3.65.36 \(\int \frac {-72+2 e^{e^x+x} x^7-25 x^8}{2 x^7} \, dx\)

Optimal. Leaf size=19 \[ 1+e^{e^x}+\frac {6}{x^6}-\frac {25 x^2}{4} \]

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Rubi [A]  time = 0.02, antiderivative size = 18, normalized size of antiderivative = 0.95, number of steps used = 7, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {12, 14, 2282, 2194} \begin {gather*} \frac {6}{x^6}-\frac {25 x^2}{4}+e^{e^x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-72 + 2*E^(E^x + x)*x^7 - 25*x^8)/(2*x^7),x]

[Out]

E^E^x + 6/x^6 - (25*x^2)/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 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 2194

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rule 2282

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{2} \int \frac {-72+2 e^{e^x+x} x^7-25 x^8}{x^7} \, dx\\ &=\frac {1}{2} \int \left (2 e^{e^x+x}+\frac {-72-25 x^8}{x^7}\right ) \, dx\\ &=\frac {1}{2} \int \frac {-72-25 x^8}{x^7} \, dx+\int e^{e^x+x} \, dx\\ &=\frac {1}{2} \int \left (-\frac {72}{x^7}-25 x\right ) \, dx+\operatorname {Subst}\left (\int e^x \, dx,x,e^x\right )\\ &=e^{e^x}+\frac {6}{x^6}-\frac {25 x^2}{4}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 18, normalized size = 0.95 \begin {gather*} e^{e^x}+\frac {6}{x^6}-\frac {25 x^2}{4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-72 + 2*E^(E^x + x)*x^7 - 25*x^8)/(2*x^7),x]

[Out]

E^E^x + 6/x^6 - (25*x^2)/4

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fricas [B]  time = 0.71, size = 31, normalized size = 1.63 \begin {gather*} \frac {{\left (4 \, x^{6} e^{\left (x + e^{x}\right )} - {\left (25 \, x^{8} - 24\right )} e^{x}\right )} e^{\left (-x\right )}}{4 \, x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(2*x^7*exp(x)*exp(exp(x))-25*x^8-72)/x^7,x, algorithm="fricas")

[Out]

1/4*(4*x^6*e^(x + e^x) - (25*x^8 - 24)*e^x)*e^(-x)/x^6

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giac [B]  time = 0.12, size = 31, normalized size = 1.63 \begin {gather*} -\frac {{\left (25 \, x^{8} e^{x} - 4 \, x^{6} e^{\left (x + e^{x}\right )} - 24 \, e^{x}\right )} e^{\left (-x\right )}}{4 \, x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(2*x^7*exp(x)*exp(exp(x))-25*x^8-72)/x^7,x, algorithm="giac")

[Out]

-1/4*(25*x^8*e^x - 4*x^6*e^(x + e^x) - 24*e^x)*e^(-x)/x^6

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maple [A]  time = 0.04, size = 15, normalized size = 0.79

method	result	size
default	\(-\frac {25 x^{2}}{4}+\frac {6}{x^{6}}+{\mathrm e}^{{\mathrm e}^{x}}\)	\(15\)
risch	\(-\frac {25 x^{2}}{4}+\frac {6}{x^{6}}+{\mathrm e}^{{\mathrm e}^{x}}\)	\(15\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/2*(2*x^7*exp(x)*exp(exp(x))-25*x^8-72)/x^7,x,method=_RETURNVERBOSE)

[Out]

-25/4*x^2+6/x^6+exp(exp(x))

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maxima [A]  time = 0.37, size = 14, normalized size = 0.74 \begin {gather*} -\frac {25}{4} \, x^{2} + \frac {6}{x^{6}} + e^{\left (e^{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(2*x^7*exp(x)*exp(exp(x))-25*x^8-72)/x^7,x, algorithm="maxima")

[Out]

-25/4*x^2 + 6/x^6 + e^(e^x)

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mupad [B]  time = 4.04, size = 14, normalized size = 0.74 \begin {gather*} {\mathrm {e}}^{{\mathrm {e}}^x}-\frac {25\,x^2}{4}+\frac {6}{x^6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((25*x^8)/2 - x^7*exp(exp(x))*exp(x) + 36)/x^7,x)

[Out]

exp(exp(x)) - (25*x^2)/4 + 6/x^6

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sympy [A]  time = 0.14, size = 15, normalized size = 0.79 \begin {gather*} - \frac {25 x^{2}}{4} + e^{e^{x}} + \frac {6}{x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(2*x**7*exp(x)*exp(exp(x))-25*x**8-72)/x**7,x)

[Out]

-25*x**2/4 + exp(exp(x)) + 6/x**6

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