Integrand size = 54, antiderivative size = 26 \[ \int \frac {15-36 x^2-8 x^3+e^{\frac {1}{3} \left (11+3 e^4+3 x\right )} \left (24 x+16 x^2+4 x^3\right )}{9+6 x+x^2} \, dx=\frac {x \left (5-4 x \left (-e^{\frac {11}{3}+e^4+x}+x\right )\right )}{3+x} \] Output:
x/(3+x)*(5-4*(x-exp(exp(4)+x+11/3))*x)
Time = 0.30 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.19 \[ \int \frac {15-36 x^2-8 x^3+e^{\frac {1}{3} \left (11+3 e^4+3 x\right )} \left (24 x+16 x^2+4 x^3\right )}{9+6 x+x^2} \, dx=\frac {93+36 x+4 e^{\frac {11}{3}+e^4+x} x^2-4 x^3}{3+x} \] Input:
Integrate[(15 - 36*x^2 - 8*x^3 + E^((11 + 3*E^4 + 3*x)/3)*(24*x + 16*x^2 + 4*x^3))/(9 + 6*x + x^2),x]
Output:
(93 + 36*x + 4*E^(11/3 + E^4 + x)*x^2 - 4*x^3)/(3 + x)
Leaf count is larger than twice the leaf count of optimal. \(73\) vs. \(2(26)=52\).
Time = 0.63 (sec) , antiderivative size = 73, normalized size of antiderivative = 2.81, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {2007, 7293, 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.
\(\displaystyle \int \frac {-8 x^3-36 x^2+e^{\frac {1}{3} \left (3 x+3 e^4+11\right )} \left (4 x^3+16 x^2+24 x\right )+15}{x^2+6 x+9} \, dx\) |
\(\Big \downarrow \) 2007 |
\(\displaystyle \int \frac {-8 x^3-36 x^2+e^{\frac {1}{3} \left (3 x+3 e^4+11\right )} \left (4 x^3+16 x^2+24 x\right )+15}{(x+3)^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {4 e^{x+e^4+\frac {11}{3}} x \left (x^2+4 x+6\right )}{(x+3)^2}+\frac {-8 x^3-36 x^2+15}{(x+3)^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -4 x^2+4 e^{x+\frac {1}{3} \left (11+3 e^4\right )} x+12 x-12 e^{x+\frac {1}{3} \left (11+3 e^4\right )}+\frac {36 e^{x+\frac {1}{3} \left (11+3 e^4\right )}}{x+3}+\frac {93}{x+3}\) |
Input:
Int[(15 - 36*x^2 - 8*x^3 + E^((11 + 3*E^4 + 3*x)/3)*(24*x + 16*x^2 + 4*x^3 ))/(9 + 6*x + x^2),x]
Output:
-12*E^((11 + 3*E^4)/3 + x) + 12*x + 4*E^((11 + 3*E^4)/3 + x)*x - 4*x^2 + 9 3/(3 + x) + (36*E^((11 + 3*E^4)/3 + x))/(3 + x)
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{a = Rt[Coeff[Px, x, 0], Expon[Px, x]], b = Rt[Coeff[Px, x, Expon[Px, x]], Expon[Px, x]]}, Int[u*(a + b*x)^(Ex pon[Px, x]*p), x] /; EqQ[Px, (a + b*x)^Expon[Px, x]]] /; IntegerQ[p] && Pol yQ[Px, x] && GtQ[Expon[Px, x], 1] && NeQ[Coeff[Px, x, 0], 0]
Time = 0.45 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.96
method | result | size |
