Optimal. Leaf size=31 \[ b \text{Unintegrable}\left (\frac{\text{sech}\left (c+d \sqrt{x}\right )}{x^{5/2}},x\right )-\frac{2 a}{3 x^{3/2}} \]
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Rubi [A] time = 0.0153306, 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 = {} \[ \int \frac{a+b \text{sech}\left (c+d \sqrt{x}\right )}{x^{5/2}} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \frac{a+b \text{sech}\left (c+d \sqrt{x}\right )}{x^{5/2}} \, dx &=\int \left (\frac{a}{x^{5/2}}+\frac{b \text{sech}\left (c+d \sqrt{x}\right )}{x^{5/2}}\right ) \, dx\\ &=-\frac{2 a}{3 x^{3/2}}+b \int \frac{\text{sech}\left (c+d \sqrt{x}\right )}{x^{5/2}} \, dx\\ \end{align*}
Mathematica [A] time = 9.21013, size = 0, normalized size = 0. \[ \int \frac{a+b \text{sech}\left (c+d \sqrt{x}\right )}{x^{5/2}} \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0.072, size = 0, normalized size = 0. \begin{align*} \int{ \left ( a+b{\rm sech} \left (c+d\sqrt{x}\right ) \right ){x}^{-{\frac{5}{2}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0., size = 0, normalized size = 0. \begin{align*} 2 \, b \int \frac{e^{\left (d \sqrt{x} + c\right )}}{x^{\frac{5}{2}} e^{\left (2 \, d \sqrt{x} + 2 \, c\right )} + x^{\frac{5}{2}}}\,{d x} - \frac{2 \, a}{3 \, x^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b \sqrt{x} \operatorname{sech}\left (d \sqrt{x} + c\right ) + a \sqrt{x}}{x^{3}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{a + b \operatorname{sech}{\left (c + d \sqrt{x} \right )}}{x^{\frac{5}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{b \operatorname{sech}\left (d \sqrt{x} + c\right ) + a}{x^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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