3.809 \(\int \frac{x^{11}}{\sqrt{a+b x^4}} \, dx\)

Optimal. Leaf size=59 \[ \frac{a^2 \sqrt{a+b x^4}}{2 b^3}+\frac{\left (a+b x^4\right )^{5/2}}{10 b^3}-\frac{a \left (a+b x^4\right )^{3/2}}{3 b^3} \]

[Out]

(a^2*Sqrt[a + b*x^4])/(2*b^3) - (a*(a + b*x^4)^(3/2))/(3*b^3) + (a + b*x^4)^(5/2
)/(10*b^3)

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Rubi [A]  time = 0.0889194, antiderivative size = 59, 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 \[ \frac{a^2 \sqrt{a+b x^4}}{2 b^3}+\frac{\left (a+b x^4\right )^{5/2}}{10 b^3}-\frac{a \left (a+b x^4\right )^{3/2}}{3 b^3} \]

Antiderivative was successfully verified.

[In]  Int[x^11/Sqrt[a + b*x^4],x]

[Out]

(a^2*Sqrt[a + b*x^4])/(2*b^3) - (a*(a + b*x^4)^(3/2))/(3*b^3) + (a + b*x^4)^(5/2
)/(10*b^3)

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Rubi in Sympy [A]  time = 10.6649, size = 49, normalized size = 0.83 \[ \frac{a^{2} \sqrt{a + b x^{4}}}{2 b^{3}} - \frac{a \left (a + b x^{4}\right )^{\frac{3}{2}}}{3 b^{3}} + \frac{\left (a + b x^{4}\right )^{\frac{5}{2}}}{10 b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**11/(b*x**4+a)**(1/2),x)

[Out]

a**2*sqrt(a + b*x**4)/(2*b**3) - a*(a + b*x**4)**(3/2)/(3*b**3) + (a + b*x**4)**
(5/2)/(10*b**3)

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Mathematica [A]  time = 0.0288769, size = 39, normalized size = 0.66 \[ \frac{\sqrt{a+b x^4} \left (8 a^2-4 a b x^4+3 b^2 x^8\right )}{30 b^3} \]

Antiderivative was successfully verified.

[In]  Integrate[x^11/Sqrt[a + b*x^4],x]

[Out]

(Sqrt[a + b*x^4]*(8*a^2 - 4*a*b*x^4 + 3*b^2*x^8))/(30*b^3)

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Maple [A]  time = 0.009, size = 36, normalized size = 0.6 \[{\frac{3\,{b}^{2}{x}^{8}-4\,ab{x}^{4}+8\,{a}^{2}}{30\,{b}^{3}}\sqrt{b{x}^{4}+a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^11/(b*x^4+a)^(1/2),x)

[Out]

1/30*(b*x^4+a)^(1/2)*(3*b^2*x^8-4*a*b*x^4+8*a^2)/b^3

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Maxima [A]  time = 1.4227, size = 63, normalized size = 1.07 \[ \frac{{\left (b x^{4} + a\right )}^{\frac{5}{2}}}{10 \, b^{3}} - \frac{{\left (b x^{4} + a\right )}^{\frac{3}{2}} a}{3 \, b^{3}} + \frac{\sqrt{b x^{4} + a} a^{2}}{2 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/10*(b*x^4 + a)^(5/2)/b^3 - 1/3*(b*x^4 + a)^(3/2)*a/b^3 + 1/2*sqrt(b*x^4 + a)*a
^2/b^3

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Fricas [A]  time = 0.247733, size = 47, normalized size = 0.8 \[ \frac{{\left (3 \, b^{2} x^{8} - 4 \, a b x^{4} + 8 \, a^{2}\right )} \sqrt{b x^{4} + a}}{30 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/30*(3*b^2*x^8 - 4*a*b*x^4 + 8*a^2)*sqrt(b*x^4 + a)/b^3

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Sympy [A]  time = 8.9373, size = 68, normalized size = 1.15 \[ \begin{cases} \frac{4 a^{2} \sqrt{a + b x^{4}}}{15 b^{3}} - \frac{2 a x^{4} \sqrt{a + b x^{4}}}{15 b^{2}} + \frac{x^{8} \sqrt{a + b x^{4}}}{10 b} & \text{for}\: b \neq 0 \\\frac{x^{12}}{12 \sqrt{a}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**11/(b*x**4+a)**(1/2),x)

[Out]

Piecewise((4*a**2*sqrt(a + b*x**4)/(15*b**3) - 2*a*x**4*sqrt(a + b*x**4)/(15*b**
2) + x**8*sqrt(a + b*x**4)/(10*b), Ne(b, 0)), (x**12/(12*sqrt(a)), True))

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GIAC/XCAS [A]  time = 0.215425, size = 58, normalized size = 0.98 \[ \frac{3 \,{\left (b x^{4} + a\right )}^{\frac{5}{2}} - 10 \,{\left (b x^{4} + a\right )}^{\frac{3}{2}} a + 15 \, \sqrt{b x^{4} + a} a^{2}}{30 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/30*(3*(b*x^4 + a)^(5/2) - 10*(b*x^4 + a)^(3/2)*a + 15*sqrt(b*x^4 + a)*a^2)/b^3