Integrand size = 15, antiderivative size = 92 \[ \int \frac {\left (a+b x^4\right )^{3/4}}{x^{20}} \, dx=-\frac {\left (a+b x^4\right )^{7/4}}{19 a x^{19}}+\frac {4 b \left (a+b x^4\right )^{7/4}}{95 a^2 x^{15}}-\frac {32 b^2 \left (a+b x^4\right )^{7/4}}{1045 a^3 x^{11}}+\frac {128 b^3 \left (a+b x^4\right )^{7/4}}{7315 a^4 x^7} \]
-1/19*(b*x^4+a)^(7/4)/a/x^19+4/95*b*(b*x^4+a)^(7/4)/a^2/x^15-32/1045*b^2*( b*x^4+a)^(7/4)/a^3/x^11+128/7315*b^3*(b*x^4+a)^(7/4)/a^4/x^7
Time = 0.27 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.58 \[ \int \frac {\left (a+b x^4\right )^{3/4}}{x^{20}} \, dx=\frac {\left (a+b x^4\right )^{7/4} \left (-385 a^3+308 a^2 b x^4-224 a b^2 x^8+128 b^3 x^{12}\right )}{7315 a^4 x^{19}} \]
((a + b*x^4)^(7/4)*(-385*a^3 + 308*a^2*b*x^4 - 224*a*b^2*x^8 + 128*b^3*x^1 2))/(7315*a^4*x^19)
Time = 0.22 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.13, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, 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.
\(\displaystyle \int \frac {\left (a+b x^4\right )^{3/4}}{x^{20}} \, dx\) |
\(\Big \downarrow \) 803 |
\(\displaystyle -\frac {12 b \int \frac {\left (b x^4+a\right )^{3/4}}{x^{16}}dx}{19 a}-\frac {\left (a+b x^4\right )^{7/4}}{19 a x^{19}}\) |
\(\Big \downarrow \) 803 |
\(\displaystyle -\frac {12 b \left (-\frac {8 b \int \frac {\left (b x^4+a\right )^{3/4}}{x^{12}}dx}{15 a}-\frac {\left (a+b x^4\right )^{7/4}}{15 a x^{15}}\right )}{19 a}-\frac {\left (a+b x^4\right )^{7/4}}{19 a x^{19}}\) |
\(\Big \downarrow \) 803 |
\(\displaystyle -\frac {12 b \left (-\frac {8 b \left (-\frac {4 b \int \frac {\left (b x^4+a\right )^{3/4}}{x^8}dx}{11 a}-\frac {\left (a+b x^4\right )^{7/4}}{11 a x^{11}}\right )}{15 a}-\frac {\left (a+b x^4\right )^{7/4}}{15 a x^{15}}\right )}{19 a}-\frac {\left (a+b x^4\right )^{7/4}}{19 a x^{19}}\) |
\(\Big \downarrow \) 796 |
\(\displaystyle -\frac {12 b \left (-\frac {8 b \left (\frac {4 b \left (a+b x^4\right )^{7/4}}{77 a^2 x^7}-\frac {\left (a+b x^4\right )^{7/4}}{11 a x^{11}}\right )}{15 a}-\frac {\left (a+b x^4\right )^{7/4}}{15 a x^{15}}\right )}{19 a}-\frac {\left (a+b x^4\right )^{7/4}}{19 a x^{19}}\) |
-1/19*(a + b*x^4)^(7/4)/(a*x^19) - (12*b*(-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)))/(19*a)
3.11.39.3.1 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 = 4.40 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.54
method | result | size |
gosper | \(-\frac {\left (b \,x^{4}+a \right )^{\frac {7}{4}} \left (-128 b^{3} x^{12}+224 a \,b^{2} x^{8}-308 a^{2} b \,x^{4}+385 a^{3}\right )}{7315 x^{19} a^{4}}\) | \(50\) |
pseudoelliptic | \(-\frac {\left (b \,x^{4}+a \right )^{\frac {7}{4}} \left (-128 b^{3} x^{12}+224 a \,b^{2} x^{8}-308 a^{2} b \,x^{4}+385 a^{3}\right )}{7315 x^{19} a^{4}}\) | \(50\) |
trager | \(-\frac {\left (-128 x^{16} b^{4}+96 a \,b^{3} x^{12}-84 a^{2} b^{2} x^{8}+77 a^{3} b \,x^{4}+385 a^{4}\right ) \left (b \,x^{4}+a \right )^{\frac {3}{4}}}{7315 x^{19} a^{4}}\) | \(61\) |
risch | \(-\frac {\left (-128 x^{16} b^{4}+96 a \,b^{3} x^{12}-84 a^{2} b^{2} x^{8}+77 a^{3} b \,x^{4}+385 a^{4}\right ) \left (b \,x^{4}+a \right )^{\frac {3}{4}}}{7315 x^{19} a^{4}}\) | \(61\) |
Time = 0.32 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.65 \[ \int \frac {\left (a+b x^4\right )^{3/4}}{x^{20}} \, dx=\frac {{\left (128 \, b^{4} x^{16} - 96 \, a b^{3} x^{12} + 84 \, a^{2} b^{2} x^{8} - 77 \, a^{3} b x^{4} - 385 \, a^{4}\right )} {\left (b x^{4} + a\right )}^{\frac {3}{4}}}{7315 \, a^{4} x^{19}} \]
1/7315*(128*b^4*x^16 - 96*a*b^3*x^12 + 84*a^2*b^2*x^8 - 77*a^3*b*x^4 - 385 *a^4)*(b*x^4 + a)^(3/4)/(a^4*x^19)
Leaf count of result is larger than twice the leaf count of optimal. 847 vs. \(2 (85) = 170\).
