3.94.91 \(\int \frac {1}{8} e^{2 x} (729 x+1053 x^2-184 x^3-280 x^4+16 x^5+16 x^6) \, dx\)

Optimal. Leaf size=21 \[ e^{2 x} x^2 \left (\frac {27}{4}+x-x^2\right )^2 \]

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Rubi [B]  time = 0.33, antiderivative size = 56, normalized size of antiderivative = 2.67, number of steps used = 30, number of rules used = 4, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {12, 2196, 2176, 2194} \begin {gather*} e^{2 x} x^6-2 e^{2 x} x^5-\frac {25}{2} e^{2 x} x^4+\frac {27}{2} e^{2 x} x^3+\frac {729}{16} e^{2 x} x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^(2*x)*(729*x + 1053*x^2 - 184*x^3 - 280*x^4 + 16*x^5 + 16*x^6))/8,x]

[Out]

(729*E^(2*x)*x^2)/16 + (27*E^(2*x)*x^3)/2 - (25*E^(2*x)*x^4)/2 - 2*E^(2*x)*x^5 + E^(2*x)*x^6

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}{8} \int e^{2 x} \left (729 x+1053 x^2-184 x^3-280 x^4+16 x^5+16 x^6\right ) \, dx\\ &=\frac {1}{8} \int \left (729 e^{2 x} x+1053 e^{2 x} x^2-184 e^{2 x} x^3-280 e^{2 x} x^4+16 e^{2 x} x^5+16 e^{2 x} x^6\right ) \, dx\\ &=2 \int e^{2 x} x^5 \, dx+2 \int e^{2 x} x^6 \, dx-23 \int e^{2 x} x^3 \, dx-35 \int e^{2 x} x^4 \, dx+\frac {729}{8} \int e^{2 x} x \, dx+\frac {1053}{8} \int e^{2 x} x^2 \, dx\\ &=\frac {729}{16} e^{2 x} x+\frac {1053}{16} e^{2 x} x^2-\frac {23}{2} e^{2 x} x^3-\frac {35}{2} e^{2 x} x^4+e^{2 x} x^5+e^{2 x} x^6-5 \int e^{2 x} x^4 \, dx-6 \int e^{2 x} x^5 \, dx+\frac {69}{2} \int e^{2 x} x^2 \, dx-\frac {729}{16} \int e^{2 x} \, dx+70 \int e^{2 x} x^3 \, dx-\frac {1053}{8} \int e^{2 x} x \, dx\\ &=-\frac {729 e^{2 x}}{32}-\frac {81}{4} e^{2 x} x+\frac {1329}{16} e^{2 x} x^2+\frac {47}{2} e^{2 x} x^3-20 e^{2 x} x^4-2 e^{2 x} x^5+e^{2 x} x^6+10 \int e^{2 x} x^3 \, dx+15 \int e^{2 x} x^4 \, dx-\frac {69}{2} \int e^{2 x} x \, dx+\frac {1053}{16} \int e^{2 x} \, dx-105 \int e^{2 x} x^2 \, dx\\ &=\frac {81 e^{2 x}}{8}-\frac {75}{2} e^{2 x} x+\frac {489}{16} e^{2 x} x^2+\frac {57}{2} e^{2 x} x^3-\frac {25}{2} e^{2 x} x^4-2 e^{2 x} x^5+e^{2 x} x^6-15 \int e^{2 x} x^2 \, dx+\frac {69}{4} \int e^{2 x} \, dx-30 \int e^{2 x} x^3 \, dx+105 \int e^{2 x} x \, dx\\ &=\frac {75 e^{2 x}}{4}+15 e^{2 x} x+\frac {369}{16} e^{2 x} x^2+\frac {27}{2} e^{2 x} x^3-\frac {25}{2} e^{2 x} x^4-2 e^{2 x} x^5+e^{2 x} x^6+15 \int e^{2 x} x \, dx+45 \int e^{2 x} x^2 \, dx-\frac {105}{2} \int e^{2 x} \, dx\\ &=-\frac {15 e^{2 x}}{2}+\frac {45}{2} e^{2 x} x+\frac {729}{16} e^{2 x} x^2+\frac {27}{2} e^{2 x} x^3-\frac {25}{2} e^{2 x} x^4-2 e^{2 x} x^5+e^{2 x} x^6-\frac {15}{2} \int e^{2 x} \, dx-45 \int e^{2 x} x \, dx\\ &=-\frac {45 e^{2 x}}{4}+\frac {729}{16} e^{2 x} x^2+\frac {27}{2} e^{2 x} x^3-\frac {25}{2} e^{2 x} x^4-2 e^{2 x} x^5+e^{2 x} x^6+\frac {45}{2} \int e^{2 x} \, dx\\ &=\frac {729}{16} e^{2 x} x^2+\frac {27}{2} e^{2 x} x^3-\frac {25}{2} e^{2 x} x^4-2 e^{2 x} x^5+e^{2 x} x^6\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.08, size = 24, normalized size = 1.14 \begin {gather*} \frac {1}{16} e^{2 x} x^2 \left (27+4 x-4 x^2\right )^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(2*x)*(729*x + 1053*x^2 - 184*x^3 - 280*x^4 + 16*x^5 + 16*x^6))/8,x]

