Optimal. Leaf size=736 \[ \frac{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}} \sqrt{\frac{c (f+g x)}{2 c f-g \left (\sqrt{b^2-4 a c}+b\right )}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right ),\frac{2 g \sqrt{b^2-4 a c}}{g \left (\sqrt{b^2-4 a c}+b\right )-2 c f}\right )}{e^2 \sqrt{f+g x} \sqrt{a+x (b+c x)}}-\frac{\sqrt{2 c f-g \left (b-\sqrt{b^2-4 a c}\right )} \sqrt{1-\frac{2 c (f+g x)}{2 c f-g \left (b-\sqrt{b^2-4 a c}\right )}} \sqrt{1-\frac{2 c (f+g x)}{2 c f-g \left (\sqrt{b^2-4 a c}+b\right )}} \left (e^2 (b f-a g)-c d (2 e f-d g)\right ) \Pi \left (\frac{e \left (2 c f-b g+\sqrt{b^2-4 a c} g\right )}{2 c (e f-d g)};\sin ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{f+g x}}{\sqrt{2 c f-\left (b-\sqrt{b^2-4 a c}\right ) g}}\right )|\frac{b-\sqrt{b^2-4 a c}-\frac{2 c f}{g}}{b+\sqrt{b^2-4 a c}-\frac{2 c f}{g}}\right )}{\sqrt{2} \sqrt{c} e^2 \sqrt{a+b x+c x^2} (e f-d g)^2}+\frac{\sqrt{b^2-4 a c} \sqrt{f+g x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} g}{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g}\right )}{\sqrt{2} e \sqrt{a+b x+c x^2} (e f-d g) \sqrt{\frac{c (f+g x)}{2 c f-g \left (\sqrt{b^2-4 a c}+b\right )}}}-\frac{\sqrt{f+g x} \sqrt{a+b x+c x^2}}{(d+e x) (e f-d g)} \]
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Rubi [A] time = 3.23004, antiderivative size = 957, normalized size of antiderivative = 1.3, number of steps used = 15, number of rules used = 10, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.323, Rules used = {924, 6742, 718, 419, 843, 424, 934, 169, 538, 537} \[ \frac{\sqrt{b^2-4 a c} \sqrt{f+g x} \sqrt{-\frac{c \left (c x^2+b x+a\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} g}{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g}\right )}{\sqrt{2} e (e f-d g) \sqrt{\frac{c (f+g x)}{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g}} \sqrt{c x^2+b x+a}}+\frac{\sqrt{2} \sqrt{b^2-4 a c} (2 e f-d g) \sqrt{\frac{c (f+g x)}{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g}} \sqrt{-\frac{c \left (c x^2+b x+a\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} g}{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g}\right )}{e^2 (e f-d g) \sqrt{f+g x} \sqrt{c x^2+b x+a}}-\frac{\sqrt{2} \sqrt{b^2-4 a c} f \sqrt{\frac{c (f+g x)}{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g}} \sqrt{-\frac{c \left (c x^2+b x+a\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} g}{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g}\right )}{e (e f-d g) \sqrt{f+g x} \sqrt{c x^2+b x+a}}-\frac{\sqrt{2 c f-\left (b-\sqrt{b^2-4 a c}\right ) g} \left (e^2 (b f-a g)-c d (2 e f-d g)\right ) \sqrt{1-\frac{2 c (f+g x)}{2 c f-\left (b-\sqrt{b^2-4 a c}\right ) g}} \sqrt{1-\frac{2 c (f+g x)}{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g}} \Pi \left (\frac{e \left (2 c f-b g+\sqrt{b^2-4 a c} g\right )}{2 c (e f-d g)};\sin ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{f+g x}}{\sqrt{2 c f-\left (b-\sqrt{b^2-4 a c}\right ) g}}\right )|\frac{b-\sqrt{b^2-4 a c}-\frac{2 c f}{g}}{b+\sqrt{b^2-4 a c}-\frac{2 c f}{g}}\right )}{\sqrt{2} \sqrt{c} e^2 (e f-d g)^2 \sqrt{c x^2+b x+a}}-\frac{\sqrt{f+g x} \sqrt{c x^2+b x+a}}{(e f-d g) (d+e x)} \]
Antiderivative was successfully verified.
