Optimal. Leaf size=650 \[ -\frac{\sqrt{-a} \sqrt{c} \sqrt{\frac{c x^2}{a}+1} (2 e f-3 d g) \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right ),-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right )}{e^3 \sqrt{a+c x^2} \sqrt{f+g x}}+\frac{3 \sqrt{-a} \sqrt{c} f \sqrt{\frac{c x^2}{a}+1} \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right ),-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right )}{e^2 \sqrt{a+c x^2} \sqrt{f+g x}}-\frac{\sqrt{\frac{c x^2}{a}+1} \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}} \left (a e^2 g-c d (2 e f-3 d g)\right ) \Pi \left (\frac{2 e}{\frac{\sqrt{c} d}{\sqrt{-a}}+e};\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|\frac{2 \sqrt{-a} g}{\sqrt{c} f+\sqrt{-a} g}\right )}{e^3 \sqrt{a+c x^2} \sqrt{f+g x} \left (\frac{\sqrt{c} d}{\sqrt{-a}}+e\right )}-\frac{\sqrt{a+c x^2} \sqrt{f+g x}}{e (d+e x)}-\frac{3 \sqrt{-a} \sqrt{c} \sqrt{\frac{c x^2}{a}+1} \sqrt{f+g x} E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right )}{e^2 \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}}} \]
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Rubi [A] time = 1.66141, antiderivative size = 650, 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 = {917, 6742, 719, 419, 844, 424, 933, 168, 538, 537} \[ -\frac{\sqrt{-a} \sqrt{c} \sqrt{\frac{c x^2}{a}+1} (2 e f-3 d g) \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right )}{e^3 \sqrt{a+c x^2} \sqrt{f+g x}}-\frac{\sqrt{\frac{c x^2}{a}+1} \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}} \left (a e^2 g-c d (2 e f-3 d g)\right ) \Pi \left (\frac{2 e}{\frac{\sqrt{c} d}{\sqrt{-a}}+e};\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|\frac{2 \sqrt{-a} g}{\sqrt{c} f+\sqrt{-a} g}\right )}{e^3 \sqrt{a+c x^2} \sqrt{f+g x} \left (\frac{\sqrt{c} d}{\sqrt{-a}}+e\right )}-\frac{\sqrt{a+c x^2} \sqrt{f+g x}}{e (d+e x)}+\frac{3 \sqrt{-a} \sqrt{c} f \sqrt{\frac{c x^2}{a}+1} \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right )}{e^2 \sqrt{a+c x^2} \sqrt{f+g x}}-\frac{3 \sqrt{-a} \sqrt{c} \sqrt{\frac{c x^2}{a}+1} \sqrt{f+g x} E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right )}{e^2 \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}}} \]
Antiderivative was successfully verified.
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Rule 917
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} \sqrt{a+c x^2}}{(d+e x)^2} \, dx &=-\frac{\sqrt{f+g x} \sqrt{a+c x^2}}{e (d+e x)}+\frac{\int \frac{a g+2 c f x+3 c g x^2}{(d+e x) \sqrt{f+g x} \sqrt{a+c x^2}} \, dx}{2 e}\\ &=-\frac{\sqrt{f+g x} \sqrt{a+c x^2}}{e (d+e x)}+\frac{\int \left (\frac{c (2 e f-3 d g)}{e^2 \sqrt{f+g x} \sqrt{a+c x^2}}+\frac{3 c g x}{e \sqrt{f+g x} \sqrt{a+c x^2}}+\frac{a e^2 g-c d (2 e f-3 d g)}{e^2 (d+e x) \sqrt{f+g x} \sqrt{a+c x^2}}\right ) \, dx}{2 e}\\ &=-\frac{\sqrt{f+g x} \sqrt{a+c x^2}}{e (d+e x)}+\frac{(3 c g) \int \frac{x}{\sqrt{f+g x} \sqrt{a+c x^2}} \, dx}{2 e^2}+\frac{(c (2 e f-3 d g)) \int \frac{1}{\sqrt{f+g x} \sqrt{a+c x^2}} \, dx}{2 e^3}+\frac{\left (a e^2 g-c d (2 e f-3 d g)\right ) \int \frac{1}{(d+e x) \sqrt{f+g x} \sqrt{a+c x^2}} \, dx}{2 e^3}\\ &=-\frac{\sqrt{f+g x} \sqrt{a+c x^2}}{e (d+e x)}+\frac{(3 c) \int \frac{\sqrt{f+g x}}{\sqrt{a+c x^2}} \, dx}{2 e^2}-\frac{(3 c f) \int \frac{1}{\sqrt{f+g x} \sqrt{a+c x^2}} \, dx}{2 e^2}+\frac{\left (\left (a e^2 g-c d (2 e f-3 