3.37.34 \(\int \frac {1}{18} e^{4 x} (x-4 x^2-12 x^3+26 x^4+51 x^5+18 x^6) \, dx\)

Optimal. Leaf size=23 \[ \frac {1}{4} e^{4 x} \left (-\frac {1}{3}+x\right )^2 \left (x+x^2\right )^2 \]

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Rubi [B]  time = 0.29, antiderivative size = 61, normalized size of antiderivative = 2.65, number of steps used = 30, number of rules used = 4, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {12, 2196, 2176, 2194} \begin {gather*} \frac {1}{4} e^{4 x} x^6+\frac {1}{3} e^{4 x} x^5-\frac {1}{18} e^{4 x} x^4-\frac {1}{9} e^{4 x} x^3+\frac {1}{36} e^{4 x} x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^(4*x)*(x - 4*x^2 - 12*x^3 + 26*x^4 + 51*x^5 + 18*x^6))/18,x]

[Out]

(E^(4*x)*x^2)/36 - (E^(4*x)*x^3)/9 - (E^(4*x)*x^4)/18 + (E^(4*x)*x^5)/3 + (E^(4*x)*x^6)/4

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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]

Rule 2196

Int[(F_)^((c_.)*(v_))*(u_), x_Symbol] :> Int[ExpandIntegrand[F^(c*ExpandToSum[v, x]), u, x], x] /; FreeQ[{F, c
}, x] && PolynomialQ[u, x] && LinearQ[v, x] &&  !$UseGamma === True

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{18} \int e^{4 x} \left (x-4 x^2-12 x^3+26 x^4+51 x^5+18 x^6\right ) \, dx\\ &=\frac {1}{18} \int \left (e^{4 x} x-4 e^{4 x} x^2-12 e^{4 x} x^3+26 e^{4 x} x^4+51 e^{4 x} x^5+18 e^{4 x} x^6\right ) \, dx\\ &=\frac {1}{18} \int e^{4 x} x \, dx-\frac {2}{9} \int e^{4 x} x^2 \, dx-\frac {2}{3} \int e^{4 x} x^3 \, dx+\frac {13}{9} \int e^{4 x} x^4 \, dx+\frac {17}{6} \int e^{4 x} x^5 \, dx+\int e^{4 x} x^6 \, dx\\ &=\frac {1}{72} e^{4 x} x-\frac {1}{18} e^{4 x} x^2-\frac {1}{6} e^{4 x} x^3+\frac {13}{36} e^{4 x} x^4+\frac {17}{24} e^{4 x} x^5+\frac {1}{4} e^{4 x} x^6-\frac {1}{72} \int e^{4 x} \, dx+\frac {1}{9} \int e^{4 x} x \, dx+\frac {1}{2} \int e^{4 x} x^2 \, dx-\frac {13}{9} \int e^{4 x} x^3 \, dx-\frac {3}{2} \int e^{4 x} x^5 \, dx-\frac {85}{24} \int e^{4 x} x^4 \, dx\\ &=-\frac {e^{4 x}}{288}+\frac {1}{24} e^{4 x} x+\frac {5}{72} e^{4 x} x^2-\frac {19}{36} e^{4 x} x^3-\frac {151}{288} e^{4 x} x^4+\frac {1}{3} e^{4 x} x^5+\frac {1}{4} e^{4 x} x^6-\frac {1}{36} \int e^{4 x} \, dx-\frac {1}{4} \int e^{4 x} x \, dx+\frac {13}{12} \int e^{4 x} x^2 \, dx+\frac {15}{8} \int e^{4 x} x^4 \, dx+\frac {85}{24} \int e^{4 x} x^3 \, dx\\ &=-\frac {e^{4 x}}{96}-\frac {1}{48} e^{4 x} x+\frac {49}{144} e^{4 x} x^2+\frac {103}{288} e^{4 x} x^3-\frac {1}{18} e^{4 x} x^4+\frac {1}{3} e^{4 x} x^5+\frac {1}{4} e^{4 x} x^6+\frac {1}{16} \int e^{4 x} \, dx-\frac {13}{24} \int e^{4 x} x \, dx-\frac {15}{8} \int e^{4 x} x^3 \, dx-\frac {85}{32} \int e^{4 x} x^2 \, dx\\ &=\frac {e^{4 x}}{192}-\frac {5}{32} e^{4 x} x-\frac {373 e^{4 x} x^2}{1152}-\frac {1}{9} e^{4 x} x^3-\frac {1}{18} e^{4 x} x^4+\frac {1}{3} e^{4 x} x^5+\frac {1}{4} e^{4 x} x^6+\frac {13}{96} \int e^{4 x} \, dx+\frac {85}{64} \int e^{4 x} x \, dx+\frac {45}{32} \int e^{4 x} x^2 \, dx\\ &=\frac {5 e^{4 x}}{128}+\frac {45}{256} e^{4 x} x+\frac {1}{36} e^{4 x} x^2-\frac {1}{9} e^{4 x} x^3-\frac {1}{18} e^{4 x} x^4+\frac {1}{3} e^{4 x} x^5+\frac {1}{4} e^{4 x} x^6-\frac {85}{256} \int e^{4 x} \, dx-\frac {45}{64} \int e^{4 x} x \, dx\\ &=-\frac {45 e^{4 x}}{1024}+\frac {1}{36} e^{4 x} x^2-\frac {1}{9} e^{4 x} x^3-\frac {1}{18} e^{4 x} x^4+\frac {1}{3} e^{4 x} x^5+\frac {1}{4} e^{4 x} x^6+\frac {45}{256} \int e^{4 x} \, dx\\ &=\frac {1}{36} e^{4 x} x^2-\frac {1}{9} e^{4 x} x^3-\frac {1}{18} e^{4 x} x^4+\frac {1}{3} e^{4 x} x^5+\frac {1}{4} e^{4 x} x^6\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.08, size = 24, normalized size = 1.04 \begin {gather*} \frac {1}{36} e^{4 x} x^2 \left (-1+2 x+3 x^2\right )^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(4*x)*(x - 4*x^2 - 12*x^3 + 26*x^4 + 51*x^5 + 18*x^6))/18,x]

