Optimal. Leaf size=354 \[ -\frac{e \sqrt{f+g x}}{\sqrt{-a} \sqrt{d+e x} \left (\sqrt{c} d-\sqrt{-a} e\right ) (e f-d g)}+\frac{e \sqrt{f+g x}}{\sqrt{-a} \sqrt{d+e x} \left (\sqrt{-a} e+\sqrt{c} d\right ) (e f-d g)}+\frac{\sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{d+e x} \sqrt{\sqrt{c} f-\sqrt{-a} g}}{\sqrt{f+g x} \sqrt{\sqrt{c} d-\sqrt{-a} e}}\right )}{\sqrt{-a} \left (\sqrt{c} d-\sqrt{-a} e\right )^{3/2} \sqrt{\sqrt{c} f-\sqrt{-a} g}}-\frac{\sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{d+e x} \sqrt{\sqrt{-a} g+\sqrt{c} f}}{\sqrt{f+g x} \sqrt{\sqrt{-a} e+\sqrt{c} d}}\right )}{\sqrt{-a} \left (\sqrt{-a} e+\sqrt{c} d\right )^{3/2} \sqrt{\sqrt{-a} g+\sqrt{c} f}} \]
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Rubi [A] time = 0.613599, 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} \[ -\frac{e \sqrt{f+g x}}{\sqrt{-a} \sqrt{d+e x} \left (\sqrt{c} d-\sqrt{-a} e\right ) (e f-d g)}+\frac{e \sqrt{f+g x}}{\sqrt{-a} \sqrt{d+e x} \left (\sqrt{-a} e+\sqrt{c} d\right ) (e f-d g)}+\frac{\sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{d+e x} \sqrt{\sqrt{c} f-\sqrt{-a} g}}{\sqrt{f+g x} \sqrt{\sqrt{c} d-\sqrt{-a} e}}\right )}{\sqrt{-a} \left (\sqrt{c} d-\sqrt{-a} e\right )^{3/2} \sqrt{\sqrt{c} f-\sqrt{-a} g}}-\frac{\sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{d+e x} \sqrt{\sqrt{-a} g+\sqrt{c} f}}{\sqrt{f+g x} \sqrt{\sqrt{-a} e+\sqrt{c} d}}\right )}{\sqrt{-a} \left (\sqrt{-a} e+\sqrt{c} d\right )^{3/2} \sqrt{\sqrt{-a} g+\sqrt{c} f}} \]
Antiderivative was successfully verified.
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Rule 912
Rule 96
Rule 93
Rule 208
Rubi steps
\begin{align*} \int \frac{1}{(d+e x)^{3/2} \sqrt{f+g x} \left (a+c x^2\right )} \, dx &=\int \left (\frac{\sqrt{-a}}{2 a \left (\sqrt{-a}-\sqrt{c} x\right ) (d+e x)^{3/2} \sqrt{f+g x}}+\frac{\sqrt{-a}}{2 a \left (\sqrt{-a}+\sqrt{c} x\right ) (d+e x)^{3/2} \sqrt{f+g x}}\right ) \, dx\\ &=-\frac{\int \frac{1}{\left (\sqrt{-a}-\sqrt{c} x\right ) (d+e x)^{3/2} \sqrt{f+g x}} \, dx}{2 \sqrt{-a}}-\frac{\int \frac{1}{\left (\sqrt{-a}+\sqrt{c} x\right ) (d+e x)^{3/2} \sqrt{f+g x}} \, dx}{2 \sqrt{-a}}\\ &=-\frac{e \sqrt{f+g x}}{\sqrt{-a} \left (\sqrt{c} d-\sqrt{-a} e\right ) (e f-d g) \sqrt{d+e x}}+\frac{e \sqrt{f+g x}}{\sqrt{-a} \left (\sqrt{c} d+\sqrt{-a} e\right ) (e f-d g) \sqrt{d+e 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} d-a e\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} d+a e\right )}\\ &=-\frac{e \sqrt{f+g x}}{\sqrt{-a} \left (\sqrt{c} d-\sqrt{-a} e\right ) (e f-d g) \sqrt{d+e x}}+\frac{e \sqrt{f+g x}}{\sqrt{-a} \left (\sqrt{c} d+\sqrt{-a} e\right ) (e f-d g) \sqrt{d+e 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} d-a e}-\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} d+a e}\\ &=-\frac{e \sqrt{f+g x}}{\sqrt{-a} \left (\sqrt{c} d-\sqrt{-a} e\right ) (e f-d g) \sqrt{d+e x}}+\frac{e \sqrt{f+g x}}{\sqrt{-a} \left (\sqrt{c} d+\sqrt{-a} e\right ) (e f-d g) \sqrt{d+e 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} \left (\sqrt{c} d-\sqrt{-a} e\right )^{3/2} \sqrt{\sqrt{c} f-\sqrt{-a} g}}-\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} \left (\sqrt{c} d+\sqrt{-a} e\right )^{3/2} \sqrt{\sqrt{c} f+\sqrt{-a} g}}\\ \end{align*}
Mathematica [A] time = 0.86199, size = 291, normalized size = 0.82 \[ \frac{\frac{2 \sqrt{-a} e^2 \sqrt{f+g x}}{\sqrt{d+e x} \left (a e^2+c d^2\right ) (d g-e f)}-\frac{\sqrt{c} \tan ^{-1}\left (\frac{\sqrt{d+e x} \sqrt{-\sqrt{-a} g-\sqrt{c} f}}{\sqrt{f+g x} \sqrt{\sqrt{-a} e+\sqrt{c} d}}\right )}{\left (\sqrt{-a} e+\sqrt{c} d\right )^{3/2} \sqrt{-\sqrt{-a} g-\sqrt{c} f}}+\frac{\sqrt{c} \tan ^{-1}\left (\frac{\sqrt{d+e x} \sqrt{\sqrt{c} f-\sqrt{-a} g}}{\sqrt{f+g x} \sqrt{\sqrt{-a} e-\sqrt{c} d}}\right )}{\left (\sqrt{-a} e-\sqrt{c} d\right )^{3/2} \sqrt{\sqrt{c} f-\sqrt{-a} g}}}{\sqrt{-a}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.451, 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 )}{\left (e x + d\right )}^{\frac{3}{2}} \sqrt{g x + f}}\,{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 ) \left (d + e x\right )^{\frac{3}{2}} \sqrt{f + g x}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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