3.22.62 \(\int \frac {3-3 x+e^{\frac {2}{3} (-10+2 x-x^2+3 \log (\frac {1}{5} (x-x^2)))} (3-5 x-8 x^2+4 x^3)}{-3 x^2+3 x^3} \, dx\)

Optimal. Leaf size=38 \[ \frac {1-\frac {1}{25} e^{\frac {2}{3} \left (-2+2 (-4+x)-x^2\right )} (1-x)^2 x^2}{x} \]

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Rubi [B]  time = 1.75, antiderivative size = 77, normalized size of antiderivative = 2.03, number of steps used = 43, number of rules used = 6, integrand size = 65, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.092, Rules used = {1593, 6742, 2234, 2205, 2240, 2241} \begin {gather*} \frac {2}{25} e^{-\frac {2 x^2}{3}+\frac {4 x}{3}-\frac {20}{3}} x^2-\frac {1}{25} e^{-\frac {2 x^2}{3}+\frac {4 x}{3}-\frac {20}{3}} x-\frac {1}{25} e^{-\frac {2 x^2}{3}+\frac {4 x}{3}-\frac {20}{3}} x^3+\frac {1}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(3 - 3*x + E^((2*(-10 + 2*x - x^2 + 3*Log[(x - x^2)/5]))/3)*(3 - 5*x - 8*x^2 + 4*x^3))/(-3*x^2 + 3*x^3),x]

[Out]

x^(-1) - (E^(-20/3 + (4*x)/3 - (2*x^2)/3)*x)/25 + (2*E^(-20/3 + (4*x)/3 - (2*x^2)/3)*x^2)/25 - (E^(-20/3 + (4*
x)/3 - (2*x^2)/3)*x^3)/25

Rule 1593

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

Rule 2205

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[(F^a*Sqrt[Pi]*Erf[(c + d*x)*Rt[-(b*Log[F]),
2]])/(2*d*Rt[-(b*Log[F]), 2]), x] /; FreeQ[{F, a, b, c, d}, x] && NegQ[b]

Rule 2234

Int[(F_)^((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[F^(a - b^2/(4*c)), Int[F^((b + 2*c*x)^2/(4*c))
, x], x] /; FreeQ[{F, a, b, c}, x]

Rule 2240

Int[(F_)^((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)*((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[(e*F^(a + b*x + c*x^2))/(
2*c*Log[F]), x] - Dist[(b*e - 2*c*d)/(2*c), Int[F^(a + b*x + c*x^2), x], x] /; FreeQ[{F, a, b, c, d, e}, x] &&
NeQ[b*e - 2*c*d, 0]

Rule 2241

Int[(F_)^((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)*((d_.) + (e_.)*(x_))^(m_), x_Symbol] :> Simp[(e*(d + e*x)^(m - 1)
*F^(a + b*x + c*x^2))/(2*c*Log[F]), x] + (-Dist[(b*e - 2*c*d)/(2*c), Int[(d + e*x)^(m - 1)*F^(a + b*x + c*x^2)
, x], x] - Dist[((m - 1)*e^2)/(2*c*Log[F]), Int[(d + e*x)^(m - 2)*F^(a + b*x + c*x^2), x], x]) /; FreeQ[{F, a,
b, c, d, e}, x] && NeQ[b*e - 2*c*d, 0] && GtQ[m, 1]