[Out]

(E^(4*x)*x^2*(-1 + 2*x + 3*x^2)^2)/36

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fricas [A]  time = 0.69, size = 30, normalized size = 1.30 \begin {gather*} \frac {1}{36} \, {\left (9 \, x^{6} + 12 \, x^{5} - 2 \, x^{4} - 4 \, x^{3} + x^{2}\right )} e^{\left (4 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/18*(18*x^6+51*x^5+26*x^4-12*x^3-4*x^2+x)*exp(x)^4,x, algorithm="fricas")

[Out]

1/36*(9*x^6 + 12*x^5 - 2*x^4 - 4*x^3 + x^2)*e^(4*x)

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giac [A]  time = 0.18, size = 30, normalized size = 1.30 \begin {gather*} \frac {1}{36} \, {\left (9 \, x^{6} + 12 \, x^{5} - 2 \, x^{4} - 4 \, x^{3} + x^{2}\right )} e^{\left (4 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/18*(18*x^6+51*x^5+26*x^4-12*x^3-4*x^2+x)*exp(x)^4,x, algorithm="giac")

[Out]

1/36*(9*x^6 + 12*x^5 - 2*x^4 - 4*x^3 + x^2)*e^(4*x)

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maple [A]  time = 0.04, size = 28, normalized size = 1.22

