Integrand size = 175, antiderivative size = 28 \begin {dmath*} \int \frac {-1875-675 x+30 x^2+e^6 (-75+3 x)+e^{\frac {2}{3} (-3 x+\log (25-x))} (-75+3 x)+e^3 \left (-750-120 x+6 x^2\right )+e^{\frac {1}{3} (-3 x+\log (25-x))} \left (-750-120 x-70 x^2+3 x^3+e^3 (-150+6 x)\right )}{-1875+75 x+e^6 (-75+3 x)+e^{\frac {2}{3} (-3 x+\log (25-x))} (-75+3 x)+e^3 (-750+30 x)+e^{\frac {1}{3} (-3 x+\log (25-x))} \left (-750+30 x+e^3 (-150+6 x)\right )} \, dx=x+\frac {x^2}{5+e^3+e^{-x} \sqrt [3]{25-x}} \end {dmath*}
Leaf count is larger than twice the leaf count of optimal. \(138\) vs. \(2(28)=56\).
Time = 11.22 (sec) , antiderivative size = 138, normalized size of antiderivative = 4.93 \begin {dmath*} \int \frac {-1875-675 x+30 x^2+e^6 (-75+3 x)+e^{\frac {2}{3} (-3 x+\log (25-x))} (-75+3 x)+e^3 \left (-750-120 x+6 x^2\right )+e^{\frac {1}{3} (-3 x+\log (25-x))} \left (-750-120 x-70 x^2+3 x^3+e^3 (-150+6 x)\right )}{-1875+75 x+e^6 (-75+3 x)+e^{\frac {2}{3} (-3 x+\log (25-x))} (-75+3 x)+e^3 (-750+30 x)+e^{\frac {1}{3} (-3 x+\log (25-x))} \left (-750+30 x+e^3 (-150+6 x)\right )} \, dx=\frac {x \left (25+e^{9+3 x}-x-5 e^{2 x} \sqrt [3]{25-x} x-e^{3+2 x} \sqrt [3]{25-x} x+e^x (25-x)^{2/3} x+25 e^{3 x} (5+x)+e^{6+3 x} (15+x)+5 e^{3+3 x} (15+2 x)\right )}{25+125 e^{3 x}+75 e^{3+3 x}+15 e^{6+3 x}+e^{9+3 x}-x} \end {dmath*}
Integrate[(-1875 - 675*x + 30*x^2 + E^6*(-75 + 3*x) + E^((2*(-3*x + Log[25 - x]))/3)*(-75 + 3*x) + E^3*(-750 - 120*x + 6*x^2) + E^((-3*x + Log[25 - x])/3)*(-750 - 120*x - 70*x^2 + 3*x^3 + E^3*(-150 + 6*x)))/(-1875 + 75*x + E^6*(-75 + 3*x) + E^((2*(-3*x + Log[25 - x]))/3)*(-75 + 3*x) + E^3*(-750 + 30*x) + E^((-3*x + Log[25 - x])/3)*(-750 + 30*x + E^3*(-150 + 6*x))),x]
(x*(25 + E^(9 + 3*x) - x - 5*E^(2*x)*(25 - x)^(1/3)*x - E^(3 + 2*x)*(25 - x)^(1/3)*x + E^x*(25 - x)^(2/3)*x + 25*E^(3*x)*(5 + x) + E^(6 + 3*x)*(15 + x) + 5*E^(3 + 3*x)*(15 + 2*x)))/(25 + 125*E^(3*x) + 75*E^(3 + 3*x) + 15*E ^(6 + 3*x) + E^(9 + 3*x) - x)
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.
