3.98.72 \(\int (1650+838 x+108 x^2+4 x^3+e^4 (6+2 x)+e^3 (30+2 e^2+2 x)+e^2 (198+84 x+6 x^2)) \, dx\)

Optimal. Leaf size=20 \[ \left (e^3+(3+x)^2\right ) \left (5+\left (15+e^2+x\right )^2\right ) \]

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Rubi [B]  time = 0.02, antiderivative size = 60, normalized size of antiderivative = 3.00, number of steps used = 2, number of rules used = 0, integrand size = 52, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} x^4+2 e^2 x^3+36 x^3+42 e^2 x^2+419 x^2+198 e^2 x+1650 x+e^4 (x+3)^2+e^3 \left (x+e^2+15\right )^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1650 + 838*x + 108*x^2 + 4*x^3 + E^4*(6 + 2*x) + E^3*(30 + 2*E^2 + 2*x) + E^2*(198 + 84*x + 6*x^2),x]

[Out]

1650*x + 198*E^2*x + 419*x^2 + 42*E^2*x^2 + 36*x^3 + 2*E^2*x^3 + x^4 + E^4*(3 + x)^2 + E^3*(15 + E^2 + x)^2

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=1650 x+419 x^2+36 x^3+x^4+e^4 (3+x)^2+e^3 \left (15+e^2+x\right )^2+e^2 \int \left (198+84 x+6 x^2\right ) \, dx\\ &=1650 x+198 e^2 x+419 x^2+42 e^2 x^2+36 x^3+2 e^2 x^3+x^4+e^4 (3+x)^2+e^3 \left (15+e^2+x\right )^2\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.01, size = 83, normalized size = 4.15 \begin {gather*} 2 \left (825 x+99 e^2 x+15 e^3 x+3 e^4 x+e^5 x+\frac {419 x^2}{2}+21 e^2 x^2+\frac {e^3 x^2}{2}+\frac {e^4 x^2}{2}+18 x^3+e^2 x^3+\frac {x^4}{2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1650 + 838*x + 108*x^2 + 4*x^3 + E^4*(6 + 2*x) + E^3*(30 + 2*E^2 + 2*x) + E^2*(198 + 84*x + 6*x^2),x
]

[Out]

2*(825*x + 99*E^2*x + 15*E^3*x + 3*E^4*x + E^5*x + (419*x^2)/2 + 21*E^2*x^2 + (E^3*x^2)/2 + (E^4*x^2)/2 + 18*x
^3 + E^2*x^3 + x^4/2)

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fricas [B]  time = 0.68, size = 58, normalized size = 2.90 \begin {gather*} x^{4} + 36 \, x^{3} + 419 \, x^{2} + 2 \, x e^{5} + {\left (x^{2} + 6 \, x\right )} e^{4} + {\left (x^{2} + 30 \, x\right )} e^{3} + 2 \, {\left (x^{3} + 21 \, x^{2} + 99 \, x\right )} e^{2} + 1650 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*exp(2)+2*x+30)*exp(3)+(2*x+6)*exp(2)^2+(6*x^2+84*x+198)*exp(2)+4*x^3+108*x^2+838*x+1650,x, algori
thm="fricas")

[Out]

x^4 + 36*x^3 + 419*x^2 + 2*x*e^5 + (x^2 + 6*x)*e^4 + (x^2 + 30*x)*e^3 + 2*(x^3 + 21*x^2 + 99*x)*e^2 + 1650*x

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giac [B]  time = 0.16, size = 58, normalized size = 2.90 \begin {gather*} x^{4} + 36 \, x^{3} + 419 \, x^{2} + {\left (x^{2} + 6 \, x\right )} e^{4} + {\left (x^{2} + 2 \, x e^{2} + 30 \, x\right )} e^{3} + 2 \, {\left (x^{3} + 21 \, x^{2} + 99 \, x\right )} e^{2} + 1650 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*exp(2)+2*x+30)*exp(3)+(2*x+6)*exp(2)^2+(6*x^2+84*x+198)*exp(2)+4*x^3+108*x^2+838*x+1650,x, algori
thm="giac")

[Out]

x^4 + 36*x^3 + 419*x^2 + (x^2 + 6*x)*e^4 + (x^2 + 2*x*e^2 + 30*x)*e^3 + 2*(x^3 + 21*x^2 + 99*x)*e^2 + 1650*x

