Optimal. Leaf size=1049 \[ -\frac{\sqrt{b^2-4 a c} \sqrt{f+g x} \sqrt{-\frac{c \left (c x^2+b x+a\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} g}{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g}\right ) (c d (2 e f+d g)-e (b e f+2 b d g-3 a e g))}{4 \sqrt{2} e \left (c d^2-b e d+a e^2\right ) (e f-d g)^2 \sqrt{\frac{c (f+g x)}{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g}} \sqrt{c x^2+b x+a}}+\frac{\sqrt{f+g x} \sqrt{c x^2+b x+a} (c d (2 e f+d g)-e (b e f+2 b d g-3 a e g))}{4 \left (c d^2-b e d+a e^2\right ) (e f-d g)^2 (d+e x)}-\frac{\sqrt{b^2-4 a c} \left ((b f-a g) e^2+c d (d g-2 e f)\right ) \sqrt{\frac{c (f+g x)}{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g}} \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right ),\frac{2 \sqrt{b^2-4 a c} g}{\left (b+\sqrt{b^2-4 a c}\right ) g-2 c f}\right )}{2 \sqrt{2} e^2 \left (c d^2+e (a e-b d)\right ) (e f-d g) \sqrt{f+g x} \sqrt{a+x (b+c x)}}-\frac{\sqrt{2 c f-b g+\sqrt{b^2-4 a c} g} \left (3 a^2 g^2 e^4+b^2 f (4 d g-e f) e^3+2 a c \left (2 e^2 f^2-2 d e g f+3 d^2 g^2\right ) e^2-2 b g \left (3 c f d^2+a e (e f+2 d g)\right ) e^2+c^2 d^3 g (4 e f-d g)\right ) \sqrt{\frac{g \left (-b-2 c x+\sqrt{b^2-4 a c}\right )}{2 c f+\left (\sqrt{b^2-4 a c}-b\right ) g}} \sqrt{\frac{g \left (b+2 c x+\sqrt{b^2-4 a c}\right )}{\left (b+\sqrt{b^2-4 a c}\right ) g-2 c f}} \Pi \left (\frac{2 c e f-b e g+\sqrt{b^2-4 a c} e g}{2 c e f-2 c d g};\sin ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{f+g x}}{\sqrt{2 c f-b g+\sqrt{b^2-4 a c} g}}\right )|\frac{2 c f+\left (\sqrt{b^2-4 a c}-b\right ) g}{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g}\right )}{4 \sqrt{2} \sqrt{c} e^2 \left (c d^2+e (a e-b d)\right ) (e f-d g)^3 \sqrt{a+x (b+c x)}}-\frac{\sqrt{f+g x} \sqrt{c x^2+b x+a}}{2 (e f-d g) (d+e x)^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 8.18444, antiderivative size = 1747, normalized size of antiderivative = 1.67, number of steps used = 25, number of rules used = 11, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.355, Rules used = {924, 6742, 718, 419, 939, 843, 424, 934, 169, 538, 537} \[ \text{result too large to display} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 924
Rule 6742
Rule 718
Rule 419
Rule 939
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)^3 \sqrt{f+g x}} \, dx &=-\frac{\sqrt{f+g x} \sqrt{a+b x+c x^2}}{2 (e f-d g) (d+e x)^2}+\frac{\int \frac{b f-3 a g+2 (c f-b g) x-c g x^2}{(d+e x)^2 \sqrt{f+g x} \sqrt{a+b x+c x^2}} \, dx}{4 (e f-d g)}\\ &=-\frac{\sqrt{f+g x} \sqrt{a+b x+c x^2}}{2 (e f-d g) (d+e x)^2}+\frac{\int \left (-\frac{c g}{e^2 \sqrt{f+g x} \sqrt{a+b x+c x^2}}+\frac{-c d (2 e f+d g)+e (b e f+2 b d g-3 a e g)}{e^2 (d+e x)^2 \sqrt{f+g x} \sqrt{a+b x+c x^2}}+\frac{2 (c e f+c d g-b e g)}{e^2 (d+e x) \sqrt{f+g x} \sqrt{a+b x+c x^2}}\right ) \, dx}{4 (e f-d g)}\\ &=-\frac{\sqrt{f+g x} \sqrt{a+b x+c x^2}}{2 (e f-d g) (d+e x)^2}-\frac{(c g) \int \frac{1}{\sqrt{f+g x} \sqrt{a+b x+c x^2}} \, dx}{4 e^2 (e f-d g)}+\frac{(c e f+c d g-b e g) \int \frac{1}{(d+e x) \sqrt{f+g x} \sqrt{a+b x+c x^2}} \, dx}{2 e^2 (e f-d g)}-\frac{(c d (2 e f+d g)-e (b e f+2 b d g-3 a e g)) \int \frac{1}{(d+e x)^2 \sqrt{f+g x} \sqrt{a+b x+c