norman | \(\frac {-4 x^{3}+4 \,{\mathrm e}^{{\mathrm e}^{4}+x +\frac {11}{3}} x^{2}-15}{3+x}\) | \(25\) |
parallelrisch | \(-\frac {4 x^{3}-4 \,{\mathrm e}^{{\mathrm e}^{4}+x +\frac {11}{3}} x^{2}+15}{3+x}\) | \(26\) |
risch | \(-4 x^{2}+12 x +\frac {93}{3+x}+\frac {4 x^{2} {\mathrm e}^{{\mathrm e}^{4}+x +\frac {11}{3}}}{3+x}\) | \(33\) |
orering | \(\frac {\left (4 x^{6}+20 x^{5}+40 x^{4}-33 x^{3}+75 x^{2}+150 x +90\right ) \left (\left (4 x^{3}+16 x^{2}+24 x \right ) {\mathrm e}^{{\mathrm e}^{4}+x +\frac {11}{3}}-8 x^{3}-36 x^{2}+15\right )}{\left (8 x^{5}+44 x^{4}+64 x^{3}-87 x^{2}-60 x -30\right ) \left (x^{2}+6 x +9\right )}-\frac {x \left (4 x^{4}-4 x^{3}+15 x +30\right ) \left (3+x \right ) \left (\frac {\left (12 x^{2}+32 x +24\right ) {\mathrm e}^{{\mathrm e}^{4}+x +\frac {11}{3}}+\left (4 x^{3}+16 x^{2}+24 x \right ) {\mathrm e}^{{\mathrm e}^{4}+x +\frac {11}{3}}-24 x^{2}-72 x}{x^{2}+6 x +9}-\frac {\left (\left (4 x^{3}+16 x^{2}+24 x \right ) {\mathrm e}^{{\mathrm e}^{4}+x +\frac {11}{3}}-8 x^{3}-36 x^{2}+15\right ) \left (2 x +6\right )}{\left (x^{2}+6 x +9\right )^{2}}\right )}{8 x^{5}+44 x^{4}+64 x^{3}-87 x^{2}-60 x -30}\) | \(260\) |
parts | \(-4 x^{2}+12 x +\frac {93}{3+x}-\frac {1892 \,{\mathrm e}^{{\mathrm e}^{4}+x +\frac {11}{3}}}{9 \left (-3 x -9\right )}+\frac {1892 \,{\mathrm e}^{{\mathrm e}^{4}+\frac {2}{3}} \operatorname {expIntegral}_{1}\left (-3-x \right )}{27}+\frac {68 \,{\mathrm e}^{{\mathrm e}^{4}+x +\frac {11}{3}} \left (2+3 \,{\mathrm e}^{4}\right )}{-3 x -9}-612 \left (\frac {5}{27}+\frac {{\mathrm e}^{4}}{9}\right ) {\mathrm e}^{{\mathrm e}^{4}+\frac {2}{3}} \operatorname {expIntegral}_{1}\left (-3-x \right )-28 \,{\mathrm e}^{{\mathrm e}^{4}+x +\frac {11}{3}}-\frac {28 \,{\mathrm e}^{{\mathrm e}^{4}+x +\frac {11}{3}} \left (9 \,{\mathrm e}^{8}+4+12 \,{\mathrm e}^{4}\right )}{3 \left (-3 x -9\right )}+28 \left ({\mathrm e}^{8}+\frac {10 \,{\mathrm e}^{4}}{3}+\frac {16}{9}\right ) {\mathrm e}^{{\mathrm e}^{4}+\frac {2}{3}} \operatorname {expIntegral}_{1}\left (-3-x \right )+\frac {4 \left (9 \,{\mathrm e}^{4}+3 x +12\right ) {\mathrm e}^{{\mathrm e}^{4}+x +\frac {11}{3}}}{3}+\frac {4 \,{\mathrm e}^{{\mathrm e}^{4}+x +\frac {11}{3}} \left (27 \,{\mathrm e}^{12}+54 \,{\mathrm e}^{8}+36 \,{\mathrm e}^{4}+8\right )}{9 \left (-3 x -9\right )}-\frac {4 \left (3 \,{\mathrm e}^{12}+15 \,{\mathrm e}^{8}+16 \,{\mathrm e}^{4}+\frac {44}{9}\right ) {\mathrm e}^{{\mathrm