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Rule 924
Rule 6742
Rule 718
Rule 419
Rule 843
Rule 424
Rule 934
Rule 169
Rule 538
Rule 537
Rubi steps
\begin{align*} \int \frac{\sqrt{a+b x+c x^2}}{(d+e x)^2 \sqrt{f+g x}} \, dx &=-\frac{\sqrt{f+g x} \sqrt{a+b x+c x^2}}{(e f-d g) (d+e x)}+\frac{\int \frac{b f-a g+2 c f x+c g x^2}{(d+e x) \sqrt{f+g x} \sqrt{a+b x+c x^2}} \, dx}{2 (e f-d g)}\\ &=-\frac{\sqrt{f+g x} \sqrt{a+b x+c x^2}}{(e f-d g) (d+e x)}+\frac{\int \left (\frac{c (2 e f-d g)}{e^2 \sqrt{f+g x} \sqrt{a+b x+c x^2}}+\frac{c g x}{e \sqrt{f+g x} \sqrt{a+b x+c x^2}}+\frac{e^2 (b f-a g)-c d (2 e f-d g)}{e^2 (d+e x) \sqrt{f+g x} \sqrt{a+b x+c x^2}}\right ) \, dx}{2 (e f-d g)}\\ &=-\frac{\sqrt{f+g x} \sqrt{a+b x+c x^2}}{(e f-d g) (d+e x)}+\frac{(c g) \int \frac{x}{\sqrt{f+g x} \sqrt{a+b x+c x^2}} \, dx}{2 e (e f-d g)}+\frac{(c (2 e f-d g)) \int \frac{1}{\sqrt{f+g x} \sqrt{a+b x+c x^2}} \, dx}{2 e^2 (e f-d g)}+\frac{\left (b f-a g-\frac{c d (2 e f-d g)}{e^2}\right ) \int \frac{1}{(d+e x) \sqrt{f+g x} \sqrt{a+b x+c x^2}} \, dx}{2 (e f-d g)}\\ &=-\frac{\sqrt{f+g x} \sqrt{a+b x+c x^2}}{(e f-d g) (d+e x)}+\frac{c \int \frac{\sqrt{f+g x}}{\sqrt{a+b x+c x^2}} \, dx}{2 e (e f-d g)}-\frac{(c f) \int \frac{1}{\sqrt{f+g x} \sqrt{a+b x+c x^2}} \, dx}{2 e (e f-d g)}+\frac{\left (\left (b f-a g-\frac{c d (2 e f-d g)}{e^2}\right ) \sqrt{b-\sqrt{b^2-4 a c}+2 c x} \sqrt{b+\sqrt{b^2-4 a c}+2 c x}\right ) \int \frac{1}{\sqrt{b-\sqrt{b^2-4 a c}+2 c x} \sqrt{b+\sqrt{b^2-4 a c}+2 c x} (d+e x) \sqrt{f+g x}} \, dx}{2 (e f-d g) \sqrt{a+b x+c x^2}}+\frac{\left (\sqrt{2} \sqrt{b^2-4 a c} (2 e f-d g) \sqrt{\frac{c (f+g x)}{2 c f-b g-\sqrt{b^2-4 a c} g}} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x^2} \sqrt{1+\frac{2 \sqrt{b^2-4 a c} g x^2}{2 c f-b g-\sqrt{b^2-4 a c} g}}} \, dx,x,\frac{\sqrt{\frac{b+\sqrt{b^2-4 a c}+2 c x}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )}{e^2 (e f-d g) \sqrt{f+g x} \sqrt{a+b x+c x^2}}\\ &=-\frac{\sqrt{f+g x} \sqrt{a+b x+c x^2}}{(e f-d g) (d+e x)}+\frac{\sqrt{2} \sqrt{b^2-4 a c} (2 e f-d g) \sqrt{\frac{c (f+g x)}{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g}} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+\sqrt{b^2-4 a c}+2 c x}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} g}{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g}\right )}{e^2 (e f-d g) \sqrt{f+g x} \sqrt{a+b x+c x^2}}-\frac{\left (\left (b f-a g-\frac{c d (2 e f-d g)}{e^2}\right ) \sqrt{b-\sqrt{b^2-4 a c}+2 c x} \sqrt{b+\sqrt{b^2-4 a c}+2 c x}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (e f-d g-e x^2\right ) \sqrt{b-\sqrt{b^2-4 a c}-\frac{2 c f}{g}+\frac{2 c x^2}{g}} \sqrt{b+\sqrt{b^2-4 a c}-\frac{2 c f}{g}+\frac{2 c x^2}{g}}} \, dx,x,\sqrt{f+g x}\right )}{(e f-d g) \sqrt{a+b x+c x^2}}+\frac{\left (\sqrt{b^2-4 a c} \sqrt{f+g x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1+\frac{2 \sqrt{b^2-4 a c} g x^2}{2 c f-b g-\sqrt{b^2-4 a c} g}}}{\sqrt{1-x^2}} \, dx,x,\frac{\sqrt{\frac{b+\sqrt{b^2-4 a c}+2 c