3.22.100 \(\int e^{3 x} (3 e^4+2 x+9 x^2+10 x^3+3 x^4+e^3 (4+12 x)+e^2 (2+18 x+18 x^2)+e (8 x+24 x^2+12 x^3)) \, dx\)

Optimal. Leaf size=15 \[ e^{3 x} \left (x+(e+x)^2\right )^2 \]

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Rubi [B]  time = 0.38, antiderivative size = 109, normalized size of antiderivative = 7.27, number of steps used = 38, number of rules used = 3, integrand size = 69, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {2196, 2194, 2176} \begin {gather*} e^{3 x} x^4+2 e^{3 x} x^3+4 e^{3 x+1} x^3+e^{3 x} x^2+4 e^{3 x+1} x^2+6 e^{3 x+2} x^2+2 e^{3 x+2} x-\frac {4}{3} e^{3 x+3}+e^{3 x+4}+\frac {4}{3} e^{3 x+3} (3 x+1) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^(3*x)*(3*E^4 + 2*x + 9*x^2 + 10*x^3 + 3*x^4 + E^3*(4 + 12*x) + E^2*(2 + 18*x + 18*x^2) + E*(8*x + 24*x^2
+ 12*x^3)),x]

[Out]

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

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 (3 e^{4+3 x}+2 e^{3 x} x+9 e^{3 x} x^2+10 e^{3 x} x^3+3 e^{3 x} x^4+4 e^{3+3 x} (1+3 x)+4 e^{1+3 x} x \left (2+6 x+3 x^2\right )+2 e^{2+3 x} \left (1+9 x+9 x^2\right )\right ) \, dx\\ &=2 \int e^{3 x} x \, dx+2 \int e^{2+3 x} \left (1+9 x+9 x^2\right ) \, dx+3 \int e^{4+3 x} \, dx+3 \int e^{3 x} x^4 \, dx+4 \int e^{3+3 x} (1+3 x) \, dx+4 \int e^{1+3 x} x \left (2+6 x+3 x^2\right ) \, dx+9 \int e^{3 x} x^2 \, dx+10 \int e^{3 x} x^3 \, dx\\ &=e^{4+3 x}+\frac {2}{3} e^{3 x} x+3 e^{3 x} x^2+\frac {10}{3} e^{3 x} x^3+e^{3 x} x^4+\frac {4}{3} e^{3+3 x} (1+3 x)-\frac {2}{3} \int e^{3 x} \, dx+2 \int \left (e^{2+3 x}+9 e^{2+3 x} x+9 e^{2+3 x} x^2\right ) \, dx-4 \int e^{3+3 x} \, dx-4 \int e^{3 x} x^3 \, dx+4 \int \left (2 e^{1+3 x} x+6 e^{1+3 x} x^2+3 e^{1+3 x} x^3\right ) \, dx-6 \int e^{3 x} x \, dx-10 \int e^{3 x} x^2 \, dx\\ &=-\frac {2 e^{3 x}}{9}-\frac {4}{3} e^{3+3 x}+e^{4+3 x}-\frac {4}{3} e^{3 x} x-\frac {1}{3} e^{3 x} x^2+2 e^{3 x} x^3+e^{3 x} x^4+\frac {4}{3} e^{3+3 x} (1+3 x)+2 \int e^{3 x} \, dx+2 \int e^{2+3 x} \, dx+4 \int e^{3 x} x^2 \, dx+\frac {20}{3} \int e^{3 x} x \, dx+8 \int e^{1+3 x} x \, dx+12 \int e^{1+3 x} x^3 \, dx+18 \int e^{2+3 x} x \, dx+18 \int e^{2+3 x} x^2 \, dx+24 \int e^{1+3 x} x^2 \, dx\\ &=\frac {4 e^{3 x}}{9}+\frac {2}{3} e^{2+3 x}-\frac {4}{3} e^{3+3 x}+e^{4+3 x}+\frac {8}{9} e^{3 x} x+\frac {8}{3} e^{1+3 x} x+6 e^{2+3 x} x+e^{3 x} x^2+8 e^{1+3 x} x^2+6 e^{2+3 x} x^2+2 e^{3 x} x^3+4 e^{1+3 x} x^3+e^{3 x} x^4+\frac {4}{3} e^{3+3 x} (1+3 x)-\frac {20}{9} \int e^{3 x} \, dx-\frac {8}{3} \int e^{1+3 x} \, dx-\frac {8}{3} \int e^{3 x} x \, dx-6 \int e^{2+3 x} \, dx-12 \int e^{2+3 x} x \, dx-12 \int e^{1+3 x} x^2 \, dx-16 \int e^{1+3 x} x \, dx\\ &=-\frac {8 e^{3 x}}{27}-\frac {8}{9} e^{1+3 x}-\frac {4}{3} e^{2+3 x}-\frac {4}{3} e^{3+3 x}+e^{4+3 x}-\frac {8}{3} e^{1+3 x} x+2 e^{2+3 x} x+e^{3 x} x^2+4 e^{1+3 x} x^2+6 e^{2+3 x} x^2+2 e^{3 x} x^3+4 e^{1+3 x} x^3+e^{3 x} x^4+\frac {4}{3} e^{3+3 x} (1+3 x)+\frac {8}{9} \int e^{3 x} \, dx+4 \int e^{2+3 x} \, dx+\frac {16}{3} \int e^{1+3 x} \, dx+8 \int e^{1+3 x} x \, dx\\ &=\frac {8}{9} e^{1+3 x}-\frac {4}{3} e^{3+3 x}+e^{4+3 x}+2 e^{2+3 x} x+e^{3 x} x^2+4 e^{1+3 x} x^2+6 e^{2+3 x} x^2+2 e^{3 x} x^3+4 e^{1+3 x} x^3+e^{3 x} x^4+\frac {4}{3} e^{3+3 x} (1+3 x)-\frac {8}{3} \int e^{1+3 x} \, dx\\ &=-\frac {4}{3} e^{3+3 x}+e^{4+3 x}+2 e^{2+3 x} x+e^{3 x} x^2+4 e^{1+3 x} x^2+6 e^{2+3 x} x^2+2 e^{3 x} x^3+4 e^{1+3 x} x^3+e^{3 x} x^4+\frac {4}{3} e^{3+3 x} (1+3 x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.25, size = 20, normalized size = 1.33 \begin {gather*} e^{3 x} \left (e^2+x+2 e x+x^2\right )^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^(3*x)*(3*E^4 + 2*x + 9*x^2 + 10*x^3 + 3*x^4 + E^3*(4 + 12*x) + E^2*(2 + 18*x + 18*x^2) + E*(8*x +
24*x^2 + 12*x^3)),x]

[Out]

E^(3*x)*(E^2 + x + 2*E*x + x^2)^2

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fricas [B]  time = 0.70, size = 46, normalized size = 3.07 \begin {gather*} {\left (x^{4} + 2 \, x^{3} + x^{2} + 4 \, x e^{3} + 2 \, {\left (3 \, x^{2} + x\right )} e^{2} + 4 \, {\left (x^{3} + x^{2}\right )} e + e^{4}\right )} e^{\left (3 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*exp(1)^4+(12*x+4)*exp(1)^3+(18*x^2+18*x+2)*exp(1)^2+(12*x^3+24*x^2+8*x)*exp(1)+3*x^4+10*x^3+9*x^2
+2*x)*exp(3*x),x, algorithm="fricas")

[Out]

(x^4 + 2*x^3 + x^2 + 4*x*e^3 + 2*(3*x^2 + x)*e^2 + 4*(x^3 + x^2)*e + e^4)*e^(3*x)

