Integrand size = 119, antiderivative size = 25 \[ \int \frac {-x+x^2+\sqrt [4]{e} (-6160+12320 x) \left (x-x^2\right )^{\sqrt [4]{e}}+\sqrt [4]{e} (-4740+9480 x) \left (x-x^2\right )^{2 \sqrt [4]{e}}+\sqrt [4]{e} (-1200+2400 x) \left (x-x^2\right )^{3 \sqrt [4]{e}}+\sqrt [4]{e} (-100+200 x) \left (x-x^2\right )^{4 \sqrt [4]{e}}}{-x+x^2} \, dx=x+\left (3-5 \left (4+\left (x-x^2\right )^{\sqrt [4]{e}}\right )^2\right )^2 \] Output:
x+(-5*(exp(exp(1/4)*ln(-x^2+x))+4)^2+3)^2
Leaf count is larger than twice the leaf count of optimal. \(64\) vs. \(2(25)=50\).
Time = 3.50 (sec) , antiderivative size = 64, normalized size of antiderivative = 2.56 \[ \int \frac {-x+x^2+\sqrt [4]{e} (-6160+12320 x) \left (x-x^2\right )^{\sqrt [4]{e}}+\sqrt [4]{e} (-4740+9480 x) \left (x-x^2\right )^{2 \sqrt [4]{e}}+\sqrt [4]{e} (-1200+2400 x) \left (x-x^2\right )^{3 \sqrt [4]{e}}+\sqrt [4]{e} (-100+200 x) \left (x-x^2\right )^{4 \sqrt [4]{e}}}{-x+x^2} \, dx=x+6160 (-((-1+x) x))^{\sqrt [4]{e}}+2370 (-((-1+x) x))^{2 \sqrt [4]{e}}+400 (-((-1+x) x))^{3 \sqrt [4]{e}}+25 (-((-1+x) x))^{4 \sqrt [4]{e}} \] Input:
Integrate[(-x + x^2 + E^(1/4)*(-6160 + 12320*x)*(x - x^2)^E^(1/4) + E^(1/4 )*(-4740 + 9480*x)*(x - x^2)^(2*E^(1/4)) + E^(1/4)*(-1200 + 2400*x)*(x - x ^2)^(3*E^(1/4)) + E^(1/4)*(-100 + 200*x)*(x - x^2)^(4*E^(1/4)))/(-x + x^2) ,x]
Output:
x + 6160*(-((-1 + x)*x))^E^(1/4) + 2370*(-((-1 + x)*x))^(2*E^(1/4)) + 400* (-((-1 + x)*x))^(3*E^(1/4)) + 25*(-((-1 + x)*x))^(4*E^(1/4))
Leaf count is larger than twice the leaf count of optimal. \(68\) vs. \(2(25)=50\).
Time = 1.27 (sec) , antiderivative size = 68, normalized size of antiderivative = 2.72, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.025, Rules used = {2026, 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.
\(\displaystyle \int \frac {\sqrt [4]{e} (200 x-100) \left (x-x^2\right )^{4 \sqrt [4]{e}}+\sqrt [4]{e} (2400 x-1200) \left (x-x^2\right )^{3 \sqrt [4]{e}}+\sqrt [4]{e} (9480 x-4740) \left (x-x^2\right )^{2 \sqrt [4]{e}}+\sqrt [4]{e} (12320 x-6160) \left (x-x^2\right )^{\sqrt [4]{e}}+x^2-x}{x^2-x} \, dx\) |
\(\Big \downarrow \) 2026 |
\(\displaystyle \int \frac {\sqrt [4]{e} (200 x-100) \left (x-x^2\right )^{4 \sqrt [4]{e}}+\sqrt [4]{e} (2400 x-1200) \left (x-x^2\right )^{3 \sqrt [4]{e}}+\sqrt [4]{e} (9480 x-4740) \left (x-x^2\right )^{2 \sqrt [4]{e}}+\sqrt [4]{e} (12320 x-6160) \left (x-x^2\right )^{\sqrt [4]{e}}+x^2-x}{(x-1) x}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {100 \sqrt [4]{e} (1-2 x) \left (x-x^2\right )^{4 \sqrt [4]{e}}}{(1-x) x}+\frac {1200 \sqrt [4]{e} (1-2 x) \left (x-x^2\right )^{3 \sqrt [4]{e}}}{(1-x) x}+\frac {4740 \sqrt [4]{e} (1-2 x) \left (x-x^2\right )^{2 \sqrt [4]{e}}}{(1-x) x}+\frac {6160 \sqrt [4]{e} (1-2 x) \left (x-x^2\right )^{\sqrt [4]{e}}}{(1-x) x}+1\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 25 \left (x-x^2\right )^{4 \sqrt [4]{e}}+400 \left (x-x^2\right )^{3 \sqrt [4]{e}}+2370 \left (x-x^2\right )^{2 \sqrt [4]{e}}+6160 \left (x-x^2\right )^{\sqrt [4]{e}}+x\) |
