Optimal. Leaf size=601 \[ -\frac{8 b x^{3/2} \text{PolyLog}\left (2,-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{b^2-a^2}}\right )}{a d^2 \sqrt{b^2-a^2}}+\frac{8 b x^{3/2} \text{PolyLog}\left (2,-\frac{a e^{c+d \sqrt{x}}}{\sqrt{b^2-a^2}+b}\right )}{a d^2 \sqrt{b^2-a^2}}+\frac{24 b x \text{PolyLog}\left (3,-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{b^2-a^2}}\right )}{a d^3 \sqrt{b^2-a^2}}-\frac{24 b x \text{PolyLog}\left (3,-\frac{a e^{c+d \sqrt{x}}}{\sqrt{b^2-a^2}+b}\right )}{a d^3 \sqrt{b^2-a^2}}-\frac{48 b \sqrt{x} \text{PolyLog}\left (4,-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{b^2-a^2}}\right )}{a d^4 \sqrt{b^2-a^2}}+\frac{48 b \sqrt{x} \text{PolyLog}\left (4,-\frac{a e^{c+d \sqrt{x}}}{\sqrt{b^2-a^2}+b}\right )}{a d^4 \sqrt{b^2-a^2}}+\frac{48 b \text{PolyLog}\left (5,-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{b^2-a^2}}\right )}{a d^5 \sqrt{b^2-a^2}}-\frac{48 b \text{PolyLog}\left (5,-\frac{a e^{c+d \sqrt{x}}}{\sqrt{b^2-a^2}+b}\right )}{a d^5 \sqrt{b^2-a^2}}-\frac{2 b x^2 \log \left (\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{b^2-a^2}}+1\right )}{a d \sqrt{b^2-a^2}}+\frac{2 b x^2 \log \left (\frac{a e^{c+d \sqrt{x}}}{\sqrt{b^2-a^2}+b}+1\right )}{a d \sqrt{b^2-a^2}}+\frac{2 x^{5/2}}{5 a} \]
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Rubi [A] time = 0.968948, antiderivative size = 601, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 9, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.409, Rules used = {5436, 4191, 3320, 2264, 2190, 2531, 6609, 2282, 6589} \[ -\frac{8 b x^{3/2} \text{PolyLog}\left (2,-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{b^2-a^2}}\right )}{a d^2 \sqrt{b^2-a^2}}+\frac{8 b x^{3/2} \text{PolyLog}\left (2,-\frac{a e^{c+d \sqrt{x}}}{\sqrt{b^2-a^2}+b}\right )}{a d^2 \sqrt{b^2-a^2}}+\frac{24 b x \text{PolyLog}\left (3,-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{b^2-a^2}}\right )}{a d^3 \sqrt{b^2-a^2}}-\frac{24 b x \text{PolyLog}\left (3,-\frac{a e^{c+d \sqrt{x}}}{\sqrt{b^2-a^2}+b}\right )}{a d^3 \sqrt{b^2-a^2}}-\frac{48 b \sqrt{x} \text{PolyLog}\left (4,-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{b^2-a^2}}\right )}{a d^4 \sqrt{b^2-a^2}}+\frac{48 b \sqrt{x} \text{PolyLog}\left (4,-\frac{a e^{c+d \sqrt{x}}}{\sqrt{b^2-a^2}+b}\right )}{a d^4 \sqrt{b^2-a^2}}+\frac{48 b \text{PolyLog}\left (5,-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{b^2-a^2}}\right )}{a d^5 \sqrt{b^2-a^2}}-\frac{48 b \text{PolyLog}\left (5,-\frac{a e^{c+d \sqrt{x}}}{\sqrt{b^2-a^2}+b}\right )}{a d^5 \sqrt{b^2-a^2}}-\frac{2 b x^2 \log \left (\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{b^2-a^2}}+1\right )}{a d \sqrt{b^2-a^2}}+\frac{2 b x^2 \log \left (\frac{a e^{c+d \sqrt{x}}}{\sqrt{b^2-a^2}+b}+1\right )}{a d \sqrt{b^2-a^2}}+\frac{2 x^{5/2}}{5 a} \]
Antiderivative was successfully verified.
