3.43.88 \(\int (-8 x-75 x^4+18 x^5+e^x (e (-1-x)+2 x+x^2)+e (4+60 x^3-15 x^4)) \, dx\)

Optimal. Leaf size=20 \[ x (-e+x) \left (-4+e^x+3 (-5+x) x^3\right ) \]

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Rubi [B]  time = 0.06, antiderivative size = 54, normalized size of antiderivative = 2.70, number of steps used = 11, number of rules used = 3, integrand size = 46, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {2196, 2176, 2194} \begin {gather*} 3 x^6-3 e x^5-15 x^5+15 e x^4+e^x x^2-4 x^2+4 e x+e^{x+1}-e^{x+1} (x+1) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[-8*x - 75*x^4 + 18*x^5 + E^x*(E*(-1 - x) + 2*x + x^2) + E*(4 + 60*x^3 - 15*x^4),x]

[Out]

E^(1 + x) + 4*E*x - 4*x^2 + E^x*x^2 + 15*E*x^4 - 15*x^5 - 3*E*x^5 + 3*x^6 - E^(1 + x)*(1 + x)

Rule 2176

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^m
*(b*F^(g*(e + f*x)))^n)/(f*g*n*Log[F]), x] - Dist[(d*m)/(f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*(b*F^(g*(e + f*x
)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && GtQ[m, 0] && IntegerQ[2*m] &&  !$UseGamma === True

Rule 2194

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rule 2196

Int[(F_)^((c_.)*(v_))*(u_), x_Symbol] :> Int[ExpandIntegrand[F^(c*ExpandToSum[v, x]), u, x], x] /; FreeQ[{F, c
}, x] && PolynomialQ[u, x] && LinearQ[v, x] &&  !$UseGamma === True

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=-4 x^2-15 x^5+3 x^6+e \int \left (4+60 x^3-15 x^4\right ) \, dx+\int e^x \left (e (-1-x)+2 x+x^2\right ) \, dx\\ &=4 e x-4 x^2+15 e x^4-15 x^5-3 e x^5+3 x^6+\int \left (2 e^x x+e^x x^2-e^{1+x} (1+x)\right ) \, dx\\ &=4 e x-4 x^2+15 e x^4-15 x^5-3 e x^5+3 x^6+2 \int e^x x \, dx+\int e^x x^2 \, dx-\int e^{1+x} (1+x) \, dx\\ &=4 e x+2 e^x x-4 x^2+e^x x^2+15 e x^4-15 x^5-3 e x^5+3 x^6-e^{1+x} (1+x)-2 \int e^x \, dx-2 \int e^x x \, dx+\int e^{1+x} \, dx\\ &=-2 e^x+e^{1+x}+4 e x-4 x^2+e^x x^2+15 e x^4-15 x^5-3 e x^5+3 x^6-e^{1+x} (1+x)+2 \int e^x \, dx\\ &=e^{1+x}+4 e x-4 x^2+e^x x^2+15 e x^4-15 x^5-3 e x^5+3 x^6-e^{1+x} (1+x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.04, size = 23, normalized size = 1.15 \begin {gather*} -\left ((e-x) x \left (-4+e^x-15 x^3+3 x^4\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[-8*x - 75*x^4 + 18*x^5 + E^x*(E*(-1 - x) + 2*x + x^2) + E*(4 + 60*x^3 - 15*x^4),x]

[Out]

-((E - x)*x*(-4 + E^x - 15*x^3 + 3*x^4))

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fricas [B]  time = 0.76, size = 46, normalized size = 2.30 \begin {gather*} 3 \, x^{6} - 15 \, x^{5} - 4 \, x^{2} - {\left (3 \, x^{5} - 15 \, x^{4} - 4 \, x\right )} e + {\left (x^{2} - x e\right )} e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x-1)*exp(1)+x^2+2*x)*exp(x)+(-15*x^4+60*x^3+4)*exp(1)+18*x^5-75*x^4-8*x,x, algorithm="fricas")

[Out]