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maple [B]  time = 0.03, size = 55, normalized size = 2.75

method	result	size
norman	\(x^{4}+\left (2 \,{\mathrm e}^{2}+36\right ) x^{3}+\left ({\mathrm e}^{4}+42 \,{\mathrm e}^{2}+{\mathrm e}^{3}+419\right ) x^{2}+\left (6 \,{\mathrm e}^{4}+2 \,{\mathrm e}^{2} {\mathrm e}^{3}+198 \,{\mathrm e}^{2}+30 \,{\mathrm e}^{3}+1650\right ) x\)	\(55\)
gosper	\(x \left (x \,{\mathrm e}^{4}+2 x^{2} {\mathrm e}^{2}+x^{3}+6 \,{\mathrm e}^{4}+2 \,{\mathrm e}^{2} {\mathrm e}^{3}+42 \,{\mathrm e}^{2} x +x \,{\mathrm e}^{3}+36 x^{2}+198 \,{\mathrm e}^{2}+30 \,{\mathrm e}^{3}+419 x +1650\right )\)	\(58\)
default	\({\mathrm e}^{3} \left (2 \,{\mathrm e}^{2} x +x^{2}+30 x \right )+{\mathrm e}^{4} \left (x^{2}+6 x \right )+{\mathrm e}^{2} \left (2 x^{3}+42 x^{2}+198 x \right )+x^{4}+36 x^{3}+419 x^{2}+1650 x\)	\(62\)
risch	\(2 x \,{\mathrm e}^{5}+x^{2} {\mathrm e}^{3}+30 x \,{\mathrm e}^{3}+x^{2} {\mathrm e}^{4}+6 x \,{\mathrm e}^{4}+2 x^{3} {\mathrm e}^{2}+42 x^{2} {\mathrm e}^{2}+198 \,{\mathrm e}^{2} x +x^{4}+36 x^{3}+419 x^{2}+1650 x\)	\(64\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*exp(2)+2*x+30)*exp(3)+(2*x+6)*exp(2)^2+(6*x^2+84*x+198)*exp(2)+4*x^3+108*x^2+838*x+1650,x,method=_RETUR
NVERBOSE)

[Out]

x^4+(2*exp(2)+36)*x^3+(exp(2)^2+42*exp(2)+exp(3)+419)*x^2+(6*exp(2)^2+2*exp(2)*exp(3)+198*exp(2)+30*exp(3)+165
0)*x

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maxima [B]  time = 0.39, size = 58, normalized size = 2.90 \begin {gather*} x^{4} + 36 \, x^{3} + 419 \, x^{2} + {\left (x^{2} + 6 \, x\right )} e^{4} + {\left (x^{2} + 2 \, x e^{2} + 30 \, x\right )} e^{3} + 2 \, {\left (x^{3} + 21 \, x^{2} + 99 \, x\right )} e^{2} + 1650 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*exp(2)+2*x+30)*exp(3)+(2*x+6)*exp(2)^2+(6*x^2+84*x+198)*exp(2)+4*x^3+108*x^2+838*x+1650,x, algori
thm="maxima")

[Out]

x^4 + 36*x^3 + 419*x^2 + (x^2 + 6*x)*e^4 + (x^2 + 2*x*e^2 + 30*x)*e^3 + 2*(x^3 + 21*x^2 + 99*x)*e^2 + 1650*x

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mupad [B]  time = 0.07, size = 49, normalized size = 2.45 \begin {gather*} x^4+\left (2\,{\mathrm {e}}^2+36\right )\,x^3+\left (42\,{\mathrm {e}}^2+{\mathrm {e}}^3+{\mathrm {e}}^4+419\right )\,x^2+\left (198\,{\mathrm {e}}^2+6\,{\mathrm {e}}^4+{\mathrm {e}}^3\,\left (2\,{\mathrm {e}}^2+30\right )+1650\right )\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(838*x + exp(2)*(84*x + 6*x^2 + 198) + exp(3)*(2*x + 2*exp(2) + 30) + 108*x^2 + 4*x^3 + exp(4)*(2*x + 6) +
1650,x)

[Out]

x^2*(42*exp(2) + exp(3) + exp(4) + 419) + x^3*(2*exp(2) + 36) + x*(198*exp(2) + 6*exp(4) + exp(3)*(2*exp(2) +
30) + 1650) + x^4

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sympy [B]  time = 0.07, size = 53, normalized size = 2.65 \begin {gather*} x^{4} + x^{3} \left (2 e^{2} + 36\right ) + x^{2} \left (e^{3} + e^{4} + 42 e^{2} + 419\right ) + x \left (2 e^{5} + 6 e^{4} + 30 e^{3} + 198 e^{2} + 1650\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*exp(2)+2*x+30)*exp(3)+(2*x+6)*exp(2)**2+(6*x**2+84*x+198)*exp(2)+4*x**3+108*x**2+838*x+1650,x)

[Out]

x**4 + x**3*(2*exp(2) + 36) + x**2*(exp(3) + exp(4) + 42*exp(2) + 419) + x*(2*exp(5) + 6*exp(4) + 30*exp(3) +
198*exp(2) + 1650)

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