4}}{385 a^3 x^7}-\frac {128 b^3 \left (a-b x^4\right )^{3/4}}{1155 a^4 x^3} \] Output:
-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 = 0.28 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.56 \[ \int \frac {1}{x^{16} \sqrt [4]{a-b x^4}} \, dx=\frac {\left (a-b x^4\right )^{3/4} \left (-77 a^3-84 a^2 b x^4-96 a b^2 x^8-128 b^3 x^{12}\right )}{1155 a^4 x^{15}} \] Input:
Integrate[1/(x^16*(a - b*x^4)^(1/4)),x]
Output:
((a - b*x^4)^(3/4)*(-77*a^3 - 84*a^2*b*x^4 - 96*a*b^2*x^8 - 128*b^3*x^12)) /(1155*a^4*x^15)
Time = 0.23 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.12, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, 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 |
\(\displaystyle \frac {4 b \int \frac {1}{x^{12} \sqrt [4]{a-b x^4}}dx}{5 a}-\frac {\left (a-b x^4\right )^{3/4}}{15 a x^{15}}\) |
\(\Big \downarrow \) 803 |
\(\displaystyle \frac {4 b \left (\frac {8 b \int \frac {1}{x^8 \sqrt [4]{a-b x^4}}dx}{11 a}-\frac {\left (a-b x^4\right )^{3/4}}{11 a x^{11}}\right )}{5 a}-\frac {\left (a-b x^4\right )^{3/4}}{15 a x^{15}}\) |
\(\Big \downarrow \) 803 |
\(\displaystyle \frac {4 b \left (\frac {8 b \left (\frac {4 b \int \frac {1}{x^4 \sqrt [4]{a-b x^4}}dx}{7 a}-\frac {\left (a-b x^4\right )^{3/4}}{7 a x^7}\right )}{11 a}-\frac {\left (a-b x^4\right )^{3/4}}{11 a x^{11}}\right )}{5 a}-\frac {\left (a-b x^4\right )^{3/4}}{15 a x^{15}}\) |
\(\Big \downarrow \) 796 |
\(\displaystyle \frac {4 b \left (\frac {8 b \left (-\frac {4 b \left (a-b x^4\right )^{3/4}}{21 a^2 x^3}-\frac {\left (a-b x^4\right )^{3/4}}{7 a x^7}\right )}{11 a}-\frac {\left (a-b x^4\right )^{3/4}}{11 a x^{11}}\right )}{5 a}-\frac {\left (a-b x^4\right )^{3/4}}{15 a x^{15}}\) |
Input:
Int[1/(x^16*(a - b*x^4)^(1/4)),x]
Output:
-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)
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 = 0.73 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.53
method | result | size |
gosper | \(-\frac {\left (-b \,x^{4}+a \right )^{\frac {3}{4}} \left (128 b^{3} x^{12}+96 a \,b^{2} x^{8}+84 a^{2} b \,x^{4}+77 a^{3}\right )}{1155 x^{15} a^{4}}\) | \(51\) |
trager | \(-\frac {\left (-b \,x^{4}+a \right )^{\frac {3}{4}} \left (128 b^{3} x^{12}+96 a \,b^{2} x^{8}+84 a^{2} b \,x^{4}+77 a^{3}\right )}{1155 x^{15} a^{4}}\) | \(51\) |
risch | \(-\frac {\left (-b \,x^{4}+a \right )^{\frac {3}{4}} \left (128 b^{3} x^{12}+96 a \,b^{2} x^{8}+84 a^{2} b \,x^{4}+77 a^{3}\right )}{1155 x^{15} a^{4}}\) | \(51\) |
pseudoelliptic | \(-\frac {\left (-b \,x^{4}+a \right )^{\frac {3}{4}} \left (128 b^{3} x^{12}+96 a \,b^{2} x^{8}+84 a^{2} b \,x^{4}+77 a^{3}\right )}{1155 x^{15} a^{4}}\) | \(51\) |
orering | \(-\frac {\left (-b \,x^{4}+a \right )^{\frac {3}{4}} \left (128 b^{3} x^{12}+96 a \,b^{2} x^{8}+84 a^{2} b \,x^{4}+77 a^{3}\right )}{1155 x^{15} a^{4}}\) | \(51\) |
Input:
int(1/x^16/(-b*x^4+a)^(1/4),x,method=_RETURNVERBOSE)
Output:
-1/1155*(-b*x^4+a)^(3/4)*(128*b^3*x^12+96*a*b^2*x^8+84*a^2*b*x^4+77*a^3)/x ^15/a^4
Time = 0.08 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.52 \[ \int \frac {1}{x^{16} \sqrt [4]{a-b x^4}} \, dx=-\frac {{\left (128 \, b^{3} x^{12} + 96 \, a b^{2} x^{8} + 84 \, a^{2} b x^{4} + 77 \, a^{3}\right )} {\left (-b x^{4} + a\right )}^{\frac {3}{4}}}{1155 \, a^{4} x^{15}} \] Input:
integrate(1/x^16/(-b*x^4+a)^(1/4),x, algorithm="fricas")
Output:
-1/1155*(128*b^3*x^12 + 96*a*b^2*x^8 + 84*a^2*b*x^4 + 77*a^3)*(-b*x^4 + a) ^(3/4)/(a^4*x^15)
Result contains complex when optimal does not.
