3.1194 \(\int \frac{\sqrt [4]{a-b x^4}}{x^{14}} \, dx\)

Optimal. Leaf size=71 \[ -\frac{32 b^2 \left (a-b x^4\right )^{5/4}}{585 a^3 x^5}-\frac{8 b \left (a-b x^4\right )^{5/4}}{117 a^2 x^9}-\frac{\left (a-b x^4\right )^{5/4}}{13 a x^{13}} \]

[Out]

-(a - b*x^4)^(5/4)/(13*a*x^13) - (8*b*(a - b*x^4)^(5/4))/(117*a^2*x^9) - (32*b^2*(a - b*x^4)^(5/4))/(585*a^3*x
^5)

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Rubi [A]  time = 0.0190207, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {271, 264} \[ -\frac{32 b^2 \left (a-b x^4\right )^{5/4}}{585 a^3 x^5}-\frac{8 b \left (a-b x^4\right )^{5/4}}{117 a^2 x^9}-\frac{\left (a-b x^4\right )^{5/4}}{13 a x^{13}} \]

Antiderivative was successfully verified.

[In]

Int[(a - b*x^4)^(1/4)/x^14,x]

[Out]

-(a - b*x^4)^(5/4)/(13*a*x^13) - (8*b*(a - b*x^4)^(5/4))/(117*a^2*x^9) - (32*b^2*(a - b*x^4)^(5/4))/(585*a^3*x
^5)

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{\sqrt [4]{a-b x^4}}{x^{14}} \, dx &=-\frac{\left (a-b x^4\right )^{5/4}}{13 a x^{13}}+\frac{(8 b) \int \frac{\sqrt [4]{a-b x^4}}{x^{10}} \, dx}{13 a}\\ &=-\frac{\left (a-b x^4\right )^{5/4}}{13 a x^{13}}-\frac{8 b \left (a-b x^4\right )^{5/4}}{117 a^2 x^9}+\frac{\left (32 b^2\right ) \int \frac{\sqrt [4]{a-b x^4}}{x^6} \, dx}{117 a^2}\\ &=-\frac{\left (a-b x^4\right )^{5/4}}{13 a x^{13}}-\frac{8 b \left (a-b x^4\right )^{5/4}}{117 a^2 x^9}-\frac{32 b^2 \left (a-b x^4\right )^{5/4}}{585 a^3 x^5}\\ \end{align*}

Mathematica [A]  time = 0.0086026, size = 43, normalized size = 0.61 \[ -\frac{\left (a-b x^4\right )^{5/4} \left (45 a^2+40 a b x^4+32 b^2 x^8\right )}{585 a^3 x^{13}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a - b*x^4)^(1/4)/x^14,x]

[Out]

-((a - b*x^4)^(5/4)*(45*a^2 + 40*a*b*x^4 + 32*b^2*x^8))/(585*a^3*x^13)

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Maple [A]  time = 0.005, size = 40, normalized size = 0.6 \begin{align*} -{\frac{32\,{b}^{2}{x}^{8}+40\,ab{x}^{4}+45\,{a}^{2}}{585\,{x}^{13}{a}^{3}} \left ( -b{x}^{4}+a \right ) ^{{\frac{5}{4}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-b*x^4+a)^(1/4)/x^14,x)

[Out]

-1/585*(-b*x^4+a)^(5/4)*(32*b^2*x^8+40*a*b*x^4+45*a^2)/x^13/a^3

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Maxima [A]  time = 0.990212, size = 74, normalized size = 1.04 \begin{align*} -\frac{\frac{117 \,{\left (-b x^{4} + a\right )}^{\frac{5}{4}} b^{2}}{x^{5}} + \frac{130 \,{\left (-b x^{4} + a\right )}^{\frac{9}{4}} b}{x^{9}} + \frac{45 \,{\left (-b x^{4} + a\right )}^{\frac{13}{4}}}{x^{13}}}{585 \, a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-b*x^4+a)^(1/4)/x^14,x, algorithm="maxima")

[Out]

-1/585*(117*(-b*x^4 + a)^(5/4)*b^2/x^5 + 130*(-b*x^4 + a)^(9/4)*b/x^9 + 45*(-b*x^4 + a)^(13/4)/x^13)/a^3

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Fricas [A]  time = 1.70793, size = 119, normalized size = 1.68 \begin{align*} \frac{{\left (32 \, b^{3} x^{12} + 8 \, a b^{2} x^{8} + 5 \, a^{2} b x^{4} - 45 \, a^{3}\right )}{\left (-b x^{4} + a\right )}^{\frac{1}{4}}}{585 \, a^{3} x^{13}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-b*x^4+a)^(1/4)/x^14,x, algorithm="fricas")

[Out]

1/585*(32*b^3*x^12 + 8*a*b^2*x^8 + 5*a^2*b*x^4 - 45*a^3)*(-b*x^4 + a)^(1/4)/(a^3*x^13)

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Sympy [B]  time = 3.90162, size = 1093, normalized size = 15.39 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-b*x**4+a)**(1/4)/x**14,x)

[Out]

