Optimal. Leaf size=955 \[ \frac{\sqrt{2} \sqrt{b^2-4 a c} (a d+b e) \sqrt{a+\frac{b}{x}+\frac{c}{x^2}} \sqrt{d+e x} \sqrt{-\frac{a \left (a x^2+b x+c\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 a x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 a d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right ) x}{3 a e \sqrt{\frac{a (d+e x)}{2 a d-\left (b+\sqrt{b^2-4 a c}\right ) e}} \left (a x^2+b x+c\right )}-\frac{2 \sqrt{2} \sqrt{b^2-4 a c} d (a d+b e) \sqrt{a+\frac{b}{x}+\frac{c}{x^2}} \sqrt{\frac{a (d+e x)}{2 a d-\left (b+\sqrt{b^2-4 a c}\right ) e}} \sqrt{-\frac{a \left (a x^2+b x+c\right )}{b^2-4 a c}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 a x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right ),-\frac{2 \sqrt{b^2-4 a c} e}{2 a d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right ) x}{3 a e \sqrt{d+e x} \left (a x^2+b x+c\right )}+\frac{4 \sqrt{2} \sqrt{b^2-4 a c} (b d+c e) \sqrt{a+\frac{b}{x}+\frac{c}{x^2}} \sqrt{\frac{a (d+e x)}{2 a d-\left (b+\sqrt{b^2-4 a c}\right ) e}} \sqrt{-\frac{a \left (a x^2+b x+c\right )}{b^2-4 a c}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 a x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right ),-\frac{2 \sqrt{b^2-4 a c} e}{2 a d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right ) x}{3 a \sqrt{d+e x} \left (a x^2+b x+c\right )}-\frac{\sqrt{2} c \sqrt{2 a d-\left (b-\sqrt{b^2-4 a c}\right ) e} \sqrt{a+\frac{b}{x}+\frac{c}{x^2}} \sqrt{1-\frac{2 a (d+e x)}{2 a d-\left (b-\sqrt{b^2-4 a c}\right ) e}} \sqrt{1-\frac{2 a (d+e x)}{2 a d-\left (b+\sqrt{b^2-4 a c}\right ) e}} \Pi \left (\frac{2 a d-b e+\sqrt{b^2-4 a c} e}{2 a d};\sin ^{-1}\left (\frac{\sqrt{2} \sqrt{a} \sqrt{d+e x}}{\sqrt{2 a d-\left (b-\sqrt{b^2-4 a c}\right ) e}}\right )|\frac{b-\sqrt{b^2-4 a c}-\frac{2 a d}{e}}{b+\sqrt{b^2-4 a c}-\frac{2 a d}{e}}\right ) x}{\sqrt{a} \left (a x^2+b x+c\right )}+\frac{2}{3} \sqrt{a+\frac{b}{x}+\frac{c}{x^2}} \sqrt{d+e x} x \]
[Out]
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Rubi [A] time = 3.31272, antiderivative size = 955, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 11, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.423, Rules used = {1449, 918, 6742, 718, 419, 934, 169, 538, 537, 843, 424} \[ \frac{\sqrt{2} \sqrt{b^2-4 a c} (a d+b e) \sqrt{a+\frac{b}{x}+\frac{c}{x^2}} \sqrt{d+e x} \sqrt{-\frac{a \left (a x^2+b x+c\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 a x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 a d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right ) x}{3 a e \sqrt{\frac{a (d+e x)}{2 a d-\left (b+\sqrt{b^2-4 a c}\right ) e}} \left (a x^2+b x+c\right )}-\frac{2 \sqrt{2} \sqrt{b^2-4 a c} d (a d+b e) \sqrt{a+\frac{b}{x}+\frac{c}{x^2}} \sqrt{\frac{a (d+e x)}{2 a d-\left (b+\sqrt{b^2-4 a c}\right ) e}} \sqrt{-\frac{a \left (a x^2+b x+c\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 a x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 a d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right ) x}{3 a e \sqrt{d+e x} \left (a x^2+b x+c\right )}+\frac{4 \sqrt{2} \sqrt{b^2-4 a c} (b d+c e) \sqrt{a+\frac{b}{x}+\frac{c}{x^2}} \sqrt{\frac{a (d+e x)}{2 a d-\left (b+\sqrt{b^2-4 a c}\right ) e}} \sqrt{-\frac{a \left (a x^2+b x+c\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 a x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 a d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right ) x}{3 a \sqrt{d+e x} \left (a x^2+b x+c\right )}-\frac{\sqrt{2} c \sqrt{2 a d-\left (b-\sqrt{b^2-4 a c}\right ) e} \sqrt{a+\frac{b}{x}+\frac{c}{x^2}} \sqrt{1-\frac{2 