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

Optimal. Leaf size=87 \[ \frac {6 \sqrt {3} (2-e x)^{11/2}}{11 e}-\frac {8 \sqrt {3} (2-e x)^{9/2}}{e}+\frac {288 \sqrt {3} (2-e x)^{7/2}}{7 e}-\frac {384 \sqrt {3} (2-e x)^{5/2}}{5 e} \]

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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 {6 \sqrt {3} (2-e x)^{11/2}}{11 e}-\frac {8 \sqrt {3} (2-e x)^{9/2}}{e}+\frac {288 \sqrt {3} (2-e x)^{7/2}}{7 e}-\frac {384 \sqrt {3} (2-e x)^{5/2}}{5 e} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-384*Sqrt[3]*(2 - e*x)^(5/2))/(5*e) + (288*Sqrt[3]*(2 - e*x)^(7/2))/(7*e) - (8*Sqrt[3]*(2 - e*x)^(9/2))/e + (
6*Sqrt[3]*(2 - e*x)^(11/2))/(11*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 (2+e x)^{3/2} \left (12-3 e^2 x^2\right )^{3/2} \, dx &=\int (6-3 e x)^{3/2} (2+e x)^3 \, dx\\ &=\int \left (64 (6-3 e x)^{3/2}-16 (6-3 e x)^{5/2}+\frac {4}{3} (6-3 e x)^{7/2}-\frac {1}{27} (6-3 e x)^{9/2}\right ) \, dx\\ &=-\frac {384 \sqrt {3} (2-e x)^{5/2}}{5 e}+\frac {288 \sqrt {3} (2-e x)^{7/2}}{7 e}-\frac {8 \sqrt {3} (2-e x)^{9/2}}{e}+\frac {6 \sqrt {3} (2-e x)^{11/2}}{11 e}\\ \end {align*}

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Mathematica [A]  time = 0.06, size = 59, normalized size = 0.68 \begin {gather*} -\frac {2 (e x-2)^2 \sqrt {12-3 e^2 x^2} \left (105 e^3 x^3+910 e^2 x^2+3020 e x+4264\right )}{385 e \sqrt {e x+2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(-2 + e*x)^2*Sqrt[12 - 3*e^2*x^2]*(4264 + 3020*e*x + 910*e^2*x^2 + 105*e^3*x^3))/(385*e*Sqrt[2 + e*x])

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IntegrateAlgebraic [A]  time = 0.29, size = 85, normalized size = 0.98 \begin {gather*} -\frac {2 \left (4 (e x+2)-(e x+2)^2\right )^{5/2} \left (105 \sqrt {3} (e x+2)^3+280 \sqrt {3} (e x+2)^2+640 \sqrt {3} (e x+2)+1024 \sqrt {3}\right )}{385 e (e x+2)^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(4*(2 + e*x) - (2 + e*x)^2)^(5/2)*(1024*Sqrt[3] + 640*Sqrt[3]*(2 + e*x) + 280*Sqrt[3]*(2 + e*x)^2 + 105*Sq
rt[3]*(2 + e*x)^3))/(385*e*(2 + e*x)^(5/2))

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fricas [A]  time = 0.40, size = 70, normalized size = 0.80 \begin {gather*} -\frac {2 \, {\left (105 \, e^{5} x^{5} + 490 \, e^{4} x^{4} - 200 \, e^{3} x^{3} - 4176 \, e^{2} x^{2} - 4976 \, e x + 17056\right )} \sqrt {-3 \, e^{2} x^{2} + 12} \sqrt {e x + 2}}{385 \, {\left (e^{2} x + 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)^(3/2),x, algorithm="fricas")

[Out]

-2/385*(105*e^5*x^5 + 490*e^4*x^4 - 200*e^3*x^3 - 4176*e^2*x^2 - 4976*e*x + 17056)*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 \begin {gather*} \int {\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac {3}{2}} {\left (e x + 2\right )}^{\frac {3}{2}}\,{d x} \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)^(3/2),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.04, size = 52, normalized size = 0.60 \begin {gather*} \frac {2 \left (e x -2\right ) \left (105 e^{3} x^{3}+910 e^{2} x^{2}+3020 e x +4264\right ) \left (-3 e^{2} x^{2}+12\right )^{\frac {3}{2}}}{1155 \left (e x +2\right )^{\frac {3}{2}} e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/1155*(e*x-2)*(105*e^3*x^3+910*e^2*x^2+3020*e*x+4264)*(-3*e^2*x^2+12)^(3/2)/e/(e*x+2)^(3/2)

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maxima [C]  time = 3.19, size = 82, normalized size = 0.94 \begin {gather*} -\frac {{\left (210 i \, \sqrt {3} e^{5} x^{5} + 980 i \, \sqrt {3} e^{4} x^{4} - 400 i \, \sqrt {3} e^{3} x^{3} - 8352 i \, \sqrt {3} e^{2} x^{2} - 9952 i \, \sqrt {3} e x + 34112 i \, \sqrt {3}\right )} {\left (e x + 2\right )} \sqrt {e x - 2}}{385 \, {\left (e^{2} x + 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)^(3/2),x, algorithm="maxima")

[Out]

-1/385*(210*I*sqrt(3)*e^5*x^5 + 980*I*sqrt(3)*e^4*x^4 - 400*I*sqrt(3)*e^3*x^3 - 8352*I*sqrt(3)*e^2*x^2 - 9952*
I*sqrt(3)*e*x + 34112*I*sqrt(3))*(e*x + 2)*sqrt(e*x - 2)/(e^2*x + 2*e)

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mupad [B]  time = 0.59, size = 53, normalized size = 0.61 \begin {gather*} -\frac {2\,\sqrt {12-3\,e^2\,x^2}\,{\left (e\,x-2\right )}^2\,\left (105\,e^3\,x^3+910\,e^2\,x^2+3020\,e\,x+4264\right )}{385\,e\,\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)^(3/2),x)

[Out]

-(2*(12 - 3*e^2*x^2)^(1/2)*(e*x - 2)^2*(3020*e*x + 910*e^2*x^2 + 105*e^3*x^3 + 4264))/(385*e*(e*x + 2)^(1/2))

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

[Out]

Timed out

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