Integrand size = 69, antiderivative size = 15 \[ \int e^{3 x} \left (3 e^4+2 x+9 x^2+10 x^3+3 x^4+e^3 (4+12 x)+e^2 \left (2+18 x+18 x^2\right )+e \left (8 x+24 x^2+12 x^3\right )\right ) \, dx=e^{3 x} \left (x+(e+x)^2\right )^2 \] Output:
exp(3*x)*((x+exp(1))^2+x)^2
Time = 5.51 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.33 \[ \int e^{3 x} \left (3 e^4+2 x+9 x^2+10 x^3+3 x^4+e^3 (4+12 x)+e^2 \left (2+18 x+18 x^2\right )+e \left (8 x+24 x^2+12 x^3\right )\right ) \, dx=e^{3 x} \left (e^2+x+2 e x+x^2\right )^2 \] Input:
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]
Output:
E^(3*x)*(E^2 + x + 2*E*x + x^2)^2
Leaf count is larger than twice the leaf count of optimal. \(109\) vs. \(2(15)=30\).
Time = 0.54 (sec) , antiderivative size = 109, normalized size of antiderivative = 7.27, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.029, Rules used = {2626, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int e^{3 x} \left (3 x^4+10 x^3+9 x^2+e^2 \left (18 x^2+18 x+2\right )+e \left (12 x^3+24 x^2+8 x\right )+2 x+e^3 (12 x+4)+3 e^4\right ) \, dx\) |
\(\Big \downarrow \) 2626 |
\(\displaystyle \int \left (3 e^{3 x} x^4+10 e^{3 x} x^3+9 e^{3 x} x^2+4 e^{3 x+1} \left (3 x^2+6 x+2\right ) x+2 e^{3 x+2} \left (9 x^2+9 x+1\right )+2 e^{3 x} x+3 e^{3 x+4}+4 e^{3 x+3} (3 x+1)\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle e^{3 x} x^4+2 e^{3 x} x^3+4 e^{3 x+1} x^3+e^{3 x} x^2+4 e^{3 x+1} x^2+6 e^{3 x+2} x^2+2 e^{3 x+2} x-\frac {4}{3} e^{3 x+3}+e^{3 x+4}+\frac {4}{3} e^{3 x+3} (3 x+1)\) |
Input:
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]
Output:
(-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
Int[(F_)^(v_)*(Px_), x_Symbol] :> Int[ExpandIntegrand[F^v, Px, x], x] /; Fr eeQ[F, x] && PolynomialQ[Px, x] && LinearQ[v, x] && !TrueQ[$UseGamma]
Leaf count of result is larger than twice the leaf count of optimal. \(50\) vs. \(2(15)=30\).
Time = 0.30 (sec) , antiderivative size = 51, normalized size of antiderivative = 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 | \({\mathrm e}^{3 x} x^{4}+{\mathrm e}^{3 x} {\mathrm e}^{4}+\left (4 \,{\mathrm e}^{3}+2 \,{\mathrm e}^{2}\right ) x \,{\mathrm e}^{3 x}+\left (4 \,{\mathrm e}+2\right ) x^{3} {\mathrm e}^{3 x}+\left (6 \,{\mathrm e}^{2}+4 \,{\mathrm e}+1\right ) x^{2} {\mathrm e}^{3 x}\) | \(72\) |
parallelrisch | \({\mathrm e}^{3 x} {\mathrm e}^{4}+4 \,{\mathrm e}^{3 x} {\mathrm e}^{3} x +6 \,{\mathrm e}^{2} {\mathrm e}^{3 x} x^{2}+4 \,{\mathrm e} x^{3} {\mathrm e}^{3 x}+{\mathrm e}^{3 x} x^{4}+2 \,{\mathrm e}^{2} {\mathrm e}^{3 x} x +4 \,{\mathrm e} x^{2} {\mathrm e}^{3 x}+2 \,{\mathrm e}^{3 x} x^{3}+x^{2} {\mathrm e}^{3 x}\) | \(93\) |
