Optimal. Leaf size=178 \[ -\frac{\left (3 a e^2 g^2+c \left (3 d^2 g^2-8 d e f g+8 e^2 f^2\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right )}{4 e^{5/2} (e f-d g)^{5/2}}-\frac{\sqrt{f+g x} \left (a+\frac{c d^2}{e^2}\right )}{2 (d+e x)^2 (e f-d g)}+\frac{\sqrt{f+g x} \left (3 a e^2 g+c d (8 e f-5 d g)\right )}{4 e^2 (d+e x) (e f-d g)^2} \]
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Rubi [A] time = 0.30305, antiderivative size = 178, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {898, 1157, 385, 208} \[ -\frac{\left (3 a e^2 g^2+c \left (3 d^2 g^2-8 d e f g+8 e^2 f^2\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right )}{4 e^{5/2} (e f-d g)^{5/2}}-\frac{\sqrt{f+g x} \left (a+\frac{c d^2}{e^2}\right )}{2 (d+e x)^2 (e f-d g)}+\frac{\sqrt{f+g x} \left (3 a e^2 g+c d (8 e f-5 d g)\right )}{4 e^2 (d+e x) (e f-d g)^2} \]
Antiderivative was successfully verified.
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Rule 898
Rule 1157
Rule 385
Rule 208
Rubi steps
\begin{align*} \int \frac{a+c x^2}{(d+e x)^3 \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}}{\left (\frac{-e f+d g}{g}+\frac{e x^2}{g}\right )^3} \, dx,x,\sqrt{f+g x}\right )}{g}\\ &=-\frac{\left (a+\frac{c d^2}{e^2}\right ) \sqrt{f+g x}}{2 (e f-d g) (d+e x)^2}+\frac{\operatorname{Subst}\left (\int \frac{-3 a+\frac{c d^2}{e^2}-\frac{4 c f^2}{g^2}+\frac{4 c (e f-d g) x^2}{e g^2}}{\left (\frac{-e f+d g}{g}+\frac{e x^2}{g}\right )^2} \, dx,x,\sqrt{f+g x}\right )}{2 (e f-d g)}\\ &=-\frac{\left (a+\frac{c d^2}{e^2}\right ) \sqrt{f+g x}}{2 (e f-d g) (d+e x)^2}+\frac{\left (3 a e^2 g+c d (8 e f-5 d g)\right ) \sqrt{f+g x}}{4 e^2 (e f-d g)^2 (d+e x)}+\frac{\left (3 a e^2 g^2+c \left (8 e^2 f^2-8 d e f g+3 d^2 g^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{-e f+d g}{g}+\frac{e x^2}{g}} \, dx,x,\sqrt{f+g x}\right )}{4 e^2 g (e f-d g)^2}\\ &=-\frac{\left (a+\frac{c d^2}{e^2}\right ) \sqrt{f+g x}}{2 (e f-d g) (d+e x)^2}+\frac{\left (3 a e^2 g+c d (8 e f-5 d g)\right ) \sqrt{f+g x}}{4 e^2 (e f-d g)^2 (d+e x)}-\frac{\left (3 a e^2 g^2+c \left (8 e^2 f^2-8 d e f g+3 d^2 g^2\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right )}{4 e^{5/2} (e f-d g)^{5/2}}\\ \end{align*}
Mathematica [C] time = 0.885958, size = 207, normalized size = 1.16 \[ \frac{2 \left (\frac{\sqrt{e} g^2 \sqrt{f+g x} \left (a e^2+c d^2\right ) \, _2F_1\left (\frac{1}{2},3;\frac{3}{2};\frac{e (f+g x)}{e f-d g}\right )}{(d g-e f)^3}-\frac{c d \left (\sqrt{e} \sqrt{f+g x} (d g-e f)+g (d+e x) \sqrt{d g-e f} \tan ^{-1}\left (\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{d g-e f}}\right )\right )}{(d+e x) (e f-d g)^2}-\frac{c \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right )}{\sqrt{e f-d g}}\right )}{e^{5/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.219, size = 384, normalized size = 2.2 \begin{align*} 2\,{\frac{1}{ \left ( e \left ( gx+f \right ) +dg-ef \right ) ^{2}} \left ( 1/8\,{\frac{g \left ( 3\,a{e}^{2}g-5\,c{d}^{2}g+8\,cdef \right ) \left ( gx+f \right ) ^{3/2}}{e \left ({d}^{2}{g}^{2}-2\,defg+{e}^{2}{f}^{2} \right ) }}+1/8\,{\frac{ \left ( 5\,a{e}^{2}g-3\,c{d}^{2}g+8\,cdef \right ) g\sqrt{gx+f}}{{e}^{2} \left ( dg-ef \right ) }} \right ) }+{\frac{3\,a{g}^{2}}{4\,{d}^{2}{g}^{2}-8\,defg+4\,{e}^{2}{f}^{2}}\arctan \left ({e\sqrt{gx+f}{\frac{1}{\sqrt{ \left ( dg-ef \right ) e}}}} \right ){\frac{1}{\sqrt{ \left ( dg-ef \right ) e}}}}+{\frac{3\,c{d}^{2}{g}^{2}}{ \left ( 4\,{d}^{2}{g}^{2}-8\,defg+4\,{e}^{2}{f}^{2} \right ){e}^{2}}\arctan \left ({e\sqrt{gx+f}{\frac{1}{\sqrt{ \left ( dg-ef \right ) e}}}} \right ){\frac{1}{\sqrt{ \left ( dg-ef \right ) e}}}}-2\,{\frac{cdfg}{e \left ({d}^{2}{g}^{2}-2\,defg+{e}^{2}{f}^{2} \right ) \sqrt{ \left ( dg-ef \right ) e}}\arctan \left ({\frac{e\sqrt{gx+f}}{\sqrt{ \left ( dg-ef \right ) e}}} \right ) }+2\,{\frac{c{f}^{2}}{ \left ({d}^{2}{g}^{2}-2\,defg+{e}^{2}{f}^{2} \right ) \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 [B] time = 2.00828, size = 1835, normalized size = 10.31 \begin{align*} \text{result too large to display} \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 [A] time = 1.14981, size = 375, normalized size = 2.11 \begin{align*} \frac{{\left (3 \, c d^{2} g^{2} - 8 \, c d f g e + 8 \, c f^{2} e^{2} + 3 \, a g^{2} e^{2}\right )} \arctan \left (\frac{\sqrt{g x + f} e}{\sqrt{d g e - f e^{2}}}\right )}{4 \,{\left (d^{2} g^{2} e^{2} - 2 \, d f g e^{3} + f^{2} e^{4}\right )} \sqrt{d g e - f e^{2}}} - \frac{3 \, \sqrt{g x + f} c d^{3} g^{3} + 5 \,{\left (g x + f\right )}^{\frac{3}{2}} c d^{2} g^{2} e - 11 \, \sqrt{g x + f} c d^{2} f g^{2} e - 8 \,{\left (g x + f\right )}^{\frac{3}{2}} c d f g e^{2} + 8 \, \sqrt{g x + f} c d f^{2} g e^{2} - 5 \, \sqrt{g x + f} a d g^{3} e^{2} - 3 \,{\left (g x + f\right )}^{\frac{3}{2}} a g^{2} e^{3} + 5 \, \sqrt{g x + f} a f g^{2} e^{3}}{4 \,{\left (d^{2} g^{2} e^{2} - 2 \, d f g e^{3} + f^{2} e^{4}\right )}{\left (d g +{\left (g x + f\right )} e - f e\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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