Time = 1.95 (sec) , antiderivative size = 847, normalized size of antiderivative = 9.21 \[ \int \frac {\left (a+b x^4\right )^{3/4}}{x^{20}} \, dx=- \frac {1155 a^{7} b^{\frac {39}{4}} \left (\frac {a}{b x^{4}} + 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {19}{4}\right )}{256 a^{7} b^{9} x^{16} \Gamma \left (- \frac {3}{4}\right ) + 768 a^{6} b^{10} x^{20} \Gamma \left (- \frac {3}{4}\right ) + 768 a^{5} b^{11} x^{24} \Gamma \left (- \frac {3}{4}\right ) + 256 a^{4} b^{12} x^{28} \Gamma \left (- \frac {3}{4}\right )} - \frac {3696 a^{6} b^{\frac {43}{4}} x^{4} \left (\frac {a}{b x^{4}} + 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {19}{4}\right )}{256 a^{7} b^{9} x^{16} \Gamma \left (- \frac {3}{4}\right ) + 768 a^{6} b^{10} x^{20} \Gamma \left (- \frac {3}{4}\right ) + 768 a^{5} b^{11} x^{24} \Gamma \left (- \frac {3}{4}\right ) + 256 a^{4} b^{12} x^{28} \Gamma \left (- \frac {3}{4}\right )} - \frac {3906 a^{5} b^{\frac {47}{4}} x^{8} \left (\frac {a}{b x^{4}} + 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {19}{4}\right )}{256 a^{7} b^{9} x^{16} \Gamma \left (- \frac {3}{4}\right ) + 768 a^{6} b^{10} x^{20} \Gamma \left (- \frac {3}{4}\right ) + 768 a^{5} b^{11} x^{24} \Gamma \left (- \frac {3}{4}\right ) + 256 a^{4} b^{12} x^{28} \Gamma \left (- \frac {3}{4}\right )} - \frac {1380 a^{4} b^{\frac {51}{4}} x^{12} \left (\frac {a}{b x^{4}} + 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {19}{4}\right )}{256 a^{7} b^{9} x^{16} \Gamma \left (- \frac {3}{4}\right ) + 768 a^{6} b^{10} x^{20} \Gamma \left (- \frac {3}{4}\right ) + 768 a^{5} b^{11} x^{24} \Gamma \left (- \frac {3}{4}\right ) + 256 a^{4} b^{12} x^{28} \Gamma \left (- \frac {3}{4}\right )} + \frac {45 a^{3} b^{\frac {55}{4}} x^{16} \left (\frac {a}{b x^{4}} + 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {19}{4}\right )}{256 a^{7} b^{9} x^{16} \Gamma \left (- \frac {3}{4}\right ) + 768 a^{6} b^{10} x^{20} \Gamma \left (- \frac {3}{4}\right ) + 768 a^{5} b^{11} x^{24} \Gamma \left (- \frac {3}{4}\right ) + 256 a^{4} b^{12} x^{28} \Gamma \left (- \frac {3}{4}\right )} + \frac {540 a^{2} b^{\frac {59}{4}} x^{20} \left (\frac {a}{b x^{4}} + 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {19}{4}\right )}{256 a^{7} b^{9} x^{16} \Gamma \left (- \frac {3}{4}\right ) + 768 a^{6} b^{10} x^{20} \Gamma \left (- \frac {3}{4}\right ) + 768 a^{5} b^{11} x^{24} \Gamma \left (- \frac {3}{4}\right ) + 256 a^{4} b^{12} x^{28} \Gamma \left (- \frac {3}{4}\right )} + \frac {864 a b^{\frac {63}{4}} x^{24} \left (\frac {a}{b x^{4}} + 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {19}{4}\right )}{256 a^{7} b^{9} x^{16} \Gamma \left (- \frac {3}{4}\right ) + 768 a^{6} b^{10} x^{20} \Gamma \left (- \frac {3}{4}\right ) + 768 a^{5} b^{11} x^{24} \Gamma \left (- \frac {3}{4}\right ) + 256 a^{4} b^{12} x^{28} \Gamma \left (- \frac {3}{4}\right )} + \frac {384 b^{\frac {67}{4}} x^{28} \left (\frac {a}{b x^{4}} + 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {19}{4}\right )}{256 a^{7} b^{9} x^{16} \Gamma \left (- \frac {3}{4}\right ) + 768 a^{6} b^{10} x^{20} \Gamma \left (- \frac {3}{4}\right ) + 768 a^{5} b^{11} x^{24} \Gamma \left (- \frac {3}{4}\right ) + 256 a^{4} b^{12} x^{28} \Gamma \left (- \frac {3}{4}\right )} \]