meijerg | \(-\frac {8}{27}+\frac {\left (405 x^{4}-540 x^{3}+540 x^{2}-360 x +120\right ) {\mathrm e}^{3 x}}{405}-\frac {\left (-\frac {4 \,{\mathrm e}}{9}-\frac {10}{27}\right ) \left (6-\frac {\left (-108 x^{3}+108 x^{2}-72 x +24\right ) {\mathrm e}^{3 x}}{4}\right )}{3}-\frac {\left (2 \,{\mathrm e}^{2}+\frac {8 \,{\mathrm e}}{3}+1\right ) \left (2-\frac {\left (27 x^{2}-18 x +6\right ) {\mathrm e}^{3 x}}{3}\right )}{3}-\frac {\left (-4 \,{\mathrm e}^{3}-6 \,{\mathrm e}^{2}-\frac {8 \,{\mathrm e}}{3}-\frac {2}{3}\right ) \left (1-\frac {\left (-6 x +2\right ) {\mathrm e}^{3 x}}{2}\right )}{3}-{\mathrm e}^{4} \left (1-{\mathrm e}^{3 x}\right )-\frac {4 \,{\mathrm e}^{3} \left (1-{\mathrm e}^{3 x}\right )}{3}-\frac {2 \,{\mathrm e}^{2} \left (1-{\mathrm e}^{3 x}\right )}{3}\) | \(155\) |
parts | \({\mathrm e}^{3 x} {\mathrm e}^{4}+4 \,{\mathrm e}^{3 x} {\mathrm e}^{3} x +6 \,{\mathrm e}^{2} {\mathrm e}^{3 x} x^{2}+4 \,{\mathrm e} x^{3} {\mathrm e}^{3 x}+{\mathrm e}^{3 x} x^{4}+6 \,{\mathrm e}^{2} {\mathrm e}^{3 x} x +8 \,{\mathrm e} x^{2} {\mathrm e}^{3 x}+2 \,{\mathrm e}^{3 x} x^{3}-\frac {4 \,{\mathrm e}^{2} {\mathrm e}^{3 x}}{3}+\frac {8 \,{\mathrm e}^{3 x} {\mathrm e} x}{3}+x^{2} {\mathrm e}^{3 x}-\frac {8 \,{\mathrm e}^{3 x} {\mathrm e}}{9}-\frac {16 \,{\mathrm e} \left (3 x \,{\mathrm e}^{3 x}-{\mathrm e}^{3 x}\right )}{9}-\frac {4 \,{\mathrm e} \left (9 x^{2} {\mathrm e}^{3 x}-6 x \,{\mathrm e}^{3 x}+2 \,{\mathrm e}^{3 x}\right )}{9}-\frac {4 \,{\mathrm e}^{2} \left (3 x \,{\mathrm e}^{3 x}-{\mathrm e}^{3 x}\right )}{3}\) | \(185\) |
orering | \(\frac {\left ({\mathrm e}^{4}+4 x \,{\mathrm e}^{3}+6 x^{2} {\mathrm e}^{2}+4 x^{3} {\mathrm e}+x^{4}+2 \,{\mathrm e}^{2} x +4 x^{2} {\mathrm e}+2 x^{3}+x^{2}\right ) \left (3 \,{\mathrm e}^{4}+\left (12 x +4\right ) {\mathrm e}^{3}+\left (18 x^{2}+18 x +2\right ) {\mathrm e}^{2}+\left (12 x^{3}+24 x^{2}+8 x \right ) {\mathrm e}+3 x^{4}+10 x^{3}+9 x^{2}+2 x \right ) {\mathrm e}^{3 x}}{3 \,{\mathrm e}^{4}+12 x \,{\mathrm e}^{3}+18 x^{2} {\mathrm e}^{2}+12 x^{3} {\mathrm e}+3 x^{4}+4 \,{\mathrm e}^{3}+18 \,{\mathrm e}^{2} x +24 x^{2} {\mathrm e}+10 x^{3}+2 \,{\mathrm e}^{2}+8 x \,{\mathrm e}+9 x^{2}+2 x}\) | \(187\) |
derivativedivides | \({\mathrm e}^{3 x} x^{4}+2 \,{\mathrm e}^{3 x} x^{3}+x^{2} {\mathrm e}^{3 x}+\frac {2 \,{\mathrm e}^{2} {\mathrm e}^{3 x}}{3}+\frac {4 \,{\mathrm e}^{3 x} {\mathrm e}^{3}}{3}+{\mathrm e}^{3 x} {\mathrm e}^{4}+\frac {8 \,{\mathrm e} \left (3 x \,{\mathrm e}^{3 x}-{\mathrm e}^{3 x}\right )}{9}+\frac {8 \,{\mathrm e} \left (9 x^{2} {\mathrm e}^{3 x}-6 x \,{\mathrm e}^{3 x}+2 \,{\mathrm e}^{3 x}\right )}{9}+\frac {4 \,{\mathrm e} \left (27 \,{\mathrm e}^{3 x} x^{3}-27 x^{2} {\mathrm e}^{3 x}+18 x \,{\mathrm e}^{3 x}-6 \,{\mathrm e}^{3 x}\right )}{27}+2 \,{\mathrm e}^{2} \left (3 x \,{\mathrm e}^{3 x}-{\mathrm e}^{3 x}\right )+\frac {2 \,{\mathrm e}^{2} \left (9 x^{2} {\mathrm e}^{3 x}-6 x \,{\mathrm e}^{3 x}+2 \,{\mathrm e}^{3 x}\right )}{3}+\frac {4 \,{\mathrm e}^{3} \left (3 x \,{\mathrm e}^{3 x}-{\mathrm e}^{3 x}\right )}{3}\) | \(206\) |
