\(\int \frac {1}{x^{16} \sqrt [4]{a+b x^4}} \, dx\) [1101]

Optimal result

Integrand size = 15, antiderivative size = 92 \[ \int \frac {1}{x^{16} \sqrt [4]{a+b x^4}} \, dx=-\frac {\left (a+b x^4\right )^{3/4}}{15 a x^{15}}+\frac {4 b \left (a+b x^4\right )^{3/4}}{55 a^2 x^{11}}-\frac {32 b^2 \left (a+b x^4\right )^{3/4}}{385 a^3 x^7}+\frac {128 b^3 \left (a+b x^4\right )^{3/4}}{1155 a^4 x^3} \]

[Out]

-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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {277, 270} \[ \int \frac {1}{x^{16} \sqrt [4]{a+b x^4}} \, dx=\frac {128 b^3 \left (a+b x^4\right )^{3/4}}{1155 a^4 x^3}-\frac {32 b^2 \left (a+b x^4\right )^{3/4}}{385 a^3 x^7}+\frac {4 b \left (a+b x^4\right )^{3/4}}{55 a^2 x^{11}}-\frac {\left (a+b x^4\right )^{3/4}}{15 a x^{15}} \]

[In]

Int[1/(x^16*(a + b*x^4)^(1/4)),x]

[Out]

-1/15*(a + b*x^4)^(3/4)/(a*x^15) + (4*b*(a + b*x^4)^(3/4))/(55*a^2*x^11) - (32*b^2*(a + b*x^4)^(3/4))/(385*a^3
*x^7) + (128*b^3*(a + b*x^4)^(3/4))/(1155*a^4*x^3)

Rule 270

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]

Rule 277

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x^(m + 1)*((a + b*x^n)^(p + 1)/(a*(m + 1))), x]
- Dist[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]
&& ILtQ[Simplify[(m + 1)/n + p + 1], 0] && NeQ[m, -1]

Rubi steps \begin{align*} \text {integral}& = -\frac {\left (a+b x^4\right )^{3/4}}{15 a x^{15}}-\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}}+\frac {4 b \left (a+b x^4\right )^{3/4}}{55 a^2 x^{11}}+\frac {\left (32 b^2\right ) \int \frac {1}{x^8 \sqrt [4]{a+b x^4}} \, dx}{55 a^2} \\ & = -\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 {\left (128 b^3\right ) \int \frac {1}{x^4 \sqrt [4]{a+b x^4}} \, dx}{385 a^3} \\ & = -\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} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.29 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.58 \[ \int \frac {1}{x^{16} \sqrt [4]{a+b x^4}} \, dx=\frac {\left (a+b x^4\right )^{3/4} \left (-77 a^3+84 a^2 b x^4-96 a b^2 x^8+128 b^3 x^{12}\right )}{1155 a^4 x^{15}} \]

[In]

Integrate[1/(x^16*(a + b*x^4)^(1/4)),x]

[Out]

((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)

Maple [A] (verified)

Time = 4.38 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.54

method	result	size
gosper	\(-\frac {\left (b \,x^{4}+a \right )^{\frac {3}{4}} \left (-128 b^{3} x^{12}+96 a \,b^{2} x^{8}-84 a^{2} b \,x^{4}+77 a^{3}\right )}{1155 x^{15} a^{4}}\)	\(50\)
trager	\(-\frac {\left (b \,x^{4}+a \right )^{\frac {3}{4}} \left (-128 b^{3} x^{12}+96 a \,b^{2} x^{8}-84 a^{2} b \,x^{4}+77 a^{3}\right )}{1155 x^{15} a^{4}}\)	\(50\)
risch	\(-\frac {\left (b \,x^{4}+a \right )^{\frac {3}{4}} \left (-128 b^{3} x^{12}+96 a \,b^{2} x^{8}-84 a^{2} b \,x^{4}+77 a^{3}\right )}{1155 x^{15} a^{4}}\)	\(50\)
pseudoelliptic	\(-\frac {\left (b \,x^{4}+a \right )^{\frac {3}{4}} \left (-128 b^{3} x^{12}+96 a \,b^{2} x^{8}-84 a^{2} b \,x^{4}+77 a^{3}\right )}{1155 x^{15} a^{4}}\)	\(50\)

[In]

int(1/x^16/(b*x^4+a)^(1/4),x,method=_RETURNVERBOSE)

[Out]

-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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.53 \[ \int \frac {1}{x^{16} \sqrt [4]{a+b x^4}} \, dx=\frac {{\left (128 \, b^{3} x^{12} - 96 \, a b^{2} x^{8} + 84 \, a^{2} b x^{4} - 77 \, a^{3}\right )} {\left (b x^{4} + a\right )}^{\frac {3}{4}}}{1155 \, a^{4} x^{15}} \]

[In]

integrate(1/x^16/(b*x^4+a)^(1/4),x, algorithm="fricas")

[Out]

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)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 692 vs. \(2 (85) = 170\).