Rule 6742

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {3-3 x+\exp \left (\frac {2}{3} \left (-10+2 x-x^2+3 \log \left (\frac {1}{5} \left (x-x^2\right )\right )\right )\right ) \left (3-5 x-8 x^2+4 x^3\right )}{x^2 (-3+3 x)} \, dx\\ &=\int \left (-\frac {1}{x^2}+\frac {1}{75} e^{-\frac {20}{3}+\frac {4 x}{3}-\frac {2 x^2}{3}} (-1+x) \left (3-5 x-8 x^2+4 x^3\right )\right ) \, dx\\ &=\frac {1}{x}+\frac {1}{75} \int e^{-\frac {20}{3}+\frac {4 x}{3}-\frac {2 x^2}{3}} (-1+x) \left (3-5 x-8 x^2+4 x^3\right ) \, dx\\ &=\frac {1}{x}+\frac {1}{75} \int \left (-3 e^{-\frac {20}{3}+\frac {4 x}{3}-\frac {2 x^2}{3}}+8 e^{-\frac {20}{3}+\frac {4 x}{3}-\frac {2 x^2}{3}} x+3 e^{-\frac {20}{3}+\frac {4 x}{3}-\frac {2 x^2}{3}} x^2-12 e^{-\frac {20}{3}+\frac {4 x}{3}-\frac {2 x^2}{3}} x^3+4 e^{-\frac {20}{3}+\frac {4 x}{3}-\frac {2 x^2}{3}} x^4\right ) \, dx\\ &=\frac {1}{x}-\frac {1}{25} \int e^{-\frac {20}{3}+\frac {4 x}{3}-\frac {2 x^2}{3}} \, dx+\frac {1}{25} \int e^{-\frac {20}{3}+\frac {4 x}{3}-\frac {2 x^2}{3}} x^2 \, dx+\frac {4}{75} \int e^{-\frac {20}{3}+\frac {4 x}{3}-\frac {2 x^2}{3}} x^4 \, dx+\frac {8}{75} \int e^{-\frac {20}{3}+\frac {4 x}{3}-\frac {2 x^2}{3}} x \, dx-\frac {4}{25} \int e^{-\frac {20}{3}+\frac {4 x}{3}-\frac {2 x^2}{3}} x^3 \, dx\\ &=-\frac {2}{25} e^{-\frac {20}{3}+\frac {4 x}{3}-\frac {2 x^2}{3}}+\frac {1}{x}-\frac {3}{100} e^{-\frac {20}{3}+\frac {4 x}{3}-\frac {2 x^2}{3}} x+\frac {3}{25} e^{-\frac {20}{3}+\frac {4 x}{3}-\frac {2 x^2}{3}} x^2-\frac {1}{25} e^{-\frac {20}{3}+\frac {4 x}{3}-\frac {2 x^2}{3}} x^3+\frac {3}{100} \int e^{-\frac {20}{3}+\frac {4 x}{3}-\frac {2 x^2}{3}} \, dx+\frac {1}{25} \int e^{-\frac {20}{3}+\frac {4 x}{3}-\frac {2 x^2}{3}} x \, dx+\frac {4}{75} \int e^{-\frac {20}{3}+\frac {4 x}{3}-\frac {2 x^2}{3}} x^3 \, dx+\frac {8}{75} \int e^{-\frac {20}{3}+\frac {4 x}{3}-\frac {2 x^2}{3}} \, dx+\frac {3}{25} \int e^{-\frac {20}{3}+\frac {4 x}{3}-\frac {2 x^2}{3}} x^2 \, dx-\frac {4}{25} \int e^{-\frac {20}{3}+\frac {4 x}{3}-\frac {2 x^2}{3}} x^2 \, dx-\frac {6}{25} \int e^{-\frac {20}{3}+\frac {4 x}{3}-\frac {2 x^2}{3}} x \, dx-\frac {\int e^{-\frac {3}{8} \left (\frac {4}{3}-\frac {4 x}{3}\right )^2} \, dx}{25 e^6}\\ &=\frac {7}{100} e^{-\frac {20}{3}+\frac {4 x}{3}-\frac {2 x^2}{3}}+\frac {1}{x}+\frac {2}{25} e^{-\frac {20}{3}+\frac {4 x}{3}-\frac {2 x^2}{3}} x^2-\frac {1}{25} e^{-\frac {20}{3}+\frac {4 x}{3}-\frac {2 x^2}{3}} x^3+\frac {\sqrt {\frac {3 \pi }{2}} \text {erf}\left (\sqrt {\frac {2}{3}} (1-x)\right )}{50 e^6}+\frac {1}{25} \int e^{-\frac {20}{3}+\frac {4 x}{3}-\frac {2 x^2}{3}} \, dx+\frac {4}{75} \int e^{-\frac {20}{3}+\frac {4 x}{3}-\frac {2 x^2}{3}} x^2 \, dx+\frac {2}{25} \int e^{-\frac {20}{3}+\frac {4 x}{3}-\frac {2 x^2}{3}} x \, dx+\frac {9}{100} \int e^{-\frac {20}{3}+\frac {4 x}{3}-\frac {2 x^2}{3}} \, dx-\frac {3}{25} \int e^{-\frac {20}{3}+\frac {4 x}{3}-\frac {2 x^2}{3}} \, dx+\frac {3}{25} \int e^{-\frac {20}{3}+\frac {4 x}{3}-\frac {2 x^2}{3}} x \, dx-\frac {4}{25} \int e^{-\frac {20}{3}+\frac {4 x}{3}-\frac {2 x^2}{3}} x \, dx-\frac {6}{25} \int e^{-\frac {20}{3}+\frac {4 x}{3}-\frac {2 x^2}{3}} \, dx+\frac {3 \int e^{-\frac {3}{8} \left (\frac {4}{3}-\frac {4 x}{3}\right )^2} \, dx}{100 e^6}+\frac {8 \int e^{-\frac {3}{8} \left (\frac {4}{3}-\frac {4 x}{3}\right )^2} \, dx}{75 e^6}\\ &=\frac {1}{25} e^{-\frac {20}{3}+\frac {4 x}{3}-\frac {2 x^2}{3}}+\frac {1}{x}-\frac {1}{25} e^{-\frac {20}{3}+\frac {4 x}{3}-\frac {2 x^2}{3}} x+\frac {2}{25} e^{-\frac {20}{3}+\frac {4 x}{3}-\frac {2 x^2}{3}} x^2-\frac {1}{25} e^{-\frac {20}{3}+\frac {4 x}{3}-\frac {2 x^2}{3}} x^3-\frac {2 \sqrt {\frac {2 \pi }{3}} \text {erf}\left (\sqrt {\frac {2}{3}} (1-x)\right )}{25 e^6}+\frac {\sqrt {\frac {3 \pi }{2}} \text {erf}\left (\sqrt {\frac {2}{3}} (1-x)\right )}{200 e^6}+\frac {1}{25} \int e^{-\frac {20}{3}+\frac {4 x}{3}-\frac {2 x^2}{3}} \, dx+\frac {4}{75} \int e^{-\frac {20}{3}+\frac {4 x}{3}-\frac {2 x^2}{3}} x \, dx+\frac {2}{25} \int e^{-\frac {20}{3}+\frac {4 x}{3}-\frac {2 x^2}{3}} \, dx+\frac {3}{25} \int e^{-\frac {20}{3}+\frac {4 x}{3}-\frac {2 x^2}{3}} \, dx-\frac {4}{25} \int e^{-\frac {20}{3}+\frac {4 x}{3}-\frac {2 x^2}{3}} \, dx+\frac {\int e^{-\frac {3}{8} \left (\frac {4}{3}-\frac {4 x}{3}\right )^2} \, dx}{25 e^6}+\frac {9 \int e^{-\frac {3}{8} \left (\frac {4}{3}-\frac {4 x}{3}\right )^2} \, dx}{100 e^6}-\frac {3 \int e^{-\frac {3}{8} \left (\frac {4}{3}-\frac {4 x}{3}\right )^2} \, dx}{25 e^6}-\frac {6 \int e^{-\frac {3}{8} \left (\frac {4}{3}-\frac {4 x}{3}\right )^2} \, dx}{25 e^6}\\ &=\frac {1}{x}-\frac {1}{25} e^{-\frac {20}{3}+\frac {4 x}{3}-\frac {2 x^2}{3}} x+\frac {2}{25} e^{-\frac {20}{3}+\frac {4 x}{3}-\frac {2 x^2}{3}} x^2-\frac {1}{25} e^{-\frac {20}{3}+\frac {4 x}{3}-\frac {2 x^2}{3}} x^3-\frac {2 \sqrt {\frac {2 \pi }{3}} \text {erf}\left (\sqrt {\frac {2}{3}} (1-x)\right )}{25 e^6}+\frac {3 \sqrt {\frac {3 \pi }{2}} \text {erf}\left (\sqrt {\frac {2}{3}} (1-x)\right )}{25 e^6}+\frac {4}{75} \int e^{-\frac {20}{3}+\frac {4 x}{3}-\frac {2 x^2}{3}} \, dx+\frac {\int e^{-\frac {3}{8} \left (\frac {4}{3}-\frac {4 x}{3}\right )^2} \, dx}{25 e^6}+\frac {2 \int e^{-\frac {3}{8} \left (\frac {4}{3}-\frac {4 x}{3}\right )^2} \, dx}{25 e^6}+\frac {3 \int e^{-\frac {3}{8} \left (\frac {4}{3}-\frac {4 x}{3}\right )^2} \, dx}{25 e^6}-\frac {4 \int e^{-\frac {3}{8} \left (\frac {4}{3}-\frac {4 x}{3}\right )^2} \, dx}{25 e^6}\\ &=\frac {1}{x}-\frac {1}{25} e^{-\frac {20}{3}+\frac {4 x}{3}-\frac {2 x^2}{3}} x+\frac {2}{25} e^{-\frac {20}{3}+\frac {4 x}{3}-\frac {2 x^2}{3}} x^2-\frac {1}{25} e^{-\frac {20}{3}+\frac {4 x}{3}-\frac {2 x^2}{3}} x^3-\frac {2 \sqrt {\frac {2 \pi }{3}} \text {erf}\left (\sqrt {\frac {2}{3}} (1-x)\right )}{25 e^6}+\frac {\sqrt {6 \pi } \text {erf}\left (\sqrt {\frac {2}{3}} (1-x)\right )}{25 e^6}+\frac {4 \int e^{-\frac {3}{8} \left (\frac {4}{3}-\frac {4 x}{3}\right )^2} \, dx}{75 e^6}\\ &=\frac {1}{x}-\frac {1}{25} e^{-\frac {20}{3}+\frac {4 x}{3}-\frac {2 x^2}{3}} x+\frac {2}{25} e^{-\frac {20}{3}+\frac {4 x}{3}-\frac {2 x^2}{3}} x^2-\frac {1}{25} e^{-\frac {20}{3}+\frac {4 x}{3}-\frac {2 x^2}{3}} x^3\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.31, size = 49, normalized size = 1.29 \begin {gather*} \frac {e^{-\frac {2}{3} \left (10+x^2\right )} \left (75 e^{\frac {2}{3} \left (10+x^2\right )}-3 e^{4 x/3} (-1+x)^2 x^2\right )}{75 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(3 - 3*x + E^((2*(-10 + 2*x - x^2 + 3*Log[(x - x^2)/5]))/3)*(3 - 5*x - 8*x^2 + 4*x^3))/(-3*x^2 + 3*x
^3),x]