Input:
Int[(-x + x^2 + E^(1/4)*(-6160 + 12320*x)*(x - x^2)^E^(1/4) + E^(1/4)*(-47 40 + 9480*x)*(x - x^2)^(2*E^(1/4)) + E^(1/4)*(-1200 + 2400*x)*(x - x^2)^(3 *E^(1/4)) + E^(1/4)*(-100 + 200*x)*(x - x^2)^(4*E^(1/4)))/(-x + x^2),x]
Output:
x + 6160*(x - x^2)^E^(1/4) + 2370*(x - x^2)^(2*E^(1/4)) + 400*(x - x^2)^(3 *E^(1/4)) + 25*(x - x^2)^(4*E^(1/4))
Int[(Fx_.)*(Px_)^(p_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Int[x^(p *r)*ExpandToSum[Px/x^r, x]^p*Fx, x] /; IGtQ[r, 0]] /; PolyQ[Px, x] && Integ erQ[p] && !MonomialQ[Px, x] && (ILtQ[p, 0] || !PolyQ[u, x])
Leaf count of result is larger than twice the leaf count of optimal. \(56\) vs. \(2(24)=48\).
Time = 3.69 (sec) , antiderivative size = 57, normalized size of antiderivative = 2.28
method | result | size |
risch | \(x +2370 \left (-x^{2}+x \right )^{2 \,{\mathrm e}^{\frac {1}{4}}}+400 \left (-x^{2}+x \right )^{3 \,{\mathrm e}^{\frac {1}{4}}}+25 \left (-x^{2}+x \right )^{4 \,{\mathrm e}^{\frac {1}{4}}}+6160 \left (-x^{2}+x \right )^{{\mathrm e}^{\frac {1}{4}}}\) | \(57\) |
default | \(x +2370 \,{\mathrm e}^{2 \,{\mathrm e}^{\frac {1}{4}} \ln \left (-x^{2}+x \right )}+400 \,{\mathrm e}^{3 \,{\mathrm e}^{\frac {1}{4}} \ln \left (-x^{2}+x \right )}+25 \,{\mathrm e}^{4 \,{\mathrm e}^{\frac {1}{4}} \ln \left (-x^{2}+x \right )}+6160 \,{\mathrm e}^{{\mathrm e}^{\frac {1}{4}} \ln \left (-x^{2}+x \right )}\) | \(65\) |
parts | \(x +2370 \,{\mathrm e}^{2 \,{\mathrm e}^{\frac {1}{4}} \ln \left (-x^{2}+x \right )}+400 \,{\mathrm e}^{3 \,{\mathrm e}^{\frac {1}{4}} \ln \left (-x^{2}+x \right )}+25 \,{\mathrm e}^{4 \,{\mathrm e}^{\frac {1}{4}} \ln \left (-x^{2}+x \right )}+6160 \,{\mathrm e}^{{\mathrm e}^{\frac {1}{4}} \ln \left (-x^{2}+x \right )}\) | \(65\) |
parallelrisch | \(25 \,{\mathrm e}^{4 \,{\mathrm e}^{\frac {1}{4}} \ln \left (-x^{2}+x \right )}+400 \,{\mathrm e}^{3 \,{\mathrm e}^{\frac {1}{4}} \ln \left (-x^{2}+x \right )}+2+2370 \,{\mathrm e}^{2 \,{\mathrm e}^{\frac {1}{4}} \ln \left (-x^{2}+x \right )}+x +6160 \,{\mathrm e}^{{\mathrm e}^{\frac {1}{4}} \ln \left (-x^{2}+x \right )}\) | \(66\) |
Input:
int(((200*x-100)*exp(1/4)*exp(exp(1/4)*ln(-x^2+x))^4+(2400*x-1200)*exp(1/4 )*exp(exp(1/4)*ln(-x^2+x))^3+(9480*x-4740)*exp(1/4)*exp(exp(1/4)*ln(-x^2+x ))^2+(12320*x-6160)*exp(1/4)*exp(exp(1/4)*ln(-x^2+x))+x^2-x)/(x^2-x),x,met hod=_RETURNVERBOSE)
Output:
x+2370*((-x^2+x)^exp(1/4))^2+400*((-x^2+x)^exp(1/4))^3+25*((-x^2+x)^exp(1/ 4))^4+6160*(-x^2+x)^exp(1/4)
Leaf count of result is larger than twice the leaf count of optimal. 56 vs. \(2 (22) = 44\).