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^3 \sqrt{a+c x^2}}+\frac{\left (a \sqrt{c} (2 e f-3 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^3 \sqrt{f+g x} \sqrt{a+c x^2}}\\ &=-\frac{\sqrt{f+g x} \sqrt{a+c x^2}}{e (d+e x)}-\frac{\sqrt{-a} \sqrt{c} (2 e f-3 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^3 \sqrt{f+g x} \sqrt{a+c x^2}}-\frac{\left (\left (a e^2 g-c d (2 e f-3 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^3 \sqrt{a+c x^2}}+\frac{\left (3 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} e^2 \sqrt{\frac{c (f+g x)}{c f-\frac{a \sqrt{c} g}{\sqrt{-a}}}} \sqrt{a+c x^2}}-\frac{\left (3 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} e^2 \sqrt{f+g x} \sqrt{a+c x^2}}\\ &=-\frac{\sqrt{f+g x} \sqrt{a+c x^2}}{e (d+e x)}-\frac{3 \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 )}{e^2 \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{c} f+\sqrt{-a} g}} \sqrt{a+c x^2}}+\frac{3 \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 )}{e^2 \sqrt{f+g x} \sqrt{a+c x^2}}-\frac{\sqrt{-a} \sqrt{c} (2 e f-3 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^3 \sqrt{f+g x} \sqrt{a+c x^2}}-\frac{\left (\left (a e^2 g-c d (2 e f-3 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^3 \sqrt{f+g x} \sqrt{a+c x^2}}\\ &=-\frac{\sqrt{f+g x} \sqrt{a+c x^2}}{e (d+e x)}-\frac{3 \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 )}{e^2 \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{c} f+\sqrt{-a} g}} \sqrt{a+c x^2}}+\frac{3 \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 )}{e^2 \sqrt{f+g x} \sqrt{a+c x^2}}-\frac{\sqrt{-a} \sqrt{c} (2 e f-3 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^3 \sqrt{f+g x} \sqrt{a+c x^2}}-\frac{\left (a e^2 g-c d (2 e f-3 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^3 \left (\frac{\sqrt{c} d}{\sqrt{-a}}+e\right ) \sqrt{f+g x} \sqrt{a+c x^2}}\\ \end{align*}
Mathematica [C] time = 7.17079, size = 1331, normalized size = 2.05 \[ \frac{\sqrt{f+g x} \left (-\frac{\left (c x^2+a\right ) e^2}{d+e x}-\frac{-3 c e^2 \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}} f^3+6 c e^2 \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}} (f+g x) f^2+3 c d e g \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}} f^2-3 c e^2 \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}} (f+g x)^2 f-6 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-3 a e^2 g^2 \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}} f+3 c d e g \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}} (f+g x)^2+3 \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 (\sqrt{c} f+i \sqrt{a} g\right ) \left (\sqrt{a} e g-i \sqrt{c} (2 e f-3 d 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 )-3 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 )+3 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 )}{e^3 \sqrt{c x^2+a}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.293, size = 6044, normalized size = 9.3 \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{c x^{2} + a} \sqrt{g x + f}}{{\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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{a + c x^{2}} \sqrt{f + g x}}{\left (d + e x\right )^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [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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