3.8.82 \(\int \frac {1}{(2+e x)^{3/2} \sqrt [4]{12-3 e^2 x^2}} \, dx\)

Optimal. Leaf size=35 \[ -\frac {\left (4-e^2 x^2\right )^{3/4}}{3 \sqrt [4]{3} e (e x+2)^{3/2}} \]

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Rubi [A]  time = 0.01, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {651} \begin {gather*} -\frac {\left (4-e^2 x^2\right )^{3/4}}{3 \sqrt [4]{3} e (e x+2)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((2 + e*x)^(3/2)*(12 - 3*e^2*x^2)^(1/4)),x]

[Out]

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

Rule 651

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^m*(a + c*x^2)^(p + 1))
/(2*c*d*(p + 1)), x] /; FreeQ[{a, c, d, e, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] && EqQ[m + 2*p
+ 2, 0]

Rubi steps

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

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Mathematica [A]  time = 0.04, size = 35, normalized size = 1.00 \begin {gather*} \frac {e x-2}{3 e \sqrt {e x+2} \sqrt [4]{12-3 e^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((2 + e*x)^(3/2)*(12 - 3*e^2*x^2)^(1/4)),x]

[Out]

(-2 + e*x)/(3*e*Sqrt[2 + e*x]*(12 - 3*e^2*x^2)^(1/4))

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IntegrateAlgebraic [A]  time = 0.30, size = 42, normalized size = 1.20 \begin {gather*} -\frac {\left (4 (e x+2)-(e x+2)^2\right )^{3/4}}{3 \sqrt [4]{3} e (e x+2)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[1/((2 + e*x)^(3/2)*(12 - 3*e^2*x^2)^(1/4)),x]

[Out]

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

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fricas [A]  time = 0.39, size = 40, normalized size = 1.14 \begin {gather*} -\frac {{\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac {3}{4}} \sqrt {e x + 2}}{9 \, {\left (e^{3} x^{2} + 4 \, e^{2} x + 4 \, e\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+2)^(3/2)/(-3*e^2*x^2+12)^(1/4),x, algorithm="fricas")

[Out]

-1/9*(-3*e^2*x^2 + 12)^(3/4)*sqrt(e*x + 2)/(e^3*x^2 + 4*e^2*x + 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(1/(e*x+2)^(3/2)/(-3*e^2*x^2+12)^(1/4),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 = 30, normalized size = 0.86 \begin {gather*} \frac {e x -2}{3 \sqrt {e x +2}\, \left (-3 e^{2} x^{2}+12\right )^{\frac {1}{4}} e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac {1}{4}} {\left (e x + 2\right )}^{\frac {3}{2}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+2)^(3/2)/(-3*e^2*x^2+12)^(1/4),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.67, size = 24, normalized size = 0.69 \begin {gather*} -\frac {{\left (12-3\,e^2\,x^2\right )}^{3/4}}{9\,e\,{\left (e\,x+2\right )}^{3/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {3^{\frac {3}{4}} \int \frac {1}{e x \sqrt {e x + 2} \sqrt [4]{- e^{2} x^{2} + 4} + 2 \sqrt {e x + 2} \sqrt [4]{- e^{2} x^{2} + 4}}\, dx}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+2)**(3/2)/(-3*e**2*x**2+12)**(1/4),x)

[Out]

3**(3/4)*Integral(1/(e*x*sqrt(e*x + 2)*(-e**2*x**2 + 4)**(1/4) + 2*sqrt(e*x + 2)*(-e**2*x**2 + 4)**(1/4)), x)/
3

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