3.69 \(\int \frac{\sqrt{c+d x}}{(a+b e^x)^3} \, dx\)

Optimal. Leaf size=21 \[ \text{Unintegrable}\left (\frac{\sqrt{c+d x}}{\left (a+b e^x\right )^3},x\right ) \]

[Out]

Unintegrable[Sqrt[c + d*x]/(a + b*E^x)^3, x]

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Rubi [A]  time = 0.0387193, 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 = {} \[ \int \frac{\sqrt{c+d x}}{\left (a+b e^x\right )^3} \, dx \]

Verification is Not applicable to the result.

[In]

Int[Sqrt[c + d*x]/(a + b*E^x)^3,x]

[Out]

Defer[Int][Sqrt[c + d*x]/(a + b*E^x)^3, x]

Rubi steps

\begin{align*} \int \frac{\sqrt{c+d x}}{\left (a+b e^x\right )^3} \, dx &=\int \frac{\sqrt{c+d x}}{\left (a+b e^x\right )^3} \, dx\\ \end{align*}

Mathematica [A]  time = 2.26549, size = 0, normalized size = 0. \[ \int \frac{\sqrt{c+d x}}{\left (a+b e^x\right )^3} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[Sqrt[c + d*x]/(a + b*E^x)^3,x]

[Out]

Integrate[Sqrt[c + d*x]/(a + b*E^x)^3, x]

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Maple [A]  time = 0.066, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( a+b{{\rm e}^{x}} \right ) ^{3}}\sqrt{dx+c}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^(1/2)/(a+b*exp(x))^3,x)

[Out]

int((d*x+c)^(1/2)/(a+b*exp(x))^3,x)

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{d x + c}}{{\left (b e^{x} + a\right )}^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^(1/2)/(a+b*exp(x))^3,x, algorithm="maxima")

[Out]

integrate(sqrt(d*x + c)/(b*e^x + a)^3, x)

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{d x + c}}{b^{3} e^{\left (3 \, x\right )} + 3 \, a b^{2} e^{\left (2 \, x\right )} + 3 \, a^{2} b e^{x} + a^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^(1/2)/(a+b*exp(x))^3,x, algorithm="fricas")

[Out]

integral(sqrt(d*x + c)/(b^3*e^(3*x) + 3*a*b^2*e^(2*x) + 3*a^2*b*e^x + a^3), x)

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{c + d x}}{\left (a + b e^{x}\right )^{3}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)**(1/2)/(a+b*exp(x))**3,x)

[Out]

Integral(sqrt(c + d*x)/(a + b*exp(x))**3, x)

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{d x + c}}{{\left (b e^{x} + a\right )}^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^(1/2)/(a+b*exp(x))^3,x, algorithm="giac")

[Out]

integrate(sqrt(d*x + c)/(b*e^x + a)^3, x)