\(\int \frac {x^7}{(1+x^4)^{3/2}} \, dx\) [938]

Optimal result

Integrand size = 13, antiderivative size = 27 \[ \int \frac {x^7}{\left (1+x^4\right )^{3/2}} \, dx=\frac {1}{2 \sqrt {1+x^4}}+\frac {\sqrt {1+x^4}}{2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {272, 45} \[ \int \frac {x^7}{\left (1+x^4\right )^{3/2}} \, dx=\frac {\sqrt {x^4+1}}{2}+\frac {1}{2 \sqrt {x^4+1}} \]

[In]

Int[x^7/(1 + x^4)^(3/2),x]

[Out]

1/(2*Sqrt[1 + x^4]) + Sqrt[1 + x^4]/2

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 272

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

Rubi steps \begin{align*} \text {integral}& = \frac {1}{4} \text {Subst}\left (\int \frac {x}{(1+x)^{3/2}} \, dx,x,x^4\right ) \\ & = \frac {1}{4} \text {Subst}\left (\int \left (-\frac {1}{(1+x)^{3/2}}+\frac {1}{\sqrt {1+x}}\right ) \, dx,x,x^4\right ) \\ & = \frac {1}{2 \sqrt {1+x^4}}+\frac {\sqrt {1+x^4}}{2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.67 \[ \int \frac {x^7}{\left (1+x^4\right )^{3/2}} \, dx=\frac {2+x^4}{2 \sqrt {1+x^4}} \]

[In]

Integrate[x^7/(1 + x^4)^(3/2),x]

[Out]

(2 + x^4)/(2*Sqrt[1 + x^4])

Maple [A] (verified)

Time = 4.16 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.56

method	result	size
gosper	\(\frac {x^{4}+2}{2 \sqrt {x^{4}+1}}\)	\(15\)
default	\(\frac {x^{4}+2}{2 \sqrt {x^{4}+1}}\)	\(15\)
trager	\(\frac {x^{4}+2}{2 \sqrt {x^{4}+1}}\)	\(15\)
risch	\(\frac {x^{4}+2}{2 \sqrt {x^{4}+1}}\)	\(15\)
elliptic	\(\frac {x^{4}+2}{2 \sqrt {x^{4}+1}}\)	\(15\)
pseudoelliptic	\(\frac {x^{4}+2}{2 \sqrt {x^{4}+1}}\)	\(15\)
meijerg	\(\frac {-2 \sqrt {\pi }+\frac {\sqrt {\pi }\, \left (4 x^{4}+8\right )}{4 \sqrt {x^{4}+1}}}{2 \sqrt {\pi }}\)	\(31\)

[In]

int(x^7/(x^4+1)^(3/2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.52 \[ \int \frac {x^7}{\left (1+x^4\right )^{3/2}} \, dx=\frac {x^{4} + 2}{2 \, \sqrt {x^{4} + 1}} \]

[In]

integrate(x^7/(x^4+1)^(3/2),x, algorithm="fricas")

[Out]

1/2*(x^4 + 2)/sqrt(x^4 + 1)

Sympy [A] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.81 \[ \int \frac {x^7}{\left (1+x^4\right )^{3/2}} \, dx=\frac {x^{4}}{2 \sqrt {x^{4} + 1}} + \frac {1}{\sqrt {x^{4} + 1}} \]

[In]

integrate(x**7/(x**4+1)**(3/2),x)

[Out]

x**4/(2*sqrt(x**4 + 1)) + 1/sqrt(x**4 + 1)

Maxima [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.70 \[ \int \frac {x^7}{\left (1+x^4\right )^{3/2}} \, dx=\frac {1}{2} \, \sqrt {x^{4} + 1} + \frac {1}{2 \, \sqrt {x^{4} + 1}} \]

[In]

integrate(x^7/(x^4+1)^(3/2),x, algorithm="maxima")

[Out]

1/2*sqrt(x^4 + 1) + 1/2/sqrt(x^4 + 1)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.70 \[ \int \frac {x^7}{\left (1+x^4\right )^{3/2}} \, dx=\frac {1}{2} \, \sqrt {x^{4} + 1} + \frac {1}{2 \, \sqrt {x^{4} + 1}} \]

[In]

integrate(x^7/(x^4+1)^(3/2),x, algorithm="giac")

[Out]

1/2*sqrt(x^4 + 1) + 1/2/sqrt(x^4 + 1)

Mupad [B] (verification not implemented)

Time = 5.81 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.52 \[ \int \frac {x^7}{\left (1+x^4\right )^{3/2}} \, dx=\frac {x^4+2}{2\,\sqrt {x^4+1}} \]

[In]

int(x^7/(x^4 + 1)^(3/2),x)

[Out]

(x^4 + 2)/(2*(x^4 + 1)^(1/2))