| \(\displaystyle \int \frac {30 x^2+e^3 \left (6 x^2-120 x-750\right )+\left (3 x^3-70 x^2-120 x+e^3 (6 x-150)-750\right ) e^{\frac {1}{3} (\log (25-x)-3 x)}-675 x+e^6 (3 x-75)+(3 x-75) e^{\frac {2}{3} (\log (25-x)-3 x)}-1875}{75 x+e^6 (3 x-75)+e^3 (30 x-750)+(3 x-75) e^{\frac {2}{3} (\log (25-x)-3 x)}+\left (30 x+e^3 (6 x-150)-750\right ) e^{\frac {1}{3} (\log (25-x)-3 x)}-1875} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {e^{2 x} \left (-30 x^2-e^3 \left (6 x^2-120 x-750\right )-\left (3 x^3-70 x^2-120 x+e^3 (6 x-150)-750\right ) e^{\frac {1}{3} (\log (25-x)-3 x)}+675 x-e^6 (3 x-75)-(3 x-75) e^{\frac {2}{3} (\log (25-x)-3 x)}+1875\right )}{3 \left (\sqrt [3]{25-x}+5 \left (1+\frac {e^3}{5}\right ) e^x\right )^2 (25-x)}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{3} \int \frac {e^{2 x} \left (-30 x^2+675 x+3 e^{-2 x} (25-x)^{5/3}+3 e^6 (25-x)+6 e^3 \left (-x^2+20 x+125\right )+e^{-x} \sqrt [3]{25-x} \left (-3 x^3+70 x^2+120 x+6 e^3 (25-x)+750\right )+1875\right )}{\left (\sqrt [3]{25-x}+e^x \left (5+e^3\right )\right )^2 (25-x)}dx\) |
| \(\Big \downarrow \) 7267 |
| \(\displaystyle -\int \frac {3 e^{2 (25-x)} (25-x)^{4/3}+6 e^{53-x} (25-x)+6 e^{53} (x+5) (25-x)^{2/3}+3 e^{56} (25-x)^{2/3}+15 e^{50} \left (55 (25-x)^{2/3}-2 (25-x)^{5/3}\right )+e^{50-x} \left (3 (25-x)^3-155 (25-x)^2+2005 (25-x)+625\right )}{\left (e^{25-x} \sqrt [3]{25-x}+e^{25} \left (5+e^3\right )\right )^2}d\sqrt [3]{25-x}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\int \left (\frac {e^{50} \left (5+e^3\right ) (-3 (25-x)-1) x^2}{\left (e^{25-x} \sqrt [3]{25-x}+5 e^{25} \left (1+\frac {e^3}{5}\right )\right )^2 \sqrt [3]{25-x}}+3 (25-x)^{2/3}+\frac {e^{25} \left (3 (25-x)^3-155 (25-x)^2+1975 (25-x)+625\right )}{\left (e^{25-x} \sqrt [3]{25-x}+5 e^{25} \left (1+\frac {e^3}{5}\right )\right ) \sqrt [3]{25-x}}\right )d\sqrt [3]{25-x}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 625 e^{50} \left (5+e^3\right ) \int \frac {1}{\left (e^{25-x} \sqrt [3]{25-x}+5 e^{25} \left (1+\frac {e^3}{5}\right )\right )^2 \sqrt [3]{25-x}}d\sqrt [3]{25-x}-625 e^{25} \int \frac {1}{\left (e^{25-x} \sqrt [3]{25-x}+5 e^{25} \left (1+\frac {e^3}{5}\right )\right ) \sqrt [3]{25-x}}d\sqrt [3]{25-x}+1825 e^{50} \left (5+e^3\right ) \int \frac {(25-x)^{2/3}}{\left (e^{25-x} \sqrt [3]{25-x}+5 e^{25} \left (1+\frac {e^3}{5}\right )\right )^2}d\sqrt [3]{25-x}-1975 e^{25} \int \frac {(25-x)^{2/3}}{e^{25-x} \sqrt [3]{25-x}+5 e^{25} \left (1+\frac {e^3}{5}\right )}d\sqrt [3]{25-x}-155 e^{25} \int \frac {(25-x)^{5/3}}{-e^{25-x} \sqrt [3]{25-x}-5 e^{25} \left (1+\frac {e^3}{5}\right )}d\sqrt [3]{25-x}-149 e^{50} \left (5+e^3\right ) \int \frac {(25-x)^{5/3}}{\left (e^{25-x} \sqrt [3]{25-x}+5 e^{25} \left (1+\frac {e^3}{5}\right )\right )^2}d\sqrt [3]{25-x}+3 e^{50} \left (5+e^3\right ) \int \frac {(25-x)^{8/3}}{\left (e^{25-x} \sqrt [3]{25-x}+5 e^{25} \left (1+\frac {e^3}{5}\right )\right )^2}d\sqrt [3]{25-x}-3 e^{25} \int \frac {(25-x)^{8/3}}{e^{25-x} \sqrt [3]{25-x}+5 e^{25} \left (1+\frac {e^3}{5}\right )}d\sqrt [3]{25-x}+x-25\) |
