3.26.10 \(\int \frac {-450+1200 x+115 x^2-2840 x^3+2920 x^4-1080 x^5+135 x^6+e (750-2000 x+750 x^2)}{243 x^2-648 x^3+594 x^4-216 x^5+27 x^6} \, dx\)

Optimal. Leaf size=30 \[ 5 \left (-\frac {10 \left (-1+\frac {5 e}{(1-x) (3-x)}\right )}{27 x}+x\right ) \]

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Rubi [A]  time = 0.12, antiderivative size = 40, normalized size of antiderivative = 1.33, number of steps used = 2, number of rules used = 1, integrand size = 71, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.014, Rules used = {2074} \begin {gather*} 5 x-\frac {125 e}{27 (1-x)}+\frac {125 e}{81 (3-x)}+\frac {50 (3-5 e)}{81 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-450 + 1200*x + 115*x^2 - 2840*x^3 + 2920*x^4 - 1080*x^5 + 135*x^6 + E*(750 - 2000*x + 750*x^2))/(243*x^2
- 648*x^3 + 594*x^4 - 216*x^5 + 27*x^6),x]

[Out]

(-125*E)/(27*(1 - x)) + (125*E)/(81*(3 - x)) + (50*(3 - 5*E))/(81*x) + 5*x

Rule 2074

Int[(P_)^(p_)*(Q_)^(q_.), x_Symbol] :> With[{PP = Factor[P]}, Int[ExpandIntegrand[PP^p*Q^q, x], x] /;  !SumQ[N
onfreeFactors[PP, x]]] /; FreeQ[q, x] && PolyQ[P, x] && PolyQ[Q, x] && IntegerQ[p] && NeQ[P, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (5+\frac {125 e}{81 (-3+x)^2}-\frac {125 e}{27 (-1+x)^2}+\frac {50 (-3+5 e)}{81 x^2}\right ) \, dx\\ &=-\frac {125 e}{27 (1-x)}+\frac {125 e}{81 (3-x)}+\frac {50 (3-5 e)}{81 x}+5 x\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.04, size = 33, normalized size = 1.10 \begin {gather*} \frac {5}{81} \left (\frac {30-50 e}{x}+81 x+\frac {50 e (-4+x)}{3-4 x+x^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-450 + 1200*x + 115*x^2 - 2840*x^3 + 2920*x^4 - 1080*x^5 + 135*x^6 + E*(750 - 2000*x + 750*x^2))/(2
43*x^2 - 648*x^3 + 594*x^4 - 216*x^5 + 27*x^6),x]

[Out]

(5*((30 - 50*E)/x + 81*x + (50*E*(-4 + x))/(3 - 4*x + x^2)))/81

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fricas [A]  time = 0.95, size = 40, normalized size = 1.33 \begin {gather*} \frac {5 \, {\left (27 \, x^{4} - 108 \, x^{3} + 91 \, x^{2} - 40 \, x - 50 \, e + 30\right )}}{27 \, {\left (x^{3} - 4 \, x^{2} + 3 \, x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((750*x^2-2000*x+750)*exp(1)+135*x^6-1080*x^5+2920*x^4-2840*x^3+115*x^2+1200*x-450)/(27*x^6-216*x^5+
594*x^4-648*x^3+243*x^2),x, algorithm="fricas")

[Out]

5/27*(27*x^4 - 108*x^3 + 91*x^2 - 40*x - 50*e + 30)/(x^3 - 4*x^2 + 3*x)

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giac [A]  time = 0.20, size = 32, normalized size = 1.07 \begin {gather*} 5 \, x + \frac {50 \, {\left (x^{2} - 4 \, x - 5 \, e + 3\right )}}{27 \, {\left (x^{3} - 4 \, x^{2} + 3 \, x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((750*x^2-2000*x+750)*exp(1)+135*x^6-1080*x^5+2920*x^4-2840*x^3+115*x^2+1200*x-450)/(27*x^6-216*x^5+
594*x^4-648*x^3+243*x^2),x, algorithm="giac")