[Out]

(E^(2*x)*x^2*(27 + 4*x - 4*x^2)^2)/16

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fricas [A]  time = 0.87, size = 32, normalized size = 1.52 \begin {gather*} \frac {1}{16} \, {\left (16 \, x^{6} - 32 \, x^{5} - 200 \, x^{4} + 216 \, x^{3} + 729 \, x^{2}\right )} e^{\left (2 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/8*(16*x^6+16*x^5-280*x^4-184*x^3+1053*x^2+729*x)*exp(x)^2,x, algorithm="fricas")

[Out]

1/16*(16*x^6 - 32*x^5 - 200*x^4 + 216*x^3 + 729*x^2)*e^(2*x)

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giac [A]  time = 0.12, size = 32, normalized size = 1.52 \begin {gather*} \frac {1}{16} \, {\left (16 \, x^{6} - 32 \, x^{5} - 200 \, x^{4} + 216 \, x^{3} + 729 \, x^{2}\right )} e^{\left (2 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/8*(16*x^6+16*x^5-280*x^4-184*x^3+1053*x^2+729*x)*exp(x)^2,x, algorithm="giac")

[Out]

1/16*(16*x^6 - 32*x^5 - 200*x^4 + 216*x^3 + 729*x^2)*e^(2*x)

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maple [A]  time = 0.04, size = 22, normalized size = 1.05

method	result	size
gosper	\(\frac {x^{2} \left (4 x^{2}-4 x -27\right )^{2} {\mathrm e}^{2 x}}{16}\)	\(22\)
risch	\(\frac {\left (8 x^{6}-16 x^{5}-100 x^{4}+108 x^{3}+\frac {729}{2} x^{2}\right ) {\mathrm e}^{2 x}}{8}\)	\(33\)
default	\(-2 x^{5} {\mathrm e}^{2 x}-\frac {25 \,{\mathrm e}^{2 x} x^{4}}{2}+\frac {27 \,{\mathrm e}^{2 x} x^{3}}{2}+\frac {729 \,{\mathrm e}^{2 x} x^{2}}{16}+x^{6} {\mathrm e}^{2 x}\)	\(46\)
norman	\(-2 x^{5} {\mathrm e}^{2 x}-\frac {25 \,{\mathrm e}^{2 x} x^{4}}{2}+\frac {27 \,{\mathrm e}^{2 x} x^{3}}{2}+\frac {729 \,{\mathrm e}^{2 x} x^{2}}{16}+x^{6} {\mathrm e}^{2 x}\)	\(46\)
meijerg	\(\frac {\left (448 x^{6}-1344 x^{5}+3360 x^{4}-6720 x^{3}+10080 x^{2}-10080 x +5040\right ) {\mathrm e}^{2 x}}{448}-\frac {\left (-192 x^{5}+480 x^{4}-960 x^{3}+1440 x^{2}-1440 x +720\right ) {\mathrm e}^{2 x}}{192}-\frac {7 \left (80 x^{4}-160 x^{3}+240 x^{2}-240 x +120\right ) {\mathrm e}^{2 x}}{32}+\frac {23 \left (-32 x^{3}+48 x^{2}-48 x +24\right ) {\mathrm e}^{2 x}}{64}+\frac {351 \left (12 x^{2}-12 x +6\right ) {\mathrm e}^{2 x}}{64}-\frac {729 \left (-4 x +2\right ) {\mathrm e}^{2 x}}{64}\)	\(143\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/8*(16*x^6+16*x^5-280*x^4-184*x^3+1053*x^2+729*x)*exp(x)^2,x,method=_RETURNVERBOSE)

