\(\int \frac {-1-10 e^{\frac {x^3}{3}} x^2+2 e^{\frac {2 x^3}{3}} x^2}{25-10 e^{\frac {x^3}{3}}+e^{\frac {2 x^3}{3}}-x} \, dx\) [2917]

Optimal result

Integrand size = 58, antiderivative size = 18 \[ \int \frac {-1-10 e^{\frac {x^3}{3}} x^2+2 e^{\frac {2 x^3}{3}} x^2}{25-10 e^{\frac {x^3}{3}}+e^{\frac {2 x^3}{3}}-x} \, dx=\log \left (\left (-5+e^{\frac {x^3}{3}}\right )^2-x\right ) \]

[Out]

ln((-5+exp(1/3*x^3))^2-x)

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.44, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.017, Rules used = {6816} \[ \int \frac {-1-10 e^{\frac {x^3}{3}} x^2+2 e^{\frac {2 x^3}{3}} x^2}{25-10 e^{\frac {x^3}{3}}+e^{\frac {2 x^3}{3}}-x} \, dx=\log \left (-10 e^{\frac {x^3}{3}}+e^{\frac {2 x^3}{3}}-x+25\right ) \]

[In]

Int[(-1 - 10*E^(x^3/3)*x^2 + 2*E^((2*x^3)/3)*x^2)/(25 - 10*E^(x^3/3) + E^((2*x^3)/3) - x),x]

[Out]

Log[25 - 10*E^(x^3/3) + E^((2*x^3)/3) - x]

Rule 6816

Int[(u_)/(y_), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[q*Log[RemoveContent[y, x]], x] /;  !Fa
lseQ[q]]

Rubi steps \begin{align*} \text {integral}& = \log \left (25-10 e^{\frac {x^3}{3}}+e^{\frac {2 x^3}{3}}-x\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.44 \[ \int \frac {-1-10 e^{\frac {x^3}{3}} x^2+2 e^{\frac {2 x^3}{3}} x^2}{25-10 e^{\frac {x^3}{3}}+e^{\frac {2 x^3}{3}}-x} \, dx=\log \left (25-10 e^{\frac {x^3}{3}}+e^{\frac {2 x^3}{3}}-x\right ) \]

[In]

Integrate[(-1 - 10*E^(x^3/3)*x^2 + 2*E^((2*x^3)/3)*x^2)/(25 - 10*E^(x^3/3) + E^((2*x^3)/3) - x),x]

[Out]

Log[25 - 10*E^(x^3/3) + E^((2*x^3)/3) - x]

Maple [A] (verified)

Time = 0.17 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.17

method	result	size
risch	\(\ln \left ({\mathrm e}^{\frac {2 x^{3}}{3}}-10 \,{\mathrm e}^{\frac {x^{3}}{3}}-x +25\right )\)	\(21\)
derivativedivides	\(\ln \left ({\mathrm e}^{\frac {2 x^{3}}{3}}-10 \,{\mathrm e}^{\frac {x^{3}}{3}}-x +25\right )\)	\(23\)
default	\(\ln \left ({\mathrm e}^{\frac {2 x^{3}}{3}}-10 \,{\mathrm e}^{\frac {x^{3}}{3}}-x +25\right )\)	\(23\)
norman	\(\ln \left (-{\mathrm e}^{\frac {2 x^{3}}{3}}+x +10 \,{\mathrm e}^{\frac {x^{3}}{3}}-25\right )\)	\(23\)
parallelrisch	\(\ln \left (-{\mathrm e}^{\frac {2 x^{3}}{3}}+x +10 \,{\mathrm e}^{\frac {x^{3}}{3}}-25\right )\)	\(23\)

[In]

int((2*x^2*exp(1/3*x^3)^2-10*x^2*exp(1/3*x^3)-1)/(exp(1/3*x^3)^2-10*exp(1/3*x^3)-x+25),x,method=_RETURNVERBOSE
)

