\(\int \frac {1}{x^{12} \sqrt [4]{a+b x^4}} \, dx\) [1100]

Optimal result

Integrand size = 15, antiderivative size = 68 \[ \int \frac {1}{x^{12} \sqrt [4]{a+b x^4}} \, dx=-\frac {\left (a+b x^4\right )^{3/4}}{11 a x^{11}}+\frac {8 b \left (a+b x^4\right )^{3/4}}{77 a^2 x^7}-\frac {32 b^2 \left (a+b x^4\right )^{3/4}}{231 a^3 x^3} \]

[Out]

-1/11*(b*x^4+a)^(3/4)/a/x^11+8/77*b*(b*x^4+a)^(3/4)/a^2/x^7-32/231*b^2*(b*x^4+a)^(3/4)/a^3/x^3

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {277, 270} \[ \int \frac {1}{x^{12} \sqrt [4]{a+b x^4}} \, dx=-\frac {32 b^2 \left (a+b x^4\right )^{3/4}}{231 a^3 x^3}+\frac {8 b \left (a+b x^4\right )^{3/4}}{77 a^2 x^7}-\frac {\left (a+b x^4\right )^{3/4}}{11 a x^{11}} \]

[In]

Int[1/(x^12*(a + b*x^4)^(1/4)),x]

[Out]

-1/11*(a + b*x^4)^(3/4)/(a*x^11) + (8*b*(a + b*x^4)^(3/4))/(77*a^2*x^7) - (32*b^2*(a + b*x^4)^(3/4))/(231*a^3*
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}}{11 a x^{11}}-\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}}+\frac {8 b \left (a+b x^4\right )^{3/4}}{77 a^2 x^7}+\frac {\left (32 b^2\right ) \int \frac {1}{x^4 \sqrt [4]{a+b x^4}} \, dx}{77 a^2} \\ & = -\frac {\left (a+b x^4\right )^{3/4}}{11 a x^{11}}+\frac {8 b \left (a+b x^4\right )^{3/4}}{77 a^2 x^7}-\frac {32 b^2 \left (a+b x^4\right )^{3/4}}{231 a^3 x^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.26 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.62 \[ \int \frac {1}{x^{12} \sqrt [4]{a+b x^4}} \, dx=\frac {\left (a+b x^4\right )^{3/4} \left (-21 a^2+24 a b x^4-32 b^2 x^8\right )}{231 a^3 x^{11}} \]

[In]

Integrate[1/(x^12*(a + b*x^4)^(1/4)),x]

[Out]

((a + b*x^4)^(3/4)*(-21*a^2 + 24*a*b*x^4 - 32*b^2*x^8))/(231*a^3*x^11)

Maple [A] (verified)

Time = 4.35 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.57

method	result	size
gosper	\(-\frac {\left (b \,x^{4}+a \right )^{\frac {3}{4}} \left (32 b^{2} x^{8}-24 a b \,x^{4}+21 a^{2}\right )}{231 a^{3} x^{11}}\)	\(39\)
trager	\(-\frac {\left (b \,x^{4}+a \right )^{\frac {3}{4}} \left (32 b^{2} x^{8}-24 a b \,x^{4}+21 a^{2}\right )}{231 a^{3} x^{11}}\)	\(39\)
risch	\(-\frac {\left (b \,x^{4}+a \right )^{\frac {3}{4}} \left (32 b^{2} x^{8}-24 a b \,x^{4}+21 a^{2}\right )}{231 a^{3} x^{11}}\)	\(39\)
pseudoelliptic	\(-\frac {\left (b \,x^{4}+a \right )^{\frac {3}{4}} \left (32 b^{2} x^{8}-24 a b \,x^{4}+21 a^{2}\right )}{231 a^{3} x^{11}}\)	\(39\)

[In]

int(1/x^12/(b*x^4+a)^(1/4),x,method=_RETURNVERBOSE)

[Out]

-1/231*(b*x^4+a)^(3/4)*(32*b^2*x^8-24*a*b*x^4+21*a^2)/a^3/x^11

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.56 \[ \int \frac {1}{x^{12} \sqrt [4]{a+b x^4}} \, dx=-\frac {{\left (32 \, b^{2} x^{8} - 24 \, a b x^{4} + 21 \, a^{2}\right )} {\left (b x^{4} + a\right )}^{\frac {3}{4}}}{231 \, a^{3} x^{11}} \]

[In]

integrate(1/x^12/(b*x^4+a)^(1/4),x, algorithm="fricas")

[Out]

-1/231*(32*b^2*x^8 - 24*a*b*x^4 + 21*a^2)*(b*x^4 + a)^(3/4)/(a^3*x^11)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 406 vs. \(2 (61) = 122\).

