\(\int (2+e x)^{3/2} (12-3 e^2 x^2)^{3/2} \, dx\) [902]

Optimal result

Integrand size = 24, antiderivative size = 87 \[ \int (2+e x)^{3/2} \left (12-3 e^2 x^2\right )^{3/2} \, dx=-\frac {384 \sqrt {3} (2-e x)^{5/2}}{5 e}+\frac {288 \sqrt {3} (2-e x)^{7/2}}{7 e}-\frac {8 \sqrt {3} (2-e x)^{9/2}}{e}+\frac {6 \sqrt {3} (2-e x)^{11/2}}{11 e} \]

[Out]

-384/5*(-e*x+2)^(5/2)*3^(1/2)/e+288/7*(-e*x+2)^(7/2)*3^(1/2)/e-8*(-e*x+2)^(9/2)/e*3^(1/2)+6/11*(-e*x+2)^(11/2)
*3^(1/2)/e

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {641, 45} \[ \int (2+e x)^{3/2} \left (12-3 e^2 x^2\right )^{3/2} \, dx=\frac {6 \sqrt {3} (2-e x)^{11/2}}{11 e}-\frac {8 \sqrt {3} (2-e x)^{9/2}}{e}+\frac {288 \sqrt {3} (2-e x)^{7/2}}{7 e}-\frac {384 \sqrt {3} (2-e x)^{5/2}}{5 e} \]

[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 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*} \text {integral}& = \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*}

Mathematica [A] (verified)

Time = 0.37 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.68 \[ \int (2+e x)^{3/2} \left (12-3 e^2 x^2\right )^{3/2} \, dx=-\frac {2 (-2+e x)^2 \sqrt {12-3 e^2 x^2} \left (4264+3020 e x+910 e^2 x^2+105 e^3 x^3\right )}{385 e \sqrt {2+e x}} \]

[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])

Maple [A] (verified)

Time = 2.22 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.60

method	result	size
gosper	\(\frac {2 \left (e x -2\right ) \left (105 e^{3} x^{3}+910 x^{2} e^{2}+3020 e x +4264\right ) \left (-3 x^{2} e^{2}+12\right )^{\frac {3}{2}}}{1155 e \left (e x +2\right )^{\frac {3}{2}}}\)	\(52\)
default	\(-\frac {2 \sqrt {-3 x^{2} e^{2}+12}\, \left (e x -2\right )^{2} \left (105 e^{3} x^{3}+910 x^{2} e^{2}+3020 e x +4264\right )}{385 \sqrt {e x +2}\, e}\)	\(54\)
risch	\(\frac {6 \sqrt {\frac {-3 x^{2} e^{2}+12}{e x +2}}\, \sqrt {e x +2}\, \left (105 e^{5} x^{5}+490 e^{4} x^{4}-200 e^{3} x^{3}-4176 x^{2} e^{2}-4976 e x +17056\right ) \left (e x -2\right )}{385 \sqrt {-3 x^{2} e^{2}+12}\, e \sqrt {-3 e x +6}}\)	\(96\)

[In]

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

[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)

Fricas [A] (verification not implemented)

none

Time = 0.36 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.80 \[ \int (2+e x)^{3/2} \left (12-3 e^2 x^2\right )^{3/2} \, dx=-\frac {2 \, {\left (105 \, e^{5} x^{5} + 490 \, e^{4} x^{4} - 200 \, e^{3} x^{3} - 4176 \, e^{2} x^{2} - 4976 \, e x + 17056\right )} \sqrt {-3 \, e^{2} x^{2} + 12} \sqrt {e x + 2}}{385 \, {\left (e^{2} x + 2 \, e\right )}} \]

[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)

Sympy [F]

\[ \int (2+e x)^{3/2} \left (12-3 e^2 x^2\right )^{3/2} \, dx=3 \sqrt {3} \left (\int 8 \sqrt {e x + 2} \sqrt {- e^{2} x^{2} + 4}\, dx + \int 4 e x \sqrt {e x + 2} \sqrt {- e^{2} x^{2} + 4}\, dx + \int \left (- 2 e^{2} x^{2} \sqrt {e x + 2} \sqrt {- e^{2} x^{2} + 4}\right )\, dx + \int \left (- e^{3} x^{3} \sqrt {e x + 2} \sqrt {- e^{2} x^{2} + 4}\right )\, dx\right ) \]

[In]

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

[Out]