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giac [B]  time = 1.33, size = 63, normalized size = 4.20 \begin {gather*} {\left (x^{4} + 2 \, x^{3} + x^{2}\right )} e^{\left (3 \, x\right )} + 4 \, x e^{\left (3 \, x + 3\right )} + 2 \, {\left (3 \, x^{2} + x\right )} e^{\left (3 \, x + 2\right )} + 4 \, {\left (x^{3} + x^{2}\right )} e^{\left (3 \, x + 1\right )} + e^{\left (3 \, x + 4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*exp(1)^4+(12*x+4)*exp(1)^3+(18*x^2+18*x+2)*exp(1)^2+(12*x^3+24*x^2+8*x)*exp(1)+3*x^4+10*x^3+9*x^2
+2*x)*exp(3*x),x, algorithm="giac")

[Out]

(x^4 + 2*x^3 + x^2)*e^(3*x) + 4*x*e^(3*x + 3) + 2*(3*x^2 + x)*e^(3*x + 2) + 4*(x^3 + x^2)*e^(3*x + 1) + e^(3*x
+ 4)

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maple [B]  time = 0.08, size = 51, normalized size = 3.40

method	result	size
risch	\(\left ({\mathrm e}^{4}+4 x \,{\mathrm e}^{3}+6 x^{2} {\mathrm e}^{2}+4 x^{3} {\mathrm e}+x^{4}+2 \,{\mathrm e}^{2} x +4 x^{2} {\mathrm e}+2 x^{3}+x^{2}\right ) {\mathrm e}^{3 x}\)	\(51\)
gosper	\(\left ({\mathrm e}^{4}+4 x \,{\mathrm e}^{3}+6 x^{2} {\mathrm e}^{2}+4 x^{3} {\mathrm e}+x^{4}+2 \,{\mathrm e}^{2} x +4 x^{2} {\mathrm e}+2 x^{3}+x^{2}\right ) {\mathrm e}^{3 x}\)	\(59\)
norman	\({\mathrm e}^{3 x} x^{4}+{\mathrm e}^{3 x} {\mathrm e}^{4}+\left (4 \,{\mathrm e}^{3}+2 \,{\mathrm e}^{2}\right ) x \,{\mathrm e}^{3 x}+\left (4 \,{\mathrm e}+2\right ) x^{3} {\mathrm e}^{3 x}+\left (6 \,{\mathrm e}^{2}+4 \,{\mathrm e}+1\right ) x^{2} {\mathrm e}^{3 x}\)	\(72\)
meijerg	\(-\frac {\left (-\frac {4 \,{\mathrm e}}{9}-\frac {10}{27}\right ) \left (6-\frac {\left (-108 x^{3}+108 x^{2}-72 x +24\right ) {\mathrm e}^{3 x}}{4}\right )}{3}-\frac {\left (-4 \,{\mathrm e}^{3}-6 \,{\mathrm e}^{2}-\frac {8 \,{\mathrm e}}{3}-\frac {2}{3}\right ) \left (1-\frac {\left (-6 x +2\right ) {\mathrm e}^{3 x}}{2}\right )}{3}-\frac {8}{27}+\frac {\left (405 x^{4}-540 x^{3}+540 x^{2}-360 x +120\right ) {\mathrm e}^{3 x}}{405}-\frac {\left (2 \,{\mathrm e}^{2}+\frac {8 \,{\mathrm e}}{3}+1\right ) \left (2-\frac {\left (27 x^{2}-18 x +6\right ) {\mathrm e}^{3 x}}{3}\right )}{3}-{\mathrm e}^{4} \left (1-{\mathrm e}^{3 x}\right )-\frac {4 \,{\mathrm e}^{3} \left (1-{\mathrm e}^{3 x}\right )}{3}-\frac {2 \,{\mathrm e}^{2} \left (1-{\mathrm e}^{3 x}\right )}{3}\)	\(155\)
derivativedivides	\({\mathrm e}^{3 x} x^{4}+2 x^{3} {\mathrm e}^{3 x}+x^{2} {\mathrm e}^{3 x}+\frac {2 \,{\mathrm e}^{3 x} {\mathrm e}^{2}}{3}+\frac {4 \,{\mathrm e}^{3 x} {\mathrm e}^{3}}{3}+{\mathrm e}^{3 x} {\mathrm e}^{4}+\frac {8 \,{\mathrm e} \left (3 x \,{\mathrm e}^{3 x}-{\mathrm e}^{3 x}\right )}{9}+\frac {8 \,{\mathrm e} \left (9 x^{2} {\mathrm e}^{3 x}-6 x \,{\mathrm e}^{3 x}+2 \,{\mathrm e}^{3 x}\right )}{9}+\frac {4 \,{\mathrm e} \left (27 x^{3} {\mathrm e}^{3 x}-27 x^{2} {\mathrm e}^{3 x}+18 x \,{\mathrm e}^{3 x}-6 \,{\mathrm e}^{3 x}\right )}{27}+2 \,{\mathrm e}^{2} \left (3 x \,{\mathrm e}^{3 x}-{\mathrm e}^{3 x}\right )+\frac {2 \,{\mathrm e}^{2} \left (9 x^{2} {\mathrm e}^{3 x}-6 x \,{\mathrm e}^{3 x}+2 \,{\mathrm e}^{3 x}\right )}{3}+\frac {4 \,{\mathrm e}^{3} \left (3 x \,{\mathrm e}^{3 x}-{\mathrm e}^{3 x}\right )}{3}\)	\(206\)
