3.13.17 \(\int (-6+14 x+2 e^2 x+12 x^2-4 x^3+10 x^4+6 x^5+e (6-4 x+6 x^2+8 x^3)+e^{4 x} (4 x^3+4 x^4)+e^{2 x} (-12 x-6 x^2-4 x^3-14 x^4-4 x^5+e (-6 x^2-4 x^3))) \, dx\)

Optimal. Leaf size=25 \[ \left (3-x-e^{2 x} x^2+x \left (e+x+x^2\right )\right )^2 \]

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Rubi [B]  time = 0.49, antiderivative size = 114, normalized size of antiderivative = 4.56, number of steps used = 46, number of rules used = 5, integrand size = 108, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.046, Rules used = {6, 1593, 2196, 2176, 2194} \begin {gather*} x^6-2 e^{2 x} x^5+2 x^5-2 e^{2 x} x^4+e^{4 x} x^4+2 e x^4-x^4+2 e^{2 x} x^3-2 e^{2 x+1} x^3+2 e x^3+4 x^3-6 e^{2 x} x^2+\left (7+e^2\right ) x^2-2 e x^2+6 e x-6 x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[-6 + 14*x + 2*E^2*x + 12*x^2 - 4*x^3 + 10*x^4 + 6*x^5 + E*(6 - 4*x + 6*x^2 + 8*x^3) + E^(4*x)*(4*x^3 + 4*x
^4) + E^(2*x)*(-12*x - 6*x^2 - 4*x^3 - 14*x^4 - 4*x^5 + E*(-6*x^2 - 4*x^3)),x]

[Out]

-6*x + 6*E*x - 2*E*x^2 - 6*E^(2*x)*x^2 + (7 + E^2)*x^2 + 4*x^3 + 2*E*x^3 + 2*E^(2*x)*x^3 - 2*E^(1 + 2*x)*x^3 -
x^4 + 2*E*x^4 - 2*E^(2*x)*x^4 + E^(4*x)*x^4 + 2*x^5 - 2*E^(2*x)*x^5 + x^6

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, x]

