Optimal. Leaf size=1114 \[ \text{result too large to display} \]
[Out]
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Rubi [A] time = 8.01181, antiderivative size = 1762, normalized size of antiderivative = 1.58, number of steps used = 25, number of rules used = 10, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.323, Rules used = {939, 6742, 718, 419, 843, 424, 934, 169, 538, 537} \[ \text{result too large to display} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 939
Rule 6742
Rule 718
Rule 419
Rule 843
Rule 424
Rule 934
Rule 169
Rule 538
Rule 537
Rubi steps
\begin{align*} \int \frac{1}{(d+e x)^3 \sqrt{f+g x} \sqrt{a+b x+c x^2}} \, dx &=-\frac{e^2 \sqrt{f+g x} \sqrt{a+b x+c x^2}}{2 \left (c d^2-b d e+a e^2\right ) (e f-d g) (d+e x)^2}-\frac{\int \frac{3 e^2 (b f+a g)-4 d (c e f-c d g+b e g)+2 e (c e f-2 c d g+b e g) x+c e^2 g x^2}{(d+e x)^2 \sqrt{f+g x} \sqrt{a+b x+c x^2}} \, dx}{4 \left (c d^2-b d e+a e^2\right ) (e f-d g)}\\ &=-\frac{e^2 \sqrt{f+g x} \sqrt{a+b x+c x^2}}{2 \left (c d^2-b d e+a e^2\right ) (e f-d g) (d+e x)^2}-\frac{\int \left (\frac{c g}{\sqrt{f+g x} \sqrt{a+b x+c x^2}}+\frac{3 (-c d (2 e f-3 d g)+e (b e f-2 b d g+a e g))}{(d+e x)^2 \sqrt{f+g x} \sqrt{a+b x+c x^2}}+\frac{2 (c e f-3 c d g+b e g)}{(d+e x) \sqrt{f+g x} \sqrt{a+b x+c x^2}}\right ) \, dx}{4 \left (c d^2-b d e+a e^2\right ) (e f-d g)}\\ &=-\frac{e^2 \sqrt{f+g x} \sqrt{a+b x+c x^2}}{2 \left (c d^2-b d e+a e^2\right ) (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 \left (c d^2-b d e+a e^2\right ) (e f-d g)}-\frac{(c e f-3 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 \left (c d^2-b d e+a e^2\right ) (e f-d g)}+\frac{(3 (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)^2 \sqrt{f+g x} \sqrt{a+b x+c x^2}} \, dx}{4 \left (c d^2-b d e+a e^2\right ) (e f-d g)}\\ &=-\frac{e^2 \sqrt{f+g x} \sqrt{a+b x+c x^2}}{2 \left (c d^2-b d e+a e^2\right ) (e f-d g) (d+e x)^2}-\frac{3 e^2 (c d (2 e f-3 d g)-e (b e f-2 b d g+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 )^2 (e f-d g)^2 (d+e x)}-\frac{(3 (c d (2 e f-3 d g)-e (b e f-2 b d g+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 \left (c d^2-b d e+a e^2\right )^2 (e f-d g)^2}-\frac{\left ((c e f-3 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 \left (c d^2-b d e+a e^2\right ) (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} \left (c d^2-b d e+a e^2\right ) (e f-d g) \sqrt{f+g x} \sqrt{a+b x+c x^2}}\\ &=-\frac{e^2 \sqrt{f+g x} \sqrt{a+b x+c x^2}}{2 \left (c d^2-b d e+a e^2\right ) (e f-d g) (d+e x)^2}-\frac{3 e^2 (c d (2 e f-3 d g)-e (b e f-2 b d g+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 )^2 (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} \left (c d^2-b d e+a e^2\right ) (e f-d g) \sqrt{f+g x} \sqrt{a+b x+c x^2}}-\frac{(3 (c d (2 e f-3 d g)-e (b e f-2 b d g+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 \left (c d^2-b d e+a e^2\right )^2 (e f-d g)^2}+\frac{\left ((c e f-3 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 )}{\left (c d^2-b d e+a e^2\right ) (e f-d g) \sqrt{a+b x+c x^2}}\\ &=-\frac{e^2 \sqrt{f+g x} \sqrt{a+b x+c x^2}}{2 \left (c d^2-b d e+a e^2\right ) (e f-d g) (d+e x)^2}-\frac{3 e^2 (c d (2 e f-3 d g)-e (b e f-2 b d g+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 )^2 (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} \left (c d^2-b d e+a e^2\right ) (e f-d g) \sqrt{f+g x} \sqrt{a+b x+c x^2}}+\frac{(3 c d g (c d (2 e f-3 d g)-e (b e f-2 b d g+a e g))) \int \frac{1}{\sqrt{f+g x} \sqrt{a+b x+c x^2}} \, dx}{8 \left (c d^2-b d e+a e^2\right )^2 (e f-d g)^2}+\frac{(3 c e g (c d (2 e f-3 d g)-e (b e f-2 b d g+a e g))) \int \frac{x}{\sqrt{f+g x} \sqrt{a+b x+c x^2}} \, dx}{8 \left (c d^2-b d e+a e^2\right )^2 (e f-d g)^2}+\frac{\left (3 (c d (2 e f-3 d g)-e (b e f-2 b d g+a e g))^2\right ) \int \frac{1}{(d+e x) \sqrt{f+g x} \sqrt{a+b x+c x^2}} \, dx}{8 \left (c d^2-b d e+a e^2\right )^2 (e f-d g)^2}+\frac{\left ((c e f-3 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 )}{\left (c d^2-b d e+a e^2\right ) (e f-d g) \sqrt{a+b x+c x^2}}\\ &=-\frac{e^2 \sqrt{f+g x} \sqrt{a+b x+c x^2}}{2 \left (c d^2-b d e+a e^2\right ) (e f-d g) (d+e x)^2}-\frac{3 e^2 (c d (2 e f-3 d g)-e (b e f-2 b d g+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 )^2 (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} \left (c d^2-b d e+a e^2\right ) (e f-d g) \sqrt{f+g x} \sqrt{a+b x+c x^2}}+\frac{(3 c e (c d (2 e f-3 d g)-e (b e f-2 b d g+a e g))) \int \frac{\sqrt{f+g x}}{\sqrt{a+b x+c x^2}} \, dx}{8 \left (c d^2-b d e+a e^2\right )^2 (e f-d g)^2}-\frac{(3 c e f (c d (2 e f-3 d g)-e (b e f-2 b d g+a e g))) \int \frac{1}{\sqrt{f+g x} \sqrt{a+b x+c x^2}} \, dx}{8 \left (c d^2-b d e+a e^2\right )^2 (e f-d g)^2}+\frac{\left (3 (c d (2 e f-3 d g)-e (b e f-2 b d g+a e g))^2 \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 \left (c d^2-b d e+a e^2\right )^2 (e f-d g)^2 \sqrt{a+b x+c x^2}}+\frac{\left (3 \sqrt{b^2-4 a c} d g (c d (2 e f-3 d g)-e (b e f-2 b d g+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} \left (c d^2-b d e+a e^2\right )^2 (e f-d g)^2 \sqrt{f+g x} \sqrt{a+b x+c x^2}}+\frac{\left ((c e f-3 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 )}{\left (c d^2-b d e+a e^2\right ) (e f-d g) \sqrt{a+b x+c x^2}}\\ &=-\frac{e^2 \sqrt{f+g x} \sqrt{a+b x+c x^2}}{2 \left (c d^2-b d e+a e^2\right ) (e f-d g) (d+e x)^2}-\frac{3 e^2 (c d (2 e f-3 d g)-e (b e f-2 b d g+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 )^2 (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} \left (c d^2-b d e+a e^2\right ) (e f-d g) \sqrt{f+g x} \sqrt{a+b x+c x^2}}+\frac{3 \sqrt{b^2-4 a c} d g (c d (2 e f-3 d g)-e (b e f-2 b d g+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} \left (c d^2-b d e+a e^2\right )^2 (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-3 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} \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 (3 (c d (2 e f-3 d g)-e (b e f-2 b d g+a e g))^2 \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 \left (c d^2-b d e+a e^2\right )^2 (e f-d g)^2 \sqrt{a+b x+c x^2}}+\frac{\left (3 \sqrt{b^2-4 a c} e (c d (2 e f-3 d g)-e (b e f-2 b d g+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} \left (c d^2-b d e+a e^2\right )^2 (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 (3 \sqrt{b^2-4 a c} e f (c d (2 e f-3 d g)-e (b e f-2 b d g+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} \left (c d^2-b d e+a e^2\right )^2 (e f-d g)^2 \sqrt{f+g x} \sqrt{a+b x+c x^2}}\\ &=-\frac{e^2 \sqrt{f+g x} \sqrt{a+b x+c x^2}}{2 \left (c d^2-b d e+a e^2\right ) (e f-d g) (d+e x)^2}-\frac{3 e^2 (c d (2 e f-3 d g)-e (b e f-2 b d g+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 )^2 (e f-d g)^2 (d+e x)}+\frac{3 \sqrt{b^2-4 a c} e (c d (2 e f-3 d g)-e (b e f-2 b d g+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} \left (c d^2-b d e+a e^2\right )^2 (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} \left (c d^2-b d e+a e^2\right ) (e f-d g) \sqrt{f+g x} \sqrt{a+b x+c x^2}}-\frac{3 \sqrt{b^2-4 a c} e f (c d (2 e f-3 d g)-e (b e f-2 b d