3.90.13 \(\int (10 x+27 x^2+12 x^3+e^{4+x} (-150 x-75 x^2)) \, dx\)

Optimal. Leaf size=21 \[ x^2 \left (5+3 \left (-25 e^{4+x}+x (3+x)\right )\right ) \]

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Rubi [A]  time = 0.07, antiderivative size = 26, normalized size of antiderivative = 1.24, number of steps used = 9, number of rules used = 4, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {1593, 2196, 2176, 2194} \begin {gather*} 3 x^4+9 x^3-75 e^{x+4} x^2+5 x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[10*x + 27*x^2 + 12*x^3 + E^(4 + x)*(-150*x - 75*x^2),x]

[Out]

5*x^2 - 75*E^(4 + x)*x^2 + 9*x^3 + 3*x^4

Rule 1593

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

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} &=5 x^2+9 x^3+3 x^4+\int e^{4+x} \left (-150 x-75 x^2\right ) \, dx\\ &=5 x^2+9 x^3+3 x^4+\int e^{4+x} (-150-75 x) x \, dx\\ &=5 x^2+9 x^3+3 x^4+\int \left (-150 e^{4+x} x-75 e^{4+x} x^2\right ) \, dx\\ &=5 x^2+9 x^3+3 x^4-75 \int e^{4+x} x^2 \, dx-150 \int e^{4+x} x \, dx\\ &=-150 e^{4+x} x+5 x^2-75 e^{4+x} x^2+9 x^3+3 x^4+150 \int e^{4+x} \, dx+150 \int e^{4+x} x \, dx\\ &=150 e^{4+x}+5 x^2-75 e^{4+x} x^2+9 x^3+3 x^4-150 \int e^{4+x} \, dx\\ &=5 x^2-75 e^{4+x} x^2+9 x^3+3 x^4\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 21, normalized size = 1.00 \begin {gather*} x^2 \left (5-75 e^{4+x}+9 x+3 x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[10*x + 27*x^2 + 12*x^3 + E^(4 + x)*(-150*x - 75*x^2),x]

[Out]

x^2*(5 - 75*E^(4 + x) + 9*x + 3*x^2)

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fricas [A]  time = 0.49, size = 25, normalized size = 1.19 \begin {gather*} 3 \, x^{4} + 9 \, x^{3} - 75 \, x^{2} e^{\left (x + 4\right )} + 5 \, x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-75*x^2-150*x)*exp(3)*exp(x+1)+12*x^3+27*x^2+10*x,x, algorithm="fricas")

[Out]

3*x^4 + 9*x^3 - 75*x^2*e^(x + 4) + 5*x^2

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giac [A]  time = 0.15, size = 25, normalized size = 1.19 \begin {gather*} 3 \, x^{4} + 9 \, x^{3} - 75 \, x^{2} e^{\left (x + 4\right )} + 5 \, x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-75*x^2-150*x)*exp(3)*exp(x+1)+12*x^3+27*x^2+10*x,x, algorithm="giac")

[Out]

3*x^4 + 9*x^3 - 75*x^2*e^(x + 4) + 5*x^2

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maple [A]  time = 0.05, size = 26, normalized size = 1.24

method	result	size
risch	\(5 x^{2}+9 x^{3}+3 x^{4}-75 x^{2} {\mathrm e}^{4+x}\)	\(26\)
norman	\(5 x^{2}+9 x^{3}+3 x^{4}-75 \,{\mathrm e}^{3} {\mathrm e}^{x +1} x^{2}\)	\(28\)
default	\(75 \,{\mathrm e}^{3} \left (-{\mathrm e}^{x +1} \left (x +1\right )^{2}+2 \left (x +1\right ) {\mathrm e}^{x +1}-{\mathrm e}^{x +1}\right )+5 x^{2}+9 x^{3}+3 x^{4}\)	\(48\)
derivativedivides	\(-10 x -10+75 \,{\mathrm e}^{3} \left (-{\mathrm e}^{x +1} \left (x +1\right )^{2}+2 \left (x +1\right ) {\mathrm e}^{x +1}-{\mathrm e}^{x +1}\right )+5 \left (x +1\right )^{2}+9 x^{3}+3 x^{4}\)	\(54\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-75*x^2-150*x)*exp(3)*exp(x+1)+12*x^3+27*x^2+10*x,x,method=_RETURNVERBOSE)

[Out]

5*x^2+9*x^3+3*x^4-75*x^2*exp(4+x)

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maxima [A]  time = 0.36, size = 25, normalized size = 1.19 \begin {gather*} 3 \, x^{4} + 9 \, x^{3} - 75 \, x^{2} e^{\left (x + 4\right )} + 5 \, x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-75*x^2-150*x)*exp(3)*exp(x+1)+12*x^3+27*x^2+10*x,x, algorithm="maxima")

[Out]

3*x^4 + 9*x^3 - 75*x^2*e^(x + 4) + 5*x^2

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mupad [B]  time = 0.05, size = 20, normalized size = 0.95 \begin {gather*} x^2\,\left (9\,x-75\,{\mathrm {e}}^{x+4}+3\,x^2+5\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(10*x + 27*x^2 + 12*x^3 - exp(x + 1)*exp(3)*(150*x + 75*x^2),x)

[Out]

x^2*(9*x - 75*exp(x + 4) + 3*x^2 + 5)

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sympy [A]  time = 0.10, size = 27, normalized size = 1.29 \begin {gather*} 3 x^{4} + 9 x^{3} - 75 x^{2} e^{3} e^{x + 1} + 5 x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-75*x**2-150*x)*exp(3)*exp(x+1)+12*x**3+27*x**2+10*x,x)

[Out]

3*x**4 + 9*x**3 - 75*x**2*exp(3)*exp(x + 1) + 5*x**2

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