Rule 1593

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - 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} &=\int \left (-6+\left (14+2 e^2\right ) x+12 x^2-4 x^3+10 x^4+6 x^5+e \left (6-4 x+6 x^2+8 x^3\right )+e^{4 x} \left (4 x^3+4 x^4\right )+e^{2 x} \left (-12 x-6 x^2-4 x^3-14 x^4-4 x^5+e \left (-6 x^2-4 x^3\right )\right )\right ) \, dx\\ &=-6 x+\left (7+e^2\right ) x^2+4 x^3-x^4+2 x^5+x^6+e \int \left (6-4 x+6 x^2+8 x^3\right ) \, dx+\int e^{4 x} \left (4 x^3+4 x^4\right ) \, dx+\int e^{2 x} \left (-12 x-6 x^2-4 x^3-14 x^4-4 x^5+e \left (-6 x^2-4 x^3\right )\right ) \, dx\\ &=-6 x+6 e x-2 e x^2+\left (7+e^2\right ) x^2+4 x^3+2 e x^3-x^4+2 e x^4+2 x^5+x^6+\int e^{4 x} x^3 (4+4 x) \, dx+\int \left (-12 e^{2 x} x-6 e^{2 x} x^2-4 e^{2 x} x^3-14 e^{2 x} x^4-4 e^{2 x} x^5-2 e^{1+2 x} x^2 (3+2 x)\right ) \, dx\\ &=-6 x+6 e x-2 e x^2+\left (7+e^2\right ) x^2+4 x^3+2 e x^3-x^4+2 e x^4+2 x^5+x^6-2 \int e^{1+2 x} x^2 (3+2 x) \, dx-4 \int e^{2 x} x^3 \, dx-4 \int e^{2 x} x^5 \, dx-6 \int e^{2 x} x^2 \, dx-12 \int e^{2 x} x \, dx-14 \int e^{2 x} x^4 \, dx+\int \left (4 e^{4 x} x^3+4 e^{4 x} x^4\right ) \, dx\\ &=-6 x+6 e x-6 e^{2 x} x-2 e x^2-3 e^{2 x} x^2+\left (7+e^2\right ) x^2+4 x^3+2 e x^3-2 e^{2 x} x^3-x^4+2 e x^4-7 e^{2 x} x^4+2 x^5-2 e^{2 x} x^5+x^6-2 \int \left (3 e^{1+2 x} x^2+2 e^{1+2 x} x^3\right ) \, dx+4 \int e^{4 x} x^3 \, dx+4 \int e^{4 x} x^4 \, dx+6 \int e^{2 x} \, dx+6 \int e^{2 x} x \, dx+6 \int e^{2 x} x^2 \, dx+10 \int e^{2 x} x^4 \, dx+28 \int e^{2 x} x^3 \, dx\\ &=3 e^{2 x}-6 x+6 e x-3 e^{2 x} x-2 e x^2+\left (7+e^2\right ) x^2+4 x^3+2 e x^3+12 e^{2 x} x^3+e^{4 x} x^3-x^4+2 e x^4-2 e^{2 x} x^4+e^{4 x} x^4+2 x^5-2 e^{2 x} x^5+x^6-3 \int e^{2 x} \, dx-3 \int e^{4 x} x^2 \, dx-4 \int e^{4 x} x^3 \, dx-4 \int e^{1+2 x} x^3 \, dx-6 \int e^{2 x} x \, dx-6 \int e^{1+2 x} x^2 \, dx-20 \int e^{2 x} x^3 \, dx-42 \int e^{2 x} x^2 \, dx\\ &=\frac {3 e^{2 x}}{2}-6 x+6 e x-6 e^{2 x} x-2 e x^2-21 e^{2 x} x^2-\frac {3}{4} e^{4 x} x^2-3 e^{1+2 x} x^2+\left (7+e^2\right ) x^2+4 x^3+2 e x^3+2 e^{2 x} x^3-2 e^{1+2 x} x^3-x^4+2 e x^4-2 e^{2 x} x^4+e^{4 x} x^4+2 x^5-2 e^{2 x} x^5+x^6+\frac {3}{2} \int e^{4 x} x \, dx+3 \int e^{2 x} \, dx+3 \int e^{4 x} x^2 \, dx+6 \int e^{1+2 x} x \, dx+6 \int e^{1+2 x} x^2 \, dx+30 \int e^{2 x} x^2 \, dx+42 \int e^{2 x} x \, dx\\ &=3 e^{2 x}-6 x+6 e x+15 e^{2 x} x+\frac {3}{8} e^{4 x} x+3 e^{1+2 x} x-2 e x^2-6 e^{2 x} x^2+\left (7+e^2\right ) x^2+4 x^3+2 e x^3+2 e^{2 x} x^3-2 e^{1+2 x} x^3-x^4+2 e x^4-2 e^{2 x} x^4+e^{4 x} x^4+2 x^5-2 e^{2 x} x^5+x^6-\frac {3}{8} \int e^{4 x} \, dx-\frac {3}{2} \int e^{4 x} x \, dx-3 \int e^{1+2 x} \, dx-6 \int e^{1+2 x} x \, dx-21 \int e^{2 x} \, dx-30 \int e^{2 x} x \, dx\\ &=-\frac {15 e^{2 x}}{2}-\frac {3 e^{4 x}}{32}-\frac {3}{2} e^{1+2 x}-6 x+6 e x-2 e x^2-6 e^{2 x} x^2+\left (7+e^2\right ) x^2+4 x^3+2 e x^3+2 e^{2 x} x^3-2 e^{1+2 x} x^3-x^4+2 e x^4-2 e^{2 x} x^4+e^{4 x} x^4+2 x^5-2 e^{2 x} x^5+x^6+\frac {3}{8} \int e^{4 x} \, dx+3 \int e^{1+2 x} \, dx+15 \int e^{2 x} \, dx\\ &=-6 x+6 e x-2 e x^2-6 e^{2 x} x^2+\left (7+e^2\right ) x^2+4 x^3+2 e x^3+2 e^{2 x} x^3-2 e^{1+2 x} x^3-x^4+2 e x^4-2 e^{2 x} x^4+e^{4 x} x^4+2 x^5-2 e^{2 x} x^5+x^6\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.11, size = 91, normalized size = 3.64 \begin {gather*} -6 x+6 e x+7 x^2-2 e x^2+e^2 x^2+4 x^3+2 e x^3-x^4+2 e x^4+e^{4 x} x^4+2 x^5+x^6-2 e^{2 x} \left (3 x^2+(-1+e) x^3+x^4+x^5\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[-6 + 14*x + 2*E^2*x + 12*x^2 - 4*x^3 + 10*x^4 + 6*x^5 + E*(6 - 4*x + 6*x^2 + 8*x^3) + E^(4*x)*(4*x^3
+ 4*x^4) + E^(2*x)*(-12*x - 6*x^2 - 4*x^3 - 14*x^4 - 4*x^5 + E*(-6*x^2 - 4*x^3)),x]

