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

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

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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} \begin {gather*} \frac {2 (2-e x)^{3/2}}{3 \sqrt {3} e}-\frac {8 \sqrt {2-e x}}{\sqrt {3} e} \end {gather*}

Antiderivative was successfully verified.

[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 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 \frac {(2+e x)^{3/2}}{\sqrt {12-3 e^2 x^2}} \, dx &=\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*}

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Mathematica [A]  time = 0.05, size = 40, normalized size = 0.93 \begin {gather*} \frac {2 (e x-2) \sqrt {e x+2} (e x+10)}{3 e \sqrt {12-3 e^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(-2 + e*x)*Sqrt[2 + e*x]*(10 + e*x))/(3*e*Sqrt[12 - 3*e^2*x^2])

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IntegrateAlgebraic [A]  time = 0.28, size = 56, normalized size = 1.30 \begin {gather*} -\frac {2 \left (\sqrt {3} (e x+2)+8 \sqrt {3}\right ) \sqrt {4 (e x+2)-(e x+2)^2}}{9 e \sqrt {e x+2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(8*Sqrt[3] + Sqrt[3]*(2 + e*x))*Sqrt[4*(2 + e*x) - (2 + e*x)^2])/(9*e*Sqrt[2 + e*x])

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fricas [A]  time = 0.40, size = 37, normalized size = 0.86 \begin {gather*} -\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 )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (e x + 2\right )}^{\frac {3}{2}}}{\sqrt {-3 \, e^{2} x^{2} + 12}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.04, size = 35, normalized size = 0.81 \begin {gather*} \frac {2 \left (e x -2\right ) \left (e x +10\right ) \sqrt {e x +2}}{3 \sqrt {-3 e^{2} x^{2}+12}\, e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [C]  time = 3.09, size = 28, normalized size = 0.65 \begin {gather*} -\frac {2 i \, \sqrt {3} {\left (e^{2} x^{2} + 8 \, e x - 20\right )}}{9 \, \sqrt {e x - 2} e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [B]  time = 0.14, size = 49, normalized size = 1.14 \begin {gather*} -\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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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