\(\int \frac {\sqrt [4]{12-3 e^2 x^2}}{(2+e x)^{7/2}} \, dx\) [934]

Optimal result

Integrand size = 24, antiderivative size = 71 \[ \int \frac {\sqrt [4]{12-3 e^2 x^2}}{(2+e x)^{7/2}} \, dx=-\frac {\left (4-e^2 x^2\right )^{5/4}}{3\ 3^{3/4} e (2+e x)^{7/2}}-\frac {\left (4-e^2 x^2\right )^{5/4}}{15\ 3^{3/4} e (2+e x)^{5/2}} \]

[Out]

-1/9*(-e^2*x^2+4)^(5/4)*3^(1/4)/e/(e*x+2)^(7/2)-1/45*3^(1/4)*(-e^2*x^2+4)^(5/4)/e/(e*x+2)^(5/2)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {673, 665} \[ \int \frac {\sqrt [4]{12-3 e^2 x^2}}{(2+e x)^{7/2}} \, dx=-\frac {\left (4-e^2 x^2\right )^{5/4}}{15\ 3^{3/4} e (e x+2)^{5/2}}-\frac {\left (4-e^2 x^2\right )^{5/4}}{3\ 3^{3/4} e (e x+2)^{7/2}} \]

[In]

Int[(12 - 3*e^2*x^2)^(1/4)/(2 + e*x)^(7/2),x]

[Out]

-1/3*(4 - e^2*x^2)^(5/4)/(3^(3/4)*e*(2 + e*x)^(7/2)) - (4 - e^2*x^2)^(5/4)/(15*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 {\left (4-e^2 x^2\right )^{5/4}}{3\ 3^{3/4} e (2+e x)^{7/2}}+\frac {1}{9} \int \frac {\sqrt [4]{12-3 e^2 x^2}}{(2+e x)^{5/2}} \, dx \\ & = -\frac {\left (4-e^2 x^2\right )^{5/4}}{3\ 3^{3/4} e (2+e x)^{7/2}}-\frac {\left (4-e^2 x^2\right )^{5/4}}{15\ 3^{3/4} e (2+e x)^{5/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.39 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.66 \[ \int \frac {\sqrt [4]{12-3 e^2 x^2}}{(2+e x)^{7/2}} \, dx=-\frac {(7+e x) \left (4 (2+e x)-(2+e x)^2\right )^{5/4}}{15\ 3^{3/4} e (2+e x)^{7/2}} \]

[In]

Integrate[(12 - 3*e^2*x^2)^(1/4)/(2 + e*x)^(7/2),x]

[Out]

-1/15*((7 + e*x)*(4*(2 + e*x) - (2 + e*x)^2)^(5/4))/(3^(3/4)*e*(2 + e*x)^(7/2))

Maple [A] (verified)

Time = 2.32 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.49

method	result	size
gosper	\(\frac {\left (e x -2\right ) \left (e x +7\right ) \left (-3 x^{2} e^{2}+12\right )^{\frac {1}{4}}}{45 \left (e x +2\right )^{\frac {5}{2}} e}\)	\(35\)

[In]

int((-3*e^2*x^2+12)^(1/4)/(e*x+2)^(7/2),x,method=_RETURNVERBOSE)

[Out]

1/45*(e*x-2)*(e*x+7)*(-3*e^2*x^2+12)^(1/4)/(e*x+2)^(5/2)/e

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.86 \[ \int \frac {\sqrt [4]{12-3 e^2 x^2}}{(2+e x)^{7/2}} \, dx=\frac {{\left (e^{2} x^{2} + 5 \, e x - 14\right )} {\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac {1}{4}} \sqrt {e x + 2}}{45 \, {\left (e^{4} x^{3} + 6 \, e^{3} x^{2} + 12 \, e^{2} x + 8 \, e\right )}} \]

[In]

integrate((-3*e^2*x^2+12)^(1/4)/(e*x+2)^(7/2),x, algorithm="fricas")

[Out]

1/45*(e^2*x^2 + 5*e*x - 14)*(-3*e^2*x^2 + 12)^(1/4)*sqrt(e*x + 2)/(e^4*x^3 + 6*e^3*x^2 + 12*e^2*x + 8*e)

Sympy [F]

\[ \int \frac {\sqrt [4]{12-3 e^2 x^2}}{(2+e x)^{7/2}} \, dx=\sqrt [4]{3} \int \frac {\sqrt [4]{- e^{2} x^{2} + 4}}{e^{3} x^{3} \sqrt {e x + 2} + 6 e^{2} x^{2} \sqrt {e x + 2} + 12 e x \sqrt {e x + 2} + 8 \sqrt {e x + 2}}\, dx \]

[In]

integrate((-3*e**2*x**2+12)**(1/4)/(e*x+2)**(7/2),x)

[Out]

3**(1/4)*Integral((-e**2*x**2 + 4)**(1/4)/(e**3*x**3*sqrt(e*x + 2) + 6*e**2*x**2*sqrt(e*x + 2) + 12*e*x*sqrt(e
*x + 2) + 8*sqrt(e*x + 2)), x)

Maxima [F]

\[ \int \frac {\sqrt [4]{12-3 e^2 x^2}}{(2+e x)^{7/2}} \, dx=\int { \frac {{\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac {1}{4}}}{{\left (e x + 2\right )}^{\frac {7}{2}}} \,d x } \]

[In]

integrate((-3*e^2*x^2+12)^(1/4)/(e*x+2)^(7/2),x, algorithm="maxima")

[Out]

integrate((-3*e^2*x^2 + 12)^(1/4)/(e*x + 2)^(7/2), x)

Giac [F(-2)]

Exception generated. \[ \int \frac {\sqrt [4]{12-3 e^2 x^2}}{(2+e x)^{7/2}} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate((-3*e^2*x^2+12)^(1/4)/(e*x+2)^(7/2),x, algorithm="giac")

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [B] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.52 \[ \int \frac {\sqrt [4]{12-3 e^2 x^2}}{(2+e x)^{7/2}} \, dx=\frac {{\left (12-3\,e^2\,x^2\right )}^{1/4}\,\left (e^2\,x^2+5\,e\,x-14\right )}{45\,e\,{\left (e\,x+2\right )}^{5/2}} \]

[In]

int((12 - 3*e^2*x^2)^(1/4)/(e*x + 2)^(7/2),x)

[Out]

((12 - 3*e^2*x^2)^(1/4)*(5*e*x + e^2*x^2 - 14))/(45*e*(e*x + 2)^(5/2))