3.31.33 \(\int (3 x^2+36 x^3+120 x^4+96 x^5+e^x (-8 x^3-52 x^4-58 x^5-8 x^6)+e^{2 x} (5 x^4+8 x^5+2 x^6)) \, dx\)

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

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Rubi [B]  time = 0.46, antiderivative size = 61, normalized size of antiderivative = 2.90, number of steps used = 46, number of rules used = 4, integrand size = 68, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {2196, 2176, 2194, 1594} \begin {gather*} -8 e^x x^6+e^{2 x} x^6+16 x^6-10 e^x x^5+e^{2 x} x^5+24 x^5-2 e^x x^4+9 x^4+x^3 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[3*x^2 + 36*x^3 + 120*x^4 + 96*x^5 + E^x*(-8*x^3 - 52*x^4 - 58*x^5 - 8*x^6) + E^(2*x)*(5*x^4 + 8*x^5 + 2*x^
6),x]

[Out]

x^3 + 9*x^4 - 2*E^x*x^4 + 24*x^5 - 10*E^x*x^5 + E^(2*x)*x^5 + 16*x^6 - 8*E^x*x^6 + E^(2*x)*x^6

Rule 1594

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.) + (c_.)*(x_)^(r_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^
(q - p) + c*x^(r - p))^n, x] /; FreeQ[{a, b, c, p, q, r}, x] && IntegerQ[n] && PosQ[q - p] && PosQ[r - p]

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} &=x^3+9 x^4+24 x^5+16 x^6+\int e^x \left (-8 x^3-52 x^4-58 x^5-8 x^6\right ) \, dx+\int e^{2 x} \left (5 x^4+8 x^5+2 x^6\right ) \, dx\\ &=x^3+9 x^4+24 x^5+16 x^6+\int e^{2 x} x^4 \left (5+8 x+2 x^2\right ) \, dx+\int \left (-8 e^x x^3-52 e^x x^4-58 e^x x^5-8 e^x x^6\right ) \, dx\\ &=x^3+9 x^4+24 x^5+16 x^6-8 \int e^x x^3 \, dx-8 \int e^x x^6 \, dx-52 \int e^x x^4 \, dx-58 \int e^x x^5 \, dx+\int \left (5 e^{2 x} x^4+8 e^{2 x} x^5+2 e^{2 x} x^6\right ) \, dx\\ &=x^3-8 e^x x^3+9 x^4-52 e^x x^4+24 x^5-58 e^x x^5+16 x^6-8 e^x x^6+2 \int e^{2 x} x^6 \, dx+5 \int e^{2 x} x^4 \, dx+8 \int e^{2 x} x^5 \, dx+24 \int e^x x^2 \, dx+48 \int e^x x^5 \, dx+208 \int e^x x^3 \, dx+290 \int e^x x^4 \, dx\\ &=24 e^x x^2+x^3+200 e^x x^3+9 x^4+238 e^x x^4+\frac {5}{2} e^{2 x} x^4+24 x^5-10 e^x x^5+4 e^{2 x} x^5+16 x^6-8 e^x x^6+e^{2 x} x^6-6 \int e^{2 x} x^5 \, dx-10 \int e^{2 x} x^3 \, dx-20 \int e^{2 x} x^4 \, dx-48 \int e^x x \, dx-240 \int e^x x^4 \, dx-624 \int e^x x^2 \, dx-1160 \int e^x x^3 \, dx\\ &=-48 e^x x-600 e^x x^2+x^3-960 e^x x^3-5 e^{2 x} x^3+9 x^4-2 e^x x^4-\frac {15}{2} e^{2 x} x^4+24 x^5-10 e^x x^5+e^{2 x} x^5+16 x^6-8 e^x x^6+e^{2 x} x^6+15 \int e^{2 x} x^2 \, dx+15 \int e^{2 x} x^4 \, dx+40 \int e^{2 x} x^3 \, dx+48 \int e^x \, dx+960 \int e^x x^3 \, dx+1248 \int e^x x \, dx+3480 \int e^x x^2 \, dx\\ &=48 e^x+1200 e^x x+2880 e^x x^2+\frac {15}{2} e^{2 x} x^2+x^3+15 e^{2 x} x^3+9 x^4-2 e^x x^4+24 x^5-10 e^x x^5+e^{2 x} x^5+16 x^6-8 e^x x^6+e^{2 x} x^6-15 \int e^{2 x} x \, dx-30 \int e^{2 x} x^3 \, dx-60 \int e^{2 x} x^2 \, dx-1248 \int e^x \, dx-2880 \int e^x x^2 \, dx-6960 \int e^x x \, dx\\ &=-1200 e^x-5760 e^x x-\frac {15}{2} e^{2 x} x-\frac {45}{2} e^{2 x} x^2+x^3+9 x^4-2 e^x x^4+24 x^5-10 e^x x^5+e^{2 x} x^5+16 x^6-8 e^x x^6+e^{2 x} x^6+\frac {15}{2} \int e^{2 x} \, dx+45 \int e^{2 x} x^2 \, dx+60 \int e^{2 x} x \, dx+5760 \int e^x x \, dx+6960 \int e^x \, dx\\ &=5760 e^x+\frac {15 e^{2 x}}{4}+\frac {45}{2} e^{2 x} x+x^3+9 x^4-2 e^x x^4+24 x^5-10 e^x x^5+e^{2 x} x^5+16 x^6-8 e^x x^6+e^{2 x} x^6-30 \int e^{2 x} \, dx-45 \int e^{2 x} x \, dx-5760 \int e^x \, dx\\ &=-\frac {45 e^{2 x}}{4}+x^3+9 x^4-2 e^x x^4+24 x^5-10 e^x x^5+e^{2 x} x^5+16 x^6-8 e^x x^6+e^{2 x} x^6+\frac {45}{2} \int e^{2 x} \, dx\\ &=x^3+9 x^4-2 e^x x^4+24 x^5-10 e^x x^5+e^{2 x} x^5+16 x^6-8 e^x x^6+e^{2 x} x^6\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.12, size = 18, normalized size = 0.86 \begin {gather*} x^3 (1+x) \left (-1+\left (-4+e^x\right ) x\right )^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[3*x^2 + 36*x^3 + 120*x^4 + 96*x^5 + E^x*(-8*x^3 - 52*x^4 - 58*x^5 - 8*x^6) + E^(2*x)*(5*x^4 + 8*x^5
+ 2*x^6),x]