e}^{4}+\frac {2}{3}} \operatorname {expIntegral}_{1}\left (-3-x \right )}{3}-612 \,{\mathrm e}^{4} \left (\frac {{\mathrm e}^{{\mathrm e}^{4}+x +\frac {11}{3}}}{-9 x -27}-\frac {{\mathrm e}^{{\mathrm e}^{4}+\frac {2}{3}} \operatorname {expIntegral}_{1}\left (-3-x \right )}{9}\right )-252 \,{\mathrm e}^{8} \left (\frac {{\mathrm e}^{{\mathrm e}^{4}+x +\frac {11}{3}}}{-9 x -27}-\frac {{\mathrm e}^{{\mathrm e}^{4}+\frac {2}{3}} \operatorname {expIntegral}_{1}\left (-3-x \right )}{9}\right )-36 \,{\mathrm e}^{12} \left (\frac {{\mathrm e}^{{\mathrm e}^{4}+x +\frac {11}{3}}}{-9 x -27}-\frac {{\mathrm e}^{{\mathrm e}^{4}+\frac {2}{3}} \operatorname {expIntegral}_{1}\left (-3-x \right )}{9}\right )+504 \,{\mathrm e}^{4} \left (\frac {{\mathrm e}^{{\mathrm e}^{4}+x +\frac {11}{3}} \left (2+3 \,{\mathrm e}^{4}\right )}{-27 x -81}-\left (\frac {5}{27}+\frac {{\mathrm e}^{4}}{9}\right ) {\mathrm e}^{{\mathrm e}^{4}+\frac {2}{3}} \operatorname {expIntegral}_{1}\left (-3-x \right )\right )+108 \,{\mathrm e}^{8} \left (\frac {{\mathrm e}^{{\mathrm e}^{4}+x +\frac {11}{3}} \left (2+3 \,{\mathrm e}^{4}\right )}{-27 x -81}-\left (\frac {5}{27}+\frac {{\mathrm e}^{4}}{9}\right ) {\mathrm e}^{{\mathrm e}^{4}+\frac {2}{3}} \operatorname {expIntegral}_{1}\left (-3-x \right )\right )-108 \,{\mathrm e}^{4} \left (\frac {{\mathrm e}^{{\mathrm e}^{4}+x +\frac {11}{3}}}{9}+\frac {{\mathrm e}^{{\mathrm e}^{4}+x +\frac {11}{3}} \left (9 \,{\mathrm e}^{8}+4+12 \,{\mathrm e}^{4}\right )}{-81 x -243}-\frac {\left ({\mathrm e}^{8}+\frac {10 \,{\mathrm e}^{4}}{3}+\frac {16}{9}\right ) {\mathrm e}^{{\mathrm e}^{4}+\frac {2}{3}} \operatorname {expIntegral}_{1}\left (-3-x \right )}{9}\right )\) | \(484\) |
derivativedivides | \(\text {Expression too large to display}\) | \(763\) |
default | \(\text {Expression too large to display}\) | \(763\) |
Input:
int(((4*x^3+16*x^2+24*x)*exp(exp(4)+x+11/3)-8*x^3-36*x^2+15)/(x^2+6*x+9),x ,method=_RETURNVERBOSE)
Output:
(-4*x^3+4*exp(exp(4)+x+11/3)*x^2-15)/(3+x)
Time = 0.09 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.08 \[ \int \frac {15-36 x^2-8 x^3+e^{\frac {1}{3} \left (11+3 e^4+3 x\right )} \left (24 x+16 x^2+4 x^3\right )}{9+6 x+x^2} \, dx=-\frac {4 \, x^{3} - 4 \, x^{2} e^{\left (x + e^{4} + \frac {11}{3}\right )} - 36 \, x - 93}{x + 3} \] Input:
integrate(((4*x^3+16*x^2+24*x)*exp(exp(4)+x+11/3)-8*x^3-36*x^2+15)/(x^2+6* x+9),x, algorithm="fricas")