-1155*a**7*b**(39/4)*(a/(b*x**4) + 1)**(3/4)*gamma(-19/4)/(256*a**7*b**9*x **16*gamma(-3/4) + 768*a**6*b**10*x**20*gamma(-3/4) + 768*a**5*b**11*x**24 *gamma(-3/4) + 256*a**4*b**12*x**28*gamma(-3/4)) - 3696*a**6*b**(43/4)*x** 4*(a/(b*x**4) + 1)**(3/4)*gamma(-19/4)/(256*a**7*b**9*x**16*gamma(-3/4) + 768*a**6*b**10*x**20*gamma(-3/4) + 768*a**5*b**11*x**24*gamma(-3/4) + 256* a**4*b**12*x**28*gamma(-3/4)) - 3906*a**5*b**(47/4)*x**8*(a/(b*x**4) + 1)* *(3/4)*gamma(-19/4)/(256*a**7*b**9*x**16*gamma(-3/4) + 768*a**6*b**10*x**2 0*gamma(-3/4) + 768*a**5*b**11*x**24*gamma(-3/4) + 256*a**4*b**12*x**28*ga mma(-3/4)) - 1380*a**4*b**(51/4)*x**12*(a/(b*x**4) + 1)**(3/4)*gamma(-19/4 )/(256*a**7*b**9*x**16*gamma(-3/4) + 768*a**6*b**10*x**20*gamma(-3/4) + 76 8*a**5*b**11*x**24*gamma(-3/4) + 256*a**4*b**12*x**28*gamma(-3/4)) + 45*a* *3*b**(55/4)*x**16*(a/(b*x**4) + 1)**(3/4)*gamma(-19/4)/(256*a**7*b**9*x** 16*gamma(-3/4) + 768*a**6*b**10*x**20*gamma(-3/4) + 768*a**5*b**11*x**24*g amma(-3/4) + 256*a**4*b**12*x**28*gamma(-3/4)) + 540*a**2*b**(59/4)*x**20* (a/(b*x**4) + 1)**(3/4)*gamma(-19/4)/(256*a**7*b**9*x**16*gamma(-3/4) + 76 8*a**6*b**10*x**20*gamma(-3/4) + 768*a**5*b**11*x**24*gamma(-3/4) + 256*a* *4*b**12*x**28*gamma(-3/4)) + 864*a*b**(63/4)*x**24*(a/(b*x**4) + 1)**(3/4 )*gamma(-19/4)/(256*a**7*b**9*x**16*gamma(-3/4) + 768*a**6*b**10*x**20*gam ma(-3/4) + 768*a**5*b**11*x**24*gamma(-3/4) + 256*a**4*b**12*x**28*gamma(- 3/4)) + 384*b**(67/4)*x**28*(a/(b*x**4) + 1)**(3/4)*gamma(-19/4)/(256*a...
Time = 0.25 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.75 \[ \int \frac {\left (a+b x^4\right )^{3/4}}{x^{20}} \, dx=\frac {\frac {1045 \, {\left (b x^{4} + a\right )}^{\frac {7}{4}} b^{3}}{x^{7}} - \frac {1995 \, {\left (b x^{4} + a\right )}^{\frac {11}{4}} b^{2}}{x^{11}} + \frac {1463 \, {\left (b x^{4} + a\right )}^{\frac {15}{4}} b}{x^{15}} - \frac {385 \, {\left (b x^{4} + a\right )}^{\frac {19}{4}}}{x^{19}}}{7315 \, a^{4}} \]
1/7315*(1045*(b*x^4 + a)^(7/4)*b^3/x^7 - 1995*(b*x^4 + a)^(11/4)*b^2/x^11 + 1463*(b*x^4 + a)^(15/4)*b/x^15 - 385*(b*x^4 + a)^(19/4)/x^19)/a^4
\[ \int \frac {\left (a+b x^4\right )^{3/4}}{x^{20}} \, dx=\int { \frac {{\left (b x^{4} + a\right )}^{\frac {3}{4}}}{x^{20}} \,d x } \]
Time = 6.34 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.01 \[ \int \frac {\left (a+b x^4\right )^{3/4}}{x^{20}} \, dx=\frac {128\,b^4\,{\left (b\,x^4+a\right )}^{3/4}}{7315\,a^4\,x^3}-\frac {b\,{\left (b\,x^4+a\right )}^{3/4}}{95\,a\,x^{15}}-\frac {{\left (b\,x^4+a\right )}^{3/4}}{19\,x^{19}}-\frac {96\,b^3\,{\left (b\,x^4+a\right )}^{3/4}}{7315\,a^3\,x^7}+\frac {12\,b^2\,{\left (b\,x^4+a\right )}^{3/4}}{1045\,a^2\,x^{11}} \]