x}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )}{\sqrt{2} e (e f-d g) \sqrt{\frac{c (f+g x)}{2 c f-b g-\sqrt{b^2-4 a c} g}} \sqrt{a+b x+c x^2}}-\frac{\left (\sqrt{2} \sqrt{b^2-4 a c} f \sqrt{\frac{c (f+g x)}{2 c f-b g-\sqrt{b^2-4 a c} g}} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x^2} \sqrt{1+\frac{2 \sqrt{b^2-4 a c} g x^2}{2 c f-b g-\sqrt{b^2-4 a c} g}}} \, dx,x,\frac{\sqrt{\frac{b+\sqrt{b^2-4 a c}+2 c x}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )}{e (e f-d g) \sqrt{f+g x} \sqrt{a+b x+c x^2}}\\ &=-\frac{\sqrt{f+g x} \sqrt{a+b x+c x^2}}{(e f-d g) (d+e x)}+\frac{\sqrt{b^2-4 a c} \sqrt{f+g x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+\sqrt{b^2-4 a c}+2 c x}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} g}{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g}\right )}{\sqrt{2} e (e f-d g) \sqrt{\frac{c (f+g x)}{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g}} \sqrt{a+b x+c x^2}}-\frac{\sqrt{2} \sqrt{b^2-4 a c} f \sqrt{\frac{c (f+g x)}{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g}} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+\sqrt{b^2-4 a c}+2 c x}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} g}{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g}\right )}{e (e f-d g) \sqrt{f+g x} \sqrt{a+b x+c x^2}}+\frac{\sqrt{2} \sqrt{b^2-4 a c} (2 e f-d g) \sqrt{\frac{c (f+g x)}{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g}} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+\sqrt{b^2-4 a c}+2 c x}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} g}{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g}\right )}{e^2 (e f-d g) \sqrt{f+g x} \sqrt{a+b x+c x^2}}-\frac{\left (\left (b f-a g-\frac{c d (2 e f-d g)}{e^2}\right ) \sqrt{b+\sqrt{b^2-4 a c}+2 c x} \sqrt{1+\frac{2 c (f+g x)}{\left (b-\sqrt{b^2-4 a c}-\frac{2 c f}{g}\right ) g}}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (e f-d g-e x^2\right ) \sqrt{b+\sqrt{b^2-4 a c}-\frac{2 c f}{g}+\frac{2 c x^2}{g}} \sqrt{1+\frac{2 c x^2}{\left (b-\sqrt{b^2-4 a c}-\frac{2 c f}{g}\right ) g}}} \, dx,x,\sqrt{f+g x}\right )}{(e f-d g) \sqrt{a+b x+c x^2}}\\ &=-\frac{\sqrt{f+g x} \sqrt{a+b x+c x^2}}{(e f-d g) (d+e x)}+\frac{\sqrt{b^2-4 a c} \sqrt{f+g x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+\sqrt{b^2-4 a c}+2 c x}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} g}{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g}\right )}{\sqrt{2} e (e f-d g) \sqrt{\frac{c (f+g x)}{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g}} \sqrt{a+b x+c x^2}}-\frac{\sqrt{2} \sqrt{b^2-4 a c} f \sqrt{\frac{c (f+g x)}{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g}} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+\sqrt{b^2-4 a c}+2 c x}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} g}{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g}\right )}{e (e f-d g) \sqrt{f+g x} \sqrt{a+b x+c x^2}}+\frac{\sqrt{2} \sqrt{b^2-4 a c} (2 e f-d g) \sqrt{\frac{c (f+g