Time = 0.12 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.24 \[ \int \frac {-x+x^2+\sqrt [4]{e} (-6160+12320 x) \left (x-x^2\right )^{\sqrt [4]{e}}+\sqrt [4]{e} (-4740+9480 x) \left (x-x^2\right )^{2 \sqrt [4]{e}}+\sqrt [4]{e} (-1200+2400 x) \left (x-x^2\right )^{3 \sqrt [4]{e}}+\sqrt [4]{e} (-100+200 x) \left (x-x^2\right )^{4 \sqrt [4]{e}}}{-x+x^2} \, dx=25 \, {\left (-x^{2} + x\right )}^{4 \, e^{\frac {1}{4}}} + 400 \, {\left (-x^{2} + x\right )}^{3 \, e^{\frac {1}{4}}} + 2370 \, {\left (-x^{2} + x\right )}^{2 \, e^{\frac {1}{4}}} + 6160 \, {\left (-x^{2} + x\right )}^{e^{\frac {1}{4}}} + x \] Input:
integrate(((200*x-100)*exp(1/4)*exp(exp(1/4)*log(-x^2+x))^4+(2400*x-1200)* exp(1/4)*exp(exp(1/4)*log(-x^2+x))^3+(9480*x-4740)*exp(1/4)*exp(exp(1/4)*l og(-x^2+x))^2+(12320*x-6160)*exp(1/4)*exp(exp(1/4)*log(-x^2+x))+x^2-x)/(x^ 2-x),x, algorithm="fricas")
Output:
25*(-x^2 + x)^(4*e^(1/4)) + 400*(-x^2 + x)^(3*e^(1/4)) + 2370*(-x^2 + x)^( 2*e^(1/4)) + 6160*(-x^2 + x)^e^(1/4) + x
Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (19) = 38\).
Time = 1.37 (sec) , antiderivative size = 53, normalized size of antiderivative = 2.12 \[ \int \frac {-x+x^2+\sqrt [4]{e} (-6160+12320 x) \left (x-x^2\right )^{\sqrt [4]{e}}+\sqrt [4]{e} (-4740+9480 x) \left (x-x^2\right )^{2 \sqrt [4]{e}}+\sqrt [4]{e} (-1200+2400 x) \left (x-x^2\right )^{3 \sqrt [4]{e}}+\sqrt [4]{e} (-100+200 x) \left (x-x^2\right )^{4 \sqrt [4]{e}}}{-x+x^2} \, dx=x + 25 \left (- x^{2} + x\right )^{4 e^{\frac {1}{4}}} + 400 \left (- x^{2} + x\right )^{3 e^{\frac {1}{4}}} + 2370 \left (- x^{2} + x\right )^{2 e^{\frac {1}{4}}} + 6160 \left (- x^{2} + x\right )^{e^{\frac {1}{4}}} \] Input:
integrate(((200*x-100)*exp(1/4)*exp(exp(1/4)*ln(-x**2+x))**4+(2400*x-1200) *exp(1/4)*exp(exp(1/4)*ln(-x**2+x))**3+(9480*x-4740)*exp(1/4)*exp(exp(1/4) *ln(-x**2+x))**2+(12320*x-6160)*exp(1/4)*exp(exp(1/4)*ln(-x**2+x))+x**2-x) /(x**2-x),x)
Output:
x + 25*(-x**2 + x)**(4*exp(1/4)) + 400*(-x**2 + x)**(3*exp(1/4)) + 2370*(- x**2 + x)**(2*exp(1/4)) + 6160*(-x**2 + x)**exp(1/4)
Leaf count of result is larger than twice the leaf count of optimal. 80 vs. \(2 (22) = 44\).
