3.36.41 \(\int \frac {e^{\frac {625}{25 x^3-10 x^4+x^5}} (-9375+3125 x)+e^x (125 x^2-200 x^3+90 x^4-16 x^5+x^6)}{-125 x^4+75 x^5-15 x^6+x^7} \, dx\)

Optimal. Leaf size=24 \[ -e^{\frac {625}{(5-x)^2 x^3}}+\frac {e^x}{x} \]

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Rubi [A]  time = 0.37, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 77, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.039, Rules used = {6688, 6706, 2197} \begin {gather*} \frac {e^x}{x}-e^{\frac {625}{(5-x)^2 x^3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^(625/(25*x^3 - 10*x^4 + x^5))*(-9375 + 3125*x) + E^x*(125*x^2 - 200*x^3 + 90*x^4 - 16*x^5 + x^6))/(-125
*x^4 + 75*x^5 - 15*x^6 + x^7),x]

[Out]

-E^(625/((5 - x)^2*x^3)) + E^x/x

Rule 2197

Int[(F_)^((c_.)*(v_))*(u_)^(m_.)*(w_), x_Symbol] :> With[{b = Coefficient[v, x, 1], d = Coefficient[u, x, 0],
e = Coefficient[u, x, 1], f = Coefficient[w, x, 0], g = Coefficient[w, x, 1]}, Simp[(g*u^(m + 1)*F^(c*v))/(b*c
*e*Log[F]), x] /; EqQ[e*g*(m + 1) - b*c*(e*f - d*g)*Log[F], 0]] /; FreeQ[{F, c, m}, x] && LinearQ[{u, v, w}, x
]

Rule 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, 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} &=\int \left (\frac {3125 e^{\frac {625}{(-5+x)^2 x^3}} (-3+x)}{(-5+x)^3 x^4}+\frac {e^x (-1+x)}{x^2}\right ) \, dx\\ &=3125 \int \frac {e^{\frac {625}{(-5+x)^2 x^3}} (-3+x)}{(-5+x)^3 x^4} \, dx+\int \frac {e^x (-1+x)}{x^2} \, dx\\ &=-e^{\frac {625}{(5-x)^2 x^3}}+\frac {e^x}{x}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.14, size = 22, normalized size = 0.92 \begin {gather*} -e^{\frac {625}{(-5+x)^2 x^3}}+\frac {e^x}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(625/(25*x^3 - 10*x^4 + x^5))*(-9375 + 3125*x) + E^x*(125*x^2 - 200*x^3 + 90*x^4 - 16*x^5 + x^6))
/(-125*x^4 + 75*x^5 - 15*x^6 + x^7),x]

[Out]

-E^(625/((-5 + x)^2*x^3)) + E^x/x

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fricas [A]  time = 1.16, size = 31, normalized size = 1.29 \begin {gather*} -\frac {x e^{\left (\frac {625}{x^{5} - 10 \, x^{4} + 25 \, x^{3}}\right )} - e^{x}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^6-16*x^5+90*x^4-200*x^3+125*x^2)*exp(x)+(3125*x-9375)*exp(625/(x^5-10*x^4+25*x^3)))/(x^7-15*x^6+
75*x^5-125*x^4),x, algorithm="fricas")

[Out]

-(x*e^(625/(x^5 - 10*x^4 + 25*x^3)) - e^x)/x

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giac [A]  time = 0.23, size = 28, normalized size = 1.17 \begin {gather*} \frac {e^{x}}{x} - e^{\left (\frac {625}{x^{5} - 10 \, x^{4} + 25 \, x^{3}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^6-16*x^5+90*x^4-200*x^3+125*x^2)*exp(x)+(3125*x-9375)*exp(625/(x^5-10*x^4+25*x^3)))/(x^7-15*x^6+
75*x^5-125*x^4),x, algorithm="giac")

[Out]

e^x/x - e^(625/(x^5 - 10*x^4 + 25*x^3))

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maple [A]  time = 0.16, size = 21, normalized size = 0.88

method	result	size
risch	\(\frac {{\mathrm e}^{x}}{x}-{\mathrm e}^{\frac {625}{x^{3} \left (x -5\right )^{2}}}\)	\(21\)
norman	\(\frac {{\mathrm e}^{x} x^{4}-25 x^{3} {\mathrm e}^{\frac {625}{x^{5}-10 x^{4}+25 x^{3}}}+10 x^{4} {\mathrm e}^{\frac {625}{x^{5}-10 x^{4}+25 x^{3}}}-x^{5} {\mathrm e}^{\frac {625}{x^{5}-10 x^{4}+25 x^{3}}}+25 \,{\mathrm e}^{x} x^{2}-10 \,{\mathrm e}^{x} x^{3}}{x^{3} \left (x -5\right )^{2}}\)	\(103\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^6-16*x^5+90*x^4-200*x^3+125*x^2)*exp(x)+(3125*x-9375)*exp(625/(x^5-10*x^4+25*x^3)))/(x^7-15*x^6+75*x^5
-125*x^4),x,method=_RETURNVERBOSE)

[Out]

exp(x)/x-exp(625/x^3/(x-5)^2)

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maxima [B]  time = 1.92, size = 57, normalized size = 2.38 \begin {gather*} -\frac {{\left (x e^{\left (\frac {5}{x^{2} - 10 \, x + 25} + \frac {3}{x} + \frac {10}{x^{2}} + \frac {25}{x^{3}}\right )} - e^{\left (x + \frac {3}{x - 5}\right )}\right )} e^{\left (-\frac {3}{x - 5}\right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^6-16*x^5+90*x^4-200*x^3+125*x^2)*exp(x)+(3125*x-9375)*exp(625/(x^5-10*x^4+25*x^3)))/(x^7-15*x^6+
75*x^5-125*x^4),x, algorithm="maxima")

[Out]

-(x*e^(5/(x^2 - 10*x + 25) + 3/x + 10/x^2 + 25/x^3) - e^(x + 3/(x - 5)))*e^(-3/(x - 5))/x

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mupad [B]  time = 2.32, size = 28, normalized size = 1.17 \begin {gather*} \frac {{\mathrm {e}}^x}{x}-{\mathrm {e}}^{\frac {625}{x^5-10\,x^4+25\,x^3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(x)*(125*x^2 - 200*x^3 + 90*x^4 - 16*x^5 + x^6) + exp(625/(25*x^3 - 10*x^4 + x^5))*(3125*x - 9375))/(
125*x^4 - 75*x^5 + 15*x^6 - x^7),x)

[Out]

exp(x)/x - exp(625/(25*x^3 - 10*x^4 + x^5))

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sympy [A]  time = 0.32, size = 20, normalized size = 0.83 \begin {gather*} - e^{\frac {625}{x^{5} - 10 x^{4} + 25 x^{3}}} + \frac {e^{x}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x**6-16*x**5+90*x**4-200*x**3+125*x**2)*exp(x)+(3125*x-9375)*exp(625/(x**5-10*x**4+25*x**3)))/(x**
7-15*x**6+75*x**5-125*x**4),x)

[Out]

-exp(625/(x**5 - 10*x**4 + 25*x**3)) + exp(x)/x

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