3.1175 \(\int x^{11} \sqrt [4]{a-b x^4} \, dx\)

Optimal. Leaf size=62 \[ -\frac{a^2 \left (a-b x^4\right )^{5/4}}{5 b^3}-\frac{\left (a-b x^4\right )^{13/4}}{13 b^3}+\frac{2 a \left (a-b x^4\right )^{9/4}}{9 b^3} \]

[Out]

-(a^2*(a - b*x^4)^(5/4))/(5*b^3) + (2*a*(a - b*x^4)^(9/4))/(9*b^3) - (a - b*x^4)
^(13/4)/(13*b^3)

_______________________________________________________________________________________

Rubi [A]  time = 0.0872046, antiderivative size = 62, 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 \[ -\frac{a^2 \left (a-b x^4\right )^{5/4}}{5 b^3}-\frac{\left (a-b x^4\right )^{13/4}}{13 b^3}+\frac{2 a \left (a-b x^4\right )^{9/4}}{9 b^3} \]

Antiderivative was successfully verified.

[In]  Int[x^11*(a - b*x^4)^(1/4),x]

[Out]

-(a^2*(a - b*x^4)^(5/4))/(5*b^3) + (2*a*(a - b*x^4)^(9/4))/(9*b^3) - (a - b*x^4)
^(13/4)/(13*b^3)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 11.682, size = 51, normalized size = 0.82 \[ - \frac{a^{2} \left (a - b x^{4}\right )^{\frac{5}{4}}}{5 b^{3}} + \frac{2 a \left (a - b x^{4}\right )^{\frac{9}{4}}}{9 b^{3}} - \frac{\left (a - b x^{4}\right )^{\frac{13}{4}}}{13 b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**11*(-b*x**4+a)**(1/4),x)

[Out]

-a**2*(a - b*x**4)**(5/4)/(5*b**3) + 2*a*(a - b*x**4)**(9/4)/(9*b**3) - (a - b*x
**4)**(13/4)/(13*b**3)

_______________________________________________________________________________________

Mathematica [A]  time = 0.028642, size = 51, normalized size = 0.82 \[ -\frac{\sqrt [4]{a-b x^4} \left (32 a^3+8 a^2 b x^4+5 a b^2 x^8-45 b^3 x^{12}\right )}{585 b^3} \]

Antiderivative was successfully verified.

[In]  Integrate[x^11*(a - b*x^4)^(1/4),x]

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.008, size = 37, normalized size = 0.6 \[ -{\frac{45\,{b}^{2}{x}^{8}+40\,ab{x}^{4}+32\,{a}^{2}}{585\,{b}^{3}} \left ( -b{x}^{4}+a \right ) ^{{\frac{5}{4}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^11*(-b*x^4+a)^(1/4),x)

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 1.43782, size = 68, normalized size = 1.1 \[ -\frac{{\left (-b x^{4} + a\right )}^{\frac{13}{4}}}{13 \, b^{3}} + \frac{2 \,{\left (-b x^{4} + a\right )}^{\frac{9}{4}} a}{9 \, b^{3}} - \frac{{\left (-b x^{4} + a\right )}^{\frac{5}{4}} a^{2}}{5 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-b*x^4 + a)^(1/4)*x^11,x, algorithm="maxima")

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.248306, size = 63, normalized size = 1.02 \[ \frac{{\left (45 \, b^{3} x^{12} - 5 \, a b^{2} x^{8} - 8 \, a^{2} b x^{4} - 32 \, a^{3}\right )}{\left (-b x^{4} + a\right )}^{\frac{1}{4}}}{585 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-b*x^4 + a)^(1/4)*x^11,x, algorithm="fricas")

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 15.2017, size = 87, normalized size = 1.4 \[ \begin{cases} - \frac{32 a^{3} \sqrt [4]{a - b x^{4}}}{585 b^{3}} - \frac{8 a^{2} x^{4} \sqrt [4]{a - b x^{4}}}{585 b^{2}} - \frac{a x^{8} \sqrt [4]{a - b x^{4}}}{117 b} + \frac{x^{12} \sqrt [4]{a - b x^{4}}}{13} & \text{for}\: b \neq 0 \\\frac{\sqrt [4]{a} x^{12}}{12} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**11*(-b*x**4+a)**(1/4),x)

[Out]

Piecewise((-32*a**3*(a - b*x**4)**(1/4)/(585*b**3) - 8*a**2*x**4*(a - b*x**4)**(
1/4)/(585*b**2) - a*x**8*(a - b*x**4)**(1/4)/(117*b) + x**12*(a - b*x**4)**(1/4)
/13, Ne(b, 0)), (a**(1/4)*x**12/12, True))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.252277, size = 92, normalized size = 1.48 \[ \frac{45 \,{\left (b x^{4} - a\right )}^{3}{\left (-b x^{4} + a\right )}^{\frac{1}{4}} + 130 \,{\left (b x^{4} - a\right )}^{2}{\left (-b x^{4} + a\right )}^{\frac{1}{4}} a - 117 \,{\left (-b x^{4} + a\right )}^{\frac{5}{4}} a^{2}}{585 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-b*x^4 + a)^(1/4)*x^11,x, algorithm="giac")

[Out]

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