Integrand size = 15, antiderivative size = 68 \[ \int \frac {\sqrt [4]{a+b x^4}}{x^{14}} \, dx=-\frac {\left (a+b x^4\right )^{5/4}}{13 a x^{13}}+\frac {8 b \left (a+b x^4\right )^{5/4}}{117 a^2 x^9}-\frac {32 b^2 \left (a+b x^4\right )^{5/4}}{585 a^3 x^5} \] Output:
-1/13*(b*x^4+a)^(5/4)/a/x^13+8/117*b*(b*x^4+a)^(5/4)/a^2/x^9-32/585*b^2*(b *x^4+a)^(5/4)/a^3/x^5
Time = 0.16 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.78 \[ \int \frac {\sqrt [4]{a+b x^4}}{x^{14}} \, dx=\frac {\sqrt [4]{a+b x^4} \left (-45 a^3-5 a^2 b x^4+8 a b^2 x^8-32 b^3 x^{12}\right )}{585 a^3 x^{13}} \] Input:
Integrate[(a + b*x^4)^(1/4)/x^14,x]
Output:
((a + b*x^4)^(1/4)*(-45*a^3 - 5*a^2*b*x^4 + 8*a*b^2*x^8 - 32*b^3*x^12))/(5 85*a^3*x^13)
Time = 0.32 (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.
\(\displaystyle \int \frac {\sqrt [4]{a+b x^4}}{x^{14}} \, dx\) |
\(\Big \downarrow \) 803 |
\(\displaystyle -\frac {8 b \int \frac {\sqrt [4]{b x^4+a}}{x^{10}}dx}{13 a}-\frac {\left (a+b x^4\right )^{5/4}}{13 a x^{13}}\) |
\(\Big \downarrow \) 803 |
\(\displaystyle -\frac {8 b \left (-\frac {4 b \int \frac {\sqrt [4]{b x^4+a}}{x^6}dx}{9 a}-\frac {\left (a+b x^4\right )^{5/4}}{9 a x^9}\right )}{13 a}-\frac {\left (a+b x^4\right )^{5/4}}{13 a x^{13}}\) |
\(\Big \downarrow \) 796 |
\(\displaystyle -\frac {8 b \left (\frac {4 b \left (a+b x^4\right )^{5/4}}{45 a^2 x^5}-\frac {\left (a+b x^4\right )^{5/4}}{9 a x^9}\right )}{13 a}-\frac {\left (a+b x^4\right )^{5/4}}{13 a x^{13}}\) |
Input:
Int[(a + b*x^4)^(1/4)/x^14,x]
Output:
-1/13*(a + b*x^4)^(5/4)/(a*x^13) - (8*b*(-1/9*(a + b*x^4)^(5/4)/(a*x^9) + (4*b*(a + b*x^4)^(5/4))/(45*a^2*x^5)))/(13*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 = 0.59 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.57
method | result | size |
gosper | \(-\frac {\left (b \,x^{4}+a \right )^{\frac {5}{4}} \left (32 b^{2} x^{8}-40 a b \,x^{4}+45 a^{2}\right )}{585 x^{13} a^{3}}\) | \(39\) |
pseudoelliptic | \(-\frac {\left (b \,x^{4}+a \right )^{\frac {5}{4}} \left (32 b^{2} x^{8}-40 a b \,x^{4}+45 a^{2}\right )}{585 x^{13} a^{3}}\) | \(39\) |
orering | \(-\frac {\left (b \,x^{4}+a \right )^{\frac {5}{4}} \left (32 b^{2} x^{8}-40 a b \,x^{4}+45 a^{2}\right )}{585 x^{13} a^{3}}\) | \(39\) |
trager | \(-\frac {\left (32 b^{3} x^{12}-8 a \,b^{2} x^{8}+5 a^{2} b \,x^{4}+45 a^{3}\right ) \left (b \,x^{4}+a \right )^{\frac {1}{4}}}{585 x^{13} a^{3}}\) | \(50\) |
risch | \(-\frac {\left (32 b^{3} x^{12}-8 a \,b^{2} x^{8}+5 a^{2} b \,x^{4}+45 a^{3}\right ) \left (b \,x^{4}+a \right )^{\frac {1}{4}}}{585 x^{13} a^{3}}\) | \(50\) |
Input:
int((b*x^4+a)^(1/4)/x^14,x,method=_RETURNVERBOSE)
Output:
-1/585*(b*x^4+a)^(5/4)*(32*b^2*x^8-40*a*b*x^4+45*a^2)/x^13/a^3
Time = 0.08 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.72 \[ \int \frac {\sqrt [4]{a+b x^4}}{x^{14}} \, dx=-\frac {{\left (32 \, b^{3} x^{12} - 8 \, a b^{2} x^{8} + 5 \, a^{2} b x^{4} + 45 \, a^{3}\right )} {\left (b x^{4} + a\right )}^{\frac {1}{4}}}{585 \, a^{3} x^{13}} \] Input:
integrate((b*x^4+a)^(1/4)/x^14,x, algorithm="fricas")
Output:
-1/585*(32*b^3*x^12 - 8*a*b^2*x^8 + 5*a^2*b*x^4 + 45*a^3)*(b*x^4 + a)^(1/4 )/(a^3*x^13)
Leaf count of result is larger than twice the leaf count of optimal. 520 vs. \(2 (61) = 122\).
