3.1100 \(\int \frac{1}{x^{12} \sqrt [4]{a+b x^4}} \, dx\)

Optimal. Leaf size=68 \[ -\frac{32 b^2 \left (a+b x^4\right )^{3/4}}{231 a^3 x^3}+\frac{8 b \left (a+b x^4\right )^{3/4}}{77 a^2 x^7}-\frac{\left (a+b x^4\right )^{3/4}}{11 a x^{11}} \]

[Out]

-(a + b*x^4)^(3/4)/(11*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)

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Rubi [A]  time = 0.0179517, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {271, 264} \[ -\frac{32 b^2 \left (a+b x^4\right )^{3/4}}{231 a^3 x^3}+\frac{8 b \left (a+b x^4\right )^{3/4}}{77 a^2 x^7}-\frac{\left (a+b x^4\right )^{3/4}}{11 a x^{11}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^12*(a + b*x^4)^(1/4)),x]

[Out]

-(a + b*x^4)^(3/4)/(11*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 271

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]

Rule 264

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]

Rubi steps

\begin{align*} \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) \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]  time = 0.0175305, size = 42, normalized size = 0.62 \[ -\frac{\left (a+b x^4\right )^{3/4} \left (21 a^2-24 a b x^4+32 b^2 x^8\right )}{231 a^3 x^{11}} \]

Antiderivative was successfully verified.

[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)

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Maple [A]  time = 0.004, size = 39, normalized size = 0.6 \begin{align*} -{\frac{32\,{b}^{2}{x}^{8}-24\,ab{x}^{4}+21\,{a}^{2}}{231\,{x}^{11}{a}^{3}} \left ( b{x}^{4}+a \right ) ^{{\frac{3}{4}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.990395, size = 70, normalized size = 1.03 \begin{align*} -\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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Fricas [A]  time = 1.47486, size = 97, normalized size = 1.43 \begin{align*} -\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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [B]  time = 3.91318, size = 406, normalized size = 5.97 \begin{align*} \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 )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x^{4} + a\right )}^{\frac{1}{4}} x^{12}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)