3.8.10 \(\int e^{6 x} \sqrt {9-e^{2 x}} \, dx\) [710]

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

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 50, normalized size of antiderivative = 1.00, 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 = {2280, 45} \begin {gather*} -\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} \end {gather*}

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 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 2280

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]

Rubi steps

\begin {align*} \int e^{6 x} \sqrt {9-e^{2 x}} \, dx &=\frac {1}{2} \text {Subst}\left (\int \sqrt {9-x} x^2 \, dx,x,e^{2 x}\right )\\ &=\frac {1}{2} \text {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*}

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Mathematica [A]
time = 0.03, size = 33, normalized size = 0.66 \begin {gather*} -\frac {1}{35} \left (9-e^{2 x}\right )^{3/2} \left (216+36 e^{2 x}+5 e^{4 x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/35*((9 - E^(2*x))^(3/2)*(216 + 36*E^(2*x) + 5*E^(4*x)))

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Maple [A]
time = 0.02, size = 46, normalized size = 0.92

method	result	size
risch	\(-\frac {\left (5 \,{\mathrm e}^{6 x}-9 \,{\mathrm e}^{4 x}-108 \,{\mathrm e}^{2 x}-1944\right ) \left (-9+{\mathrm e}^{2 x}\right )}{35 \sqrt {9-{\mathrm e}^{2 x}}}\)	\(39\)
default	\(-\frac {{\mathrm e}^{4 x} \left (9-{\mathrm e}^{2 x}\right )^{\frac {3}{2}}}{7}-\frac {36 \,{\mathrm e}^{2 x} \left (9-{\mathrm e}^{2 x}\right )^{\frac {3}{2}}}{35}-\frac {216 \left (9-{\mathrm e}^{2 x}\right )^{\frac {3}{2}}}{35}\)	\(46\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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 = 0.28, size = 37, normalized size = 0.74 \begin {gather*} -\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 {gather*}

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 = 0.36, size = 32, normalized size = 0.64 \begin {gather*} \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 {gather*}

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 = 1.56, size = 44, normalized size = 0.88 \begin {gather*} \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} > -3 \wedge e^{x} < 3 \end {cases} \end {gather*}

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) > -3) & (
exp(x) < 3)))

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Giac [A]
time = 3.72, size = 53, normalized size = 1.06 \begin {gather*} \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 {gather*}

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)

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Mupad [B]
time = 3.57, size = 32, normalized size = 0.64 \begin {gather*} -\sqrt {9-{\mathrm {e}}^{2\,x}}\,\left (\frac {108\,{\mathrm {e}}^{2\,x}}{35}+\frac {9\,{\mathrm {e}}^{4\,x}}{35}-\frac {{\mathrm {e}}^{6\,x}}{7}+\frac {1944}{35}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(6*x)*(9 - exp(2*x))^(1/2),x)

[Out]

-(9 - exp(2*x))^(1/2)*((108*exp(2*x))/35 + (9*exp(4*x))/35 - exp(6*x)/7 + 1944/35)

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