default | \({\mathrm e}^{3 x} x^{4}+2 \,{\mathrm e}^{3 x} x^{3}+x^{2} {\mathrm e}^{3 x}+\frac {2 \,{\mathrm e}^{2} {\mathrm e}^{3 x}}{3}+\frac {4 \,{\mathrm e}^{3 x} {\mathrm e}^{3}}{3}+{\mathrm e}^{3 x} {\mathrm e}^{4}+\frac {8 \,{\mathrm e} \left (3 x \,{\mathrm e}^{3 x}-{\mathrm e}^{3 x}\right )}{9}+\frac {8 \,{\mathrm e} \left (9 x^{2} {\mathrm e}^{3 x}-6 x \,{\mathrm e}^{3 x}+2 \,{\mathrm e}^{3 x}\right )}{9}+\frac {4 \,{\mathrm e} \left (27 \,{\mathrm e}^{3 x} x^{3}-27 x^{2} {\mathrm e}^{3 x}+18 x \,{\mathrm e}^{3 x}-6 \,{\mathrm e}^{3 x}\right )}{27}+2 \,{\mathrm e}^{2} \left (3 x \,{\mathrm e}^{3 x}-{\mathrm e}^{3 x}\right )+\frac {2 \,{\mathrm e}^{2} \left (9 x^{2} {\mathrm e}^{3 x}-6 x \,{\mathrm e}^{3 x}+2 \,{\mathrm e}^{3 x}\right )}{3}+\frac {4 \,{\mathrm e}^{3} \left (3 x \,{\mathrm e}^{3 x}-{\mathrm e}^{3 x}\right )}{3}\) | \(206\) |
Input:
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)
Output:
(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)
Leaf count of result is larger than twice the leaf count of optimal. 46 vs. \(2 (15) = 30\).
Time = 0.06 (sec) , antiderivative size = 46, normalized size of antiderivative = 3.07 \[ \int e^{3 x} \left (3 e^4+2 x+9 x^2+10 x^3+3 x^4+e^3 (4+12 x)+e^2 \left (2+18 x+18 x^2\right )+e \left (8 x+24 x^2+12 x^3\right )\right ) \, dx={\left (x^{4} + 2 \, x^{3} + x^{2} + 4 \, x e^{3} + 2 \, {\left (3 \, x^{2} + x\right )} e^{2} + 4 \, {\left (x^{3} + x^{2}\right )} e + e^{4}\right )} e^{\left (3 \, x\right )} \] Input:
integrate((3*exp(1)^4+(12*x+4)*exp(1)^3+(18*x^2+18*x+2)*exp(1)^2+(12*x^3+2 4*x^2+8*x)*exp(1)+3*x^4+10*x^3+9*x^2+2*x)*exp(3*x),x, algorithm="fricas")
Output:
(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)
Leaf count of result is larger than twice the leaf count of optimal. 58 vs. \(2 (14) = 28\).
Time = 0.08 (sec) , antiderivative size = 58, normalized size of antiderivative = 3.87 \[ \int e^{3 x} \left (3 e^4+2 x+9 x^2+10 x^3+3 x^4+e^3 (4+12 x)+e^2 \left (2+18 x+18 x^2\right )+e \left (8 x+24 x^2+12 x^3\right )\right ) \, dx=\left (x^{4} + 2 x^{3} + 4 e x^{3} + x^{2} + 4 e x^{2} + 6 x^{2} e^{2} + 2 x e^{2} + 4 x e^{3} + e^{4}\right ) e^{3 x} \] Input:
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)
Output:
(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)
Leaf count of result is larger than twice the leaf count of optimal. 221 vs. \(2 (15) = 30\).