Time = 0.98 (sec) , antiderivative size = 520, normalized size of antiderivative = 7.65 \[ \int \frac {\sqrt [4]{a+b x^4}}{x^{14}} \, dx=\frac {45 a^{5} b^{\frac {17}{4}} \sqrt [4]{\frac {a}{b x^{4}} + 1} \Gamma \left (- \frac {13}{4}\right )}{64 a^{5} b^{4} x^{12} \Gamma \left (- \frac {1}{4}\right ) + 128 a^{4} b^{5} x^{16} \Gamma \left (- \frac {1}{4}\right ) + 64 a^{3} b^{6} x^{20} \Gamma \left (- \frac {1}{4}\right )} + \frac {95 a^{4} b^{\frac {21}{4}} x^{4} \sqrt [4]{\frac {a}{b x^{4}} + 1} \Gamma \left (- \frac {13}{4}\right )}{64 a^{5} b^{4} x^{12} \Gamma \left (- \frac {1}{4}\right ) + 128 a^{4} b^{5} x^{16} \Gamma \left (- \frac {1}{4}\right ) + 64 a^{3} b^{6} x^{20} \Gamma \left (- \frac {1}{4}\right )} + \frac {47 a^{3} b^{\frac {25}{4}} x^{8} \sqrt [4]{\frac {a}{b x^{4}} + 1} \Gamma \left (- \frac {13}{4}\right )}{64 a^{5} b^{4} x^{12} \Gamma \left (- \frac {1}{4}\right ) + 128 a^{4} b^{5} x^{16} \Gamma \left (- \frac {1}{4}\right ) + 64 a^{3} b^{6} x^{20} \Gamma \left (- \frac {1}{4}\right )} + \frac {21 a^{2} b^{\frac {29}{4}} x^{12} \sqrt [4]{\frac {a}{b x^{4}} + 1} \Gamma \left (- \frac {13}{4}\right )}{64 a^{5} b^{4} x^{12} \Gamma \left (- \frac {1}{4}\right ) + 128 a^{4} b^{5} x^{16} \Gamma \left (- \frac {1}{4}\right ) + 64 a^{3} b^{6} x^{20} \Gamma \left (- \frac {1}{4}\right )} + \frac {56 a b^{\frac {33}{4}} x^{16} \sqrt [4]{\frac {a}{b x^{4}} + 1} \Gamma \left (- \frac {13}{4}\right )}{64 a^{5} b^{4} x^{12} \Gamma \left (- \frac {1}{4}\right ) + 128 a^{4} b^{5} x^{16} \Gamma \left (- \frac {1}{4}\right ) + 64 a^{3} b^{6} x^{20} \Gamma \left (- \frac {1}{4}\right )} + \frac {32 b^{\frac {37}{4}} x^{20} \sqrt [4]{\frac {a}{b x^{4}} + 1} \Gamma \left (- \frac {13}{4}\right )}{64 a^{5} b^{4} x^{12} \Gamma \left (- \frac {1}{4}\right ) + 128 a^{4} b^{5} x^{16} \Gamma \left (- \frac {1}{4}\right ) + 64 a^{3} b^{6} x^{20} \Gamma \left (- \frac {1}{4}\right )} \] Input:
integrate((b*x**4+a)**(1/4)/x**14,x)
Output:
45*a**5*b**(17/4)*(a/(b*x**4) + 1)**(1/4)*gamma(-13/4)/(64*a**5*b**4*x**12 *gamma(-1/4) + 128*a**4*b**5*x**16*gamma(-1/4) + 64*a**3*b**6*x**20*gamma( -1/4)) + 95*a**4*b**(21/4)*x**4*(a/(b*x**4) + 1)**(1/4)*gamma(-13/4)/(64*a **5*b**4*x**12*gamma(-1/4) + 128*a**4*b**5*x**16*gamma(-1/4) + 64*a**3*b** 6*x**20*gamma(-1/4)) + 47*a**3*b**(25/4)*x**8*(a/(b*x**4) + 