3.8.52 \(\int \frac {(2+e x)^{7/2}}{\sqrt {12-3 e^2 x^2}} \, dx\)

Optimal. Leaf size=85 \[ \frac {2 (2-e x)^{7/2}}{7 \sqrt {3} e}-\frac {8 \sqrt {3} (2-e x)^{5/2}}{5 e}+\frac {32 (2-e x)^{3/2}}{\sqrt {3} e}-\frac {128 \sqrt {2-e x}}{\sqrt {3} e} \]

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Rubi [A]  time = 0.02, antiderivative size = 85, 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)^{7/2}}{7 \sqrt {3} e}-\frac {8 \sqrt {3} (2-e x)^{5/2}}{5 e}+\frac {32 (2-e x)^{3/2}}{\sqrt {3} e}-\frac {128 \sqrt {2-e x}}{\sqrt {3} e} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(2 + e*x)^(7/2)/Sqrt[12 - 3*e^2*x^2],x]

[Out]

(-128*Sqrt[2 - e*x])/(Sqrt[3]*e) + (32*(2 - e*x)^(3/2))/(Sqrt[3]*e) - (8*Sqrt[3]*(2 - e*x)^(5/2))/(5*e) + (2*(
2 - e*x)^(7/2))/(7*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)^{7/2}}{\sqrt {12-3 e^2 x^2}} \, dx &=\int \frac {(2+e x)^3}{\sqrt {6-3 e x}} \, dx\\ &=\int \left (\frac {64}{\sqrt {6-3 e x}}-16 \sqrt {6-3 e x}+\frac {4}{3} (6-3 e x)^{3/2}-\frac {1}{27} (6-3 e x)^{5/2}\right ) \, dx\\ &=-\frac {128 \sqrt {2-e x}}{\sqrt {3} e}+\frac {32 (2-e x)^{3/2}}{\sqrt {3} e}-\frac {8 \sqrt {3} (2-e x)^{5/2}}{5 e}+\frac {2 (2-e x)^{7/2}}{7 \sqrt {3} e}\\ \end {align*}

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Mathematica [A]  time = 0.08, size = 57, normalized size = 0.67 \begin {gather*} \frac {2 (e x-2) \sqrt {e x+2} \left (5 e^3 x^3+54 e^2 x^2+284 e x+1416\right )}{35 e \sqrt {12-3 e^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2 + e*x)^(7/2)/Sqrt[12 - 3*e^2*x^2],x]

[Out]

(2*(-2 + e*x)*Sqrt[2 + e*x]*(1416 + 284*e*x + 54*e^2*x^2 + 5*e^3*x^3))/(35*e*Sqrt[12 - 3*e^2*x^2])

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IntegrateAlgebraic [A]  time = 0.31, size = 85, normalized size = 1.00 \begin {gather*} -\frac {2 \sqrt {4 (e x+2)-(e x+2)^2} \left (5 \sqrt {3} (e x+2)^3+24 \sqrt {3} (e x+2)^2+128 \sqrt {3} (e x+2)+1024 \sqrt {3}\right )}{105 e \sqrt {e x+2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(2 + e*x)^(7/2)/Sqrt[12 - 3*e^2*x^2],x]

[Out]

(-2*Sqrt[4*(2 + e*x) - (2 + e*x)^2]*(1024*Sqrt[3] + 128*Sqrt[3]*(2 + e*x) + 24*Sqrt[3]*(2 + e*x)^2 + 5*Sqrt[3]
*(2 + e*x)^3))/(105*e*Sqrt[2 + e*x])

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fricas [A]  time = 0.41, size = 54, normalized size = 0.64 \begin {gather*} -\frac {2 \, {\left (5 \, e^{3} x^{3} + 54 \, e^{2} x^{2} + 284 \, e x + 1416\right )} \sqrt {-3 \, e^{2} x^{2} + 12} \sqrt {e x + 2}}{105 \, {\left (e^{2} x + 2 \, e\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+2)^(7/2)/(-3*e^2*x^2+12)^(1/2),x, algorithm="fricas")

[Out]

-2/105*(5*e^3*x^3 + 54*e^2*x^2 + 284*e*x + 1416)*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 \frac {{\left (e x + 2\right )}^{\frac {7}{2}}}{\sqrt {-3 \, e^{2} x^{2} + 12}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+2)^(7/2)/(-3*e^2*x^2+12)^(1/2),x, algorithm="giac")

[Out]

integrate((e*x + 2)^(7/2)/sqrt(-3*e^2*x^2 + 12), x)

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maple [A]  time = 0.05, size = 52, normalized size = 0.61 \begin {gather*} \frac {2 \left (e x -2\right ) \left (5 e^{3} x^{3}+54 e^{2} x^{2}+284 e x +1416\right ) \sqrt {e x +2}}{35 \sqrt {-3 e^{2} x^{2}+12}\, e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+2)^(7/2)/(-3*e^2*x^2+12)^(1/2),x)

[Out]

2/35*(e*x-2)*(5*e^3*x^3+54*e^2*x^2+284*e*x+1416)*(e*x+2)^(1/2)/e/(-3*e^2*x^2+12)^(1/2)

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maxima [C]  time = 3.18, size = 45, normalized size = 0.53 \begin {gather*} -\frac {2 i \, \sqrt {3} {\left (5 \, e^{4} x^{4} + 44 \, e^{3} x^{3} + 176 \, e^{2} x^{2} + 848 \, e x - 2832\right )}}{105 \, \sqrt {e x - 2} e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+2)^(7/2)/(-3*e^2*x^2+12)^(1/2),x, algorithm="maxima")

[Out]

-2/105*I*sqrt(3)*(5*e^4*x^4 + 44*e^3*x^3 + 176*e^2*x^2 + 848*e*x - 2832)/(sqrt(e*x - 2)*e)

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mupad [B]  time = 0.22, size = 74, normalized size = 0.87 \begin {gather*} -\frac {\sqrt {12-3\,e^2\,x^2}\,\left (\frac {944\,\sqrt {e\,x+2}}{35\,e^2}+\frac {36\,x^2\,\sqrt {e\,x+2}}{35}+\frac {568\,x\,\sqrt {e\,x+2}}{105\,e}+\frac {2\,e\,x^3\,\sqrt {e\,x+2}}{21}\right )}{x+\frac {2}{e}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x + 2)^(7/2)/(12 - 3*e^2*x^2)^(1/2),x)

[Out]

-((12 - 3*e^2*x^2)^(1/2)*((944*(e*x + 2)^(1/2))/(35*e^2) + (36*x^2*(e*x + 2)^(1/2))/35 + (568*x*(e*x + 2)^(1/2
))/(105*e) + (2*e*x^3*(e*x + 2)^(1/2))/21))/(x + 2/e)

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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)**(7/2)/(-3*e**2*x**2+12)**(1/2),x)

[Out]

Timed out

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