[Out]

-6*x + 6*E*x + 7*x^2 - 2*E*x^2 + E^2*x^2 + 4*x^3 + 2*E*x^3 - x^4 + 2*E*x^4 + E^(4*x)*x^4 + 2*x^5 + x^6 - 2*E^(
2*x)*(3*x^2 + (-1 + E)*x^3 + x^4 + x^5)

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fricas [B]  time = 0.60, size = 89, normalized size = 3.56 \begin {gather*} x^{6} + 2 \, x^{5} + x^{4} e^{\left (4 \, x\right )} - x^{4} + 4 \, x^{3} + x^{2} e^{2} + 7 \, x^{2} + 2 \, {\left (x^{4} + x^{3} - x^{2} + 3 \, x\right )} e - 2 \, {\left (x^{5} + x^{4} + x^{3} e - x^{3} + 3 \, x^{2}\right )} e^{\left (2 \, x\right )} - 6 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^4+4*x^3)*exp(x)^4+((-4*x^3-6*x^2)*exp(1)-4*x^5-14*x^4-4*x^3-6*x^2-12*x)*exp(x)^2+2*x*exp(1)^2+(
8*x^3+6*x^2-4*x+6)*exp(1)+6*x^5+10*x^4-4*x^3+12*x^2+14*x-6,x, algorithm="fricas")

[Out]

x^6 + 2*x^5 + x^4*e^(4*x) - x^4 + 4*x^3 + x^2*e^2 + 7*x^2 + 2*(x^4 + x^3 - x^2 + 3*x)*e - 2*(x^5 + x^4 + x^3*e
- x^3 + 3*x^2)*e^(2*x) - 6*x

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giac [B]  time = 0.25, size = 94, normalized size = 3.76 \begin {gather*} x^{6} + 2 \, x^{5} + x^{4} e^{\left (4 \, x\right )} - x^{4} - 2 \, x^{3} e^{\left (2 \, x + 1\right )} + 4 \, x^{3} + x^{2} e^{2} + 7 \, x^{2} + 2 \, {\left (x^{4} + x^{3} - x^{2} + 3 \, x\right )} e - 2 \, {\left (x^{5} + x^{4} - x^{3} + 3 \, x^{2}\right )} e^{\left (2 \, x\right )} - 6 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^4+4*x^3)*exp(x)^4+((-4*x^3-6*x^2)*exp(1)-4*x^5-14*x^4-4*x^3-6*x^2-12*x)*exp(x)^2+2*x*exp(1)^2+(
8*x^3+6*x^2-4*x+6)*exp(1)+6*x^5+10*x^4-4*x^3+12*x^2+14*x-6,x, algorithm="giac")

