3.9.95 \(\int (2+e x)^{3/2} \sqrt {12-3 e^2 x^2} \, dx\) [895]

Optimal. Leaf size=65 \[ -\frac {32 (2-e x)^{3/2}}{\sqrt {3} e}+\frac {16 \sqrt {3} (2-e x)^{5/2}}{5 e}-\frac {2 \sqrt {3} (2-e x)^{7/2}}{7 e} \]

[Out]

-32/3*(-e*x+2)^(3/2)/e*3^(1/2)+16/5*(-e*x+2)^(5/2)*3^(1/2)/e-2/7*(-e*x+2)^(7/2)*3^(1/2)/e

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Rubi [A]
time = 0.01, antiderivative size = 65, 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 {2 \sqrt {3} (2-e x)^{7/2}}{7 e}+\frac {16 \sqrt {3} (2-e x)^{5/2}}{5 e}-\frac {32 (2-e x)^{3/2}}{\sqrt {3} e} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-32*(2 - e*x)^(3/2))/(Sqrt[3]*e) + (16*Sqrt[3]*(2 - e*x)^(5/2))/(5*e) - (2*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 (2+e x)^{3/2} \sqrt {12-3 e^2 x^2} \, dx &=\int \sqrt {6-3 e x} (2+e x)^2 \, dx\\ &=\int \left (16 \sqrt {6-3 e x}-\frac {8}{3} (6-3 e x)^{3/2}+\frac {1}{9} (6-3 e x)^{5/2}\right ) \, dx\\ &=-\frac {32 (2-e x)^{3/2}}{\sqrt {3} e}+\frac {16 \sqrt {3} (2-e x)^{5/2}}{5 e}-\frac {2 \sqrt {3} (2-e x)^{7/2}}{7 e}\\ \end {align*}

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Mathematica [A]
time = 0.16, size = 50, normalized size = 0.77 \begin {gather*} \frac {2 (-2+e x) \sqrt {4-e^2 x^2} \left (284+108 e x+15 e^2 x^2\right )}{35 e \sqrt {6+3 e x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(-2 + e*x)*Sqrt[4 - e^2*x^2]*(284 + 108*e*x + 15*e^2*x^2))/(35*e*Sqrt[6 + 3*e*x])

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Maple [A]
time = 0.47, size = 44, normalized size = 0.68

method	result	size
gosper	\(\frac {2 \left (e x -2\right ) \left (15 e^{2} x^{2}+108 e x +284\right ) \sqrt {-3 e^{2} x^{2}+12}}{105 \sqrt {e x +2}\, e}\)	\(44\)
default	\(\frac {2 \left (e x -2\right ) \left (15 e^{2} x^{2}+108 e x +284\right ) \sqrt {-3 e^{2} x^{2}+12}}{105 \sqrt {e x +2}\, e}\)	\(44\)
risch	\(-\frac {2 \sqrt {\frac {-3 e^{2} x^{2}+12}{e x +2}}\, \sqrt {e x +2}\, \left (15 e^{3} x^{3}+78 e^{2} x^{2}+68 e x -568\right ) \left (e x -2\right )}{35 \sqrt {-3 e^{2} x^{2}+12}\, e \sqrt {-3 e x +6}}\)	\(80\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/105*(e*x-2)*(15*e^2*x^2+108*e*x+284)*(-3*e^2*x^2+12)^(1/2)/(e*x+2)^(1/2)/e

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Maxima [C] Result contains complex when optimal does not.
time = 0.55, size = 61, normalized size = 0.94 \begin {gather*} -\frac {2 \, {\left (-15 i \, \sqrt {3} x^{3} e^{3} - 78 i \, \sqrt {3} x^{2} e^{2} - 68 i \, \sqrt {3} x e + 568 i \, \sqrt {3}\right )} {\left (x e + 2\right )} \sqrt {x e - 2}}{105 \, {\left (x e^{2} + 2 \, e\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/105*(-15*I*sqrt(3)*x^3*e^3 - 78*I*sqrt(3)*x^2*e^2 - 68*I*sqrt(3)*x*e + 568*I*sqrt(3))*(x*e + 2)*sqrt(x*e -
2)/(x*e^2 + 2*e)

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Fricas [A]
time = 1.78, size = 53, normalized size = 0.82 \begin {gather*} \frac {2 \, {\left (15 \, x^{3} e^{3} + 78 \, x^{2} e^{2} + 68 \, x e - 568\right )} \sqrt {-3 \, x^{2} e^{2} + 12} \sqrt {x e + 2}}{105 \, {\left (x e^{2} + 2 \, e\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/105*(15*x^3*e^3 + 78*x^2*e^2 + 68*x*e - 568)*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*} \sqrt {3} \left (\int 2 \sqrt {e x + 2} \sqrt {- e^{2} x^{2} + 4}\, dx + \int e x \sqrt {e x + 2} \sqrt {- e^{2} x^{2} + 4}\, dx\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 100 vs. \(2 (49) = 98\).
time = 1.70, size = 100, normalized size = 1.54 \begin {gather*} \frac {2}{105} \, \sqrt {3} {\left (84 \, {\left (x e - 2\right )}^{2} \sqrt {-x e + 2} + {\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} - 420 \, {\left (-x e + 2\right )}^{\frac {3}{2}} + 672\right )} e^{\left (-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/105*sqrt(3)*(84*(x*e - 2)^2*sqrt(-x*e + 2) + ((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 - 420*(-x*e + 2)^(3/2) + 672)*e^(-1)

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Mupad [B]
time = 0.52, size = 73, normalized size = 1.12 \begin {gather*} \frac {\sqrt {12-3\,e^2\,x^2}\,\left (\frac {52\,x^2\,\sqrt {e\,x+2}}{35}-\frac {1136\,\sqrt {e\,x+2}}{105\,e^2}+\frac {136\,x\,\sqrt {e\,x+2}}{105\,e}+\frac {2\,e\,x^3\,\sqrt {e\,x+2}}{7}\right )}{x+\frac {2}{e}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((12 - 3*e^2*x^2)^(1/2)*((52*x^2*(e*x + 2)^(1/2))/35 - (1136*(e*x + 2)^(1/2))/(105*e^2) + (136*x*(e*x + 2)^(1/
2))/(105*e) + (2*e*x^3*(e*x + 2)^(1/2))/7))/(x + 2/e)

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