Int[(-1875 - 675*x + 30*x^2 + E^6*(-75 + 3*x) + E^((2*(-3*x + Log[25 - x]) )/3)*(-75 + 3*x) + E^3*(-750 - 120*x + 6*x^2) + E^((-3*x + Log[25 - x])/3) *(-750 - 120*x - 70*x^2 + 3*x^3 + E^3*(-150 + 6*x)))/(-1875 + 75*x + E^6*( -75 + 3*x) + E^((2*(-3*x + Log[25 - x]))/3)*(-75 + 3*x) + E^3*(-750 + 30*x ) + E^((-3*x + Log[25 - x])/3)*(-750 + 30*x + E^3*(-150 + 6*x))),x]
3.12.99.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[u_, x_Symbol] :> With[{lst = SubstForFractionalPowerOfLinear[u, x]}, Si mp[lst[[2]]*lst[[4]] Subst[Int[lst[[1]], x], x, lst[[3]]^(1/lst[[2]])], x ] /; !FalseQ[lst] && SubstForFractionalPowerQ[u, lst[[3]], x]]
Time = 2.35 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.89
| method | result | size |
| risch | \(\frac {x^{2}}{\left (-x +25\right )^{\frac {1}{3}} {\mathrm e}^{-x}+{\mathrm e}^{3}+5}+x\) | \(25\) |
| norman | \(\frac {x^{2}+\left ({\mathrm e}^{3}+5\right ) x +{\mathrm e}^{\frac {\ln \left (-x +25\right )}{3}-x} x}{{\mathrm e}^{\frac {\ln \left (-x +25\right )}{3}-x}+{\mathrm e}^{3}+5}\) | \(46\) |
| parallelrisch | \(\frac {750+3 x \,{\mathrm e}^{3}+3 x^{2}+3 \,{\mathrm e}^{\frac {\ln \left (-x +25\right )}{3}-x} x +150 \,{\mathrm e}^{3}+15 x +150 \,{\mathrm e}^{\frac {\ln \left (-x +25\right )}{3}-x}}{3 \,{\mathrm e}^{\frac {\ln \left (-x +25\right )}{3}-x}+3 \,{\mathrm e}^{3}+15}\) | \(72\) |
int(((3*x-75)*exp(1/3*ln(-x+25)-x)^2+((6*x-150)*exp(3)+3*x^3-70*x^2-120*x- 750)*exp(1/3*ln(-x+25)-x)+(3*x-75)*exp(3)^2+(6*x^2-120*x-750)*exp(3)+30*x^ 2-675*x-1875)/((3*x-75)*exp(1/3*ln(-x+25)-x)^2+((6*x-150)*exp(3)+30*x-750) *exp(1/3*ln(-x+25)-x)+(3*x-75)*exp(3)^2+(30*x-750)*exp(3)+75*x-1875),x,met hod=_RETURNVERBOSE)
Time = 0.27 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.64 \begin {dmath*} \int \frac {-1875-675 x+30 x^2+e^6 (-75+3 x)+e^{\frac {2}{3} (-3 x+\log (25-x))} (-75+3 x)+e^3 \left (-750-120 x+6 x^2\right )+e^{\frac {1}{3} (-3 x+\log (25-x))} \left (-750-120 x-70 x^2+3 x^3+e^3 (-150+6 x)\right )}{-1875+75 x+e^6 (-75+3 x)+e^{\frac {2}{3} (-3 x+\log (25-x))} (-75+3 x)+e^3 (-750+30 x)+e^{\frac {1}{3} (-3 x+\log (25-x))} \left (-750+30 x+e^3 (-150+6 x)\right )} \, dx=\frac {x^{2} + x e^{3} + x e^{\left (-x + \frac {1}{3} \, \log \left (-x + 25\right )\right )} + 5 \, x}{e^{3} + e^{\left (-x + \frac {1}{3} \, \log \left (-x + 25\right )\right )} + 5} \end {dmath*}