[Out]

1/16*x^2*(4*x^2-4*x-27)^2*exp(x)^2

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maxima [B]  time = 0.36, size = 142, normalized size = 6.76 \begin {gather*} \frac {1}{4} \, {\left (4 \, x^{6} - 12 \, x^{5} + 30 \, x^{4} - 60 \, x^{3} + 90 \, x^{2} - 90 \, x + 45\right )} e^{\left (2 \, x\right )} + \frac {1}{4} \, {\left (4 \, x^{5} - 10 \, x^{4} + 20 \, x^{3} - 30 \, x^{2} + 30 \, x - 15\right )} e^{\left (2 \, x\right )} - \frac {35}{4} \, {\left (2 \, x^{4} - 4 \, x^{3} + 6 \, x^{2} - 6 \, x + 3\right )} e^{\left (2 \, x\right )} - \frac {23}{8} \, {\left (4 \, x^{3} - 6 \, x^{2} + 6 \, x - 3\right )} e^{\left (2 \, x\right )} + \frac {1053}{32} \, {\left (2 \, x^{2} - 2 \, x + 1\right )} e^{\left (2 \, x\right )} + \frac {729}{32} \, {\left (2 \, x - 1\right )} e^{\left (2 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/8*(16*x^6+16*x^5-280*x^4-184*x^3+1053*x^2+729*x)*exp(x)^2,x, algorithm="maxima")

[Out]

1/4*(4*x^6 - 12*x^5 + 30*x^4 - 60*x^3 + 90*x^2 - 90*x + 45)*e^(2*x) + 1/4*(4*x^5 - 10*x^4 + 20*x^3 - 30*x^2 +
30*x - 15)*e^(2*x) - 35/4*(2*x^4 - 4*x^3 + 6*x^2 - 6*x + 3)*e^(2*x) - 23/8*(4*x^3 - 6*x^2 + 6*x - 3)*e^(2*x) +
1053/32*(2*x^2 - 2*x + 1)*e^(2*x) + 729/32*(2*x - 1)*e^(2*x)

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mupad [B]  time = 8.74, size = 21, normalized size = 1.00 \begin {gather*} \frac {x^2\,{\mathrm {e}}^{2\,x}\,{\left (-4\,x^2+4\,x+27\right )}^2}{16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(2*x)*(729*x + 1053*x^2 - 184*x^3 - 280*x^4 + 16*x^5 + 16*x^6))/8,x)

[Out]

(x^2*exp(2*x)*(4*x - 4*x^2 + 27)^2)/16

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sympy [A]  time = 0.13, size = 31, normalized size = 1.48 \begin {gather*} \frac {\left (16 x^{6} - 32 x^{5} - 200 x^{4} + 216 x^{3} + 729 x^{2}\right ) e^{2 x}}{16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/8*(16*x**6+16*x**5-280*x**4-184*x**3+1053*x**2+729*x)*exp(x)**2,x)

[Out]

(16*x**6 - 32*x**5 - 200*x**4 + 216*x**3 + 729*x**2)*exp(2*x)/16

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