3.8.76 \(\int \frac {\sqrt [4]{12-3 e^2 x^2}}{(2+e x)^{9/2}} \, dx\)

Optimal. Leaf size=106 \[ -\frac {2 \left (4-e^2 x^2\right )^{5/4}}{195\ 3^{3/4} e (e x+2)^{5/2}}-\frac {2 \left (4-e^2 x^2\right )^{5/4}}{39\ 3^{3/4} e (e x+2)^{7/2}}-\frac {\sqrt [4]{3} \left (4-e^2 x^2\right )^{5/4}}{13 e (e x+2)^{9/2}} \]

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Rubi [A]  time = 0.04, antiderivative size = 106, 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 = {659, 651} \begin {gather*} -\frac {2 \left (4-e^2 x^2\right )^{5/4}}{195\ 3^{3/4} e (e x+2)^{5/2}}-\frac {2 \left (4-e^2 x^2\right )^{5/4}}{39\ 3^{3/4} e (e x+2)^{7/2}}-\frac {\sqrt [4]{3} \left (4-e^2 x^2\right )^{5/4}}{13 e (e x+2)^{9/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(12 - 3*e^2*x^2)^(1/4)/(2 + e*x)^(9/2),x]

[Out]

-(3^(1/4)*(4 - e^2*x^2)^(5/4))/(13*e*(2 + e*x)^(9/2)) - (2*(4 - e^2*x^2)^(5/4))/(39*3^(3/4)*e*(2 + e*x)^(7/2))
- (2*(4 - e^2*x^2)^(5/4))/(195*3^(3/4)*e*(2 + e*x)^(5/2))

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]

Rule 659

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*(m + p + 1)), x] + Dist[Simplify[m + 2*p + 2]/(2*d*(m + p + 1)), Int[(d + e*x)^(m + 1)*(a + c*x^2)^p,
x], x] /; FreeQ[{a, c, d, e, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] && ILtQ[Simplify[m + 2*p + 2
], 0]

Rubi steps

\begin {align*} \int \frac {\sqrt [4]{12-3 e^2 x^2}}{(2+e x)^{9/2}} \, dx &=-\frac {\sqrt [4]{3} \left (4-e^2 x^2\right )^{5/4}}{13 e (2+e x)^{9/2}}+\frac {2}{13} \int \frac {\sqrt [4]{12-3 e^2 x^2}}{(2+e x)^{7/2}} \, dx\\ &=-\frac {\sqrt [4]{3} \left (4-e^2 x^2\right )^{5/4}}{13 e (2+e x)^{9/2}}-\frac {2 \left (4-e^2 x^2\right )^{5/4}}{39\ 3^{3/4} e (2+e x)^{7/2}}+\frac {2}{117} \int \frac {\sqrt [4]{12-3 e^2 x^2}}{(2+e x)^{5/2}} \, dx\\ &=-\frac {\sqrt [4]{3} \left (4-e^2 x^2\right )^{5/4}}{13 e (2+e x)^{9/2}}-\frac {2 \left (4-e^2 x^2\right )^{5/4}}{39\ 3^{3/4} e (2+e x)^{7/2}}-\frac {2 \left (4-e^2 x^2\right )^{5/4}}{195\ 3^{3/4} e (2+e x)^{5/2}}\\ \end {align*}

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Mathematica [A]  time = 0.07, size = 57, normalized size = 0.54 \begin {gather*} \frac {\sqrt [4]{4-e^2 x^2} \left (2 e^3 x^3+14 e^2 x^2+37 e x-146\right )}{195\ 3^{3/4} e (e x+2)^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(12 - 3*e^2*x^2)^(1/4)/(2 + e*x)^(9/2),x]

[Out]

((4 - e^2*x^2)^(1/4)*(-146 + 37*e*x + 14*e^2*x^2 + 2*e^3*x^3))/(195*3^(3/4)*e*(2 + e*x)^(7/2))

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IntegrateAlgebraic [A]  time = 0.33, size = 60, normalized size = 0.57 \begin {gather*} -\frac {\left (4 (e x+2)-(e x+2)^2\right )^{5/4} \left (2 (e x+2)^2+10 (e x+2)+45\right )}{195\ 3^{3/4} e (e x+2)^{9/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(12 - 3*e^2*x^2)^(1/4)/(2 + e*x)^(9/2),x]

