3.8.42 \(\int (2+e x)^{5/2} (12-3 e^2 x^2)^{3/2} \, dx\)

Optimal. Leaf size=109 \[ -\frac {6 \sqrt {3} (2-e x)^{13/2}}{13 e}+\frac {96 \sqrt {3} (2-e x)^{11/2}}{11 e}-\frac {64 \sqrt {3} (2-e x)^{9/2}}{e}+\frac {1536 \sqrt {3} (2-e x)^{7/2}}{7 e}-\frac {1536 \sqrt {3} (2-e x)^{5/2}}{5 e} \]

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Rubi [A]  time = 0.03, antiderivative size = 109, 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)^{13/2}}{13 e}+\frac {96 \sqrt {3} (2-e x)^{11/2}}{11 e}-\frac {64 \sqrt {3} (2-e x)^{9/2}}{e}+\frac {1536 \sqrt {3} (2-e x)^{7/2}}{7 e}-\frac {1536 \sqrt {3} (2-e x)^{5/2}}{5 e} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(2 + e*x)^(5/2)*(12 - 3*e^2*x^2)^(3/2),x]

[Out]

(-1536*Sqrt[3]*(2 - e*x)^(5/2))/(5*e) + (1536*Sqrt[3]*(2 - e*x)^(7/2))/(7*e) - (64*Sqrt[3]*(2 - e*x)^(9/2))/e
+ (96*Sqrt[3]*(2 - e*x)^(11/2))/(11*e) - (6*Sqrt[3]*(2 - e*x)^(13/2))/(13*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)^{5/2} \left (12-3 e^2 x^2\right )^{3/2} \, dx &=\int (6-3 e x)^{3/2} (2+e x)^4 \, dx\\ &=\int \left (256 (6-3 e x)^{3/2}-\frac {256}{3} (6-3 e x)^{5/2}+\frac {32}{3} (6-3 e x)^{7/2}-\frac {16}{27} (6-3 e x)^{9/2}+\frac {1}{81} (6-3 e x)^{11/2}\right ) \, dx\\ &=-\frac {1536 \sqrt {3} (2-e x)^{5/2}}{5 e}+\frac {1536 \sqrt {3} (2-e x)^{7/2}}{7 e}-\frac {64 \sqrt {3} (2-e x)^{9/2}}{e}+\frac {96 \sqrt {3} (2-e x)^{11/2}}{11 e}-\frac {6 \sqrt {3} (2-e x)^{13/2}}{13 e}\\ \end {align*}

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Mathematica [A]  time = 0.07, size = 67, normalized size = 0.61 \begin {gather*} -\frac {2 (e x-2)^2 \sqrt {12-3 e^2 x^2} \left (1155 e^4 x^4+12600 e^3 x^3+56840 e^2 x^2+133600 e x+154928\right )}{5005 e \sqrt {e x+2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2 + e*x)^(5/2)*(12 - 3*e^2*x^2)^(3/2),x]

[Out]

(-2*(-2 + e*x)^2*Sqrt[12 - 3*e^2*x^2]*(154928 + 133600*e*x + 56840*e^2*x^2 + 12600*e^3*x^3 + 1155*e^4*x^4))/(5
005*e*Sqrt[2 + e*x])

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IntegrateAlgebraic [A]  time = 0.31, size = 99, normalized size = 0.91 \begin {gather*} -\frac {2 \left (4 (e x+2)-(e x+2)^2\right )^{5/2} \left (1155 \sqrt {3} (e x+2)^4+3360 \sqrt {3} (e x+2)^3+8960 \sqrt {3} (e x+2)^2+20480 \sqrt {3} (e x+2)+32768 \sqrt {3}\right )}{5005 e (e x+2)^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(2 + e*x)^(5/2)*(12 - 3*e^2*x^2)^(3/2),x]

[Out]

(-2*(4*(2 + e*x) - (2 + e*x)^2)^(5/2)*(32768*Sqrt[3] + 20480*Sqrt[3]*(2 + e*x) + 8960*Sqrt[3]*(2 + e*x)^2 + 33
60*Sqrt[3]*(2 + e*x)^3 + 1155*Sqrt[3]*(2 + e*x)^4))/(5005*e*(2 + e*x)^(5/2))

