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

Optimal. Leaf size=45 \[ \frac {6 \sqrt {3} (2-e x)^{7/2}}{7 e}-\frac {24 \sqrt {3} (2-e x)^{5/2}}{5 e} \]

[Out]

-24/5*(-e*x+2)^(5/2)*3^(1/2)/e+6/7*(-e*x+2)^(7/2)*3^(1/2)/e

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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} \[ \frac {6 \sqrt {3} (2-e x)^{7/2}}{7 e}-\frac {24 \sqrt {3} (2-e x)^{5/2}}{5 e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-24*Sqrt[3]*(2 - e*x)^(5/2))/(5*e) + (6*Sqrt[3]*(2 - e*x)^(7/2))/(7*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 {\left (12-3 e^2 x^2\right )^{3/2}}{\sqrt {2+e x}} \, dx &=\int (6-3 e x)^{3/2} (2+e x) \, dx\\ &=\int \left (4 (6-3 e x)^{3/2}-\frac {1}{3} (6-3 e x)^{5/2}\right ) \, dx\\ &=-\frac {24 \sqrt {3} (2-e x)^{5/2}}{5 e}+\frac {6 \sqrt {3} (2-e x)^{7/2}}{7 e}\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 43, normalized size = 0.96 \[ -\frac {6 (e x-2)^2 (5 e x+18) \sqrt {12-3 e^2 x^2}}{35 e \sqrt {e x+2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-6*(-2 + e*x)^2*(18 + 5*e*x)*Sqrt[12 - 3*e^2*x^2])/(35*e*Sqrt[2 + e*x])

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fricas [A]  time = 0.99, size = 54, normalized size = 1.20 \[ -\frac {6 \, {\left (5 \, e^{3} x^{3} - 2 \, e^{2} x^{2} - 52 \, e x + 72\right )} \sqrt {-3 \, e^{2} x^{2} + 12} \sqrt {e x + 2}}{35 \, {\left (e^{2} x + 2 \, e\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-6/35*(5*e^3*x^3 - 2*e^2*x^2 - 52*e*x + 72)*sqrt(-3*e^2*x^2 + 12)*sqrt(e*x + 2)/(e^2*x + 2*e)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.04, size = 36, normalized size = 0.80 \[ \frac {2 \left (e x -2\right ) \left (5 e x +18\right ) \left (-3 e^{2} x^{2}+12\right )^{\frac {3}{2}}}{35 \left (e x +2\right )^{\frac {3}{2}} e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/35*(e*x-2)*(5*e*x+18)*(-3*e^2*x^2+12)^(3/2)/e/(e*x+2)^(3/2)

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maxima [C]  time = 2.97, size = 47, normalized size = 1.04 \[ -\frac {{\left (30 i \, \sqrt {3} e^{3} x^{3} - 12 i \, \sqrt {3} e^{2} x^{2} - 312 i \, \sqrt {3} e x + 432 i \, \sqrt {3}\right )} \sqrt {e x - 2}}{35 \, e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/35*(30*I*sqrt(3)*e^3*x^3 - 12*I*sqrt(3)*e^2*x^2 - 312*I*sqrt(3)*e*x + 432*I*sqrt(3))*sqrt(e*x - 2)/e

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mupad [B]  time = 0.17, size = 43, normalized size = 0.96 \[ \frac {\sqrt {12-3\,e^2\,x^2}\,\left (\frac {312\,x}{35}+\frac {12\,e\,x^2}{35}-\frac {432}{35\,e}-\frac {6\,e^2\,x^3}{7}\right )}{\sqrt {e\,x+2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((12 - 3*e^2*x^2)^(1/2)*((312*x)/35 + (12*e*x^2)/35 - 432/(35*e) - (6*e^2*x^3)/7))/(e*x + 2)^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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