\(\int \frac {x^7}{\sqrt {1+x^8}} \, dx\) [1517]

Optimal result

Integrand size = 13, antiderivative size = 13 \[ \int \frac {x^7}{\sqrt {1+x^8}} \, dx=\frac {\sqrt {1+x^8}}{4} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {267} \[ \int \frac {x^7}{\sqrt {1+x^8}} \, dx=\frac {\sqrt {x^8+1}}{4} \]

[In]

Int[x^7/Sqrt[1 + x^8],x]

[Out]

Sqrt[1 + x^8]/4

Rule 267

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*} \text {integral}& = \frac {\sqrt {1+x^8}}{4} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int \frac {x^7}{\sqrt {1+x^8}} \, dx=\frac {\sqrt {1+x^8}}{4} \]

[In]

Integrate[x^7/Sqrt[1 + x^8],x]

[Out]

Sqrt[1 + x^8]/4

Maple [A] (verified)

Time = 3.37 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.77

method	result	size
gosper	\(\frac {\sqrt {x^{8}+1}}{4}\)	\(10\)
derivativedivides	\(\frac {\sqrt {x^{8}+1}}{4}\)	\(10\)
default	\(\frac {\sqrt {x^{8}+1}}{4}\)	\(10\)
trager	\(\frac {\sqrt {x^{8}+1}}{4}\)	\(10\)
risch	\(\frac {\sqrt {x^{8}+1}}{4}\)	\(10\)
pseudoelliptic	\(\frac {\sqrt {x^{8}+1}}{4}\)	\(10\)
meijerg	\(\frac {-2 \sqrt {\pi }+2 \sqrt {\pi }\, \sqrt {x^{8}+1}}{8 \sqrt {\pi }}\)	\(24\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.69 \[ \int \frac {x^7}{\sqrt {1+x^8}} \, dx=\frac {1}{4} \, \sqrt {x^{8} + 1} \]

[In]

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

[Out]

1/4*sqrt(x^8 + 1)

Sympy [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.62 \[ \int \frac {x^7}{\sqrt {1+x^8}} \, dx=\frac {\sqrt {x^{8} + 1}}{4} \]

[In]

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

[Out]

sqrt(x**8 + 1)/4

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.69 \[ \int \frac {x^7}{\sqrt {1+x^8}} \, dx=\frac {1}{4} \, \sqrt {x^{8} + 1} \]

[In]

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

[Out]

1/4*sqrt(x^8 + 1)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.69 \[ \int \frac {x^7}{\sqrt {1+x^8}} \, dx=\frac {1}{4} \, \sqrt {x^{8} + 1} \]

[In]

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

[Out]

1/4*sqrt(x^8 + 1)

Mupad [B] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.69 \[ \int \frac {x^7}{\sqrt {1+x^8}} \, dx=\frac {\sqrt {x^8+1}}{4} \]

[In]

int(x^7/(x^8 + 1)^(1/2),x)

[Out]

(x^8 + 1)^(1/2)/4