3.1204 \(\int \frac{x^{11}}{\sqrt [4]{a-b x^4}} \, dx\)

Optimal. Leaf size=62 \[ -\frac{a^2 \left (a-b x^4\right )^{3/4}}{3 b^3}-\frac{\left (a-b x^4\right )^{11/4}}{11 b^3}+\frac{2 a \left (a-b x^4\right )^{7/4}}{7 b^3} \]

[Out]

-(a^2*(a - b*x^4)^(3/4))/(3*b^3) + (2*a*(a - b*x^4)^(7/4))/(7*b^3) - (a - b*x^4)
^(11/4)/(11*b^3)

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Rubi [A]  time = 0.0892385, 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 \[ -\frac{a^2 \left (a-b x^4\right )^{3/4}}{3 b^3}-\frac{\left (a-b x^4\right )^{11/4}}{11 b^3}+\frac{2 a \left (a-b x^4\right )^{7/4}}{7 b^3} \]

Antiderivative was successfully verified.

[In]  Int[x^11/(a - b*x^4)^(1/4),x]

[Out]

-(a^2*(a - b*x^4)^(3/4))/(3*b^3) + (2*a*(a - b*x^4)^(7/4))/(7*b^3) - (a - b*x^4)
^(11/4)/(11*b^3)

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Rubi in Sympy [A]  time = 11.5577, size = 51, normalized size = 0.82 \[ - \frac{a^{2} \left (a - b x^{4}\right )^{\frac{3}{4}}}{3 b^{3}} + \frac{2 a \left (a - b x^{4}\right )^{\frac{7}{4}}}{7 b^{3}} - \frac{\left (a - b x^{4}\right )^{\frac{11}{4}}}{11 b^{3}} \]

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)**(3/4)/(3*b**3) + 2*a*(a - b*x**4)**(7/4)/(7*b**3) - (a - b*x
**4)**(11/4)/(11*b**3)

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Mathematica [A]  time = 0.0314233, size = 40, normalized size = 0.65 \[ -\frac{\left (a-b x^4\right )^{3/4} \left (32 a^2+24 a b x^4+21 b^2 x^8\right )}{231 b^3} \]

Antiderivative was successfully verified.

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

[Out]

-((a - b*x^4)^(3/4)*(32*a^2 + 24*a*b*x^4 + 21*b^2*x^8))/(231*b^3)

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Maple [A]  time = 0.009, size = 37, normalized size = 0.6 \[ -{\frac{21\,{b}^{2}{x}^{8}+24\,ab{x}^{4}+32\,{a}^{2}}{231\,{b}^{3}} \left ( -b{x}^{4}+a \right ) ^{{\frac{3}{4}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^11/(-b*x^4+a)^(1/4),x)

[Out]

-1/231*(-b*x^4+a)^(3/4)*(21*b^2*x^8+24*a*b*x^4+32*a^2)/b^3

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Maxima [A]  time = 1.43841, size = 68, normalized size = 1.1 \[ -\frac{{\left (-b x^{4} + a\right )}^{\frac{11}{4}}}{11 \, b^{3}} + \frac{2 \,{\left (-b x^{4} + a\right )}^{\frac{7}{4}} a}{7 \, b^{3}} - \frac{{\left (-b x^{4} + a\right )}^{\frac{3}{4}} a^{2}}{3 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/11*(-b*x^4 + a)^(11/4)/b^3 + 2/7*(-b*x^4 + a)^(7/4)*a/b^3 - 1/3*(-b*x^4 + a)^
(3/4)*a^2/b^3

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Fricas [A]  time = 0.230098, size = 49, normalized size = 0.79 \[ -\frac{{\left (21 \, b^{2} x^{8} + 24 \, a b x^{4} + 32 \, a^{2}\right )}{\left (-b x^{4} + a\right )}^{\frac{3}{4}}}{231 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/231*(21*b^2*x^8 + 24*a*b*x^4 + 32*a^2)*(-b*x^4 + a)^(3/4)/b^3

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Sympy [A]  time = 11.0313, size = 70, normalized size = 1.13 \[ \begin{cases} - \frac{32 a^{2} \left (a - b x^{4}\right )^{\frac{3}{4}}}{231 b^{3}} - \frac{8 a x^{4} \left (a - b x^{4}\right )^{\frac{3}{4}}}{77 b^{2}} - \frac{x^{8} \left (a - b x^{4}\right )^{\frac{3}{4}}}{11 b} & \text{for}\: b \neq 0 \\\frac{x^{12}}{12 \sqrt [4]{a}} & \text{otherwise} \end{cases} \]

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**2*(a - b*x**4)**(3/4)/(231*b**3) - 8*a*x**4*(a - b*x**4)**(3/4
)/(77*b**2) - x**8*(a - b*x**4)**(3/4)/(11*b), Ne(b, 0)), (x**12/(12*a**(1/4)),
True))

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GIAC/XCAS [A]  time = 0.216756, size = 77, normalized size = 1.24 \[ -\frac{21 \,{\left (b x^{4} - a\right )}^{2}{\left (-b x^{4} + a\right )}^{\frac{3}{4}} - 66 \,{\left (-b x^{4} + a\right )}^{\frac{7}{4}} a + 77 \,{\left (-b x^{4} + a\right )}^{\frac{3}{4}} a^{2}}{231 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/231*(21*(b*x^4 - a)^2*(-b*x^4 + a)^(3/4) - 66*(-b*x^4 + a)^(7/4)*a + 77*(-b*x
^4 + a)^(3/4)*a^2)/b^3