3.71.42 \(\int \frac {217500+1064450 x+1250 e^{16} x+95700 x^2+1936 x^3+e^8 (-7500-72500 x-3300 x^2)}{5625} \, dx\)

Optimal. Leaf size=30 \[ x^2 \left (10+\frac {2}{x}-\frac {x}{25}+\frac {1}{3} \left (-1-e^8+x\right )\right )^2 \]

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Rubi [B]  time = 0.02, antiderivative size = 62, normalized size of antiderivative = 2.07, number of steps used = 4, number of rules used = 2, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.051, Rules used = {6, 12} \begin {gather*} \frac {484 x^4}{5625}-\frac {44 e^8 x^3}{225}+\frac {1276 x^3}{225}+\frac {1}{225} \left (21289+25 e^{16}\right ) x^2-\frac {58 e^8 x^2}{9}-\frac {4 e^8 x}{3}+\frac {116 x}{3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(217500 + 1064450*x + 1250*E^16*x + 95700*x^2 + 1936*x^3 + E^8*(-7500 - 72500*x - 3300*x^2))/5625,x]

[Out]

(116*x)/3 - (4*E^8*x)/3 - (58*E^8*x^2)/9 + ((21289 + 25*E^16)*x^2)/225 + (1276*x^3)/225 - (44*E^8*x^3)/225 + (
484*x^4)/5625

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 {217500+\left (1064450+1250 e^{16}\right ) x+95700 x^2+1936 x^3+e^8 \left (-7500-72500 x-3300 x^2\right )}{5625} \, dx\\ &=\frac {\int \left (217500+\left (1064450+1250 e^{16}\right ) x+95700 x^2+1936 x^3+e^8 \left (-7500-72500 x-3300 x^2\right )\right ) \, dx}{5625}\\ &=\frac {116 x}{3}+\frac {1}{225} \left (21289+25 e^{16}\right ) x^2+\frac {1276 x^3}{225}+\frac {484 x^4}{5625}+\frac {e^8 \int \left (-7500-72500 x-3300 x^2\right ) \, dx}{5625}\\ &=\frac {116 x}{3}-\frac {4 e^8 x}{3}-\frac {58 e^8 x^2}{9}+\frac {1}{225} \left (21289+25 e^{16}\right ) x^2+\frac {1276 x^3}{225}-\frac {44 e^8 x^3}{225}+\frac {484 x^4}{5625}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 57, normalized size = 1.90 \begin {gather*} \frac {2 \left (108750 x-3750 e^8 x+\frac {532225 x^2}{2}-18125 e^8 x^2+\frac {625 e^{16} x^2}{2}+15950 x^3-550 e^8 x^3+242 x^4\right )}{5625} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(217500 + 1064450*x + 1250*E^16*x + 95700*x^2 + 1936*x^3 + E^8*(-7500 - 72500*x - 3300*x^2))/5625,x]

[Out]

(2*(108750*x - 3750*E^8*x + (532225*x^2)/2 - 18125*E^8*x^2 + (625*E^16*x^2)/2 + 15950*x^3 - 550*E^8*x^3 + 242*
x^4))/5625

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fricas [B]  time = 0.65, size = 44, normalized size = 1.47 \begin {gather*} \frac {484}{5625} \, x^{4} + \frac {1276}{225} \, x^{3} + \frac {1}{9} \, x^{2} e^{16} + \frac {21289}{225} \, x^{2} - \frac {2}{225} \, {\left (22 \, x^{3} + 725 \, x^{2} + 150 \, x\right )} e^{8} + \frac {116}{3} \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2/9*x*exp(8)^2+1/5625*(-3300*x^2-72500*x-7500)*exp(8)+1936/5625*x^3+1276/75*x^2+42578/225*x+116/3,x,
algorithm="fricas")

[Out]

484/5625*x^4 + 1276/225*x^3 + 1/9*x^2*e^16 + 21289/225*x^2 - 2/225*(22*x^3 + 725*x^2 + 150*x)*e^8 + 116/3*x

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giac [B]  time = 0.14, size = 44, normalized size = 1.47 \begin {gather*} \frac {484}{5625} \, x^{4} + \frac {1276}{225} \, x^{3} + \frac {1}{9} \, x^{2} e^{16} + \frac {21289}{225} \, x^{2} - \frac {2}{225} \, {\left (22 \, x^{3} + 725 \, x^{2} + 150 \, x\right )} e^{8} + \frac {116}{3} \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2/9*x*exp(8)^2+1/5625*(-3300*x^2-72500*x-7500)*exp(8)+1936/5625*x^3+1276/75*x^2+42578/225*x+116/3,x,
algorithm="giac")

[Out]

484/5625*x^4 + 1276/225*x^3 + 1/9*x^2*e^16 + 21289/225*x^2 - 2/225*(22*x^3 + 725*x^2 + 150*x)*e^8 + 116/3*x