integrate(((3*x-75)*exp(1/3*log(-x+25)-x)^2+((6*x-150)*exp(3)+3*x^3-70*x^2 -120*x-750)*exp(1/3*log(-x+25)-x)+(3*x-75)*exp(3)^2+(6*x^2-120*x-750)*exp( 3)+30*x^2-675*x-1875)/((3*x-75)*exp(1/3*log(-x+25)-x)^2+((6*x-150)*exp(3)+ 30*x-750)*exp(1/3*log(-x+25)-x)+(3*x-75)*exp(3)^2+(30*x-750)*exp(3)+75*x-1 875),x, algorithm=\
Timed out. \begin {dmath*} \int \frac {-1875-675 x+30 x^2+e^6 (-75+3 x)+e^{\frac {2}{3} (-3 x+\log (25-x))} (-75+3 x)+e^3 \left (-750-120 x+6 x^2\right )+e^{\frac {1}{3} (-3 x+\log (25-x))} \left (-750-120 x-70 x^2+3 x^3+e^3 (-150+6 x)\right )}{-1875+75 x+e^6 (-75+3 x)+e^{\frac {2}{3} (-3 x+\log (25-x))} (-75+3 x)+e^3 (-750+30 x)+e^{\frac {1}{3} (-3 x+\log (25-x))} \left (-750+30 x+e^3 (-150+6 x)\right )} \, dx=\text {Timed out} \end {dmath*}
integrate(((3*x-75)*exp(1/3*ln(-x+25)-x)**2+((6*x-150)*exp(3)+3*x**3-70*x* *2-120*x-750)*exp(1/3*ln(-x+25)-x)+(3*x-75)*exp(3)**2+(6*x**2-120*x-750)*e xp(3)+30*x**2-675*x-1875)/((3*x-75)*exp(1/3*ln(-x+25)-x)**2+((6*x-150)*exp (3)+30*x-750)*exp(1/3*ln(-x+25)-x)+(3*x-75)*exp(3)**2+(30*x-750)*exp(3)+75 *x-1875),x)
Leaf count of result is larger than twice the leaf count of optimal. 63 vs. \(2 (25) = 50\).
Time = 0.24 (sec) , antiderivative size = 63, normalized size of antiderivative = 2.25 \begin {dmath*} \int \frac {-1875-675 x+30 x^2+e^6 (-75+3 x)+e^{\frac {2}{3} (-3 x+\log (25-x))} (-75+3 x)+e^3 \left (-750-120 x+6 x^2\right )+e^{\frac {1}{3} (-3 x+\log (25-x))} \left (-750-120 x-70 x^2+3 x^3+e^3 (-150+6 x)\right )}{-1875+75 x+e^6 (-75+3 x)+e^{\frac {2}{3} (-3 x+\log (25-x))} (-75+3 x)+e^3 (-750+30 x)+e^{\frac {1}{3} (-3 x+\log (25-x))} \left (-750+30 x+e^3 (-150+6 x)\right )} \, dx=\frac {{\left (x - 25\right )}^{2} e^{25} - {\left (-x + 25\right )}^{\frac {4}{3}} e^{\left (-x + 25\right )} + {\left (x - 25\right )} {\left (e^{28} + 55 \, e^{25}\right )} + 625 \, e^{25}}{{\left (-x + 25\right )}^{\frac {1}{3}} e^{\left (-x + 25\right )} + e^{28} + 5 \, e^{25}} \end {dmath*}
integrate(((3*x-75)*exp(1/3*log(-x+25)-x)^2+((6*x-150)*exp(3)+3*x^3-70*x^2 -120*x-750)*exp(1/3*log(-x+25)-x)+(3*x-75)*exp(3)^2+(6*x^2-120*x-750)*exp( 3)+30*x^2-675*x-1875)/((3*x-75)*exp(1/3*log(-x+25)-x)^2+((6*x-150)*exp(3)+ 30*x-750)*exp(1/3*log(-x+25)-x)+(3*x-75)*exp(3)^2+(30*x-750)*exp(3)+75*x-1 875),x, algorithm=\
((x - 25)^2*e^25 - (-x + 25)^(4/3)*e^(-x + 25) + (x - 25)*(e^28 + 55*e^25) + 625*e^25)/((-x + 25)^(1/3)*e^(-x + 25) + e^28 + 5*e^25)
