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

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 = 0.0941681, 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 \[ -\frac{3^{2/3} e^{-2 x} \left (e^{7 x}-3\right )^{2/3} \, _2F_1\left (-\frac{2}{3},-\frac{2}{7};\frac{5}{7};\frac{e^{7 x}}{3}\right )}{2 \left (3-e^{7 x}\right )^{2/3}} \]

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))

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Rubi in Sympy [A]  time = 6.56238, size = 46, normalized size = 1.24 \[ - \frac{\left (e^{7 x} - 3\right )^{\frac{2}{3}} e^{- 2 x}{{}_{2}F_{1}\left (\begin{matrix} - \frac{2}{3}, - \frac{2}{7} \\ \frac{5}{7} \end{matrix}\middle |{\frac{e^{7 x}}{3}} \right )}}{2 \left (- \frac{e^{7 x}}{3} + 1\right )^{\frac{2}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-3+exp(7*x))**(2/3)/exp(2*x),x)

[Out]

-(exp(7*x) - 3)**(2/3)*exp(-2*x)*hyper((-2/3, -2/7), (5/7,), exp(7*x)/3)/(2*(-ex
p(7*x)/3 + 1)**(2/3))

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Mathematica [A]  time = 0.0560825, size = 68, normalized size = 1.84 \[ \frac{e^{-2 x} \left (7\ 3^{2/3} \sqrt [3]{3-e^{7 x}} \, _2F_1\left (-\frac{2}{7},\frac{1}{3};\frac{5}{7};\frac{e^{7 x}}{3}\right )+3 e^{7 x}-9\right )}{8 \sqrt [3]{e^{7 x}-3}} \]

Antiderivative was successfully verified.

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

[Out]

(-9 + 3*E^(7*x) + 7*3^(2/3)*(3 - E^(7*x))^(1/3)*Hypergeometric2F1[-2/7, 1/3, 5/7
, E^(7*x)/3])/(8*E^(2*x)*(-3 + E^(7*x))^(1/3))

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Maple [F]  time = 0.033, size = 0, normalized size = 0. \[ \int{\frac{1}{{{\rm e}^{2\,x}}} \left ( -3+{{\rm e}^{7\,x}} \right ) ^{{\frac{2}{3}}}}\, dx \]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e^(7*x) - 3)^(2/3)*e^(-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. \[{\rm integral}\left ({\left (e^{\left (7 \, x\right )} - 3\right )}^{\frac{2}{3}} e^{\left (-2 \, x\right )}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e^(7*x) - 3)^(2/3)*e^(-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. \[ \int \left (e^{7 x} - 3\right )^{\frac{2}{3}} e^{- 2 x}\, dx \]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e^(7*x) - 3)^(2/3)*e^(-2*x),x, algorithm="giac")

[Out]

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