[Out]

x^3*(1 + x)*(-1 + (-4 + E^x)*x)^2

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fricas [B]  time = 0.81, size = 49, normalized size = 2.33 \begin {gather*} 16 \, x^{6} + 24 \, x^{5} + 9 \, x^{4} + x^{3} + {\left (x^{6} + x^{5}\right )} e^{\left (2 \, x\right )} - 2 \, {\left (4 \, x^{6} + 5 \, x^{5} + x^{4}\right )} e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^6+8*x^5+5*x^4)*exp(x)^2+(-8*x^6-58*x^5-52*x^4-8*x^3)*exp(x)+96*x^5+120*x^4+36*x^3+3*x^2,x, algo
rithm="fricas")

[Out]

16*x^6 + 24*x^5 + 9*x^4 + x^3 + (x^6 + x^5)*e^(2*x) - 2*(4*x^6 + 5*x^5 + x^4)*e^x

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giac [B]  time = 0.34, size = 49, normalized size = 2.33 \begin {gather*} 16 \, x^{6} + 24 \, x^{5} + 9 \, x^{4} + x^{3} + {\left (x^{6} + x^{5}\right )} e^{\left (2 \, x\right )} - 2 \, {\left (4 \, x^{6} + 5 \, x^{5} + x^{4}\right )} e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^6+8*x^5+5*x^4)*exp(x)^2+(-8*x^6-58*x^5-52*x^4-8*x^3)*exp(x)+96*x^5+120*x^4+36*x^3+3*x^2,x, algo
rithm="giac")