default	\({\mathrm e}^{3 x} x^{4}+2 x^{3} {\mathrm e}^{3 x}+x^{2} {\mathrm e}^{3 x}+\frac {2 \,{\mathrm e}^{3 x} {\mathrm e}^{2}}{3}+\frac {4 \,{\mathrm e}^{3 x} {\mathrm e}^{3}}{3}+{\mathrm e}^{3 x} {\mathrm e}^{4}+\frac {8 \,{\mathrm e} \left (3 x \,{\mathrm e}^{3 x}-{\mathrm e}^{3 x}\right )}{9}+\frac {8 \,{\mathrm e} \left (9 x^{2} {\mathrm e}^{3 x}-6 x \,{\mathrm e}^{3 x}+2 \,{\mathrm e}^{3 x}\right )}{9}+\frac {4 \,{\mathrm e} \left (27 x^{3} {\mathrm e}^{3 x}-27 x^{2} {\mathrm e}^{3 x}+18 x \,{\mathrm e}^{3 x}-6 \,{\mathrm e}^{3 x}\right )}{27}+2 \,{\mathrm e}^{2} \left (3 x \,{\mathrm e}^{3 x}-{\mathrm e}^{3 x}\right )+\frac {2 \,{\mathrm e}^{2} \left (9 x^{2} {\mathrm e}^{3 x}-6 x \,{\mathrm e}^{3 x}+2 \,{\mathrm e}^{3 x}\right )}{3}+\frac {4 \,{\mathrm e}^{3} \left (3 x \,{\mathrm e}^{3 x}-{\mathrm e}^{3 x}\right )}{3}\)	\(206\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*exp(1)^4+(12*x+4)*exp(1)^3+(18*x^2+18*x+2)*exp(1)^2+(12*x^3+24*x^2+8*x)*exp(1)+3*x^4+10*x^3+9*x^2+2*x)*
exp(3*x),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [B]  time = 0.52, size = 221, normalized size = 14.73 \begin {gather*} \frac {1}{27} \, {\left (27 \, x^{4} - 36 \, x^{3} + 36 \, x^{2} - 24 \, x + 8\right )} e^{\left (3 \, x\right )} + \frac {4}{9} \, {\left (9 \, x^{3} e - 9 \, x^{2} e + 6 \, x e - 2 \, e\right )} e^{\left (3 \, x\right )} + \frac {10}{27} \, {\left (9 \, x^{3} - 9 \, x^{2} + 6 \, x - 2\right )} e^{\left (3 \, x\right )} + \frac {2}{3} \, {\left (9 \, x^{2} e^{2} - 6 \, x e^{2} + 2 \, e^{2}\right )} e^{\left (3 \, x\right )} + \frac {8}{9} \, {\left (9 \, x^{2} e - 6 \, x e + 2 \, e\right )} e^{\left (3 \, x\right )} + \frac {1}{3} \, {\left (9 \, x^{2} - 6 \, x + 2\right )} e^{\left (3 \, x\right )} + \frac {4}{3} \, {\left (3 \, x e^{3} - e^{3}\right )} e^{\left (3 \, x\right )} + 2 \, {\left (3 \, x e^{2} - e^{2}\right )} e^{\left (3 \, x\right )} + \frac {8}{9} \, {\left (3 \, x e - e\right )} e^{\left (3 \, x\right )} + \frac {2}{9} \, {\left (3 \, x - 1\right )} e^{\left (3 \, x\right )} + e^{\left (3 \, x + 4\right )} + \frac {4}{3} \, e^{\left (3 \, x + 3\right )} + \frac {2}{3} \, e^{\left (3 \, x + 2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*exp(1)^4+(12*x+4)*exp(1)^3+(18*x^2+18*x+2)*exp(1)^2+(12*x^3+24*x^2+8*x)*exp(1)+3*x^4+10*x^3+9*x^2
+2*x)*exp(3*x),x, algorithm="maxima")

