3.84.4 \(\int \frac {-x+x^2+4 e (80-120 x) (x^2-x^3)+16 e^2 (-16+24 x) (x^2-x^3)^2}{-x+x^2} \, dx\)

Optimal. Leaf size=22 \[ -5+x+4 \left (5-4 e (1-x) x^2\right )^2+\log (2) \]

________________________________________________________________________________________

Rubi [A]  time = 0.03, antiderivative size = 38, normalized size of antiderivative = 1.73, number of steps used = 2, number of rules used = 1, integrand size = 55, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.018, Rules used = {1586} \begin {gather*} 64 e^2 x^6-128 e^2 x^5+64 e^2 x^4+160 e x^3-160 e x^2+x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-x + x^2 + 4*E*(80 - 120*x)*(x^2 - x^3) + 16*E^2*(-16 + 24*x)*(x^2 - x^3)^2)/(-x + x^2),x]

[Out]

x - 160*E*x^2 + 160*E*x^3 + 64*E^2*x^4 - 128*E^2*x^5 + 64*E^2*x^6

Rule 1586

Int[(u_.)*(Px_)^(p_.)*(Qx_)^(q_.), x_Symbol] :> Int[u*PolynomialQuotient[Px, Qx, x]^p*Qx^(p + q), x] /; FreeQ[
q, x] && PolyQ[Px, x] && PolyQ[Qx, x] && EqQ[PolynomialRemainder[Px, Qx, x], 0] && IntegerQ[p] && LtQ[p*q, 0]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (1-320 e x+480 e x^2+256 e^2 x^3-640 e^2 x^4+384 e^2 x^5\right ) \, dx\\ &=x-160 e x^2+160 e x^3+64 e^2 x^4-128 e^2 x^5+64 e^2 x^6\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 38, normalized size = 1.73 \begin {gather*} x-160 e x^2+160 e x^3+64 e^2 x^4-128 e^2 x^5+64 e^2 x^6 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-x + x^2 + 4*E*(80 - 120*x)*(x^2 - x^3) + 16*E^2*(-16 + 24*x)*(x^2 - x^3)^2)/(-x + x^2),x]

[Out]

x - 160*E*x^2 + 160*E*x^3 + 64*E^2*x^4 - 128*E^2*x^5 + 64*E^2*x^6

________________________________________________________________________________________

fricas [A]  time = 1.10, size = 41, normalized size = 1.86 \begin {gather*} 4 \, {\left (x^{6} - 2 \, x^{5} + x^{4}\right )} e^{\left (4 \, \log \relax (2) + 2\right )} + 40 \, {\left (x^{3} - x^{2}\right )} e^{\left (2 \, \log \relax (2) + 1\right )} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((24*x-16)*exp(log(-x^3+x^2)+1+2*log(2))^2+(-120*x+80)*exp(log(-x^3+x^2)+1+2*log(2))+x^2-x)/(x^2-x),
x, algorithm="fricas")

[Out]

4*(x^6 - 2*x^5 + x^4)*e^(4*log(2) + 2) + 40*(x^3 - x^2)*e^(2*log(2) + 1) + x

________________________________________________________________________________________

giac [A]  time = 0.23, size = 37, normalized size = 1.68 \begin {gather*} 64 \, x^{6} e^{2} - 128 \, x^{5} e^{2} + 64 \, x^{4} e^{2} + 160 \, x^{3} e - 160 \, x^{2} e + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((24*x-16)*exp(log(-x^3+x^2)+1+2*log(2))^2+(-120*x+80)*exp(log(-x^3+x^2)+1+2*log(2))+x^2-x)/(x^2-x),
x, algorithm="giac")

[Out]

64*x^6*e^2 - 128*x^5*e^2 + 64*x^4*e^2 + 160*x^3*e - 160*x^2*e + x

________________________________________________________________________________________

maple [A]  time = 0.35, size = 36, normalized size = 1.64

method	result	size
default	\(x +128 \,{\mathrm e}^{2} \left (\frac {1}{2} x^{6}-x^{5}+\frac {1}{2} x^{4}\right )-160 \,{\mathrm e} \left (-x^{3}+x^{2}\right )\)	\(36\)
risch	\(x -160 x^{2} {\mathrm e}+160 x^{3} {\mathrm e}+64 x^{4} {\mathrm e}^{2}-128 \,{\mathrm e}^{2} x^{5}+64 x^{6} {\mathrm e}^{2}\)	\(38\)
norman	\(x -160 x^{2} {\mathrm e}+160 x^{3} {\mathrm e}+64 x^{4} {\mathrm e}^{2}-128 \,{\mathrm e}^{2} x^{5}+64 x^{6} {\mathrm e}^{2}\)	\(44\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((24*x-16)*exp(ln(-x^3+x^2)+1+2*ln(2))^2+(-120*x+80)*exp(ln(-x^3+x^2)+1+2*ln(2))+x^2-x)/(x^2-x),x,method=_
RETURNVERBOSE)

[Out]

x+128*exp(2)*(1/2*x^6-x^5+1/2*x^4)-160*exp(1)*(-x^3+x^2)

________________________________________________________________________________________

maxima [A]  time = 0.38, size = 37, normalized size = 1.68 \begin {gather*} 64 \, x^{6} e^{2} - 128 \, x^{5} e^{2} + 64 \, x^{4} e^{2} + 160 \, x^{3} e - 160 \, x^{2} e + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((24*x-16)*exp(log(-x^3+x^2)+1+2*log(2))^2+(-120*x+80)*exp(log(-x^3+x^2)+1+2*log(2))+x^2-x)/(x^2-x),
x, algorithm="maxima")

[Out]

64*x^6*e^2 - 128*x^5*e^2 + 64*x^4*e^2 + 160*x^3*e - 160*x^2*e + x

________________________________________________________________________________________

mupad [B]  time = 5.10, size = 37, normalized size = 1.68 \begin {gather*} 64\,{\mathrm {e}}^2\,x^6-128\,{\mathrm {e}}^2\,x^5+64\,{\mathrm {e}}^2\,x^4+160\,\mathrm {e}\,x^3-160\,\mathrm {e}\,x^2+x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x + exp(2*log(2) + log(x^2 - x^3) + 1)*(120*x - 80) - exp(4*log(2) + 2*log(x^2 - x^3) + 2)*(24*x - 16) -
x^2)/(x - x^2),x)

[Out]

x - 160*x^2*exp(1) + 160*x^3*exp(1) + 64*x^4*exp(2) - 128*x^5*exp(2) + 64*x^6*exp(2)

________________________________________________________________________________________

sympy [B]  time = 0.09, size = 42, normalized size = 1.91 \begin {gather*} 64 x^{6} e^{2} - 128 x^{5} e^{2} + 64 x^{4} e^{2} + 160 e x^{3} - 160 e x^{2} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((24*x-16)*exp(ln(-x**3+x**2)+1+2*ln(2))**2+(-120*x+80)*exp(ln(-x**3+x**2)+1+2*ln(2))+x**2-x)/(x**2-
x),x)

[Out]

64*x**6*exp(2) - 128*x**5*exp(2) + 64*x**4*exp(2) + 160*E*x**3 - 160*E*x**2 + x

________________________________________________________________________________________