3.25.87 \(\int \frac {e^{\frac {125+50 x+5 x^2+153 x^6+x^7}{5 x^6}} (-750-250 x-20 x^2+x^7)}{5 x^7} \, dx\)

Optimal. Leaf size=20 \[ e^{30+\frac {3+x}{5}+\frac {(5+x)^2}{x^6}} \]

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Rubi [A]  time = 0.31, antiderivative size = 27, normalized size of antiderivative = 1.35, number of steps used = 2, number of rules used = 2, integrand size = 47, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {12, 6706} \begin {gather*} e^{\frac {x^7+153 x^6+5 x^2+50 x+125}{5 x^6}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^((125 + 50*x + 5*x^2 + 153*x^6 + x^7)/(5*x^6))*(-750 - 250*x - 20*x^2 + x^7))/(5*x^7),x]

[Out]

E^((125 + 50*x + 5*x^2 + 153*x^6 + x^7)/(5*x^6))

Rule 12

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

Rule 6706

Int[(F_)^(v_)*(u_), x_Symbol] :> With[{q = DerivativeDivides[v, u, x]}, Simp[(q*F^v)/Log[F], x] /;  !FalseQ[q]
] /; FreeQ[F, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{5} \int \frac {e^{\frac {125+50 x+5 x^2+153 x^6+x^7}{5 x^6}} \left (-750-250 x-20 x^2+x^7\right )}{x^7} \, dx\\ &=e^{\frac {125+50 x+5 x^2+153 x^6+x^7}{5 x^6}}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.14, size = 24, normalized size = 1.20 \begin {gather*} e^{\frac {153}{5}+\frac {25}{x^6}+\frac {10}{x^5}+\frac {1}{x^4}+\frac {x}{5}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((125 + 50*x + 5*x^2 + 153*x^6 + x^7)/(5*x^6))*(-750 - 250*x - 20*x^2 + x^7))/(5*x^7),x]

[Out]

E^(153/5 + 25/x^6 + 10/x^5 + x^(-4) + x/5)

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fricas [A]  time = 0.51, size = 24, normalized size = 1.20 \begin {gather*} e^{\left (\frac {x^{7} + 153 \, x^{6} + 5 \, x^{2} + 50 \, x + 125}{5 \, x^{6}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5*(x^7-20*x^2-250*x-750)*exp(1/5*(x^7+153*x^6+5*x^2+50*x+125)/x^6)/x^7,x, algorithm="fricas")

[Out]

e^(1/5*(x^7 + 153*x^6 + 5*x^2 + 50*x + 125)/x^6)

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giac [A]  time = 0.17, size = 19, normalized size = 0.95 \begin {gather*} e^{\left (\frac {1}{5} \, x + \frac {1}{x^{4}} + \frac {10}{x^{5}} + \frac {25}{x^{6}} + \frac {153}{5}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5*(x^7-20*x^2-250*x-750)*exp(1/5*(x^7+153*x^6+5*x^2+50*x+125)/x^6)/x^7,x, algorithm="giac")

[Out]

e^(1/5*x + 1/x^4 + 10/x^5 + 25/x^6 + 153/5)

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maple [A]  time = 0.06, size = 25, normalized size = 1.25

method	result	size
gosper	\({\mathrm e}^{\frac {x^{7}+153 x^{6}+5 x^{2}+50 x +125}{5 x^{6}}}\)	\(25\)
norman	\({\mathrm e}^{\frac {x^{7}+153 x^{6}+5 x^{2}+50 x +125}{5 x^{6}}}\)	\(25\)
risch	\({\mathrm e}^{\frac {x^{7}+153 x^{6}+5 x^{2}+50 x +125}{5 x^{6}}}\)	\(25\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/5*(x^7-20*x^2-250*x-750)*exp(1/5*(x^7+153*x^6+5*x^2+50*x+125)/x^6)/x^7,x,method=_RETURNVERBOSE)

[Out]

exp(1/5*(x^7+153*x^6+5*x^2+50*x+125)/x^6)

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maxima [A]  time = 0.51, size = 19, normalized size = 0.95 \begin {gather*} e^{\left (\frac {1}{5} \, x + \frac {1}{x^{4}} + \frac {10}{x^{5}} + \frac {25}{x^{6}} + \frac {153}{5}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5*(x^7-20*x^2-250*x-750)*exp(1/5*(x^7+153*x^6+5*x^2+50*x+125)/x^6)/x^7,x, algorithm="maxima")

[Out]

e^(1/5*x + 1/x^4 + 10/x^5 + 25/x^6 + 153/5)

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mupad [B]  time = 1.43, size = 23, normalized size = 1.15 \begin {gather*} {\mathrm {e}}^{x/5}\,{\mathrm {e}}^{\frac {1}{x^4}}\,{\mathrm {e}}^{153/5}\,{\mathrm {e}}^{\frac {10}{x^5}}\,{\mathrm {e}}^{\frac {25}{x^6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp((10*x + x^2 + (153*x^6)/5 + x^7/5 + 25)/x^6)*(250*x + 20*x^2 - x^7 + 750))/(5*x^7),x)

[Out]

exp(x/5)*exp(1/x^4)*exp(153/5)*exp(10/x^5)*exp(25/x^6)

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sympy [A]  time = 0.16, size = 24, normalized size = 1.20 \begin {gather*} e^{\frac {\frac {x^{7}}{5} + \frac {153 x^{6}}{5} + x^{2} + 10 x + 25}{x^{6}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5*(x**7-20*x**2-250*x-750)*exp(1/5*(x**7+153*x**6+5*x**2+50*x+125)/x**6)/x**7,x)

[Out]

exp((x**7/5 + 153*x**6/5 + x**2 + 10*x + 25)/x**6)

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