3.1513 \(\int \frac{x^{15}}{\sqrt{1+x^8}} \, dx\)

Optimal. Leaf size=27 \[ \frac{1}{12} \left (x^8+1\right )^{3/2}-\frac{\sqrt{x^8+1}}{4} \]

[Out]

-Sqrt[1 + x^8]/4 + (1 + x^8)^(3/2)/12

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Rubi [A]  time = 0.0352397, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{1}{12} \left (x^8+1\right )^{3/2}-\frac{\sqrt{x^8+1}}{4} \]

Antiderivative was successfully verified.

[In]  Int[x^15/Sqrt[1 + x^8],x]

[Out]

-Sqrt[1 + x^8]/4 + (1 + x^8)^(3/2)/12

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Rubi in Sympy [A]  time = 3.55767, size = 19, normalized size = 0.7 \[ \frac{\left (x^{8} + 1\right )^{\frac{3}{2}}}{12} - \frac{\sqrt{x^{8} + 1}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**15/(x**8+1)**(1/2),x)

[Out]

(x**8 + 1)**(3/2)/12 - sqrt(x**8 + 1)/4

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Mathematica [A]  time = 0.0094315, size = 18, normalized size = 0.67 \[ \frac{1}{12} \left (x^8-2\right ) \sqrt{x^8+1} \]

Antiderivative was successfully verified.

[In]  Integrate[x^15/Sqrt[1 + x^8],x]

[Out]

((-2 + x^8)*Sqrt[1 + x^8])/12

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Maple [A]  time = 0.007, size = 15, normalized size = 0.6 \[{\frac{{x}^{8}-2}{12}\sqrt{{x}^{8}+1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^15/(x^8+1)^(1/2),x)

[Out]

1/12*(x^8+1)^(1/2)*(x^8-2)

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Maxima [A]  time = 1.4369, size = 26, normalized size = 0.96 \[ \frac{1}{12} \,{\left (x^{8} + 1\right )}^{\frac{3}{2}} - \frac{1}{4} \, \sqrt{x^{8} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^15/sqrt(x^8 + 1),x, algorithm="maxima")

[Out]

1/12*(x^8 + 1)^(3/2) - 1/4*sqrt(x^8 + 1)

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Fricas [A]  time = 0.221203, size = 19, normalized size = 0.7 \[ \frac{1}{12} \, \sqrt{x^{8} + 1}{\left (x^{8} - 2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^15/sqrt(x^8 + 1),x, algorithm="fricas")

[Out]

1/12*sqrt(x^8 + 1)*(x^8 - 2)

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Sympy [A]  time = 10.2816, size = 22, normalized size = 0.81 \[ \frac{x^{8} \sqrt{x^{8} + 1}}{12} - \frac{\sqrt{x^{8} + 1}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**15/(x**8+1)**(1/2),x)

[Out]

x**8*sqrt(x**8 + 1)/12 - sqrt(x**8 + 1)/6

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GIAC/XCAS [A]  time = 0.223406, size = 26, normalized size = 0.96 \[ \frac{1}{12} \,{\left (x^{8} + 1\right )}^{\frac{3}{2}} - \frac{1}{4} \, \sqrt{x^{8} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^15/sqrt(x^8 + 1),x, algorithm="giac")

[Out]

1/12*(x^8 + 1)^(3/2) - 1/4*sqrt(x^8 + 1)