\(\int \frac {\sqrt [4]{12-3 e^2 x^2}}{(2+e x)^{9/2}} \, dx\) [935]

Optimal result

Integrand size = 24, antiderivative size = 106 \[ \int \frac {\sqrt [4]{12-3 e^2 x^2}}{(2+e x)^{9/2}} \, dx=-\frac {\sqrt [4]{3} \left (4-e^2 x^2\right )^{5/4}}{13 e (2+e x)^{9/2}}-\frac {2 \left (4-e^2 x^2\right )^{5/4}}{39\ 3^{3/4} e (2+e x)^{7/2}}-\frac {2 \left (4-e^2 x^2\right )^{5/4}}{195\ 3^{3/4} e (2+e x)^{5/2}} \]

[Out]

-1/13*3^(1/4)*(-e^2*x^2+4)^(5/4)/e/(e*x+2)^(9/2)-2/117*(-e^2*x^2+4)^(5/4)*3^(1/4)/e/(e*x+2)^(7/2)-2/585*3^(1/4
)*(-e^2*x^2+4)^(5/4)/e/(e*x+2)^(5/2)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {673, 665} \[ \int \frac {\sqrt [4]{12-3 e^2 x^2}}{(2+e x)^{9/2}} \, dx=-\frac {2 \left (4-e^2 x^2\right )^{5/4}}{195\ 3^{3/4} e (e x+2)^{5/2}}-\frac {2 \left (4-e^2 x^2\right )^{5/4}}{39\ 3^{3/4} e (e x+2)^{7/2}}-\frac {\sqrt [4]{3} \left (4-e^2 x^2\right )^{5/4}}{13 e (e x+2)^{9/2}} \]

[In]

Int[(12 - 3*e^2*x^2)^(1/4)/(2 + e*x)^(9/2),x]

[Out]

-1/13*(3^(1/4)*(4 - e^2*x^2)^(5/4))/(e*(2 + e*x)^(9/2)) - (2*(4 - e^2*x^2)^(5/4))/(39*3^(3/4)*e*(2 + e*x)^(7/2
)) - (2*(4 - e^2*x^2)^(5/4))/(195*3^(3/4)*e*(2 + e*x)^(5/2))

Rule 665

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[e*(d + e*x)^m*((a + c*x^2)^(p + 1)/
(2*c*d*(p + 1))), x] /; FreeQ[{a, c, d, e, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] && EqQ[m + 2*p
+ 2, 0]

Rule 673

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-e)*(d + e*x)^m*((a + c*x^2)^(p +
1)/(2*c*d*(m + p + 1))), x] + Dist[Simplify[m + 2*p + 2]/(2*d*(m + p + 1)), Int[(d + e*x)^(m + 1)*(a + c*x^2)^
p, x], x] /; FreeQ[{a, c, d, e, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] && ILtQ[Simplify[m + 2*p +
2], 0]

Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt [4]{3} \left (4-e^2 x^2\right )^{5/4}}{13 e (2+e x)^{9/2}}+\frac {2}{13} \int \frac {\sqrt [4]{12-3 e^2 x^2}}{(2+e x)^{7/2}} \, dx \\ & = -\frac {\sqrt [4]{3} \left (4-e^2 x^2\right )^{5/4}}{13 e (2+e x)^{9/2}}-\frac {2 \left (4-e^2 x^2\right )^{5/4}}{39\ 3^{3/4} e (2+e x)^{7/2}}+\frac {2}{117} \int \frac {\sqrt [4]{12-3 e^2 x^2}}{(2+e x)^{5/2}} \, dx \\ & = -\frac {\sqrt [4]{3} \left (4-e^2 x^2\right )^{5/4}}{13 e (2+e x)^{9/2}}-\frac {2 \left (4-e^2 x^2\right )^{5/4}}{39\ 3^{3/4} e (2+e x)^{7/2}}-\frac {2 \left (4-e^2 x^2\right )^{5/4}}{195\ 3^{3/4} e (2+e x)^{5/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.43 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.51 \[ \int \frac {\sqrt [4]{12-3 e^2 x^2}}{(2+e x)^{9/2}} \, dx=\frac {(-2+e x) \sqrt [4]{4-e^2 x^2} \left (73+18 e x+2 e^2 x^2\right )}{195\ 3^{3/4} e (2+e x)^{7/2}} \]

[In]

Integrate[(12 - 3*e^2*x^2)^(1/4)/(2 + e*x)^(9/2),x]

