3.78.10 \(\int \frac {e^{-\frac {625+x^8 \log (x)}{2 x^7}} (4375+2 x^7+2 e^{\frac {625+x^8 \log (x)}{2 x^7}} x^7-x^8-x^8 \log (x))}{2 x^7} \, dx\)

Optimal. Leaf size=19 \[ \left (1+e^{-\frac {1}{2} x \left (\frac {625}{x^8}+\log (x)\right )}\right ) x \]

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Rubi [B]  time = 1.45, antiderivative size = 67, normalized size of antiderivative = 3.53, number of steps used = 6, number of rules used = 5, integrand size = 65, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {12, 6742, 6688, 8, 2288} \begin {gather*} x-\frac {e^{-\frac {625}{2 x^7}} x^{-\frac {x}{2}-7} \left (-x^8+x^8 (-\log (x))+4375\right )}{\frac {x^7+8 x^7 \log (x)}{x^7}-\frac {7 \left (x^8 \log (x)+625\right )}{x^8}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(4375 + 2*x^7 + 2*E^((625 + x^8*Log[x])/(2*x^7))*x^7 - x^8 - x^8*Log[x])/(2*E^((625 + x^8*Log[x])/(2*x^7))
*x^7),x]

[Out]

x - (x^(-7 - x/2)*(4375 - x^8 - x^8*Log[x]))/(E^(625/(2*x^7))*((x^7 + 8*x^7*Log[x])/x^7 - (7*(625 + x^8*Log[x]
))/x^8))

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 12

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

Rule 2288

Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = (v*y)/(Log[F]*D[u, x])}, Simp[F^u*z, x] /; EqQ[D[z,
x], w*y]] /; FreeQ[F, x]

Rule 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{2} \int \frac {e^{-\frac {625+x^8 \log (x)}{2 x^7}} \left (4375+2 x^7+2 e^{\frac {625+x^8 \log (x)}{2 x^7}} x^7-x^8-x^8 \log (x)\right )}{x^7} \, dx\\ &=\frac {1}{2} \int \left (2 e^{\frac {625}{2 x^7}-\frac {625+x^8 \log (x)}{2 x^7}} x^{x/2}+\frac {e^{-\frac {625+x^8 \log (x)}{2 x^7}} \left (4375+2 x^7-x^8-x^8 \log (x)\right )}{x^7}\right ) \, dx\\ &=\frac {1}{2} \int \frac {e^{-\frac {625+x^8 \log (x)}{2 x^7}} \left (4375+2 x^7-x^8-x^8 \log (x)\right )}{x^7} \, dx+\int e^{\frac {625}{2 x^7}-\frac {625+x^8 \log (x)}{2 x^7}} x^{x/2} \, dx\\ &=-\frac {e^{-\frac {625}{2 x^7}} x^{-7-\frac {x}{2}} \left (4375-x^8-x^8 \log (x)\right )}{\frac {x^7+8 x^7 \log (x)}{x^7}-\frac {7 \left (625+x^8 \log (x)\right )}{x^8}}+\int 1 \, dx\\ &=x-\frac {e^{-\frac {625}{2 x^7}} x^{-7-\frac {x}{2}} \left (4375-x^8-x^8 \log (x)\right )}{\frac {x^7+8 x^7 \log (x)}{x^7}-\frac {7 \left (625+x^8 \log (x)\right )}{x^8}}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.35, size = 28, normalized size = 1.47 \begin {gather*} \frac {1}{2} \left (2 x+2 e^{-\frac {625}{2 x^7}} x^{1-\frac {x}{2}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(4375 + 2*x^7 + 2*E^((625 + x^8*Log[x])/(2*x^7))*x^7 - x^8 - x^8*Log[x])/(2*E^((625 + x^8*Log[x])/(2
*x^7))*x^7),x]

[Out]

