\(\int \frac {(2+e x)^{7/2}}{(12-3 e^2 x^2)^{3/2}} \, dx\) [920]

Optimal result

Integrand size = 24, antiderivative size = 67 \[ \int \frac {(2+e x)^{7/2}}{\left (12-3 e^2 x^2\right )^{3/2}} \, dx=\frac {32}{3 \sqrt {3} e \sqrt {2-e x}}+\frac {16 \sqrt {2-e x}}{3 \sqrt {3} e}-\frac {2 (2-e x)^{3/2}}{9 \sqrt {3} e} \]

[Out]

-2/27*(-e*x+2)^(3/2)/e*3^(1/2)+32/9/e*3^(1/2)/(-e*x+2)^(1/2)+16/9*3^(1/2)*(-e*x+2)^(1/2)/e

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 67, 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 = {641, 45} \[ \int \frac {(2+e x)^{7/2}}{\left (12-3 e^2 x^2\right )^{3/2}} \, dx=-\frac {2 (2-e x)^{3/2}}{9 \sqrt {3} e}+\frac {16 \sqrt {2-e x}}{3 \sqrt {3} e}+\frac {32}{3 \sqrt {3} e \sqrt {2-e x}} \]

[In]

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

[Out]

32/(3*Sqrt[3]*e*Sqrt[2 - e*x]) + (16*Sqrt[2 - e*x])/(3*Sqrt[3]*e) - (2*(2 - e*x)^(3/2))/(9*Sqrt[3]*e)

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 641

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

Rubi steps \begin{align*} \text {integral}& = \int \frac {(2+e x)^2}{(6-3 e x)^{3/2}} \, dx \\ & = \int \left (\frac {16}{(6-3 e x)^{3/2}}-\frac {8}{3 \sqrt {6-3 e x}}+\frac {1}{9} \sqrt {6-3 e x}\right ) \, dx \\ & = \frac {32}{3 \sqrt {3} e \sqrt {2-e x}}+\frac {16 \sqrt {2-e x}}{3 \sqrt {3} e}-\frac {2 (2-e x)^{3/2}}{9 \sqrt {3} e} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.49 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.76 \[ \int \frac {(2+e x)^{7/2}}{\left (12-3 e^2 x^2\right )^{3/2}} \, dx=\frac {2 \sqrt {4-e^2 x^2} \left (-92+20 e x+e^2 x^2\right )}{9 e (-2+e x) \sqrt {6+3 e x}} \]

[In]

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

[Out]

(2*Sqrt[4 - e^2*x^2]*(-92 + 20*e*x + e^2*x^2))/(9*e*(-2 + e*x)*Sqrt[6 + 3*e*x])

Maple [A] (verified)

Time = 2.42 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.64

method	result	size
gosper	\(\frac {2 \left (e x -2\right ) \left (x^{2} e^{2}+20 e x -92\right ) \left (e x +2\right )^{\frac {3}{2}}}{3 e \left (-3 x^{2} e^{2}+12\right )^{\frac {3}{2}}}\)	\(43\)
default	\(\frac {2 \sqrt {-3 x^{2} e^{2}+12}\, \left (x^{2} e^{2}+20 e x -92\right )}{27 \sqrt {e x +2}\, \left (e x -2\right ) e}\)	\(45\)
risch	\(-\frac {2 \left (e x +22\right ) \left (e x -2\right ) \sqrt {\frac {-3 x^{2} e^{2}+12}{e x +2}}\, \sqrt {e x +2}}{9 e \sqrt {-3 e x +6}\, \sqrt {-3 x^{2} e^{2}+12}}+\frac {32 \sqrt {\frac {-3 x^{2} e^{2}+12}{e x +2}}\, \sqrt {e x +2}}{3 e \sqrt {-3 e x +6}\, \sqrt {-3 x^{2} e^{2}+12}}\)	\(116\)

[In]

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

[Out]

2/3*(e*x-2)*(e^2*x^2+20*e*x-92)*(e*x+2)^(3/2)/e/(-3*e^2*x^2+12)^(3/2)

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.70 \[ \int \frac {(2+e x)^{7/2}}{\left (12-3 e^2 x^2\right )^{3/2}} \, dx=\frac {2 \, {\left (e^{2} x^{2} + 20 \, e x - 92\right )} \sqrt {-3 \, e^{2} x^{2} + 12} \sqrt {e x + 2}}{27 \, {\left (e^{3} x^{2} - 4 \, e\right )}} \]

[In]

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

[Out]

2/27*(e^2*x^2 + 20*e*x - 92)*sqrt(-3*e^2*x^2 + 12)*sqrt(e*x + 2)/(e^3*x^2 - 4*e)

Sympy [F(-1)]

Timed out. \[ \int \frac {(2+e x)^{7/2}}{\left (12-3 e^2 x^2\right )^{3/2}} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.28 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.54 \[ \int \frac {(2+e x)^{7/2}}{\left (12-3 e^2 x^2\right )^{3/2}} \, dx=-\frac {2 \, {\left (-i \, \sqrt {3} e^{2} x^{2} - 20 i \, \sqrt {3} e x + 92 i \, \sqrt {3}\right )}}{27 \, \sqrt {e x - 2} e} \]

[In]

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

[Out]

-2/27*(-I*sqrt(3)*e^2*x^2 - 20*I*sqrt(3)*e*x + 92*I*sqrt(3))/(sqrt(e*x - 2)*e)

Giac [F(-2)]

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

[In]

integrate((e*x+2)^(7/2)/(-3*e^2*x^2+12)^(3/2),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 = 10.25 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.72 \[ \int \frac {(2+e x)^{7/2}}{\left (12-3 e^2 x^2\right )^{3/2}} \, dx=\frac {2\,\sqrt {12-3\,e^2\,x^2}\,\sqrt {e\,x+2}\,\left (e^2\,x^2+20\,e\,x-92\right )}{27\,e\,\left (e^2\,x^2-4\right )} \]

[In]

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

[Out]

(2*(12 - 3*e^2*x^2)^(1/2)*(e*x + 2)^(1/2)*(20*e*x + e^2*x^2 - 92))/(27*e*(e^2*x^2 - 4))