a (d+e x)}{2 a d-\left (b-\sqrt{b^2-4 a c}\right ) e}} \sqrt{1-\frac{2 a (d+e x)}{2 a d-\left (b+\sqrt{b^2-4 a c}\right ) e}} \Pi \left (\frac{2 a d-b e+\sqrt{b^2-4 a c} e}{2 a d};\sin ^{-1}\left (\frac{\sqrt{2} \sqrt{a} \sqrt{d+e x}}{\sqrt{2 a d-\left (b-\sqrt{b^2-4 a c}\right ) e}}\right )|\frac{b-\sqrt{b^2-4 a c}-\frac{2 a d}{e}}{b+\sqrt{b^2-4 a c}-\frac{2 a d}{e}}\right ) x}{\sqrt{a} \left (a x^2+b x+c\right )}+\frac{2}{3} \sqrt{a+\frac{b}{x}+\frac{c}{x^2}} \sqrt{d+e x} x \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1449
Rule 918
Rule 6742
Rule 718
Rule 419
Rule 934
Rule 169
Rule 538
Rule 537
Rule 843
Rule 424
Rubi steps
\begin{align*} \int \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} \sqrt{d+e x} \, dx &=\frac{\left (\sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x\right ) \int \frac{\sqrt{d+e x} \sqrt{c+b x+a x^2}}{x} \, dx}{\sqrt{c+b x+a x^2}}\\ &=\frac{2}{3} \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x \sqrt{d+e x}-\frac{\left (\sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x\right ) \int \frac{-3 c d-2 (b d+c e) x-(a d+b e) x^2}{x \sqrt{d+e x} \sqrt{c+b x+a x^2}} \, dx}{3 \sqrt{c+b x+a x^2}}\\ &=\frac{2}{3} \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x \sqrt{d+e x}-\frac{\left (\sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x\right ) \int \left (-\frac{2 (b d+c e)}{\sqrt{d+e x} \sqrt{c+b x+a x^2}}-\frac{3 c d}{x \sqrt{d+e x} \sqrt{c+b x+a x^2}}-\frac{(a d+b e) x}{\sqrt{d+e x} \sqrt{c+b x+a x^2}}\right ) \, dx}{3 \sqrt{c+b x+a x^2}}\\ &=\frac{2}{3} \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x \sqrt{d+e x}+\frac{\left (c d \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x\right ) \int \frac{1}{x \sqrt{d+e x} \sqrt{c+b x+a x^2}} \, dx}{\sqrt{c+b x+a x^2}}-\frac{\left ((-a d-b e) \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x\right ) \int \frac{x}{\sqrt{d+e x} \sqrt{c+b x+a x^2}} \, dx}{3 \sqrt{c+b x+a x^2}}+\frac{\left (2 (b d+c e) \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x\right ) \int \frac{1}{\sqrt{d+e x} \sqrt{c+b x+a x^2}} \, dx}{3 \sqrt{c+b x+a x^2}}\\ &=\frac{2}{3} \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x \sqrt{d+e x}+\frac{\left (c d \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x \sqrt{b-\sqrt{b^2-4 a c}+2 a x} \sqrt{b+\sqrt{b^2-4 a c}+2 a x}\right ) \int \frac{1}{x \sqrt{b-\sqrt{b^2-4 a c}+2 a x} \sqrt{b+\sqrt{b^2-4 a c}+2 a x} \sqrt{d+e x}} \, dx}{c+b x+a x^2}-\frac{\left ((-a d-b e) \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x\right ) \int \frac{\sqrt{d+e x}}{\sqrt{c+b x+a x^2}} \, dx}{3 e \sqrt{c+b x+a x^2}}+\frac{\left (d (-a d-b e) \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x\right ) \int \frac{1}{\sqrt{d+e x} \sqrt{c+b x+a x^2}} \, dx}{3 e \sqrt{c+b x+a x^2}}+\frac{\left (4 \sqrt{2} \sqrt{b^2-4 a c} (b d+c e) \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x \sqrt{\frac{a (d+e x)}{2 a d-b e-\sqrt{b^2-4 a c} e}} \sqrt{-\frac{a \left (c+b x+a 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} e x^2}{2 a d-b e-\sqrt{b^2-4 a c} e}}} \, dx,x,\frac{\sqrt{\frac{b+\sqrt{b^2-4 a c}+2 a x}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )}{3 a \sqrt{d+e x} \left (c+b x+a x^2\right )}\\ &=\frac{2}{3} \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x \sqrt{d+e x}+\frac{4 \sqrt{2} \sqrt{b^2-4 a c} (b d+c e) \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x \sqrt{\frac{a (d+e x)}{2 a d-\left (b+\sqrt{b^2-4 a c}\right ) e}} \sqrt{-\frac{a \left (c+b x+a x^2\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+\sqrt{b^2-4 