3.92.37 \(\int (-13824+4608 x+e^{4 x} (4320-3168 x+576 x^2)+e^{3 x} (24192-18432 x+3456 x^2)+e^x (13824-18432 x+4608 x^2)+e^{2 x} (41472-34560 x+6912 x^2)) \, dx\)

Optimal. Leaf size=19 \[ 9 (-3+x)^2 \left (4+\log \left (e^{2 e^x}\right )\right )^4 \]

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Rubi [B]  time = 0.19, antiderivative size = 103, normalized size of antiderivative = 5.42, number of steps used = 33, number of rules used = 3, integrand size = 67, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {2196, 2194, 2176} \begin {gather*} 4608 e^x x^2+3456 e^{2 x} x^2+1152 e^{3 x} x^2+144 e^{4 x} x^2+2304 x^2-27648 e^x x-20736 e^{2 x} x-6912 e^{3 x} x-864 e^{4 x} x-13824 x+41472 e^x+31104 e^{2 x}+10368 e^{3 x}+1296 e^{4 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[-13824 + 4608*x + E^(4*x)*(4320 - 3168*x + 576*x^2) + E^(3*x)*(24192 - 18432*x + 3456*x^2) + E^x*(13824 -
18432*x + 4608*x^2) + E^(2*x)*(41472 - 34560*x + 6912*x^2),x]

[Out]

41472*E^x + 31104*E^(2*x) + 10368*E^(3*x) + 1296*E^(4*x) - 13824*x - 27648*E^x*x - 20736*E^(2*x)*x - 6912*E^(3
*x)*x - 864*E^(4*x)*x + 2304*x^2 + 4608*E^x*x^2 + 3456*E^(2*x)*x^2 + 1152*E^(3*x)*x^2 + 144*E^(4*x)*x^2

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} &=-13824 x+2304 x^2+\int e^{4 x} \left (4320-3168 x+576 x^2\right ) \, dx+\int e^{3 x} \left (24192-18432 x+3456 x^2\right ) \, dx+\int e^x \left (13824-18432 x+4608 x^2\right ) \, dx+\int e^{2 x} \left (41472-34560 x+6912 x^2\right ) \, dx\\ &=-13824 x+2304 x^2+\int \left (13824 e^x-18432 e^x x+4608 e^x x^2\right ) \, dx+\int \left (41472 e^{2 x}-34560 e^{2 x} x+6912 e^{2 x} x^2\right ) \, dx+\int \left (24192 e^{3 x}-18432 e^{3 x} x+3456 e^{3 x} x^2\right ) \, dx+\int \left (4320 e^{4 x}-3168 e^{4 x} x+576 e^{4 x} x^2\right ) \, dx\\ &=-13824 x+2304 x^2+576 \int e^{4 x} x^2 \, dx-3168 \int e^{4 x} x \, dx+3456 \int e^{3 x} x^2 \, dx+4320 \int e^{4 x} \, dx+4608 \int e^x x^2 \, dx+6912 \int e^{2 x} x^2 \, dx+13824 \int e^x \, dx-18432 \int e^x x \, dx-18432 \int e^{3 x} x \, dx+24192 \int e^{3 x} \, dx-34560 \int e^{2 x} x \, dx+41472 \int e^{2 x} \, dx\\ &=13824 e^x+20736 e^{2 x}+8064 e^{3 x}+1080 e^{4 x}-13824 x-18432 e^x x-17280 e^{2 x} x-6144 e^{3 x} x-792 e^{4 x} x+2304 x^2+4608 e^x x^2+3456 e^{2 x} x^2+1152 e^{3 x} x^2+144 e^{4 x} x^2-288 \int e^{4 x} x \, dx+792 \int e^{4 x} \, dx-2304 \int e^{3 x} x \, dx+6144 \int e^{3 x} \, dx-6912 \int e^{2 x} x \, dx-9216 \int e^x x \, dx+17280 \int e^{2 x} \, dx+18432 \int e^x \, dx\\ &=32256 e^x+29376 e^{2 x}+10112 e^{3 x}+1278 e^{4 x}-13824 x-27648 e^x x-20736 e^{2 x} x-6912 e^{3 x} x-864 e^{4 x} x+2304 x^2+4608 e^x x^2+3456 e^{2 x} x^2+1152 e^{3 x} x^2+144 e^{4 x} x^2+72 \int e^{4 x} \, dx+768 \int e^{3 x} \, dx+3456 \int e^{2 x} \, dx+9216 \int e^x \, dx\\ &=41472 e^x+31104 e^{2 x}+10368 e^{3 x}+1296 e^{4 x}-13824 x-27648 e^x x-20736 e^{2 x} x-6912 e^{3 x} x-864 e^{4 x} x+2304 x^2+4608 e^x x^2+3456 e^{2 x} x^2+1152 e^{3 x} x^2+144 e^{4 x} x^2\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.13, size = 57, normalized size = 3.00 \begin {gather*} 288 \left (16 e^x (-3+x)^2+12 e^{2 x} (-3+x)^2+4 e^{3 x} (-3+x)^2+\frac {1}{2} e^{4 x} (-3+x)^2+8 (-6+x) x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[-13824 + 4608*x + E^(4*x)*(4320 - 3168*x + 576*x^2) + E^(3*x)*(24192 - 18432*x + 3456*x^2) + E^x*(13
824 - 18432*x + 4608*x^2) + E^(2*x)*(41472 - 34560*x + 6912*x^2),x]

