Optimal. Leaf size=322 \[ -\frac{4 \sqrt{-a} \sqrt{\frac{c x^2}{a}+1} \left (a g^2+c f^2\right ) \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right ),-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right )}{3 \sqrt{c} g^2 \sqrt{a+c x^2} \sqrt{f+g x}}+\frac{4 \sqrt{-a} \sqrt{c} f \sqrt{\frac{c x^2}{a}+1} \sqrt{f+g x} E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right )}{3 g^2 \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}}}+\frac{2 \sqrt{a+c x^2} \sqrt{f+g x}}{3 g} \]
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Rubi [A] time = 0.204486, antiderivative size = 322, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {735, 844, 719, 424, 419} \[ -\frac{4 \sqrt{-a} \sqrt{\frac{c x^2}{a}+1} \left (a g^2+c f^2\right ) \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right )}{3 \sqrt{c} g^2 \sqrt{a+c x^2} \sqrt{f+g x}}+\frac{4 \sqrt{-a} \sqrt{c} f \sqrt{\frac{c x^2}{a}+1} \sqrt{f+g x} E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right )}{3 g^2 \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}}}+\frac{2 \sqrt{a+c x^2} \sqrt{f+g x}}{3 g} \]
Antiderivative was successfully verified.
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Rule 735
Rule 844
Rule 719
Rule 424
Rule 419
Rubi steps
\begin{align*} \int \frac{\sqrt{a+c x^2}}{\sqrt{f+g x}} \, dx &=\frac{2 \sqrt{f+g x} \sqrt{a+c x^2}}{3 g}+\frac{2 \int \frac{a g-c f x}{\sqrt{f+g x} \sqrt{a+c x^2}} \, dx}{3 g}\\ &=\frac{2 \sqrt{f+g x} \sqrt{a+c x^2}}{3 g}+\frac{1}{3} \left (2 \left (a+\frac{c f^2}{g^2}\right )\right ) \int \frac{1}{\sqrt{f+g x} \sqrt{a+c x^2}} \, dx-\frac{(2 c f) \int \frac{\sqrt{f+g x}}{\sqrt{a+c x^2}} \, dx}{3 g^2}\\ &=\frac{2 \sqrt{f+g x} \sqrt{a+c x^2}}{3 g}-\frac{\left (4 a \sqrt{c} f \sqrt{f+g x} \sqrt{1+\frac{c x^2}{a}}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1+\frac{2 a \sqrt{c} g x^2}{\sqrt{-a} \left (c f-\frac{a \sqrt{c} g}{\sqrt{-a}}\right )}}}{\sqrt{1-x^2}} \, dx,x,\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )}{3 \sqrt{-a} g^2 \sqrt{\frac{c (f+g x)}{c f-\frac{a \sqrt{c} g}{\sqrt{-a}}}} \sqrt{a+c x^2}}+\frac{\left (4 a \left (a+\frac{c f^2}{g^2}\right ) \sqrt{\frac{c (f+g x)}{c f-\frac{a \sqrt{c} g}{\sqrt{-a}}}} \sqrt{1+\frac{c x^2}{a}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x^2} \sqrt{1+\frac{2 a \sqrt{c} g x^2}{\sqrt{-a} \left (c f-\frac{a \sqrt{c} g}{\sqrt{-a}}\right )}}} \, dx,x,\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )}{3 \sqrt{-a} \sqrt{c} \sqrt{f+g x} \sqrt{a+c x^2}}\\ &=\frac{2 \sqrt{f+g x} \sqrt{a+c x^2}}{3 g}+\frac{4 \sqrt{-a} \sqrt{c} f \sqrt{f+g x} \sqrt{1+\frac{c x^2}{a}} E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right )}{3 g^2 \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{c} f+\sqrt{-a} g}} \sqrt{a+c x^2}}-\frac{4 \sqrt{-a} \left (a+\frac{c f^2}{g^2}\right ) \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{c} f+\sqrt{-a} g}} \sqrt{1+\frac{c x^2}{a}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right )}{3 \sqrt{c} \sqrt{f+g x} \sqrt{a+c x^2}}\\ \end{align*}
