Optimal. Leaf size=698 \[ \frac{\sqrt{-a} \sqrt{c} f \sqrt{\frac{c x^2}{a}+1} \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right ),-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right )}{\sqrt{a+c x^2} \sqrt{f+g x} \left (a e^2+c d^2\right )}-\frac{\sqrt{-a} \sqrt{c} d g \sqrt{\frac{c x^2}{a}+1} \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right ),-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right )}{e \sqrt{a+c x^2} \sqrt{f+g x} \left (a e^2+c d^2\right )}-\frac{e \sqrt{a+c x^2} \sqrt{f+g x}}{(d+e x) \left (a e^2+c d^2\right )}-\frac{\sqrt{-a} \sqrt{c} \sqrt{\frac{c x^2}{a}+1} \sqrt{f+g x} E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right )}{\sqrt{a+c x^2} \left (a e^2+c d^2\right ) \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}}}-\frac{\sqrt{\frac{c x^2}{a}+1} \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}} \left (a e^2 g+c d (2 e f-d g)\right ) \Pi \left (\frac{2 e}{\frac{\sqrt{c} d}{\sqrt{-a}}+e};\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|\frac{2 \sqrt{-a} g}{\sqrt{c} f+\sqrt{-a} g}\right )}{e \sqrt{a+c x^2} \sqrt{f+g x} \left (\frac{\sqrt{c} d}{\sqrt{-a}}+e\right ) \left (a e^2+c d^2\right )} \]
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Rubi [A] time = 1.93269, antiderivative size = 698, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 10, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.357, Rules used = {946, 6742, 719, 419, 844, 424, 933, 168, 538, 537} \[ -\frac{e \sqrt{a+c x^2} \sqrt{f+g x}}{(d+e x) \left (a e^2+c d^2\right )}+\frac{\sqrt{-a} \sqrt{c} f \sqrt{\frac{c x^2}{a}+1} \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right )}{\sqrt{a+c x^2} \sqrt{f+g x} \left (a e^2+c d^2\right )}-\frac{\sqrt{-a} \sqrt{c} d g \sqrt{\frac{c x^2}{a}+1} \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right )}{e \sqrt{a+c x^2} \sqrt{f+g x} \left (a e^2+c d^2\right )}-\frac{\sqrt{-a} \sqrt{c} \sqrt{\frac{c x^2}{a}+1} \sqrt{f+g x} E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right )}{\sqrt{a+c x^2} \left (a e^2+c d^2\right ) \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}}}-\frac{\sqrt{\frac{c x^2}{a}+1} \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}} \left (a e^2 g+c d (2 e f-d g)\right ) \Pi \left (\frac{2 e}{\frac{\sqrt{c} d}{\sqrt{-a}}+e};\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|\frac{2 \sqrt{-a} g}{\sqrt{c} f+\sqrt{-a} g}\right )}{e \sqrt{a+c x^2} \sqrt{f+g x} \left (\frac{\sqrt{c} d}{\sqrt{-a}}+e\right ) \left (a e^2+c d^2\right )} \]
Antiderivative was successfully verified.
