3.1252 \(\int \frac{1}{x^{10} \left (a-b x^4\right )^{3/4}} \, dx\)

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 = 0.0696853, 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 \[ -\frac{32 b^2 \sqrt [4]{a-b x^4}}{45 a^3 x}-\frac{8 b \sqrt [4]{a-b x^4}}{45 a^2 x^5}-\frac{\sqrt [4]{a-b x^4}}{9 a x^9} \]

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)

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Rubi in Sympy [A]  time = 7.69216, size = 61, normalized size = 0.86 \[ - \frac{\sqrt [4]{a - b x^{4}}}{9 a x^{9}} - \frac{8 b \sqrt [4]{a - b x^{4}}}{45 a^{2} x^{5}} - \frac{32 b^{2} \sqrt [4]{a - b x^{4}}}{45 a^{3} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**10/(-b*x**4+a)**(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)

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Mathematica [A]  time = 0.0325752, 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 = 0.006, size = 40, normalized size = 0.6 \[ -{\frac{32\,{b}^{2}{x}^{8}+8\,ab{x}^{4}+5\,{a}^{2}}{45\,{a}^{3}{x}^{9}}\sqrt [4]{-b{x}^{4}+a}} \]

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 = 1.44039, size = 74, normalized size = 1.04 \[ -\frac{\frac{45 \,{\left (-b x^{4} + a\right )}^{\frac{1}{4}} b^{2}}{x} + \frac{18 \,{\left (-b x^{4} + a\right )}^{\frac{5}{4}} b}{x^{5}} + \frac{5 \,{\left (-b x^{4} + a\right )}^{\frac{9}{4}}}{x^{9}}}{45 \, a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-b*x^4 + a)^(3/4)*x^10),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 = 0.226916, size = 53, normalized size = 0.75 \[ -\frac{{\left (32 \, b^{2} x^{8} + 8 \, a b x^{4} + 5 \, a^{2}\right )}{\left (-b x^{4} + a\right )}^{\frac{1}{4}}}{45 \, a^{3} x^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-b*x^4 + a)^(3/4)*x^10),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 [A]  time = 11.2171, size = 864, normalized size = 12.17 \[ \text{result too large to display} \]

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)*gamma(-9/4)/(64*a**5*b**4*x*
*8*gamma(3/4) - 128*a**4*b**5*x**12*gamma(3/4) + 64*a**3*b**6*x**16*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*g
amma(3/4) - 128*a**4*b**5*x**12*gamma(3/4) + 64*a**3*b**6*x**16*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*gamm
a(3/4) - 128*a**4*b**5*x**12*gamma(3/4) + 64*a**3*b**6*x**16*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*gamma(3/4
) - 128*a**4*b**5*x**12*gamma(3/4) + 64*a**3*b**6*x**16*gamma(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*gamma(3/4) - 128
*a**4*b**5*x**12*gamma(3/4) + 64*a**3*b**6*x**16*gamma(3/4)), Abs(a/(b*x**4)) >
1), (-5*a**4*b**(17/4)*(-a/(b*x**4) + 1)**(1/4)*exp(13*I*pi/4)*gamma(-9/4)/(64*a
**5*b**4*x**8*gamma(3/4) - 128*a**4*b**5*x**12*gamma(3/4) + 64*a**3*b**6*x**16*g
amma(3/4)) + 2*a**3*b**(21/4)*x**4*(-a/(b*x**4) + 1)**(1/4)*exp(13*I*pi/4)*gamma
(-9/4)/(64*a**5*b**4*x**8*gamma(3/4) - 128*a**4*b**5*x**12*gamma(3/4) + 64*a**3*
b**6*x**16*gamma(3/4)) - 21*a**2*b**(25/4)*x**8*(-a/(b*x**4) + 1)**(1/4)*exp(13*
I*pi/4)*gamma(-9/4)/(64*a**5*b**4*x**8*gamma(3/4) - 128*a**4*b**5*x**12*gamma(3/
4) + 64*a**3*b**6*x**16*gamma(3/4)) + 56*a*b**(29/4)*x**12*(-a/(b*x**4) + 1)**(1
/4)*exp(13*I*pi/4)*gamma(-9/4)/(64*a**5*b**4*x**8*gamma(3/4) - 128*a**4*b**5*x**
12*gamma(3/4) + 64*a**3*b**6*x**16*gamma(3/4)) - 32*b**(33/4)*x**16*(-a/(b*x**4)
+ 1)**(1/4)*exp(13*I*pi/4)*gamma(-9/4)/(64*a**5*b**4*x**8*gamma(3/4) - 128*a**4
*b**5*x**12*gamma(3/4) + 64*a**3*b**6*x**16*gamma(3/4)), True))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (-b x^{4} + a\right )}^{\frac{3}{4}} x^{10}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-b*x^4 + a)^(3/4)*x^10),x, algorithm="giac")

[Out]

integrate(1/((-b*x^4 + a)^(3/4)*x^10), x)