3.1234 \(\int \frac{x^{11}}{\left (a-b x^4\right )^{3/4}} \, dx\)

Optimal. Leaf size=60 \[ -\frac{a^2 \sqrt [4]{a-b x^4}}{b^3}-\frac{\left (a-b x^4\right )^{9/4}}{9 b^3}+\frac{2 a \left (a-b x^4\right )^{5/4}}{5 b^3} \]

[Out]

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

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Rubi [A]  time = 0.0909222, antiderivative size = 60, 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 \[ -\frac{a^2 \sqrt [4]{a-b x^4}}{b^3}-\frac{\left (a-b x^4\right )^{9/4}}{9 b^3}+\frac{2 a \left (a-b x^4\right )^{5/4}}{5 b^3} \]

Antiderivative was successfully verified.

[In]  Int[x^11/(a - b*x^4)^(3/4),x]

[Out]

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

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Rubi in Sympy [A]  time = 11.9874, size = 49, normalized size = 0.82 \[ - \frac{a^{2} \sqrt [4]{a - b x^{4}}}{b^{3}} + \frac{2 a \left (a - b x^{4}\right )^{\frac{5}{4}}}{5 b^{3}} - \frac{\left (a - b x^{4}\right )^{\frac{9}{4}}}{9 b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**11/(-b*x**4+a)**(3/4),x)

[Out]

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

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Mathematica [A]  time = 0.0298941, size = 40, normalized size = 0.67 \[ -\frac{\sqrt [4]{a-b x^4} \left (32 a^2+8 a b x^4+5 b^2 x^8\right )}{45 b^3} \]

Antiderivative was successfully verified.

[In]  Integrate[x^11/(a - b*x^4)^(3/4),x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^11/(-b*x^4+a)^(3/4),x)

[Out]

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

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Maxima [A]  time = 1.4394, size = 68, normalized size = 1.13 \[ -\frac{{\left (-b x^{4} + a\right )}^{\frac{9}{4}}}{9 \, b^{3}} + \frac{2 \,{\left (-b x^{4} + a\right )}^{\frac{5}{4}} a}{5 \, b^{3}} - \frac{{\left (-b x^{4} + a\right )}^{\frac{1}{4}} a^{2}}{b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^11/(-b*x^4 + a)^(3/4),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.220632, size = 49, normalized size = 0.82 \[ -\frac{{\left (5 \, b^{2} x^{8} + 8 \, a b x^{4} + 32 \, a^{2}\right )}{\left (-b x^{4} + a\right )}^{\frac{1}{4}}}{45 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^11/(-b*x^4 + a)^(3/4),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 10.2231, size = 70, normalized size = 1.17 \[ \begin{cases} - \frac{32 a^{2} \sqrt [4]{a - b x^{4}}}{45 b^{3}} - \frac{8 a x^{4} \sqrt [4]{a - b x^{4}}}{45 b^{2}} - \frac{x^{8} \sqrt [4]{a - b x^{4}}}{9 b} & \text{for}\: b \neq 0 \\\frac{x^{12}}{12 a^{\frac{3}{4}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**11/(-b*x**4+a)**(3/4),x)

[Out]

Piecewise((-32*a**2*(a - b*x**4)**(1/4)/(45*b**3) - 8*a*x**4*(a - b*x**4)**(1/4)
/(45*b**2) - x**8*(a - b*x**4)**(1/4)/(9*b), Ne(b, 0)), (x**12/(12*a**(3/4)), Tr
ue))

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GIAC/XCAS [A]  time = 0.215399, size = 77, normalized size = 1.28 \[ -\frac{5 \,{\left (b x^{4} - a\right )}^{2}{\left (-b x^{4} + a\right )}^{\frac{1}{4}} - 18 \,{\left (-b x^{4} + a\right )}^{\frac{5}{4}} a + 45 \,{\left (-b x^{4} + a\right )}^{\frac{1}{4}} a^{2}}{45 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^11/(-b*x^4 + a)^(3/4),x, algorithm="giac")

[Out]

-1/45*(5*(b*x^4 - a)^2*(-b*x^4 + a)^(1/4) - 18*(-b*x^4 + a)^(5/4)*a + 45*(-b*x^4
+ a)^(1/4)*a^2)/b^3