Optimal. Leaf size=362 \[ -\frac{4 \sqrt{-a} f \sqrt{\frac{c x^2}{a}+1} \left (a g^2+c f^2\right ) \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right ),-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right )}{15 \sqrt{c} g^2 \sqrt{a+c x^2} \sqrt{f+g x}}+\frac{4 \sqrt{-a} \sqrt{\frac{c x^2}{a}+1} \sqrt{f+g x} \left (c f^2-3 a g^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right )}{15 \sqrt{c} g^2 \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}}}+\frac{2 \sqrt{a+c x^2} (f+g x)^{3/2}}{5 g}-\frac{4 f \sqrt{a+c x^2} \sqrt{f+g x}}{15 g} \]
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Rubi [A] time = 0.317375, antiderivative size = 362, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {735, 833, 844, 719, 424, 419} \[ -\frac{4 \sqrt{-a} f \sqrt{\frac{c x^2}{a}+1} \left (a g^2+c f^2\right ) \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right )}{15 \sqrt{c} g^2 \sqrt{a+c x^2} \sqrt{f+g x}}+\frac{4 \sqrt{-a} \sqrt{\frac{c x^2}{a}+1} \sqrt{f+g x} \left (c f^2-3 a g^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right )}{15 \sqrt{c} g^2 \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}}}+\frac{2 \sqrt{a+c x^2} (f+g x)^{3/2}}{5 g}-\frac{4 f \sqrt{a+c x^2} \sqrt{f+g x}}{15 g} \]
Antiderivative was successfully verified.
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Rule 735
Rule 833
Rule 844
Rule 719
Rule 424
Rule 419
Rubi steps
\begin{align*} \int \sqrt{f+g x} \sqrt{a+c x^2} \, dx &=\frac{2 (f+g x)^{3/2} \sqrt{a+c x^2}}{5 g}+\frac{2 \int \frac{(a g-c f x) \sqrt{f+g x}}{\sqrt{a+c x^2}} \, dx}{5 g}\\ &=-\frac{4 f \sqrt{f+g x} \sqrt{a+c x^2}}{15 g}+\frac{2 (f+g x)^{3/2} \sqrt{a+c x^2}}{5 g}+\frac{4 \int \frac{2 a c f g-\frac{1}{2} c \left (c f^2-3 a g^2\right ) x}{\sqrt{f+g x} \sqrt{a+c x^2}} \, dx}{15 c g}\\ &=-\frac{4 f \sqrt{f+g x} \sqrt{a+c x^2}}{15 g}+\frac{2 (f+g x)^{3/2} \sqrt{a+c x^2}}{5 g}+\frac{1}{15} \left (2 \left (3 a-\frac{c f^2}{g^2}\right )\right ) \int \frac{\sqrt{f+g x}}{\sqrt{a+c x^2}} \, dx+\frac{1}{15} \left (2 f \left (a+\frac{c f^2}{g^2}\right )\right ) \int \frac{1}{\sqrt{f+g x} \sqrt{a+c x^2}} \, dx\\ &=-\frac{4 f \sqrt{f+g x} \sqrt{a+c x^2}}{15 g}+\frac{2 (f+g x)^{3/2} \sqrt{a+c x^2}}{5 g}+\frac{\left (4 a \left (3 a-\frac{c f^2}{g^2}\right ) \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 )}{15 \sqrt{-a} \sqrt{c} \sqrt{\frac{c (f+g x)}{c f-\frac{a \sqrt{c} g}{\sqrt{-a}}}} \sqrt{a+c x^2}}+\frac{\left (4 a f \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 )}{15 \sqrt{-a} \sqrt{c} \sqrt{f+g x} \sqrt{a+c x^2}}\\ &=-\frac{4 f \sqrt{f+g x} \sqrt{a+c x^2}}{15 g}+\frac{2 (f+g x)^{3/2} \sqrt{a+c x^2}}{5 g}-\frac{4 \sqrt{-a} \left (3 a-\frac{c f^2}{g^2}\right ) \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 )}{15 \sqrt{c} \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{c} f+\sqrt{-a} g}} \sqrt{a+c x^2}}-\frac{4 \sqrt{-a} f \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 )}{15 \sqrt{c} \sqrt{f+g x} \sqrt{a+c x^2}}\\ \end{align*}
Mathematica [C] time = 2.96802, size = 536, normalized size = 1.48 \[ \frac{\sqrt{f+g x} \left (\frac{2 \left (a+c x^2\right ) (f+3 g x)}{g}-\frac{4 \left (-\sqrt{a} \sqrt{c} g (f+g x)^{3/2} \left (4 i \sqrt{a} \sqrt{c} f g-3 a g^2+c f^2\right ) \sqrt{\frac{g \left (x+\frac{i \sqrt{a}}{\sqrt{c}}\right )}{f+g x}} \sqrt{-\frac{-g x+\frac{i \sqrt{a} g}{\sqrt{c}}}{f+g x}} \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}{\sqrt{f+g x}}\right ),\frac{\sqrt{c} f-i \sqrt{a} g}{\sqrt{c} f+i \sqrt{a} g}\right )+g^2 \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}} \left (-3 a^2 g^2+a c \left (f^2-3 g^2 x^2\right )+c^2 f^2 x^2\right )+\sqrt{c} (f+g x)^{3/2} \left (-3 a^{3/2} g^3+\sqrt{a} c f^2 g+3 i a \sqrt{c} f g^2-i c^{3/2} f^3\right ) \sqrt{\frac{g \left (x+\frac{i \sqrt{a}}{\sqrt{c}}\right )}{f+g x}} \sqrt{-\frac{-g x+\frac{i \sqrt{a} g}{\sqrt{c}}}{f+g x}} E\left (i \sinh ^{-1}\left (\frac{\sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}{\sqrt{f+g x}}\right )|\frac{\sqrt{c} f-i \sqrt{a} g}{\sqrt{c} f+i \sqrt{a} g}\right )\right )}{c g^3 (f+g x) \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}\right )}{15 \sqrt{a+c x^2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.29, size = 1162, normalized size = 3.2 \begin{align*} \text{result 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 \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 (\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 \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 \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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