3.94.81 \(\int \frac {1}{25} (25-1350 x-1000 e^4 x-300 e^8 x-40 e^{12} x-2 e^{16} x-75 x^2) \, dx\)

Optimal. Leaf size=22 \[ 4+x-x^2 \left (2+\frac {1}{25} \left (5+e^4\right )^4+x\right ) \]

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Rubi [A]  time = 0.02, antiderivative size = 34, normalized size of antiderivative = 1.55, number of steps used = 6, number of rules used = 2, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {6, 12} \begin {gather*} -x^3-\frac {1}{25} \left (675+500 e^4+150 e^8+20 e^{12}+e^{16}\right ) x^2+x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(25 - 1350*x - 1000*E^4*x - 300*E^8*x - 40*E^12*x - 2*E^16*x - 75*x^2)/25,x]

[Out]

x - ((675 + 500*E^4 + 150*E^8 + 20*E^12 + E^16)*x^2)/25 - x^3

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, x]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {1}{25} \left (25-300 e^8 x-40 e^{12} x-2 e^{16} x+\left (-1350-1000 e^4\right ) x-75 x^2\right ) \, dx\\ &=\int \frac {1}{25} \left (25-2 e^{16} x+\left (-1350-1000 e^4\right ) x+\left (-300 e^8-40 e^{12}\right ) x-75 x^2\right ) \, dx\\ &=\int \frac {1}{25} \left (25+\left (-300 e^8-40 e^{12}\right ) x+\left (-1350-1000 e^4-2 e^{16}\right ) x-75 x^2\right ) \, dx\\ &=\int \frac {1}{25} \left (25+\left (-1350-1000 e^4-300 e^8-40 e^{12}-2 e^{16}\right ) x-75 x^2\right ) \, dx\\ &=\frac {1}{25} \int \left (25+\left (-1350-1000 e^4-300 e^8-40 e^{12}-2 e^{16}\right ) x-75 x^2\right ) \, dx\\ &=x-\frac {1}{25} \left (675+500 e^4+150 e^8+20 e^{12}+e^{16}\right ) x^2-x^3\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 34, normalized size = 1.55 \begin {gather*} x-\frac {1}{25} \left (675+500 e^4+150 e^8+20 e^{12}+e^{16}\right ) x^2-x^3 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(25 - 1350*x - 1000*E^4*x - 300*E^8*x - 40*E^12*x - 2*E^16*x - 75*x^2)/25,x]

[Out]

x - ((675 + 500*E^4 + 150*E^8 + 20*E^12 + E^16)*x^2)/25 - x^3

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fricas [B]  time = 0.54, size = 40, normalized size = 1.82 \begin {gather*} -x^{3} - \frac {1}{25} \, x^{2} e^{16} - \frac {4}{5} \, x^{2} e^{12} - 6 \, x^{2} e^{8} - 20 \, x^{2} e^{4} - 27 \, x^{2} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-2/25*x*exp(4)^4-8/5*x*exp(4)^3-12*x*exp(4)^2-40*x*exp(4)-3*x^2-54*x+1,x, algorithm="fricas")

[Out]

-x^3 - 1/25*x^2*e^16 - 4/5*x^2*e^12 - 6*x^2*e^8 - 20*x^2*e^4 - 27*x^2 + x

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giac [B]  time = 0.14, size = 40, normalized size = 1.82 \begin {gather*} -x^{3} - \frac {1}{25} \, x^{2} e^{16} - \frac {4}{5} \, x^{2} e^{12} - 6 \, x^{2} e^{8} - 20 \, x^{2} e^{4} - 27 \, x^{2} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-2/25*x*exp(4)^4-8/5*x*exp(4)^3-12*x*exp(4)^2-40*x*exp(4)-3*x^2-54*x+1,x, algorithm="giac")

[Out]

-x^3 - 1/25*x^2*e^16 - 4/5*x^2*e^12 - 6*x^2*e^8 - 20*x^2*e^4 - 27*x^2 + x

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maple [A]  time = 0.02, size = 36, normalized size = 1.64

method	result	size
norman	\(x +\left (-\frac {{\mathrm e}^{16}}{25}-\frac {4 \,{\mathrm e}^{12}}{5}-6 \,{\mathrm e}^{8}-20 \,{\mathrm e}^{4}-27\right ) x^{2}-x^{3}\)	\(36\)
gosper	\(-\frac {x \left (x \,{\mathrm e}^{16}+20 x \,{\mathrm e}^{12}+150 x \,{\mathrm e}^{8}+500 x \,{\mathrm e}^{4}+25 x^{2}+675 x -25\right )}{25}\)	\(39\)
risch	\(-\frac {x^{2} {\mathrm e}^{16}}{25}-\frac {4 x^{2} {\mathrm e}^{12}}{5}-6 x^{2} {\mathrm e}^{8}-20 x^{2} {\mathrm e}^{4}-x^{3}-27 x^{2}+x\)	\(41\)
default	\(-\frac {x^{2} {\mathrm e}^{16}}{25}-\frac {4 x^{2} {\mathrm e}^{12}}{5}-6 x^{2} {\mathrm e}^{8}-20 x^{2} {\mathrm e}^{4}-x^{3}-27 x^{2}+x\)	\(47\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-2/25*x*exp(4)^4-8/5*x*exp(4)^3-12*x*exp(4)^2-40*x*exp(4)-3*x^2-54*x+1,x,method=_RETURNVERBOSE)

[Out]

x+(-1/25*exp(4)^4-4/5*exp(4)^3-6*exp(4)^2-20*exp(4)-27)*x^2-x^3

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maxima [B]  time = 0.36, size = 40, normalized size = 1.82 \begin {gather*} -x^{3} - \frac {1}{25} \, x^{2} e^{16} - \frac {4}{5} \, x^{2} e^{12} - 6 \, x^{2} e^{8} - 20 \, x^{2} e^{4} - 27 \, x^{2} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-2/25*x*exp(4)^4-8/5*x*exp(4)^3-12*x*exp(4)^2-40*x*exp(4)-3*x^2-54*x+1,x, algorithm="maxima")

[Out]

-x^3 - 1/25*x^2*e^16 - 4/5*x^2*e^12 - 6*x^2*e^8 - 20*x^2*e^4 - 27*x^2 + x

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mupad [B]  time = 0.04, size = 30, normalized size = 1.36 \begin {gather*} -x^3+\left (-20\,{\mathrm {e}}^4-6\,{\mathrm {e}}^8-\frac {4\,{\mathrm {e}}^{12}}{5}-\frac {{\mathrm {e}}^{16}}{25}-27\right )\,x^2+x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1 - 40*x*exp(4) - 12*x*exp(8) - (8*x*exp(12))/5 - (2*x*exp(16))/25 - 3*x^2 - 54*x,x)

[Out]

x - x^2*(20*exp(4) + 6*exp(8) + (4*exp(12))/5 + exp(16)/25 + 27) - x^3

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sympy [A]  time = 0.07, size = 32, normalized size = 1.45 \begin {gather*} - x^{3} + x^{2} \left (- \frac {e^{16}}{25} - \frac {4 e^{12}}{5} - 6 e^{8} - 20 e^{4} - 27\right ) + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-2/25*x*exp(4)**4-8/5*x*exp(4)**3-12*x*exp(4)**2-40*x*exp(4)-3*x**2-54*x+1,x)

[Out]

-x**3 + x**2*(-exp(16)/25 - 4*exp(12)/5 - 6*exp(8) - 20*exp(4) - 27) + x

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