[Out]

x^6 + 2*x^5 + x^4*e^(4*x) - x^4 - 2*x^3*e^(2*x + 1) + 4*x^3 + x^2*e^2 + 7*x^2 + 2*(x^4 + x^3 - x^2 + 3*x)*e -
2*(x^5 + x^4 - x^3 + 3*x^2)*e^(2*x) - 6*x

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maple [B]  time = 0.06, size = 101, normalized size = 4.04

method	result	size
norman	\(x^{6}+x^{4} {\mathrm e}^{4 x}+\left (2 \,{\mathrm e}-1\right ) x^{4}+\left (2 \,{\mathrm e}+4\right ) x^{3}+\left (6 \,{\mathrm e}-6\right ) x +\left ({\mathrm e}^{2}-2 \,{\mathrm e}+7\right ) x^{2}+\left (-2 \,{\mathrm e}+2\right ) x^{3} {\mathrm e}^{2 x}+2 x^{5}-2 x^{5} {\mathrm e}^{2 x}-6 \,{\mathrm e}^{2 x} x^{2}-2 \,{\mathrm e}^{2 x} x^{4}\)	\(101\)
risch	\(x^{4} {\mathrm e}^{4 x}+\left (-2 x^{5}-2 x^{3} {\mathrm e}-2 x^{4}+2 x^{3}-6 x^{2}\right ) {\mathrm e}^{2 x}+x^{2} {\mathrm e}^{2}+2 x^{4} {\mathrm e}+2 x^{3} {\mathrm e}-2 x^{2} {\mathrm e}+6 x \,{\mathrm e}+x^{6}+2 x^{5}-x^{4}+4 x^{3}+7 x^{2}-6 x\)	\(101\)
default	\(-6 x -2 \,{\mathrm e}^{2 x} x^{4}+2 \,{\mathrm e}^{2 x} x^{3}-6 \,{\mathrm e}^{2 x} x^{2}-2 x^{5} {\mathrm e}^{2 x}-6 \,{\mathrm e} \left (\frac {{\mathrm e}^{2 x} x^{2}}{2}-\frac {x \,{\mathrm e}^{2 x}}{2}+\frac {{\mathrm e}^{2 x}}{4}\right )-4 \,{\mathrm e} \left (\frac {{\mathrm e}^{2 x} x^{3}}{2}-\frac {3 \,{\mathrm e}^{2 x} x^{2}}{4}+\frac {3 x \,{\mathrm e}^{2 x}}{4}-\frac {3 \,{\mathrm e}^{2 x}}{8}\right )+x^{4} {\mathrm e}^{4 x}+{\mathrm e} \left (2 x^{4}+2 x^{3}-2 x^{2}+6 x \right )+7 x^{2}+4 x^{3}-x^{4}+2 x^{5}+x^{6}+x^{2} {\mathrm e}^{2}\)	\(165\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*x^4+4*x^3)*exp(x)^4+((-4*x^3-6*x^2)*exp(1)-4*x^5-14*x^4-4*x^3-6*x^2-12*x)*exp(x)^2+2*x*exp(1)^2+(8*x^3+
6*x^2-4*x+6)*exp(1)+6*x^5+10*x^4-4*x^3+12*x^2+14*x-6,x,method=_RETURNVERBOSE)

[Out]

x^6+x^4*exp(x)^4+(2*exp(1)-1)*x^4+(2*exp(1)+4)*x^3+(6*exp(1)-6)*x+(exp(1)^2-2*exp(1)+7)*x^2+(-2*exp(1)+2)*x^3*
exp(x)^2+2*x^5-2*x^5*exp(x)^2-6*exp(x)^2*x^2-2*exp(x)^2*x^4

