3.710 \(\int e^{6 x} \sqrt{9-e^{2 x}} \, dx\)

Optimal. Leaf size=50 \[ -\frac{1}{7} \left (9-e^{2 x}\right )^{7/2}+\frac{18}{5} \left (9-e^{2 x}\right )^{5/2}-27 \left (9-e^{2 x}\right )^{3/2} \]

[Out]

-27*(9 - E^(2*x))^(3/2) + (18*(9 - E^(2*x))^(5/2))/5 - (9 - E^(2*x))^(7/2)/7

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Rubi [A]  time = 0.0362776, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {2248, 43} \[ -\frac{1}{7} \left (9-e^{2 x}\right )^{7/2}+\frac{18}{5} \left (9-e^{2 x}\right )^{5/2}-27 \left (9-e^{2 x}\right )^{3/2} \]

Antiderivative was successfully verified.

[In]

Int[E^(6*x)*Sqrt[9 - E^(2*x)],x]

[Out]

-27*(9 - E^(2*x))^(3/2) + (18*(9 - E^(2*x))^(5/2))/5 - (9 - E^(2*x))^(7/2)/7

Rule 2248

Int[((a_) + (b_.)*(F_)^((e_.)*((c_.) + (d_.)*(x_))))^(p_.)*(G_)^((h_.)*((f_.) + (g_.)*(x_))), x_Symbol] :> Wit
h[{m = FullSimplify[(g*h*Log[G])/(d*e*Log[F])]}, Dist[(Denominator[m]*G^(f*h - (c*g*h)/d))/(d*e*Log[F]), Subst
[Int[x^(Numerator[m] - 1)*(a + b*x^Denominator[m])^p, x], x, F^((e*(c + d*x))/Denominator[m])], x] /; LeQ[m, -
1] || GeQ[m, 1]] /; FreeQ[{F, G, a, b, c, d, e, f, g, h, p}, x]

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])

Rubi steps

\begin{align*} \int e^{6 x} \sqrt{9-e^{2 x}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \sqrt{9-x} x^2 \, dx,x,e^{2 x}\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (81 \sqrt{9-x}-18 (9-x)^{3/2}+(9-x)^{5/2}\right ) \, dx,x,e^{2 x}\right )\\ &=-27 \left (9-e^{2 x}\right )^{3/2}+\frac{18}{5} \left (9-e^{2 x}\right )^{5/2}-\frac{1}{7} \left (9-e^{2 x}\right )^{7/2}\\ \end{align*}

Mathematica [A]  time = 0.0196505, size = 33, normalized size = 0.66 \[ -\frac{1}{35} \left (9-e^{2 x}\right )^{3/2} \left (36 e^{2 x}+5 e^{4 x}+216\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[E^(6*x)*Sqrt[9 - E^(2*x)],x]

[Out]

-((9 - E^(2*x))^(3/2)*(216 + 36*E^(2*x) + 5*E^(4*x)))/35

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Maple [A]  time = 0.021, size = 46, normalized size = 0.9 \begin{align*} -{\frac{ \left ({{\rm e}^{x}} \right ) ^{4}}{7} \left ( 9- \left ({{\rm e}^{x}} \right ) ^{2} \right ) ^{{\frac{3}{2}}}}-{\frac{36\, \left ({{\rm e}^{x}} \right ) ^{2}}{35} \left ( 9- \left ({{\rm e}^{x}} \right ) ^{2} \right ) ^{{\frac{3}{2}}}}-{\frac{216}{35} \left ( 9- \left ({{\rm e}^{x}} \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(6*x)*(9-exp(2*x))^(1/2),x)

[Out]

-1/7*exp(x)^4*(9-exp(x)^2)^(3/2)-36/35*exp(x)^2*(9-exp(x)^2)^(3/2)-216/35*(9-exp(x)^2)^(3/2)

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Maxima [A]  time = 0.974887, size = 50, normalized size = 1. \begin{align*} -\frac{1}{7} \,{\left (-e^{\left (2 \, x\right )} + 9\right )}^{\frac{7}{2}} + \frac{18}{5} \,{\left (-e^{\left (2 \, x\right )} + 9\right )}^{\frac{5}{2}} - 27 \,{\left (-e^{\left (2 \, x\right )} + 9\right )}^{\frac{3}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(6*x)*(9-exp(2*x))^(1/2),x, algorithm="maxima")

[Out]

-1/7*(-e^(2*x) + 9)^(7/2) + 18/5*(-e^(2*x) + 9)^(5/2) - 27*(-e^(2*x) + 9)^(3/2)

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Fricas [A]  time = 0.872713, size = 95, normalized size = 1.9 \begin{align*} \frac{1}{35} \,{\left (5 \, e^{\left (6 \, x\right )} - 9 \, e^{\left (4 \, x\right )} - 108 \, e^{\left (2 \, x\right )} - 1944\right )} \sqrt{-e^{\left (2 \, x\right )} + 9} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(6*x)*(9-exp(2*x))^(1/2),x, algorithm="fricas")

[Out]

1/35*(5*e^(6*x) - 9*e^(4*x) - 108*e^(2*x) - 1944)*sqrt(-e^(2*x) + 9)

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Sympy [A]  time = 3.56376, size = 41, normalized size = 0.82 \begin{align*} \begin{cases} - \frac{\left (9 - e^{2 x}\right )^{\frac{7}{2}}}{7} + \frac{18 \left (9 - e^{2 x}\right )^{\frac{5}{2}}}{5} - 27 \left (9 - e^{2 x}\right )^{\frac{3}{2}} & \text{for}\: e^{x} < \log{\left (3 \right )} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(6*x)*(9-exp(2*x))**(1/2),x)

[Out]

Piecewise((-(9 - exp(2*x))**(7/2)/7 + 18*(9 - exp(2*x))**(5/2)/5 - 27*(9 - exp(2*x))**(3/2), exp(x) < log(3)))

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Giac [A]  time = 1.31294, size = 72, normalized size = 1.44 \begin{align*} \frac{1}{7} \,{\left (e^{\left (2 \, x\right )} - 9\right )}^{3} \sqrt{-e^{\left (2 \, x\right )} + 9} + \frac{18}{5} \,{\left (e^{\left (2 \, x\right )} - 9\right )}^{2} \sqrt{-e^{\left (2 \, x\right )} + 9} - 27 \,{\left (-e^{\left (2 \, x\right )} + 9\right )}^{\frac{3}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(6*x)*(9-exp(2*x))^(1/2),x, algorithm="giac")

[Out]

1/7*(e^(2*x) - 9)^3*sqrt(-e^(2*x) + 9) + 18/5*(e^(2*x) - 9)^2*sqrt(-e^(2*x) + 9) - 27*(-e^(2*x) + 9)^(3/2)