Integrand size = 15, antiderivative size = 68 \[ \int \frac {\left (a+b x^4\right )^{3/4}}{x^{16}} \, dx=-\frac {\left (a+b x^4\right )^{7/4}}{15 a x^{15}}+\frac {8 b \left (a+b x^4\right )^{7/4}}{165 a^2 x^{11}}-\frac {32 b^2 \left (a+b x^4\right )^{7/4}}{1155 a^3 x^7} \] Output:
-1/15*(b*x^4+a)^(7/4)/a/x^15+8/165*b*(b*x^4+a)^(7/4)/a^2/x^11-32/1155*b^2* (b*x^4+a)^(7/4)/a^3/x^7
Time = 0.18 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.62 \[ \int \frac {\left (a+b x^4\right )^{3/4}}{x^{16}} \, dx=\frac {\left (a+b x^4\right )^{7/4} \left (-77 a^2+56 a b x^4-32 b^2 x^8\right )}{1155 a^3 x^{15}} \] Input:
Integrate[(a + b*x^4)^(3/4)/x^16,x]
Output:
((a + b*x^4)^(7/4)*(-77*a^2 + 56*a*b*x^4 - 32*b^2*x^8))/(1155*a^3*x^15)
Time = 0.30 (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.
\(\displaystyle \int \frac {\left (a+b x^4\right )^{3/4}}{x^{16}} \, dx\) |
\(\Big \downarrow \) 803 |
\(\displaystyle -\frac {8 b \int \frac {\left (b x^4+a\right )^{3/4}}{x^{12}}dx}{15 a}-\frac {\left (a+b x^4\right )^{7/4}}{15 a x^{15}}\) |
\(\Big \downarrow \) 803 |
\(\displaystyle -\frac {8 b \left (-\frac {4 b \int \frac {\left (b x^4+a\right )^{3/4}}{x^8}dx}{11 a}-\frac {\left (a+b x^4\right )^{7/4}}{11 a x^{11}}\right )}{15 a}-\frac {\left (a+b x^4\right )^{7/4}}{15 a x^{15}}\) |
\(\Big \downarrow \) 796 |
\(\displaystyle -\frac {8 b \left (\frac {4 b \left (a+b x^4\right )^{7/4}}{77 a^2 x^7}-\frac {\left (a+b x^4\right )^{7/4}}{11 a x^{11}}\right )}{15 a}-\frac {\left (a+b x^4\right )^{7/4}}{15 a x^{15}}\) |
Input:
Int[(a + b*x^4)^(3/4)/x^16,x]
Output:
-1/15*(a + b*x^4)^(7/4)/(a*x^15) - (8*b*(-1/11*(a + b*x^4)^(7/4)/(a*x^11) + (4*b*(a + b*x^4)^(7/4))/(77*a^2*x^7)))/(15*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 = 0.60 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.57
method | result | size |
gosper | \(-\frac {\left (b \,x^{4}+a \right )^{\frac {7}{4}} \left (32 b^{2} x^{8}-56 a b \,x^{4}+77 a^{2}\right )}{1155 a^{3} x^{15}}\) | \(39\) |
pseudoelliptic | \(-\frac {\left (b \,x^{4}+a \right )^{\frac {7}{4}} \left (32 b^{2} x^{8}-56 a b \,x^{4}+77 a^{2}\right )}{1155 a^{3} x^{15}}\) | \(39\) |
orering | \(-\frac {\left (b \,x^{4}+a \right )^{\frac {7}{4}} \left (32 b^{2} x^{8}-56 a b \,x^{4}+77 a^{2}\right )}{1155 a^{3} x^{15}}\) | \(39\) |
trager | \(-\frac {\left (32 b^{3} x^{12}-24 a \,b^{2} x^{8}+21 a^{2} b \,x^{4}+77 a^{3}\right ) \left (b \,x^{4}+a \right )^{\frac {3}{4}}}{1155 a^{3} x^{15}}\) | \(50\) |
risch | \(-\frac {\left (32 b^{3} x^{12}-24 a \,b^{2} x^{8}+21 a^{2} b \,x^{4}+77 a^{3}\right ) \left (b \,x^{4}+a \right )^{\frac {3}{4}}}{1155 a^{3} x^{15}}\) | \(50\) |
Input:
int((b*x^4+a)^(3/4)/x^16,x,method=_RETURNVERBOSE)
Output:
-1/1155*(b*x^4+a)^(7/4)*(32*b^2*x^8-56*a*b*x^4+77*a^2)/a^3/x^15
Time = 0.10 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.72 \[ \int \frac {\left (a+b x^4\right )^{3/4}}{x^{16}} \, dx=-\frac {{\left (32 \, b^{3} x^{12} - 24 \, a b^{2} x^{8} + 21 \, a^{2} b x^{4} + 77 \, a^{3}\right )} {\left (b x^{4} + a\right )}^{\frac {3}{4}}}{1155 \, a^{3} x^{15}} \] Input:
integrate((b*x^4+a)^(3/4)/x^16,x, algorithm="fricas")
Output:
-1/1155*(32*b^3*x^12 - 24*a*b^2*x^8 + 21*a^2*b*x^4 + 77*a^3)*(b*x^4 + a)^( 3/4)/(a^3*x^15)
Leaf count of result is larger than twice the leaf count of optimal. 520 vs. \(2 (61) = 122\).