[Out]

288*(16*E^x*(-3 + x)^2 + 12*E^(2*x)*(-3 + x)^2 + 4*E^(3*x)*(-3 + x)^2 + (E^(4*x)*(-3 + x)^2)/2 + 8*(-6 + x)*x)

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fricas [B]  time = 0.78, size = 63, normalized size = 3.32 \begin {gather*} 2304 \, x^{2} + 144 \, {\left (x^{2} - 6 \, x + 9\right )} e^{\left (4 \, x\right )} + 1152 \, {\left (x^{2} - 6 \, x + 9\right )} e^{\left (3 \, x\right )} + 3456 \, {\left (x^{2} - 6 \, x + 9\right )} e^{\left (2 \, x\right )} + 4608 \, {\left (x^{2} - 6 \, x + 9\right )} e^{x} - 13824 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((576*x^2-3168*x+4320)*exp(x)^4+(3456*x^2-18432*x+24192)*exp(x)^3+(6912*x^2-34560*x+41472)*exp(x)^2+(
4608*x^2-18432*x+13824)*exp(x)+4608*x-13824,x, algorithm="fricas")

[Out]

2304*x^2 + 144*(x^2 - 6*x + 9)*e^(4*x) + 1152*(x^2 - 6*x + 9)*e^(3*x) + 3456*(x^2 - 6*x + 9)*e^(2*x) + 4608*(x
^2 - 6*x + 9)*e^x - 13824*x

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giac [B]  time = 0.16, size = 63, normalized size = 3.32 \begin {gather*} 2304 \, x^{2} + 144 \, {\left (x^{2} - 6 \, x + 9\right )} e^{\left (4 \, x\right )} + 1152 \, {\left (x^{2} - 6 \, x + 9\right )} e^{\left (3 \, x\right )} + 3456 \, {\left (x^{2} - 6 \, x + 9\right )} e^{\left (2 \, x\right )} + 4608 \, {\left (x^{2} - 6 \, x + 9\right )} e^{x} - 13824 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((576*x^2-3168*x+4320)*exp(x)^4+(3456*x^2-18432*x+24192)*exp(x)^3+(6912*x^2-34560*x+41472)*exp(x)^2+(
4608*x^2-18432*x+13824)*exp(x)+4608*x-13824,x, algorithm="giac")

[Out]

2304*x^2 + 144*(x^2 - 6*x + 9)*e^(4*x) + 1152*(x^2 - 6*x + 9)*e^(3*x) + 3456*(x^2 - 6*x + 9)*e^(2*x) + 4608*(x
^2 - 6*x + 9)*e^x - 13824*x

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maple [B]  time = 0.07, size = 68, normalized size = 3.58

method	result	size
risch	\(\left (144 x^{2}-864 x +1296\right ) {\mathrm e}^{4 x}+\left (1152 x^{2}-6912 x +10368\right ) {\mathrm e}^{3 x}+\left (3456 x^{2}-20736 x +31104\right ) {\mathrm e}^{2 x}+\left (4608 x^{2}-27648 x +41472\right ) {\mathrm e}^{x}+2304 x^{2}-13824 x\)	\(68\)
default	\(-13824 x +1296 \,{\mathrm e}^{4 x}+144 x^{2} {\mathrm e}^{4 x}-864 x \,{\mathrm e}^{4 x}+10368 \,{\mathrm e}^{3 x}-6912 x \,{\mathrm e}^{3 x}+1152 x^{2} {\mathrm e}^{3 x}+4608 \,{\mathrm e}^{x} x^{2}-27648 \,{\mathrm e}^{x} x +41472 \,{\mathrm e}^{x}+3456 \,{\mathrm e}^{2 x} x^{2}-20736 x \,{\mathrm e}^{2 x}+31104 \,{\mathrm e}^{2 x}+2304 x^{2}\)	\(92\)
norman	\(-13824 x +1296 \,{\mathrm e}^{4 x}+144 x^{2} {\mathrm e}^{4 x}-864 x \,{\mathrm e}^{4 x}+10368 \,{\mathrm e}^{3 x}-6912 x \,{\mathrm e}^{3 x}+1152 x^{2} {\mathrm e}^{3 x}+4608 \,{\mathrm e}^{x} x^{2}-27648 \,{\mathrm e}^{x} x +41472 \,{\mathrm e}^{x}+3456 \,{\mathrm e}^{2 x} x^{2}-20736 x \,{\mathrm e}^{2 x}+31104 \,{\mathrm e}^{2 x}+2304 x^{2}\)	\(92\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((576*x^2-3168*x+4320)*exp(x)^4+(3456*x^2-18432*x+24192)*exp(x)^3+(6912*x^2-34560*x+41472)*exp(x)^2+(4608*x
^2-18432*x+13824)*exp(x)+4608*x-13824,x,method=_RETURNVERBOSE)