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Rule 5436
Rule 4191
Rule 3320
Rule 2264
Rule 2190
Rule 2531
Rule 6609
Rule 2282
Rule 6589
Rubi steps
\begin{align*} \int \frac{x^{3/2}}{a+b \text{sech}\left (c+d \sqrt{x}\right )} \, dx &=2 \operatorname{Subst}\left (\int \frac{x^4}{a+b \text{sech}(c+d x)} \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (\frac{x^4}{a}-\frac{b x^4}{a (b+a \cosh (c+d x))}\right ) \, dx,x,\sqrt{x}\right )\\ &=\frac{2 x^{5/2}}{5 a}-\frac{(2 b) \operatorname{Subst}\left (\int \frac{x^4}{b+a \cosh (c+d x)} \, dx,x,\sqrt{x}\right )}{a}\\ &=\frac{2 x^{5/2}}{5 a}-\frac{(4 b) \operatorname{Subst}\left (\int \frac{e^{c+d x} x^4}{a+2 b e^{c+d x}+a e^{2 (c+d x)}} \, dx,x,\sqrt{x}\right )}{a}\\ &=\frac{2 x^{5/2}}{5 a}-\frac{(4 b) \operatorname{Subst}\left (\int \frac{e^{c+d x} x^4}{2 b-2 \sqrt{-a^2+b^2}+2 a e^{c+d x}} \, dx,x,\sqrt{x}\right )}{\sqrt{-a^2+b^2}}+\frac{(4 b) \operatorname{Subst}\left (\int \frac{e^{c+d x} x^4}{2 b+2 \sqrt{-a^2+b^2}+2 a e^{c+d x}} \, dx,x,\sqrt{x}\right )}{\sqrt{-a^2+b^2}}\\ &=\frac{2 x^{5/2}}{5 a}-\frac{2 b x^2 \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d}+\frac{2 b x^2 \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d}+\frac{(8 b) \operatorname{Subst}\left (\int x^3 \log \left (1+\frac{2 a e^{c+d x}}{2 b-2 \sqrt{-a^2+b^2}}\right ) \, dx,x,\sqrt{x}\right )}{a \sqrt{-a^2+b^2} d}-\frac{(8 b) \operatorname{Subst}\left (\int x^3 \log \left (1+\frac{2 a e^{c+d x}}{2 b+2 \sqrt{-a^2+b^2}}\right ) \, dx,x,\sqrt{x}\right )}{a \sqrt{-a^2+b^2} d}\\ &=\frac{2 x^{5/2}}{5 a}-\frac{2 b x^2 \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d}+\frac{2 b x^2 \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d}-\frac{8 b x^{3/2} \text{Li}_2\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^2}+\frac{8 b x^{3/2} \text{Li}_2\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^2}+\frac{(24 b) \operatorname{Subst}\left (\int x^2 \text{Li}_2\left (-\frac{2 a e^{c+d x}}{2 b-2 \sqrt{-a^2+b^2}}\right ) \, dx,x,\sqrt{x}\right )}{a \sqrt{-a^2+b^2} d^2}-\frac{(24 b) \operatorname{Subst}\left (\int x^2 \text{Li}_2\left (-\frac{2 a e^{c+d x}}{2 b+2 \sqrt{-a^2+b^2}}\right ) \, dx,x,\sqrt{x}\right )}{a \sqrt{-a^2+b^2} d^2}\\ &=\frac{2 x^{5/2}}{5 a}-\frac{2 b x^2 \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d}+\frac{2 b x^2 \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d}-\frac{8 b x^{3/2} \text{Li}_2\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^2}+\frac{8 b x^{3/2} \text{Li}_2\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^2}+\frac{24 b x \text{Li}_3\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^3}-\frac{24 b x \text{Li}_3\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^3}-\frac{(48 b) \operatorname{Subst}\left (\int x \text{Li}_3\left (-\frac{2 a e^{c+d x}}{2 b-2 \sqrt{-a^2+b^2}}\right ) \, dx,x,\sqrt{x}\right )}{a \sqrt{-a^2+b^2} d^3}+\frac{(48 b) \operatorname{Subst}\left (\int x \text{Li}_3\left (-\frac{2 a e^{c+d x}}{2 b+2 \sqrt{-a^2+b^2}}\right ) \, dx,x,\sqrt{x}\right )}{a \sqrt{-a^2+b^2} d^3}\\ &=\frac{2 x^{5/2}}{5 a}-\frac{2 b x^2 \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d}+\frac{2 b x^2 \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d}-\frac{8 b x^{3/2} \text{Li}_2\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^2}+\frac{8 b x^{3/2} \text{Li}_2\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^2}+\frac{24 b x \text{Li}_3\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^3}-\frac{24 b x \text{Li}_3\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^3}-\frac{48 b \sqrt{x} \text{Li}_4\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^4}+\frac{48 b \sqrt{x} \text{Li}_4\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^4}+\frac{(48 b) \operatorname{Subst}\left (\int \text{Li}_4\left (-\frac{2 a e^{c+d x}}{2 b-2 \sqrt{-a^2+b^2}}\right ) \, dx,x,\sqrt{x}\right )}{a \sqrt{-a^2+b^2} d^4}-\frac{(48 b) \operatorname{Subst}\left (\int \text{Li}_4\left (-\frac{2 a e^{c+d x}}{2 b+2 \sqrt{-a^2+b^2}}\right ) \, dx,x,\sqrt{x}\right )}{a \sqrt{-a^2+b^2} d^4}\\ &=\frac{2 x^{5/2}}{5 a}-\frac{2 b x^2 \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d}+\frac{2 b x^2 \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d}-\frac{8 b x^{3/2} \text{Li}_2\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^2}+\frac{8 b x^{3/2} \text{Li}_2\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^2}+\frac{24 b x \text{Li}_3\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^3}-\frac{24 b x \text{Li}_3\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^3}-\frac{48 b \sqrt{x} \text{Li}_4\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^4}+\frac{48 b \sqrt{x} \text{Li}_4\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^4}+\frac{(48 b) \operatorname{Subst}\left (\int \frac{\text{Li}_4\left (\frac{a x}{-b+\sqrt{-a^2+b^2}}\right )}{x} \, dx,x,e^{c+d \sqrt{x}}\right )}{a \sqrt{-a^2+b^2} d^5}-\frac{(48 b) \operatorname{Subst}\left (\int \frac{\text{Li}_4\left (-\frac{a x}{b+\sqrt{-a^2+b^2}}\right )}{x} \, dx,x,e^{c+d \sqrt{x}}\right )}{a \sqrt{-a^2+b^2} d^5}\\ &=\frac{2 x^{5/2}}{5 a}-\frac{2 b x^2 \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d}+\frac{2 b x^2 \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d}-\frac{8 b x^{3/2} \text{Li}_2\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^2}+\frac{8 b x^{3/2} \text{Li}_2\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^2}+\frac{24 b x \text{Li}_3\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^3}-\frac{24 b x \text{Li}_3\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^3}-\frac{48 b \sqrt{x} \text{Li}_4\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^4}+\frac{48 b \sqrt{x} \text{Li}_4\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^4}+\frac{48 b \text{Li}_5\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^5}-\frac{48 b \text{Li}_5\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^5}\\ \end{align*}
Mathematica [A] time = 2.17163, size = 626, normalized size = 1.04 \[ \frac{2 \left (-20 b e^c d^3 x^{3/2} \text{PolyLog}\left (2,-\frac{a e^{2 c+d \sqrt{x}}}{b e^c-\sqrt{e^{2 c} \left (b^2-a^2\right )}}\right )+20 b e^c d^3 x^{3/2} \text{PolyLog}\left (2,-\frac{a e^{2 c+d \sqrt{x}}}{\sqrt{e^{2 c} \left (b^2-a^2\right )}+b e^c}\right )+60 b e^c d^2 x \text{PolyLog}\left (3,-\frac{a e^{2 c+d \sqrt{x}}}{b e^c-\sqrt{e^{2 c} \left (b^2-a^2\right )}}\right )-60 b e^c d^2 x \text{PolyLog}\left (3,-\frac{a e^{2 c+d \sqrt{x}}}{\sqrt{e^{2 c} \left (b^2-a^2\right )}+b e^c}\right )-120 b e^c d \sqrt{x} \text{PolyLog}\left (4,-\frac{a e^{2 c+d \sqrt{x}}}{b e^c-\sqrt{e^{2 c} \left (b^2-a^2\right )}}\right )+120 b e^c d \sqrt{x} \text{PolyLog}\left (4,-\frac{a e^{2 c+d \sqrt{x}}}{\sqrt{e^{2 c} \left (b^2-a^2\right )}+b e^c}\right )+120 b e^c \text{PolyLog}\left (5,-\frac{a e^{2 c+d \sqrt{x}}}{b e^c-\sqrt{e^{2 c} \left (b^2-a^2\right )}}\right )-120 b e^c \text{PolyLog}\left (5,-\frac{a e^{2 c+d \sqrt{x}}}{\sqrt{e^{2 c} \left (b^2-a^2\right )}+b e^c}\right )+d^5 x^{5/2} \sqrt{e^{2 c} \left (b^2-a^2\right )}-5 b e^c d^4 x^2 \log \left (\frac{a e^{2 c+d \sqrt{x}}}{b e^c-\sqrt{e^{2 c} \left (b^2-a^2\right )}}+1\right )+5 b e^c d^4 x^2 \log \left (\frac{a e^{2 c+d \sqrt{x}}}{\sqrt{e^{2 c} \left (b^2-a^2\right )}+b e^c}+1\right )\right )}{5 a d^5 \sqrt{e^{2 c} \left (b^2-a^2\right )}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.083, size = 0, normalized size = 0. \begin{align*} \int{{x}^{{\frac{3}{2}}} \left ( a+b{\rm sech} \left (c+d\sqrt{x}\right ) \right ) ^{-1}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{x^{\frac{3}{2}}}{b \operatorname{sech}\left (d \sqrt{x} + c\right ) + a}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{\frac{3}{2}}}{a + b \operatorname{sech}{\left (c + d \sqrt{x} \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{\frac{3}{2}}}{b \operatorname{sech}\left (d \sqrt{x} + c\right ) + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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