3.8.60 \(\int \frac {(2+e x)^{9/2}}{(12-3 e^2 x^2)^{3/2}} \, dx\)

Optimal. Leaf size=87 \[ \frac {2 (2-e x)^{5/2}}{15 \sqrt {3} e}-\frac {8 (2-e x)^{3/2}}{3 \sqrt {3} e}+\frac {32 \sqrt {2-e x}}{\sqrt {3} e}+\frac {128}{3 \sqrt {3} e \sqrt {2-e x}} \]

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Rubi [A]  time = 0.02, antiderivative size = 87, 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} \begin {gather*} \frac {2 (2-e x)^{5/2}}{15 \sqrt {3} e}-\frac {8 (2-e x)^{3/2}}{3 \sqrt {3} e}+\frac {32 \sqrt {2-e x}}{\sqrt {3} e}+\frac {128}{3 \sqrt {3} e \sqrt {2-e x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

128/(3*Sqrt[3]*e*Sqrt[2 - e*x]) + (32*Sqrt[2 - e*x])/(Sqrt[3]*e) - (8*(2 - e*x)^(3/2))/(3*Sqrt[3]*e) + (2*(2 -
e*x)^(5/2))/(15*Sqrt[3]*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 {(2+e x)^{9/2}}{\left (12-3 e^2 x^2\right )^{3/2}} \, dx &=\int \frac {(2+e x)^3}{(6-3 e x)^{3/2}} \, dx\\ &=\int \left (\frac {64}{(6-3 e x)^{3/2}}-\frac {16}{\sqrt {6-3 e x}}+\frac {4}{3} \sqrt {6-3 e x}-\frac {1}{27} (6-3 e x)^{3/2}\right ) \, dx\\ &=\frac {128}{3 \sqrt {3} e \sqrt {2-e x}}+\frac {32 \sqrt {2-e x}}{\sqrt {3} e}-\frac {8 (2-e x)^{3/2}}{3 \sqrt {3} e}+\frac {2 (2-e x)^{5/2}}{15 \sqrt {3} e}\\ \end {align*}

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Mathematica [A]  time = 0.08, size = 51, normalized size = 0.59 \begin {gather*} -\frac {2 \sqrt {e x+2} \left (e^3 x^3+14 e^2 x^2+172 e x-728\right )}{15 e \sqrt {12-3 e^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[2 + e*x]*(-728 + 172*e*x + 14*e^2*x^2 + e^3*x^3))/(15*e*Sqrt[12 - 3*e^2*x^2])

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IntegrateAlgebraic [A]  time = 0.42, size = 91, normalized size = 1.05 \begin {gather*} \frac {2 \sqrt {4 (e x+2)-(e x+2)^2} \left (\sqrt {3} (e x+2)^3+8 \sqrt {3} (e x+2)^2+128 \sqrt {3} (e x+2)-1024 \sqrt {3}\right )}{45 e (e x-2) \sqrt {e x+2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[4*(2 + e*x) - (2 + e*x)^2]*(-1024*Sqrt[3] + 128*Sqrt[3]*(2 + e*x) + 8*Sqrt[3]*(2 + e*x)^2 + Sqrt[3]*(2
+ e*x)^3))/(45*e*(-2 + e*x)*Sqrt[2 + e*x])

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fricas [A]  time = 0.40, size = 55, normalized size = 0.63 \begin {gather*} \frac {2 \, {\left (e^{3} x^{3} + 14 \, e^{2} x^{2} + 172 \, e x - 728\right )} \sqrt {-3 \, e^{2} x^{2} + 12} \sqrt {e x + 2}}{45 \, {\left (e^{3} x^{2} - 4 \, e\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/45*(e^3*x^3 + 14*e^2*x^2 + 172*e*x - 728)*sqrt(-3*e^2*x^2 + 12)*sqrt(e*x + 2)/(e^3*x^2 - 4*e)

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Erro
r index.cc index_gcd Error: Bad Argument ValueError index.cc index_gcd Error: Bad Argument Value

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maple [A]  time = 0.05, size = 51, normalized size = 0.59 \begin {gather*} \frac {2 \left (e x -2\right ) \left (e^{3} x^{3}+14 e^{2} x^{2}+172 e x -728\right ) \left (e x +2\right )^{\frac {3}{2}}}{5 \left (-3 e^{2} x^{2}+12\right )^{\frac {3}{2}} e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/5*(e*x-2)*(e^3*x^3+14*e^2*x^2+172*e*x-728)*(e*x+2)^(3/2)/e/(-3*e^2*x^2+12)^(3/2)

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maxima [C]  time = 3.07, size = 36, normalized size = 0.41 \begin {gather*} \frac {2 i \, \sqrt {3} {\left (e^{3} x^{3} + 14 \, e^{2} x^{2} + 172 \, e x - 728\right )}}{45 \, \sqrt {e x - 2} e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/45*I*sqrt(3)*(e^3*x^3 + 14*e^2*x^2 + 172*e*x - 728)/(sqrt(e*x - 2)*e)

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mupad [B]  time = 0.62, size = 80, normalized size = 0.92 \begin {gather*} -\frac {\sqrt {12-3\,e^2\,x^2}\,\left (\frac {2\,x^3\,\sqrt {e\,x+2}}{45}-\frac {1456\,\sqrt {e\,x+2}}{45\,e^3}+\frac {344\,x\,\sqrt {e\,x+2}}{45\,e^2}+\frac {28\,x^2\,\sqrt {e\,x+2}}{45\,e}\right )}{\frac {4}{e^2}-x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-((12 - 3*e^2*x^2)^(1/2)*((2*x^3*(e*x + 2)^(1/2))/45 - (1456*(e*x + 2)^(1/2))/(45*e^3) + (344*x*(e*x + 2)^(1/2
))/(45*e^2) + (28*x^2*(e*x + 2)^(1/2))/(45*e)))/(4/e^2 - x^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((e*x+2)**(9/2)/(-3*e**2*x**2+12)**(3/2),x)

[Out]

Timed out

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