3.96.27 \(\int \frac {25+14 x^2+(-75-14 x^2) \log (x)}{-9 e^e x^4+(25 x+14 x^3) \log (x)} \, dx\)

Optimal. Leaf size=26 \[ \log \left (3 e^e+\frac {\left (-5+\frac {1}{3} \left (1-\frac {25}{x^2}\right )\right ) \log (x)}{x}\right ) \]

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Rubi [F]  time = 2.23, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {25+14 x^2+\left (-75-14 x^2\right ) \log (x)}{-9 e^e x^4+\left (25 x+14 x^3\right ) \log (x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(25 + 14*x^2 + (-75 - 14*x^2)*Log[x])/(-9*E^E*x^4 + (25*x + 14*x^3)*Log[x]),x]

[Out]

-3*Log[x] + Log[25 + 14*x^2] + (225*E^E*Defer[Int][(9*E^E*x^3 - 25*Log[x] - 14*x^2*Log[x])^(-1), x])/7 - 25*De
fer[Int][1/(x*(9*E^E*x^3 - 25*Log[x] - 14*x^2*Log[x])), x] - 14*Defer[Int][x/(9*E^E*x^3 - 25*Log[x] - 14*x^2*L
og[x]), x] + 9*E^E*Defer[Int][x^2/(9*E^E*x^3 - 25*Log[x] - 14*x^2*Log[x]), x] - ((1125*I)/14)*E^E*Defer[Int][1
/((5*I - Sqrt[14]*x)*(9*E^E*x^3 - 25*Log[x] - 14*x^2*Log[x])), x] - ((1125*I)/14)*E^E*Defer[Int][1/((5*I + Sqr
t[14]*x)*(9*E^E*x^3 - 25*Log[x] - 14*x^2*Log[x])), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {-75-14 x^2}{x \left (25+14 x^2\right )}+\frac {-625-700 x^2+675 e^e x^3-196 x^4+126 e^e x^5}{x \left (25+14 x^2\right ) \left (9 e^e x^3-25 \log (x)-14 x^2 \log (x)\right )}\right ) \, dx\\ &=\int \frac {-75-14 x^2}{x \left (25+14 x^2\right )} \, dx+\int \frac {-625-700 x^2+675 e^e x^3-196 x^4+126 e^e x^5}{x \left (25+14 x^2\right ) \left (9 e^e x^3-25 \log (x)-14 x^2 \log (x)\right )} \, dx\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {-75-14 x}{x (25+14 x)} \, dx,x,x^2\right )+\int \left (\frac {225 e^e}{7 \left (9 e^e x^3-25 \log (x)-14 x^2 \log (x)\right )}-\frac {25}{x \left (9 e^e x^3-25 \log (x)-14 x^2 \log (x)\right )}-\frac {14 x}{9 e^e x^3-25 \log (x)-14 x^2 \log (x)}+\frac {9 e^e x^2}{9 e^e x^3-25 \log (x)-14 x^2 \log (x)}-\frac {5625 e^e}{7 \left (25+14 x^2\right ) \left (9 e^e x^3-25 \log (x)-14 x^2 \log (x)\right )}\right ) \, dx\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (-\frac {3}{x}+\frac {28}{25+14 x}\right ) \, dx,x,x^2\right )-14 \int \frac {x}{9 e^e x^3-25 \log (x)-14 x^2 \log (x)} \, dx-25 \int \frac {1}{x \left (9 e^e x^3-25 \log (x)-14 x^2 \log (x)\right )} \, dx+\left (9 e^e\right ) \int \frac {x^2}{9 e^e x^3-25 \log (x)-14 x^2 \log (x)} \, dx+\frac {1}{7} \left (225 e^e\right ) \int \frac {1}{9 e^e x^3-25 \log (x)-14 x^2 \log (x)} \, dx-\frac {1}{7} \left (5625 e^e\right ) \int \frac {1}{\left (25+14 x^2\right ) \left (9 e^e x^3-25 \log (x)-14 x^2 \log (x)\right )} \, dx\\ &=-3 \log (x)+\log \left (25+14 x^2\right )-14 \int \frac {x}{9 e^e x^3-25 \log (x)-14 x^2 \log (x)} \, dx-25 \int \frac {1}{x \left (9 e^e x^3-25 \log (x)-14 x^2 \log (x)\right )} \, dx+\left (9 e^e\right ) \int \frac {x^2}{9 e^e x^3-25 \log (x)-14 x^2 \log (x)} \, dx+\frac {1}{7} \left (225 e^e\right ) \int \frac {1}{9 e^e x^3-25 \log (x)-14 x^2 \log (x)} \, dx-\frac {1}{7} \left (5625 e^e\right ) \int \left (\frac {i}{10 \left (5 i-\sqrt {14} x\right ) \left (9 e^e x^3-25 \log (x)-14 x^2 \log (x)\right )}+\frac {i}{10 \left (5 i+\sqrt {14} x\right ) \left (9 e^e x^3-25 \log (x)-14 x^2 \log (x)\right )}\right ) \, dx\\ &=-3 \log (x)+\log \left (25+14 x^2\right )-14 \int \frac {x}{9 e^e x^3-25 \log (x)-14 x^2 \log (x)} \, dx-25 \int \frac {1}{x \left (9 e^e x^3-25 \log (x)-14 x^2 \log (x)\right )} \, dx-\frac {1}{14} \left (1125 i e^e\right ) \int \frac {1}{\left (5 i-\sqrt {14} x\right ) \left (9 e^e x^3-25 \log (x)-14 x^2 \log (x)\right )} \, dx-\frac {1}{14} \left (1125 i e^e\right ) \int \frac {1}{\left (5 i+\sqrt {14} x\right ) \left (9 e^e x^3-25 \log (x)-14 x^2 \log (x)\right )} \, dx+\left (9 e^e\right ) \int \frac {x^2}{9 e^e x^3-25 \log (x)-14 x^2 \log (x)} \, dx+\frac {1}{7} \left (225 e^e\right ) \int \frac {1}{9 e^e x^3-25 \log (x)-14 x^2 \log (x)} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.33, size = 26, normalized size = 1.00 \begin {gather*} -3 \log (x)+\log \left (9 e^e x^3-\left (25+14 x^2\right ) \log (x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(25 + 14*x^2 + (-75 - 14*x^2)*Log[x])/(-9*E^E*x^4 + (25*x + 14*x^3)*Log[x]),x]

