Optimal. Leaf size=351 \[ -\frac{2 g \sqrt{d+e x}}{\sqrt{f+g x} \left (a g^2+c f^2\right )}+\frac{\left (-\sqrt{-a} \sqrt{c} (e f-d g)+a e g+c d f\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x} \sqrt{\sqrt{c} f-\sqrt{-a} g}}{\sqrt{f+g x} \sqrt{\sqrt{c} d-\sqrt{-a} e}}\right )}{\sqrt{-a} \sqrt{\sqrt{c} d-\sqrt{-a} e} \sqrt{\sqrt{c} f-\sqrt{-a} g} \left (a g^2+c f^2\right )}-\frac{\left (\sqrt{-a} \sqrt{c} (e f-d g)+a e g+c d f\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x} \sqrt{\sqrt{-a} g+\sqrt{c} f}}{\sqrt{f+g x} \sqrt{\sqrt{-a} e+\sqrt{c} d}}\right )}{\sqrt{-a} \sqrt{\sqrt{-a} e+\sqrt{c} d} \sqrt{\sqrt{-a} g+\sqrt{c} f} \left (a g^2+c f^2\right )} \]
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Rubi [A] time = 1.80615, antiderivative size = 351, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179, Rules used = {908, 37, 6725, 93, 208} \[ -\frac{2 g \sqrt{d+e x}}{\sqrt{f+g x} \left (a g^2+c f^2\right )}+\frac{\left (-\sqrt{-a} \sqrt{c} (e f-d g)+a e g+c d f\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x} \sqrt{\sqrt{c} f-\sqrt{-a} g}}{\sqrt{f+g x} \sqrt{\sqrt{c} d-\sqrt{-a} e}}\right )}{\sqrt{-a} \sqrt{\sqrt{c} d-\sqrt{-a} e} \sqrt{\sqrt{c} f-\sqrt{-a} g} \left (a g^2+c f^2\right )}-\frac{\left (\sqrt{-a} \sqrt{c} (e f-d g)+a e g+c d f\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x} \sqrt{\sqrt{-a} g+\sqrt{c} f}}{\sqrt{f+g x} \sqrt{\sqrt{-a} e+\sqrt{c} d}}\right )}{\sqrt{-a} \sqrt{\sqrt{-a} e+\sqrt{c} d} \sqrt{\sqrt{-a} g+\sqrt{c} f} \left (a g^2+c f^2\right )} \]
Antiderivative was successfully verified.
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Rule 908
Rule 37
Rule 6725
Rule 93
Rule 208
Rubi steps
\begin{align*} \int \frac{\sqrt{d+e x}}{(f+g x)^{3/2} \left (a+c x^2\right )} \, dx &=\frac{\int \frac{c d f+a e g+c (e f-d g) x}{\sqrt{d+e x} \sqrt{f+g x} \left (a+c x^2\right )} \, dx}{c f^2+a g^2}-\frac{(g (e f-d g)) \int \frac{1}{\sqrt{d+e x} (f+g x)^{3/2}} \, dx}{c f^2+a g^2}\\ &=-\frac{2 g \sqrt{d+e x}}{\left (c f^2+a g^2\right ) \sqrt{f+g x}}+\frac{\int \left (\frac{-a \sqrt{c} (e f-d g)+\sqrt{-a} (c d f+a e g)}{2 a \left (\sqrt{-a}-\sqrt{c} x\right ) \sqrt{d+e x} \sqrt{f+g x}}+\frac{a \sqrt{c} (e f-d g)+\sqrt{-a} (c d f+a e g)}{2 a \left (\sqrt{-a}+\sqrt{c} x\right ) \sqrt{d+e x} \sqrt{f+g x}}\right ) \, dx}{c f^2+a g^2}\\ &=-\frac{2 g \sqrt{d+e x}}{\left (c f^2+a g^2\right ) \sqrt{f+g x}}-\frac{\left (c d f+a e g-\sqrt{-a} \sqrt{c} (e f-d g)\right ) \int \frac{1}{\left (\sqrt{-a}+\sqrt{c} x\right ) \sqrt{d+e x} \sqrt{f+g x}} \, dx}{2 \sqrt{-a} \left (c f^2+a g^2\right )}-\frac{\left (c d f+a e g+\sqrt{-a} \sqrt{c} (e f-d g)\right ) \int \frac{1}{\left (\sqrt{-a}-\sqrt{c} x\right ) \sqrt{d+e x} \sqrt{f+g x}} \, dx}{2 \sqrt{-a} \left (c f^2+a g^2\right )}\\ &=-\frac{2 g \sqrt{d+e x}}{\left (c f^2+a g^2\right ) \sqrt{f+g x}}-\frac{\left (c d f+a e g-\sqrt{-a} \sqrt{c} (e f-d g)\right ) \operatorname{Subst}\left (\int \frac{1}{-\sqrt{c} d+\sqrt{-a} e-\left (-\sqrt{c} f+\sqrt{-a} g\right ) x^2} \, dx,x,\frac{\sqrt{d+e x}}{\sqrt{f+g x}}\right )}{\sqrt{-a} \left (c f^2+a g^2\right )}-\frac{\left (c d f+a e g+\sqrt{-a} \sqrt{c} (e f-d g)\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{c} d+\sqrt{-a} e-\left (\sqrt{c} f+\sqrt{-a} g\right ) x^2} \, dx,x,\frac{\sqrt{d+e x}}{\sqrt{f+g x}}\right )}{\sqrt{-a} \left (c f^2+a g^2\right )}\\ &=-\frac{2 g \sqrt{d+e x}}{\left (c f^2+a g^2\right ) \sqrt{f+g x}}+\frac{\left (c d f+a e g-\sqrt{-a} \sqrt{c} (e f-d g)\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{c} f-\sqrt{-a} g} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{-a} e} \sqrt{f+g x}}\right )}{\sqrt{-a} \sqrt{\sqrt{c} d-\sqrt{-a} e} \sqrt{\sqrt{c} f-\sqrt{-a} g} \left (c f^2+a g^2\right )}-\frac{\left (c d f+a e g+\sqrt{-a} \sqrt{c} (e f-d g)\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{c} f+\sqrt{-a} g} \sqrt{d+e x}}{\sqrt{\sqrt{c} d+\sqrt{-a} e} \sqrt{f+g x}}\right )}{\sqrt{-a} \sqrt{\sqrt{c} d+\sqrt{-a} e} \sqrt{\sqrt{c} f+\sqrt{-a} g} \left (c f^2+a g^2\right )}\\ \end{align*}
Mathematica [A] time = 0.596949, size = 267, normalized size = 0.76 \[ -\frac{2 g \sqrt{d+e x}}{\sqrt{f+g x} \left (a g^2+c f^2\right )}+\frac{\sqrt{\sqrt{-a} e+\sqrt{c} d} \tan ^{-1}\left (\frac{\sqrt{d+e x} \sqrt{-\sqrt{-a} g-\sqrt{c} f}}{\sqrt{f+g x} \sqrt{\sqrt{-a} e+\sqrt{c} d}}\right )}{\sqrt{-a} \left (-\sqrt{-a} g-\sqrt{c} f\right )^{3/2}}+\frac{\sqrt{\sqrt{-a} e-\sqrt{c} d} \tan ^{-1}\left (\frac{\sqrt{d+e x} \sqrt{\sqrt{c} f-\sqrt{-a} g}}{\sqrt{f+g x} \sqrt{\sqrt{-a} e-\sqrt{c} d}}\right )}{\sqrt{-a} \left (\sqrt{c} f-\sqrt{-a} g\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.426, size = 5383, normalized size = 15.3 \begin{align*} \text{output too large to display} \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*} \int \frac{\sqrt{e x + d}}{{\left (c x^{2} + a\right )}{\left (g x + f\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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