[Out]

-1/195*((4*(2 + e*x) - (2 + e*x)^2)^(5/4)*(45 + 10*(2 + e*x) + 2*(2 + e*x)^2))/(3^(3/4)*e*(2 + e*x)^(9/2))

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fricas [A]  time = 0.40, size = 78, normalized size = 0.74 \begin {gather*} \frac {{\left (2 \, e^{3} x^{3} + 14 \, e^{2} x^{2} + 37 \, e x - 146\right )} {\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac {1}{4}} \sqrt {e x + 2}}{585 \, {\left (e^{5} x^{4} + 8 \, e^{4} x^{3} + 24 \, e^{3} x^{2} + 32 \, e^{2} x + 16 \, e\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3*e^2*x^2+12)^(1/4)/(e*x+2)^(9/2),x, algorithm="fricas")

[Out]

1/585*(2*e^3*x^3 + 14*e^2*x^2 + 37*e*x - 146)*(-3*e^2*x^2 + 12)^(1/4)*sqrt(e*x + 2)/(e^5*x^4 + 8*e^4*x^3 + 24*
e^3*x^2 + 32*e^2*x + 16*e)

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giac [A]  time = 0.39, size = 146, normalized size = 1.38 \begin {gather*} -\frac {1}{9360} \cdot 3^{\frac {1}{4}} {\left (\frac {117 \, {\left (-{\left (x e + 2\right )}^{2} + 4 \, x e + 8\right )}^{\frac {1}{4}} {\left (\frac {4}{x e + 2} - 1\right )}}{\sqrt {x e + 2}} + \frac {130 \, {\left ({\left (x e + 2\right )}^{2} - 8 \, x e\right )} {\left (-{\left (x e + 2\right )}^{2} + 4 \, x e + 8\right )}^{\frac {1}{4}}}{{\left (x e + 2\right )}^{\frac {5}{2}}} - \frac {45 \, {\left ({\left (x e + 2\right )}^{3} - 12 \, {\left (x e + 2\right )}^{2} + 48 \, x e + 32\right )} {\left (-{\left (x e + 2\right )}^{2} + 4 \, x e + 8\right )}^{\frac {1}{4}}}{{\left (x e + 2\right )}^{\frac {7}{2}}}\right )} e^{\left (-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3*e^2*x^2+12)^(1/4)/(e*x+2)^(9/2),x, algorithm="giac")

[Out]

-1/9360*3^(1/4)*(117*(-(x*e + 2)^2 + 4*x*e + 8)^(1/4)*(4/(x*e + 2) - 1)/sqrt(x*e + 2) + 130*((x*e + 2)^2 - 8*x
*e)*(-(x*e + 2)^2 + 4*x*e + 8)^(1/4)/(x*e + 2)^(5/2) - 45*((x*e + 2)^3 - 12*(x*e + 2)^2 + 48*x*e + 32)*(-(x*e
+ 2)^2 + 4*x*e + 8)^(1/4)/(x*e + 2)^(7/2))*e^(-1)

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maple [A]  time = 0.05, size = 44, normalized size = 0.42 \begin {gather*} \frac {\left (e x -2\right ) \left (2 e^{2} x^{2}+18 e x +73\right ) \left (-3 e^{2} x^{2}+12\right )^{\frac {1}{4}}}{585 \left (e x +2\right )^{\frac {7}{2}} e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/585*(e*x-2)*(2*e^2*x^2+18*e*x+73)*(-3*e^2*x^2+12)^(1/4)/(e*x+2)^(7/2)/e

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac {1}{4}}}{{\left (e x + 2\right )}^{\frac {9}{2}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3*e^2*x^2+12)^(1/4)/(e*x+2)^(9/2),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.70, size = 46, normalized size = 0.43 \begin {gather*} \frac {{\left (12-3\,e^2\,x^2\right )}^{1/4}\,\left (2\,e^3\,x^3+14\,e^2\,x^2+37\,e\,x-146\right )}{585\,e\,{\left (e\,x+2\right )}^{7/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((12 - 3*e^2*x^2)^(1/4)*(37*e*x + 14*e^2*x^2 + 2*e^3*x^3 - 146))/(585*e*(e*x + 2)^(7/2))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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