3.1517 \(\int \frac{x^7}{\sqrt{1+x^8}} \, dx\)

Optimal. Leaf size=13 \[ \frac{\sqrt{x^8+1}}{4} \]

[Out]

Sqrt[1 + x^8]/4

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Rubi [A]  time = 0.0025551, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {261} \[ \frac{\sqrt{x^8+1}}{4} \]

Antiderivative was successfully verified.

[In]

Int[x^7/Sqrt[1 + x^8],x]

[Out]

Sqrt[1 + x^8]/4

Rule 261

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rubi steps

\begin{align*} \int \frac{x^7}{\sqrt{1+x^8}} \, dx &=\frac{\sqrt{1+x^8}}{4}\\ \end{align*}

Mathematica [A]  time = 0.001798, size = 13, normalized size = 1. \[ \frac{\sqrt{x^8+1}}{4} \]

Antiderivative was successfully verified.

[In]

Integrate[x^7/Sqrt[1 + x^8],x]

[Out]

Sqrt[1 + x^8]/4

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Maple [A]  time = 0.003, size = 10, normalized size = 0.8 \begin{align*}{\frac{1}{4}\sqrt{{x}^{8}+1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^7/(x^8+1)^(1/2),x)

[Out]

1/4*(x^8+1)^(1/2)

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Maxima [A]  time = 0.955257, size = 12, normalized size = 0.92 \begin{align*} \frac{1}{4} \, \sqrt{x^{8} + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7/(x^8+1)^(1/2),x, algorithm="maxima")

[Out]

1/4*sqrt(x^8 + 1)

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Fricas [A]  time = 1.24983, size = 26, normalized size = 2. \begin{align*} \frac{1}{4} \, \sqrt{x^{8} + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7/(x^8+1)^(1/2),x, algorithm="fricas")

[Out]

1/4*sqrt(x^8 + 1)

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Sympy [A]  time = 0.417179, size = 8, normalized size = 0.62 \begin{align*} \frac{\sqrt{x^{8} + 1}}{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**7/(x**8+1)**(1/2),x)

[Out]

sqrt(x**8 + 1)/4

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Giac [A]  time = 1.14609, size = 12, normalized size = 0.92 \begin{align*} \frac{1}{4} \, \sqrt{x^{8} + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7/(x^8+1)^(1/2),x, algorithm="giac")

[Out]

1/4*sqrt(x^8 + 1)