3.8.7 \(\int e^{7 x} x^3 \, dx\) [707]

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

[Out]

-6/2401*exp(7*x)+6/343*exp(7*x)*x-3/49*exp(7*x)*x^2+1/7*exp(7*x)*x^3

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Rubi [A]
time = 0.02, antiderivative size = 44, normalized size of antiderivative = 1.00, 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 = {2207, 2225} \begin {gather*} \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} \end {gather*}

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 2207

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] &&  !TrueQ[$UseGamma]

Rule 2225

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*}

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Mathematica [A]
time = 0.01, size = 24, normalized size = 0.55 \begin {gather*} \frac {e^{7 x} \left (-6+42 x-147 x^2+343 x^3\right )}{2401} \end {gather*}

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 = 0.02, size = 33, normalized size = 0.75

method	result	size
risch	\(\left (\frac {1}{7} x^{3}-\frac {3}{49} x^{2}+\frac {6}{343} x -\frac {6}{2401}\right ) {\mathrm e}^{7 x}\)	\(21\)
gosper	\(\frac {\left (343 x^{3}-147 x^{2}+42 x -6\right ) {\mathrm e}^{7 x}}{2401}\)	\(22\)
meijerg	\(\frac {6}{2401}-\frac {\left (-1372 x^{3}+588 x^{2}-168 x +24\right ) {\mathrm e}^{7 x}}{9604}\)	\(24\)
derivativedivides	\(-\frac {6 \,{\mathrm e}^{7 x}}{2401}+\frac {6 \,{\mathrm e}^{7 x} x}{343}-\frac {3 \,{\mathrm e}^{7 x} x^{2}}{49}+\frac {{\mathrm e}^{7 x} x^{3}}{7}\)	\(33\)
default	\(-\frac {6 \,{\mathrm e}^{7 x}}{2401}+\frac {6 \,{\mathrm e}^{7 x} x}{343}-\frac {3 \,{\mathrm e}^{7 x} x^{2}}{49}+\frac {{\mathrm e}^{7 x} x^{3}}{7}\)	\(33\)
norman	\(-\frac {6 \,{\mathrm e}^{7 x}}{2401}+\frac {6 \,{\mathrm e}^{7 x} x}{343}-\frac {3 \,{\mathrm e}^{7 x} x^{2}}{49}+\frac {{\mathrm e}^{7 x} x^{3}}{7}\)	\(33\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(7*x)*x^3,x,method=_RETURNVERBOSE)

[Out]

-6/2401*exp(7*x)+6/343*exp(7*x)*x-3/49*exp(7*x)*x^2+1/7*exp(7*x)*x^3

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Maxima [A]
time = 0.28, size = 21, normalized size = 0.48 \begin {gather*} \frac {1}{2401} \, {\left (343 \, x^{3} - 147 \, x^{2} + 42 \, x - 6\right )} e^{\left (7 \, x\right )} \end {gather*}

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 = 0.35, size = 21, normalized size = 0.48 \begin {gather*} \frac {1}{2401} \, {\left (343 \, x^{3} - 147 \, x^{2} + 42 \, x - 6\right )} e^{\left (7 \, x\right )} \end {gather*}

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 = 0.02, size = 20, normalized size = 0.45 \begin {gather*} \frac {\left (343 x^{3} - 147 x^{2} + 42 x - 6\right ) e^{7 x}}{2401} \end {gather*}

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 = 5.29, size = 21, normalized size = 0.48 \begin {gather*} \frac {1}{2401} \, {\left (343 \, x^{3} - 147 \, x^{2} + 42 \, x - 6\right )} e^{\left (7 \, x\right )} \end {gather*}

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)

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Mupad [B]
time = 0.03, size = 21, normalized size = 0.48 \begin {gather*} \frac {{\mathrm {e}}^{7\,x}\,\left (343\,x^3-147\,x^2+42\,x-6\right )}{2401} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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