3.1253 \(\int \frac{1}{x^{14} \left (a-b x^4\right )^{3/4}} \, dx\)

Optimal. Leaf size=96 \[ -\frac{128 b^3 \sqrt [4]{a-b x^4}}{195 a^4 x}-\frac{32 b^2 \sqrt [4]{a-b x^4}}{195 a^3 x^5}-\frac{4 b \sqrt [4]{a-b x^4}}{39 a^2 x^9}-\frac{\sqrt [4]{a-b x^4}}{13 a x^{13}} \]

[Out]

-(a - b*x^4)^(1/4)/(13*a*x^13) - (4*b*(a - b*x^4)^(1/4))/(39*a^2*x^9) - (32*b^2*
(a - b*x^4)^(1/4))/(195*a^3*x^5) - (128*b^3*(a - b*x^4)^(1/4))/(195*a^4*x)

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Rubi [A]  time = 0.0977074, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ -\frac{128 b^3 \sqrt [4]{a-b x^4}}{195 a^4 x}-\frac{32 b^2 \sqrt [4]{a-b x^4}}{195 a^3 x^5}-\frac{4 b \sqrt [4]{a-b x^4}}{39 a^2 x^9}-\frac{\sqrt [4]{a-b x^4}}{13 a x^{13}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^14*(a - b*x^4)^(3/4)),x]

[Out]

-(a - b*x^4)^(1/4)/(13*a*x^13) - (4*b*(a - b*x^4)^(1/4))/(39*a^2*x^9) - (32*b^2*
(a - b*x^4)^(1/4))/(195*a^3*x^5) - (128*b^3*(a - b*x^4)^(1/4))/(195*a^4*x)

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Rubi in Sympy [A]  time = 11.1506, size = 85, normalized size = 0.89 \[ - \frac{\sqrt [4]{a - b x^{4}}}{13 a x^{13}} - \frac{4 b \sqrt [4]{a - b x^{4}}}{39 a^{2} x^{9}} - \frac{32 b^{2} \sqrt [4]{a - b x^{4}}}{195 a^{3} x^{5}} - \frac{128 b^{3} \sqrt [4]{a - b x^{4}}}{195 a^{4} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**14/(-b*x**4+a)**(3/4),x)

[Out]

-(a - b*x**4)**(1/4)/(13*a*x**13) - 4*b*(a - b*x**4)**(1/4)/(39*a**2*x**9) - 32*
b**2*(a - b*x**4)**(1/4)/(195*a**3*x**5) - 128*b**3*(a - b*x**4)**(1/4)/(195*a**
4*x)

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Mathematica [A]  time = 0.0414992, size = 54, normalized size = 0.56 \[ -\frac{\sqrt [4]{a-b x^4} \left (15 a^3+20 a^2 b x^4+32 a b^2 x^8+128 b^3 x^{12}\right )}{195 a^4 x^{13}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^14*(a - b*x^4)^(3/4)),x]

[Out]

-((a - b*x^4)^(1/4)*(15*a^3 + 20*a^2*b*x^4 + 32*a*b^2*x^8 + 128*b^3*x^12))/(195*
a^4*x^13)

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Maple [A]  time = 0.009, size = 51, normalized size = 0.5 \[ -{\frac{128\,{b}^{3}{x}^{12}+32\,a{b}^{2}{x}^{8}+20\,{a}^{2}b{x}^{4}+15\,{a}^{3}}{195\,{x}^{13}{a}^{4}}\sqrt [4]{-b{x}^{4}+a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^14/(-b*x^4+a)^(3/4),x)

[Out]

-1/195*(-b*x^4+a)^(1/4)*(128*b^3*x^12+32*a*b^2*x^8+20*a^2*b*x^4+15*a^3)/x^13/a^4

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Maxima [A]  time = 1.44402, size = 99, normalized size = 1.03 \[ -\frac{\frac{195 \,{\left (-b x^{4} + a\right )}^{\frac{1}{4}} b^{3}}{x} + \frac{117 \,{\left (-b x^{4} + a\right )}^{\frac{5}{4}} b^{2}}{x^{5}} + \frac{65 \,{\left (-b x^{4} + a\right )}^{\frac{9}{4}} b}{x^{9}} + \frac{15 \,{\left (-b x^{4} + a\right )}^{\frac{13}{4}}}{x^{13}}}{195 \, a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-b*x^4 + a)^(3/4)*x^14),x, algorithm="maxima")

[Out]

-1/195*(195*(-b*x^4 + a)^(1/4)*b^3/x + 117*(-b*x^4 + a)^(5/4)*b^2/x^5 + 65*(-b*x
^4 + a)^(9/4)*b/x^9 + 15*(-b*x^4 + a)^(13/4)/x^13)/a^4

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Fricas [A]  time = 0.229411, size = 68, normalized size = 0.71 \[ -\frac{{\left (128 \, b^{3} x^{12} + 32 \, a b^{2} x^{8} + 20 \, a^{2} b x^{4} + 15 \, a^{3}\right )}{\left (-b x^{4} + a\right )}^{\frac{1}{4}}}{195 \, a^{4} x^{13}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-b*x^4 + a)^(3/4)*x^14),x, algorithm="fricas")

[Out]

-1/195*(128*b^3*x^12 + 32*a*b^2*x^8 + 20*a^2*b*x^4 + 15*a^3)*(-b*x^4 + a)^(1/4)/
(a^4*x^13)

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Sympy [A]  time = 23.7669, size = 1452, normalized size = 15.12 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**14/(-b*x**4+a)**(3/4),x)

[Out]

