3.8.62 \(\int \frac {(2+e x)^{5/2}}{(12-3 e^2 x^2)^{3/2}} \, dx\)

Optimal. Leaf size=45 \[ \frac {2 \sqrt {2-e x}}{3 \sqrt {3} e}+\frac {8}{3 \sqrt {3} e \sqrt {2-e x}} \]

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Rubi [A]  time = 0.02, antiderivative size = 45, 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 \sqrt {2-e x}}{3 \sqrt {3} e}+\frac {8}{3 \sqrt {3} e \sqrt {2-e x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(2 + e*x)^(5/2)/(12 - 3*e^2*x^2)^(3/2),x]

[Out]

8/(3*Sqrt[3]*e*Sqrt[2 - e*x]) + (2*Sqrt[2 - e*x])/(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)^{5/2}}{\left (12-3 e^2 x^2\right )^{3/2}} \, dx &=\int \frac {2+e x}{(6-3 e x)^{3/2}} \, dx\\ &=\int \left (\frac {4}{(6-3 e x)^{3/2}}-\frac {1}{3 \sqrt {6-3 e x}}\right ) \, dx\\ &=\frac {8}{3 \sqrt {3} e \sqrt {2-e x}}+\frac {2 \sqrt {2-e x}}{3 \sqrt {3} e}\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 35, normalized size = 0.78 \begin {gather*} -\frac {2 (e x-6) \sqrt {e x+2}}{3 e \sqrt {12-3 e^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2 + e*x)^(5/2)/(12 - 3*e^2*x^2)^(3/2),x]

[Out]

(-2*(-6 + e*x)*Sqrt[2 + e*x])/(3*e*Sqrt[12 - 3*e^2*x^2])

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IntegrateAlgebraic [A]  time = 0.37, size = 63, normalized size = 1.40 \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 (e x-2) \sqrt {e x+2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(2 + e*x)^(5/2)/(12 - 3*e^2*x^2)^(3/2),x]

[Out]

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

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fricas [A]  time = 0.39, size = 39, normalized size = 0.87 \begin {gather*} \frac {2 \, \sqrt {-3 \, e^{2} x^{2} + 12} \sqrt {e x + 2} {\left (e x - 6\right )}}{9 \, {\left (e^{3} x^{2} - 4 \, e\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+2)^(5/2)/(-3*e^2*x^2+12)^(3/2),x, algorithm="fricas")

[Out]

2/9*sqrt(-3*e^2*x^2 + 12)*sqrt(e*x + 2)*(e*x - 6)/(e^3*x^2 - 4*e)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+2)^(5/2)/(-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,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 = 35, normalized size = 0.78 \begin {gather*} \frac {2 \left (e x -2\right ) \left (e x -6\right ) \left (e x +2\right )^{\frac {3}{2}}}{\left (-3 e^{2} x^{2}+12\right )^{\frac {3}{2}} e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+2)^(5/2)/(-3*e^2*x^2+12)^(3/2),x)

[Out]

2*(e*x-2)*(e*x-6)*(e*x+2)^(3/2)/e/(-3*e^2*x^2+12)^(3/2)

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maxima [C]  time = 3.01, size = 20, normalized size = 0.44 \begin {gather*} \frac {2 i \, \sqrt {3} {\left (e x - 6\right )}}{9 \, \sqrt {e x - 2} e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+2)^(5/2)/(-3*e^2*x^2+12)^(3/2),x, algorithm="maxima")

[Out]

2/9*I*sqrt(3)*(e*x - 6)/(sqrt(e*x - 2)*e)

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mupad [B]  time = 0.57, size = 52, normalized size = 1.16 \begin {gather*} \frac {\left (\frac {4\,\sqrt {e\,x+2}}{3\,e^3}-\frac {2\,x\,\sqrt {e\,x+2}}{9\,e^2}\right )\,\sqrt {12-3\,e^2\,x^2}}{\frac {4}{e^2}-x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x + 2)^(5/2)/(12 - 3*e^2*x^2)^(3/2),x)

[Out]

(((4*(e*x + 2)^(1/2))/(3*e^3) - (2*x*(e*x + 2)^(1/2))/(9*e^2))*(12 - 3*e^2*x^2)^(1/2))/(4/e^2 - x^2)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\sqrt {3} \left (\int \frac {4 \sqrt {e x + 2}}{- e^{2} x^{2} \sqrt {- e^{2} x^{2} + 4} + 4 \sqrt {- e^{2} x^{2} + 4}}\, dx + \int \frac {4 e x \sqrt {e x + 2}}{- e^{2} x^{2} \sqrt {- e^{2} x^{2} + 4} + 4 \sqrt {- e^{2} x^{2} + 4}}\, dx + \int \frac {e^{2} x^{2} \sqrt {e x + 2}}{- e^{2} x^{2} \sqrt {- e^{2} x^{2} + 4} + 4 \sqrt {- e^{2} x^{2} + 4}}\, dx\right )}{9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+2)**(5/2)/(-3*e**2*x**2+12)**(3/2),x)

[Out]

sqrt(3)*(Integral(4*sqrt(e*x + 2)/(-e**2*x**2*sqrt(-e**2*x**2 + 4) + 4*sqrt(-e**2*x**2 + 4)), x) + Integral(4*
e*x*sqrt(e*x + 2)/(-e**2*x**2*sqrt(-e**2*x**2 + 4) + 4*sqrt(-e**2*x**2 + 4)), x) + Integral(e**2*x**2*sqrt(e*x
+ 2)/(-e**2*x**2*sqrt(-e**2*x**2 + 4) + 4*sqrt(-e**2*x**2 + 4)), x))/9

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