3.18.6 \(\int \frac {-378+150 e}{-48+24 x-3 x^2+e (16-8 x+x^2)} \, dx\)

Optimal. Leaf size=29 \[ \frac {3 \left (2+3 x-4 \left (-9+\frac {2+x}{3-e}\right )\right )}{4-x} \]

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Rubi [A]  time = 0.02, antiderivative size = 21, normalized size of antiderivative = 0.72, number of steps used = 5, number of rules used = 4, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {12, 1981, 27, 32} \begin {gather*} \frac {6 (63-25 e)}{(3-e) (4-x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-378 + 150*E)/(-48 + 24*x - 3*x^2 + E*(16 - 8*x + x^2)),x]

[Out]

(6*(63 - 25*E))/((3 - E)*(4 - x))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rule 1981

Int[(u_)^(p_), x_Symbol] :> Int[ExpandToSum[u, x]^p, x] /; FreeQ[p, x] && QuadraticQ[u, x] &&  !QuadraticMatch
Q[u, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=-\left ((6 (63-25 e)) \int \frac {1}{-48+24 x-3 x^2+e \left (16-8 x+x^2\right )} \, dx\right )\\ &=-\left ((6 (63-25 e)) \int \frac {1}{-16 (3-e)+8 (3-e) x-(3-e) x^2} \, dx\right )\\ &=-\left ((6 (63-25 e)) \int \frac {1}{(-3+e) (-4+x)^2} \, dx\right )\\ &=\frac {(6 (63-25 e)) \int \frac {1}{(-4+x)^2} \, dx}{3-e}\\ &=\frac {6 (63-25 e)}{(3-e) (4-x)}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 17, normalized size = 0.59 \begin {gather*} -\frac {6 (-63+25 e)}{(-3+e) (-4+x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-378 + 150*E)/(-48 + 24*x - 3*x^2 + E*(16 - 8*x + x^2)),x]

[Out]

(-6*(-63 + 25*E))/((-3 + E)*(-4 + x))

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fricas [A]  time = 0.58, size = 21, normalized size = 0.72 \begin {gather*} -\frac {6 \, {\left (25 \, e - 63\right )}}{{\left (x - 4\right )} e - 3 \, x + 12} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((150*exp(1)-378)/((x^2-8*x+16)*exp(1)-3*x^2+24*x-48),x, algorithm="fricas")

[Out]

-6*(25*e - 63)/((x - 4)*e - 3*x + 12)

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giac [A]  time = 0.21, size = 19, normalized size = 0.66 \begin {gather*} -\frac {6 \, {\left (25 \, e - 63\right )}}{{\left (x - 4\right )} {\left (e - 3\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((150*exp(1)-378)/((x^2-8*x+16)*exp(1)-3*x^2+24*x-48),x, algorithm="giac")

[Out]

-6*(25*e - 63)/((x - 4)*(e - 3))

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maple [A]  time = 0.34, size = 20, normalized size = 0.69

method	result	size
default	\(-\frac {150 \,{\mathrm e}-378}{\left ({\mathrm e}-3\right ) \left (x -4\right )}\)	\(20\)
norman	\(-\frac {6 \left (25 \,{\mathrm e}-63\right )}{\left ({\mathrm e}-3\right ) \left (x -4\right )}\)	\(20\)
gosper	\(-\frac {6 \left (25 \,{\mathrm e}-63\right )}{x \,{\mathrm e}-4 \,{\mathrm e}-3 x +12}\)	\(24\)
meijerg	\(-\frac {189 x}{8 \left ({\mathrm e}-3\right ) \left (-\frac {x}{4}+1\right )}+\frac {75 \,{\mathrm e} x}{8 \left ({\mathrm e}-3\right ) \left (-\frac {x}{4}+1\right )}\)	\(36\)
risch	\(-\frac {150 \,{\mathrm e}}{x \,{\mathrm e}-4 \,{\mathrm e}-3 x +12}+\frac {378}{x \,{\mathrm e}-4 \,{\mathrm e}-3 x +12}\)	\(38\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((150*exp(1)-378)/((x^2-8*x+16)*exp(1)-3*x^2+24*x-48),x,method=_RETURNVERBOSE)

[Out]

-(150*exp(1)-378)/(exp(1)-3)/(x-4)

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maxima [A]  time = 0.40, size = 22, normalized size = 0.76 \begin {gather*} -\frac {6 \, {\left (25 \, e - 63\right )}}{x {\left (e - 3\right )} - 4 \, e + 12} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((150*exp(1)-378)/((x^2-8*x+16)*exp(1)-3*x^2+24*x-48),x, algorithm="maxima")

[Out]

-6*(25*e - 63)/(x*(e - 3) - 4*e + 12)

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mupad [B]  time = 1.09, size = 19, normalized size = 0.66 \begin {gather*} -\frac {6\,\left (25\,\mathrm {e}-63\right )}{\left (\mathrm {e}-3\right )\,\left (x-4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((150*exp(1) - 378)/(24*x + exp(1)*(x^2 - 8*x + 16) - 3*x^2 - 48),x)

[Out]

-(6*(25*exp(1) - 63))/((exp(1) - 3)*(x - 4))

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sympy [A]  time = 0.17, size = 20, normalized size = 0.69 \begin {gather*} - \frac {-378 + 150 e}{x \left (-3 + e\right ) - 4 e + 12} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((150*exp(1)-378)/((x**2-8*x+16)*exp(1)-3*x**2+24*x-48),x)

[Out]

-(-378 + 150*E)/(x*(-3 + E) - 4*E + 12)

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