Time = 0.09 (sec) , antiderivative size = 80, normalized size of antiderivative = 3.20 \[ \int \frac {-x+x^2+\sqrt [4]{e} (-6160+12320 x) \left (x-x^2\right )^{\sqrt [4]{e}}+\sqrt [4]{e} (-4740+9480 x) \left (x-x^2\right )^{2 \sqrt [4]{e}}+\sqrt [4]{e} (-1200+2400 x) \left (x-x^2\right )^{3 \sqrt [4]{e}}+\sqrt [4]{e} (-100+200 x) \left (x-x^2\right )^{4 \sqrt [4]{e}}}{-x+x^2} \, dx=x + 25 \, e^{\left (4 \, e^{\frac {1}{4}} \log \left (x\right ) + 4 \, e^{\frac {1}{4}} \log \left (-x + 1\right )\right )} + 400 \, e^{\left (3 \, e^{\frac {1}{4}} \log \left (x\right ) + 3 \, e^{\frac {1}{4}} \log \left (-x + 1\right )\right )} + 2370 \, e^{\left (2 \, e^{\frac {1}{4}} \log \left (x\right ) + 2 \, e^{\frac {1}{4}} \log \left (-x + 1\right )\right )} + 6160 \, e^{\left (e^{\frac {1}{4}} \log \left (x\right ) + e^{\frac {1}{4}} \log \left (-x + 1\right )\right )} \] Input:
integrate(((200*x-100)*exp(1/4)*exp(exp(1/4)*log(-x^2+x))^4+(2400*x-1200)* exp(1/4)*exp(exp(1/4)*log(-x^2+x))^3+(9480*x-4740)*exp(1/4)*exp(exp(1/4)*l og(-x^2+x))^2+(12320*x-6160)*exp(1/4)*exp(exp(1/4)*log(-x^2+x))+x^2-x)/(x^ 2-x),x, algorithm="maxima")
Output:
x + 25*e^(4*e^(1/4)*log(x) + 4*e^(1/4)*log(-x + 1)) + 400*e^(3*e^(1/4)*log (x) + 3*e^(1/4)*log(-x + 1)) + 2370*e^(2*e^(1/4)*log(x) + 2*e^(1/4)*log(-x + 1)) + 6160*e^(e^(1/4)*log(x) + e^(1/4)*log(-x + 1))
\[ \int \frac {-x+x^2+\sqrt [4]{e} (-6160+12320 x) \left (x-x^2\right )^{\sqrt [4]{e}}+\sqrt [4]{e} (-4740+9480 x) \left (x-x^2\right )^{2 \sqrt [4]{e}}+\sqrt [4]{e} (-1200+2400 x) \left (x-x^2\right )^{3 \sqrt [4]{e}}+\sqrt [4]{e} (-100+200 x) \left (x-x^2\right )^{4 \sqrt [4]{e}}}{-x+x^2} \, dx=\int { \frac {100 \, {\left (-x^{2} + x\right )}^{4 \, e^{\frac {1}{4}}} {\left (2 \, x - 1\right )} e^{\frac {1}{4}} + 1200 \, {\left (-x^{2} + x\right )}^{3 \, e^{\frac {1}{4}}} {\left (2 \, x - 1\right )} e^{\frac {1}{4}} + 4740 \, {\left (-x^{2} + x\right )}^{2 \, e^{\frac {1}{4}}} {\left (2 \, x - 1\right )} e^{\frac {1}{4}} + 6160 \, {\left (-x^{2} + x\right )}^{e^{\frac {1}{4}}} {\left (2 \, x - 1\right )} e^{\frac {1}{4}} + x^{2} - x}{x^{2} - x} \,d x } \] Input:
integrate(((200*x-100)*exp(1/4)*exp(exp(1/4)*log(-x^2+x))^4+(2400*x-1200)* exp(1/4)*exp(exp(1/4)*log(-x^2+x))^3+(9480*x-4740)*exp(1/4)*exp(exp(1/4)*l og(-x^2+x))^2+(12320*x-6160)*exp(1/4)*exp(exp(1/4)*log(-x^2+x))+x^2-x)/(x^ 2-x),x, algorithm="giac")