Time = 1.33 (sec) , antiderivative size = 1821, normalized size of antiderivative = 18.97 \[ \int \frac {1}{x^{16} \sqrt [4]{a-b x^4}} \, dx=\text {Too large to display} \] Input:
integrate(1/x**16/(-b*x**4+a)**(1/4),x)
Output:
Piecewise((231*a**6*b**(39/4)*(a/(b*x**4) - 1)**(3/4)*exp(I*pi/4)*gamma(-1 5/4)/(-256*a**7*b**9*x**12*exp(I*pi/4)*gamma(1/4) + 768*a**6*b**10*x**16*e xp(I*pi/4)*gamma(1/4) - 768*a**5*b**11*x**20*exp(I*pi/4)*gamma(1/4) + 256* a**4*b**12*x**24*exp(I*pi/4)*gamma(1/4)) - 441*a**5*b**(43/4)*x**4*(a/(b*x **4) - 1)**(3/4)*exp(I*pi/4)*gamma(-15/4)/(-256*a**7*b**9*x**12*exp(I*pi/4 )*gamma(1/4) + 768*a**6*b**10*x**16*exp(I*pi/4)*gamma(1/4) - 768*a**5*b**1 1*x**20*exp(I*pi/4)*gamma(1/4) + 256*a**4*b**12*x**24*exp(I*pi/4)*gamma(1/ 4)) + 225*a**4*b**(47/4)*x**8*(a/(b*x**4) - 1)**(3/4)*exp(I*pi/4)*gamma(-1 5/4)/(-256*a**7*b**9*x**12*exp(I*pi/4)*gamma(1/4) + 768*a**6*b**10*x**16*e xp(I*pi/4)*gamma(1/4) - 768*a**5*b**11*x**20*exp(I*pi/4)*gamma(1/4) + 256* a**4*b**12*x**24*exp(I*pi/4)*gamma(1/4)) + 45*a**3*b**(51/4)*x**12*(a/(b*x **4) - 1)**(3/4)*exp(I*pi/4)*gamma(-15/4)/(-256*a**7*b**9*x**12*exp(I*pi/4 )*gamma(1/4) + 768*a**6*b**10*x**16*exp(I*pi/4)*gamma(1/4) - 768*a**5*b**1 1*x**20*exp(I*pi/4)*gamma(1/4) + 256*a**4*b**12*x**24*exp(I*pi/4)*gamma(1/ 4)) - 540*a**2*b**(55/4)*x**16*(a/(b*x**4) - 1)**(3/4)*exp(I*pi/4)*gamma(- 15/4)/(-256*a**7*b**9*x**12*exp(I*pi/4)*gamma(1/4) + 768*a**6*b**10*x**16* exp(I*pi/4)*gamma(1/4) - 768*a**5*b**11*x**20*exp(I*pi/4)*gamma(1/4) + 256 *a**4*b**12*x**24*exp(I*pi/4)*gamma(1/4)) + 864*a*b**(59/4)*x**20*(a/(b*x* *4) - 1)**(3/4)*exp(I*pi/4)*gamma(-15/4)/(-256*a**7*b**9*x**12*exp(I*pi/4) *gamma(1/4) + 768*a**6*b**10*x**16*exp(I*pi/4)*gamma(1/4) - 768*a**5*b*...
Time = 0.04 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.76 \[ \int \frac {1}{x^{16} \sqrt [4]{a-b x^4}} \, dx=-\frac {\frac {385 \, {\left (-b x^{4} + a\right )}^{\frac {3}{4}} b^{3}}{x^{3}} + \frac {495 \, {\left (-b x^{4} + a\right )}^{\frac {7}{4}} b^{2}}{x^{7}} + \frac {315 \, {\left (-b x^{4} + a\right )}^{\frac {11}{4}} b}{x^{11}} + \frac {77 \, {\left (-b x^{4} + a\right )}^{\frac {15}{4}}}{x^{15}}}{1155 \, a^{4}} \] Input:
integrate(1/x^16/(-b*x^4+a)^(1/4),x, algorithm="maxima")
Output:
-1/1155*(385*(-b*x^4 + a)^(3/4)*b^3/x^3 + 495*(-b*x^4 + a)^(7/4)*b^2/x^7 + 315*(-b*x^4 + a)^(11/4)*b/x^11 + 77*(-b*x^4 + a)^(15/4)/x^15)/a^4
\[ \int \frac {1}{x^{16} \sqrt [4]{a-b x^4}} \, dx=\int { \frac {1}{{\left (-b x^{4} + a\right )}^{\frac {1}{4}} x^{16}} \,d x } \] Input:
integrate(1/x^16/(-b*x^4+a)^(1/4),x, algorithm="giac")
Output:
integrate(1/((-b*x^4 + a)^(1/4)*x^16), x)
Time = 0.57 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.83 \[ \int \frac {1}{x^{16} \sqrt [4]{a-b x^4}} \, dx=-\frac {{\left (a-b\,x^4\right )}^{3/4}}{15\,a\,x^{15}}-\frac {4\,b\,{\left (a-b\,x^4\right )}^{3/4}}{55\,a^2\,x^{11}}-\frac {128\,b^3\,{\left (a-b\,x^4\right )}^{3/4}}{1155\,a^4\,x^3}-\frac {32\,b^2\,{\left (a-b\,x^4\right )}^{3/4}}{385\,a^3\,x^7} \] Input:
int(1/(x^16*(a - b*x^4)^(1/4)),x)
Output:
- (a - b*x^4)^(3/4)/(15*a*x^15) - (4*b*(a - b*x^4)^(3/4))/(55*a^2*x^11) - (128*b^3*(a - b*x^4)^(3/4))/(1155*a^4*x^3) - (32*b^2*(a - b*x^4)^(3/4))/(3 85*a^3*x^7)
\[ \int \frac {1}{x^{16} \sqrt [4]{a-b x^4}} \, dx=\int \frac {1}{\left (-b \,x^{4}+a \right )^{\frac {1}{4}} x^{16}}d x \] Input:
int(1/x^16/(-b*x^4+a)^(1/4),x)
Output:
int(1/((a - b*x**4)**(1/4)*x**16),x)