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

Optimal. Leaf size=22 \[ \text {Int}\left (\frac {\sqrt {c+d x}}{\left (a+b e^x\right )^2},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.04, 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 = {} \[ \int \frac {\sqrt {c+d x}}{\left (a+b e^x\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

Int[Sqrt[c + d*x]/(a + b*E^x)^2,x]

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 1.27, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {c+d x}}{\left (a+b e^x\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {d x + c}}{b^{2} e^{\left (2 \, x\right )} + 2 \, a b e^{x} + a^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.09, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {d x +c}}{\left (b \,{\mathrm e}^{x}+a \right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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