\(\int \frac {(\frac {2+x^2}{x^2})^{7/9}}{(2+x^2)^{3/2}} \, dx\) [473]

Optimal result

Integrand size = 23, antiderivative size = 25 \[ \int \frac {\left (\frac {2+x^2}{x^2}\right )^{7/9}}{\left (2+x^2\right )^{3/2}} \, dx=-\frac {9 \left (1+\frac {2}{x^2}\right )^{7/9} x}{10 \sqrt {2+x^2}} \]

[Out]

-9/10*(1+2/x^2)^(7/9)*x/(x^2+2)^(1/2)

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {2016, 446, 270} \[ \int \frac {\left (\frac {2+x^2}{x^2}\right )^{7/9}}{\left (2+x^2\right )^{3/2}} \, dx=-\frac {9 \left (\frac {2}{x^2}+1\right )^{7/9} x}{10 \sqrt {x^2+2}} \]

[In]

Int[((2 + x^2)/x^2)^(7/9)/(2 + x^2)^(3/2),x]

[Out]

(-9*(1 + 2/x^2)^(7/9)*x)/(10*Sqrt[2 + x^2])

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*
c*(m + 1))), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rule 446

Int[((c_) + (d_.)*(x_)^(mn_.))^(q_)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Dist[x^(n*FracPart[q])*((c +
d/x^n)^FracPart[q]/(d + c*x^n)^FracPart[q]), Int[(a + b*x^n)^p*((d + c*x^n)^q/x^(n*q)), x], x] /; FreeQ[{a, b,
c, d, n, p, q}, x] && EqQ[mn, -n] &&  !IntegerQ[q] &&  !IntegerQ[p]

Rule 2016

Int[(u_)^(q_.)*(v_)^(p_.), x_Symbol] :> Int[ExpandToSum[u, x]^q*ExpandToSum[v, x]^p, x] /; FreeQ[{p, q}, x] &&
BinomialQ[u, x] && BinomialQ[v, x] &&  !(BinomialMatchQ[u, x] && BinomialMatchQ[v, x])

Rubi steps \begin{align*} \text {integral}& = \int \frac {\left (1+\frac {2}{x^2}\right )^{7/9}}{\left (2+x^2\right )^{3/2}} \, dx \\ & = \frac {\left (\left (1+\frac {2}{x^2}\right )^{7/9} x^{14/9}\right ) \int \frac {1}{x^{14/9} \left (2+x^2\right )^{13/18}} \, dx}{\left (2+x^2\right )^{7/9}} \\ & = -\frac {9 \left (1+\frac {2}{x^2}\right )^{7/9} x}{10 \sqrt {2+x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 6.49 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {\left (\frac {2+x^2}{x^2}\right )^{7/9}}{\left (2+x^2\right )^{3/2}} \, dx=-\frac {9 \left (1+\frac {2}{x^2}\right )^{7/9} x}{10 \sqrt {2+x^2}} \]

[In]

Integrate[((2 + x^2)/x^2)^(7/9)/(2 + x^2)^(3/2),x]

[Out]

(-9*(1 + 2/x^2)^(7/9)*x)/(10*Sqrt[2 + x^2])

Maple [A] (verified)

Time = 0.26 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.88

method	result	size
gosper	\(-\frac {9 x \left (\frac {x^{2}+2}{x^{2}}\right )^{\frac {7}{9}}}{10 \sqrt {x^{2}+2}}\)	\(22\)
risch	\(-\frac {9 x \left (\frac {x^{2}+2}{x^{2}}\right )^{\frac {7}{9}}}{10 \sqrt {x^{2}+2}}\)	\(22\)

[In]

int(((x^2+2)/x^2)^(7/9)/(x^2+2)^(3/2),x,method=_RETURNVERBOSE)

[Out]

-9/10*x/(x^2+2)^(1/2)*((x^2+2)/x^2)^(7/9)

Fricas [A] (verification not implemented)

none

Time = 0.39 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.84 \[ \int \frac {\left (\frac {2+x^2}{x^2}\right )^{7/9}}{\left (2+x^2\right )^{3/2}} \, dx=-\frac {9 \, x \left (\frac {x^{2} + 2}{x^{2}}\right )^{\frac {7}{9}}}{10 \, \sqrt {x^{2} + 2}} \]

[In]

integrate(((x^2+2)/x^2)^(7/9)/(x^2+2)^(3/2),x, algorithm="fricas")

[Out]

-9/10*x*((x^2 + 2)/x^2)^(7/9)/sqrt(x^2 + 2)

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (\frac {2+x^2}{x^2}\right )^{7/9}}{\left (2+x^2\right )^{3/2}} \, dx=\text {Timed out} \]

[In]

integrate(((x**2+2)/x**2)**(7/9)/(x**2+2)**(3/2),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {\left (\frac {2+x^2}{x^2}\right )^{7/9}}{\left (2+x^2\right )^{3/2}} \, dx=\int { \frac {\left (\frac {x^{2} + 2}{x^{2}}\right )^{\frac {7}{9}}}{{\left (x^{2} + 2\right )}^{\frac {3}{2}}} \,d x } \]

[In]

integrate(((x^2+2)/x^2)^(7/9)/(x^2+2)^(3/2),x, algorithm="maxima")

[Out]

integrate(((x^2 + 2)/x^2)^(7/9)/(x^2 + 2)^(3/2), x)

Giac [F]

\[ \int \frac {\left (\frac {2+x^2}{x^2}\right )^{7/9}}{\left (2+x^2\right )^{3/2}} \, dx=\int { \frac {\left (\frac {x^{2} + 2}{x^{2}}\right )^{\frac {7}{9}}}{{\left (x^{2} + 2\right )}^{\frac {3}{2}}} \,d x } \]

[In]

integrate(((x^2+2)/x^2)^(7/9)/(x^2+2)^(3/2),x, algorithm="giac")

[Out]

integrate(((x^2 + 2)/x^2)^(7/9)/(x^2 + 2)^(3/2), x)

Mupad [B] (verification not implemented)

Time = 0.50 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.60 \[ \int \frac {\left (\frac {2+x^2}{x^2}\right )^{7/9}}{\left (2+x^2\right )^{3/2}} \, dx=-\frac {9\,x\,{\left (x^2+2\right )}^{5/18}\,{\left (\frac {1}{x^2}\right )}^{7/9}}{10} \]

[In]

int(((x^2 + 2)/x^2)^(7/9)/(x^2 + 2)^(3/2),x)

[Out]

-(9*x*(x^2 + 2)^(5/18)*(1/x^2)^(7/9))/10