Optimal. Leaf size=994 \[ -\frac{\sqrt{f+g x} \sqrt{c x^2+b x+a} e}{\left (c d^2-b e d+a e^2\right ) (d+e x)}+\frac{\sqrt{b^2-4 a c} \sqrt{f+g x} \sqrt{-\frac{c \left (c x^2+b x+a\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} g}{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g}\right )}{\sqrt{2} \left (c d^2-b e d+a e^2\right ) \sqrt{\frac{c (f+g x)}{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g}} \sqrt{c x^2+b x+a}}-\frac{\sqrt{2} \sqrt{b^2-4 a c} f \sqrt{\frac{c (f+g x)}{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g}} \sqrt{-\frac{c \left (c x^2+b x+a\right )}{b^2-4 a c}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right ),-\frac{2 \sqrt{b^2-4 a c} g}{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g}\right )}{\left (c d^2-b e d+a e^2\right ) \sqrt{f+g x} \sqrt{c x^2+b x+a}}+\frac{\sqrt{2} \sqrt{b^2-4 a c} d g \sqrt{\frac{c (f+g x)}{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g}} \sqrt{-\frac{c \left (c x^2+b x+a\right )}{b^2-4 a c}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right ),-\frac{2 \sqrt{b^2-4 a c} g}{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g}\right )}{\left (c d^2-b e d+a e^2\right ) \sqrt{f+g x} \sqrt{c x^2+b x+a} e}+\frac{\sqrt{2 c f-\left (b-\sqrt{b^2-4 a c}\right ) g} \left (e^2 (b f-a g)-c d (2 e f-d g)\right ) \sqrt{1-\frac{2 c (f+g x)}{2 c f-\left (b-\sqrt{b^2-4 a c}\right ) g}} \sqrt{1-\frac{2 c (f+g x)}{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g}} \Pi \left (\frac{e \left (2 c f-b g+\sqrt{b^2-4 a c} g\right )}{2 c (e f-d g)};\sin ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{f+g x}}{\sqrt{2 c f-\left (b-\sqrt{b^2-4 a c}\right ) g}}\right )|\frac{b-\sqrt{b^2-4 a c}-\frac{2 c f}{g}}{b+\sqrt{b^2-4 a c}-\frac{2 c f}{g}}\right )}{\sqrt{2} \sqrt{c} \left (c d^2-b e d+a e^2\right ) (e f-d g) \sqrt{c x^2+b x+a} e} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 3.70064, antiderivative size = 994, normalized size of antiderivative = 1., 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 = {945, 6742, 718, 419, 843, 424, 934, 169, 538, 537} \[ -\frac{\sqrt{f+g x} \sqrt{c x^2+b x+a} e}{\left (c d^2-b e d+a e^2\right ) (d+e x)}+\frac{\sqrt{b^2-4 a c} \sqrt{f+g x} \sqrt{-\frac{c \left (c x^2+b x+a\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} g}{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g}\right )}{\sqrt{2} \left (c d^2-b e d+a e^2\right ) \sqrt{\frac{c (f+g x)}{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g}} \sqrt{c x^2+b x+a}}-\frac{\sqrt{2} \sqrt{b^2-4 a c} f \sqrt{\frac{c (f+g x)}{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g}} \sqrt{-\frac{c \left (c x^2+b x+a\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} g}{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g}\right )}{\left (c d^2-b e d+a e^2\right ) \sqrt{f+g x} \sqrt{c x^2+b x+a}}+\frac{\sqrt{2} \sqrt{b^2-4 a c} d g \sqrt{\frac{c (f+g x)}{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g}} \sqrt{-\frac{c \left (c x^2+b x+a\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} g}{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g}\right )}{\left (c d^2-b e d+a e^2\right ) \sqrt{f+g x} \sqrt{c x^2+b x+a} e}+\frac{\sqrt{2 c f-\left (b-\sqrt{b^2-4 a c}\right ) g} \left (e^2 (b f-a g)-c d (2 e f-d g)\right ) \sqrt{1-\frac{2 c (f+g x)}{2 c f-\left (b-\sqrt{b^2-4 a c}\right ) g}} \sqrt{1-\frac{2 c (f+g x)}{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g}} \Pi \left (\frac{e \left (2 c f-b g+\sqrt{b^2-4 a c} g\right )}{2 c (e f-d g)};\sin ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{f+g x}}{\sqrt{2 c f-\left (b-\sqrt{b^2-4 a c}\right ) g}}\right )|\frac{b-\sqrt{b^2-4 a c}-\frac{2 c f}{g}}{b+\sqrt{b^2-4 a c}-\frac{2 c f}{g}}\right )}{\sqrt{2} \sqrt{c} \left (c d^2-b e d+a e^2\right ) (e f-d g) \sqrt{c x^2+b x+a} e} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 945