x)}{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g}} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+\sqrt{b^2-4 a c}+2 c x}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} g}{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g}\right )}{e^2 (e f-d g) \sqrt{f+g x} \sqrt{a+b x+c x^2}}-\frac{\left (\left (b f-a g-\frac{c d (2 e f-d g)}{e^2}\right ) \sqrt{1+\frac{2 c (f+g x)}{\left (b-\sqrt{b^2-4 a c}-\frac{2 c f}{g}\right ) g}} \sqrt{1+\frac{2 c (f+g x)}{\left (b+\sqrt{b^2-4 a c}-\frac{2 c f}{g}\right ) g}}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (e f-d g-e x^2\right ) \sqrt{1+\frac{2 c x^2}{\left (b-\sqrt{b^2-4 a c}-\frac{2 c f}{g}\right ) g}} \sqrt{1+\frac{2 c x^2}{\left (b+\sqrt{b^2-4 a c}-\frac{2 c f}{g}\right ) g}}} \, dx,x,\sqrt{f+g x}\right )}{(e f-d g) \sqrt{a+b x+c x^2}}\\ &=-\frac{\sqrt{f+g x} \sqrt{a+b x+c x^2}}{(e f-d g) (d+e x)}+\frac{\sqrt{b^2-4 a c} \sqrt{f+g x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+\sqrt{b^2-4 a c}+2 c x}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} g}{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g}\right )}{\sqrt{2} e (e f-d g) \sqrt{\frac{c (f+g x)}{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g}} \sqrt{a+b x+c x^2}}-\frac{\sqrt{2} \sqrt{b^2-4 a c} f \sqrt{\frac{c (f+g x)}{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g}} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+\sqrt{b^2-4 a c}+2 c x}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} g}{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g}\right )}{e (e f-d g) \sqrt{f+g x} \sqrt{a+b x+c x^2}}+\frac{\sqrt{2} \sqrt{b^2-4 a c} (2 e f-d g) \sqrt{\frac{c (f+g x)}{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g}} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+\sqrt{b^2-4 a c}+2 c x}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} g}{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g}\right )}{e^2 (e f-d g) \sqrt{f+g x} \sqrt{a+b x+c x^2}}-\frac{\sqrt{2 c f-\left (b-\sqrt{b^2-4 a c}\right ) g} \left (e^2 (b f-a g)-c d (2 e f-d g)\right ) \sqrt{1-\frac{2 c (f+g x)}{2 c f-\left (b-\sqrt{b^2-4 a c}\right ) g}} \sqrt{1-\frac{2 c (f+g x)}{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g}} \Pi \left (\frac{e \left (2 c f-b g+\sqrt{b^2-4 a c} g\right )}{2 c (e f-d g)};\sin ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{f+g x}}{\sqrt{2 c f-\left (b-\sqrt{b^2-4 a c}\right ) g}}\right )|\frac{b-\sqrt{b^2-4 a c}-\frac{2 c f}{g}}{b+\sqrt{b^2-4 a c}-\frac{2 c f}{g}}\right )}{\sqrt{2} \sqrt{c} e^2 (e f-d g)^2 \sqrt{a+b x+c x^2}}\\ \end{align*}
Mathematica [C] time = 13.476, size = 6911, normalized size = 9.39 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.423, size = 13874, normalized size = 18.9 \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} + b x + a}}{{\left (e x + d\right )}^{2} \sqrt{g x + f}}\,{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(-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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