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fricas [A]  time = 0.40, size = 78, normalized size = 0.72 \begin {gather*} -\frac {2 \, {\left (1155 \, e^{6} x^{6} + 7980 \, e^{5} x^{5} + 11060 \, e^{4} x^{4} - 43360 \, e^{3} x^{3} - 152112 \, e^{2} x^{2} - 85312 \, e x + 619712\right )} \sqrt {-3 \, e^{2} x^{2} + 12} \sqrt {e x + 2}}{5005 \, {\left (e^{2} x + 2 \, e\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+2)^(5/2)*(-3*e^2*x^2+12)^(3/2),x, algorithm="fricas")

[Out]

-2/5005*(1155*e^6*x^6 + 7980*e^5*x^5 + 11060*e^4*x^4 - 43360*e^3*x^3 - 152112*e^2*x^2 - 85312*e*x + 619712)*sq
rt(-3*e^2*x^2 + 12)*sqrt(e*x + 2)/(e^2*x + 2*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)^(5/2)*(-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 = 0.04, size = 60, normalized size = 0.55 \begin {gather*} \frac {2 \left (e x -2\right ) \left (1155 e^{4} x^{4}+12600 e^{3} x^{3}+56840 e^{2} x^{2}+133600 e x +154928\right ) \left (-3 e^{2} x^{2}+12\right )^{\frac {3}{2}}}{15015 \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)^(5/2)*(-3*e^2*x^2+12)^(3/2),x)

[Out]

2/15015*(e*x-2)*(1155*e^4*x^4+12600*e^3*x^3+56840*e^2*x^2+133600*e*x+154928)*(-3*e^2*x^2+12)^(3/2)/e/(e*x+2)^(
3/2)

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maxima [C]  time = 3.09, size = 93, normalized size = 0.85 \begin {gather*} -\frac {{\left (2310 i \, \sqrt {3} e^{6} x^{6} + 15960 i \, \sqrt {3} e^{5} x^{5} + 22120 i \, \sqrt {3} e^{4} x^{4} - 86720 i \, \sqrt {3} e^{3} x^{3} - 304224 i \, \sqrt {3} e^{2} x^{2} - 170624 i \, \sqrt {3} e x + 1239424 i \, \sqrt {3}\right )} {\left (e x + 2\right )} \sqrt {e x - 2}}{5005 \, {\left (e^{2} x + 2 \, e\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+2)^(5/2)*(-3*e^2*x^2+12)^(3/2),x, algorithm="maxima")

[Out]

-1/5005*(2310*I*sqrt(3)*e^6*x^6 + 15960*I*sqrt(3)*e^5*x^5 + 22120*I*sqrt(3)*e^4*x^4 - 86720*I*sqrt(3)*e^3*x^3
- 304224*I*sqrt(3)*e^2*x^2 - 170624*I*sqrt(3)*e*x + 1239424*I*sqrt(3))*(e*x + 2)*sqrt(e*x - 2)/(e^2*x + 2*e)

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mupad [B]  time = 0.27, size = 87, normalized size = 0.80 \begin {gather*} \frac {2\,\sqrt {12-3\,e^2\,x^2}\,\sqrt {e\,x+2}\,\left (-1155\,e^5\,x^5-5670\,e^4\,x^4+280\,e^3\,x^3+42800\,e^2\,x^2+66512\,e\,x-47712\right )}{5005\,e}-\frac {1048576\,\sqrt {12-3\,e^2\,x^2}}{5005\,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)^(5/2),x)

[Out]

(2*(12 - 3*e^2*x^2)^(1/2)*(e*x + 2)^(1/2)*(66512*e*x + 42800*e^2*x^2 + 280*e^3*x^3 - 5670*e^4*x^4 - 1155*e^5*x
^5 - 47712))/(5005*e) - (1048576*(12 - 3*e^2*x^2)^(1/2))/(5005*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)**(5/2)*(-3*e**2*x**2+12)**(3/2),x)

[Out]

Timed out

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