[Out]

16*x^6 + 24*x^5 + 9*x^4 + x^3 + (x^6 + x^5)*e^(2*x) - 2*(4*x^6 + 5*x^5 + x^4)*e^x

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maple [B]  time = 0.04, size = 51, normalized size = 2.43

method	result	size
risch	\(\left (x^{6}+x^{5}\right ) {\mathrm e}^{2 x}+\left (-8 x^{6}-10 x^{5}-2 x^{4}\right ) {\mathrm e}^{x}+16 x^{6}+24 x^{5}+9 x^{4}+x^{3}\)	\(51\)
default	\(x^{5} {\mathrm e}^{2 x}+x^{6} {\mathrm e}^{2 x}-10 x^{5} {\mathrm e}^{x}-2 \,{\mathrm e}^{x} x^{4}-8 x^{6} {\mathrm e}^{x}+x^{3}+9 x^{4}+24 x^{5}+16 x^{6}\)	\(57\)
norman	\(x^{5} {\mathrm e}^{2 x}+x^{6} {\mathrm e}^{2 x}-10 x^{5} {\mathrm e}^{x}-2 \,{\mathrm e}^{x} x^{4}-8 x^{6} {\mathrm e}^{x}+x^{3}+9 x^{4}+24 x^{5}+16 x^{6}\)	\(57\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x^6+8*x^5+5*x^4)*exp(x)^2+(-8*x^6-58*x^5-52*x^4-8*x^3)*exp(x)+96*x^5+120*x^4+36*x^3+3*x^2,x,method=_RET
URNVERBOSE)

[Out]

(x^6+x^5)*exp(2*x)+(-8*x^6-10*x^5-2*x^4)*exp(x)+16*x^6+24*x^5+9*x^4+x^3

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maxima [B]  time = 0.40, size = 49, normalized size = 2.33 \begin {gather*} 16 \, x^{6} + 24 \, x^{5} + 9 \, x^{4} + x^{3} + {\left (x^{6} + x^{5}\right )} e^{\left (2 \, x\right )} - 2 \, {\left (4 \, x^{6} + 5 \, x^{5} + x^{4}\right )} e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^6+8*x^5+5*x^4)*exp(x)^2+(-8*x^6-58*x^5-52*x^4-8*x^3)*exp(x)+96*x^5+120*x^4+36*x^3+3*x^2,x, algo
rithm="maxima")

[Out]

16*x^6 + 24*x^5 + 9*x^4 + x^3 + (x^6 + x^5)*e^(2*x) - 2*(4*x^6 + 5*x^5 + x^4)*e^x

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mupad [B]  time = 1.79, size = 19, normalized size = 0.90 \begin {gather*} x^3\,\left (x+1\right )\,{\left (4\,x-x\,{\mathrm {e}}^x+1\right )}^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(2*x)*(5*x^4 + 8*x^5 + 2*x^6) - exp(x)*(8*x^3 + 52*x^4 + 58*x^5 + 8*x^6) + 3*x^2 + 36*x^3 + 120*x^4 + 9
6*x^5,x)

[Out]

x^3*(x + 1)*(4*x - x*exp(x) + 1)^2

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sympy [B]  time = 0.12, size = 49, normalized size = 2.33 \begin {gather*} 16 x^{6} + 24 x^{5} + 9 x^{4} + x^{3} + \left (x^{6} + x^{5}\right ) e^{2 x} + \left (- 8 x^{6} - 10 x^{5} - 2 x^{4}\right ) e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x**6+8*x**5+5*x**4)*exp(x)**2+(-8*x**6-58*x**5-52*x**4-8*x**3)*exp(x)+96*x**5+120*x**4+36*x**3+3*
x**2,x)

[Out]

16*x**6 + 24*x**5 + 9*x**4 + x**3 + (x**6 + x**5)*exp(2*x) + (-8*x**6 - 10*x**5 - 2*x**4)*exp(x)

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