Piecewise((45*a**6*b**(37/4)*(a/(b*x**4) - 1)**(1/4)*gamma(-13/4)/(-256*a**7*b**
9*x**12*gamma(3/4) + 768*a**6*b**10*x**16*gamma(3/4) - 768*a**5*b**11*x**20*gamm
a(3/4) + 256*a**4*b**12*x**24*gamma(3/4)) - 75*a**5*b**(41/4)*x**4*(a/(b*x**4) -
1)**(1/4)*gamma(-13/4)/(-256*a**7*b**9*x**12*gamma(3/4) + 768*a**6*b**10*x**16*
gamma(3/4) - 768*a**5*b**11*x**20*gamma(3/4) + 256*a**4*b**12*x**24*gamma(3/4))
+ 51*a**4*b**(45/4)*x**8*(a/(b*x**4) - 1)**(1/4)*gamma(-13/4)/(-256*a**7*b**9*x*
*12*gamma(3/4) + 768*a**6*b**10*x**16*gamma(3/4) - 768*a**5*b**11*x**20*gamma(3/
4) + 256*a**4*b**12*x**24*gamma(3/4)) + 231*a**3*b**(49/4)*x**12*(a/(b*x**4) - 1
)**(1/4)*gamma(-13/4)/(-256*a**7*b**9*x**12*gamma(3/4) + 768*a**6*b**10*x**16*ga
mma(3/4) - 768*a**5*b**11*x**20*gamma(3/4) + 256*a**4*b**12*x**24*gamma(3/4)) -
924*a**2*b**(53/4)*x**16*(a/(b*x**4) - 1)**(1/4)*gamma(-13/4)/(-256*a**7*b**9*x*
*12*gamma(3/4) + 768*a**6*b**10*x**16*gamma(3/4) - 768*a**5*b**11*x**20*gamma(3/
4) + 256*a**4*b**12*x**24*gamma(3/4)) + 1056*a*b**(57/4)*x**20*(a/(b*x**4) - 1)*
*(1/4)*gamma(-13/4)/(-256*a**7*b**9*x**12*gamma(3/4) + 768*a**6*b**10*x**16*gamm
a(3/4) - 768*a**5*b**11*x**20*gamma(3/4) + 256*a**4*b**12*x**24*gamma(3/4)) - 38
4*b**(61/4)*x**24*(a/(b*x**4) - 1)**(1/4)*gamma(-13/4)/(-256*a**7*b**9*x**12*gam
ma(3/4) + 768*a**6*b**10*x**16*gamma(3/4) - 768*a**5*b**11*x**20*gamma(3/4) + 25
6*a**4*b**12*x**24*gamma(3/4)), Abs(a/(b*x**4)) > 1), (-45*a**6*b**(37/4)*(-a/(b
*x**4) + 1)**(1/4)*exp(13*I*pi/4)*gamma(-13/4)/(-256*a**7*b**9*x**12*gamma(3/4)
+ 768*a**6*b**10*x**16*gamma(3/4) - 768*a**5*b**11*x**20*gamma(3/4) + 256*a**4*b
**12*x**24*gamma(3/4)) + 75*a**5*b**(41/4)*x**4*(-a/(b*x**4) + 1)**(1/4)*exp(13*
I*pi/4)*gamma(-13/4)/(-256*a**7*b**9*x**12*gamma(3/4) + 768*a**6*b**10*x**16*gam
ma(3/4) - 768*a**5*b**11*x**20*gamma(3/4) + 256*a**4*b**12*x**24*gamma(3/4)) - 5
1*a**4*b**(45/4)*x**8*(-a/(b*x**4) + 1)**(1/4)*exp(13*I*pi/4)*gamma(-13/4)/(-256
*a**7*b**9*x**12*gamma(3/4) + 768*a**6*b**10*x**16*gamma(3/4) - 768*a**5*b**11*x
**20*gamma(3/4) + 256*a**4*b**12*x**24*gamma(3/4)) - 231*a**3*b**(49/4)*x**12*(-
a/(b*x**4) + 1)**(1/4)*exp(13*I*pi/4)*gamma(-13/4)/(-256*a**7*b**9*x**12*gamma(3
/4) + 768*a**6*b**10*x**16*gamma(3/4) - 768*a**5*b**11*x**20*gamma(3/4) + 256*a*
*4*b**12*x**24*gamma(3/4)) + 924*a**2*b**(53/4)*x**16*(-a/(b*x**4) + 1)**(1/4)*e
xp(13*I*pi/4)*gamma(-13/4)/(-256*a**7*b**9*x**12*gamma(3/4) + 768*a**6*b**10*x**
16*gamma(3/4) - 768*a**5*b**11*x**20*gamma(3/4) + 256*a**4*b**12*x**24*gamma(3/4
)) - 1056*a*b**(57/4)*x**20*(-a/(b*x**4) + 1)**(1/4)*exp(13*I*pi/4)*gamma(-13/4)
/(-256*a**7*b**9*x**12*gamma(3/4) + 768*a**6*b**10*x**16*gamma(3/4) - 768*a**5*b
**11*x**20*gamma(3/4) + 256*a**4*b**12*x**24*gamma(3/4)) + 384*b**(61/4)*x**24*(
-a/(b*x**4) + 1)**(1/4)*exp(13*I*pi/4)*gamma(-13/4)/(-256*a**7*b**9*x**12*gamma(
3/4) + 768*a**6*b**10*x**16*gamma(3/4) - 768*a**5*b**11*x**20*gamma(3/4) + 256*a
**4*b**12*x**24*gamma(3/4)), True))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (-b x^{4} + a\right )}^{\frac{3}{4}} x^{14}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-b*x^4 + a)^(3/4)*x^14),x, algorithm="giac")

[Out]

integrate(1/((-b*x^4 + a)^(3/4)*x^14), x)