x^2}} \, dx}{4 e^2 (e f-d g)}\\ &=-\frac{\sqrt{f+g x} \sqrt{a+b x+c x^2}}{2 (e f-d g) (d+e x)^2}+\frac{(c d (2 e f+d g)-e (b e f+2 b d g-3 a e g)) \sqrt{f+g x} \sqrt{a+b x+c x^2}}{4 \left (c d^2-b d e+a e^2\right ) (e f-d g)^2 (d+e x)}+\frac{(c d (2 e f+d g)-e (b e f+2 b d g-3 a e g)) \int \frac{-2 c d (e f-d g)+e (b e f-2 b d g+a e g)-2 c d e g x-c e^2 g x^2}{(d+e x) \sqrt{f+g x} \sqrt{a+b x+c x^2}} \, dx}{8 e^2 \left (c d^2-b d e+a e^2\right ) (e f-d g)^2}+\frac{\left ((c e f+c d g-b e g) \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^2 (e f-d g) \sqrt{a+b x+c x^2}}-\frac{\left (\sqrt{b^2-4 a c} 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 )}{\sqrt{2} 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}}{2 (e f-d g) (d+e x)^2}+\frac{(c d (2 e f+d g)-e (b e f+2 b d g-3 a e g)) \sqrt{f+g x} \sqrt{a+b x+c x^2}}{4 \left (c d^2-b d e+a e^2\right ) (e f-d g)^2 (d+e x)}-\frac{\sqrt{b^2-4 a c} 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 )}{\sqrt{2} e^2 (e f-d g) \sqrt{f+g x} \sqrt{a+b x+c x^2}}+\frac{(c d (2 e f+d g)-e (b e f+2 b d g-3 a e g)) \int \left (-\frac{c d g}{\sqrt{f+g x} \sqrt{a+b x+c x^2}}-\frac{c e g x}{\sqrt{f+g x} \sqrt{a+b x+c x^2}}+\frac{-c d (2 e f-3 d g)+e (b e f-2 b d g+a e g)}{(d+e x) \sqrt{f+g x} \sqrt{a+b x+c x^2}}\right ) \, dx}{8 e^2 \left (c d^2-b d e+a e^2\right ) (e f-d g)^2}-\frac{\left ((c e f+c d g-b e g) \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^2 (e f-d g) \sqrt{a+b x+c x^2}}\\ &=-\frac{\sqrt{f+g x} \sqrt{a+b x+c x^2}}{2 (e f-d g) (d+e x)^2}+\frac{(c d (2 e f+d g)-e (b e f+2 b d g-3 a e g)) \sqrt{f+g x} \sqrt{a+b x+c x^2}}{4 \left (c d^2-b d e+a e^2\right ) (e f-d g)^2 (d+e x)}-\frac{\sqrt{b^2-4 a c} 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 )}{\sqrt{2} e^2 (e f-d g) \sqrt{f+g x} \sqrt{a+b x+c x^2}}-\frac{(c d g (c d (2 e f+d g)-e (b e f+2 b d g-3 a e g))) \int \frac{1}{\sqrt{f+g x} \sqrt{a+b x+c x^2}} \, dx}{8 e^2 \left (c d^2-b d e+a e^2\right ) (e f-d g)^2}-\frac{(c g (c d (2 e f+d g)-e (b e f+2 b d g-3 a e g))) \int \frac{x}{\sqrt{f+g x} \sqrt{a+b x+c x^2}} \, dx}{8 e \left (c d^2-b d e+a e^2\right ) (e f-d g)^2}-\frac{((c d (2 e f+d g)-e (b e f+2 b d g-3 a e g)) (c d (2 e f-3 d g)-e (b e f-2 b d g+a e g))) \int \frac{1}{(d+e x) \sqrt{f+g x} \sqrt{a+b x+c x^2}} \, dx}{8 e^2 \left (c d^2-b d e+a e^2\right ) (e f-d g)^2}-\frac{\left ((c e f+c d g-b e g) \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^2 (e f-d g) \sqrt{a+b x+c x^2}}\\ &=-\frac{\sqrt{f+g x} \sqrt{a+b x+c x^2}}{2 (e f-d g) (d+e x)^2}+\frac{(c d (2 e f+d g)-e (b e f+2 b d g-3 a e g)) \sqrt{f+g x} \sqrt{a+b x+c x^2}}{4 \left (c d^2-b d e+a e^2\right ) (e f-d g)^2 (d+e x)}-\frac{\sqrt{b^2-4 a c} 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 )}{\sqrt{2} e^2 (e f-d g) \sqrt{f+g x} \sqrt{a+b x+c x^2}}-\frac{(c (c d (2 e f+d g)-e (b e f+2 b d g-3 a e g))) \int \frac{\sqrt{f+g x}}{\sqrt{a+b x+c x^2}} \, dx}{8 e \left (c d^2-b d e+a e^2\right ) (e f-d g)^2}+\frac{(c f (c d (2 e f+d g)-e (b e f+2 b d g-3 a e g))) \int \frac{1}{\sqrt{f+g x} \sqrt{a+b x+c x^2}} \, dx}{8 e \left (c d^2-b d e+a e^2\right ) (e f-d g)^2}-\frac{\left ((c d (2 e f+d g)-e (b e f+2 b d g-3 a e g)) (c d (2 e f-3 d g)-e (b e f-2 b d g+a e g)) \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}{8 e^2 \left (c d^2-b d e+a e^2\right ) (e f-d g)^2 \sqrt{a+b x+c x^2}}-\frac{\left (\sqrt{b^2-4 a c} d g (c d (2 e f+d g)-e (b e f+2 b d g-3 a e 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 )}{2 \sqrt{2} e^2 \left (c d^2-b d e+a e^2\right ) (e f-d g)^2 \sqrt{f+g x} \sqrt{a+b x+c x^2}}-\frac{\left ((c e f+c d g-b e g) \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^2 (e f-d g) \sqrt{a+b x+c x^2}}\\ &=-\frac{\sqrt{f+g x} \sqrt{a+b x+c x^2}}{2 (e f-d g) (d+e x)^2}+\frac{(c d (2 e f+d g)-e (b e f+2 b d g-3 a e g)) \sqrt{f+g x} \sqrt{a+b x+c x^2}}{4 \left (c d^2-b d e+a e^2\right ) (e f-d g)^2 (d+e x)}-\frac{\sqrt{b^2-4 a c} 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 )}{\sqrt{2} e^2 (e f-d g) \sqrt{f+g x} \sqrt{a+b x+c x^2}}-\frac{\sqrt{b^2-4 a c} d g (c d (2 e f+d g)-e (b e f+2 b d g-3 a e 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 )}{2 \sqrt{2} e^2 \left (c d^2-b d e+a e^2\right ) (e f-d g)^2 \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} (c e f+c d g-b e g) \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}}+\frac{\left ((c d (2 e f+d g)-e (b e f+2 b d g-3 a e g)) (c d (2 e f-3 d g)-e (b e f-2 b d g+a e g)) \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 )}{4 e^2 \left (c d^2-b d e+a e^2\right ) (e f-d g)^2 \sqrt{a+b x+c x^2}}-\frac{\left (\sqrt{b^2-4 a c} (c d (2 e f+d g)-e (b e f+2 b d g-3 a e g)) \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 )}{4 \sqrt{2} e \left (c d^2-b d e+a e^2\right ) (e f-d g)^2 \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{b^2-4 a c} f (c d (2 e f+d g)-e (b e f+2 b d g-3 a e 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 )}{2 \sqrt{2} e \left (c d^2-b d e+a e^2\right ) (e f-d g)^2 \sqrt{f+g x} \sqrt{a+b x+c x^2}}\\ &=-\frac{\sqrt{f+g x} \sqrt{a+b x+c x^2}}{2 (e f-d g) (d+e x)^2}+\frac{(c d (2 e f+d g)-e (b e f+2 b d g-3 a e g)) \sqrt{f+g x} \sqrt{a+b x+c x^2}}{4 \left (c d^2-b d e+a e^2\right ) (e f-d g)^2 (d+e x)}-\frac{\sqrt{b^2-4 a c} (c d (2 e f+d g)-e (b e f+2 b d g-3 a e g)) \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 )}{4 \sqrt{2} e \left (c d^2-b d e+a e^2\right ) (e f-d g)^2 \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{b^2-4 a c} 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 )}{\sqrt{2} e^2 (e f-d g) \sqrt{f+g x} \sqrt{a+b x+c x^2}}+\frac{\sqrt{b^2-4 a c} f (c d (2 e f+d g)-e (b e f+2 b d g-3 a e 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 )}{2 \sqrt{2} e \left (c d^2-b d e+a e^2\right ) (e f-d g)^2 \sqrt{f+g x} \sqrt{a+b x+c x^2}}-\frac{\sqrt{b^2-4 a c} d g (c d (2 e f+d g)-e (b e f+2 b d g-3 a e 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 )}{2 \sqrt{2} e^2 \left (c d^2-b d e+a e^2\right ) (e f-d g)^2 \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} (c e f+c d g-b e g) \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}}+\frac{\left ((c d (2 e f+d g)-e (b e f+2 b d g-3 a e g)) (c d (2 e f-3 d g)-e (b e f-2 b d g+a e g)) \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 )}{4 e^2 \left (c d^2-b d e+a e^2\right ) (e f-d g)^2 \sqrt{a+b x+c x^2}}\\ &=-\frac{\sqrt{f+g x} \sqrt{a+b x+c