Optimal. Leaf size=104 \[ -\frac{2 \left (a e^2+c d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right )}{e^{5/2} \sqrt{e f-d g}}-\frac{2 c \sqrt{f+g x} (d g+e f)}{e^2 g^2}+\frac{2 c (f+g x)^{3/2}}{3 e g^2} \]
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Rubi [A] time = 0.123759, antiderivative size = 104, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {898, 1153, 208} \[ -\frac{2 \left (a e^2+c d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right )}{e^{5/2} \sqrt{e f-d g}}-\frac{2 c \sqrt{f+g x} (d g+e f)}{e^2 g^2}+\frac{2 c (f+g x)^{3/2}}{3 e g^2} \]
Antiderivative was successfully verified.
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Rule 898
Rule 1153
Rule 208
Rubi steps
\begin{align*} \int \frac{a+c x^2}{(d+e x) \sqrt{f+g x}} \, dx &=\frac{2 \operatorname{Subst}\left (\int \frac{\frac{c f^2+a g^2}{g^2}-\frac{2 c f x^2}{g^2}+\frac{c x^4}{g^2}}{\frac{-e f+d g}{g}+\frac{e x^2}{g}} \, dx,x,\sqrt{f+g x}\right )}{g}\\ &=\frac{2 \operatorname{Subst}\left (\int \left (-\frac{c (e f+d g)}{e^2 g}+\frac{c x^2}{e g}+\frac{c d^2+a e^2}{e^2 \left (d-\frac{e f}{g}+\frac{e x^2}{g}\right )}\right ) \, dx,x,\sqrt{f+g x}\right )}{g}\\ &=-\frac{2 c (e f+d g) \sqrt{f+g x}}{e^2 g^2}+\frac{2 c (f+g x)^{3/2}}{3 e g^2}+\frac{\left (2 \left (a+\frac{c d^2}{e^2}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{d-\frac{e f}{g}+\frac{e x^2}{g}} \, dx,x,\sqrt{f+g x}\right )}{g}\\ &=-\frac{2 c (e f+d g) \sqrt{f+g x}}{e^2 g^2}+\frac{2 c (f+g x)^{3/2}}{3 e g^2}-\frac{2 \left (c d^2+a e^2\right ) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right )}{e^{5/2} \sqrt{e f-d g}}\\ \end{align*}
Mathematica [A] time = 0.169567, size = 92, normalized size = 0.88 \[ \frac{2 c \sqrt{f+g x} (-3 d g-2 e f+e g x)}{3 e^2 g^2}-\frac{2 \left (a e^2+c d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right )}{e^{5/2} \sqrt{e f-d g}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.077, size = 132, normalized size = 1.3 \begin{align*}{\frac{2\,c}{3\,e{g}^{2}} \left ( gx+f \right ) ^{{\frac{3}{2}}}}-2\,{\frac{cd\sqrt{gx+f}}{g{e}^{2}}}-2\,{\frac{cf\sqrt{gx+f}}{e{g}^{2}}}+2\,{\frac{a}{\sqrt{ \left ( dg-ef \right ) e}}\arctan \left ({\frac{e\sqrt{gx+f}}{\sqrt{ \left ( dg-ef \right ) e}}} \right ) }+2\,{\frac{c{d}^{2}}{{e}^{2}\sqrt{ \left ( dg-ef \right ) e}}\arctan \left ({\frac{e\sqrt{gx+f}}{\sqrt{ \left ( dg-ef \right ) e}}} \right ) } \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 [A] time = 1.83007, size = 628, normalized size = 6.04 \begin{align*} \left [\frac{3 \,{\left (c d^{2} + a e^{2}\right )} \sqrt{e^{2} f - d e g} g^{2} \log \left (\frac{e g x + 2 \, e f - d g - 2 \, \sqrt{e^{2} f - d e g} \sqrt{g x + f}}{e x + d}\right ) - 2 \,{\left (2 \, c e^{3} f^{2} + c d e^{2} f g - 3 \, c d^{2} e g^{2} -{\left (c e^{3} f g - c d e^{2} g^{2}\right )} x\right )} \sqrt{g x + f}}{3 \,{\left (e^{4} f g^{2} - d e^{3} g^{3}\right )}}, \frac{2 \,{\left (3 \,{\left (c d^{2} + a e^{2}\right )} \sqrt{-e^{2} f + d e g} g^{2} \arctan \left (\frac{\sqrt{-e^{2} f + d e g} \sqrt{g x + f}}{e g x + e f}\right ) -{\left (2 \, c e^{3} f^{2} + c d e^{2} f g - 3 \, c d^{2} e g^{2} -{\left (c e^{3} f g - c d e^{2} g^{2}\right )} x\right )} \sqrt{g x + f}\right )}}{3 \,{\left (e^{4} f g^{2} - d e^{3} g^{3}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 20.3544, size = 100, normalized size = 0.96 \begin{align*} \frac{2 c \left (f + g x\right )^{\frac{3}{2}}}{3 e g^{2}} - \frac{2 c \sqrt{f + g x} \left (d g + e f\right )}{e^{2} g^{2}} - \frac{2 \left (a e^{2} + c d^{2}\right ) \operatorname{atan}{\left (\frac{1}{\sqrt{\frac{e}{d g - e f}} \sqrt{f + g x}} \right )}}{e^{2} \sqrt{\frac{e}{d g - e f}} \left (d g - e f\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18318, size = 144, normalized size = 1.38 \begin{align*} \frac{2 \,{\left (c d^{2} + a e^{2}\right )} \arctan \left (\frac{\sqrt{g x + f} e}{\sqrt{d g e - f e^{2}}}\right ) e^{\left (-2\right )}}{\sqrt{d g e - f e^{2}}} - \frac{2 \,{\left (3 \, \sqrt{g x + f} c d g^{5} e -{\left (g x + f\right )}^{\frac{3}{2}} c g^{4} e^{2} + 3 \, \sqrt{g x + f} c f g^{4} e^{2}\right )} e^{\left (-3\right )}}{3 \, g^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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