3.98.57 \(\int \frac {-48+e^3 (-175 x^6+240 x^7+99 x^8-120 x^9-44 x^{10})}{80 e^3} \, dx\)

Optimal. Leaf size=36 \[ 3-\frac {1}{5} x \left (\frac {3}{e^3}+\frac {1}{16} x^4 (5+2 x)^2 \left (-x+x^2\right )^2\right ) \]

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Rubi [A]  time = 0.02, antiderivative size = 44, normalized size of antiderivative = 1.22, 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 = {12} \begin {gather*} -\frac {x^{11}}{20}-\frac {3 x^{10}}{20}+\frac {11 x^9}{80}+\frac {3 x^8}{8}-\frac {5 x^7}{16}-\frac {3 x}{5 e^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-48 + E^3*(-175*x^6 + 240*x^7 + 99*x^8 - 120*x^9 - 44*x^10))/(80*E^3),x]

[Out]

(-3*x)/(5*E^3) - (5*x^7)/16 + (3*x^8)/8 + (11*x^9)/80 - (3*x^10)/20 - x^11/20

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} &=\frac {\int \left (-48+e^3 \left (-175 x^6+240 x^7+99 x^8-120 x^9-44 x^{10}\right )\right ) \, dx}{80 e^3}\\ &=-\frac {3 x}{5 e^3}+\frac {1}{80} \int \left (-175 x^6+240 x^7+99 x^8-120 x^9-44 x^{10}\right ) \, dx\\ &=-\frac {3 x}{5 e^3}-\frac {5 x^7}{16}+\frac {3 x^8}{8}+\frac {11 x^9}{80}-\frac {3 x^{10}}{20}-\frac {x^{11}}{20}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 44, normalized size = 1.22 \begin {gather*} -\frac {3 x}{5 e^3}-\frac {5 x^7}{16}+\frac {3 x^8}{8}+\frac {11 x^9}{80}-\frac {3 x^{10}}{20}-\frac {x^{11}}{20} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-48 + E^3*(-175*x^6 + 240*x^7 + 99*x^8 - 120*x^9 - 44*x^10))/(80*E^3),x]

[Out]

(-3*x)/(5*E^3) - (5*x^7)/16 + (3*x^8)/8 + (11*x^9)/80 - (3*x^10)/20 - x^11/20

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fricas [A]  time = 0.54, size = 37, normalized size = 1.03 \begin {gather*} -\frac {1}{80} \, {\left ({\left (4 \, x^{11} + 12 \, x^{10} - 11 \, x^{9} - 30 \, x^{8} + 25 \, x^{7}\right )} e^{3} + 48 \, x\right )} e^{\left (-3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/80*((-44*x^10-120*x^9+99*x^8+240*x^7-175*x^6)*exp(3)-48)/exp(3),x, algorithm="fricas")

[Out]

-1/80*((4*x^11 + 12*x^10 - 11*x^9 - 30*x^8 + 25*x^7)*e^3 + 48*x)*e^(-3)

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giac [A]  time = 0.14, size = 37, normalized size = 1.03 \begin {gather*} -\frac {1}{80} \, {\left ({\left (4 \, x^{11} + 12 \, x^{10} - 11 \, x^{9} - 30 \, x^{8} + 25 \, x^{7}\right )} e^{3} + 48 \, x\right )} e^{\left (-3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/80*((-44*x^10-120*x^9+99*x^8+240*x^7-175*x^6)*exp(3)-48)/exp(3),x, algorithm="giac")

[Out]

-1/80*((4*x^11 + 12*x^10 - 11*x^9 - 30*x^8 + 25*x^7)*e^3 + 48*x)*e^(-3)

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maple [A]  time = 0.04, size = 32, normalized size = 0.89

method	result	size
risch	\(-\frac {5 x^{7}}{16}+\frac {3 x^{8}}{8}+\frac {11 x^{9}}{80}-\frac {3 x^{10}}{20}-\frac {x^{11}}{20}-\frac {3 x \,{\mathrm e}^{-3}}{5}\)	\(32\)
norman	\(-\frac {5 x^{7}}{16}+\frac {3 x^{8}}{8}+\frac {11 x^{9}}{80}-\frac {3 x^{10}}{20}-\frac {x^{11}}{20}-\frac {3 x \,{\mathrm e}^{-3}}{5}\)	\(34\)
default	\(\frac {{\mathrm e}^{-3} \left ({\mathrm e}^{3} \left (-4 x^{11}-12 x^{10}+11 x^{9}+30 x^{8}-25 x^{7}\right )-48 x \right )}{80}\)	\(40\)
gosper	\(-\frac {x \left (4 \,{\mathrm e}^{3} x^{10}+12 \,{\mathrm e}^{3} x^{9}-11 \,{\mathrm e}^{3} x^{8}-30 x^{7} {\mathrm e}^{3}+25 x^{6} {\mathrm e}^{3}+48\right ) {\mathrm e}^{-3}}{80}\)	\(45\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/80*((-44*x^10-120*x^9+99*x^8+240*x^7-175*x^6)*exp(3)-48)/exp(3),x,method=_RETURNVERBOSE)

[Out]

-5/16*x^7+3/8*x^8+11/80*x^9-3/20*x^10-1/20*x^11-3/5*x*exp(-3)

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maxima [A]  time = 0.40, size = 37, normalized size = 1.03 \begin {gather*} -\frac {1}{80} \, {\left ({\left (4 \, x^{11} + 12 \, x^{10} - 11 \, x^{9} - 30 \, x^{8} + 25 \, x^{7}\right )} e^{3} + 48 \, x\right )} e^{\left (-3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/80*((-44*x^10-120*x^9+99*x^8+240*x^7-175*x^6)*exp(3)-48)/exp(3),x, algorithm="maxima")

[Out]

-1/80*((4*x^11 + 12*x^10 - 11*x^9 - 30*x^8 + 25*x^7)*e^3 + 48*x)*e^(-3)

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mupad [B]  time = 0.04, size = 31, normalized size = 0.86 \begin {gather*} -\frac {x^{11}}{20}-\frac {3\,x^{10}}{20}+\frac {11\,x^9}{80}+\frac {3\,x^8}{8}-\frac {5\,x^7}{16}-\frac {3\,{\mathrm {e}}^{-3}\,x}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-exp(-3)*((exp(3)*(175*x^6 - 240*x^7 - 99*x^8 + 120*x^9 + 44*x^10))/80 + 3/5),x)

[Out]

(3*x^8)/8 - (5*x^7)/16 - (3*x*exp(-3))/5 + (11*x^9)/80 - (3*x^10)/20 - x^11/20

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sympy [A]  time = 0.07, size = 39, normalized size = 1.08 \begin {gather*} - \frac {x^{11}}{20} - \frac {3 x^{10}}{20} + \frac {11 x^{9}}{80} + \frac {3 x^{8}}{8} - \frac {5 x^{7}}{16} - \frac {3 x}{5 e^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/80*((-44*x**10-120*x**9+99*x**8+240*x**7-175*x**6)*exp(3)-48)/exp(3),x)

[Out]

-x**11/20 - 3*x**10/20 + 11*x**9/80 + 3*x**8/8 - 5*x**7/16 - 3*x*exp(-3)/5

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