[Out]

5*x + 50/27*(x^2 - 4*x - 5*e + 3)/(x^3 - 4*x^2 + 3*x)

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maple [A]  time = 0.06, size = 33, normalized size = 1.10

method	result	size
risch	\(5 x +\frac {\frac {50 x^{2}}{27}-\frac {200 x}{27}+\frac {50}{9}-\frac {250 \,{\mathrm e}}{27}}{x \left (x^{2}-4 x +3\right )}\)	\(33\)
default	\(5 x +\frac {125 \,{\mathrm e}}{27 \left (x -1\right )}-\frac {5 \left (\frac {50 \,{\mathrm e}}{3}-10\right )}{27 x}-\frac {125 \,{\mathrm e}}{81 \left (x -3\right )}\)	\(34\)
norman	\(\frac {-\frac {1705 x^{2}}{27}+\frac {1420 x}{27}+5 x^{4}+\frac {50}{9}-\frac {250 \,{\mathrm e}}{27}}{x \left (x^{2}-4 x +3\right )}\)	\(34\)
gosper	\(-\frac {5 \left (-27 x^{4}+341 x^{2}+50 \,{\mathrm e}-284 x -30\right )}{27 x \left (x^{2}-4 x +3\right )}\)	\(35\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((750*x^2-2000*x+750)*exp(1)+135*x^6-1080*x^5+2920*x^4-2840*x^3+115*x^2+1200*x-450)/(27*x^6-216*x^5+594*x^
4-648*x^3+243*x^2),x,method=_RETURNVERBOSE)

[Out]

5*x+(50/27*x^2-200/27*x+50/9-250/27*exp(1))/x/(x^2-4*x+3)

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maxima [A]  time = 0.44, size = 32, normalized size = 1.07 \begin {gather*} 5 \, x + \frac {50 \, {\left (x^{2} - 4 \, x - 5 \, e + 3\right )}}{27 \, {\left (x^{3} - 4 \, x^{2} + 3 \, x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((750*x^2-2000*x+750)*exp(1)+135*x^6-1080*x^5+2920*x^4-2840*x^3+115*x^2+1200*x-450)/(27*x^6-216*x^5+
594*x^4-648*x^3+243*x^2),x, algorithm="maxima")

[Out]

5*x + 50/27*(x^2 - 4*x - 5*e + 3)/(x^3 - 4*x^2 + 3*x)

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mupad [B]  time = 1.40, size = 33, normalized size = 1.10 \begin {gather*} 5\,x-\frac {-\frac {50\,x^2}{27}+\frac {200\,x}{27}+\frac {250\,\mathrm {e}}{27}-\frac {50}{9}}{x\,\left (x^2-4\,x+3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1200*x + exp(1)*(750*x^2 - 2000*x + 750) + 115*x^2 - 2840*x^3 + 2920*x^4 - 1080*x^5 + 135*x^6 - 450)/(243
*x^2 - 648*x^3 + 594*x^4 - 216*x^5 + 27*x^6),x)

[Out]

5*x - ((200*x)/27 + (250*exp(1))/27 - (50*x^2)/27 - 50/9)/(x*(x^2 - 4*x + 3))

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sympy [A]  time = 0.60, size = 31, normalized size = 1.03 \begin {gather*} 5 x + \frac {50 x^{2} - 200 x - 250 e + 150}{27 x^{3} - 108 x^{2} + 81 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((750*x**2-2000*x+750)*exp(1)+135*x**6-1080*x**5+2920*x**4-2840*x**3+115*x**2+1200*x-450)/(27*x**6-2
16*x**5+594*x**4-648*x**3+243*x**2),x)

[Out]

5*x + (50*x**2 - 200*x - 250*E + 150)/(27*x**3 - 108*x**2 + 81*x)

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