3.59.81 \(\int \frac {-128 x+e (196608-1536 x+3 x^2)}{65536 x-512 x^2+x^3} \, dx\)

Optimal. Leaf size=23 \[ -4-\frac {x}{2 (256-x)}-\log (5)+3 e \log (x) \]

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Rubi [A]  time = 0.04, antiderivative size = 15, normalized size of antiderivative = 0.65, number of steps used = 4, number of rules used = 3, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {1594, 27, 1820} \begin {gather*} 3 e \log (x)-\frac {128}{256-x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-128*x + E*(196608 - 1536*x + 3*x^2))/(65536*x - 512*x^2 + x^3),x]

[Out]

-128/(256 - x) + 3*E*Log[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 1594

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

Rule 1820

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*Pq*(a +
b*x^n)^p, x], x] /; FreeQ[{a, b, c, m, n}, x] && PolyQ[Pq, x] && (IGtQ[p, 0] || EqQ[n, 1])

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-128 x+e \left (196608-1536 x+3 x^2\right )}{x \left (65536-512 x+x^2\right )} \, dx\\ &=\int \frac {-128 x+e \left (196608-1536 x+3 x^2\right )}{(-256+x)^2 x} \, dx\\ &=\int \left (-\frac {128}{(-256+x)^2}+\frac {3 e}{x}\right ) \, dx\\ &=-\frac {128}{256-x}+3 e \log (x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 13, normalized size = 0.57 \begin {gather*} \frac {128}{-256+x}+3 e \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-128*x + E*(196608 - 1536*x + 3*x^2))/(65536*x - 512*x^2 + x^3),x]

[Out]

128/(-256 + x) + 3*E*Log[x]

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fricas [A]  time = 0.57, size = 17, normalized size = 0.74 \begin {gather*} \frac {3 \, {\left (x - 256\right )} e \log \relax (x) + 128}{x - 256} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x^2-1536*x+196608)*exp(1)-128*x)/(x^3-512*x^2+65536*x),x, algorithm="fricas")

[Out]

(3*(x - 256)*e*log(x) + 128)/(x - 256)

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giac [A]  time = 0.14, size = 15, normalized size = 0.65 \begin {gather*} 3 \, e \log \left ({\left | x \right |}\right ) + \frac {128}{x - 256} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x^2-1536*x+196608)*exp(1)-128*x)/(x^3-512*x^2+65536*x),x, algorithm="giac")

[Out]

3*e*log(abs(x)) + 128/(x - 256)

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maple [A]  time = 0.05, size = 15, normalized size = 0.65

method	result	size
default	\(\frac {128}{x -256}+3 \,{\mathrm e} \ln \relax (x )\)	\(15\)
norman	\(\frac {128}{x -256}+3 \,{\mathrm e} \ln \relax (x )\)	\(15\)
risch	\(\frac {128}{x -256}+3 \,{\mathrm e} \ln \left (-x \right )\)	\(17\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((3*x^2-1536*x+196608)*exp(1)-128*x)/(x^3-512*x^2+65536*x),x,method=_RETURNVERBOSE)

[Out]

128/(x-256)+3*exp(1)*ln(x)

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maxima [A]  time = 0.36, size = 14, normalized size = 0.61 \begin {gather*} 3 \, e \log \relax (x) + \frac {128}{x - 256} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x^2-1536*x+196608)*exp(1)-128*x)/(x^3-512*x^2+65536*x),x, algorithm="maxima")

[Out]

3*e*log(x) + 128/(x - 256)

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mupad [B]  time = 0.09, size = 14, normalized size = 0.61 \begin {gather*} \frac {128}{x-256}+3\,\mathrm {e}\,\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(128*x - exp(1)*(3*x^2 - 1536*x + 196608))/(65536*x - 512*x^2 + x^3),x)

[Out]

128/(x - 256) + 3*exp(1)*log(x)

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sympy [A]  time = 0.16, size = 12, normalized size = 0.52 \begin {gather*} 3 e \log {\relax (x )} + \frac {128}{x - 256} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x**2-1536*x+196608)*exp(1)-128*x)/(x**3-512*x**2+65536*x),x)

[Out]

3*E*log(x) + 128/(x - 256)

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