3.18.93 \(\int \frac {e^{\frac {5 x^6}{5-x+5 x^4}} (150 x^5-25 x^6+50 x^9)}{25-10 x+x^2+50 x^4-10 x^5+25 x^8} \, dx\)

Optimal. Leaf size=19 \[ e^{\frac {5 x^2}{5-\frac {-5+x}{x^4}}} \]

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Rubi [A]  time = 0.74, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 61, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.066, Rules used = {1594, 6688, 12, 6706} \begin {gather*} e^{\frac {5 x^6}{5 x^4-x+5}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^((5*x^6)/(5 - x + 5*x^4))*(150*x^5 - 25*x^6 + 50*x^9))/(25 - 10*x + x^2 + 50*x^4 - 10*x^5 + 25*x^8),x]

[Out]

E^((5*x^6)/(5 - x + 5*x^4))

Rule 12

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

Rule 1594

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.) + (c_.)*(x_)^(r_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^
(q - p) + c*x^(r - p))^n, x] /; FreeQ[{a, b, c, p, q, r}, x] && IntegerQ[n] && PosQ[q - p] && PosQ[r - p]

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 \frac {e^{\frac {5 x^6}{5-x+5 x^4}} x^5 \left (150-25 x+50 x^4\right )}{25-10 x+x^2+50 x^4-10 x^5+25 x^8} \, dx\\ &=\int \frac {25 e^{\frac {5 x^6}{5-x+5 x^4}} x^5 \left (6-x+2 x^4\right )}{\left (5-x+5 x^4\right )^2} \, dx\\ &=25 \int \frac {e^{\frac {5 x^6}{5-x+5 x^4}} x^5 \left (6-x+2 x^4\right )}{\left (5-x+5 x^4\right )^2} \, dx\\ &=e^{\frac {5 x^6}{5-x+5 x^4}}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.29, size = 19, normalized size = 1.00 \begin {gather*} e^{\frac {5 x^6}{5-x+5 x^4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((5*x^6)/(5 - x + 5*x^4))*(150*x^5 - 25*x^6 + 50*x^9))/(25 - 10*x + x^2 + 50*x^4 - 10*x^5 + 25*x^
8),x]

[Out]

E^((5*x^6)/(5 - x + 5*x^4))

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fricas [A]  time = 0.86, size = 18, normalized size = 0.95 \begin {gather*} e^{\left (\frac {5 \, x^{6}}{5 \, x^{4} - x + 5}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((50*x^9-25*x^6+150*x^5)*exp(5*x^6/(5*x^4-x+5))/(25*x^8-10*x^5+50*x^4+x^2-10*x+25),x, algorithm="fric
as")

[Out]

e^(5*x^6/(5*x^4 - x + 5))

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giac [A]  time = 0.30, size = 18, normalized size = 0.95 \begin {gather*} e^{\left (\frac {5 \, x^{6}}{5 \, x^{4} - x + 5}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((50*x^9-25*x^6+150*x^5)*exp(5*x^6/(5*x^4-x+5))/(25*x^8-10*x^5+50*x^4+x^2-10*x+25),x, algorithm="giac
")

[Out]

e^(5*x^6/(5*x^4 - x + 5))

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maple [A]  time = 0.09, size = 19, normalized size = 1.00

method	result	size
gosper	\({\mathrm e}^{\frac {5 x^{6}}{5 x^{4}-x +5}}\)	\(19\)
risch	\({\mathrm e}^{\frac {5 x^{6}}{5 x^{4}-x +5}}\)	\(19\)
norman	\(\frac {-x \,{\mathrm e}^{\frac {5 x^{6}}{5 x^{4}-x +5}}+5 x^{4} {\mathrm e}^{\frac {5 x^{6}}{5 x^{4}-x +5}}+5 \,{\mathrm e}^{\frac {5 x^{6}}{5 x^{4}-x +5}}}{5 x^{4}-x +5}\)	\(79\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((50*x^9-25*x^6+150*x^5)*exp(5*x^6/(5*x^4-x+5))/(25*x^8-10*x^5+50*x^4+x^2-10*x+25),x,method=_RETURNVERBOSE)

[Out]

exp(5*x^6/(5*x^4-x+5))

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maxima [B]  time = 0.63, size = 38, normalized size = 2.00 \begin {gather*} e^{\left (x^{2} + \frac {x^{3}}{5 \, x^{4} - x + 5} - \frac {5 \, x^{2}}{5 \, x^{4} - x + 5}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((50*x^9-25*x^6+150*x^5)*exp(5*x^6/(5*x^4-x+5))/(25*x^8-10*x^5+50*x^4+x^2-10*x+25),x, algorithm="maxi
ma")

[Out]

e^(x^2 + x^3/(5*x^4 - x + 5) - 5*x^2/(5*x^4 - x + 5))

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mupad [B]  time = 1.25, size = 18, normalized size = 0.95 \begin {gather*} {\mathrm {e}}^{\frac {5\,x^6}{5\,x^4-x+5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp((5*x^6)/(5*x^4 - x + 5))*(150*x^5 - 25*x^6 + 50*x^9))/(x^2 - 10*x + 50*x^4 - 10*x^5 + 25*x^8 + 25),x)

[Out]

exp((5*x^6)/(5*x^4 - x + 5))

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sympy [A]  time = 0.23, size = 14, normalized size = 0.74 \begin {gather*} e^{\frac {5 x^{6}}{5 x^{4} - x + 5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((50*x**9-25*x**6+150*x**5)*exp(5*x**6/(5*x**4-x+5))/(25*x**8-10*x**5+50*x**4+x**2-10*x+25),x)

[Out]

exp(5*x**6/(5*x**4 - x + 5))

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