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maple [A]  time = 0.02, size = 41, normalized size = 1.37

method	result	size
norman	\(\left (-\frac {44 \,{\mathrm e}^{8}}{225}+\frac {1276}{225}\right ) x^{3}+\left (-\frac {4 \,{\mathrm e}^{8}}{3}+\frac {116}{3}\right ) x +\left (\frac {{\mathrm e}^{16}}{9}-\frac {58 \,{\mathrm e}^{8}}{9}+\frac {21289}{225}\right ) x^{2}+\frac {484 x^{4}}{5625}\)	\(41\)
gosper	\(\frac {x \left (625 x \,{\mathrm e}^{16}-1100 x^{2} {\mathrm e}^{8}+484 x^{3}-36250 x \,{\mathrm e}^{8}+31900 x^{2}-7500 \,{\mathrm e}^{8}+532225 x +217500\right )}{5625}\)	\(42\)
risch	\(\frac {x^{2} {\mathrm e}^{16}}{9}-\frac {44 x^{3} {\mathrm e}^{8}}{225}-\frac {58 x^{2} {\mathrm e}^{8}}{9}-\frac {4 x \,{\mathrm e}^{8}}{3}+\frac {484 x^{4}}{5625}+\frac {1276 x^{3}}{225}+\frac {21289 x^{2}}{225}+\frac {116 x}{3}\)	\(46\)
default	\(\frac {x^{2} {\mathrm e}^{16}}{9}+\frac {{\mathrm e}^{8} \left (-1100 x^{3}-36250 x^{2}-7500 x \right )}{5625}+\frac {484 x^{4}}{5625}+\frac {1276 x^{3}}{225}+\frac {21289 x^{2}}{225}+\frac {116 x}{3}\)	\(47\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(2/9*x*exp(8)^2+1/5625*(-3300*x^2-72500*x-7500)*exp(8)+1936/5625*x^3+1276/75*x^2+42578/225*x+116/3,x,method
=_RETURNVERBOSE)

[Out]

(-44/225*exp(8)+1276/225)*x^3+(-4/3*exp(8)+116/3)*x+(1/9*exp(8)^2-58/9*exp(8)+21289/225)*x^2+484/5625*x^4

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maxima [B]  time = 0.38, size = 44, normalized size = 1.47 \begin {gather*} \frac {484}{5625} \, x^{4} + \frac {1276}{225} \, x^{3} + \frac {1}{9} \, x^{2} e^{16} + \frac {21289}{225} \, x^{2} - \frac {2}{225} \, {\left (22 \, x^{3} + 725 \, x^{2} + 150 \, x\right )} e^{8} + \frac {116}{3} \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2/9*x*exp(8)^2+1/5625*(-3300*x^2-72500*x-7500)*exp(8)+1936/5625*x^3+1276/75*x^2+42578/225*x+116/3,x,
algorithm="maxima")

[Out]

484/5625*x^4 + 1276/225*x^3 + 1/9*x^2*e^16 + 21289/225*x^2 - 2/225*(22*x^3 + 725*x^2 + 150*x)*e^8 + 116/3*x

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mupad [B]  time = 4.09, size = 40, normalized size = 1.33 \begin {gather*} \frac {484\,x^4}{5625}+\left (\frac {1276}{225}-\frac {44\,{\mathrm {e}}^8}{225}\right )\,x^3+\left (\frac {{\mathrm {e}}^{16}}{9}-\frac {58\,{\mathrm {e}}^8}{9}+\frac {21289}{225}\right )\,x^2+\left (\frac {116}{3}-\frac {4\,{\mathrm {e}}^8}{3}\right )\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((42578*x)/225 - (exp(8)*(72500*x + 3300*x^2 + 7500))/5625 + (2*x*exp(16))/9 + (1276*x^2)/75 + (1936*x^3)/5
625 + 116/3,x)

[Out]

x^2*(exp(16)/9 - (58*exp(8))/9 + 21289/225) - x^3*((44*exp(8))/225 - 1276/225) + (484*x^4)/5625 - x*((4*exp(8)
)/3 - 116/3)

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sympy [B]  time = 0.07, size = 49, normalized size = 1.63 \begin {gather*} \frac {484 x^{4}}{5625} + x^{3} \left (\frac {1276}{225} - \frac {44 e^{8}}{225}\right ) + x^{2} \left (- \frac {58 e^{8}}{9} + \frac {21289}{225} + \frac {e^{16}}{9}\right ) + x \left (\frac {116}{3} - \frac {4 e^{8}}{3}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2/9*x*exp(8)**2+1/5625*(-3300*x**2-72500*x-7500)*exp(8)+1936/5625*x**3+1276/75*x**2+42578/225*x+116/
3,x)

[Out]

484*x**4/5625 + x**3*(1276/225 - 44*exp(8)/225) + x**2*(-58*exp(8)/9 + 21289/225 + exp(16)/9) + x*(116/3 - 4*e
xp(8)/3)

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