g+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} \left (c d^2-b d e+a e^2\right )^2 (e f-d g)^2 \sqrt{f+g x} \sqrt{a+b x+c x^2}}+\frac{3 \sqrt{b^2-4 a c} d g (c d (2 e f-3 d g)-e (b e f-2 b d g+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} \left (c d^2-b d e+a e^2\right )^2 (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-3 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} \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 (3 (c d (2 e f-3 d g)-e (b e f-2 b d g+a e g))^2 \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 \left (c d^2-b d e+a e^2\right )^2 (e f-d g)^2 \sqrt{a+b x+c x^2}}\\ &=-\frac{e^2 \sqrt{f+g x} \sqrt{a+b x+c x^2}}{2 \left (c d^2-b d e+a e^2\right ) (e f-d g) (d+e x)^2}-\frac{3 e^2 (c d (2 e f-3 d g)-e (b e f-2 b d g+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 )^2 (e f-d g)^2 (d+e x)}+\frac{3 \sqrt{b^2-4 a c} e (c d (2 e f-3 d g)-e (b e f-2 b d g+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} \left (c d^2-b d e+a e^2\right )^2 (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} \left (c d^2-b d e+a e^2\right ) (e f-d g) \sqrt{f+g x} \sqrt{a+b x+c x^2}}-\frac{3 \sqrt{b^2-4 a c} e f (c d (2 e f-3 d g)-e (b e f-2 b d g+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} \left (c d^2-b d e+a e^2\right )^2 (e f-d g)^2 \sqrt{f+g x} \sqrt{a+b x+c x^2}}+\frac{3 \sqrt{b^2-4 a c} d g (c d (2 e f-3 d g)-e (b e f-2 b d g+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} \left (c d^2-b d e+a e^2\right )^2 (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-3 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} \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 (3 (c d (2 e f-3 d g)-e (b e f-2 b d g+a e g))^2 \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 \left (c d^2-b d e+a e^2\right )^2 (e f-d g)^2 \sqrt{a+b x+c x^2}}\\ &=-\frac{e^2 \sqrt{f+g x} \sqrt{a+b x+c x^2}}{2 \left (c d^2-b d e+a e^2\right ) (e f-d g) (d+e x)^2}-\frac{3 e^2 (c d (2 e f-3 d g)-e (b e f-2 b d g+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 )^2 (e f-d g)^2 (d+e x)}+\frac{3 \sqrt{b^2-4 a c} e (c d (2 e f-3 d g)-e (b e f-2 b d g+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} \left (c d^2-b d e+a e^2\right )^2 (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} \left (c d^2-b d e+a e^2\right ) (e f-d g) \sqrt{f+g x} \sqrt{a+b x+c x^2}}-\frac{3 \sqrt{b^2-4 a c} e f (c d (2 e f-3 d g)-e (b e f-2 b d g+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} \left (c d^2-b d e+a e^2\right )^2 (e f-d g)^2 \sqrt{f+g x} \sqrt{a+b x+c x^2}}+\frac{3 \sqrt{b^2-4 a c} d g (c d (2 e f-3 d g)-e (b e f-2 b d g+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} \left (c d^2-b d e+a e^2\right )^2 (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-3 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} \left (c d^2-b d e+a e^2\right ) (e f-d g)^2 \sqrt{a+b x+c x^2}}-\frac{3 \sqrt{2 c f-\left (b-\sqrt{b^2-4 a c}\right ) g} (c d (2 e f-3 d g)-e (b e f-2 b d g+a e g))^2 \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} \left (c d^2-b d e+a e^2\right )^2 (e f-d g)^3 \sqrt{a+b x+c x^2}}\\ \end{align*}
Mathematica [C] time = 18.2261, size = 40396, normalized size = 36.26 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.616, size = 64947, normalized size = 58.3 \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{1}{\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(-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{1}{\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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