∫ −x+x2+ 4√_e (−6160+12320x)(x−x2 ) 4 √ _e + 4√_e (−4740+9480x)(x−x2 )2 4√ _e + 4√_e (−1200+2400x)(x−x2 )3 4√ _e + 4√_e (−100+200x)(x−x2 )4 4√ _e ______________________________________________________________________________________________________________________________________________________________________

Optimal. Leaf size=25 \[ x+\left (3-5 \left (4+\left (x-x^2\right )^{\sqrt [4]{e}}\right )^2\right )^2 \]

________________________________________________________________________________________

Rubi [B] time = 1.91, antiderivative size = 68, normalized size of antiderivative = 2.72, number of steps used = 11, number of rules used = 4, integrand size = 119, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.034, Rules used = {1593, 6742, 6688, 629} \begin {gather*} 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 \end {gather*}

Int[(-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]

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[((d_) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d*(a + b*x + c*x^2)^(p +

1))/(b*(p + 1)), x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[2*c*d - b*e, 0] && NeQ[p, -1]

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

\begin {gather*} \begin {aligned} \text {integral} &=\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}}}{(-1+x) x} \, dx\\ &=\int \left (1+\frac {6160 \sqrt [4]{e} (1-2 x) \left (x-x^2\right )^{\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 {1200 \sqrt [4]{e} (1-2 x) \left (x-x^2\right )^{3 \sqrt [4]{e}}}{(1-x) x}+\frac {100 \sqrt [4]{e} (1-2 x) \left (x-x^2\right )^{4 \sqrt [4]{e}}}{(1-x) x}\right ) \, dx\\ &=x+\left (100 \sqrt [4]{e}\right ) \int \frac {(1-2 x) \left (x-x^2\right )^{4 \sqrt [4]{e}}}{(1-x) x} \, dx+\left (1200 \sqrt [4]{e}\right ) \int \frac {(1-2 x) \left (x-x^2\right )^{3 \sqrt [4]{e}}}{(1-x) x} \, dx+\left (4740 \sqrt [4]{e}\right ) \int \frac {(1-2 x) \left (x-x^2\right )^{2 \sqrt [4]{e}}}{(1-x) x} \, dx+\left (6160 \sqrt [4]{e}\right ) \int \frac {(1-2 x) \left (x-x^2\right )^{\sqrt [4]{e}}}{(1-x) x} \, dx\\ &=x+\left (100 \sqrt [4]{e}\right ) \int (1-2 x) \left (x-x^2\right )^{-1+4 \sqrt [4]{e}} \, dx+\left (1200 \sqrt [4]{e}\right ) \int (1-2 x) \left (x-x^2\right )^{-1+3 \sqrt [4]{e}} \, dx+\left (4740 \sqrt [4]{e}\right ) \int (1-2 x) \left (x-x^2\right )^{-1+2 \sqrt [4]{e}} \, dx+\left (6160 \sqrt [4]{e}\right ) \int (1-2 x) \left (x-x^2\right )^{-1+\sqrt [4]{e}} \, dx\\ &=x+6160 \left (x-x^2\right )^{\sqrt [4]{e}}+2370 \left (x-x^2\right )^{2 \sqrt [4]{e}}+400 \left (x-x^2\right )^{3 \sqrt [4]{e}}+25 \left (x-x^2\right )^{4 \sqrt [4]{e}}\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [B] time = 0.13, size = 64, normalized size = 2.56 \begin {gather*} 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}} \end {gather*}

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^

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*(-(

________________________________________________________________________________________

fricas [B] time = 0.66, size = 56, normalized size = 2.24 \begin {gather*} 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 \end {gather*}

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)*log(-x^2+x))^2+(12320*x-6160)*exp(1/4)*exp(exp(1/4)*log(-x^2+x))+x^2-x)/(

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)

________________________________________________________________________________________

giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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} \end {gather*}

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)*log(-x^2+x))^2+(12320*x-6160)*exp(1/4)*exp(exp(1/4)*log(-x^2+x))+x^2-x)/(

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)

________________________________________________________________________________________


method	result	size

risch	\(25 \left (-x^{2}+x \right )^{4 \,{\mathrm e}^{\frac {1}{4}}}+400 \left (-x^{2}+x \right )^{3 \,{\mathrm e}^{\frac {1}{4}}}+2370 \left (-x^{2}+x \right )^{2 \,{\mathrm e}^{\frac {1}{4}}}+x +6160 \left (-x^{2}+x \right )^{{\mathrm e}^{\frac {1}{4}}}\)	\(57\)

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,m

25*((-x^2+x)^exp(1/4))^4+400*((-x^2+x)^exp(1/4))^3+2370*((-x^2+x)^exp(1/4))^2+x+6160*(-x^2+x)^exp(1/4)

________________________________________________________________________________________

maxima [B] time = 0.41, size = 80, normalized size = 3.20 \begin {gather*} x + 25 \, e^{\left (4 \, e^{\frac {1}{4}} \log \relax (x) + 4 \, e^{\frac {1}{4}} \log \left (-x + 1\right )\right )} + 400 \, e^{\left (3 \, e^{\frac {1}{4}} \log \relax (x) + 3 \, e^{\frac {1}{4}} \log \left (-x + 1\right )\right )} + 2370 \, e^{\left (2 \, e^{\frac {1}{4}} \log \relax (x) + 2 \, e^{\frac {1}{4}} \log \left (-x + 1\right )\right )} + 6160 \, e^{\left (e^{\frac {1}{4}} \log \relax (x) + e^{\frac {1}{4}} \log \left (-x + 1\right )\right )} \end {gather*}

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)*log(-x^2+x))^2+(12320*x-6160)*exp(1/4)*exp(exp(1/4)*log(-x^2+x))+x^2-x)/(

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))

________________________________________________________________________________________

mupad [B] time = 6.08, size = 56, normalized size = 2.24 \begin {gather*} 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}} \end {gather*}

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))*(9480*x - 4740) + exp(1/4)*(x - x^2)^exp(1/4)*(12320*x - 6160))/(x - x^2),

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(

________________________________________________________________________________________

sympy [B] time = 7.61, size = 53, normalized size = 2.12 \begin {gather*} 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}}} \end {gather*}

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 + 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**

________________________________________________________________________________________

3.98.23 \(\int \frac {-x+x^2+\sqrt [4]{e} (-6160+12320 x) (x-x^2)^{\sqrt [4]{e}}+\sqrt [4]{e} (-4740+9480 x) (x-x^2)^{2 \sqrt [4]{e}}+\sqrt [4]{e} (-1200+2400 x) (x-x^2)^{3 \sqrt [4]{e}}+\sqrt [4]{e} (-100+200 x) (x-x^2)^{4 \sqrt [4]{e}}}{-x+x^2} \, dx\)