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maxima [B]  time = 0.39, size = 86, normalized size = 3.44 \begin {gather*} x^{6} + 2 \, x^{5} + x^{4} e^{\left (4 \, x\right )} - x^{4} + 4 \, x^{3} + x^{2} e^{2} + 7 \, x^{2} + 2 \, {\left (x^{4} + x^{3} - x^{2} + 3 \, x\right )} e - 2 \, {\left (x^{5} + x^{4} + x^{3} {\left (e - 1\right )} + 3 \, x^{2}\right )} e^{\left (2 \, x\right )} - 6 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^4+4*x^3)*exp(x)^4+((-4*x^3-6*x^2)*exp(1)-4*x^5-14*x^4-4*x^3-6*x^2-12*x)*exp(x)^2+2*x*exp(1)^2+(
8*x^3+6*x^2-4*x+6)*exp(1)+6*x^5+10*x^4-4*x^3+12*x^2+14*x-6,x, algorithm="maxima")

[Out]

x^6 + 2*x^5 + x^4*e^(4*x) - x^4 + 4*x^3 + x^2*e^2 + 7*x^2 + 2*(x^4 + x^3 - x^2 + 3*x)*e - 2*(x^5 + x^4 + x^3*(
e - 1) + 3*x^2)*e^(2*x) - 6*x

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mupad [B]  time = 0.13, size = 99, normalized size = 3.96 \begin {gather*} x^4\,\left (2\,\mathrm {e}-1\right )+x^3\,\left (2\,\mathrm {e}+4\right )-6\,x^2\,{\mathrm {e}}^{2\,x}-2\,x^4\,{\mathrm {e}}^{2\,x}-2\,x^5\,{\mathrm {e}}^{2\,x}+x^4\,{\mathrm {e}}^{4\,x}+x^2\,\left ({\mathrm {e}}^2-2\,\mathrm {e}+7\right )+2\,x^5+x^6+x\,\left (6\,\mathrm {e}-6\right )-x^3\,{\mathrm {e}}^{2\,x}\,\left (2\,\mathrm {e}-2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(14*x + 2*x*exp(2) - exp(2*x)*(12*x + exp(1)*(6*x^2 + 4*x^3) + 6*x^2 + 4*x^3 + 14*x^4 + 4*x^5) + exp(4*x)*(
4*x^3 + 4*x^4) + exp(1)*(6*x^2 - 4*x + 8*x^3 + 6) + 12*x^2 - 4*x^3 + 10*x^4 + 6*x^5 - 6,x)

[Out]

x^4*(2*exp(1) - 1) + x^3*(2*exp(1) + 4) - 6*x^2*exp(2*x) - 2*x^4*exp(2*x) - 2*x^5*exp(2*x) + x^4*exp(4*x) + x^
2*(exp(2) - 2*exp(1) + 7) + 2*x^5 + x^6 + x*(6*exp(1) - 6) - x^3*exp(2*x)*(2*exp(1) - 2)

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sympy [B]  time = 0.18, size = 92, normalized size = 3.68 \begin {gather*} x^{6} + 2 x^{5} + x^{4} e^{4 x} + x^{4} \left (-1 + 2 e\right ) + x^{3} \left (4 + 2 e\right ) + x^{2} \left (- 2 e + 7 + e^{2}\right ) + x \left (-6 + 6 e\right ) + \left (- 2 x^{5} - 2 x^{4} - 2 e x^{3} + 2 x^{3} - 6 x^{2}\right ) e^{2 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x**4+4*x**3)*exp(x)**4+((-4*x**3-6*x**2)*exp(1)-4*x**5-14*x**4-4*x**3-6*x**2-12*x)*exp(x)**2+2*x*
exp(1)**2+(8*x**3+6*x**2-4*x+6)*exp(1)+6*x**5+10*x**4-4*x**3+12*x**2+14*x-6,x)

[Out]

x**6 + 2*x**5 + x**4*exp(4*x) + x**4*(-1 + 2*E) + x**3*(4 + 2*E) + x**2*(-2*E + 7 + exp(2)) + x*(-6 + 6*E) + (
-2*x**5 - 2*x**4 - 2*E*x**3 + 2*x**3 - 6*x**2)*exp(2*x)

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