a c}+2 a x}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 a d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{3 a \sqrt{d+e x} \left (c+b x+a x^2\right )}-\frac{\left (2 c d \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x \sqrt{b-\sqrt{b^2-4 a c}+2 a x} \sqrt{b+\sqrt{b^2-4 a c}+2 a x}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (d-x^2\right ) \sqrt{b-\sqrt{b^2-4 a c}-\frac{2 a d}{e}+\frac{2 a x^2}{e}} \sqrt{b+\sqrt{b^2-4 a c}-\frac{2 a d}{e}+\frac{2 a x^2}{e}}} \, dx,x,\sqrt{d+e x}\right )}{c+b x+a x^2}-\frac{\left (\sqrt{2} \sqrt{b^2-4 a c} (-a d-b e) \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x \sqrt{d+e x} \sqrt{-\frac{a \left (c+b x+a x^2\right )}{b^2-4 a c}}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1+\frac{2 \sqrt{b^2-4 a c} e x^2}{2 a d-b e-\sqrt{b^2-4 a c} e}}}{\sqrt{1-x^2}} \, dx,x,\frac{\sqrt{\frac{b+\sqrt{b^2-4 a c}+2 a x}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )}{3 a e \sqrt{\frac{a (d+e x)}{2 a d-b e-\sqrt{b^2-4 a c} e}} \left (c+b x+a x^2\right )}+\frac{\left (2 \sqrt{2} \sqrt{b^2-4 a c} d (-a d-b e) \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x \sqrt{\frac{a (d+e x)}{2 a d-b e-\sqrt{b^2-4 a c} e}} \sqrt{-\frac{a \left (c+b x+a 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} e x^2}{2 a d-b e-\sqrt{b^2-4 a c} e}}} \, dx,x,\frac{\sqrt{\frac{b+\sqrt{b^2-4 a c}+2 a x}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )}{3 a e \sqrt{d+e x} \left (c+b x+a x^2\right )}\\ &=\frac{2}{3} \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x \sqrt{d+e x}+\frac{\sqrt{2} \sqrt{b^2-4 a c} (a d+b e) \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x \sqrt{d+e x} \sqrt{-\frac{a \left (c+b x+a x^2\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+\sqrt{b^2-4 a c}+2 a x}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 a d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{3 a e \sqrt{\frac{a (d+e x)}{2 a d-\left (b+\sqrt{b^2-4 a c}\right ) e}} \left (c+b x+a x^2\right )}-\frac{2 \sqrt{2} \sqrt{b^2-4 a c} d (a d+b e) \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x \sqrt{\frac{a (d+e x)}{2 a d-\left (b+\sqrt{b^2-4 a c}\right ) e}} \sqrt{-\frac{a \left (c+b x+a x^2\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+\sqrt{b^2-4 a c}+2 a x}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 a d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{3 a e \sqrt{d+e x} \left (c+b x+a x^2\right )}+\frac{4 \sqrt{2} \sqrt{b^2-4 a c} (b d+c e) \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x \sqrt{\frac{a (d+e x)}{2 a d-\left (b+\sqrt{b^2-4 a c}\right ) e}} \sqrt{-\frac{a \left (c+b x+a x^2\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+\sqrt{b^2-4 a c}+2 a x}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 a d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{3 a \sqrt{d+e x} \left (c+b x+a x^2\right )}-\frac{\left (2 c d \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x \sqrt{b+\sqrt{b^2-4 a c}+2 a x} \sqrt{1+\frac{2 a (d+e x)}{\left (b-\sqrt{b^2-4 a c}-\frac{2 a d}{e}\right ) e}}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (d-x^2\right ) \sqrt{b+\sqrt{b^2-4 a c}-\frac{2 a d}{e}+\frac{2 a x^2}{e}} \sqrt{1+\frac{2 a x^2}{\left (b-\sqrt{b^2-4 a c}-\frac{2 a d}{e}\right ) e}}} \, dx,x,\sqrt{d+e x}\right )}{c+b x+a x^2}\\ &=\frac{2}{3} \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x \sqrt{d+e x}+\frac{\sqrt{2} \sqrt{b^2-4 a c} (a d+b e) \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x \sqrt{d+e x} \sqrt{-\frac{a \left (c+b x+a