[Out]

(144*x^2-864*x+1296)*exp(4*x)+(1152*x^2-6912*x+10368)*exp(3*x)+(3456*x^2-20736*x+31104)*exp(2*x)+(4608*x^2-276
48*x+41472)*exp(x)+2304*x^2-13824*x

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maxima [B]  time = 0.37, size = 63, normalized size = 3.32 \begin {gather*} 2304 \, x^{2} + 144 \, {\left (x^{2} - 6 \, x + 9\right )} e^{\left (4 \, x\right )} + 1152 \, {\left (x^{2} - 6 \, x + 9\right )} e^{\left (3 \, x\right )} + 3456 \, {\left (x^{2} - 6 \, x + 9\right )} e^{\left (2 \, x\right )} + 4608 \, {\left (x^{2} - 6 \, x + 9\right )} e^{x} - 13824 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((576*x^2-3168*x+4320)*exp(x)^4+(3456*x^2-18432*x+24192)*exp(x)^3+(6912*x^2-34560*x+41472)*exp(x)^2+(
4608*x^2-18432*x+13824)*exp(x)+4608*x-13824,x, algorithm="maxima")

[Out]

2304*x^2 + 144*(x^2 - 6*x + 9)*e^(4*x) + 1152*(x^2 - 6*x + 9)*e^(3*x) + 3456*(x^2 - 6*x + 9)*e^(2*x) + 4608*(x
^2 - 6*x + 9)*e^x - 13824*x

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mupad [B]  time = 6.81, size = 91, normalized size = 4.79 \begin {gather*} 31104\,{\mathrm {e}}^{2\,x}-13824\,x+10368\,{\mathrm {e}}^{3\,x}+1296\,{\mathrm {e}}^{4\,x}+41472\,{\mathrm {e}}^x-20736\,x\,{\mathrm {e}}^{2\,x}-6912\,x\,{\mathrm {e}}^{3\,x}-864\,x\,{\mathrm {e}}^{4\,x}+4608\,x^2\,{\mathrm {e}}^x+3456\,x^2\,{\mathrm {e}}^{2\,x}+1152\,x^2\,{\mathrm {e}}^{3\,x}+144\,x^2\,{\mathrm {e}}^{4\,x}-27648\,x\,{\mathrm {e}}^x+2304\,x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(4608*x + exp(4*x)*(576*x^2 - 3168*x + 4320) + exp(3*x)*(3456*x^2 - 18432*x + 24192) + exp(2*x)*(6912*x^2 -
34560*x + 41472) + exp(x)*(4608*x^2 - 18432*x + 13824) - 13824,x)

[Out]

31104*exp(2*x) - 13824*x + 10368*exp(3*x) + 1296*exp(4*x) + 41472*exp(x) - 20736*x*exp(2*x) - 6912*x*exp(3*x)
- 864*x*exp(4*x) + 4608*x^2*exp(x) + 3456*x^2*exp(2*x) + 1152*x^2*exp(3*x) + 144*x^2*exp(4*x) - 27648*x*exp(x)
+ 2304*x^2

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sympy [B]  time = 0.17, size = 66, normalized size = 3.47 \begin {gather*} 2304 x^{2} - 13824 x + \left (144 x^{2} - 864 x + 1296\right ) e^{4 x} + \left (1152 x^{2} - 6912 x + 10368\right ) e^{3 x} + \left (3456 x^{2} - 20736 x + 31104\right ) e^{2 x} + \left (4608 x^{2} - 27648 x + 41472\right ) e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((576*x**2-3168*x+4320)*exp(x)**4+(3456*x**2-18432*x+24192)*exp(x)**3+(6912*x**2-34560*x+41472)*exp(x
)**2+(4608*x**2-18432*x+13824)*exp(x)+4608*x-13824,x)

[Out]

2304*x**2 - 13824*x + (144*x**2 - 864*x + 1296)*exp(4*x) + (1152*x**2 - 6912*x + 10368)*exp(3*x) + (3456*x**2
- 20736*x + 31104)*exp(2*x) + (4608*x**2 - 27648*x + 41472)*exp(x)

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