Time = 0.89 (sec) , antiderivative size = 406, normalized size of antiderivative = 5.97 \[ \int \frac {1}{x^{12} \sqrt [4]{a+b x^4}} \, dx=\frac {21 a^{4} b^{\frac {19}{4}} \left (\frac {a}{b x^{4}} + 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {11}{4}\right )}{64 a^{5} b^{4} x^{8} \Gamma \left (\frac {1}{4}\right ) + 128 a^{4} b^{5} x^{12} \Gamma \left (\frac {1}{4}\right ) + 64 a^{3} b^{6} x^{16} \Gamma \left (\frac {1}{4}\right )} + \frac {18 a^{3} b^{\frac {23}{4}} x^{4} \left (\frac {a}{b x^{4}} + 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {11}{4}\right )}{64 a^{5} b^{4} x^{8} \Gamma \left (\frac {1}{4}\right ) + 128 a^{4} b^{5} x^{12} \Gamma \left (\frac {1}{4}\right ) + 64 a^{3} b^{6} x^{16} \Gamma \left (\frac {1}{4}\right )} + \frac {5 a^{2} b^{\frac {27}{4}} x^{8} \left (\frac {a}{b x^{4}} + 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {11}{4}\right )}{64 a^{5} b^{4} x^{8} \Gamma \left (\frac {1}{4}\right ) + 128 a^{4} b^{5} x^{12} \Gamma \left (\frac {1}{4}\right ) + 64 a^{3} b^{6} x^{16} \Gamma \left (\frac {1}{4}\right )} + \frac {40 a b^{\frac {31}{4}} x^{12} \left (\frac {a}{b x^{4}} + 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {11}{4}\right )}{64 a^{5} b^{4} x^{8} \Gamma \left (\frac {1}{4}\right ) + 128 a^{4} b^{5} x^{12} \Gamma \left (\frac {1}{4}\right ) + 64 a^{3} b^{6} x^{16} \Gamma \left (\frac {1}{4}\right )} + \frac {32 b^{\frac {35}{4}} x^{16} \left (\frac {a}{b x^{4}} + 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {11}{4}\right )}{64 a^{5} b^{4} x^{8} \Gamma \left (\frac {1}{4}\right ) + 128 a^{4} b^{5} x^{12} \Gamma \left (\frac {1}{4}\right ) + 64 a^{3} b^{6} x^{16} \Gamma \left (\frac {1}{4}\right )} \]

[In]

integrate(1/x**12/(b*x**4+a)**(1/4),x)

[Out]

21*a**4*b**(19/4)*(a/(b*x**4) + 1)**(3/4)*gamma(-11/4)/(64*a**5*b**4*x**8*gamma(1/4) + 128*a**4*b**5*x**12*gam
ma(1/4) + 64*a**3*b**6*x**16*gamma(1/4)) + 18*a**3*b**(23/4)*x**4*(a/(b*x**4) + 1)**(3/4)*gamma(-11/4)/(64*a**
5*b**4*x**8*gamma(1/4) + 128*a**4*b**5*x**12*gamma(1/4) + 64*a**3*b**6*x**16*gamma(1/4)) + 5*a**2*b**(27/4)*x*
*8*(a/(b*x**4) + 1)**(3/4)*gamma(-11/4)/(64*a**5*b**4*x**8*gamma(1/4) + 128*a**4*b**5*x**12*gamma(1/4) + 64*a*
*3*b**6*x**16*gamma(1/4)) + 40*a*b**(31/4)*x**12*(a/(b*x**4) + 1)**(3/4)*gamma(-11/4)/(64*a**5*b**4*x**8*gamma
(1/4) + 128*a**4*b**5*x**12*gamma(1/4) + 64*a**3*b**6*x**16*gamma(1/4)) + 32*b**(35/4)*x**16*(a/(b*x**4) + 1)*
*(3/4)*gamma(-11/4)/(64*a**5*b**4*x**8*gamma(1/4) + 128*a**4*b**5*x**12*gamma(1/4) + 64*a**3*b**6*x**16*gamma(
1/4))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.76 \[ \int \frac {1}{x^{12} \sqrt [4]{a+b x^4}} \, dx=-\frac {\frac {77 \, {\left (b x^{4} + a\right )}^{\frac {3}{4}} b^{2}}{x^{3}} - \frac {66 \, {\left (b x^{4} + a\right )}^{\frac {7}{4}} b}{x^{7}} + \frac {21 \, {\left (b x^{4} + a\right )}^{\frac {11}{4}}}{x^{11}}}{231 \, a^{3}} \]

[In]

integrate(1/x^12/(b*x^4+a)^(1/4),x, algorithm="maxima")

[Out]

-1/231*(77*(b*x^4 + a)^(3/4)*b^2/x^3 - 66*(b*x^4 + a)^(7/4)*b/x^7 + 21*(b*x^4 + a)^(11/4)/x^11)/a^3

Giac [F]

\[ \int \frac {1}{x^{12} \sqrt [4]{a+b x^4}} \, dx=\int { \frac {1}{{\left (b x^{4} + a\right )}^{\frac {1}{4}} x^{12}} \,d x } \]

[In]

integrate(1/x^12/(b*x^4+a)^(1/4),x, algorithm="giac")

[Out]

integrate(1/((b*x^4 + a)^(1/4)*x^12), x)

Mupad [B] (verification not implemented)

Time = 5.81 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.82 \[ \int \frac {1}{x^{12} \sqrt [4]{a+b x^4}} \, dx=-\frac {21\,a^2\,{\left (b\,x^4+a\right )}^{3/4}+32\,b^2\,x^8\,{\left (b\,x^4+a\right )}^{3/4}-24\,a\,b\,x^4\,{\left (b\,x^4+a\right )}^{3/4}}{231\,a^3\,x^{11}} \]

[In]

int(1/(x^12*(a + b*x^4)^(1/4)),x)

[Out]

-(21*a^2*(a + b*x^4)^(3/4) + 32*b^2*x^8*(a + b*x^4)^(3/4) - 24*a*b*x^4*(a + b*x^4)^(3/4))/(231*a^3*x^11)