\begin {dmath*} \int \frac {-1875-675 x+30 x^2+e^6 (-75+3 x)+e^{\frac {2}{3} (-3 x+\log (25-x))} (-75+3 x)+e^3 \left (-750-120 x+6 x^2\right )+e^{\frac {1}{3} (-3 x+\log (25-x))} \left (-750-120 x-70 x^2+3 x^3+e^3 (-150+6 x)\right )}{-1875+75 x+e^6 (-75+3 x)+e^{\frac {2}{3} (-3 x+\log (25-x))} (-75+3 x)+e^3 (-750+30 x)+e^{\frac {1}{3} (-3 x+\log (25-x))} \left (-750+30 x+e^3 (-150+6 x)\right )} \, dx=\int { \frac {30 \, x^{2} + 3 \, {\left (x - 25\right )} e^{6} + 6 \, {\left (x^{2} - 20 \, x - 125\right )} e^{3} + {\left (3 \, x^{3} - 70 \, x^{2} + 6 \, {\left (x - 25\right )} e^{3} - 120 \, x - 750\right )} e^{\left (-x + \frac {1}{3} \, \log \left (-x + 25\right )\right )} + 3 \, {\left (x - 25\right )} e^{\left (-2 \, x + \frac {2}{3} \, \log \left (-x + 25\right )\right )} - 675 \, x - 1875}{3 \, {\left ({\left (x - 25\right )} e^{6} + 10 \, {\left (x - 25\right )} e^{3} + 2 \, {\left ({\left (x - 25\right )} e^{3} + 5 \, x - 125\right )} e^{\left (-x + \frac {1}{3} \, \log \left (-x + 25\right )\right )} + {\left (x - 25\right )} e^{\left (-2 \, x + \frac {2}{3} \, \log \left (-x + 25\right )\right )} + 25 \, x - 625\right )}} \,d x } \end {dmath*}
integrate(((3*x-75)*exp(1/3*log(-x+25)-x)^2+((6*x-150)*exp(3)+3*x^3-70*x^2 -120*x-750)*exp(1/3*log(-x+25)-x)+(3*x-75)*exp(3)^2+(6*x^2-120*x-750)*exp( 3)+30*x^2-675*x-1875)/((3*x-75)*exp(1/3*log(-x+25)-x)^2+((6*x-150)*exp(3)+ 30*x-750)*exp(1/3*log(-x+25)-x)+(3*x-75)*exp(3)^2+(30*x-750)*exp(3)+75*x-1 875),x, algorithm=\
integrate(1/3*(30*x^2 + 3*(x - 25)*e^6 + 6*(x^2 - 20*x - 125)*e^3 + (3*x^3 - 70*x^2 + 6*(x - 25)*e^3 - 120*x - 750)*e^(-x + 1/3*log(-x + 25)) + 3*(x - 25)*e^(-2*x + 2/3*log(-x + 25)) - 675*x - 1875)/((x - 25)*e^6 + 10*(x - 25)*e^3 + 2*((x - 25)*e^3 + 5*x - 125)*e^(-x + 1/3*log(-x + 25)) + (x - 2 5)*e^(-2*x + 2/3*log(-x + 25)) + 25*x - 625), x)
Time = 15.13 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.14 \begin {dmath*} \int \frac {-1875-675 x+30 x^2+e^6 (-75+3 x)+e^{\frac {2}{3} (-3 x+\log (25-x))} (-75+3 x)+e^3 \left (-750-120 x+6 x^2\right )+e^{\frac {1}{3} (-3 x+\log (25-x))} \left (-750-120 x-70 x^2+3 x^3+e^3 (-150+6 x)\right )}{-1875+75 x+e^6 (-75+3 x)+e^{\frac {2}{3} (-3 x+\log (25-x))} (-75+3 x)+e^3 (-750+30 x)+e^{\frac {1}{3} (-3 x+\log (25-x))} \left (-750+30 x+e^3 (-150+6 x)\right )} \, dx=x-\frac {76\,x^2\,{\mathrm {e}}^3-3\,x^3\,{\mathrm {e}}^3+380\,x^2-15\,x^3}{\left (3\,x-76\right )\,\left ({\mathrm {e}}^3+5\right )\,\left ({\mathrm {e}}^3+{\mathrm {e}}^{-x}\,{\left (25-x\right )}^{1/3}+5\right )} \end {dmath*}
int(-(675*x + exp(3)*(120*x - 6*x^2 + 750) - exp((2*log(25 - x))/3 - 2*x)* (3*x - 75) + exp(log(25 - x)/3 - x)*(120*x + 70*x^2 - 3*x^3 - exp(3)*(6*x - 150) + 750) - 30*x^2 - exp(6)*(3*x - 75) + 1875)/(75*x + exp(log(25 - x) /3 - x)*(30*x + exp(3)*(6*x - 150) - 750) + exp((2*log(25 - x))/3 - 2*x)*( 3*x - 75) + exp(6)*(3*x - 75) + exp(3)*(30*x - 750) - 1875),x)