3.81.5 \(\int \frac {-25 e^2+x^2}{25 e^2 x+x^3} \, dx\)

Optimal. Leaf size=17 \[ \log \left (\frac {25+\frac {x^2}{e^2}}{25 x}\right ) \]

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Rubi [A]  time = 0.03, antiderivative size = 15, normalized size of antiderivative = 0.88, number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {1593, 446, 72} \begin {gather*} \log \left (x^2+25 e^2\right )-\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-25*E^2 + x^2)/(25*E^2*x + x^3),x]

[Out]

-Log[x] + Log[25*E^2 + x^2]

Rule 72

Int[((e_.) + (f_.)*(x_))^(p_.)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Int[ExpandIntegrand[(
e + f*x)^p/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IntegerQ[p]

Rule 446

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rule 1593

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

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 15, normalized size = 0.88 \begin {gather*} -\log (x)+\log \left (25 e^2+x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-25*E^2 + x^2)/(25*E^2*x + x^3),x]

[Out]

-Log[x] + Log[25*E^2 + x^2]

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fricas [A]  time = 1.23, size = 14, normalized size = 0.82 \begin {gather*} \log \left (x^{2} + 25 \, e^{2}\right ) - \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-25*exp(2)+x^2)/(25*exp(2)*x+x^3),x, algorithm="fricas")

[Out]

log(x^2 + 25*e^2) - log(x)

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giac [A]  time = 0.20, size = 16, normalized size = 0.94 \begin {gather*} \log \left (x^{2} + 25 \, e^{2}\right ) - \frac {1}{2} \, \log \left (x^{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-25*exp(2)+x^2)/(25*exp(2)*x+x^3),x, algorithm="giac")

[Out]

log(x^2 + 25*e^2) - 1/2*log(x^2)

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maple [A]  time = 0.29, size = 15, normalized size = 0.88

method	result	size
default	\(-\ln \relax (x )+\ln \left (x^{2}+25 \,{\mathrm e}^{2}\right )\)	\(15\)
norman	\(-\ln \relax (x )+\ln \left (x^{2}+25 \,{\mathrm e}^{2}\right )\)	\(15\)
risch	\(-\ln \relax (x )+\ln \left (x^{2}+25 \,{\mathrm e}^{2}\right )\)	\(15\)
meijerg	\(-\ln \relax (x )+\ln \relax (5)+1+\ln \left (1+\frac {{\mathrm e}^{-2} x^{2}}{25}\right )\)	\(19\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-25*exp(2)+x^2)/(25*exp(2)*x+x^3),x,method=_RETURNVERBOSE)

[Out]

-ln(x)+ln(x^2+25*exp(2))

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maxima [A]  time = 0.37, size = 14, normalized size = 0.82 \begin {gather*} \log \left (x^{2} + 25 \, e^{2}\right ) - \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-25*exp(2)+x^2)/(25*exp(2)*x+x^3),x, algorithm="maxima")

[Out]

log(x^2 + 25*e^2) - log(x)

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mupad [B]  time = 4.74, size = 14, normalized size = 0.82 \begin {gather*} \ln \left (x^2+25\,{\mathrm {e}}^2\right )-\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(25*exp(2) - x^2)/(25*x*exp(2) + x^3),x)

[Out]

log(25*exp(2) + x^2) - log(x)

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sympy [A]  time = 0.16, size = 12, normalized size = 0.71 \begin {gather*} - \log {\relax (x )} + \log {\left (x^{2} + 25 e^{2} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-25*exp(2)+x**2)/(25*exp(2)*x+x**3),x)

[Out]

-log(x) + log(x**2 + 25*exp(2))

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