Time = 1.23 (sec) , antiderivative size = 520, normalized size of antiderivative = 7.65 \[ \int \frac {\left (a+b x^4\right )^{3/4}}{x^{16}} \, dx=\frac {77 a^{5} b^{\frac {19}{4}} \left (\frac {a}{b x^{4}} + 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {15}{4}\right )}{64 a^{5} b^{4} x^{12} \Gamma \left (- \frac {3}{4}\right ) + 128 a^{4} b^{5} x^{16} \Gamma \left (- \frac {3}{4}\right ) + 64 a^{3} b^{6} x^{20} \Gamma \left (- \frac {3}{4}\right )} + \frac {175 a^{4} b^{\frac {23}{4}} x^{4} \left (\frac {a}{b x^{4}} + 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {15}{4}\right )}{64 a^{5} b^{4} x^{12} \Gamma \left (- \frac {3}{4}\right ) + 128 a^{4} b^{5} x^{16} \Gamma \left (- \frac {3}{4}\right ) + 64 a^{3} b^{6} x^{20} \Gamma \left (- \frac {3}{4}\right )} + \frac {95 a^{3} b^{\frac {27}{4}} x^{8} \left (\frac {a}{b x^{4}} + 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {15}{4}\right )}{64 a^{5} b^{4} x^{12} \Gamma \left (- \frac {3}{4}\right ) + 128 a^{4} b^{5} x^{16} \Gamma \left (- \frac {3}{4}\right ) + 64 a^{3} b^{6} x^{20} \Gamma \left (- \frac {3}{4}\right )} + \frac {5 a^{2} b^{\frac {31}{4}} x^{12} \left (\frac {a}{b x^{4}} + 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {15}{4}\right )}{64 a^{5} b^{4} x^{12} \Gamma \left (- \frac {3}{4}\right ) + 128 a^{4} b^{5} x^{16} \Gamma \left (- \frac {3}{4}\right ) + 64 a^{3} b^{6} x^{20} \Gamma \left (- \frac {3}{4}\right )} + \frac {40 a b^{\frac {35}{4}} x^{16} \left (\frac {a}{b x^{4}} + 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {15}{4}\right )}{64 a^{5} b^{4} x^{12} \Gamma \left (- \frac {3}{4}\right ) + 128 a^{4} b^{5} x^{16} \Gamma \left (- \frac {3}{4}\right ) + 64 a^{3} b^{6} x^{20} \Gamma \left (- \frac {3}{4}\right )} + \frac {32 b^{\frac {39}{4}} x^{20} \left (\frac {a}{b x^{4}} + 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {15}{4}\right )}{64 a^{5} b^{4} x^{12} \Gamma \left (- \frac {3}{4}\right ) + 128 a^{4} b^{5} x^{16} \Gamma \left (- \frac {3}{4}\right ) + 64 a^{3} b^{6} x^{20} \Gamma \left (- \frac {3}{4}\right )} \] Input:
integrate((b*x**4+a)**(3/4)/x**16,x)
Output:
77*a**5*b**(19/4)*(a/(b*x**4) + 1)**(3/4)*gamma(-15/4)/(64*a**5*b**4*x**12 *gamma(-3/4) + 128*a**4*b**5*x**16*gamma(-3/4) + 64*a**3*b**6*x**20*gamma( -3/4)) + 175*a**4*b**(23/4)*x**4*(a/(b*x**4) + 1)**(3/4)*gamma(-15/4)/(64* a**5*b**4*x**12*gamma(-3/4) + 128*a**4*b**5*x**16*gamma(-3/4) + 64*a**3*b* *6*x**20*gamma(-3/4)) + 95*a**3*b**(27/4)*x**8*(a/(b*x**4) + 1)**(3/4)*gam ma(-15/4)/(64*a**5*b**4*x**12*gamma(-3/4) + 128*a**4*b**5*x**16*gamma(-3/4 ) + 64*a**3*b**6*x**20*gamma(-3/4)) + 5*a**2*b**(31/4)*x**12*(a/(b*x**4) + 1)**(3/4)*gamma(-15/4)/(64*a**5*b**4*x**12*gamma(-3/4) + 128*a**4*b**5*x* *16*gamma(-3/4) + 64*a**3*b**6*x**20*gamma(-3/4)) + 40*a*b**(35/4)*x**16*( a/(b*x**4) + 1)**(3/4)*gamma(-15/4)/(64*a**5*b**4*x**12*gamma(-3/4) + 128* a**4*b**5*x**16*gamma(-3/4) + 64*a**3*b**6*x**20*gamma(-3/4)) + 32*b**(39/ 4)*x**20*(a/(b*x**4) + 1)**(3/4)*gamma(-15/4)/(64*a**5*b**4*x**12*gamma(-3 /4) + 128*a**4*b**5*x**16*gamma(-3/4) + 64*a**3*b**6*x**20*gamma(-3/4))
Time = 0.04 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.76 \[ \int \frac {\left (a+b x^4\right )^{3/4}}{x^{16}} \, dx=-\frac {\frac {165 \, {\left (b x^{4} + a\right )}^{\frac {7}{4}} b^{2}}{x^{7}} - \frac {210 \, {\left (b x^{4} + a\right )}^{\frac {11}{4}} b}{x^{11}} + \frac {77 \, {\left (b x^{4} + a\right )}^{\frac {15}{4}}}{x^{15}}}{1155 \, a^{3}} \] Input:
integrate((b*x^4+a)^(3/4)/x^16,x, algorithm="maxima")
Output:
-1/1155*(165*(b*x^4 + a)^(7/4)*b^2/x^7 - 210*(b*x^4 + a)^(11/4)*b/x^11 + 7 7*(b*x^4 + a)^(15/4)/x^15)/a^3
\[ \int \frac {\left (a+b x^4\right )^{3/4}}{x^{16}} \, dx=\int { \frac {{\left (b x^{4} + a\right )}^{\frac {3}{4}}}{x^{16}} \,d x } \] Input:
integrate((b*x^4+a)^(3/4)/x^16,x, algorithm="giac")
Output:
integrate((b*x^4 + a)^(3/4)/x^16, x)
Time = 0.72 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.07 \[ \int \frac {\left (a+b x^4\right )^{3/4}}{x^{16}} \, dx=\frac {8\,b^2\,{\left (b\,x^4+a\right )}^{3/4}}{385\,a^2\,x^7}-\frac {b\,{\left (b\,x^4+a\right )}^{3/4}}{55\,a\,x^{11}}-\frac {32\,b^3\,{\left (b\,x^4+a\right )}^{3/4}}{1155\,a^3\,x^3}-\frac {{\left (b\,x^4+a\right )}^{3/4}}{15\,x^{15}} \] Input:
int((a + b*x^4)^(3/4)/x^16,x)
Output:
(8*b^2*(a + b*x^4)^(3/4))/(385*a^2*x^7) - (b*(a + b*x^4)^(3/4))/(55*a*x^11 ) - (32*b^3*(a + b*x^4)^(3/4))/(1155*a^3*x^3) - (a + b*x^4)^(3/4)/(15*x^15 )
Time = 0.19 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.72 \[ \int \frac {\left (a+b x^4\right )^{3/4}}{x^{16}} \, dx=\frac {\left (b \,x^{4}+a \right )^{\frac {3}{4}} \left (-32 b^{3} x^{12}+24 a \,b^{2} x^{8}-21 a^{2} b \,x^{4}-77 a^{3}\right )}{1155 a^{3} x^{15}} \] Input:
int((b*x^4+a)^(3/4)/x^16,x)
Output:
((a + b*x**4)**(3/4)*( - 77*a**3 - 21*a**2*b*x**4 + 24*a*b**2*x**8 - 32*b* *3*x**12))/(1155*a**3*x**15)