\(\int \sqrt {2+e x} (12-3 e^2 x^2)^{3/2} \, dx\) [903]

Optimal result

Integrand size = 24, antiderivative size = 65 \[ \int \sqrt {2+e x} \left (12-3 e^2 x^2\right )^{3/2} \, dx=-\frac {96 \sqrt {3} (2-e x)^{5/2}}{5 e}+\frac {48 \sqrt {3} (2-e x)^{7/2}}{7 e}-\frac {2 (2-e x)^{9/2}}{\sqrt {3} e} \]

[Out]

-2/3*(-e*x+2)^(9/2)/e*3^(1/2)-96/5*(-e*x+2)^(5/2)*3^(1/2)/e+48/7*(-e*x+2)^(7/2)*3^(1/2)/e

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 65, 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} \[ \int \sqrt {2+e x} \left (12-3 e^2 x^2\right )^{3/2} \, dx=-\frac {2 (2-e x)^{9/2}}{\sqrt {3} e}+\frac {48 \sqrt {3} (2-e x)^{7/2}}{7 e}-\frac {96 \sqrt {3} (2-e x)^{5/2}}{5 e} \]

[In]

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

[Out]

(-96*Sqrt[3]*(2 - e*x)^(5/2))/(5*e) + (48*Sqrt[3]*(2 - e*x)^(7/2))/(7*e) - (2*(2 - e*x)^(9/2))/(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 (6-3 e x)^{3/2} (2+e x)^2 \, dx \\ & = \int \left (16 (6-3 e x)^{3/2}-\frac {8}{3} (6-3 e x)^{5/2}+\frac {1}{9} (6-3 e x)^{7/2}\right ) \, dx \\ & = -\frac {96 \sqrt {3} (2-e x)^{5/2}}{5 e}+\frac {48 \sqrt {3} (2-e x)^{7/2}}{7 e}-\frac {2 (2-e x)^{9/2}}{\sqrt {3} e} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.32 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.80 \[ \int \sqrt {2+e x} \left (12-3 e^2 x^2\right )^{3/2} \, dx=-\frac {2 (-2+e x)^2 \sqrt {4-e^2 x^2} \left (428+220 e x+35 e^2 x^2\right )}{35 e \sqrt {6+3 e x}} \]

[In]

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

[Out]

(-2*(-2 + e*x)^2*Sqrt[4 - e^2*x^2]*(428 + 220*e*x + 35*e^2*x^2))/(35*e*Sqrt[6 + 3*e*x])

Maple [A] (verified)

Time = 2.18 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.68

method	result	size
gosper	\(\frac {2 \left (e x -2\right ) \left (35 x^{2} e^{2}+220 e x +428\right ) \left (-3 x^{2} e^{2}+12\right )^{\frac {3}{2}}}{315 e \left (e x +2\right )^{\frac {3}{2}}}\)	\(44\)
default	\(-\frac {2 \sqrt {-3 x^{2} e^{2}+12}\, \left (e x -2\right )^{2} \left (35 x^{2} e^{2}+220 e x +428\right )}{105 \sqrt {e x +2}\, e}\)	\(46\)
risch	\(\frac {2 \sqrt {\frac {-3 x^{2} e^{2}+12}{e x +2}}\, \sqrt {e x +2}\, \left (35 e^{4} x^{4}+80 e^{3} x^{3}-312 x^{2} e^{2}-832 e x +1712\right ) \left (e x -2\right )}{35 \sqrt {-3 x^{2} e^{2}+12}\, e \sqrt {-3 e x +6}}\)	\(88\)

[In]

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

[Out]

2/315*(e*x-2)*(35*e^2*x^2+220*e*x+428)*(-3*e^2*x^2+12)^(3/2)/e/(e*x+2)^(3/2)

Fricas [A] (verification not implemented)

none

Time = 0.64 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.95 \[ \int \sqrt {2+e x} \left (12-3 e^2 x^2\right )^{3/2} \, dx=-\frac {2 \, {\left (35 \, e^{4} x^{4} + 80 \, e^{3} x^{3} - 312 \, e^{2} x^{2} - 832 \, e x + 1712\right )} \sqrt {-3 \, e^{2} x^{2} + 12} \sqrt {e x + 2}}{105 \, {\left (e^{2} x + 2 \, e\right )}} \]

[In]

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

[Out]

-2/105*(35*e^4*x^4 + 80*e^3*x^3 - 312*e^2*x^2 - 832*e*x + 1712)*sqrt(-3*e^2*x^2 + 12)*sqrt(e*x + 2)/(e^2*x + 2
*e)

Sympy [F]

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

[In]

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

[Out]

3*sqrt(3)*(Integral(4*sqrt(e*x + 2)*sqrt(-e**2*x**2 + 4), x) + Integral(-e**2*x**2*sqrt(e*x + 2)*sqrt(-e**2*x*
*2 + 4), x))

Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.28 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.09 \[ \int \sqrt {2+e x} \left (12-3 e^2 x^2\right )^{3/2} \, dx=\frac {2 \, {\left (-35 i \, \sqrt {3} e^{4} x^{4} - 80 i \, \sqrt {3} e^{3} x^{3} + 312 i \, \sqrt {3} e^{2} x^{2} + 832 i \, \sqrt {3} e x - 1712 i \, \sqrt {3}\right )} {\left (e x + 2\right )} \sqrt {e x - 2}}{105 \, {\left (e^{2} x + 2 \, e\right )}} \]

[In]

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

[Out]

2/105*(-35*I*sqrt(3)*e^4*x^4 - 80*I*sqrt(3)*e^3*x^3 + 312*I*sqrt(3)*e^2*x^2 + 832*I*sqrt(3)*e*x - 1712*I*sqrt(
3))*(e*x + 2)*sqrt(e*x - 2)/(e^2*x + 2*e)

Giac [F(-2)]

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

[In]

integrate((e*x+2)^(1/2)*(-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 = 0.18 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.09 \[ \int \sqrt {2+e x} \left (12-3 e^2 x^2\right )^{3/2} \, dx=\frac {2\,\sqrt {12-3\,e^2\,x^2}\,\sqrt {e\,x+2}\,\left (-35\,e^3\,x^3-10\,e^2\,x^2+332\,e\,x+168\right )}{105\,e}-\frac {4096\,\sqrt {12-3\,e^2\,x^2}}{105\,e\,\sqrt {e\,x+2}} \]

[In]

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

[Out]

(2*(12 - 3*e^2*x^2)^(1/2)*(e*x + 2)^(1/2)*(332*e*x - 10*e^2*x^2 - 35*e^3*x^3 + 168))/(105*e) - (4096*(12 - 3*e
^2*x^2)^(1/2))/(105*e*(e*x + 2)^(1/2))