Optimal. Leaf size=354 \[ \frac{g \sqrt{d+e x}}{\sqrt{-a} \sqrt{f+g x} \left (\sqrt{c} f-\sqrt{-a} g\right ) (e f-d g)}-\frac{g \sqrt{d+e x}}{\sqrt{-a} \sqrt{f+g x} \left (\sqrt{-a} g+\sqrt{c} f\right ) (e f-d g)}+\frac{\sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{d+e x} \sqrt{\sqrt{c} f-\sqrt{-a} g}}{\sqrt{f+g x} \sqrt{\sqrt{c} d-\sqrt{-a} e}}\right )}{\sqrt{-a} \sqrt{\sqrt{c} d-\sqrt{-a} e} \left (\sqrt{c} f-\sqrt{-a} g\right )^{3/2}}-\frac{\sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{d+e x} \sqrt{\sqrt{-a} g+\sqrt{c} f}}{\sqrt{f+g x} \sqrt{\sqrt{-a} e+\sqrt{c} d}}\right )}{\sqrt{-a} \sqrt{\sqrt{-a} e+\sqrt{c} d} \left (\sqrt{-a} g+\sqrt{c} f\right )^{3/2}} \]
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Rubi [A] time = 0.762631, antiderivative size = 354, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 4, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {912, 96, 93, 208} \[ \frac{g \sqrt{d+e x}}{\sqrt{-a} \sqrt{f+g x} \left (\sqrt{c} f-\sqrt{-a} g\right ) (e f-d g)}-\frac{g \sqrt{d+e x}}{\sqrt{-a} \sqrt{f+g x} \left (\sqrt{-a} g+\sqrt{c} f\right ) (e f-d g)}+\frac{\sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{d+e x} \sqrt{\sqrt{c} f-\sqrt{-a} g}}{\sqrt{f+g x} \sqrt{\sqrt{c} d-\sqrt{-a} e}}\right )}{\sqrt{-a} \sqrt{\sqrt{c} d-\sqrt{-a} e} \left (\sqrt{c} f-\sqrt{-a} g\right )^{3/2}}-\frac{\sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{d+e x} \sqrt{\sqrt{-a} g+\sqrt{c} f}}{\sqrt{f+g x} \sqrt{\sqrt{-a} e+\sqrt{c} d}}\right )}{\sqrt{-a} \sqrt{\sqrt{-a} e+\sqrt{c} d} \left (\sqrt{-a} g+\sqrt{c} f\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 912
Rule 96
Rule 93
Rule 208
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{d+e x} (f+g x)^{3/2} \left (a+c x^2\right )} \, dx &=\int \left (\frac{\sqrt{-a}}{2 a \left (\sqrt{-a}-\sqrt{c} x\right ) \sqrt{d+e x} (f+g x)^{3/2}}+\frac{\sqrt{-a}}{2 a \left (\sqrt{-a}+\sqrt{c} x\right ) \sqrt{d+e x} (f+g x)^{3/2}}\right ) \, dx\\ &=-\frac{\int \frac{1}{\left (\sqrt{-a}-\sqrt{c} x\right ) \sqrt{d+e x} (f+g x)^{3/2}} \, dx}{2 \sqrt{-a}}-\frac{\int \frac{1}{\left (\sqrt{-a}+\sqrt{c} x\right ) \sqrt{d+e x} (f+g x)^{3/2}} \, dx}{2 \sqrt{-a}}\\ &=\frac{g \sqrt{d+e x}}{\sqrt{-a} \left (\sqrt{c} f-\sqrt{-a} g\right ) (e f-d g) \sqrt{f+g x}}-\frac{g \sqrt{d+e x}}{\sqrt{-a} \left (\sqrt{c} f+\sqrt{-a} g\right ) (e f-d g) \sqrt{f+g x}}-\frac{\sqrt{c} \int \frac{1}{\left (\sqrt{-a}-\sqrt{c} x\right ) \sqrt{d+e x} \sqrt{f+g x}} \, dx}{2 \left (\sqrt{-a} \sqrt{c} f-a g\right )}-\frac{\sqrt{c} \int \frac{1}{\left (\sqrt{-a}+\sqrt{c} x\right ) \sqrt{d+e x} \sqrt{f+g x}} \, dx}{2 \left (\sqrt{-a} \sqrt{c} f+a g\right )}\\ &=\frac{g \sqrt{d+e x}}{\sqrt{-a} \left (\sqrt{c} f-\sqrt{-a} g\right ) (e f-d g) \sqrt{f+g x}}-\frac{g \sqrt{d+e x}}{\sqrt{-a} \left (\sqrt{c} f+\sqrt{-a} g\right ) (e f-d g) \sqrt{f+g x}}-\frac{\sqrt{c} \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} \sqrt{c} f-a g}-\frac{\sqrt{c} \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} \sqrt{c} f+a g}\\ &=\frac{g \sqrt{d+e x}}{\sqrt{-a} \left (\sqrt{c} f-\sqrt{-a} g\right ) (e f-d g) \sqrt{f+g x}}-\frac{g \sqrt{d+e x}}{\sqrt{-a} \left (\sqrt{c} f+\sqrt{-a} g\right ) (e f-d g) \sqrt{f+g x}}+\frac{\sqrt{c} \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} \left (\sqrt{c} f-\sqrt{-a} g\right )^{3/2}}-\frac{\sqrt{c} \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} \left (\sqrt{c} f+\sqrt{-a} g\right )^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.843464, size = 291, normalized size = 0.82 \[ \frac{\frac{2 \sqrt{-a} g^2 \sqrt{d+e x}}{\sqrt{f+g x} \left (a g^2+c f^2\right ) (e f-d g)}+\frac{\sqrt{c} \tan ^{-1}\left (\frac{\sqrt{d+e x} \sqrt{-\sqrt{-a} g-\sqrt{c} f}}{\sqrt{f+g x} \sqrt{\sqrt{-a} e+\sqrt{c} d}}\right )}{\sqrt{\sqrt{-a} e+\sqrt{c} d} \left (-\sqrt{-a} g-\sqrt{c} f\right )^{3/2}}-\frac{\sqrt{c} \tan ^{-1}\left (\frac{\sqrt{d+e x} \sqrt{\sqrt{c} f-\sqrt{-a} g}}{\sqrt{f+g x} \sqrt{\sqrt{-a} e-\sqrt{c} d}}\right )}{\sqrt{\sqrt{-a} e-\sqrt{c} d} \left (\sqrt{c} f-\sqrt{-a} g\right )^{3/2}}}{\sqrt{-a}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.572, size = 10977, normalized size = 31. \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{1}{{\left (c x^{2} + a\right )} \sqrt{e x + d}{\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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a + c x^{2}\right ) \sqrt{d + e x} \left (f + g x\right )^{\frac{3}{2}}}\, dx \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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