1)**(1/4)*gamm a(-13/4)/(64*a**5*b**4*x**12*gamma(-1/4) + 128*a**4*b**5*x**16*gamma(-1/4) + 64*a**3*b**6*x**20*gamma(-1/4)) + 21*a**2*b**(29/4)*x**12*(a/(b*x**4) + 1)**(1/4)*gamma(-13/4)/(64*a**5*b**4*x**12*gamma(-1/4) + 128*a**4*b**5*x* *16*gamma(-1/4) + 64*a**3*b**6*x**20*gamma(-1/4)) + 56*a*b**(33/4)*x**16*( a/(b*x**4) + 1)**(1/4)*gamma(-13/4)/(64*a**5*b**4*x**12*gamma(-1/4) + 128* a**4*b**5*x**16*gamma(-1/4) + 64*a**3*b**6*x**20*gamma(-1/4)) + 32*b**(37/ 4)*x**20*(a/(b*x**4) + 1)**(1/4)*gamma(-13/4)/(64*a**5*b**4*x**12*gamma(-1 /4) + 128*a**4*b**5*x**16*gamma(-1/4) + 64*a**3*b**6*x**20*gamma(-1/4))
Time = 0.03 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.76 \[ \int \frac {\sqrt [4]{a+b x^4}}{x^{14}} \, dx=-\frac {\frac {117 \, {\left (b x^{4} + a\right )}^{\frac {5}{4}} b^{2}}{x^{5}} - \frac {130 \, {\left (b x^{4} + a\right )}^{\frac {9}{4}} b}{x^{9}} + \frac {45 \, {\left (b x^{4} + a\right )}^{\frac {13}{4}}}{x^{13}}}{585 \, a^{3}} \] Input:
integrate((b*x^4+a)^(1/4)/x^14,x, algorithm="maxima")
Output:
-1/585*(117*(b*x^4 + a)^(5/4)*b^2/x^5 - 130*(b*x^4 + a)^(9/4)*b/x^9 + 45*( b*x^4 + a)^(13/4)/x^13)/a^3
\[ \int \frac {\sqrt [4]{a+b x^4}}{x^{14}} \, dx=\int { \frac {{\left (b x^{4} + a\right )}^{\frac {1}{4}}}{x^{14}} \,d x } \] Input:
integrate((b*x^4+a)^(1/4)/x^14,x, algorithm="giac")
Output:
integrate((b*x^4 + a)^(1/4)/x^14, x)
Time = 0.68 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.07 \[ \int \frac {\sqrt [4]{a+b x^4}}{x^{14}} \, dx=\frac {8\,b^2\,{\left (b\,x^4+a\right )}^{1/4}}{585\,a^2\,x^5}-\frac {b\,{\left (b\,x^4+a\right )}^{1/4}}{117\,a\,x^9}-\frac {32\,b^3\,{\left (b\,x^4+a\right )}^{1/4}}{585\,a^3\,x}-\frac {{\left (b\,x^4+a\right )}^{1/4}}{13\,x^{13}} \] Input:
int((a + b*x^4)^(1/4)/x^14,x)
Output:
(8*b^2*(a + b*x^4)^(1/4))/(585*a^2*x^5) - (b*(a + b*x^4)^(1/4))/(117*a*x^9 ) - (32*b^3*(a + b*x^4)^(1/4))/(585*a^3*x) - (a + b*x^4)^(1/4)/(13*x^13)
Time = 0.22 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.72 \[ \int \frac {\sqrt [4]{a+b x^4}}{x^{14}} \, dx=\frac {\left (b \,x^{4}+a \right )^{\frac {1}{4}} \left (-32 b^{3} x^{12}+8 a \,b^{2} x^{8}-5 a^{2} b \,x^{4}-45 a^{3}\right )}{585 a^{3} x^{13}} \] Input:
int((b*x^4+a)^(1/4)/x^14,x)
Output:
((a + b*x**4)**(1/4)*( - 45*a**3 - 5*a**2*b*x**4 + 8*a*b**2*x**8 - 32*b**3 *x**12))/(585*a**3*x**13)