[Out]

((-2 + e*x)*(4 - e^2*x^2)^(1/4)*(73 + 18*e*x + 2*e^2*x^2))/(195*3^(3/4)*e*(2 + e*x)^(7/2))

Maple [A] (verified)

Time = 2.31 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.42

method	result	size
gosper	\(\frac {\left (e x -2\right ) \left (2 x^{2} e^{2}+18 e x +73\right ) \left (-3 x^{2} e^{2}+12\right )^{\frac {1}{4}}}{585 \left (e x +2\right )^{\frac {7}{2}} e}\)	\(44\)

[In]

int((-3*e^2*x^2+12)^(1/4)/(e*x+2)^(9/2),x,method=_RETURNVERBOSE)

[Out]

1/585*(e*x-2)*(2*e^2*x^2+18*e*x+73)*(-3*e^2*x^2+12)^(1/4)/(e*x+2)^(7/2)/e

Fricas [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.74 \[ \int \frac {\sqrt [4]{12-3 e^2 x^2}}{(2+e x)^{9/2}} \, dx=\frac {{\left (2 \, e^{3} x^{3} + 14 \, e^{2} x^{2} + 37 \, e x - 146\right )} {\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac {1}{4}} \sqrt {e x + 2}}{585 \, {\left (e^{5} x^{4} + 8 \, e^{4} x^{3} + 24 \, e^{3} x^{2} + 32 \, e^{2} x + 16 \, e\right )}} \]

[In]

integrate((-3*e^2*x^2+12)^(1/4)/(e*x+2)^(9/2),x, algorithm="fricas")

[Out]

1/585*(2*e^3*x^3 + 14*e^2*x^2 + 37*e*x - 146)*(-3*e^2*x^2 + 12)^(1/4)*sqrt(e*x + 2)/(e^5*x^4 + 8*e^4*x^3 + 24*
e^3*x^2 + 32*e^2*x + 16*e)

Sympy [F(-1)]

Timed out. \[ \int \frac {\sqrt [4]{12-3 e^2 x^2}}{(2+e x)^{9/2}} \, dx=\text {Timed out} \]

[In]

integrate((-3*e**2*x**2+12)**(1/4)/(e*x+2)**(9/2),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {\sqrt [4]{12-3 e^2 x^2}}{(2+e x)^{9/2}} \, dx=\int { \frac {{\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac {1}{4}}}{{\left (e x + 2\right )}^{\frac {9}{2}}} \,d x } \]

[In]

integrate((-3*e^2*x^2+12)^(1/4)/(e*x+2)^(9/2),x, algorithm="maxima")

[Out]

integrate((-3*e^2*x^2 + 12)^(1/4)/(e*x + 2)^(9/2), x)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.51 \[ \int \frac {\sqrt [4]{12-3 e^2 x^2}}{(2+e x)^{9/2}} \, dx=-\frac {3^{\frac {1}{4}} {\left (45 \, {\left (\frac {4}{e x + 2} - 1\right )}^{\frac {13}{4}} + 130 \, {\left (\frac {4}{e x + 2} - 1\right )}^{\frac {9}{4}} + 117 \, {\left (\frac {4}{e x + 2} - 1\right )}^{\frac {5}{4}}\right )}}{9360 \, e} \]

[In]

integrate((-3*e^2*x^2+12)^(1/4)/(e*x+2)^(9/2),x, algorithm="giac")

[Out]

-1/9360*3^(1/4)*(45*(4/(e*x + 2) - 1)^(13/4) + 130*(4/(e*x + 2) - 1)^(9/4) + 117*(4/(e*x + 2) - 1)^(5/4))/e

Mupad [B] (verification not implemented)

Time = 10.49 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.43 \[ \int \frac {\sqrt [4]{12-3 e^2 x^2}}{(2+e x)^{9/2}} \, dx=\frac {{\left (12-3\,e^2\,x^2\right )}^{1/4}\,\left (2\,e^3\,x^3+14\,e^2\,x^2+37\,e\,x-146\right )}{585\,e\,{\left (e\,x+2\right )}^{7/2}} \]

[In]

int((12 - 3*e^2*x^2)^(1/4)/(e*x + 2)^(9/2),x)

[Out]

((12 - 3*e^2*x^2)^(1/4)*(37*e*x + 14*e^2*x^2 + 2*e^3*x^3 - 146))/(585*e*(e*x + 2)^(7/2))