3.85.86 \(\int \frac {-52+e^{3 x/4} (-16+3 x)}{64-32 x+4 x^2} \, dx\)

Optimal. Leaf size=15 \[ \frac {13+e^{3 x/4}}{-4+x} \]

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Rubi [A]  time = 0.10, antiderivative size = 26, normalized size of antiderivative = 1.73, 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 = {27, 12, 6742, 2197} \begin {gather*} -\frac {e^{3 x/4}}{4-x}-\frac {13}{4-x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-52 + E^((3*x)/4)*(-16 + 3*x))/(64 - 32*x + 4*x^2),x]

[Out]

-13/(4 - x) - E^((3*x)/4)/(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 2197

Int[(F_)^((c_.)*(v_))*(u_)^(m_.)*(w_), x_Symbol] :> With[{b = Coefficient[v, x, 1], d = Coefficient[u, x, 0],
e = Coefficient[u, x, 1], f = Coefficient[w, x, 0], g = Coefficient[w, x, 1]}, Simp[(g*u^(m + 1)*F^(c*v))/(b*c
*e*Log[F]), x] /; EqQ[e*g*(m + 1) - b*c*(e*f - d*g)*Log[F], 0]] /; FreeQ[{F, c, m}, x] && LinearQ[{u, v, w}, x
]

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 {-52+e^{3 x/4} (-16+3 x)}{4 (-4+x)^2} \, dx\\ &=\frac {1}{4} \int \frac {-52+e^{3 x/4} (-16+3 x)}{(-4+x)^2} \, dx\\ &=\frac {1}{4} \int \left (-\frac {52}{(-4+x)^2}+\frac {e^{3 x/4} (-16+3 x)}{(-4+x)^2}\right ) \, dx\\ &=-\frac {13}{4-x}+\frac {1}{4} \int \frac {e^{3 x/4} (-16+3 x)}{(-4+x)^2} \, dx\\ &=-\frac {13}{4-x}-\frac {e^{3 x/4}}{4-x}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.06, size = 15, normalized size = 1.00 \begin {gather*} \frac {13+e^{3 x/4}}{-4+x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-52 + E^((3*x)/4)*(-16 + 3*x))/(64 - 32*x + 4*x^2),x]

[Out]

(13 + E^((3*x)/4))/(-4 + x)

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fricas [A]  time = 1.27, size = 12, normalized size = 0.80 \begin {gather*} \frac {e^{\left (\frac {3}{4} \, x\right )} + 13}{x - 4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x-16)*exp(3/4*x)-52)/(4*x^2-32*x+64),x, algorithm="fricas")

[Out]

(e^(3/4*x) + 13)/(x - 4)

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giac [A]  time = 0.14, size = 12, normalized size = 0.80 \begin {gather*} \frac {e^{\left (\frac {3}{4} \, x\right )} + 13}{x - 4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x-16)*exp(3/4*x)-52)/(4*x^2-32*x+64),x, algorithm="giac")

[Out]

(e^(3/4*x) + 13)/(x - 4)

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maple [A]  time = 0.42, size = 13, normalized size = 0.87

method	result	size
norman	\(\frac {{\mathrm e}^{\frac {3 x}{4}}+13}{x -4}\)	\(13\)
risch	\(\frac {13}{x -4}+\frac {{\mathrm e}^{\frac {3 x}{4}}}{x -4}\)	\(19\)
derivativedivides	\(\frac {39}{4 \left (\frac {3 x}{4}-3\right )}+\frac {3 \,{\mathrm e}^{\frac {3 x}{4}}}{4 \left (\frac {3 x}{4}-3\right )}\)	\(24\)
default	\(\frac {39}{4 \left (\frac {3 x}{4}-3\right )}+\frac {3 \,{\mathrm e}^{\frac {3 x}{4}}}{4 \left (\frac {3 x}{4}-3\right )}\)	\(24\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((3*x-16)*exp(3/4*x)-52)/(4*x^2-32*x+64),x,method=_RETURNVERBOSE)

[Out]

(exp(3/4*x)+13)/(x-4)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {x e^{\left (\frac {3}{4} \, x\right )}}{x^{2} - 8 \, x + 16} + \frac {4 \, e^{3} E_{2}\left (-\frac {3}{4} \, x + 3\right )}{x - 4} + \frac {13}{x - 4} + \frac {3}{4} \, \int \frac {4 \, {\left (x + 4\right )} e^{\left (\frac {3}{4} \, x\right )}}{3 \, {\left (x^{3} - 12 \, x^{2} + 48 \, x - 64\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x-16)*exp(3/4*x)-52)/(4*x^2-32*x+64),x, algorithm="maxima")

[Out]

x*e^(3/4*x)/(x^2 - 8*x + 16) + 4*e^3*exp_integral_e(2, -3/4*x + 3)/(x - 4) + 13/(x - 4) + 3/4*integrate(4/3*(x
+ 4)*e^(3/4*x)/(x^3 - 12*x^2 + 48*x - 64), x)

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mupad [B]  time = 0.08, size = 12, normalized size = 0.80 \begin {gather*} \frac {{\mathrm {e}}^{\frac {3\,x}{4}}+13}{x-4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp((3*x)/4)*(3*x - 16) - 52)/(4*x^2 - 32*x + 64),x)

[Out]

(exp((3*x)/4) + 13)/(x - 4)

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sympy [A]  time = 0.13, size = 14, normalized size = 0.93 \begin {gather*} \frac {e^{\frac {3 x}{4}}}{x - 4} + \frac {13}{x - 4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x-16)*exp(3/4*x)-52)/(4*x**2-32*x+64),x)

[Out]

exp(3*x/4)/(x - 4) + 13/(x - 4)

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