Time = 1.38 (sec) , antiderivative size = 692, normalized size of antiderivative = 7.52 \[ \int \frac {1}{x^{16} \sqrt [4]{a+b x^4}} \, dx=- \frac {231 a^{6} b^{\frac {39}{4}} \left (\frac {a}{b x^{4}} + 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {15}{4}\right )}{256 a^{7} b^{9} x^{12} \Gamma \left (\frac {1}{4}\right ) + 768 a^{6} b^{10} x^{16} \Gamma \left (\frac {1}{4}\right ) + 768 a^{5} b^{11} x^{20} \Gamma \left (\frac {1}{4}\right ) + 256 a^{4} b^{12} x^{24} \Gamma \left (\frac {1}{4}\right )} - \frac {441 a^{5} b^{\frac {43}{4}} x^{4} \left (\frac {a}{b x^{4}} + 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {15}{4}\right )}{256 a^{7} b^{9} x^{12} \Gamma \left (\frac {1}{4}\right ) + 768 a^{6} b^{10} x^{16} \Gamma \left (\frac {1}{4}\right ) + 768 a^{5} b^{11} x^{20} \Gamma \left (\frac {1}{4}\right ) + 256 a^{4} b^{12} x^{24} \Gamma \left (\frac {1}{4}\right )} - \frac {225 a^{4} b^{\frac {47}{4}} x^{8} \left (\frac {a}{b x^{4}} + 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {15}{4}\right )}{256 a^{7} b^{9} x^{12} \Gamma \left (\frac {1}{4}\right ) + 768 a^{6} b^{10} x^{16} \Gamma \left (\frac {1}{4}\right ) + 768 a^{5} b^{11} x^{20} \Gamma \left (\frac {1}{4}\right ) + 256 a^{4} b^{12} x^{24} \Gamma \left (\frac {1}{4}\right )} + \frac {45 a^{3} b^{\frac {51}{4}} x^{12} \left (\frac {a}{b x^{4}} + 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {15}{4}\right )}{256 a^{7} b^{9} x^{12} \Gamma \left (\frac {1}{4}\right ) + 768 a^{6} b^{10} x^{16} \Gamma \left (\frac {1}{4}\right ) + 768 a^{5} b^{11} x^{20} \Gamma \left (\frac {1}{4}\right ) + 256 a^{4} b^{12} x^{24} \Gamma \left (\frac {1}{4}\right )} + \frac {540 a^{2} b^{\frac {55}{4}} x^{16} \left (\frac {a}{b x^{4}} + 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {15}{4}\right )}{256 a^{7} b^{9} x^{12} \Gamma \left (\frac {1}{4}\right ) + 768 a^{6} b^{10} x^{16} \Gamma \left (\frac {1}{4}\right ) + 768 a^{5} b^{11} x^{20} \Gamma \left (\frac {1}{4}\right ) + 256 a^{4} b^{12} x^{24} \Gamma \left (\frac {1}{4}\right )} + \frac {864 a b^{\frac {59}{4}} x^{20} \left (\frac {a}{b x^{4}} + 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {15}{4}\right )}{256 a^{7} b^{9} x^{12} \Gamma \left (\frac {1}{4}\right ) + 768 a^{6} b^{10} x^{16} \Gamma \left (\frac {1}{4}\right ) + 768 a^{5} b^{11} x^{20} \Gamma \left (\frac {1}{4}\right ) + 256 a^{4} b^{12} x^{24} \Gamma \left (\frac {1}{4}\right )} + \frac {384 b^{\frac {63}{4}} x^{24} \left (\frac {a}{b x^{4}} + 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {15}{4}\right )}{256 a^{7} b^{9} x^{12} \Gamma \left (\frac {1}{4}\right ) + 768 a^{6} b^{10} x^{16} \Gamma \left (\frac {1}{4}\right ) + 768 a^{5} b^{11} x^{20} \Gamma \left (\frac {1}{4}\right ) + 256 a^{4} b^{12} x^{24} \Gamma \left (\frac {1}{4}\right )} \]

[In]

integrate(1/x**16/(b*x**4+a)**(1/4),x)

[Out]