[Out]

(75*E^((2*(10 + x^2))/3) - 3*E^((4*x)/3)*(-1 + x)^2*x^2)/(75*E^((2*(10 + x^2))/3)*x)

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fricas [A]  time = 0.72, size = 30, normalized size = 0.79 \begin {gather*} -\frac {e^{\left (-\frac {2}{3} \, x^{2} + \frac {4}{3} \, x + 2 \, \log \left (-\frac {1}{5} \, x^{2} + \frac {1}{5} \, x\right ) - \frac {20}{3}\right )} - 1}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^3-8*x^2-5*x+3)*exp(log(-1/5*x^2+1/5*x)-1/3*x^2+2/3*x-10/3)^2-3*x+3)/(3*x^3-3*x^2),x, algorithm
="fricas")

[Out]

-(e^(-2/3*x^2 + 4/3*x + 2*log(-1/5*x^2 + 1/5*x) - 20/3) - 1)/x

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giac [A]  time = 0.38, size = 53, normalized size = 1.39 \begin {gather*} -\frac {x^{4} e^{\left (-\frac {2}{3} \, x^{2} + \frac {4}{3} \, x - \frac {20}{3}\right )} - 2 \, x^{3} e^{\left (-\frac {2}{3} \, x^{2} + \frac {4}{3} \, x - \frac {20}{3}\right )} + x^{2} e^{\left (-\frac {2}{3} \, x^{2} + \frac {4}{3} \, x - \frac {20}{3}\right )} - 25}{25 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^3-8*x^2-5*x+3)*exp(log(-1/5*x^2+1/5*x)-1/3*x^2+2/3*x-10/3)^2-3*x+3)/(3*x^3-3*x^2),x, algorithm
="giac")

[Out]

-1/25*(x^4*e^(-2/3*x^2 + 4/3*x - 20/3) - 2*x^3*e^(-2/3*x^2 + 4/3*x - 20/3) + x^2*e^(-2/3*x^2 + 4/3*x - 20/3) -
25)/x

