3.90.81 \(\int (250+10 x+e^{30 x-21 x^2+3 x^3} (270-378 x+81 x^2)) \, dx\)

Optimal. Leaf size=28 \[ 9 e^{6 (5-x) \left (x-\frac {x^2}{2}\right )}+5 (25+x)^2 \]

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Rubi [A]  time = 0.05, antiderivative size = 27, normalized size of antiderivative = 0.96, number of steps used = 2, number of rules used = 1, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.031, Rules used = {6706} \begin {gather*} 5 x^2+9 e^{3 x^3-21 x^2+30 x}+250 x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[250 + 10*x + E^(30*x - 21*x^2 + 3*x^3)*(270 - 378*x + 81*x^2),x]

[Out]

9*E^(30*x - 21*x^2 + 3*x^3) + 250*x + 5*x^2

Rule 6706

Int[(F_)^(v_)*(u_), x_Symbol] :> With[{q = DerivativeDivides[v, u, x]}, Simp[(q*F^v)/Log[F], x] /;  !FalseQ[q]
] /; FreeQ[F, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=250 x+5 x^2+\int e^{30 x-21 x^2+3 x^3} \left (270-378 x+81 x^2\right ) \, dx\\ &=9 e^{30 x-21 x^2+3 x^3}+250 x+5 x^2\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.07, size = 27, normalized size = 0.96 \begin {gather*} 9 e^{30 x-21 x^2+3 x^3}+250 x+5 x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[250 + 10*x + E^(30*x - 21*x^2 + 3*x^3)*(270 - 378*x + 81*x^2),x]

[Out]

9*E^(30*x - 21*x^2 + 3*x^3) + 250*x + 5*x^2

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fricas [A]  time = 0.83, size = 26, normalized size = 0.93 \begin {gather*} 5 \, x^{2} + 250 \, x + 9 \, e^{\left (3 \, x^{3} - 21 \, x^{2} + 30 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((81*x^2-378*x+270)*exp(3*x^3-21*x^2+30*x)+10*x+250,x, algorithm="fricas")

[Out]

5*x^2 + 250*x + 9*e^(3*x^3 - 21*x^2 + 30*x)

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giac [A]  time = 0.19, size = 26, normalized size = 0.93 \begin {gather*} 5 \, x^{2} + 250 \, x + 9 \, e^{\left (3 \, x^{3} - 21 \, x^{2} + 30 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((81*x^2-378*x+270)*exp(3*x^3-21*x^2+30*x)+10*x+250,x, algorithm="giac")

[Out]

5*x^2 + 250*x + 9*e^(3*x^3 - 21*x^2 + 30*x)

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maple [A]  time = 0.04, size = 22, normalized size = 0.79

method	result	size
risch	\(250 x +9 \,{\mathrm e}^{3 x \left (x -2\right ) \left (x -5\right )}+5 x^{2}\)	\(22\)
default	\(250 x +9 \,{\mathrm e}^{3 x^{3}-21 x^{2}+30 x}+5 x^{2}\)	\(27\)
norman	\(250 x +9 \,{\mathrm e}^{3 x^{3}-21 x^{2}+30 x}+5 x^{2}\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((81*x^2-378*x+270)*exp(3*x^3-21*x^2+30*x)+10*x+250,x,method=_RETURNVERBOSE)

[Out]

250*x+9*exp(3*x*(x-2)*(x-5))+5*x^2

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maxima [A]  time = 0.34, size = 26, normalized size = 0.93 \begin {gather*} 5 \, x^{2} + 250 \, x + 9 \, e^{\left (3 \, x^{3} - 21 \, x^{2} + 30 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((81*x^2-378*x+270)*exp(3*x^3-21*x^2+30*x)+10*x+250,x, algorithm="maxima")

[Out]

5*x^2 + 250*x + 9*e^(3*x^3 - 21*x^2 + 30*x)

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mupad [B]  time = 7.82, size = 27, normalized size = 0.96 \begin {gather*} 250\,x+5\,x^2+9\,{\mathrm {e}}^{30\,x}\,{\mathrm {e}}^{3\,x^3}\,{\mathrm {e}}^{-21\,x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(10*x + exp(30*x - 21*x^2 + 3*x^3)*(81*x^2 - 378*x + 270) + 250,x)

[Out]

250*x + 5*x^2 + 9*exp(30*x)*exp(3*x^3)*exp(-21*x^2)

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sympy [A]  time = 0.11, size = 24, normalized size = 0.86 \begin {gather*} 5 x^{2} + 250 x + 9 e^{3 x^{3} - 21 x^{2} + 30 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((81*x**2-378*x+270)*exp(3*x**3-21*x**2+30*x)+10*x+250,x)

[Out]

5*x**2 + 250*x + 9*exp(3*x**3 - 21*x**2 + 30*x)

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