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{f+g x}}{(d+e x)^2 \sqrt{a+b x+c x^2}} \, dx &=-\frac{e \sqrt{f+g x} \sqrt{a+b x+c x^2}}{\left (c d^2-b d e+a e^2\right ) (d+e x)}-\frac{\int \frac{-2 c d f+b e f-a e g-2 c d g x-c e g x^2}{(d+e x) \sqrt{f+g x} \sqrt{a+b x+c x^2}} \, dx}{2 \left (c d^2-b d e+a e^2\right )}\\ &=-\frac{e \sqrt{f+g x} \sqrt{a+b x+c x^2}}{\left (c d^2-b d e+a e^2\right ) (d+e x)}-\frac{\int \left (-\frac{c d g}{e \sqrt{f+g x} \sqrt{a+b x+c x^2}}-\frac{c g x}{\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 (d+e x) \sqrt{f+g x} \sqrt{a+b x+c x^2}}\right ) \, dx}{2 \left (c d^2-b d e+a e^2\right )}\\ &=-\frac{e \sqrt{f+g x} \sqrt{a+b x+c x^2}}{\left (c d^2-b d e+a e^2\right ) (d+e x)}+\frac{(c g) \int \frac{x}{\sqrt{f+g x} \sqrt{a+b x+c x^2}} \, dx}{2 \left (c d^2-b d e+a e^2\right )}+\frac{(c d g) \int \frac{1}{\sqrt{f+g x} \sqrt{a+b x+c x^2}} \, dx}{2 e \left (c d^2-b d e+a e^2\right )}-\frac{\left (e^2 (b f-a g)-c d (2 e f-d g)\right ) \int \frac{1}{(d+e x) \sqrt{f+g x} \sqrt{a+b x+c x^2}} \, dx}{2 e \left (c d^2-b d e+a e^2\right )}\\ &=-\frac{e \sqrt{f+g x} \sqrt{a+b x+c x^2}}{\left (c d^2-b d e+a e^2\right ) (d+e x)}+\frac{c \int \frac{\sqrt{f+g x}}{\sqrt{a+b x+c x^2}} \, dx}{2 \left (c d^2-b d e+a e^2\right )}-\frac{(c f) \int \frac{1}{\sqrt{f+g x} \sqrt{a+b x+c x^2}} \, dx}{2 \left (c d^2-b d e+a e^2\right )}-\frac{\left (\left (e^2 (b f-a g)-c d (2 e f-d g)\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 \left (c d^2-b d e+a e^2\right ) \sqrt{a+b x+c x^2}}+\frac{\left (\sqrt{2} \sqrt{b^2-4 a c} 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 \left (c d^2-b d e+a e^2\right ) \sqrt{f+g x} \sqrt{a+b x+c x^2}}\\ &=-\frac{e \sqrt{f+g x} \sqrt{a+b x+c x^2}}{\left (c d^2-b d e+a e^2\right ) (d+e x)}+\frac{\sqrt{2} \sqrt{b^2-4 a c} 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 \left (c d^2-b d e+a e^2\right ) \sqrt{f+g x} \sqrt{a+b x+c x^2}}+\frac{\left (\left (e^2 (b f-a g)-c d (2 e f-d g)\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 \left (c d^2-b d e+a e^2\right ) \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} \left (c d^2-b d e+a e^2\right ) \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 )}{\left (c d^2-b d e+a e^2\right ) \sqrt{f+g x} \sqrt{a+b x+c x^2}}\\ &=-\frac{e \sqrt{f+g x} \sqrt{a+b x+c x^2}}{\left (c d^2-b d e+a e^2\right ) (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} \left (c d^2-b d e+a e^2\right ) \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 )}{\left (c d^2-b d e+a e^2\right ) \sqrt{f+g x} \sqrt{a+b x+c x^2}}+\frac{\sqrt{2} \sqrt{b^2-4 a c} 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 \left (c d^2-b d e+a e^2\right ) \sqrt{f+g x} \sqrt{a+b x+c x^2}}+\frac{\left (\left (e^2 (b f-a g)-c d (2 e f-d g)\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 \left (c d^2-b d e+a e^2\right ) \sqrt{a+b x+c x^2}}\\ &=-\frac{e \sqrt{f+g x} \sqrt{a+b x+c x^2}}{\left (c d^2-b d e+a e^2\right ) (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} \left (c d^2-b d e+a e^2\right ) \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 )}{\left (c d^2-b d e+a e^2\right ) \sqrt{f+g x} \sqrt{a+b x+c x^2}}+\frac{\sqrt{2} \sqrt{b^2-4 a c} 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 \left (c d^2-b d e+a e^2\right ) \sqrt{f+g x} \sqrt{a+b x+c x^2}}+\frac{\left (\left (e^2 (b f-a g)-c d (2 e f-d g)\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 \left (c d^2-b d e+a e^2\right ) \sqrt{a+b x+c x^2}}\\ &=-\frac{e \sqrt{f+g x} \sqrt{a+b x+c x^2}}{\left (c d^2-b d e+a e^2\right ) (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} \left (c d^2-b d e+a e^2\right ) \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 )}{\left (c d^2-b d e+a e^2\right ) \sqrt{f+g x} \sqrt{a+b x+c x^2}}+\frac{\sqrt{2} \sqrt{b^2-4 a c} 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 \left (c d^2-b d e+a e^2\right ) \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 \left (c d^2-b d e+a e^2\right ) (e f-d g) \sqrt{a+b x+c x^2}}\\ \end{align*}
Mathematica [C] time = 13.8658, size = 18563, normalized size = 18.68 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.37, size = 13017, normalized size = 13.1 \begin{align*} \text{output too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{g x + f}}{\sqrt{c x^{2} + b x + a}{\left (e x + d\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]