Output:
-(4*x^3 - 4*x^2*e^(x + e^4 + 11/3) - 36*x - 93)/(x + 3)
Time = 0.06 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.19 \[ \int \frac {15-36 x^2-8 x^3+e^{\frac {1}{3} \left (11+3 e^4+3 x\right )} \left (24 x+16 x^2+4 x^3\right )}{9+6 x+x^2} \, dx=- 4 x^{2} + \frac {4 x^{2} e^{x + \frac {11}{3} + e^{4}}}{x + 3} + 12 x + \frac {93}{x + 3} \] Input:
integrate(((4*x**3+16*x**2+24*x)*exp(exp(4)+x+11/3)-8*x**3-36*x**2+15)/(x* *2+6*x+9),x)
Output:
-4*x**2 + 4*x**2*exp(x + 11/3 + exp(4))/(x + 3) + 12*x + 93/(x + 3)
Time = 0.17 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.23 \[ \int \frac {15-36 x^2-8 x^3+e^{\frac {1}{3} \left (11+3 e^4+3 x\right )} \left (24 x+16 x^2+4 x^3\right )}{9+6 x+x^2} \, dx=-4 \, x^{2} + \frac {4 \, x^{2} e^{\left (x + e^{4} + \frac {11}{3}\right )}}{x + 3} + 12 \, x + \frac {93}{x + 3} \] Input:
integrate(((4*x^3+16*x^2+24*x)*exp(exp(4)+x+11/3)-8*x^3-36*x^2+15)/(x^2+6* x+9),x, algorithm="maxima")
Output:
-4*x^2 + 4*x^2*e^(x + e^4 + 11/3)/(x + 3) + 12*x + 93/(x + 3)
Time = 0.11 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.08 \[ \int \frac {15-36 x^2-8 x^3+e^{\frac {1}{3} \left (11+3 e^4+3 x\right )} \left (24 x+16 x^2+4 x^3\right )}{9+6 x+x^2} \, dx=-\frac {4 \, x^{3} - 4 \, x^{2} e^{\left (x + e^{4} + \frac {11}{3}\right )} - 36 \, x - 93}{x + 3} \] Input:
integrate(((4*x^3+16*x^2+24*x)*exp(exp(4)+x+11/3)-8*x^3-36*x^2+15)/(x^2+6* x+9),x, algorithm="giac")
Output:
-(4*x^3 - 4*x^2*e^(x + e^4 + 11/3) - 36*x - 93)/(x + 3)
Time = 3.94 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.88 \[ \int \frac {15-36 x^2-8 x^3+e^{\frac {1}{3} \left (11+3 e^4+3 x\right )} \left (24 x+16 x^2+4 x^3\right )}{9+6 x+x^2} \, dx=\frac {x\,\left (4\,x\,{\mathrm {e}}^{x+{\mathrm {e}}^4+\frac {11}{3}}-4\,x^2+5\right )}{x+3} \] Input:
int((exp(x + exp(4) + 11/3)*(24*x + 16*x^2 + 4*x^3) - 36*x^2 - 8*x^3 + 15) /(6*x + x^2 + 9),x)
Output:
(x*(4*x*exp(x + exp(4) + 11/3) - 4*x^2 + 5))/(x + 3)
Time = 0.15 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.08 \[ \int \frac {15-36 x^2-8 x^3+e^{\frac {1}{3} \left (11+3 e^4+3 x\right )} \left (24 x+16 x^2+4 x^3\right )}{9+6 x+x^2} \, dx=\frac {x \left (4 e^{e^{4}+x +\frac {2}{3}} e^{3} x -4 x^{2}+5\right )}{x +3} \] Input:
int(((4*x^3+16*x^2+24*x)*exp(exp(4)+x+11/3)-8*x^3-36*x^2+15)/(x^2+6*x+9),x )
Output:
(x*(4*e**((3*e**4 + 3*x + 2)/3)*e**3*x - 4*x**2 + 5))/(x + 3)