3.6 \(\int \frac{1}{(a+b e^{c+d x}) x^2} \, dx\)

Optimal. Leaf size=19 \[ \text{Unintegrable}\left (\frac{1}{x^2 \left (a+b e^{c+d x}\right )},x\right ) \]

[Out]

Unintegrable[1/((a + b*E^(c + d*x))*x^2), x]

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Rubi [A]  time = 0.0453303, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{1}{\left (a+b e^{c+d x}\right ) x^2} \, dx \]

Verification is Not applicable to the result.

[In]

Int[1/((a + b*E^(c + d*x))*x^2),x]

[Out]

Defer[Int][1/((a + b*E^(c + d*x))*x^2), x]

Rubi steps

\begin{align*} \int \frac{1}{\left (a+b e^{c+d x}\right ) x^2} \, dx &=\int \frac{1}{\left (a+b e^{c+d x}\right ) x^2} \, dx\\ \end{align*}

Mathematica [A]  time = 0.113561, size = 0, normalized size = 0. \[ \int \frac{1}{\left (a+b e^{c+d x}\right ) x^2} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[1/((a + b*E^(c + d*x))*x^2),x]

[Out]

Integrate[1/((a + b*E^(c + d*x))*x^2), x]

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Maple [A]  time = 0.023, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( a+b{{\rm e}^{dx+c}} \right ){x}^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b*exp(d*x+c))/x^2,x)

[Out]

int(1/(a+b*exp(d*x+c))/x^2,x)

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b e^{\left (d x + c\right )} + a\right )} x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*exp(d*x+c))/x^2,x, algorithm="maxima")

[Out]

integrate(1/((b*e^(d*x + c) + a)*x^2), x)

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{b x^{2} e^{\left (d x + c\right )} + a x^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*exp(d*x+c))/x^2,x, algorithm="fricas")

[Out]

integral(1/(b*x^2*e^(d*x + c) + a*x^2), x)

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x^{2} \left (a + b e^{c} e^{d x}\right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*exp(d*x+c))/x**2,x)

[Out]

Integral(1/(x**2*(a + b*exp(c)*exp(d*x))), x)

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b e^{\left (d x + c\right )} + a\right )} x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*exp(d*x+c))/x^2,x, algorithm="giac")

[Out]

integrate(1/((b*e^(d*x + c) + a)*x^2), x)