3.50.81 \(\int \frac {-240 e^4-360 x+240 x^2}{9-24 x+16 x^2} \, dx\)

Optimal. Leaf size=16 \[ \frac {15 \left (e^4+x^2\right )}{-\frac {3}{4}+x} \]

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Rubi [A]  time = 0.02, antiderivative size = 22, normalized size of antiderivative = 1.38, number of steps used = 3, number of rules used = 2, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {27, 683} \begin {gather*} 15 x-\frac {15 \left (9+16 e^4\right )}{4 (3-4 x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-240*E^4 - 360*x + 240*x^2)/(9 - 24*x + 16*x^2),x]

[Out]

(-15*(9 + 16*E^4))/(4*(3 - 4*x)) + 15*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 683

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[2*c*d - b*e,
0] && IGtQ[p, 0] &&  !(EqQ[m, 3] && NeQ[p, 1])

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-240 e^4-360 x+240 x^2}{(-3+4 x)^2} \, dx\\ &=\int \left (15-\frac {15 \left (9+16 e^4\right )}{(-3+4 x)^2}\right ) \, dx\\ &=-\frac {15 \left (9+16 e^4\right )}{4 (3-4 x)}+15 x\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 24, normalized size = 1.50 \begin {gather*} -\frac {120 \left (9+8 e^4-12 x+8 x^2\right )}{48-64 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-240*E^4 - 360*x + 240*x^2)/(9 - 24*x + 16*x^2),x]

[Out]

(-120*(9 + 8*E^4 - 12*x + 8*x^2))/(48 - 64*x)

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fricas [A]  time = 0.75, size = 23, normalized size = 1.44 \begin {gather*} \frac {15 \, {\left (16 \, x^{2} - 12 \, x + 16 \, e^{4} + 9\right )}}{4 \, {\left (4 \, x - 3\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-240*exp(4)+240*x^2-360*x)/(16*x^2-24*x+9),x, algorithm="fricas")

[Out]

15/4*(16*x^2 - 12*x + 16*e^4 + 9)/(4*x - 3)

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giac [A]  time = 0.13, size = 19, normalized size = 1.19 \begin {gather*} 15 \, x + \frac {15 \, {\left (16 \, e^{4} + 9\right )}}{4 \, {\left (4 \, x - 3\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-240*exp(4)+240*x^2-360*x)/(16*x^2-24*x+9),x, algorithm="giac")

[Out]

15*x + 15/4*(16*e^4 + 9)/(4*x - 3)

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maple [A]  time = 0.19, size = 16, normalized size = 1.00

method	result	size
gosper	\(\frac {60 x^{2}+60 \,{\mathrm e}^{4}}{4 x -3}\)	\(16\)
norman	\(\frac {60 x^{2}+60 \,{\mathrm e}^{4}}{4 x -3}\)	\(19\)
default	\(15 x -\frac {120 \left (-\frac {9}{32}-\frac {{\mathrm e}^{4}}{2}\right )}{4 x -3}\)	\(20\)
risch	\(15 x +\frac {135}{16 \left (x -\frac {3}{4}\right )}+\frac {15 \,{\mathrm e}^{4}}{x -\frac {3}{4}}\)	\(21\)
meijerg	\(-\frac {80 \,{\mathrm e}^{4} x}{3 \left (1-\frac {4 x}{3}\right )}+\frac {5 x \left (6-4 x \right )}{1-\frac {4 x}{3}}-\frac {30 x}{1-\frac {4 x}{3}}\)	\(39\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-240*exp(4)+240*x^2-360*x)/(16*x^2-24*x+9),x,method=_RETURNVERBOSE)

[Out]

60*(exp(4)+x^2)/(4*x-3)

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maxima [A]  time = 0.35, size = 19, normalized size = 1.19 \begin {gather*} 15 \, x + \frac {15 \, {\left (16 \, e^{4} + 9\right )}}{4 \, {\left (4 \, x - 3\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-240*exp(4)+240*x^2-360*x)/(16*x^2-24*x+9),x, algorithm="maxima")

[Out]

15*x + 15/4*(16*e^4 + 9)/(4*x - 3)

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mupad [B]  time = 3.16, size = 18, normalized size = 1.12 \begin {gather*} 15\,x+\frac {60\,{\mathrm {e}}^4+\frac {135}{4}}{4\,x-3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(360*x + 240*exp(4) - 240*x^2)/(16*x^2 - 24*x + 9),x)

[Out]

15*x + (60*exp(4) + 135/4)/(4*x - 3)

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sympy [A]  time = 0.12, size = 14, normalized size = 0.88 \begin {gather*} 15 x + \frac {135 + 240 e^{4}}{16 x - 12} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-240*exp(4)+240*x**2-360*x)/(16*x**2-24*x+9),x)

[Out]

15*x + (135 + 240*exp(4))/(16*x - 12)

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