[Out]

-3*Log[x] + Log[9*E^E*x^3 - (25 + 14*x^2)*Log[x]]

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fricas [B]  time = 0.64, size = 45, normalized size = 1.73 \begin {gather*} \log \left (14 \, x^{2} + 25\right ) - 3 \, \log \relax (x) + \log \left (-\frac {9 \, x^{3} e^{e} - {\left (14 \, x^{2} + 25\right )} \log \relax (x)}{14 \, x^{2} + 25}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-14*x^2-75)*log(x)+14*x^2+25)/((14*x^3+25*x)*log(x)-9*x^4*exp(exp(1))),x, algorithm="fricas")

[Out]

log(14*x^2 + 25) - 3*log(x) + log(-(9*x^3*e^e - (14*x^2 + 25)*log(x))/(14*x^2 + 25))

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giac [A]  time = 0.22, size = 26, normalized size = 1.00 \begin {gather*} \log \left (-9 \, x^{3} e^{e} + 14 \, x^{2} \log \relax (x) + 25 \, \log \relax (x)\right ) - 3 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-14*x^2-75)*log(x)+14*x^2+25)/((14*x^3+25*x)*log(x)-9*x^4*exp(exp(1))),x, algorithm="giac")

[Out]

log(-9*x^3*e^e + 14*x^2*log(x) + 25*log(x)) - 3*log(x)

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maple [A]  time = 0.06, size = 27, normalized size = 1.04

method	result	size
norman	\(-3 \ln \relax (x )+\ln \left (9 x^{3} {\mathrm e}^{{\mathrm e}}-14 x^{2} \ln \relax (x )-25 \ln \relax (x )\right )\)	\(27\)
risch	\(-3 \ln \relax (x )+\ln \left (14 x^{2}+25\right )+\ln \left (\ln \relax (x )-\frac {9 x^{3} {\mathrm e}^{{\mathrm e}}}{14 x^{2}+25}\right )\)	\(35\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-14*x^2-75)*ln(x)+14*x^2+25)/((14*x^3+25*x)*ln(x)-9*x^4*exp(exp(1))),x,method=_RETURNVERBOSE)

[Out]

-3*ln(x)+ln(9*x^3*exp(exp(1))-14*x^2*ln(x)-25*ln(x))

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maxima [B]  time = 0.39, size = 45, normalized size = 1.73 \begin {gather*} \log \left (14 \, x^{2} + 25\right ) - 3 \, \log \relax (x) + \log \left (-\frac {9 \, x^{3} e^{e} - {\left (14 \, x^{2} + 25\right )} \log \relax (x)}{14 \, x^{2} + 25}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-14*x^2-75)*log(x)+14*x^2+25)/((14*x^3+25*x)*log(x)-9*x^4*exp(exp(1))),x, algorithm="maxima")

[Out]

log(14*x^2 + 25) - 3*log(x) + log(-(9*x^3*e^e - (14*x^2 + 25)*log(x))/(14*x^2 + 25))

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mupad [B]  time = 7.56, size = 26, normalized size = 1.00 \begin {gather*} \ln \left (9\,x^3\,{\mathrm {e}}^{\mathrm {e}}-25\,\ln \relax (x)-14\,x^2\,\ln \relax (x)\right )-3\,\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(14*x^2 - log(x)*(14*x^2 + 75) + 25)/(9*x^4*exp(exp(1)) - log(x)*(25*x + 14*x^3)),x)

[Out]

log(9*x^3*exp(exp(1)) - 25*log(x) - 14*x^2*log(x)) - 3*log(x)

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sympy [A]  time = 0.41, size = 34, normalized size = 1.31 \begin {gather*} - 3 \log {\relax (x )} + \log {\left (14 x^{2} + 25 \right )} + \log {\left (- \frac {9 x^{3} e^{e}}{14 x^{2} + 25} + \log {\relax (x )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-14*x**2-75)*ln(x)+14*x**2+25)/((14*x**3+25*x)*ln(x)-9*x**4*exp(exp(1))),x)

[Out]

-3*log(x) + log(14*x**2 + 25) + log(-9*x**3*exp(E)/(14*x**2 + 25) + log(x))

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