3.79.15 \(\int \frac {1}{10} (-9+54 x-108 x^2+e (-18+48 x)) \, dx\)

Optimal. Leaf size=27 \[ \frac {3}{10} \left ((1-2 x)^2+x\right ) \left (-x+\log \left (e^{2 e-2 x}\right )\right ) \]

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Rubi [A]  time = 0.01, antiderivative size = 32, normalized size of antiderivative = 1.19, number of steps used = 2, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {12} \begin {gather*} -\frac {18 x^3}{5}+\frac {27 x^2}{10}-\frac {9 x}{10}+\frac {3}{80} e (3-8 x)^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-9 + 54*x - 108*x^2 + E*(-18 + 48*x))/10,x]

[Out]

(3*E*(3 - 8*x)^2)/80 - (9*x)/10 + (27*x^2)/10 - (18*x^3)/5

Rule 12

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{10} \int \left (-9+54 x-108 x^2+e (-18+48 x)\right ) \, dx\\ &=\frac {3}{80} e (3-8 x)^2-\frac {9 x}{10}+\frac {27 x^2}{10}-\frac {18 x^3}{5}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 28, normalized size = 1.04 \begin {gather*} \frac {3}{10} \left (-3 x-6 e x+9 x^2+8 e x^2-12 x^3\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-9 + 54*x - 108*x^2 + E*(-18 + 48*x))/10,x]

[Out]

(3*(-3*x - 6*E*x + 9*x^2 + 8*E*x^2 - 12*x^3))/10

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fricas [A]  time = 0.64, size = 27, normalized size = 1.00 \begin {gather*} -\frac {18}{5} \, x^{3} + \frac {27}{10} \, x^{2} + \frac {3}{5} \, {\left (4 \, x^{2} - 3 \, x\right )} e - \frac {9}{10} \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/10*(48*x-18)*exp(1)-54/5*x^2+27/5*x-9/10,x, algorithm="fricas")

[Out]

-18/5*x^3 + 27/10*x^2 + 3/5*(4*x^2 - 3*x)*e - 9/10*x

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giac [A]  time = 0.22, size = 27, normalized size = 1.00 \begin {gather*} -\frac {18}{5} \, x^{3} + \frac {27}{10} \, x^{2} + \frac {3}{5} \, {\left (4 \, x^{2} - 3 \, x\right )} e - \frac {9}{10} \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/10*(48*x-18)*exp(1)-54/5*x^2+27/5*x-9/10,x, algorithm="giac")

[Out]

-18/5*x^3 + 27/10*x^2 + 3/5*(4*x^2 - 3*x)*e - 9/10*x

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maple [A]  time = 0.02, size = 23, normalized size = 0.85

method	result	size
gosper	\(\frac {3 x \left (8 x \,{\mathrm e}-12 x^{2}-6 \,{\mathrm e}+9 x -3\right )}{10}\)	\(23\)
norman	\(\left (-\frac {9 \,{\mathrm e}}{5}-\frac {9}{10}\right ) x +\left (\frac {12 \,{\mathrm e}}{5}+\frac {27}{10}\right ) x^{2}-\frac {18 x^{3}}{5}\)	\(25\)
risch	\(\frac {12 x^{2} {\mathrm e}}{5}-\frac {9 x \,{\mathrm e}}{5}-\frac {18 x^{3}}{5}+\frac {27 x^{2}}{10}-\frac {9 x}{10}\)	\(27\)
default	\(\frac {{\mathrm e} \left (24 x^{2}-18 x \right )}{10}-\frac {18 x^{3}}{5}+\frac {27 x^{2}}{10}-\frac {9 x}{10}\)	\(28\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/10*(48*x-18)*exp(1)-54/5*x^2+27/5*x-9/10,x,method=_RETURNVERBOSE)

[Out]

3/10*x*(8*x*exp(1)-12*x^2-6*exp(1)+9*x-3)

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maxima [A]  time = 0.37, size = 27, normalized size = 1.00 \begin {gather*} -\frac {18}{5} \, x^{3} + \frac {27}{10} \, x^{2} + \frac {3}{5} \, {\left (4 \, x^{2} - 3 \, x\right )} e - \frac {9}{10} \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/10*(48*x-18)*exp(1)-54/5*x^2+27/5*x-9/10,x, algorithm="maxima")

[Out]

-18/5*x^3 + 27/10*x^2 + 3/5*(4*x^2 - 3*x)*e - 9/10*x

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mupad [B]  time = 5.35, size = 25, normalized size = 0.93 \begin {gather*} -\frac {18\,x^3}{5}+\left (\frac {12\,\mathrm {e}}{5}+\frac {27}{10}\right )\,x^2+\left (-\frac {9\,\mathrm {e}}{5}-\frac {9}{10}\right )\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((27*x)/5 - (54*x^2)/5 + (exp(1)*(48*x - 18))/10 - 9/10,x)

[Out]

x^2*((12*exp(1))/5 + 27/10) - (18*x^3)/5 - x*((9*exp(1))/5 + 9/10)

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sympy [A]  time = 0.06, size = 32, normalized size = 1.19 \begin {gather*} - \frac {18 x^{3}}{5} + x^{2} \left (\frac {27}{10} + \frac {12 e}{5}\right ) + x \left (- \frac {9 e}{5} - \frac {9}{10}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/10*(48*x-18)*exp(1)-54/5*x**2+27/5*x-9/10,x)

[Out]

-18*x**3/5 + x**2*(27/10 + 12*E/5) + x*(-9*E/5 - 9/10)

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