(2*x + (2*x^(1 - x/2))/E^(625/(2*x^7)))/2

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fricas [B]  time = 0.56, size = 33, normalized size = 1.74 \begin {gather*} {\left (x e^{\left (\frac {x^{8} \log \relax (x) + 625}{2 \, x^{7}}\right )} + x\right )} e^{\left (-\frac {x^{8} \log \relax (x) + 625}{2 \, x^{7}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(2*x^7*exp(1/2*(x^8*log(x)+625)/x^7)-x^8*log(x)-x^8+2*x^7+4375)/x^7/exp(1/2*(x^8*log(x)+625)/x^7
),x, algorithm="fricas")

[Out]

(x*e^(1/2*(x^8*log(x) + 625)/x^7) + x)*e^(-1/2*(x^8*log(x) + 625)/x^7)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {{\left (x^{8} \log \relax (x) + x^{8} - 2 \, x^{7} e^{\left (\frac {x^{8} \log \relax (x) + 625}{2 \, x^{7}}\right )} - 2 \, x^{7} - 4375\right )} e^{\left (-\frac {x^{8} \log \relax (x) + 625}{2 \, x^{7}}\right )}}{2 \, x^{7}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(2*x^7*exp(1/2*(x^8*log(x)+625)/x^7)-x^8*log(x)-x^8+2*x^7+4375)/x^7/exp(1/2*(x^8*log(x)+625)/x^7
),x, algorithm="giac")

[Out]

integrate(-1/2*(x^8*log(x) + x^8 - 2*x^7*e^(1/2*(x^8*log(x) + 625)/x^7) - 2*x^7 - 4375)*e^(-1/2*(x^8*log(x) +
625)/x^7)/x^7, x)

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maple [A]  time = 0.04, size = 18, normalized size = 0.95

method	result	size
risch	\(x +x \,x^{-\frac {x}{2}} {\mathrm e}^{-\frac {625}{2 x^{7}}}\)	\(18\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/2*(2*x^7*exp(1/2*(x^8*ln(x)+625)/x^7)-x^8*ln(x)-x^8+2*x^7+4375)/x^7/exp(1/2*(x^8*ln(x)+625)/x^7),x,metho
d=_RETURNVERBOSE)

[Out]

x+x/(x^(1/2*x))*exp(-625/2/x^7)

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maxima [A]  time = 0.40, size = 16, normalized size = 0.84 \begin {gather*} x e^{\left (-\frac {1}{2} \, x \log \relax (x) - \frac {625}{2 \, x^{7}}\right )} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(2*x^7*exp(1/2*(x^8*log(x)+625)/x^7)-x^8*log(x)-x^8+2*x^7+4375)/x^7/exp(1/2*(x^8*log(x)+625)/x^7
),x, algorithm="maxima")

[Out]

x*e^(-1/2*x*log(x) - 625/2/x^7) + x

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mupad [B]  time = 5.34, size = 16, normalized size = 0.84 \begin {gather*} x\,\left ({\mathrm {e}}^{-\frac {x\,\ln \relax (x)}{2}-\frac {625}{2\,x^7}}+1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(-((x^8*log(x))/2 + 625/2)/x^7)*(x^7*exp(((x^8*log(x))/2 + 625/2)/x^7) - (x^8*log(x))/2 + x^7 - x^8/2
+ 4375/2))/x^7,x)

[Out]

x*(exp(- (x*log(x))/2 - 625/(2*x^7)) + 1)

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sympy [A]  time = 0.40, size = 19, normalized size = 1.00 \begin {gather*} x + x e^{- \frac {\frac {x^{8} \log {\relax (x )}}{2} + \frac {625}{2}}{x^{7}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(2*x**7*exp(1/2*(x**8*ln(x)+625)/x**7)-x**8*ln(x)-x**8+2*x**7+4375)/x**7/exp(1/2*(x**8*ln(x)+625
)/x**7),x)

[Out]

x + x*exp(-(x**8*log(x)/2 + 625/2)/x**7)

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