method	result	size
gosper	\(\frac {{\mathrm e}^{4 x} \left (3 x -1\right ) \left (3 x^{3}+5 x^{2}+x -1\right ) x^{2}}{36}\)	\(28\)
risch	\(\frac {\left (\frac {9}{2} x^{6}+6 x^{5}-x^{4}-2 x^{3}+\frac {1}{2} x^{2}\right ) {\mathrm e}^{4 x}}{18}\)	\(33\)
default	\(\frac {x^{2} {\mathrm e}^{4 x}}{36}-\frac {x^{3} {\mathrm e}^{4 x}}{9}-\frac {x^{4} {\mathrm e}^{4 x}}{18}+\frac {x^{5} {\mathrm e}^{4 x}}{3}+\frac {x^{6} {\mathrm e}^{4 x}}{4}\)	\(47\)
meijerg	\(\frac {\left (28672 x^{6}-43008 x^{5}+53760 x^{4}-53760 x^{3}+40320 x^{2}-20160 x +5040\right ) {\mathrm e}^{4 x}}{114688}-\frac {17 \left (-6144 x^{5}+7680 x^{4}-7680 x^{3}+5760 x^{2}-2880 x +720\right ) {\mathrm e}^{4 x}}{147456}+\frac {13 \left (1280 x^{4}-1280 x^{3}+960 x^{2}-480 x +120\right ) {\mathrm e}^{4 x}}{46080}+\frac {\left (-256 x^{3}+192 x^{2}-96 x +24\right ) {\mathrm e}^{4 x}}{1536}-\frac {\left (48 x^{2}-24 x +6\right ) {\mathrm e}^{4 x}}{864}-\frac {\left (-8 x +2\right ) {\mathrm e}^{4 x}}{576}\)	\(143\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/18*(18*x^6+51*x^5+26*x^4-12*x^3-4*x^2+x)*exp(x)^4,x,method=_RETURNVERBOSE)

[Out]

1/36*exp(x)^4*(3*x-1)*(3*x^3+5*x^2+x-1)*x^2

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maxima [B]  time = 0.36, size = 142, normalized size = 6.17 \begin {gather*} \frac {1}{1024} \, {\left (256 \, x^{6} - 384 \, x^{5} + 480 \, x^{4} - 480 \, x^{3} + 360 \, x^{2} - 180 \, x + 45\right )} e^{\left (4 \, x\right )} + \frac {17}{3072} \, {\left (128 \, x^{5} - 160 \, x^{4} + 160 \, x^{3} - 120 \, x^{2} + 60 \, x - 15\right )} e^{\left (4 \, x\right )} + \frac {13}{1152} \, {\left (32 \, x^{4} - 32 \, x^{3} + 24 \, x^{2} - 12 \, x + 3\right )} e^{\left (4 \, x\right )} - \frac {1}{192} \, {\left (32 \, x^{3} - 24 \, x^{2} + 12 \, x - 3\right )} e^{\left (4 \, x\right )} - \frac {1}{144} \, {\left (8 \, x^{2} - 4 \, x + 1\right )} e^{\left (4 \, x\right )} + \frac {1}{288} \, {\left (4 \, x - 1\right )} e^{\left (4 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/18*(18*x^6+51*x^5+26*x^4-12*x^3-4*x^2+x)*exp(x)^4,x, algorithm="maxima")

[Out]

1/1024*(256*x^6 - 384*x^5 + 480*x^4 - 480*x^3 + 360*x^2 - 180*x + 45)*e^(4*x) + 17/3072*(128*x^5 - 160*x^4 + 1
60*x^3 - 120*x^2 + 60*x - 15)*e^(4*x) + 13/1152*(32*x^4 - 32*x^3 + 24*x^2 - 12*x + 3)*e^(4*x) - 1/192*(32*x^3
- 24*x^2 + 12*x - 3)*e^(4*x) - 1/144*(8*x^2 - 4*x + 1)*e^(4*x) + 1/288*(4*x - 1)*e^(4*x)

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mupad [B]  time = 2.05, size = 21, normalized size = 0.91 \begin {gather*} \frac {x^2\,{\mathrm {e}}^{4\,x}\,{\left (3\,x^2+2\,x-1\right )}^2}{36} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(4*x)*(x - 4*x^2 - 12*x^3 + 26*x^4 + 51*x^5 + 18*x^6))/18,x)

[Out]

(x^2*exp(4*x)*(2*x + 3*x^2 - 1)^2)/36

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sympy [A]  time = 0.11, size = 29, normalized size = 1.26 \begin {gather*} \frac {\left (9 x^{6} + 12 x^{5} - 2 x^{4} - 4 x^{3} + x^{2}\right ) e^{4 x}}{36} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/18*(18*x**6+51*x**5+26*x**4-12*x**3-4*x**2+x)*exp(x)**4,x)

[Out]

(9*x**6 + 12*x**5 - 2*x**4 - 4*x**3 + x**2)*exp(4*x)/36

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