3.1253 \(\int \frac{1}{x^6 (a-b x^4)^{3/4}} \, dx\)

Optimal. Leaf size=46 \[ -\frac{4 b \sqrt [4]{a-b x^4}}{5 a^2 x}-\frac{\sqrt [4]{a-b x^4}}{5 a x^5} \]

[Out]

-(a - b*x^4)^(1/4)/(5*a*x^5) - (4*b*(a - b*x^4)^(1/4))/(5*a^2*x)

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Rubi [A]  time = 0.0115406, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {271, 264} \[ -\frac{4 b \sqrt [4]{a-b x^4}}{5 a^2 x}-\frac{\sqrt [4]{a-b x^4}}{5 a x^5} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^6*(a - b*x^4)^(3/4)),x]

[Out]

-(a - b*x^4)^(1/4)/(5*a*x^5) - (4*b*(a - b*x^4)^(1/4))/(5*a^2*x)

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^6 \left (a-b x^4\right )^{3/4}} \, dx &=-\frac{\sqrt [4]{a-b x^4}}{5 a x^5}+\frac{(4 b) \int \frac{1}{x^2 \left (a-b x^4\right )^{3/4}} \, dx}{5 a}\\ &=-\frac{\sqrt [4]{a-b x^4}}{5 a x^5}-\frac{4 b \sqrt [4]{a-b x^4}}{5 a^2 x}\\ \end{align*}

Mathematica [A]  time = 0.0098985, size = 30, normalized size = 0.65 \[ -\frac{\sqrt [4]{a-b x^4} \left (a+4 b x^4\right )}{5 a^2 x^5} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^6*(a - b*x^4)^(3/4)),x]

[Out]

-((a - b*x^4)^(1/4)*(a + 4*b*x^4))/(5*a^2*x^5)

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Maple [A]  time = 0.004, size = 27, normalized size = 0.6 \begin{align*} -{\frac{4\,b{x}^{4}+a}{5\,{x}^{5}{a}^{2}}\sqrt [4]{-b{x}^{4}+a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^6/(-b*x^4+a)^(3/4),x)

[Out]

-1/5*(-b*x^4+a)^(1/4)*(4*b*x^4+a)/x^5/a^2

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Maxima [A]  time = 1.07284, size = 49, normalized size = 1.07 \begin{align*} -\frac{\frac{5 \,{\left (-b x^{4} + a\right )}^{\frac{1}{4}} b}{x} + \frac{{\left (-b x^{4} + a\right )}^{\frac{5}{4}}}{x^{5}}}{5 \, a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^6/(-b*x^4+a)^(3/4),x, algorithm="maxima")

[Out]

-1/5*(5*(-b*x^4 + a)^(1/4)*b/x + (-b*x^4 + a)^(5/4)/x^5)/a^2

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Fricas [A]  time = 1.67425, size = 66, normalized size = 1.43 \begin{align*} -\frac{{\left (4 \, b x^{4} + a\right )}{\left (-b x^{4} + a\right )}^{\frac{1}{4}}}{5 \, a^{2} x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^6/(-b*x^4+a)^(3/4),x, algorithm="fricas")

[Out]

-1/5*(4*b*x^4 + a)*(-b*x^4 + a)^(1/4)/(a^2*x^5)

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Sympy [A]  time = 1.74089, size = 314, normalized size = 6.83 \begin{align*} \begin{cases} - \frac{\sqrt [4]{b} \sqrt [4]{\frac{a}{b x^{4}} - 1} \Gamma \left (- \frac{5}{4}\right )}{16 a x^{4} \Gamma \left (\frac{3}{4}\right )} - \frac{b^{\frac{5}{4}} \sqrt [4]{\frac{a}{b x^{4}} - 1} \Gamma \left (- \frac{5}{4}\right )}{4 a^{2} \Gamma \left (\frac{3}{4}\right )} & \text{for}\: \frac{\left |{a}\right |}{\left |{b}\right | \left |{x^{4}}\right |} > 1 \\- \frac{a^{2} b^{\frac{5}{4}} \sqrt [4]{- \frac{a}{b x^{4}} + 1} \Gamma \left (- \frac{5}{4}\right )}{- 16 a^{3} b x^{4} e^{\frac{3 i \pi }{4}} \Gamma \left (\frac{3}{4}\right ) + 16 a^{2} b^{2} x^{8} e^{\frac{3 i \pi }{4}} \Gamma \left (\frac{3}{4}\right )} - \frac{3 a b^{\frac{9}{4}} x^{4} \sqrt [4]{- \frac{a}{b x^{4}} + 1} \Gamma \left (- \frac{5}{4}\right )}{- 16 a^{3} b x^{4} e^{\frac{3 i \pi }{4}} \Gamma \left (\frac{3}{4}\right ) + 16 a^{2} b^{2} x^{8} e^{\frac{3 i \pi }{4}} \Gamma \left (\frac{3}{4}\right )} + \frac{4 b^{\frac{13}{4}} x^{8} \sqrt [4]{- \frac{a}{b x^{4}} + 1} \Gamma \left (- \frac{5}{4}\right )}{- 16 a^{3} b x^{4} e^{\frac{3 i \pi }{4}} \Gamma \left (\frac{3}{4}\right ) + 16 a^{2} b^{2} x^{8} e^{\frac{3 i \pi }{4}} \Gamma \left (\frac{3}{4}\right )} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**6/(-b*x**4+a)**(3/4),x)

[Out]

Piecewise((-b**(1/4)*(a/(b*x**4) - 1)**(1/4)*gamma(-5/4)/(16*a*x**4*gamma(3/4)) - b**(5/4)*(a/(b*x**4) - 1)**(
1/4)*gamma(-5/4)/(4*a**2*gamma(3/4)), Abs(a)/(Abs(b)*Abs(x**4)) > 1), (-a**2*b**(5/4)*(-a/(b*x**4) + 1)**(1/4)
*gamma(-5/4)/(-16*a**3*b*x**4*exp(3*I*pi/4)*gamma(3/4) + 16*a**2*b**2*x**8*exp(3*I*pi/4)*gamma(3/4)) - 3*a*b**
(9/4)*x**4*(-a/(b*x**4) + 1)**(1/4)*gamma(-5/4)/(-16*a**3*b*x**4*exp(3*I*pi/4)*gamma(3/4) + 16*a**2*b**2*x**8*
exp(3*I*pi/4)*gamma(3/4)) + 4*b**(13/4)*x**8*(-a/(b*x**4) + 1)**(1/4)*gamma(-5/4)/(-16*a**3*b*x**4*exp(3*I*pi/
4)*gamma(3/4) + 16*a**2*b**2*x**8*exp(3*I*pi/4)*gamma(3/4)), True))

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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{3}{4}} x^{6}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^6/(-b*x^4+a)^(3/4),x, algorithm="giac")

[Out]

integrate(1/((-b*x^4 + a)^(3/4)*x^6), x)