3.1110 \(\int \frac{x^{11}}{\left (a+b x^4\right )^{3/4}} \, dx\)

Optimal. Leaf size=56 \[ \frac{a^2 \sqrt [4]{a+b x^4}}{b^3}+\frac{\left (a+b x^4\right )^{9/4}}{9 b^3}-\frac{2 a \left (a+b x^4\right )^{5/4}}{5 b^3} \]

[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 = 0.0852534, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{a^2 \sqrt [4]{a+b x^4}}{b^3}+\frac{\left (a+b x^4\right )^{9/4}}{9 b^3}-\frac{2 a \left (a+b x^4\right )^{5/4}}{5 b^3} \]

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 = 10.5072, size = 49, normalized size = 0.88 \[ \frac{a^{2} \sqrt [4]{a + b x^{4}}}{b^{3}} - \frac{2 a \left (a + b x^{4}\right )^{\frac{5}{4}}}{5 b^{3}} + \frac{\left (a + b x^{4}\right )^{\frac{9}{4}}}{9 b^{3}} \]

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 = 0.0268645, size = 39, normalized size = 0.7 \[ \frac{\sqrt [4]{a+b x^4} \left (32 a^2-8 a b x^4+5 b^2 x^8\right )}{45 b^3} \]

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 = 36, 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 = 1.41126, size = 62, normalized size = 1.11 \[ \frac{{\left (b x^{4} + a\right )}^{\frac{9}{4}}}{9 \, b^{3}} - \frac{2 \,{\left (b x^{4} + a\right )}^{\frac{5}{4}} a}{5 \, b^{3}} + \frac{{\left (b x^{4} + a\right )}^{\frac{1}{4}} a^{2}}{b^{3}} \]

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 = 0.228284, size = 47, normalized size = 0.84 \[ \frac{{\left (5 \, b^{2} x^{8} - 8 \, a b x^{4} + 32 \, a^{2}\right )}{\left (b x^{4} + a\right )}^{\frac{1}{4}}}{45 \, b^{3}} \]

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 = 9.98532, size = 68, normalized size = 1.21 \[ \begin{cases} \frac{32 a^{2} \sqrt [4]{a + b x^{4}}}{45 b^{3}} - \frac{8 a x^{4} \sqrt [4]{a + b x^{4}}}{45 b^{2}} + \frac{x^{8} \sqrt [4]{a + b x^{4}}}{9 b} & \text{for}\: b \neq 0 \\\frac{x^{12}}{12 a^{\frac{3}{4}}} & \text{otherwise} \end{cases} \]

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)), Tru
e))

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GIAC/XCAS [A]  time = 0.214636, size = 58, normalized size = 1.04 \[ \frac{5 \,{\left (b x^{4} + a\right )}^{\frac{9}{4}} - 18 \,{\left (b x^{4} + a\right )}^{\frac{5}{4}} a + 45 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}} a^{2}}{45 \, b^{3}} \]

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)^(9/4) - 18*(b*x^4 + a)^(5/4)*a + 45*(b*x^4 + a)^(1/4)*a^2)/b
^3