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

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

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.00, antiderivative size = 13, normalized size of antiderivative = 1.00, 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 = 0.00, size = 13, normalized size = 1.00 \[ \frac {\sqrt {x^8+1}}{4} \]

Antiderivative was successfully verified.

[In]

Integrate[x^7/Sqrt[1 + x^8],x]

[Out]

Sqrt[1 + x^8]/4

________________________________________________________________________________________

fricas [A]  time = 0.86, size = 9, normalized size = 0.69 \[ \frac {1}{4} \, \sqrt {x^{8} + 1} \]

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)

________________________________________________________________________________________

giac [A]  time = 0.15, size = 9, normalized size = 0.69 \[ \frac {1}{4} \, \sqrt {x^{8} + 1} \]

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)

________________________________________________________________________________________

maple [A]  time = 0.00, size = 10, normalized size = 0.77 \[ \frac {\sqrt {x^{8}+1}}{4} \]

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)

________________________________________________________________________________________

maxima [A]  time = 0.97, size = 9, normalized size = 0.69 \[ \frac {1}{4} \, \sqrt {x^{8} + 1} \]

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)

________________________________________________________________________________________

mupad [B]  time = 0.09, size = 9, normalized size = 0.69 \[ \frac {\sqrt {x^8+1}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^7/(x^8 + 1)^(1/2),x)

[Out]

(x^8 + 1)^(1/2)/4

________________________________________________________________________________________

sympy [A]  time = 0.49, size = 8, normalized size = 0.62 \[ \frac {\sqrt {x^{8} + 1}}{4} \]

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

________________________________________________________________________________________