3.1254 \(\int \frac{1}{x^{10} (a-b x^4)^{3/4}} \, dx\)
Optimal. Leaf size=71 \[ -\frac{32 b^2 \sqrt [4]{a-b x^4}}{45 a^3 x}-\frac{8 b \sqrt [4]{a-b x^4}}{45 a^2 x^5}-\frac{\sqrt [4]{a-b x^4}}{9 a x^9} \]
[Out]
-(a - b*x^4)^(1/4)/(9*a*x^9) - (8*b*(a - b*x^4)^(1/4))/(45*a^2*x^5) - (32*b^2*(a - b*x^4)^(1/4))/(45*a^3*x)
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Rubi [A] time = 0.0193996, 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} \[ -\frac{32 b^2 \sqrt [4]{a-b x^4}}{45 a^3 x}-\frac{8 b \sqrt [4]{a-b x^4}}{45 a^2 x^5}-\frac{\sqrt [4]{a-b x^4}}{9 a x^9} \]
Antiderivative was successfully verified.
[In]
Int[1/(x^10*(a - b*x^4)^(3/4)),x]
[Out]
-(a - b*x^4)^(1/4)/(9*a*x^9) - (8*b*(a - b*x^4)^(1/4))/(45*a^2*x^5) - (32*b^2*(a - b*x^4)^(1/4))/(45*a^3*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^{10} \left (a-b x^4\right )^{3/4}} \, dx &=-\frac{\sqrt [4]{a-b x^4}}{9 a x^9}+\frac{(8 b) \int \frac{1}{x^6 \left (a-b x^4\right )^{3/4}} \, dx}{9 a}\\ &=-\frac{\sqrt [4]{a-b x^4}}{9 a x^9}-\frac{8 b \sqrt [4]{a-b x^4}}{45 a^2 x^5}+\frac{\left (32 b^2\right ) \int \frac{1}{x^2 \left (a-b x^4\right )^{3/4}} \, dx}{45 a^2}\\ &=-\frac{\sqrt [4]{a-b x^4}}{9 a x^9}-\frac{8 b \sqrt [4]{a-b x^4}}{45 a^2 x^5}-\frac{32 b^2 \sqrt [4]{a-b x^4}}{45 a^3 x}\\ \end{align*}
Mathematica [A] time = 0.014661, size = 43, normalized size = 0.61 \[ -\frac{\sqrt [4]{a-b x^4} \left (5 a^2+8 a b x^4+32 b^2 x^8\right )}{45 a^3 x^9} \]
Antiderivative was successfully verified.
[In]
Integrate[1/(x^10*(a - b*x^4)^(3/4)),x]
[Out]
-((a - b*x^4)^(1/4)*(5*a^2 + 8*a*b*x^4 + 32*b^2*x^8))/(45*a^3*x^9)
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Maple [A] time = 0.004, size = 40, normalized size = 0.6 \begin{align*} -{\frac{32\,{b}^{2}{x}^{8}+8\,ab{x}^{4}+5\,{a}^{2}}{45\,{a}^{3}{x}^{9}}\sqrt [4]{-b{x}^{4}+a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/x^10/(-b*x^4+a)^(3/4),x)
[Out]
-1/45*(-b*x^4+a)^(1/4)*(32*b^2*x^8+8*a*b*x^4+5*a^2)/a^3/x^9
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Maxima [A] time = 1.15657, size = 74, normalized size = 1.04 \begin{align*} -\frac{\frac{45 \,{\left (-b x^{4} + a\right )}^{\frac{1}{4}} b^{2}}{x} + \frac{18 \,{\left (-b x^{4} + a\right )}^{\frac{5}{4}} b}{x^{5}} + \frac{5 \,{\left (-b x^{4} + a\right )}^{\frac{9}{4}}}{x^{9}}}{45 \, a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/x^10/(-b*x^4+a)^(3/4),x, algorithm="maxima")
[Out]
-1/45*(45*(-b*x^4 + a)^(1/4)*b^2/x + 18*(-b*x^4 + a)^(5/4)*b/x^5 + 5*(-b*x^4 + a)^(9/4)/x^9)/a^3
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Fricas [A] time = 1.66108, size = 93, normalized size = 1.31 \begin{align*} -\frac{{\left (32 \, b^{2} x^{8} + 8 \, a b x^{4} + 5 \, a^{2}\right )}{\left (-b x^{4} + a\right )}^{\frac{1}{4}}}{45 \, a^{3} x^{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/x^10/(-b*x^4+a)^(3/4),x, algorithm="fricas")
