3.95.37 \(\int (1682 x+2 e^4 x+1044 x^2+144 x^3+e^2 (-116 x-36 x^2)) \, dx\) [9437]

Optimal. Leaf size=19 \[ 2+\left (x-x \left (e^2+x-7 (4+x)\right )\right )^2 \]

[Out]

2+(x-x*(-6*x-28+exp(2)))^2

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Rubi [A]
time = 0.01, antiderivative size = 36, normalized size of antiderivative = 1.89, number of steps used = 3, number of rules used = 1, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.030, Rules used = {6} \begin {gather*} 36 x^4-12 e^2 x^3+348 x^3+\left (841+e^4\right ) x^2-58 e^2 x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1682*x + 2*E^4*x + 1044*x^2 + 144*x^3 + E^2*(-116*x - 36*x^2),x]

[Out]

-58*E^2*x^2 + (841 + E^4)*x^2 + 348*x^3 - 12*E^2*x^3 + 36*x^4

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\left (1682+2 e^4\right ) x+1044 x^2+144 x^3+e^2 \left (-116 x-36 x^2\right )\right ) \, dx\\ &=\left (841+e^4\right ) x^2+348 x^3+36 x^4+e^2 \int \left (-116 x-36 x^2\right ) \, dx\\ &=-58 e^2 x^2+\left (841+e^4\right ) x^2+348 x^3-12 e^2 x^3+36 x^4\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.01, size = 14, normalized size = 0.74 \begin {gather*} \left (-29+e^2-6 x\right )^2 x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1682*x + 2*E^4*x + 1044*x^2 + 144*x^3 + E^2*(-116*x - 36*x^2),x]

[Out]

(-29 + E^2 - 6*x)^2*x^2

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Maple [A]
time = 0.42, size = 28, normalized size = 1.47

method	result	size
gosper	\(x^{2} \left ({\mathrm e}^{2}-6 x -29\right )^{2}\)	\(14\)
default	\(36 x^{4}+\frac {2 \left (-18 \,{\mathrm e}^{2}+522\right ) x^{3}}{3}+\left ({\mathrm e}^{2}-29\right )^{2} x^{2}\)	\(28\)
norman	\(\left (-12 \,{\mathrm e}^{2}+348\right ) x^{3}+\left ({\mathrm e}^{4}-58 \,{\mathrm e}^{2}+841\right ) x^{2}+36 x^{4}\)	\(31\)
risch	\(x^{2} {\mathrm e}^{4}-12 x^{3} {\mathrm e}^{2}-58 x^{2} {\mathrm e}^{2}+36 x^{4}+348 x^{3}+841 x^{2}\)	\(37\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(2*x*exp(2)^2+(-36*x^2-116*x)*exp(2)+144*x^3+1044*x^2+1682*x,x,method=_RETURNVERBOSE)

[Out]

36*x^4+2/3*(-18*exp(2)+522)*x^3+(exp(2)-29)^2*x^2

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 37 vs. \(2 (17) = 34\).
time = 0.26, size = 37, normalized size = 1.95 \begin {gather*} 36 \, x^{4} + 348 \, x^{3} + x^{2} e^{4} + 841 \, x^{2} - 2 \, {\left (6 \, x^{3} + 29 \, x^{2}\right )} e^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2*x*exp(2)^2+(-36*x^2-116*x)*exp(2)+144*x^3+1044*x^2+1682*x,x, algorithm="maxima")

[Out]

36*x^4 + 348*x^3 + x^2*e^4 + 841*x^2 - 2*(6*x^3 + 29*x^2)*e^2

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 37 vs. \(2 (17) = 34\).
time = 0.38, size = 37, normalized size = 1.95 \begin {gather*} 36 \, x^{4} + 348 \, x^{3} + x^{2} e^{4} + 841 \, x^{2} - 2 \, {\left (6 \, x^{3} + 29 \, x^{2}\right )} e^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2*x*exp(2)^2+(-36*x^2-116*x)*exp(2)+144*x^3+1044*x^2+1682*x,x, algorithm="fricas")

[Out]

36*x^4 + 348*x^3 + x^2*e^4 + 841*x^2 - 2*(6*x^3 + 29*x^2)*e^2

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Sympy [A]
time = 0.01, size = 27, normalized size = 1.42 \begin {gather*} 36 x^{4} + x^{3} \cdot \left (348 - 12 e^{2}\right ) + x^{2} \left (- 58 e^{2} + e^{4} + 841\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2*x*exp(2)**2+(-36*x**2-116*x)*exp(2)+144*x**3+1044*x**2+1682*x,x)

[Out]

36*x**4 + x**3*(348 - 12*exp(2)) + x**2*(-58*exp(2) + exp(4) + 841)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 37 vs. \(2 (17) = 34\).
time = 0.41, size = 37, normalized size = 1.95 \begin {gather*} 36 \, x^{4} + 348 \, x^{3} + x^{2} e^{4} + 841 \, x^{2} - 2 \, {\left (6 \, x^{3} + 29 \, x^{2}\right )} e^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2*x*exp(2)^2+(-36*x^2-116*x)*exp(2)+144*x^3+1044*x^2+1682*x,x, algorithm="giac")

[Out]

36*x^4 + 348*x^3 + x^2*e^4 + 841*x^2 - 2*(6*x^3 + 29*x^2)*e^2

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Mupad [B]
time = 6.92, size = 15, normalized size = 0.79 \begin {gather*} x^2\,{\left (6\,x-{\mathrm {e}}^2+29\right )}^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1682*x - exp(2)*(116*x + 36*x^2) + 2*x*exp(4) + 1044*x^2 + 144*x^3,x)

[Out]

x^2*(6*x - exp(2) + 29)^2

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