3.10.4 \(\int \frac {(12-3 e^2 x^2)^{3/2}}{\sqrt {2+e x}} \, dx\) [904]

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

[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 = 0.01, 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 = {641, 45} \begin {gather*} \frac {6 \sqrt {3} (2-e x)^{7/2}}{7 e}-\frac {24 \sqrt {3} (2-e x)^{5/2}}{5 e} \end {gather*}

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 45

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 641

Int[((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[(d + e*x)^(m + p)*(a/d + (c/e)*x)^
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 = 0.17, size = 43, normalized size = 0.96 \begin {gather*} -\frac {6 (-2+e x)^2 (18+5 e x) \sqrt {12-3 e^2 x^2}}{35 e \sqrt {2+e x}} \end {gather*}

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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Maple [A]
time = 0.47, size = 38, normalized size = 0.84

method	result	size
gosper	\(\frac {2 \left (e x -2\right ) \left (5 e x +18\right ) \left (-3 e^{2} x^{2}+12\right )^{\frac {3}{2}}}{35 e \left (e x +2\right )^{\frac {3}{2}}}\)	\(36\)
default	\(-\frac {6 \sqrt {-3 e^{2} x^{2}+12}\, \left (e x -2\right )^{2} \left (5 e x +18\right )}{35 \sqrt {e x +2}\, e}\)	\(38\)
risch	\(\frac {18 \sqrt {\frac {-3 e^{2} x^{2}+12}{e x +2}}\, \sqrt {e x +2}\, \left (5 e^{3} x^{3}-2 e^{2} x^{2}-52 e x +72\right ) \left (e x -2\right )}{35 \sqrt {-3 e^{2} x^{2}+12}\, e \sqrt {-3 e x +6}}\)	\(80\)

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,method=_RETURNVERBOSE)

[Out]

-6/35*(-3*e^2*x^2+12)^(1/2)/(e*x+2)^(1/2)*(e*x-2)^2*(5*e*x+18)/e

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Maxima [C] Result contains complex when optimal does not.
time = 0.50, size = 46, normalized size = 1.02 \begin {gather*} \frac {6}{35} \, {\left (-5 i \, \sqrt {3} x^{3} e^{3} + 2 i \, \sqrt {3} x^{2} e^{2} + 52 i \, \sqrt {3} x e - 72 i \, \sqrt {3}\right )} \sqrt {x e - 2} e^{\left (-1\right )} \end {gather*}

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]

6/35*(-5*I*sqrt(3)*x^3*e^3 + 2*I*sqrt(3)*x^2*e^2 + 52*I*sqrt(3)*x*e - 72*I*sqrt(3))*sqrt(x*e - 2)*e^(-1)

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Fricas [A]
time = 1.93, size = 53, normalized size = 1.18 \begin {gather*} -\frac {6 \, {\left (5 \, x^{3} e^{3} - 2 \, x^{2} e^{2} - 52 \, x e + 72\right )} \sqrt {-3 \, x^{2} e^{2} + 12} \sqrt {x e + 2}}{35 \, {\left (x e^{2} + 2 \, e\right )}} \end {gather*}

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} 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 ) \end {gather*}

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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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 81 vs. \(2 (33) = 66\).
time = 0.99, size = 81, normalized size = 1.80 \begin {gather*} -\frac {2}{35} \, \sqrt {3} {\left ({\left ({\left (15 \, {\left (x e - 2\right )}^{3} \sqrt {-x e + 2} + 84 \, {\left (x e - 2\right )}^{2} \sqrt {-x e + 2} - 140 \, {\left (-x e + 2\right )}^{\frac {3}{2}}\right )} e^{\left (-2\right )} + 352 \, e^{\left (-2\right )}\right )} e^{2} + 140 \, {\left (-x e + 2\right )}^{\frac {3}{2}} - 1120\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)^(3/2)/(e*x+2)^(1/2),x, algorithm="giac")

[Out]

-2/35*sqrt(3)*(((15*(x*e - 2)^3*sqrt(-x*e + 2) + 84*(x*e - 2)^2*sqrt(-x*e + 2) - 140*(-x*e + 2)^(3/2))*e^(-2)
+ 352*e^(-2))*e^2 + 140*(-x*e + 2)^(3/2) - 1120)*e^(-1)

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Mupad [B]
time = 0.17, size = 43, normalized size = 0.96 \begin {gather*} \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}} \end {gather*}

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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