3.42.67 \(\int \frac {x^3-10 x^4-15 x^5+e^{\frac {e^2}{x^4}} (-20 e^2+10 x^4)}{x^3} \, dx\)

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

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Rubi [A]  time = 0.06, antiderivative size = 26, normalized size of antiderivative = 1.04, number of steps used = 3, number of rules used = 2, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.051, Rules used = {14, 2288} \begin {gather*} -5 x^3-5 x^2+5 e^{\frac {e^2}{x^4}} x^2+x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x^3 - 10*x^4 - 15*x^5 + E^(E^2/x^4)*(-20*E^2 + 10*x^4))/x^3,x]

[Out]

x - 5*x^2 + 5*E^(E^2/x^4)*x^2 - 5*x^3

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 2288

Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = (v*y)/(Log[F]*D[u, x])}, Simp[F^u*z, x] /; EqQ[D[z,
x], w*y]] /; FreeQ[F, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (1-10 x-15 x^2+\frac {10 e^{\frac {e^2}{x^4}} \left (-2 e^2+x^4\right )}{x^3}\right ) \, dx\\ &=x-5 x^2-5 x^3+10 \int \frac {e^{\frac {e^2}{x^4}} \left (-2 e^2+x^4\right )}{x^3} \, dx\\ &=x-5 x^2+5 e^{\frac {e^2}{x^4}} x^2-5 x^3\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 26, normalized size = 1.04 \begin {gather*} x-5 x^2+5 e^{\frac {e^2}{x^4}} x^2-5 x^3 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x^3 - 10*x^4 - 15*x^5 + E^(E^2/x^4)*(-20*E^2 + 10*x^4))/x^3,x]

[Out]

x - 5*x^2 + 5*E^(E^2/x^4)*x^2 - 5*x^3

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fricas [A]  time = 0.57, size = 24, normalized size = 0.96 \begin {gather*} -5 \, x^{3} + 5 \, x^{2} e^{\left (\frac {e^{2}}{x^{4}}\right )} - 5 \, x^{2} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-20*exp(1)^2+10*x^4)*exp(exp(1)^2/x^4)-15*x^5-10*x^4+x^3)/x^3,x, algorithm="fricas")

[Out]

-5*x^3 + 5*x^2*e^(e^2/x^4) - 5*x^2 + x

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giac [B]  time = 0.22, size = 42, normalized size = 1.68 \begin {gather*} -{\left (5 \, x^{3} e^{2} + 5 \, x^{2} e^{2} - 5 \, x^{2} e^{\left (\frac {2 \, x^{4} + e^{2}}{x^{4}}\right )} - x e^{2}\right )} e^{\left (-2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-20*exp(1)^2+10*x^4)*exp(exp(1)^2/x^4)-15*x^5-10*x^4+x^3)/x^3,x, algorithm="giac")

[Out]

-(5*x^3*e^2 + 5*x^2*e^2 - 5*x^2*e^((2*x^4 + e^2)/x^4) - x*e^2)*e^(-2)

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maple [A]  time = 0.16, size = 25, normalized size = 1.00

method	result	size
risch	\(-5 x^{3}+5 x^{2} {\mathrm e}^{\frac {{\mathrm e}^{2}}{x^{4}}}-5 x^{2}+x\)	\(25\)
norman	\(\frac {x^{3}-5 x^{4}-5 x^{5}+5 \,{\mathrm e}^{\frac {{\mathrm e}^{2}}{x^{4}}} x^{4}}{x^{2}}\)	\(33\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-20*exp(1)^2+10*x^4)*exp(exp(1)^2/x^4)-15*x^5-10*x^4+x^3)/x^3,x,method=_RETURNVERBOSE)

[Out]

-5*x^3+5*x^2*exp(exp(2)/x^4)-5*x^2+x

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maxima [C]  time = 0.54, size = 66, normalized size = 2.64 \begin {gather*} \frac {5}{2} \, x^{2} \sqrt {-\frac {e^{2}}{x^{4}}} \Gamma \left (-\frac {1}{2}, -\frac {e^{2}}{x^{4}}\right ) - 5 \, x^{3} - 5 \, x^{2} + x + \frac {5 \, \sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {-\frac {e^{2}}{x^{4}}}\right ) - 1\right )} e^{2}}{x^{2} \sqrt {-\frac {e^{2}}{x^{4}}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-20*exp(1)^2+10*x^4)*exp(exp(1)^2/x^4)-15*x^5-10*x^4+x^3)/x^3,x, algorithm="maxima")

[Out]

5/2*x^2*sqrt(-e^2/x^4)*gamma(-1/2, -e^2/x^4) - 5*x^3 - 5*x^2 + x + 5*sqrt(pi)*(erf(sqrt(-e^2/x^4)) - 1)*e^2/(x
^2*sqrt(-e^2/x^4))

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mupad [B]  time = 2.93, size = 24, normalized size = 0.96 \begin {gather*} x+5\,x^2\,{\mathrm {e}}^{\frac {{\mathrm {e}}^2}{x^4}}-5\,x^2-5\,x^3 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(exp(2)/x^4)*(20*exp(2) - 10*x^4) - x^3 + 10*x^4 + 15*x^5)/x^3,x)

[Out]

x + 5*x^2*exp(exp(2)/x^4) - 5*x^2 - 5*x^3

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sympy [A]  time = 0.16, size = 24, normalized size = 0.96 \begin {gather*} - 5 x^{3} + 5 x^{2} e^{\frac {e^{2}}{x^{4}}} - 5 x^{2} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-20*exp(1)**2+10*x**4)*exp(exp(1)**2/x**4)-15*x**5-10*x**4+x**3)/x**3,x)

[Out]

-5*x**3 + 5*x**2*exp(exp(2)/x**4) - 5*x**2 + x

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