[Out]

1/27*(27*x^4 - 36*x^3 + 36*x^2 - 24*x + 8)*e^(3*x) + 4/9*(9*x^3*e - 9*x^2*e + 6*x*e - 2*e)*e^(3*x) + 10/27*(9*
x^3 - 9*x^2 + 6*x - 2)*e^(3*x) + 2/3*(9*x^2*e^2 - 6*x*e^2 + 2*e^2)*e^(3*x) + 8/9*(9*x^2*e - 6*x*e + 2*e)*e^(3*
x) + 1/3*(9*x^2 - 6*x + 2)*e^(3*x) + 4/3*(3*x*e^3 - e^3)*e^(3*x) + 2*(3*x*e^2 - e^2)*e^(3*x) + 8/9*(3*x*e - e)
*e^(3*x) + 2/9*(3*x - 1)*e^(3*x) + e^(3*x + 4) + 4/3*e^(3*x + 3) + 2/3*e^(3*x + 2)

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mupad [B]  time = 1.36, size = 19, normalized size = 1.27 \begin {gather*} {\mathrm {e}}^{3\,x}\,{\left (x+{\mathrm {e}}^2+2\,x\,\mathrm {e}+x^2\right )}^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(3*x)*(2*x + 3*exp(4) + exp(2)*(18*x + 18*x^2 + 2) + exp(1)*(8*x + 24*x^2 + 12*x^3) + 9*x^2 + 10*x^3 +
3*x^4 + exp(3)*(12*x + 4)),x)

[Out]

exp(3*x)*(x + exp(2) + 2*x*exp(1) + x^2)^2

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sympy [B]  time = 0.16, size = 58, normalized size = 3.87 \begin {gather*} \left (x^{4} + 2 x^{3} + 4 e x^{3} + x^{2} + 4 e x^{2} + 6 x^{2} e^{2} + 2 x e^{2} + 4 x e^{3} + e^{4}\right ) e^{3 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*exp(1)**4+(12*x+4)*exp(1)**3+(18*x**2+18*x+2)*exp(1)**2+(12*x**3+24*x**2+8*x)*exp(1)+3*x**4+10*x*
*3+9*x**2+2*x)*exp(3*x),x)

[Out]

(x**4 + 2*x**3 + 4*E*x**3 + x**2 + 4*E*x**2 + 6*x**2*exp(2) + 2*x*exp(2) + 4*x*exp(3) + exp(4))*exp(3*x)

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