3*x^6 - 15*x^5 - 4*x^2 - (3*x^5 - 15*x^4 - 4*x)*e + (x^2 - x*e)*e^x

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giac [B]  time = 0.12, size = 47, normalized size = 2.35 \begin {gather*} 3 \, x^{6} - 15 \, x^{5} + x^{2} e^{x} - 4 \, x^{2} - {\left (3 \, x^{5} - 15 \, x^{4} - 4 \, x\right )} e - x e^{\left (x + 1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x-1)*exp(1)+x^2+2*x)*exp(x)+(-15*x^4+60*x^3+4)*exp(1)+18*x^5-75*x^4-8*x,x, algorithm="giac")

[Out]

3*x^6 - 15*x^5 + x^2*e^x - 4*x^2 - (3*x^5 - 15*x^4 - 4*x)*e - x*e^(x + 1)

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maple [B]  time = 0.03, size = 47, normalized size = 2.35

method	result	size
norman	\(\left (-3 \,{\mathrm e}-15\right ) x^{5}+{\mathrm e}^{x} x^{2}-4 x^{2}+3 x^{6}+4 x \,{\mathrm e}+15 x^{4} {\mathrm e}-x \,{\mathrm e} \,{\mathrm e}^{x}\)	\(47\)
risch	\(\left (-x \,{\mathrm e}+x^{2}\right ) {\mathrm e}^{x}-3 x^{5} {\mathrm e}+15 x^{4} {\mathrm e}+4 x \,{\mathrm e}+3 x^{6}-15 x^{5}-4 x^{2}\)	\(48\)
default	\({\mathrm e}^{x} x^{2}-{\mathrm e} \,{\mathrm e}^{x}-{\mathrm e} \left ({\mathrm e}^{x} x -{\mathrm e}^{x}\right )+{\mathrm e} \left (-3 x^{5}+15 x^{4}+4 x \right )-4 x^{2}-15 x^{5}+3 x^{6}\)	\(59\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-x-1)*exp(1)+x^2+2*x)*exp(x)+(-15*x^4+60*x^3+4)*exp(1)+18*x^5-75*x^4-8*x,x,method=_RETURNVERBOSE)

[Out]

(-3*exp(1)-15)*x^5+exp(x)*x^2-4*x^2+3*x^6+4*x*exp(1)+15*x^4*exp(1)-x*exp(1)*exp(x)

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maxima [B]  time = 0.36, size = 46, normalized size = 2.30 \begin {gather*} 3 \, x^{6} - 15 \, x^{5} - 4 \, x^{2} - {\left (3 \, x^{5} - 15 \, x^{4} - 4 \, x\right )} e + {\left (x^{2} - x e\right )} e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x-1)*exp(1)+x^2+2*x)*exp(x)+(-15*x^4+60*x^3+4)*exp(1)+18*x^5-75*x^4-8*x,x, algorithm="maxima")

[Out]

3*x^6 - 15*x^5 - 4*x^2 - (3*x^5 - 15*x^4 - 4*x)*e + (x^2 - x*e)*e^x

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mupad [B]  time = 0.07, size = 22, normalized size = 1.10 \begin {gather*} x\,\left (x-\mathrm {e}\right )\,\left ({\mathrm {e}}^x-15\,x^3+3\,x^4-4\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(1)*(60*x^3 - 15*x^4 + 4) - 8*x - 75*x^4 + 18*x^5 + exp(x)*(2*x - exp(1)*(x + 1) + x^2),x)

[Out]

x*(x - exp(1))*(exp(x) - 15*x^3 + 3*x^4 - 4)

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sympy [B]  time = 0.11, size = 48, normalized size = 2.40 \begin {gather*} 3 x^{6} + x^{5} \left (-15 - 3 e\right ) + 15 e x^{4} - 4 x^{2} + 4 e x + \left (x^{2} - e x\right ) e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x-1)*exp(1)+x**2+2*x)*exp(x)+(-15*x**4+60*x**3+4)*exp(1)+18*x**5-75*x**4-8*x,x)

[Out]

3*x**6 + x**5*(-15 - 3*E) + 15*E*x**4 - 4*x**2 + 4*E*x + (x**2 - E*x)*exp(x)

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