3.533 \(\int e^{-2 x} (-3+e^{7 x})^{2/3} \, dx\)

Optimal. Leaf size=37 \[ \frac {1}{6} e^{-2 x} \left (e^{7 x}-3\right )^{5/3} \, _2F_1\left (1,\frac {29}{21};\frac {5}{7};\frac {e^{7 x}}{3}\right ) \]

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Rubi [A]  time = 0.05, antiderivative size = 57, normalized size of antiderivative = 1.54, number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {2249, 335, 365, 364} \[ -\frac {3^{2/3} e^{-2 x} \left (e^{7 x}-3\right )^{2/3} \text {Hypergeometric2F1}\left (-\frac {2}{3},-\frac {2}{7},\frac {5}{7},\frac {e^{7 x}}{3}\right )}{2 \left (3-e^{7 x}\right )^{2/3}} \]

Antiderivative was successfully verified.

[In]

Int[(-3 + E^(7*x))^(2/3)/E^(2*x),x]

[Out]

-(3^(2/3)*(-3 + E^(7*x))^(2/3)*Hypergeometric2F1[-2/3, -2/7, 5/7, E^(7*x)/3])/(2*E^(2*x)*(3 - E^(7*x))^(2/3))

Rule 335

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Subst[Int[(a + b/x^n)^p/x^(m + 2), x], x, 1/x] /;
FreeQ[{a, b, p}, x] && ILtQ[n, 0] && IntegerQ[m]

Rule 364

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a^p*(c*x)^(m + 1)*Hypergeometric2F1[-
p, (m + 1)/n, (m + 1)/n + 1, -((b*x^n)/a)])/(c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rule 365

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[(a^IntPart[p]*(a + b*x^n)^FracPart[p])
/(1 + (b*x^n)/a)^FracPart[p], Int[(c*x)^m*(1 + (b*x^n)/a)^p, x], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[
p, 0] &&  !(ILtQ[p, 0] || GtQ[a, 0])

Rule 2249

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

Rubi steps

\begin {align*} \int e^{-2 x} \left (-3+e^{7 x}\right )^{2/3} \, dx &=-\operatorname {Subst}\left (\int \left (-3+\frac {1}{x^7}\right )^{2/3} x \, dx,x,e^{-x}\right )\\ &=\operatorname {Subst}\left (\int \frac {\left (-3+x^7\right )^{2/3}}{x^3} \, dx,x,e^x\right )\\ &=\frac {\left (-3+e^{7 x}\right )^{2/3} \operatorname {Subst}\left (\int \frac {\left (1-\frac {x^7}{3}\right )^{2/3}}{x^3} \, dx,x,e^x\right )}{\left (1-\frac {e^{7 x}}{3}\right )^{2/3}}\\ &=-\frac {3^{2/3} e^{-2 x} \left (-3+e^{7 x}\right )^{2/3} \, _2F_1\left (-\frac {2}{3},-\frac {2}{7};\frac {5}{7};\frac {e^{7 x}}{3}\right )}{2 \left (3-e^{7 x}\right )^{2/3}}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 54, normalized size = 1.46 \[ -\frac {e^{-2 x} \left (e^{7 x}-3\right )^{2/3} \, _2F_1\left (-\frac {2}{3},-\frac {2}{7};\frac {5}{7};\frac {e^{7 x}}{3}\right )}{2 \left (1-\frac {e^{7 x}}{3}\right )^{2/3}} \]

Antiderivative was successfully verified.

[In]

Integrate[(-3 + E^(7*x))^(2/3)/E^(2*x),x]

[Out]

-1/2*((-3 + E^(7*x))^(2/3)*Hypergeometric2F1[-2/3, -2/7, 5/7, E^(7*x)/3])/(E^(2*x)*(1 - E^(7*x)/3)^(2/3))

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int e^{-2 x} \left (-3+e^{7 x}\right )^{2/3} \, dx \]

Verification is Not applicable to the result.

[In]

IntegrateAlgebraic[(-3 + E^(7*x))^(2/3)/E^(2*x),x]

[Out]

Could not integrate

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fricas [F]  time = 0.94, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (e^{\left (7 \, x\right )} - 3\right )}^{\frac {2}{3}} e^{\left (-2 \, x\right )}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((e^(7*x) - 3)^(2/3)*e^(-2*x), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (e^{\left (7 \, x\right )} - 3\right )}^{\frac {2}{3}} e^{\left (-2 \, x\right )}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 180.00, size = 0, normalized size = 0.00 \[\int \left (-3+{\mathrm e}^{7 x}\right )^{\frac {2}{3}} {\mathrm e}^{-2 x}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-3+exp(7*x))^(2/3)/exp(2*x),x)

[Out]

int((-3+exp(7*x))^(2/3)/exp(2*x),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (e^{\left (7 \, x\right )} - 3\right )}^{\frac {2}{3}} e^{\left (-2 \, x\right )}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \[ \int {\mathrm {e}}^{-2\,x}\,{\left ({\mathrm {e}}^{7\,x}-3\right )}^{2/3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(-2*x)*(exp(7*x) - 3)^(2/3),x)

[Out]

int(exp(-2*x)*(exp(7*x) - 3)^(2/3), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (e^{7 x} - 3\right )^{\frac {2}{3}} e^{- 2 x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3+exp(7*x))**(2/3)/exp(2*x),x)

[Out]

Integral((exp(7*x) - 3)**(2/3)*exp(-2*x), x)

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