[Out]
-1/45*(32*b^2*x^8 + 8*a*b*x^4 + 5*a^2)*(-b*x^4 + a)^(1/4)/(a^3*x^9)
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Sympy [C] time = 3.15512, size = 1114, normalized size = 15.69 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/x**10/(-b*x**4+a)**(3/4),x)
[Out]
Piecewise((-5*a**4*b**(17/4)*(a/(b*x**4) - 1)**(1/4)*exp(-I*pi/4)*gamma(-9/4)/(64*a**5*b**4*x**8*exp(3*I*pi/4)
*gamma(3/4) - 128*a**4*b**5*x**12*exp(3*I*pi/4)*gamma(3/4) + 64*a**3*b**6*x**16*exp(3*I*pi/4)*gamma(3/4)) + 2*
a**3*b**(21/4)*x**4*(a/(b*x**4) - 1)**(1/4)*exp(-I*pi/4)*gamma(-9/4)/(64*a**5*b**4*x**8*exp(3*I*pi/4)*gamma(3/
4) - 128*a**4*b**5*x**12*exp(3*I*pi/4)*gamma(3/4) + 64*a**3*b**6*x**16*exp(3*I*pi/4)*gamma(3/4)) - 21*a**2*b**
(25/4)*x**8*(a/(b*x**4) - 1)**(1/4)*exp(-I*pi/4)*gamma(-9/4)/(64*a**5*b**4*x**8*exp(3*I*pi/4)*gamma(3/4) - 128
*a**4*b**5*x**12*exp(3*I*pi/4)*gamma(3/4) + 64*a**3*b**6*x**16*exp(3*I*pi/4)*gamma(3/4)) + 56*a*b**(29/4)*x**1
2*(a/(b*x**4) - 1)**(1/4)*exp(-I*pi/4)*gamma(-9/4)/(64*a**5*b**4*x**8*exp(3*I*pi/4)*gamma(3/4) - 128*a**4*b**5
*x**12*exp(3*I*pi/4)*gamma(3/4) + 64*a**3*b**6*x**16*exp(3*I*pi/4)*gamma(3/4)) - 32*b**(33/4)*x**16*(a/(b*x**4
) - 1)**(1/4)*exp(-I*pi/4)*gamma(-9/4)/(64*a**5*b**4*x**8*exp(3*I*pi/4)*gamma(3/4) - 128*a**4*b**5*x**12*exp(3
*I*pi/4)*gamma(3/4) + 64*a**3*b**6*x**16*exp(3*I*pi/4)*gamma(3/4)), Abs(a)/(Abs(b)*Abs(x**4)) > 1), (-5*a**4*b
**(17/4)*(-a/(b*x**4) + 1)**(1/4)*gamma(-9/4)/(64*a**5*b**4*x**8*exp(3*I*pi/4)*gamma(3/4) - 128*a**4*b**5*x**1
2*exp(3*I*pi/4)*gamma(3/4) + 64*a**3*b**6*x**16*exp(3*I*pi/4)*gamma(3/4)) + 2*a**3*b**(21/4)*x**4*(-a/(b*x**4)
+ 1)**(1/4)*gamma(-9/4)/(64*a**5*b**4*x**8*exp(3*I*pi/4)*gamma(3/4) - 128*a**4*b**5*x**12*exp(3*I*pi/4)*gamma
(3/4) + 64*a**3*b**6*x**16*exp(3*I*pi/4)*gamma(3/4)) - 21*a**2*b**(25/4)*x**8*(-a/(b*x**4) + 1)**(1/4)*gamma(-
9/4)/(64*a**5*b**4*x**8*exp(3*I*pi/4)*gamma(3/4) - 128*a**4*b**5*x**12*exp(3*I*pi/4)*gamma(3/4) + 64*a**3*b**6
*x**16*exp(3*I*pi/4)*gamma(3/4)) + 56*a*b**(29/4)*x**12*(-a/(b*x**4) + 1)**(1/4)*gamma(-9/4)/(64*a**5*b**4*x**
8*exp(3*I*pi/4)*gamma(3/4) - 128*a**4*b**5*x**12*exp(3*I*pi/4)*gamma(3/4) + 64*a**3*b**6*x**16*exp(3*I*pi/4)*g
amma(3/4)) - 32*b**(33/4)*x**16*(-a/(b*x**4) + 1)**(1/4)*gamma(-9/4)/(64*a**5*b**4*x**8*exp(3*I*pi/4)*gamma(3/
4) - 128*a**4*b**5*x**12*exp(3*I*pi/4)*gamma(3/4) + 64*a**3*b**6*x**16*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^{10}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/x^10/(-b*x^4+a)^(3/4),x, algorithm="giac")
[Out]
integrate(1/((-b*x^4 + a)^(3/4)*x^10), x)