3.21.5 \(\int \frac {90-3 e^{\frac {1}{2} (-10-3 x)} x^2}{-90 x+2 e^{\frac {1}{2} (-10-3 x)} x^2} \, dx\)

Optimal. Leaf size=16 \[ \log \left (e^{-5-\frac {3 x}{2}}-\frac {45}{x}\right ) \]

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Rubi [F]  time = 0.37, 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 {90-3 e^{\frac {1}{2} (-10-3 x)} x^2}{-90 x+2 e^{\frac {1}{2} (-10-3 x)} x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(90 - 3*E^((-10 - 3*x)/2)*x^2)/(-90*x + 2*E^((-10 - 3*x)/2)*x^2),x]

[Out]

-Log[x] - Defer[Int][(45*E^(5 + (3*x)/2) - x)^(-1), x] + (3*Defer[Int][x/(45*E^(5 + (3*x)/2) - x), x])/2

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-\frac {1}{x}+\frac {-2+3 x}{2 \left (45 e^{5+\frac {3 x}{2}}-x\right )}\right ) \, dx\\ &=-\log (x)+\frac {1}{2} \int \frac {-2+3 x}{45 e^{5+\frac {3 x}{2}}-x} \, dx\\ &=-\log (x)+\frac {1}{2} \int \left (-\frac {2}{45 e^{5+\frac {3 x}{2}}-x}+\frac {3 x}{45 e^{5+\frac {3 x}{2}}-x}\right ) \, dx\\ &=-\log (x)+\frac {3}{2} \int \frac {x}{45 e^{5+\frac {3 x}{2}}-x} \, dx-\int \frac {1}{45 e^{5+\frac {3 x}{2}}-x} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.17, size = 26, normalized size = 1.62 \begin {gather*} -\frac {3 x}{2}+\log \left (45 e^{5+\frac {3 x}{2}}-x\right )-\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(90 - 3*E^((-10 - 3*x)/2)*x^2)/(-90*x + 2*E^((-10 - 3*x)/2)*x^2),x]

[Out]

(-3*x)/2 + Log[45*E^(5 + (3*x)/2) - x] - Log[x]

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fricas [A]  time = 0.54, size = 15, normalized size = 0.94 \begin {gather*} \log \left (\frac {x e^{\left (-\frac {3}{2} \, x - 5\right )} - 45}{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3*x^2*exp(-3/2*x-5)+90)/(2*x^2*exp(-3/2*x-5)-90*x),x, algorithm="fricas")

[Out]

log((x*e^(-3/2*x - 5) - 45)/x)

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giac [A]  time = 0.20, size = 17, normalized size = 1.06 \begin {gather*} \log \left (x e^{\left (-\frac {3}{2} \, x\right )} - 45 \, e^{5}\right ) - \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3*x^2*exp(-3/2*x-5)+90)/(2*x^2*exp(-3/2*x-5)-90*x),x, algorithm="giac")

[Out]

log(x*e^(-3/2*x) - 45*e^5) - log(x)

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maple [A]  time = 0.14, size = 16, normalized size = 1.00

method	result	size
risch	\(5+\ln \left ({\mathrm e}^{-\frac {3 x}{2}-5}-\frac {45}{x}\right )\)	\(16\)
norman	\(-\ln \relax (x )+\ln \left (x \,{\mathrm e}^{-\frac {3 x}{2}-5}-45\right )\)	\(17\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-3*x^2*exp(-3/2*x-5)+90)/(2*x^2*exp(-3/2*x-5)-90*x),x,method=_RETURNVERBOSE)

[Out]

5+ln(exp(-3/2*x-5)-45/x)

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maxima [A]  time = 0.47, size = 23, normalized size = 1.44 \begin {gather*} -\frac {3}{2} \, x + \log \left (-\frac {1}{45} \, {\left (x - 45 \, e^{\left (\frac {3}{2} \, x + 5\right )}\right )} e^{\left (-5\right )}\right ) - \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3*x^2*exp(-3/2*x-5)+90)/(2*x^2*exp(-3/2*x-5)-90*x),x, algorithm="maxima")

[Out]

-3/2*x + log(-1/45*(x - 45*e^(3/2*x + 5))*e^(-5)) - log(x)

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mupad [B]  time = 0.21, size = 16, normalized size = 1.00 \begin {gather*} \ln \left (\frac {x\,{\mathrm {e}}^{-5}}{{\left ({\mathrm {e}}^x\right )}^{3/2}}-45\right )-\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x^2*exp(- (3*x)/2 - 5) - 90)/(90*x - 2*x^2*exp(- (3*x)/2 - 5)),x)

[Out]

log((x*exp(-5))/exp(x)^(3/2) - 45) - log(x)

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sympy [A]  time = 0.17, size = 14, normalized size = 0.88 \begin {gather*} \log {\left (e^{- \frac {3 x}{2} - 5} - \frac {45}{x} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3*x**2*exp(-3/2*x-5)+90)/(2*x**2*exp(-3/2*x-5)-90*x),x)

[Out]

log(exp(-3*x/2 - 5) - 45/x)

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