Time = 0.04 (sec) , antiderivative size = 221, normalized size of antiderivative = 14.73 \[ \int e^{3 x} \left (3 e^4+2 x+9 x^2+10 x^3+3 x^4+e^3 (4+12 x)+e^2 \left (2+18 x+18 x^2\right )+e \left (8 x+24 x^2+12 x^3\right )\right ) \, dx=\frac {1}{27} \, {\left (27 \, x^{4} - 36 \, x^{3} + 36 \, x^{2} - 24 \, x + 8\right )} e^{\left (3 \, x\right )} + \frac {4}{9} \, {\left (9 \, x^{3} e - 9 \, x^{2} e + 6 \, x e - 2 \, e\right )} e^{\left (3 \, x\right )} + \frac {10}{27} \, {\left (9 \, x^{3} - 9 \, x^{2} + 6 \, x - 2\right )} e^{\left (3 \, x\right )} + \frac {2}{3} \, {\left (9 \, x^{2} e^{2} - 6 \, x e^{2} + 2 \, e^{2}\right )} e^{\left (3 \, x\right )} + \frac {8}{9} \, {\left (9 \, x^{2} e - 6 \, x e + 2 \, e\right )} e^{\left (3 \, x\right )} + \frac {1}{3} \, {\left (9 \, x^{2} - 6 \, x + 2\right )} e^{\left (3 \, x\right )} + \frac {4}{3} \, {\left (3 \, x e^{3} - e^{3}\right )} e^{\left (3 \, x\right )} + 2 \, {\left (3 \, x e^{2} - e^{2}\right )} e^{\left (3 \, x\right )} + \frac {8}{9} \, {\left (3 \, x e - e\right )} e^{\left (3 \, x\right )} + \frac {2}{9} \, {\left (3 \, x - 1\right )} e^{\left (3 \, x\right )} + e^{\left (3 \, x + 4\right )} + \frac {4}{3} \, e^{\left (3 \, x + 3\right )} + \frac {2}{3} \, e^{\left (3 \, x + 2\right )} \] Input:
integrate((3*exp(1)^4+(12*x+4)*exp(1)^3+(18*x^2+18*x+2)*exp(1)^2+(12*x^3+2 4*x^2+8*x)*exp(1)+3*x^4+10*x^3+9*x^2+2*x)*exp(3*x),x, algorithm="maxima")
Output:
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)
Leaf count of result is larger than twice the leaf count of optimal. 63 vs. \(2 (15) = 30\).
Time = 0.12 (sec) , antiderivative size = 63, normalized size of antiderivative = 4.20 \[ \int e^{3 x} \left (3 e^4+2 x+9 x^2+10 x^3+3 x^4+e^3 (4+12 x)+e^2 \left (2+18 x+18 x^2\right )+e \left (8 x+24 x^2+12 x^3\right )\right ) \, dx={\left (x^{4} + 2 \, x^{3} + x^{2}\right )} e^{\left (3 \, x\right )} + 4 \, x e^{\left (3 \, x + 3\right )} + 2 \, {\left (3 \, x^{2} + x\right )} e^{\left (3 \, x + 2\right )} + 4 \, {\left (x^{3} + x^{2}\right )} e^{\left (3 \, x + 1\right )} + e^{\left (3 \, x + 4\right )} \] Input:
integrate((3*exp(1)^4+(12*x+4)*exp(1)^3+(18*x^2+18*x+2)*exp(1)^2+(12*x^3+2 4*x^2+8*x)*exp(1)+3*x^4+10*x^3+9*x^2+2*x)*exp(3*x),x, algorithm="giac")
Output:
(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)
Time = 1.78 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.27 \[ \int e^{3 x} \left (3 e^4+2 x+9 x^2+10 x^3+3 x^4+e^3 (4+12 x)+e^2 \left (2+18 x+18 x^2\right )+e \left (8 x+24 x^2+12 x^3\right )\right ) \, dx={\mathrm {e}}^{3\,x}\,{\left (x+{\mathrm {e}}^2+2\,x\,\mathrm {e}+x^2\right )}^2 \] Input:
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)
Output:
exp(3*x)*(x + exp(2) + 2*x*exp(1) + x^2)^2
Time = 0.18 (sec) , antiderivative size = 53, normalized size of antiderivative = 3.53 \[ \int e^{3 x} \left (3 e^4+2 x+9 x^2+10 x^3+3 x^4+e^3 (4+12 x)+e^2 \left (2+18 x+18 x^2\right )+e \left (8 x+24 x^2+12 x^3\right )\right ) \, dx=e^{3 x} \left (e^{4}+4 e^{3} x +6 e^{2} x^{2}+4 e \,x^{3}+x^{4}+2 e^{2} x +4 e \,x^{2}+2 x^{3}+x^{2}\right ) \] Input:
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)
Output:
e**(3*x)*(e**4 + 4*e**3*x + 6*e**2*x**2 + 2*e**2*x + 4*e*x**3 + 4*e*x**2 + x**4 + 2*x**3 + x**2)