3.7.15 \(\int \frac {-6-12 x-3 x^2+e^4 (6+10 x)}{-6+6 e^4} \, dx\)

Optimal. Leaf size=29 \[ \frac {1}{6} \left (x+\frac {1}{4} x (4+4 x) \left (5+\frac {x}{1-e^4}\right )\right ) \]

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Rubi [B]  time = 0.01, antiderivative size = 64, normalized size of antiderivative = 2.21, number of steps used = 2, number of rules used = 1, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.034, Rules used = {12} \begin {gather*} \frac {x^3}{6 \left (1-e^4\right )}+\frac {x^2}{1-e^4}+\frac {x}{1-e^4}-\frac {e^4 (5 x+3)^2}{30 \left (1-e^4\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-6 - 12*x - 3*x^2 + E^4*(6 + 10*x))/(-6 + 6*E^4),x]

[Out]

x/(1 - E^4) + x^2/(1 - E^4) + x^3/(6*(1 - E^4)) - (E^4*(3 + 5*x)^2)/(30*(1 - E^4))

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 {\int \left (-6-12 x-3 x^2+e^4 (6+10 x)\right ) \, dx}{6 \left (1-e^4\right )}\\ &=\frac {x}{1-e^4}+\frac {x^2}{1-e^4}+\frac {x^3}{6 \left (1-e^4\right )}-\frac {e^4 (3+5 x)^2}{30 \left (1-e^4\right )}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 39, normalized size = 1.34 \begin {gather*} \frac {-6 x+6 e^4 x-6 x^2+5 e^4 x^2-x^3}{6 \left (-1+e^4\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-6 - 12*x - 3*x^2 + E^4*(6 + 10*x))/(-6 + 6*E^4),x]

[Out]

(-6*x + 6*E^4*x - 6*x^2 + 5*E^4*x^2 - x^3)/(6*(-1 + E^4))

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fricas [A]  time = 0.71, size = 33, normalized size = 1.14 \begin {gather*} -\frac {x^{3} + 6 \, x^{2} - {\left (5 \, x^{2} + 6 \, x\right )} e^{4} + 6 \, x}{6 \, {\left (e^{4} - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((10*x+6)*exp(4)-3*x^2-12*x-6)/(6*exp(4)-6),x, algorithm="fricas")

[Out]

-1/6*(x^3 + 6*x^2 - (5*x^2 + 6*x)*e^4 + 6*x)/(e^4 - 1)

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giac [A]  time = 0.40, size = 33, normalized size = 1.14 \begin {gather*} -\frac {x^{3} + 6 \, x^{2} - {\left (5 \, x^{2} + 6 \, x\right )} e^{4} + 6 \, x}{6 \, {\left (e^{4} - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((10*x+6)*exp(4)-3*x^2-12*x-6)/(6*exp(4)-6),x, algorithm="giac")

[Out]

-1/6*(x^3 + 6*x^2 - (5*x^2 + 6*x)*e^4 + 6*x)/(e^4 - 1)

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maple [A]  time = 0.03, size = 29, normalized size = 1.00

method	result	size
gosper	\(\frac {x \left (5 x \,{\mathrm e}^{4}-x^{2}+6 \,{\mathrm e}^{4}-6 x -6\right )}{6 \,{\mathrm e}^{4}-6}\)	\(29\)
norman	\(x -\frac {x^{3}}{6 \left ({\mathrm e}^{4}-1\right )}+\frac {\left (5 \,{\mathrm e}^{4}-6\right ) x^{2}}{6 \,{\mathrm e}^{4}-6}\)	\(31\)
default	\(\frac {{\mathrm e}^{4} \left (5 x^{2}+6 x \right )-x^{3}-6 x^{2}-6 x}{6 \,{\mathrm e}^{4}-6}\)	\(36\)
risch	\(\frac {5 x^{2} {\mathrm e}^{4}}{6 \,{\mathrm e}^{4}-6}-\frac {x^{3}}{6 \,{\mathrm e}^{4}-6}+\frac {6 x \,{\mathrm e}^{4}}{6 \,{\mathrm e}^{4}-6}-\frac {6 x^{2}}{6 \,{\mathrm e}^{4}-6}-\frac {6 x}{6 \,{\mathrm e}^{4}-6}\)	\(67\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((10*x+6)*exp(4)-3*x^2-12*x-6)/(6*exp(4)-6),x,method=_RETURNVERBOSE)

[Out]

1/6*x*(5*x*exp(4)-x^2+6*exp(4)-6*x-6)/(exp(4)-1)

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maxima [A]  time = 0.44, size = 33, normalized size = 1.14 \begin {gather*} -\frac {x^{3} + 6 \, x^{2} - {\left (5 \, x^{2} + 6 \, x\right )} e^{4} + 6 \, x}{6 \, {\left (e^{4} - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((10*x+6)*exp(4)-3*x^2-12*x-6)/(6*exp(4)-6),x, algorithm="maxima")

[Out]

-1/6*(x^3 + 6*x^2 - (5*x^2 + 6*x)*e^4 + 6*x)/(e^4 - 1)

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mupad [B]  time = 0.51, size = 33, normalized size = 1.14 \begin {gather*} -\frac {x^3}{6\,{\mathrm {e}}^4-6}+\frac {\left (10\,{\mathrm {e}}^4-12\right )\,x^2}{2\,\left (6\,{\mathrm {e}}^4-6\right )}+x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(12*x + 3*x^2 - exp(4)*(10*x + 6) + 6)/(6*exp(4) - 6),x)

[Out]

x - x^3/(6*exp(4) - 6) + (x^2*(10*exp(4) - 12))/(2*(6*exp(4) - 6))

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sympy [A]  time = 0.06, size = 27, normalized size = 0.93 \begin {gather*} - \frac {x^{3}}{-6 + 6 e^{4}} + \frac {x^{2} \left (-6 + 5 e^{4}\right )}{-6 + 6 e^{4}} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((10*x+6)*exp(4)-3*x**2-12*x-6)/(6*exp(4)-6),x)

[Out]

-x**3/(-6 + 6*exp(4)) + x**2*(-6 + 5*exp(4))/(-6 + 6*exp(4)) + x

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