3.8.83 \(\int \frac {1}{(2+e x)^{5/2} \sqrt [4]{12-3 e^2 x^2}} \, dx\)

Optimal. Leaf size=71 \[ -\frac {\left (4-e^2 x^2\right )^{3/4}}{21 \sqrt [4]{3} e (e x+2)^{3/2}}-\frac {\left (4-e^2 x^2\right )^{3/4}}{7 \sqrt [4]{3} e (e x+2)^{5/2}} \]

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Rubi [A]  time = 0.03, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 2, 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 {\left (4-e^2 x^2\right )^{3/4}}{21 \sqrt [4]{3} e (e x+2)^{3/2}}-\frac {\left (4-e^2 x^2\right )^{3/4}}{7 \sqrt [4]{3} e (e x+2)^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((2 + e*x)^(5/2)*(12 - 3*e^2*x^2)^(1/4)),x]

[Out]

-(4 - e^2*x^2)^(3/4)/(7*3^(1/4)*e*(2 + e*x)^(5/2)) - (4 - e^2*x^2)^(3/4)/(21*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]

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 {1}{(2+e x)^{5/2} \sqrt [4]{12-3 e^2 x^2}} \, dx &=-\frac {\left (4-e^2 x^2\right )^{3/4}}{7 \sqrt [4]{3} e (2+e x)^{5/2}}+\frac {1}{7} \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}}{7 \sqrt [4]{3} e (2+e x)^{5/2}}-\frac {\left (4-e^2 x^2\right )^{3/4}}{21 \sqrt [4]{3} e (2+e x)^{3/2}}\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 40, normalized size = 0.56 \begin {gather*} \frac {(e x-2) (e x+5)}{21 e (e x+2)^{3/2} \sqrt [4]{12-3 e^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((2 + e*x)^(5/2)*(12 - 3*e^2*x^2)^(1/4)),x]

[Out]

((-2 + e*x)*(5 + e*x))/(21*e*(2 + e*x)^(3/2)*(12 - 3*e^2*x^2)^(1/4))

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IntegrateAlgebraic [A]  time = 0.35, size = 47, normalized size = 0.66 \begin {gather*} -\frac {(e x+5) \left (4 (e x+2)-(e x+2)^2\right )^{3/4}}{21 \sqrt [4]{3} e (e x+2)^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[1/((2 + e*x)^(5/2)*(12 - 3*e^2*x^2)^(1/4)),x]

[Out]

-1/21*((5 + e*x)*(4*(2 + e*x) - (2 + e*x)^2)^(3/4))/(3^(1/4)*e*(2 + e*x)^(5/2))

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fricas [A]  time = 0.41, size = 53, normalized size = 0.75 \begin {gather*} -\frac {{\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac {3}{4}} {\left (e x + 5\right )} \sqrt {e x + 2}}{63 \, {\left (e^{4} x^{3} + 6 \, e^{3} x^{2} + 12 \, e^{2} x + 8 \, e\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+2)^(5/2)/(-3*e^2*x^2+12)^(1/4),x, algorithm="fricas")

[Out]

-1/63*(-3*e^2*x^2 + 12)^(3/4)*(e*x + 5)*sqrt(e*x + 2)/(e^4*x^3 + 6*e^3*x^2 + 12*e^2*x + 8*e)

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giac [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 {5}{2}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+2)^(5/2)/(-3*e^2*x^2+12)^(1/4),x, algorithm="giac")

[Out]

integrate(1/((-3*e^2*x^2 + 12)^(1/4)*(e*x + 2)^(5/2)), x)

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maple [A]  time = 0.06, size = 35, normalized size = 0.49 \begin {gather*} \frac {\left (e x -2\right ) \left (e x +5\right )}{21 \left (e x +2\right )^{\frac {3}{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)^(5/2)/(-3*e^2*x^2+12)^(1/4),x)

[Out]

1/21*(e*x-2)*(e*x+5)/(e*x+2)^(3/2)/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 {5}{2}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+2)^(5/2)/(-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)^(5/2)), x)

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mupad [B]  time = 0.68, size = 65, normalized size = 0.92 \begin {gather*} -\frac {\left (\frac {x}{63\,e^2}+\frac {5}{63\,e^3}\right )\,{\left (12-3\,e^2\,x^2\right )}^{3/4}}{\frac {4\,\sqrt {e\,x+2}}{e^2}+x^2\,\sqrt {e\,x+2}+\frac {4\,x\,\sqrt {e\,x+2}}{e}} \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)^(5/2)),x)

[Out]

-((x/(63*e^2) + 5/(63*e^3))*(12 - 3*e^2*x^2)^(3/4))/((4*(e*x + 2)^(1/2))/e^2 + x^2*(e*x + 2)^(1/2) + (4*x*(e*x
+ 2)^(1/2))/e)

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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^{2} x^{2} \sqrt {e x + 2} \sqrt [4]{- e^{2} x^{2} + 4} + 4 e x \sqrt {e x + 2} \sqrt [4]{- e^{2} x^{2} + 4} + 4 \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)**(5/2)/(-3*e**2*x**2+12)**(1/4),x)

[Out]

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

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