3.34.50 \(\int \frac {1}{2} (3+7 x+3 x^2+2 x^3+e^2 (1+2 x)) \, dx\)

Optimal. Leaf size=15 \[ \frac {1}{4} \left (3+e^2+x+x^2\right )^2 \]

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Rubi [B]  time = 0.01, antiderivative size = 41, normalized size of antiderivative = 2.73, number of steps used = 2, number of rules used = 1, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.036, Rules used = {12} \begin {gather*} \frac {x^4}{4}+\frac {x^3}{2}+\frac {7 x^2}{4}+\frac {3 x}{2}+\frac {1}{8} e^2 (2 x+1)^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(3 + 7*x + 3*x^2 + 2*x^3 + E^2*(1 + 2*x))/2,x]

[Out]

(3*x)/2 + (7*x^2)/4 + x^3/2 + x^4/4 + (E^2*(1 + 2*x)^2)/8

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}{2} \int \left (3+7 x+3 x^2+2 x^3+e^2 (1+2 x)\right ) \, dx\\ &=\frac {3 x}{2}+\frac {7 x^2}{4}+\frac {x^3}{2}+\frac {x^4}{4}+\frac {1}{8} e^2 (1+2 x)^2\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 19, normalized size = 1.27 \begin {gather*} \frac {1}{4} x (1+x) \left (6+2 e^2+x+x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(3 + 7*x + 3*x^2 + 2*x^3 + E^2*(1 + 2*x))/2,x]

[Out]

(x*(1 + x)*(6 + 2*E^2 + x + x^2))/4

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fricas [B]  time = 0.46, size = 28, normalized size = 1.87 \begin {gather*} \frac {1}{4} \, x^{4} + \frac {1}{2} \, x^{3} + \frac {7}{4} \, x^{2} + \frac {1}{2} \, {\left (x^{2} + x\right )} e^{2} + \frac {3}{2} \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(2*x+1)*exp(2)+x^3+3/2*x^2+7/2*x+3/2,x, algorithm="fricas")

[Out]

1/4*x^4 + 1/2*x^3 + 7/4*x^2 + 1/2*(x^2 + x)*e^2 + 3/2*x

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giac [B]  time = 0.20, size = 28, normalized size = 1.87 \begin {gather*} \frac {1}{4} \, x^{4} + \frac {1}{2} \, x^{3} + \frac {7}{4} \, x^{2} + \frac {1}{2} \, {\left (x^{2} + x\right )} e^{2} + \frac {3}{2} \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(2*x+1)*exp(2)+x^3+3/2*x^2+7/2*x+3/2,x, algorithm="giac")

[Out]

1/4*x^4 + 1/2*x^3 + 7/4*x^2 + 1/2*(x^2 + x)*e^2 + 3/2*x

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maple [B]  time = 0.02, size = 26, normalized size = 1.73

method	result	size
gosper	\(\frac {x \left (x^{3}+2 \,{\mathrm e}^{2} x +2 x^{2}+2 \,{\mathrm e}^{2}+7 x +6\right )}{4}\)	\(26\)
default	\(\frac {\left (x^{2}+x \right ) {\mathrm e}^{2}}{2}+\frac {x^{4}}{4}+\frac {x^{3}}{2}+\frac {7 x^{2}}{4}+\frac {3 x}{2}\)	\(29\)
norman	\(\left (\frac {7}{4}+\frac {{\mathrm e}^{2}}{2}\right ) x^{2}+\left (\frac {{\mathrm e}^{2}}{2}+\frac {3}{2}\right ) x +\frac {x^{3}}{2}+\frac {x^{4}}{4}\)	\(30\)
risch	\(\frac {x^{2} {\mathrm e}^{2}}{2}+\frac {{\mathrm e}^{2} x}{2}+\frac {x^{4}}{4}+\frac {x^{3}}{2}+\frac {7 x^{2}}{4}+\frac {3 x}{2}\)	\(32\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/2*(2*x+1)*exp(2)+x^3+3/2*x^2+7/2*x+3/2,x,method=_RETURNVERBOSE)

[Out]

1/4*x*(x^3+2*exp(2)*x+2*x^2+2*exp(2)+7*x+6)

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maxima [B]  time = 0.49, size = 28, normalized size = 1.87 \begin {gather*} \frac {1}{4} \, x^{4} + \frac {1}{2} \, x^{3} + \frac {7}{4} \, x^{2} + \frac {1}{2} \, {\left (x^{2} + x\right )} e^{2} + \frac {3}{2} \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(2*x+1)*exp(2)+x^3+3/2*x^2+7/2*x+3/2,x, algorithm="maxima")

[Out]

1/4*x^4 + 1/2*x^3 + 7/4*x^2 + 1/2*(x^2 + x)*e^2 + 3/2*x

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mupad [B]  time = 1.95, size = 29, normalized size = 1.93 \begin {gather*} \frac {x^4}{4}+\frac {x^3}{2}+\left (\frac {{\mathrm {e}}^2}{2}+\frac {7}{4}\right )\,x^2+\left (\frac {{\mathrm {e}}^2}{2}+\frac {3}{2}\right )\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((7*x)/2 + (3*x^2)/2 + x^3 + (exp(2)*(2*x + 1))/2 + 3/2,x)

[Out]

x^2*(exp(2)/2 + 7/4) + x^3/2 + x^4/4 + x*(exp(2)/2 + 3/2)

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sympy [B]  time = 0.06, size = 31, normalized size = 2.07 \begin {gather*} \frac {x^{4}}{4} + \frac {x^{3}}{2} + x^{2} \left (\frac {7}{4} + \frac {e^{2}}{2}\right ) + x \left (\frac {3}{2} + \frac {e^{2}}{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(2*x+1)*exp(2)+x**3+3/2*x**2+7/2*x+3/2,x)

[Out]

x**4/4 + x**3/2 + x**2*(7/4 + exp(2)/2) + x*(3/2 + exp(2)/2)

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