3.49.1 \(\int (-2 e^{2 x}-8 x+e^5 (-30 x+51 x^2+16 x^3+100 x^4+24 x^5)) \, dx\)

Optimal. Leaf size=31 \[ -e^{2 x}+x^2 \left (-4+e^5 (5+x) \left (-3+4 x \left (1+x^2\right )\right )\right ) \]

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Rubi [A]  time = 0.02, antiderivative size = 53, normalized size of antiderivative = 1.71, number of steps used = 3, number of rules used = 1, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.026, Rules used = {2194} \begin {gather*} 4 e^5 x^6+20 e^5 x^5+4 e^5 x^4+17 e^5 x^3-15 e^5 x^2-4 x^2-e^{2 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[-2*E^(2*x) - 8*x + E^5*(-30*x + 51*x^2 + 16*x^3 + 100*x^4 + 24*x^5),x]

[Out]

-E^(2*x) - 4*x^2 - 15*E^5*x^2 + 17*E^5*x^3 + 4*E^5*x^4 + 20*E^5*x^5 + 4*E^5*x^6

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]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=-4 x^2-2 \int e^{2 x} \, dx+e^5 \int \left (-30 x+51 x^2+16 x^3+100 x^4+24 x^5\right ) \, dx\\ &=-e^{2 x}-4 x^2-15 e^5 x^2+17 e^5 x^3+4 e^5 x^4+20 e^5 x^5+4 e^5 x^6\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 53, normalized size = 1.71 \begin {gather*} -e^{2 x}-4 x^2-15 e^5 x^2+17 e^5 x^3+4 e^5 x^4+20 e^5 x^5+4 e^5 x^6 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[-2*E^(2*x) - 8*x + E^5*(-30*x + 51*x^2 + 16*x^3 + 100*x^4 + 24*x^5),x]

[Out]

-E^(2*x) - 4*x^2 - 15*E^5*x^2 + 17*E^5*x^3 + 4*E^5*x^4 + 20*E^5*x^5 + 4*E^5*x^6

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fricas [A]  time = 0.65, size = 41, normalized size = 1.32 \begin {gather*} -4 \, x^{2} + {\left (4 \, x^{6} + 20 \, x^{5} + 4 \, x^{4} + 17 \, x^{3} - 15 \, x^{2}\right )} e^{5} - e^{\left (2 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-2*exp(x)^2+(24*x^5+100*x^4+16*x^3+51*x^2-30*x)*exp(5)-8*x,x, algorithm="fricas")

[Out]

-4*x^2 + (4*x^6 + 20*x^5 + 4*x^4 + 17*x^3 - 15*x^2)*e^5 - e^(2*x)

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giac [A]  time = 0.15, size = 41, normalized size = 1.32 \begin {gather*} -4 \, x^{2} + {\left (4 \, x^{6} + 20 \, x^{5} + 4 \, x^{4} + 17 \, x^{3} - 15 \, x^{2}\right )} e^{5} - e^{\left (2 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-2*exp(x)^2+(24*x^5+100*x^4+16*x^3+51*x^2-30*x)*exp(5)-8*x,x, algorithm="giac")

[Out]

-4*x^2 + (4*x^6 + 20*x^5 + 4*x^4 + 17*x^3 - 15*x^2)*e^5 - e^(2*x)

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maple [A]  time = 0.04, size = 42, normalized size = 1.35

method	result	size
default	\({\mathrm e}^{5} \left (4 x^{6}+20 x^{5}+4 x^{4}+17 x^{3}-15 x^{2}\right )-4 x^{2}-{\mathrm e}^{2 x}\)	\(42\)
norman	\(\left (-15 \,{\mathrm e}^{5}-4\right ) x^{2}-{\mathrm e}^{2 x}+17 x^{3} {\mathrm e}^{5}+4 x^{4} {\mathrm e}^{5}+20 x^{5} {\mathrm e}^{5}+4 x^{6} {\mathrm e}^{5}\)	\(46\)
risch	\(-{\mathrm e}^{2 x}+4 x^{6} {\mathrm e}^{5}+20 x^{5} {\mathrm e}^{5}+4 x^{4} {\mathrm e}^{5}+17 x^{3} {\mathrm e}^{5}-15 x^{2} {\mathrm e}^{5}-4 x^{2}\)	\(48\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-2*exp(x)^2+(24*x^5+100*x^4+16*x^3+51*x^2-30*x)*exp(5)-8*x,x,method=_RETURNVERBOSE)

[Out]

exp(5)*(4*x^6+20*x^5+4*x^4+17*x^3-15*x^2)-4*x^2-exp(x)^2

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maxima [A]  time = 0.37, size = 41, normalized size = 1.32 \begin {gather*} -4 \, x^{2} + {\left (4 \, x^{6} + 20 \, x^{5} + 4 \, x^{4} + 17 \, x^{3} - 15 \, x^{2}\right )} e^{5} - e^{\left (2 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-2*exp(x)^2+(24*x^5+100*x^4+16*x^3+51*x^2-30*x)*exp(5)-8*x,x, algorithm="maxima")

[Out]

-4*x^2 + (4*x^6 + 20*x^5 + 4*x^4 + 17*x^3 - 15*x^2)*e^5 - e^(2*x)

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mupad [B]  time = 3.41, size = 46, normalized size = 1.48 \begin {gather*} 17\,x^3\,{\mathrm {e}}^5-x^2\,\left (15\,{\mathrm {e}}^5+4\right )-{\mathrm {e}}^{2\,x}+4\,x^4\,{\mathrm {e}}^5+20\,x^5\,{\mathrm {e}}^5+4\,x^6\,{\mathrm {e}}^5 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(5)*(51*x^2 - 30*x + 16*x^3 + 100*x^4 + 24*x^5) - 2*exp(2*x) - 8*x,x)

[Out]

17*x^3*exp(5) - x^2*(15*exp(5) + 4) - exp(2*x) + 4*x^4*exp(5) + 20*x^5*exp(5) + 4*x^6*exp(5)

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sympy [A]  time = 0.10, size = 49, normalized size = 1.58 \begin {gather*} 4 x^{6} e^{5} + 20 x^{5} e^{5} + 4 x^{4} e^{5} + 17 x^{3} e^{5} + x^{2} \left (- 15 e^{5} - 4\right ) - e^{2 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-2*exp(x)**2+(24*x**5+100*x**4+16*x**3+51*x**2-30*x)*exp(5)-8*x,x)

[Out]

4*x**6*exp(5) + 20*x**5*exp(5) + 4*x**4*exp(5) + 17*x**3*exp(5) + x**2*(-15*exp(5) - 4) - exp(2*x)

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