3.5.72 \(\int (4+2 x-6 x^2+3 e^5 x^2) \, dx\)

Optimal. Leaf size=22 \[ x \left (4+x+2 x^2-\left (4-e^5\right ) x^2\right ) \]

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Rubi [A]  time = 0.01, antiderivative size = 19, normalized size of antiderivative = 0.86, number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {6} \begin {gather*} -\left (\left (2-e^5\right ) x^3\right )+x^2+4 x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[4 + 2*x - 6*x^2 + 3*E^5*x^2,x]

[Out]

4*x + x^2 - (2 - E^5)*x^3

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

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Mathematica [A]  time = 0.00, size = 19, normalized size = 0.86 \begin {gather*} 4 x+x^2-2 x^3+e^5 x^3 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[4 + 2*x - 6*x^2 + 3*E^5*x^2,x]

[Out]

4*x + x^2 - 2*x^3 + E^5*x^3

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fricas [A]  time = 0.96, size = 18, normalized size = 0.82 \begin {gather*} x^{3} e^{5} - 2 \, x^{3} + x^{2} + 4 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(3*x^2*exp(5)-6*x^2+2*x+4,x, algorithm="fricas")

[Out]

x^3*e^5 - 2*x^3 + x^2 + 4*x

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giac [A]  time = 0.43, size = 18, normalized size = 0.82 \begin {gather*} x^{3} e^{5} - 2 \, x^{3} + x^{2} + 4 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(3*x^2*exp(5)-6*x^2+2*x+4,x, algorithm="giac")

[Out]

x^3*e^5 - 2*x^3 + x^2 + 4*x

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maple [A]  time = 0.02, size = 16, normalized size = 0.73

method	result	size
norman	\(\left ({\mathrm e}^{5}-2\right ) x^{3}+x^{2}+4 x\)	\(16\)
gosper	\(x^{3} {\mathrm e}^{5}-2 x^{3}+x^{2}+4 x\)	\(19\)
default	\(x^{3} {\mathrm e}^{5}-2 x^{3}+x^{2}+4 x\)	\(19\)
risch	\(x^{3} {\mathrm e}^{5}-2 x^{3}+x^{2}+4 x\)	\(19\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(3*x^2*exp(5)-6*x^2+2*x+4,x,method=_RETURNVERBOSE)

[Out]

(exp(5)-2)*x^3+x^2+4*x

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maxima [A]  time = 0.38, size = 18, normalized size = 0.82 \begin {gather*} x^{3} e^{5} - 2 \, x^{3} + x^{2} + 4 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(3*x^2*exp(5)-6*x^2+2*x+4,x, algorithm="maxima")

[Out]

x^3*e^5 - 2*x^3 + x^2 + 4*x

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mupad [B]  time = 0.03, size = 15, normalized size = 0.68 \begin {gather*} \left ({\mathrm {e}}^5-2\right )\,x^3+x^2+4\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(2*x + 3*x^2*exp(5) - 6*x^2 + 4,x)

[Out]

4*x + x^2 + x^3*(exp(5) - 2)

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sympy [A]  time = 0.06, size = 14, normalized size = 0.64 \begin {gather*} x^{3} \left (-2 + e^{5}\right ) + x^{2} + 4 x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**3*(-2 + exp(5)) + x**2 + 4*x

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