3*sqrt(3)*(Integral(8*sqrt(e*x + 2)*sqrt(-e**2*x**2 + 4), x) + Integral(4*e*x*sqrt(e*x + 2)*sqrt(-e**2*x**2 +
4), x) + Integral(-2*e**2*x**2*sqrt(e*x + 2)*sqrt(-e**2*x**2 + 4), x) + Integral(-e**3*x**3*sqrt(e*x + 2)*sqrt
(-e**2*x**2 + 4), x))

Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.29 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.94 \[ \int (2+e x)^{3/2} \left (12-3 e^2 x^2\right )^{3/2} \, dx=\frac {2 \, {\left (-105 i \, \sqrt {3} e^{5} x^{5} - 490 i \, \sqrt {3} e^{4} x^{4} + 200 i \, \sqrt {3} e^{3} x^{3} + 4176 i \, \sqrt {3} e^{2} x^{2} + 4976 i \, \sqrt {3} e x - 17056 i \, \sqrt {3}\right )} {\left (e x + 2\right )} \sqrt {e x - 2}}{385 \, {\left (e^{2} x + 2 \, e\right )}} \]

[In]

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

[Out]

2/385*(-105*I*sqrt(3)*e^5*x^5 - 490*I*sqrt(3)*e^4*x^4 + 200*I*sqrt(3)*e^3*x^3 + 4176*I*sqrt(3)*e^2*x^2 + 4976*
I*sqrt(3)*e*x - 17056*I*sqrt(3))*(e*x + 2)*sqrt(e*x - 2)/(e^2*x + 2*e)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 207 vs. \(2 (65) = 130\).

Time = 0.29 (sec) , antiderivative size = 207, normalized size of antiderivative = 2.38 \[ \int (2+e x)^{3/2} \left (12-3 e^2 x^2\right )^{3/2} \, dx=-\frac {2 \, \sqrt {3} {\left (e^{4} {\left (\frac {315 \, {\left (e x - 2\right )}^{5} \sqrt {-e x + 2} + 3080 \, {\left (e x - 2\right )}^{4} \sqrt {-e x + 2} + 11880 \, {\left (e x - 2\right )}^{3} \sqrt {-e x + 2} + 22176 \, {\left (e x - 2\right )}^{2} \sqrt {-e x + 2} - 18480 \, {\left (-e x + 2\right )}^{\frac {3}{2}}}{e^{4}} + \frac {27008}{e^{4}}\right )} + 44 \, e^{3} {\left (\frac {35 \, {\left (e x - 2\right )}^{4} \sqrt {-e x + 2} + 270 \, {\left (e x - 2\right )}^{3} \sqrt {-e x + 2} + 756 \, {\left (e x - 2\right )}^{2} \sqrt {-e x + 2} - 840 \, {\left (-e x + 2\right )}^{\frac {3}{2}}}{e^{3}} - \frac {832}{e^{3}}\right )} - 11088 \, {\left (e x - 2\right )}^{2} \sqrt {-e x + 2} + 55440 \, {\left (-e x + 2\right )}^{\frac {3}{2}} - 88704\right )}}{1155 \, e} \]

[In]

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

[Out]

-2/1155*sqrt(3)*(e^4*((315*(e*x - 2)^5*sqrt(-e*x + 2) + 3080*(e*x - 2)^4*sqrt(-e*x + 2) + 11880*(e*x - 2)^3*sq
rt(-e*x + 2) + 22176*(e*x - 2)^2*sqrt(-e*x + 2) - 18480*(-e*x + 2)^(3/2))/e^4 + 27008/e^4) + 44*e^3*((35*(e*x
- 2)^4*sqrt(-e*x + 2) + 270*(e*x - 2)^3*sqrt(-e*x + 2) + 756*(e*x - 2)^2*sqrt(-e*x + 2) - 840*(-e*x + 2)^(3/2)
)/e^3 - 832/e^3) - 11088*(e*x - 2)^2*sqrt(-e*x + 2) + 55440*(-e*x + 2)^(3/2) - 88704)/e

Mupad [B] (verification not implemented)

Time = 10.06 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.61 \[ \int (2+e x)^{3/2} \left (12-3 e^2 x^2\right )^{3/2} \, dx=-\frac {2\,\sqrt {12-3\,e^2\,x^2}\,{\left (e\,x-2\right )}^2\,\left (105\,e^3\,x^3+910\,e^2\,x^2+3020\,e\,x+4264\right )}{385\,e\,\sqrt {e\,x+2}} \]

[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))