3.14.93 \(\int \frac {-1-18 x+2 e^2 x-3 x^2}{1-x-9 x^2+e^2 x^2-x^3} \, dx\)

Optimal. Leaf size=16 \[ \log \left (-1+x+x^2 \left (9-e^2+x\right )\right ) \]

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Rubi [A]  time = 0.03, antiderivative size = 23, normalized size of antiderivative = 1.44, number of steps used = 3, number of rules used = 2, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.049, Rules used = {6, 1587} \begin {gather*} \log \left (-x^3-\left (9-e^2\right ) x^2-x+1\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-1 - 18*x + 2*E^2*x - 3*x^2)/(1 - x - 9*x^2 + E^2*x^2 - x^3),x]

[Out]

Log[1 - x - (9 - E^2)*x^2 - x^3]

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, x]

Rule 1587

Int[(Pp_)/(Qq_), x_Symbol] :> With[{p = Expon[Pp, x], q = Expon[Qq, x]}, Simp[(Coeff[Pp, x, p]*Log[RemoveConte
nt[Qq, x]])/(q*Coeff[Qq, x, q]), x] /; EqQ[p, q - 1] && EqQ[Pp, Simplify[(Coeff[Pp, x, p]*D[Qq, x])/(q*Coeff[Q
q, x, q])]]] /; PolyQ[Pp, x] && PolyQ[Qq, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-1+\left (-18+2 e^2\right ) x-3 x^2}{1-x-9 x^2+e^2 x^2-x^3} \, dx\\ &=\int \frac {-1+\left (-18+2 e^2\right ) x-3 x^2}{1-x+\left (-9+e^2\right ) x^2-x^3} \, dx\\ &=\log \left (1-x-\left (9-e^2\right ) x^2-x^3\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 20, normalized size = 1.25 \begin {gather*} \log \left (1-x+\left (-9+e^2\right ) x^2-x^3\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-1 - 18*x + 2*E^2*x - 3*x^2)/(1 - x - 9*x^2 + E^2*x^2 - x^3),x]

[Out]

Log[1 - x + (-9 + E^2)*x^2 - x^3]

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fricas [A]  time = 0.66, size = 19, normalized size = 1.19 \begin {gather*} \log \left (x^{3} - x^{2} e^{2} + 9 \, x^{2} + x - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*exp(2)*x-3*x^2-18*x-1)/(x^2*exp(2)-x^3-9*x^2-x+1),x, algorithm="fricas")

[Out]

log(x^3 - x^2*e^2 + 9*x^2 + x - 1)

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giac [A]  time = 0.31, size = 20, normalized size = 1.25 \begin {gather*} \log \left ({\left | x^{3} - x^{2} e^{2} + 9 \, x^{2} + x - 1 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*exp(2)*x-3*x^2-18*x-1)/(x^2*exp(2)-x^3-9*x^2-x+1),x, algorithm="giac")

[Out]

log(abs(x^3 - x^2*e^2 + 9*x^2 + x - 1))

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maple [A]  time = 0.05, size = 18, normalized size = 1.12

method	result	size
risch	\(\ln \left (x^{3}+\left (-{\mathrm e}^{2}+9\right ) x^{2}+x -1\right )\)	\(18\)
derivativedivides	\(\ln \left (x^{2} {\mathrm e}^{2}-x^{3}-9 x^{2}-x +1\right )\)	\(23\)
default	\(\ln \left (x^{2} {\mathrm e}^{2}-x^{3}-9 x^{2}-x +1\right )\)	\(23\)
norman	\(\ln \left (x^{2} {\mathrm e}^{2}-x^{3}-9 x^{2}-x +1\right )\)	\(23\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*exp(2)*x-3*x^2-18*x-1)/(x^2*exp(2)-x^3-9*x^2-x+1),x,method=_RETURNVERBOSE)

[Out]

ln(x^3+(-exp(2)+9)*x^2+x-1)

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maxima [A]  time = 0.77, size = 19, normalized size = 1.19 \begin {gather*} \log \left (x^{3} - x^{2} e^{2} + 9 \, x^{2} + x - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*exp(2)*x-3*x^2-18*x-1)/(x^2*exp(2)-x^3-9*x^2-x+1),x, algorithm="maxima")

[Out]

log(x^3 - x^2*e^2 + 9*x^2 + x - 1)

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mupad [B]  time = 1.00, size = 16, normalized size = 1.00 \begin {gather*} \ln \left (x^3+\left (9-{\mathrm {e}}^2\right )\,x^2+x-1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((18*x - 2*x*exp(2) + 3*x^2 + 1)/(x - x^2*exp(2) + 9*x^2 + x^3 - 1),x)

[Out]

log(x + x^3 - x^2*(exp(2) - 9) - 1)

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sympy [A]  time = 0.29, size = 15, normalized size = 0.94 \begin {gather*} \log {\left (x^{3} + x^{2} \left (9 - e^{2}\right ) + x - 1 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*exp(2)*x-3*x**2-18*x-1)/(x**2*exp(2)-x**3-9*x**2-x+1),x)

[Out]

log(x**3 + x**2*(9 - exp(2)) + x - 1)

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