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

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

[Out]

((-3 + E^(7*x))^(5/3)*Hypergeometric2F1[1, 29/21, 5/7, E^(7*x)/3])/(6*E^(2*x))

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Rubi [A]  time = 0.0517065, 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 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]

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 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 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])

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*}

Mathematica [A]  time = 0.0163485, 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]

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

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Maple [F]  time = 0.021, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{{\rm e}^{2\,x}}} \left ( -3+{{\rm e}^{7\,x}} \right ) ^{{\frac{2}{3}}}}\, dx \end{align*}

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., size = 0, normalized size = 0. \begin{align*} \int{\left (e^{\left (7 \, x\right )} - 3\right )}^{\frac{2}{3}} e^{\left (-2 \, x\right )}\,{d x} \end{align*}

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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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (e^{\left (7 \, x\right )} - 3\right )}^{\frac{2}{3}} e^{\left (-2 \, x\right )}, x\right ) \end{align*}

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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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (e^{7 x} - 3\right )^{\frac{2}{3}} e^{- 2 x}\, dx \end{align*}

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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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (e^{\left (7 \, x\right )} - 3\right )}^{\frac{2}{3}} e^{\left (-2 \, x\right )}\,{d x} \end{align*}

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)