Piecewise((45*a**5*b**(17/4)*(a/(b*x**4) - 1)**(1/4)*gamma(-13/4)/(64*a**5*b**4*x**12*gamma(-1/4) - 128*a**4*b
**5*x**16*gamma(-1/4) + 64*a**3*b**6*x**20*gamma(-1/4)) - 95*a**4*b**(21/4)*x**4*(a/(b*x**4) - 1)**(1/4)*gamma
(-13/4)/(64*a**5*b**4*x**12*gamma(-1/4) - 128*a**4*b**5*x**16*gamma(-1/4) + 64*a**3*b**6*x**20*gamma(-1/4)) +
47*a**3*b**(25/4)*x**8*(a/(b*x**4) - 1)**(1/4)*gamma(-13/4)/(64*a**5*b**4*x**12*gamma(-1/4) - 128*a**4*b**5*x*
*16*gamma(-1/4) + 64*a**3*b**6*x**20*gamma(-1/4)) - 21*a**2*b**(29/4)*x**12*(a/(b*x**4) - 1)**(1/4)*gamma(-13/
4)/(64*a**5*b**4*x**12*gamma(-1/4) - 128*a**4*b**5*x**16*gamma(-1/4) + 64*a**3*b**6*x**20*gamma(-1/4)) + 56*a*
b**(33/4)*x**16*(a/(b*x**4) - 1)**(1/4)*gamma(-13/4)/(64*a**5*b**4*x**12*gamma(-1/4) - 128*a**4*b**5*x**16*gam
ma(-1/4) + 64*a**3*b**6*x**20*gamma(-1/4)) - 32*b**(37/4)*x**20*(a/(b*x**4) - 1)**(1/4)*gamma(-13/4)/(64*a**5*
b**4*x**12*gamma(-1/4) - 128*a**4*b**5*x**16*gamma(-1/4) + 64*a**3*b**6*x**20*gamma(-1/4)), Abs(a)/(Abs(b)*Abs
(x**4)) > 1), (45*a**5*b**(17/4)*(-a/(b*x**4) + 1)**(1/4)*exp(I*pi/4)*gamma(-13/4)/(64*a**5*b**4*x**12*gamma(-
1/4) - 128*a**4*b**5*x**16*gamma(-1/4) + 64*a**3*b**6*x**20*gamma(-1/4)) - 95*a**4*b**(21/4)*x**4*(-a/(b*x**4)
+ 1)**(1/4)*exp(I*pi/4)*gamma(-13/4)/(64*a**5*b**4*x**12*gamma(-1/4) - 128*a**4*b**5*x**16*gamma(-1/4) + 64*a
**3*b**6*x**20*gamma(-1/4)) + 47*a**3*b**(25/4)*x**8*(-a/(b*x**4) + 1)**(1/4)*exp(I*pi/4)*gamma(-13/4)/(64*a**
5*b**4*x**12*gamma(-1/4) - 128*a**4*b**5*x**16*gamma(-1/4) + 64*a**3*b**6*x**20*gamma(-1/4)) - 21*a**2*b**(29/
4)*x**12*(-a/(b*x**4) + 1)**(1/4)*exp(I*pi/4)*gamma(-13/4)/(64*a**5*b**4*x**12*gamma(-1/4) - 128*a**4*b**5*x**
16*gamma(-1/4) + 64*a**3*b**6*x**20*gamma(-1/4)) + 56*a*b**(33/4)*x**16*(-a/(b*x**4) + 1)**(1/4)*exp(I*pi/4)*g
amma(-13/4)/(64*a**5*b**4*x**12*gamma(-1/4) - 128*a**4*b**5*x**16*gamma(-1/4) + 64*a**3*b**6*x**20*gamma(-1/4)
) - 32*b**(37/4)*x**20*(-a/(b*x**4) + 1)**(1/4)*exp(I*pi/4)*gamma(-13/4)/(64*a**5*b**4*x**12*gamma(-1/4) - 128
*a**4*b**5*x**16*gamma(-1/4) + 64*a**3*b**6*x**20*gamma(-1/4)), True))

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Giac [A]  time = 1.18388, size = 151, normalized size = 2.13 \begin{align*} \frac{\frac{117 \,{\left (-b x^{4} + a\right )}^{\frac{1}{4}}{\left (b - \frac{a}{x^{4}}\right )} b^{2}}{x} - \frac{130 \,{\left (b^{2} x^{8} - 2 \, a b x^{4} + a^{2}\right )}{\left (-b x^{4} + a\right )}^{\frac{1}{4}} b}{x^{9}} + \frac{45 \,{\left (b^{3} x^{12} - 3 \, a b^{2} x^{8} + 3 \, a^{2} b x^{4} - a^{3}\right )}{\left (-b x^{4} + a\right )}^{\frac{1}{4}}}{x^{13}}}{585 \, a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-b*x^4+a)^(1/4)/x^14,x, algorithm="giac")

[Out]

1/585*(117*(-b*x^4 + a)^(1/4)*(b - a/x^4)*b^2/x - 130*(b^2*x^8 - 2*a*b*x^4 + a^2)*(-b*x^4 + a)^(1/4)*b/x^9 + 4
5*(b^3*x^12 - 3*a*b^2*x^8 + 3*a^2*b*x^4 - a^3)*(-b*x^4 + a)^(1/4)/x^13)/a^3