3.896 \(\int \sqrt {2+e x} \sqrt {12-3 e^2 x^2} \, dx\)

Optimal. Leaf size=43 \[ \frac {2 \sqrt {3} (2-e x)^{5/2}}{5 e}-\frac {8 (2-e x)^{3/2}}{\sqrt {3} e} \]

[Out]

-8/3*(-e*x+2)^(3/2)/e*3^(1/2)+2/5*(-e*x+2)^(5/2)*3^(1/2)/e

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Rubi [A]  time = 0.02, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {627, 43} \[ \frac {2 \sqrt {3} (2-e x)^{5/2}}{5 e}-\frac {8 (2-e x)^{3/2}}{\sqrt {3} e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-8*(2 - e*x)^(3/2))/(Sqrt[3]*e) + (2*Sqrt[3]*(2 - e*x)^(5/2))/(5*e)

Rule 43

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 627

Int[((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[(d + e*x)^(m + p)*(a/d + (c*x)/e)^
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*} \int \sqrt {2+e x} \sqrt {12-3 e^2 x^2} \, dx &=\int \sqrt {6-3 e x} (2+e x) \, dx\\ &=\int \left (4 \sqrt {6-3 e x}-\frac {1}{3} (6-3 e x)^{3/2}\right ) \, dx\\ &=-\frac {8 (2-e x)^{3/2}}{\sqrt {3} e}+\frac {2 \sqrt {3} (2-e x)^{5/2}}{5 e}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 42, normalized size = 0.98 \[ \frac {2 (e x-2) (3 e x+14) \sqrt {4-e^2 x^2}}{5 e \sqrt {3 e x+6}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(-2 + e*x)*(14 + 3*e*x)*Sqrt[4 - e^2*x^2])/(5*e*Sqrt[6 + 3*e*x])

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fricas [A]  time = 0.74, size = 46, normalized size = 1.07 \[ \frac {2 \, {\left (3 \, e^{2} x^{2} + 8 \, e x - 28\right )} \sqrt {-3 \, e^{2} x^{2} + 12} \sqrt {e x + 2}}{15 \, {\left (e^{2} x + 2 \, e\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15*(3*e^2*x^2 + 8*e*x - 28)*sqrt(-3*e^2*x^2 + 12)*sqrt(e*x + 2)/(e^2*x + 2*e)

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+2)^(1/2)*(-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,x):;OUTPUT:Erro
r index.cc index_gcd Error: Bad Argument ValueError index.cc index_gcd Error: Bad Argument Value

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maple [A]  time = 0.05, size = 36, normalized size = 0.84 \[ \frac {2 \left (e x -2\right ) \left (3 e x +14\right ) \sqrt {-3 e^{2} x^{2}+12}}{15 \sqrt {e x +2}\, e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15*(e*x-2)*(3*e*x+14)*(-3*e^2*x^2+12)^(1/2)/e/(e*x+2)^(1/2)

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maxima [C]  time = 3.12, size = 49, normalized size = 1.14 \[ \frac {{\left (6 i \, \sqrt {3} e^{2} x^{2} + 16 i \, \sqrt {3} e x - 56 i \, \sqrt {3}\right )} {\left (e x + 2\right )} \sqrt {e x - 2}}{15 \, {\left (e^{2} x + 2 \, e\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/15*(6*I*sqrt(3)*e^2*x^2 + 16*I*sqrt(3)*e*x - 56*I*sqrt(3))*(e*x + 2)*sqrt(e*x - 2)/(e^2*x + 2*e)

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mupad [B]  time = 0.49, size = 60, normalized size = 1.40 \[ \frac {\sqrt {12-3\,e^2\,x^2}\,\left (\frac {2\,x^2\,\sqrt {e\,x+2}}{5}-\frac {56\,\sqrt {e\,x+2}}{15\,e^2}+\frac {16\,x\,\sqrt {e\,x+2}}{15\,e}\right )}{x+\frac {2}{e}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((12 - 3*e^2*x^2)^(1/2)*((2*x^2*(e*x + 2)^(1/2))/5 - (56*(e*x + 2)^(1/2))/(15*e^2) + (16*x*(e*x + 2)^(1/2))/(1
5*e)))/(x + 2/e)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \sqrt {3} \int \sqrt {e x + 2} \sqrt {- e^{2} x^{2} + 4}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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