3.707 \(\int e^{7 x} x^3 \, dx\)

Optimal. Leaf size=44 \[ \frac{1}{7} e^{7 x} x^3-\frac{3}{49} e^{7 x} x^2+\frac{6}{343} e^{7 x} x-\frac{6 e^{7 x}}{2401} \]

[Out]

(-6*E^(7*x))/2401 + (6*E^(7*x)*x)/343 - (3*E^(7*x)*x^2)/49 + (E^(7*x)*x^3)/7

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Rubi [A]  time = 0.0348359, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {2176, 2194} \[ \frac{1}{7} e^{7 x} x^3-\frac{3}{49} e^{7 x} x^2+\frac{6}{343} e^{7 x} x-\frac{6 e^{7 x}}{2401} \]

Antiderivative was successfully verified.

[In]

Int[E^(7*x)*x^3,x]

[Out]

(-6*E^(7*x))/2401 + (6*E^(7*x)*x)/343 - (3*E^(7*x)*x^2)/49 + (E^(7*x)*x^3)/7

Rule 2176

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^m
*(b*F^(g*(e + f*x)))^n)/(f*g*n*Log[F]), x] - Dist[(d*m)/(f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*(b*F^(g*(e + f*x
)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && GtQ[m, 0] && IntegerQ[2*m] &&  !$UseGamma === True

Rule 2194

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rubi steps

\begin{align*} \int e^{7 x} x^3 \, dx &=\frac{1}{7} e^{7 x} x^3-\frac{3}{7} \int e^{7 x} x^2 \, dx\\ &=-\frac{3}{49} e^{7 x} x^2+\frac{1}{7} e^{7 x} x^3+\frac{6}{49} \int e^{7 x} x \, dx\\ &=\frac{6}{343} e^{7 x} x-\frac{3}{49} e^{7 x} x^2+\frac{1}{7} e^{7 x} x^3-\frac{6}{343} \int e^{7 x} \, dx\\ &=-\frac{6 e^{7 x}}{2401}+\frac{6}{343} e^{7 x} x-\frac{3}{49} e^{7 x} x^2+\frac{1}{7} e^{7 x} x^3\\ \end{align*}

Mathematica [A]  time = 0.0073774, size = 24, normalized size = 0.55 \[ \frac{e^{7 x} \left (343 x^3-147 x^2+42 x-6\right )}{2401} \]

Antiderivative was successfully verified.

[In]

Integrate[E^(7*x)*x^3,x]

[Out]

(E^(7*x)*(-6 + 42*x - 147*x^2 + 343*x^3))/2401

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Maple [A]  time = 0.021, size = 22, normalized size = 0.5 \begin{align*}{\frac{ \left ( 343\,{x}^{3}-147\,{x}^{2}+42\,x-6 \right ){{\rm e}^{7\,x}}}{2401}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(7*x)*x^3,x)

[Out]

1/2401*(343*x^3-147*x^2+42*x-6)*exp(7*x)

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Maxima [A]  time = 0.964457, size = 28, normalized size = 0.64 \begin{align*} \frac{1}{2401} \,{\left (343 \, x^{3} - 147 \, x^{2} + 42 \, x - 6\right )} e^{\left (7 \, x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(7*x)*x^3,x, algorithm="maxima")

[Out]

1/2401*(343*x^3 - 147*x^2 + 42*x - 6)*e^(7*x)

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Fricas [A]  time = 0.807989, size = 63, normalized size = 1.43 \begin{align*} \frac{1}{2401} \,{\left (343 \, x^{3} - 147 \, x^{2} + 42 \, x - 6\right )} e^{\left (7 \, x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(7*x)*x^3,x, algorithm="fricas")

[Out]

1/2401*(343*x^3 - 147*x^2 + 42*x - 6)*e^(7*x)

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Sympy [A]  time = 0.088803, size = 20, normalized size = 0.45 \begin{align*} \frac{\left (343 x^{3} - 147 x^{2} + 42 x - 6\right ) e^{7 x}}{2401} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(7*x)*x**3,x)

[Out]

(343*x**3 - 147*x**2 + 42*x - 6)*exp(7*x)/2401

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Giac [A]  time = 1.40771, size = 28, normalized size = 0.64 \begin{align*} \frac{1}{2401} \,{\left (343 \, x^{3} - 147 \, x^{2} + 42 \, x - 6\right )} e^{\left (7 \, x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(7*x)*x^3,x, algorithm="giac")

[Out]

1/2401*(343*x^3 - 147*x^2 + 42*x - 6)*e^(7*x)