\(\int \frac {(2+e x)^{3/2}}{\sqrt {12-3 e^2 x^2}} \, dx\) [913]

Optimal result

Integrand size = 24, antiderivative size = 43 \[ \int \frac {(2+e x)^{3/2}}{\sqrt {12-3 e^2 x^2}} \, dx=-\frac {8 \sqrt {2-e x}}{\sqrt {3} e}+\frac {2 (2-e x)^{3/2}}{3 \sqrt {3} e} \]

[Out]

2/9*(-e*x+2)^(3/2)/e*3^(1/2)-8/3*3^(1/2)*(-e*x+2)^(1/2)/e

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 43, 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} \[ \int \frac {(2+e x)^{3/2}}{\sqrt {12-3 e^2 x^2}} \, dx=\frac {2 (2-e x)^{3/2}}{3 \sqrt {3} e}-\frac {8 \sqrt {2-e x}}{\sqrt {3} e} \]

[In]

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

[Out]

(-8*Sqrt[2 - e*x])/(Sqrt[3]*e) + (2*(2 - e*x)^(3/2))/(3*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}{\sqrt {6-3 e x}} \, dx \\ & = \int \left (\frac {4}{\sqrt {6-3 e x}}-\frac {1}{3} \sqrt {6-3 e x}\right ) \, dx \\ & = -\frac {8 \sqrt {2-e x}}{\sqrt {3} e}+\frac {2 (2-e x)^{3/2}}{3 \sqrt {3} e} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.34 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.84 \[ \int \frac {(2+e x)^{3/2}}{\sqrt {12-3 e^2 x^2}} \, dx=-\frac {2 (10+e x) \sqrt {4-e^2 x^2}}{3 e \sqrt {6+3 e x}} \]

[In]

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

[Out]

(-2*(10 + e*x)*Sqrt[4 - e^2*x^2])/(3*e*Sqrt[6 + 3*e*x])

Maple [A] (verified)

Time = 2.31 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.70

method	result	size
default	\(-\frac {2 \sqrt {-3 x^{2} e^{2}+12}\, \left (e x +10\right )}{9 \sqrt {e x +2}\, e}\)	\(30\)
gosper	\(\frac {2 \left (e x -2\right ) \left (e x +10\right ) \sqrt {e x +2}}{3 e \sqrt {-3 x^{2} e^{2}+12}}\)	\(35\)
risch	\(\frac {2 \sqrt {\frac {-3 x^{2} e^{2}+12}{e x +2}}\, \sqrt {e x +2}\, \left (e x +10\right ) \left (e x -2\right )}{3 \sqrt {-3 x^{2} e^{2}+12}\, e \sqrt {-3 e x +6}}\)	\(63\)

[In]

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

[Out]

-2/9/(e*x+2)^(1/2)*(-3*e^2*x^2+12)^(1/2)*(e*x+10)/e

Fricas [A] (verification not implemented)

none

Time = 0.39 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.86 \[ \int \frac {(2+e x)^{3/2}}{\sqrt {12-3 e^2 x^2}} \, dx=-\frac {2 \, \sqrt {-3 \, e^{2} x^{2} + 12} {\left (e x + 10\right )} \sqrt {e x + 2}}{9 \, {\left (e^{2} x + 2 \, e\right )}} \]

[In]

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

[Out]

-2/9*sqrt(-3*e^2*x^2 + 12)*(e*x + 10)*sqrt(e*x + 2)/(e^2*x + 2*e)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.28 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.65 \[ \int \frac {(2+e x)^{3/2}}{\sqrt {12-3 e^2 x^2}} \, dx=-\frac {2 i \, \sqrt {3} {\left (e^{2} x^{2} + 8 \, e x - 20\right )}}{9 \, \sqrt {e x - 2} e} \]

[In]

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

[Out]

-2/9*I*sqrt(3)*(e^2*x^2 + 8*e*x - 20)/(sqrt(e*x - 2)*e)

Giac [F(-2)]

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

[In]

integrate((e*x+2)^(3/2)/(-3*e^2*x^2+12)^(1/2),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 = 10.21 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.14 \[ \int \frac {(2+e x)^{3/2}}{\sqrt {12-3 e^2 x^2}} \, dx=-\frac {\left (\frac {20\,\sqrt {e\,x+2}}{9\,e^2}+\frac {2\,x\,\sqrt {e\,x+2}}{9\,e}\right )\,\sqrt {12-3\,e^2\,x^2}}{x+\frac {2}{e}} \]

[In]

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

[Out]

-(((20*(e*x + 2)^(1/2))/(9*e^2) + (2*x*(e*x + 2)^(1/2))/(9*e))*(12 - 3*e^2*x^2)^(1/2))/(x + 2/e)