Optimal. Leaf size=92 \[ \frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{\frac{e x^2}{d}+1} \text{EllipticF}\left (\sin ^{-1}(c x),-\frac{e}{c^2 d}\right )}{c d \sqrt{d+e x^2}}+\frac{x \left (a+b \text{sech}^{-1}(c x)\right )}{d \sqrt{d+e x^2}} \]
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Rubi [A] time = 0.0786416, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {191, 6291, 12, 421, 419} \[ \frac{x \left (a+b \text{sech}^{-1}(c x)\right )}{d \sqrt{d+e x^2}}+\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{\frac{e x^2}{d}+1} F\left (\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right )}{c d \sqrt{d+e x^2}} \]
Antiderivative was successfully verified.
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Rule 191
Rule 6291
Rule 12
Rule 421
Rule 419
Rubi steps
\begin{align*} \int \frac{a+b \text{sech}^{-1}(c x)}{\left (d+e x^2\right )^{3/2}} \, dx &=\frac{x \left (a+b \text{sech}^{-1}(c x)\right )}{d \sqrt{d+e x^2}}+\left (b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{1}{d \sqrt{1-c^2 x^2} \sqrt{d+e x^2}} \, dx\\ &=\frac{x \left (a+b \text{sech}^{-1}(c x)\right )}{d \sqrt{d+e x^2}}+\frac{\left (b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{1}{\sqrt{1-c^2 x^2} \sqrt{d+e x^2}} \, dx}{d}\\ &=\frac{x \left (a+b \text{sech}^{-1}(c x)\right )}{d \sqrt{d+e x^2}}+\frac{\left (b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1+\frac{e x^2}{d}}\right ) \int \frac{1}{\sqrt{1-c^2 x^2} \sqrt{1+\frac{e x^2}{d}}} \, dx}{d \sqrt{d+e x^2}}\\ &=\frac{x \left (a+b \text{sech}^{-1}(c x)\right )}{d \sqrt{d+e x^2}}+\frac{b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1+\frac{e x^2}{d}} F\left (\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right )}{c d \sqrt{d+e x^2}}\\ \end{align*}
Mathematica [C] time = 1.32424, size = 334, normalized size = 3.63 \[ \frac{x \left (a+b \text{sech}^{-1}(c x)\right )}{d \sqrt{d+e x^2}}+\frac{2 i b \sqrt{\frac{1-c x}{c x+1}} \left (\sqrt{e} x-i \sqrt{d}\right ) \sqrt{\frac{(c x+1) \left (c \sqrt{d}+i \sqrt{e}\right )}{(c x-1) \left (c \sqrt{d}-i \sqrt{e}\right )}} \sqrt{-\frac{c \left (x+\frac{i \sqrt{d}}{\sqrt{e}}\right )+\frac{i \sqrt{e} x}{\sqrt{d}}-1}{1-c x}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{\frac{i c \sqrt{d}}{\sqrt{e}}+c (-x)+\frac{i \sqrt{e} x}{\sqrt{d}}+1}{2-2 c x}}\right ),-\frac{4 i c \sqrt{d} \sqrt{e}}{\left (c \sqrt{d}-i \sqrt{e}\right )^2}\right )}{d \left (c \sqrt{d}+i \sqrt{e}\right ) \sqrt{d+e x^2} \sqrt{\frac{\frac{i c \sqrt{d}}{\sqrt{e}}+c (-x)+\frac{i \sqrt{e} x}{\sqrt{d}}+1}{1-c x}}} \]
Antiderivative was successfully verified.
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Maple [F] time = 1.067, size = 0, normalized size = 0. \begin{align*} \int{(a+b{\rm arcsech} \left (cx\right )) \left ( e{x}^{2}+d \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} b \int \frac{\log \left (\sqrt{\frac{1}{c x} + 1} \sqrt{\frac{1}{c x} - 1} + \frac{1}{c x}\right )}{{\left (e x^{2} + d\right )}^{\frac{3}{2}}}\,{d x} + \frac{a x}{\sqrt{e x^{2} + d} d} \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{\sqrt{e x^{2} + d}{\left (b \operatorname{arsech}\left (c x\right ) + a\right )}}{e^{2} x^{4} + 2 \, d e x^{2} + d^{2}}, 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{a + b \operatorname{asech}{\left (c x \right )}}{\left (d + e x^{2}\right )^{\frac{3}{2}}}\, 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{b \operatorname{arsech}\left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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