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maple [A]  time = 0.39, size = 32, normalized size = 0.84

method	result	size
norman	\(\frac {1-\left (-\frac {1}{5} x^{2}+\frac {1}{5} x \right )^{2} {\mathrm e}^{-\frac {20}{3}-\frac {2}{3} x^{2}+\frac {4}{3} x}}{x}\)	\(32\)
default	\(-\frac {x \,{\mathrm e}^{-\frac {20}{3}-\frac {2}{3} x^{2}+\frac {4}{3} x}}{25}+\frac {2 x^{2} {\mathrm e}^{-\frac {20}{3}-\frac {2}{3} x^{2}+\frac {4}{3} x}}{25}-\frac {x^{3} {\mathrm e}^{-\frac {20}{3}-\frac {2}{3} x^{2}+\frac {4}{3} x}}{25}+\frac {1}{x}\)	\(51\)
risch	\(\frac {1}{x}+\frac {25 \left (-\frac {1}{25} x +\frac {2}{25} x^{2}-\frac {1}{25} x^{3}\right ) \left (-\frac {1}{5} x^{2}+\frac {1}{5} x \right )^{2} {\mathrm e}^{-\frac {20}{3}-\frac {2}{3} x^{2}+\frac {4}{3} x}}{x^{2} \left (x -1\right )^{2}}\)	\(51\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((4*x^3-8*x^2-5*x+3)*exp(ln(-1/5*x^2+1/5*x)-1/3*x^2+2/3*x-10/3)^2-3*x+3)/(3*x^3-3*x^2),x,method=_RETURNVER
BOSE)

[Out]

(1-exp(ln(-1/5*x^2+1/5*x)-1/3*x^2+2/3*x-10/3)^2)/x

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maxima [A]  time = 0.75, size = 35, normalized size = 0.92 \begin {gather*} -\frac {1}{25} \, {\left (x^{3} e^{\frac {1}{3}} - 2 \, x^{2} e^{\frac {1}{3}} + x e^{\frac {1}{3}}\right )} e^{\left (-\frac {2}{3} \, x^{2} + \frac {4}{3} \, x - 7\right )} + \frac {1}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^3-8*x^2-5*x+3)*exp(log(-1/5*x^2+1/5*x)-1/3*x^2+2/3*x-10/3)^2-3*x+3)/(3*x^3-3*x^2),x, algorithm
="maxima")

[Out]

-1/25*(x^3*e^(1/3) - 2*x^2*e^(1/3) + x*e^(1/3))*e^(-2/3*x^2 + 4/3*x - 7) + 1/x

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mupad [B]  time = 1.36, size = 50, normalized size = 1.32 \begin {gather*} \frac {2\,x^2\,{\mathrm {e}}^{-\frac {2\,x^2}{3}+\frac {4\,x}{3}-\frac {20}{3}}}{25}-\frac {x^3\,{\mathrm {e}}^{-\frac {2\,x^2}{3}+\frac {4\,x}{3}-\frac {20}{3}}}{25}-\frac {x\,{\mathrm {e}}^{-\frac {2\,x^2}{3}+\frac {4\,x}{3}-\frac {20}{3}}}{25}+\frac {1}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x + exp((4*x)/3 + 2*log(x/5 - x^2/5) - (2*x^2)/3 - 20/3)*(5*x + 8*x^2 - 4*x^3 - 3) - 3)/(3*x^2 - 3*x^3)
,x)

[Out]

(2*x^2*exp((4*x)/3 - (2*x^2)/3 - 20/3))/25 - (x^3*exp((4*x)/3 - (2*x^2)/3 - 20/3))/25 - (x*exp((4*x)/3 - (2*x^
2)/3 - 20/3))/25 + 1/x

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sympy [A]  time = 0.17, size = 31, normalized size = 0.82 \begin {gather*} \frac {\left (- x^{3} + 2 x^{2} - x\right ) e^{- \frac {2 x^{2}}{3} + \frac {4 x}{3} - \frac {20}{3}}}{25} + \frac {1}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x**3-8*x**2-5*x+3)*exp(ln(-1/5*x**2+1/5*x)-1/3*x**2+2/3*x-10/3)**2-3*x+3)/(3*x**3-3*x**2),x)

[Out]

(-x**3 + 2*x**2 - x)*exp(-2*x**2/3 + 4*x/3 - 20/3)/25 + 1/x

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