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Rule 946
Rule 6742
Rule 719
Rule 419
Rule 844
Rule 424
Rule 933
Rule 168
Rule 538
Rule 537
Rubi steps
\begin{align*} \int \frac{\sqrt{f+g x}}{(d+e x)^2 \sqrt{a+c x^2}} \, dx &=-\frac{e \sqrt{f+g x} \sqrt{a+c x^2}}{\left (c d^2+a e^2\right ) (d+e x)}-\frac{\int \frac{-2 c d f-a e g-2 c d g x-c e g x^2}{(d+e x) \sqrt{f+g x} \sqrt{a+c x^2}} \, dx}{2 \left (c d^2+a e^2\right )}\\ &=-\frac{e \sqrt{f+g x} \sqrt{a+c x^2}}{\left (c d^2+a e^2\right ) (d+e x)}-\frac{\int \left (-\frac{c d g}{e \sqrt{f+g x} \sqrt{a+c x^2}}-\frac{c g x}{\sqrt{f+g x} \sqrt{a+c x^2}}+\frac{-a e^2 g-c d (2 e f-d g)}{e (d+e x) \sqrt{f+g x} \sqrt{a+c x^2}}\right ) \, dx}{2 \left (c d^2+a e^2\right )}\\ &=-\frac{e \sqrt{f+g x} \sqrt{a+c x^2}}{\left (c d^2+a e^2\right ) (d+e x)}+\frac{(c g) \int \frac{x}{\sqrt{f+g x} \sqrt{a+c x^2}} \, dx}{2 \left (c d^2+a e^2\right )}+\frac{(c d g) \int \frac{1}{\sqrt{f+g x} \sqrt{a+c x^2}} \, dx}{2 e \left (c d^2+a e^2\right )}+\frac{\left (a e^2 g+c d (2 e f-d g)\right ) \int \frac{1}{(d+e x) \sqrt{f+g x} \sqrt{a+c x^2}} \, dx}{2 e \left (c d^2+a e^2\right )}\\ &=-\frac{e \sqrt{f+g x} \sqrt{a+c x^2}}{\left (c d^2+a e^2\right ) (d+e x)}+\frac{c \int \frac{\sqrt{f+g x}}{\sqrt{a+c x^2}} \, dx}{2 \left (c d^2+a e^2\right )}-\frac{(c f) \int \frac{1}{\sqrt{f+g x} \sqrt{a+c x^2}} \, dx}{2 \left (c d^2+a e^2\right )}+\frac{\left (\left (a e^2 g+c d (2 e f-d g)\right ) \sqrt{1+\frac{c x^2}{a}}\right ) \int \frac{1}{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}} \sqrt{1+\frac{\sqrt{c} x}{\sqrt{-a}}} (d+e x) \sqrt{f+g x}} \, dx}{2 e \left (c d^2+a e^2\right ) \sqrt{a+c x^2}}+\frac{\left (a \sqrt{c} d g \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 )}{\sqrt{-a} e \left (c d^2+a e^2\right ) \sqrt{f+g x} \sqrt{a+c x^2}}\\ &=-\frac{e \sqrt{f+g x} \sqrt{a+c x^2}}{\left (c d^2+a e^2\right ) (d+e x)}-\frac{\sqrt{-a} \sqrt{c} d g \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 )}{e \left (c d^2+a e^2\right ) \sqrt{f+g x} \sqrt{a+c x^2}}-\frac{\left (\left (a e^2 g+c d (2 e f-d g)\right ) \sqrt{1+\frac{c x^2}{a}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{2-x^2} \left (\frac{\sqrt{c} d}{\sqrt{-a}}+e-e x^2\right ) \sqrt{f+\frac{\sqrt{-a} g}{\sqrt{c}}-\frac{\sqrt{-a} g x^2}{\sqrt{c}}}} \, dx,x,\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}\right )}{e \left (c d^2+a e^2\right ) \sqrt{a+c x^2}}+\frac{\left (a \sqrt{c} \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 )}{\sqrt{-a} \left (c d^2+a e^2\right ) \sqrt{\frac{c (f+g x)}{c f-\frac{a \sqrt{c} g}{\sqrt{-a}}}} \sqrt{a+c x^2}}-\frac{\left (a \sqrt{c} f \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 )}{\sqrt{-a} \left (c d^2+a e^2\right ) \sqrt{f+g x} \sqrt{a+c x^2}}\\ &=-\frac{e \sqrt{f+g x} \sqrt{a+c x^2}}{\left (c d^2+a e^2\right ) (d+e x)}-\frac{\sqrt{-a} \sqrt{c} \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 )}{\left (c d^2+a e^2\right ) \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{c} f+\sqrt{-a} g}} \sqrt{a+c x^2}}+\frac{\sqrt{-a} \sqrt{c} f \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 )}{\left (c d^2+a e^2\right ) \sqrt{f+g x} \sqrt{a+c x^2}}-\frac{\sqrt{-a} \sqrt{c} d g \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 )}{e \left (c d^2+a e^2\right ) \sqrt{f+g x} \sqrt{a+c x^2}}-\frac{\left (\left (a e^2 g+c d (2 e f-d g)\right ) \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{c} f+\sqrt{-a} g}} \sqrt{1+\frac{c x^2}{a}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{2-x^2} \left (\frac{\sqrt{c} d}{\sqrt{-a}}+e-e x^2\right ) \sqrt{1-\frac{\sqrt{-a} g x^2}{\sqrt{c} \left (f+\frac{\sqrt{-a} g}{\sqrt{c}}\right )}}} \, dx,x,\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}\right )}{e \left (c d^2+a e^2\right ) \sqrt{f+g x} \sqrt{a+c x^2}}\\ &=-\frac{e \sqrt{f+g x} \sqrt{a+c x^2}}{\left (c d^2+a e^2\right ) (d+e x)}-\frac{\sqrt{-a} \sqrt{c} \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 )}{\left (c d^2+a e^2\right ) \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{c} f+\sqrt{-a} g}} \sqrt{a+c x^2}}+\frac{\sqrt{-a} \sqrt{c} f \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 )}{\left (c d^2+a e^2\right ) \sqrt{f+g x} \sqrt{a+c x^2}}-\frac{\sqrt{-a} \sqrt{c} d g \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 )}{e \left (c d^2+a e^2\right ) \sqrt{f+g x} \sqrt{a+c x^2}}-\frac{\left (a e^2 g+c d (2 e f-d g)\right ) \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{c} f+\sqrt{-a} g}} \sqrt{1+\frac{c x^2}{a}} \Pi \left (\frac{2 e}{\frac{\sqrt{c} d}{\sqrt{-a}}+e};\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|\frac{2 \sqrt{-a} g}{\sqrt{c} f+\sqrt{-a} g}\right )}{e \left (\frac{\sqrt{c} d}{\sqrt{-a}}+e\right ) \left (c d^2+a e^2\right ) \sqrt{f+g x} \sqrt{a+c x^2}}\\ \end{align*}
Mathematica [C] time = 6.17204, size = 1330, normalized size = 1.91 \[ \frac{\sqrt{f+g x} \left (-\frac{\left (c x^2+a\right ) e^2}{d+e x}-\frac{-c e^2 \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}} f^3+2 c e^2 \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}} (f+g x) f^2+c d e g \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}} f^2-c e^2 \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}} (f+g x)^2 f-2 c d e g \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}} (f+g x) f-2 i c d e g \sqrt{\frac{g \left (x+\frac{i \sqrt{a}}{\sqrt{c}}\right )}{f+g x}} \sqrt{-\frac{\frac{i \sqrt{a} g}{\sqrt{c}}-g x}{f+g x}} (f+g x)^{3/2} \Pi \left (\frac{\sqrt{c} (e f-d g)}{e \left (\sqrt{c} f+i \sqrt{a} g\right )};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-a e^2 g^2 \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}} f+c d e g \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}} (f+g x)^2+\sqrt{c} e \left (\sqrt{a} g-i \sqrt{c} f\right ) (d g-e f) \sqrt{\frac{g \left (x+\frac{i \sqrt{a}}{\sqrt{c}}\right )}{f+g x}} \sqrt{-\frac{\frac{i \sqrt{a} g}{\sqrt{c}}-g x}{f+g x}} (f+g x)^{3/2} 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 )+e \left (i \sqrt{c} d+\sqrt{a} e\right ) g \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{\frac{i \sqrt{a} g}{\sqrt{c}}-g x}{f+g x}} (f+g x)^{3/2} \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 )+i c d^2 g^2 \sqrt{\frac{g \left (x+\frac{i \sqrt{a}}{\sqrt{c}}\right )}{f+g x}} \sqrt{-\frac{\frac{i \sqrt{a} g}{\sqrt{c}}-g x}{f+g x}} (f+g x)^{3/2} \Pi \left (\frac{\sqrt{c} (e f-d g)}{e \left (\sqrt{c} f+i \sqrt{a} g\right )};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 )-i a e^2 g^2 \sqrt{\frac{g \left (x+\frac{i \sqrt{a}}{\sqrt{c}}\right )}{f+g x}} \sqrt{-\frac{\frac{i \sqrt{a} g}{\sqrt{c}}-g x}{f+g x}} (f+g x)^{3/2} \Pi \left (\frac{\sqrt{c} (e f-d g)}{e \left (\sqrt{c} f+i \sqrt{a} g\right )};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 )+a d e g^3 \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}{g \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}} (e f-d g) (f+g x)}\right )}{\left (a e^3+c d^2 e\right ) \sqrt{c x^2+a}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.263, size = 5743, normalized size = 8.2 \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{\sqrt{g x + f}}{\sqrt{c x^{2} + a}{\left (e x + d\right )}^{2}}\,{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(-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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{g x + f}}{\sqrt{c x^{2} + a}{\left (e x + d\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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