[Out]

ln(exp(2/3*x^3)-10*exp(1/3*x^3)-x+25)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {-1-10 e^{\frac {x^3}{3}} x^2+2 e^{\frac {2 x^3}{3}} x^2}{25-10 e^{\frac {x^3}{3}}+e^{\frac {2 x^3}{3}}-x} \, dx=\log \left (-x + e^{\left (\frac {2}{3} \, x^{3}\right )} - 10 \, e^{\left (\frac {1}{3} \, x^{3}\right )} + 25\right ) \]

[In]

integrate((2*x^2*exp(1/3*x^3)^2-10*x^2*exp(1/3*x^3)-1)/(exp(1/3*x^3)^2-10*exp(1/3*x^3)-x+25),x, algorithm="fri
cas")

[Out]

log(-x + e^(2/3*x^3) - 10*e^(1/3*x^3) + 25)

Sympy [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {-1-10 e^{\frac {x^3}{3}} x^2+2 e^{\frac {2 x^3}{3}} x^2}{25-10 e^{\frac {x^3}{3}}+e^{\frac {2 x^3}{3}}-x} \, dx=\log {\left (- x + e^{\frac {2 x^{3}}{3}} - 10 e^{\frac {x^{3}}{3}} + 25 \right )} \]

[In]

integrate((2*x**2*exp(1/3*x**3)**2-10*x**2*exp(1/3*x**3)-1)/(exp(1/3*x**3)**2-10*exp(1/3*x**3)-x+25),x)

[Out]

log(-x + exp(2*x**3/3) - 10*exp(x**3/3) + 25)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {-1-10 e^{\frac {x^3}{3}} x^2+2 e^{\frac {2 x^3}{3}} x^2}{25-10 e^{\frac {x^3}{3}}+e^{\frac {2 x^3}{3}}-x} \, dx=\log \left (x - e^{\left (\frac {2}{3} \, x^{3}\right )} + 10 \, e^{\left (\frac {1}{3} \, x^{3}\right )} - 25\right ) \]

[In]

integrate((2*x^2*exp(1/3*x^3)^2-10*x^2*exp(1/3*x^3)-1)/(exp(1/3*x^3)^2-10*exp(1/3*x^3)-x+25),x, algorithm="max
ima")

[Out]

log(x - e^(2/3*x^3) + 10*e^(1/3*x^3) - 25)

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {-1-10 e^{\frac {x^3}{3}} x^2+2 e^{\frac {2 x^3}{3}} x^2}{25-10 e^{\frac {x^3}{3}}+e^{\frac {2 x^3}{3}}-x} \, dx=\log \left (x - e^{\left (\frac {2}{3} \, x^{3}\right )} + 10 \, e^{\left (\frac {1}{3} \, x^{3}\right )} - 25\right ) \]

[In]

integrate((2*x^2*exp(1/3*x^3)^2-10*x^2*exp(1/3*x^3)-1)/(exp(1/3*x^3)^2-10*exp(1/3*x^3)-x+25),x, algorithm="gia
c")

[Out]

log(x - e^(2/3*x^3) + 10*e^(1/3*x^3) - 25)

Mupad [B] (verification not implemented)

Time = 8.59 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {-1-10 e^{\frac {x^3}{3}} x^2+2 e^{\frac {2 x^3}{3}} x^2}{25-10 e^{\frac {x^3}{3}}+e^{\frac {2 x^3}{3}}-x} \, dx=\ln \left (x+10\,{\mathrm {e}}^{\frac {x^3}{3}}-{\mathrm {e}}^{\frac {2\,x^3}{3}}-25\right ) \]

[In]

int((10*x^2*exp(x^3/3) - 2*x^2*exp((2*x^3)/3) + 1)/(x + 10*exp(x^3/3) - exp((2*x^3)/3) - 25),x)

[Out]

log(x + 10*exp(x^3/3) - exp((2*x^3)/3) - 25)