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

Optimal. Leaf size=20 \[ \text {Int}\left (\frac {1}{x^2 \left (a+b e^{c+d x}\right )},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.05, 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 = {} \[ \int \frac {1}{\left (a+b e^{c+d x}\right ) x^2} \, dx \]

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*}

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Mathematica [A]  time = 0.13, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (a+b e^{c+d x}\right ) x^2} \, dx \]

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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fricas [A]  time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{b x^{2} e^{\left (d x + c\right )} + a x^{2}}, x\right ) \]

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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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b e^{\left (d x + c\right )} + a\right )} x^{2}}\,{d x} \]

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)

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maple [A]  time = 0.05, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (b \,{\mathrm e}^{d x +c}+a \right ) x^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b e^{\left (d x + c\right )} + a\right )} x^{2}}\,{d x} \]

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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mupad [A]  time = 0.00, size = -1, normalized size = -0.05 \[ \int \frac {1}{x^2\,\left (a+b\,{\mathrm {e}}^{c+d\,x}\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^2*(a + b*exp(c + d*x))),x)

[Out]

int(1/(x^2*(a + b*exp(c + d*x))), x)

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^{2} \left (a + b e^{c} e^{d x}\right )}\, dx \]

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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