-231*a**6*b**(39/4)*(a/(b*x**4) + 1)**(3/4)*gamma(-15/4)/(256*a**7*b**9*x**12*gamma(1/4) + 768*a**6*b**10*x**1
6*gamma(1/4) + 768*a**5*b**11*x**20*gamma(1/4) + 256*a**4*b**12*x**24*gamma(1/4)) - 441*a**5*b**(43/4)*x**4*(a
/(b*x**4) + 1)**(3/4)*gamma(-15/4)/(256*a**7*b**9*x**12*gamma(1/4) + 768*a**6*b**10*x**16*gamma(1/4) + 768*a**
5*b**11*x**20*gamma(1/4) + 256*a**4*b**12*x**24*gamma(1/4)) - 225*a**4*b**(47/4)*x**8*(a/(b*x**4) + 1)**(3/4)*
gamma(-15/4)/(256*a**7*b**9*x**12*gamma(1/4) + 768*a**6*b**10*x**16*gamma(1/4) + 768*a**5*b**11*x**20*gamma(1/
4) + 256*a**4*b**12*x**24*gamma(1/4)) + 45*a**3*b**(51/4)*x**12*(a/(b*x**4) + 1)**(3/4)*gamma(-15/4)/(256*a**7
*b**9*x**12*gamma(1/4) + 768*a**6*b**10*x**16*gamma(1/4) + 768*a**5*b**11*x**20*gamma(1/4) + 256*a**4*b**12*x*
*24*gamma(1/4)) + 540*a**2*b**(55/4)*x**16*(a/(b*x**4) + 1)**(3/4)*gamma(-15/4)/(256*a**7*b**9*x**12*gamma(1/4
) + 768*a**6*b**10*x**16*gamma(1/4) + 768*a**5*b**11*x**20*gamma(1/4) + 256*a**4*b**12*x**24*gamma(1/4)) + 864
*a*b**(59/4)*x**20*(a/(b*x**4) + 1)**(3/4)*gamma(-15/4)/(256*a**7*b**9*x**12*gamma(1/4) + 768*a**6*b**10*x**16
*gamma(1/4) + 768*a**5*b**11*x**20*gamma(1/4) + 256*a**4*b**12*x**24*gamma(1/4)) + 384*b**(63/4)*x**24*(a/(b*x
**4) + 1)**(3/4)*gamma(-15/4)/(256*a**7*b**9*x**12*gamma(1/4) + 768*a**6*b**10*x**16*gamma(1/4) + 768*a**5*b**
11*x**20*gamma(1/4) + 256*a**4*b**12*x**24*gamma(1/4))

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.75 \[ \int \frac {1}{x^{16} \sqrt [4]{a+b x^4}} \, dx=\frac {\frac {385 \, {\left (b x^{4} + a\right )}^{\frac {3}{4}} b^{3}}{x^{3}} - \frac {495 \, {\left (b x^{4} + a\right )}^{\frac {7}{4}} b^{2}}{x^{7}} + \frac {315 \, {\left (b x^{4} + a\right )}^{\frac {11}{4}} b}{x^{11}} - \frac {77 \, {\left (b x^{4} + a\right )}^{\frac {15}{4}}}{x^{15}}}{1155 \, a^{4}} \]

[In]

integrate(1/x^16/(b*x^4+a)^(1/4),x, algorithm="maxima")

[Out]

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

Giac [F]

\[ \int \frac {1}{x^{16} \sqrt [4]{a+b x^4}} \, dx=\int { \frac {1}{{\left (b x^{4} + a\right )}^{\frac {1}{4}} x^{16}} \,d x } \]

[In]

integrate(1/x^16/(b*x^4+a)^(1/4),x, algorithm="giac")

[Out]

integrate(1/((b*x^4 + a)^(1/4)*x^16), x)

Mupad [B] (verification not implemented)

Time = 5.92 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.83 \[ \int \frac {1}{x^{16} \sqrt [4]{a+b x^4}} \, dx=\frac {4\,b\,{\left (b\,x^4+a\right )}^{3/4}}{55\,a^2\,x^{11}}-\frac {{\left (b\,x^4+a\right )}^{3/4}}{15\,a\,x^{15}}+\frac {128\,b^3\,{\left (b\,x^4+a\right )}^{3/4}}{1155\,a^4\,x^3}-\frac {32\,b^2\,{\left (b\,x^4+a\right )}^{3/4}}{385\,a^3\,x^7} \]

[In]

int(1/(x^16*(a + b*x^4)^(1/4)),x)

[Out]

(4*b*(a + b*x^4)^(3/4))/(55*a^2*x^11) - (a + b*x^4)^(3/4)/(15*a*x^15) + (128*b^3*(a + b*x^4)^(3/4))/(1155*a^4*
x^3) - (32*b^2*(a + b*x^4)^(3/4))/(385*a^3*x^7)