Output:
integrate((100*(-x^2 + x)^(4*e^(1/4))*(2*x - 1)*e^(1/4) + 1200*(-x^2 + x)^ (3*e^(1/4))*(2*x - 1)*e^(1/4) + 4740*(-x^2 + x)^(2*e^(1/4))*(2*x - 1)*e^(1 /4) + 6160*(-x^2 + x)^e^(1/4)*(2*x - 1)*e^(1/4) + x^2 - x)/(x^2 - x), x)
Time = 4.53 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.24 \[ \int \frac {-x+x^2+\sqrt [4]{e} (-6160+12320 x) \left (x-x^2\right )^{\sqrt [4]{e}}+\sqrt [4]{e} (-4740+9480 x) \left (x-x^2\right )^{2 \sqrt [4]{e}}+\sqrt [4]{e} (-1200+2400 x) \left (x-x^2\right )^{3 \sqrt [4]{e}}+\sqrt [4]{e} (-100+200 x) \left (x-x^2\right )^{4 \sqrt [4]{e}}}{-x+x^2} \, dx=x+2370\,{\left (x-x^2\right )}^{2\,{\mathrm {e}}^{1/4}}+400\,{\left (x-x^2\right )}^{3\,{\mathrm {e}}^{1/4}}+25\,{\left (x-x^2\right )}^{4\,{\mathrm {e}}^{1/4}}+6160\,{\left (x-x^2\right )}^{{\mathrm {e}}^{1/4}} \] Input:
int(-(x^2 - x + exp(1/4)*(x - x^2)^(4*exp(1/4))*(200*x - 100) + exp(1/4)*( x - x^2)^(3*exp(1/4))*(2400*x - 1200) + exp(1/4)*(x - x^2)^(2*exp(1/4))*(9 480*x - 4740) + exp(1/4)*(x - x^2)^exp(1/4)*(12320*x - 6160))/(x - x^2),x)
Output:
x + 2370*(x - x^2)^(2*exp(1/4)) + 400*(x - x^2)^(3*exp(1/4)) + 25*(x - x^2 )^(4*exp(1/4)) + 6160*(x - x^2)^exp(1/4)
Time = 0.17 (sec) , antiderivative size = 69, normalized size of antiderivative = 2.76 \[ \int \frac {-x+x^2+\sqrt [4]{e} (-6160+12320 x) \left (x-x^2\right )^{\sqrt [4]{e}}+\sqrt [4]{e} (-4740+9480 x) \left (x-x^2\right )^{2 \sqrt [4]{e}}+\sqrt [4]{e} (-1200+2400 x) \left (x-x^2\right )^{3 \sqrt [4]{e}}+\sqrt [4]{e} (-100+200 x) \left (x-x^2\right )^{4 \sqrt [4]{e}}}{-x+x^2} \, dx=25 e^{4 e^{\frac {1}{4}} \mathrm {log}\left (-x^{2}+x \right )}+400 e^{3 e^{\frac {1}{4}} \mathrm {log}\left (-x^{2}+x \right )}+2370 e^{2 e^{\frac {1}{4}} \mathrm {log}\left (-x^{2}+x \right )}+6160 e^{e^{\frac {1}{4}} \mathrm {log}\left (-x^{2}+x \right )}+x \] Input:
int(((200*x-100)*exp(1/4)*exp(exp(1/4)*log(-x^2+x))^4+(2400*x-1200)*exp(1/ 4)*exp(exp(1/4)*log(-x^2+x))^3+(9480*x-4740)*exp(1/4)*exp(exp(1/4)*log(-x^ 2+x))^2+(12320*x-6160)*exp(1/4)*exp(exp(1/4)*log(-x^2+x))+x^2-x)/(x^2-x),x )
Output:
25*e**(4*e**(1/4)*log( - x**2 + x)) + 400*e**(3*e**(1/4)*log( - x**2 + x)) + 2370*e**(2*e**(1/4)*log( - x**2 + x)) + 6160*e**(e**(1/4)*log( - x**2 + x)) + x