3.29.31 \(\int (56 x+8 e x+16 x^3) \, dx\)

Optimal. Leaf size=10 \[ \left (7+e+2 x^2\right )^2 \]

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Rubi [A]  time = 0.01, antiderivative size = 14, normalized size of antiderivative = 1.40, number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {6} \begin {gather*} 4 x^4+4 (7+e) x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[56*x + 8*E*x + 16*x^3,x]

[Out]

4*(7 + E)*x^2 + 4*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 ((56+8 e) x+16 x^3\right ) \, dx\\ &=4 (7+e) x^2+4 x^4\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 17, normalized size = 1.70 \begin {gather*} 28 x^2+4 e x^2+4 x^4 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[56*x + 8*E*x + 16*x^3,x]

[Out]

28*x^2 + 4*E*x^2 + 4*x^4

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fricas [A]  time = 0.64, size = 18, normalized size = 1.80 \begin {gather*} 4 \, x^{4} + 4 \, x^{2} e + 28 \, x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(8*x*exp(1)+16*x^3+56*x,x, algorithm="fricas")

[Out]

4*x^4 + 4*x^2*e + 28*x^2

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giac [A]  time = 0.14, size = 18, normalized size = 1.80 \begin {gather*} 4 \, x^{4} + 4 \, x^{2} e + 28 \, x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(8*x*exp(1)+16*x^3+56*x,x, algorithm="giac")

[Out]

4*x^4 + 4*x^2*e + 28*x^2

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maple [A]  time = 0.04, size = 13, normalized size = 1.30

method	result	size
gosper	\(4 x^{2} \left (x^{2}+{\mathrm e}+7\right )\)	\(13\)
norman	\(\left (4 \,{\mathrm e}+28\right ) x^{2}+4 x^{4}\)	\(17\)
default	\(4 x^{2} {\mathrm e}+4 x^{4}+28 x^{2}\)	\(19\)
risch	\(4 x^{2} {\mathrm e}+4 x^{4}+28 x^{2}\)	\(19\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(8*x*exp(1)+16*x^3+56*x,x,method=_RETURNVERBOSE)

[Out]

4*x^2*(x^2+exp(1)+7)

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maxima [A]  time = 0.38, size = 18, normalized size = 1.80 \begin {gather*} 4 \, x^{4} + 4 \, x^{2} e + 28 \, x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(8*x*exp(1)+16*x^3+56*x,x, algorithm="maxima")

[Out]

4*x^4 + 4*x^2*e + 28*x^2

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mupad [B]  time = 1.62, size = 16, normalized size = 1.60 \begin {gather*} 4\,x^4+\left (4\,\mathrm {e}+28\right )\,x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(56*x + 8*x*exp(1) + 16*x^3,x)

[Out]

x^2*(4*exp(1) + 28) + 4*x^4

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sympy [A]  time = 0.05, size = 14, normalized size = 1.40 \begin {gather*} 4 x^{4} + x^{2} \left (4 e + 28\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(8*x*exp(1)+16*x**3+56*x,x)

[Out]

4*x**4 + x**2*(4*E + 28)

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