x^2}}{2 (e f-d g) (d+e x)^2}+\frac{(c d (2 e f+d g)-e (b e f+2 b d g-3 a e g)) \sqrt{f+g x} \sqrt{a+b x+c x^2}}{4 \left (c d^2-b d e+a e^2\right ) (e f-d g)^2 (d+e x)}-\frac{\sqrt{b^2-4 a c} (c d (2 e f+d g)-e (b e f+2 b d g-3 a e g)) \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 )}{4 \sqrt{2} e \left (c d^2-b d e+a e^2\right ) (e f-d g)^2 \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{b^2-4 a c} 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 )}{\sqrt{2} e^2 (e f-d g) \sqrt{f+g x} \sqrt{a+b x+c x^2}}+\frac{\sqrt{b^2-4 a c} f (c d (2 e f+d g)-e (b e f+2 b d g-3 a e 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 )}{2 \sqrt{2} e \left (c d^2-b d e+a e^2\right ) (e f-d g)^2 \sqrt{f+g x} \sqrt{a+b x+c x^2}}-\frac{\sqrt{b^2-4 a c} d g (c d (2 e f+d g)-e (b e f+2 b d g-3 a e 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 )}{2 \sqrt{2} e^2 \left (c d^2-b d e+a e^2\right ) (e f-d g)^2 \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} (c e f+c d g-b e g) \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}}+\frac{\left ((c d (2 e f+d g)-e (b e f+2 b d g-3 a e g)) (c d (2 e f-3 d g)-e (b e f-2 b d g+a e g)) \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 )}{4 e^2 \left (c d^2-b d e+a e^2\right ) (e f-d g)^2 \sqrt{a+b x+c x^2}}\\ &=-\frac{\sqrt{f+g x} \sqrt{a+b x+c x^2}}{2 (e f-d g) (d+e x)^2}+\frac{(c d (2 e f+d g)-e (b e f+2 b d g-3 a e g)) \sqrt{f+g x} \sqrt{a+b x+c x^2}}{4 \left (c d^2-b d e+a e^2\right ) (e f-d g)^2 (d+e x)}-\frac{\sqrt{b^2-4 a c} (c d (2 e f+d g)-e (b e f+2 b d g-3 a e g)) \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 )}{4 \sqrt{2} e \left (c d^2-b d e+a e^2\right ) (e f-d g)^2 \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{b^2-4 a c} 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 )}{\sqrt{2} e^2 (e f-d g) \sqrt{f+g x} \sqrt{a+b x+c x^2}}+\frac{\sqrt{b^2-4 a c} f (c d (2 e f+d g)-e (b e f+2 b d g-3 a e 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 )}{2 \sqrt{2} e \left (c d^2-b d e+a e^2\right ) (e f-d g)^2 \sqrt{f+g x} \sqrt{a+b x+c x^2}}-\frac{\sqrt{b^2-4 a c} d g (c d (2 e f+d g)-e (b e f+2 b d g-3 a e 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 )}{2 \sqrt{2} e^2 \left (c d^2-b d e+a e^2\right ) (e f-d g)^2 \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} (c e f+c d g-b e g) \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}}+\frac{\sqrt{2 c f-\left (b-\sqrt{b^2-4 a c}\right ) g} (c d (2 e f+d g)-e (b e f+2 b d g-3 a e g)) (c d (2 e f-3 d g)-e (b e f-2 b d g+a e g)) \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 )}{4 \sqrt{2} \sqrt{c} e^2 \left (c d^2-b d e+a e^2\right ) (e f-d g)^3 \sqrt{a+b x+c x^2}}\\ \end{align*}
Mathematica [C] time = 16.9141, size = 36617, normalized size = 34.91 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.538, size = 57841, normalized size = 55.1 \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 )}^{3} \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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{a + b x + c x^{2}}}{\left (d + e x\right )^{3} \sqrt{f + g x}}\, 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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