Mathematica [C] time = 2.02741, size = 456, normalized size = 1.42 \[ \frac{2 \sqrt{f+g x} \left (g^2 \left (a+c x^2\right )-\frac{2 \left (-\sqrt{a} g (f+g x)^{3/2} \left (\sqrt{c} f+i \sqrt{a} g\right ) \sqrt{\frac{g \left (x+\frac{i \sqrt{a}}{\sqrt{c}}\right )}{f+g x}} \sqrt{-\frac{-g x+\frac{i \sqrt{a} g}{\sqrt{c}}}{f+g x}} \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}{\sqrt{f+g x}}\right ),\frac{\sqrt{c} f-i \sqrt{a} g}{\sqrt{c} f+i \sqrt{a} g}\right )+f g^2 \left (a+c x^2\right ) \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}+\sqrt{c} f (f+g x)^{3/2} \left (\sqrt{a} g-i \sqrt{c} f\right ) \sqrt{\frac{g \left (x+\frac{i \sqrt{a}}{\sqrt{c}}\right )}{f+g x}} \sqrt{-\frac{-g x+\frac{i \sqrt{a} g}{\sqrt{c}}}{f+g x}} E\left (i \sinh ^{-1}\left (\frac{\sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}{\sqrt{f+g x}}\right )|\frac{\sqrt{c} f-i \sqrt{a} g}{\sqrt{c} f+i \sqrt{a} g}\right )\right )}{(f+g x) \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}\right )}{3 g^3 \sqrt{a+c x^2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.299, size = 688, normalized size = 2.1 \begin{align*} -{\frac{2}{3\,c \left ( cg{x}^{3}+cf{x}^{2}+agx+af \right ){g}^{3}}\sqrt{gx+f}\sqrt{c{x}^{2}+a} \left ( 2\,\sqrt{-{\frac{c \left ( gx+f \right ) }{\sqrt{-ac}g-cf}}}\sqrt{{\frac{ \left ( -cx+\sqrt{-ac} \right ) g}{\sqrt{-ac}g+cf}}}\sqrt{{\frac{ \left ( cx+\sqrt{-ac} \right ) g}{\sqrt{-ac}g-cf}}}{\it EllipticF} \left ( \sqrt{-{\frac{c \left ( gx+f \right ) }{\sqrt{-ac}g-cf}}},\sqrt{-{\frac{\sqrt{-ac}g-cf}{\sqrt{-ac}g+cf}}} \right ) \sqrt{-ac}a{g}^{3}+2\,\sqrt{-{\frac{c \left ( gx+f \right ) }{\sqrt{-ac}g-cf}}}\sqrt{{\frac{ \left ( -cx+\sqrt{-ac} \right ) g}{\sqrt{-ac}g+cf}}}\sqrt{{\frac{ \left ( cx+\sqrt{-ac} \right ) g}{\sqrt{-ac}g-cf}}}{\it EllipticF} \left ( \sqrt{-{\frac{c \left ( gx+f \right ) }{\sqrt{-ac}g-cf}}},\sqrt{-{\frac{\sqrt{-ac}g-cf}{\sqrt{-ac}g+cf}}} \right ) \sqrt{-ac}c{f}^{2}g-2\,\sqrt{-{\frac{c \left ( gx+f \right ) }{\sqrt{-ac}g-cf}}}\sqrt{{\frac{ \left ( -cx+\sqrt{-ac} \right ) g}{\sqrt{-ac}g+cf}}}\sqrt{{\frac{ \left ( cx+\sqrt{-ac} \right ) g}{\sqrt{-ac}g-cf}}}{\it EllipticE} \left ( \sqrt{-{\frac{c \left ( gx+f \right ) }{\sqrt{-ac}g-cf}}},\sqrt{-{\frac{\sqrt{-ac}g-cf}{\sqrt{-ac}g+cf}}} \right ) acf{g}^{2}-2\,\sqrt{-{\frac{c \left ( gx+f \right ) }{\sqrt{-ac}g-cf}}}\sqrt{{\frac{ \left ( -cx+\sqrt{-ac} \right ) g}{\sqrt{-ac}g+cf}}}\sqrt{{\frac{ \left ( cx+\sqrt{-ac} \right ) g}{\sqrt{-ac}g-cf}}}{\it EllipticE} \left ( \sqrt{-{\frac{c \left ( gx+f \right ) }{\sqrt{-ac}g-cf}}},\sqrt{-{\frac{\sqrt{-ac}g-cf}{\sqrt{-ac}g+cf}}} \right ){c}^{2}{f}^{3}-{x}^{3}{c}^{2}{g}^{3}-{x}^{2}{c}^{2}f{g}^{2}-xac{g}^{3}-acf{g}^{2} \right ) } \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{c x^{2} + a}}{\sqrt{g x + f}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{c x^{2} + a}}{\sqrt{g x + f}}, x\right ) \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{\sqrt{a + c x^{2}}}{\sqrt{f + g x}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{c x^{2} + a}}{\sqrt{g x + f}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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