x^2\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+\sqrt{b^2-4 a c}+2 a x}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 a d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{3 a e \sqrt{\frac{a (d+e x)}{2 a d-\left (b+\sqrt{b^2-4 a c}\right ) e}} \left (c+b x+a x^2\right )}-\frac{2 \sqrt{2} \sqrt{b^2-4 a c} d (a d+b e) \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x \sqrt{\frac{a (d+e x)}{2 a d-\left (b+\sqrt{b^2-4 a c}\right ) e}} \sqrt{-\frac{a \left (c+b x+a x^2\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+\sqrt{b^2-4 a c}+2 a x}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 a d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{3 a e \sqrt{d+e x} \left (c+b x+a x^2\right )}+\frac{4 \sqrt{2} \sqrt{b^2-4 a c} (b d+c e) \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x \sqrt{\frac{a (d+e x)}{2 a d-\left (b+\sqrt{b^2-4 a c}\right ) e}} \sqrt{-\frac{a \left (c+b x+a x^2\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+\sqrt{b^2-4 a c}+2 a x}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 a d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{3 a \sqrt{d+e x} \left (c+b x+a x^2\right )}-\frac{\left (2 c d \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x \sqrt{1+\frac{2 a (d+e x)}{\left (b-\sqrt{b^2-4 a c}-\frac{2 a d}{e}\right ) e}} \sqrt{1+\frac{2 a (d+e x)}{\left (b+\sqrt{b^2-4 a c}-\frac{2 a d}{e}\right ) e}}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (d-x^2\right ) \sqrt{1+\frac{2 a x^2}{\left (b-\sqrt{b^2-4 a c}-\frac{2 a d}{e}\right ) e}} \sqrt{1+\frac{2 a x^2}{\left (b+\sqrt{b^2-4 a c}-\frac{2 a d}{e}\right ) e}}} \, dx,x,\sqrt{d+e x}\right )}{c+b x+a x^2}\\ &=\frac{2}{3} \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x \sqrt{d+e x}+\frac{\sqrt{2} \sqrt{b^2-4 a c} (a d+b e) \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x \sqrt{d+e x} \sqrt{-\frac{a \left (c+b x+a x^2\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+\sqrt{b^2-4 a c}+2 a x}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 a d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{3 a e \sqrt{\frac{a (d+e x)}{2 a d-\left (b+\sqrt{b^2-4 a c}\right ) e}} \left (c+b x+a x^2\right )}-\frac{2 \sqrt{2} \sqrt{b^2-4 a c} d (a d+b e) \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x \sqrt{\frac{a (d+e x)}{2 a d-\left (b+\sqrt{b^2-4 a c}\right ) e}} \sqrt{-\frac{a \left (c+b x+a x^2\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+\sqrt{b^2-4 a c}+2 a x}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 a d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{3 a e \sqrt{d+e x} \left (c+b x+a x^2\right )}+\frac{4 \sqrt{2} \sqrt{b^2-4 a c} (b d+c e) \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x \sqrt{\frac{a (d+e x)}{2 a d-\left (b+\sqrt{b^2-4 a c}\right ) e}} \sqrt{-\frac{a \left (c+b x+a x^2\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+\sqrt{b^2-4 a c}+2 a x}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 a d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{3 a \sqrt{d+e x} \left (c+b x+a x^2\right )}-\frac{\sqrt{2} c \sqrt{2 a d-\left (b-\sqrt{b^2-4 a c}\right ) e} \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x \sqrt{1-\frac{2 a (d+e x)}{2 a d-\left (b-\sqrt{b^2-4 a c}\right ) e}} \sqrt{1-\frac{2 a (d+e x)}{2 a d-\left (b+\sqrt{b^2-4 a c}\right ) e}} \Pi \left (\frac{2 a d-b e+\sqrt{b^2-4 a c} e}{2 a d};\sin ^{-1}\left (\frac{\sqrt{2} \sqrt{a} \sqrt{d+e x}}{\sqrt{2 a d-\left (b-\sqrt{b^2-4 a c}\right ) e}}\right )|\frac{b-\sqrt{b^2-4 a c}-\frac{2 a d}{e}}{b+\sqrt{b^2-4 a c}-\frac{2 a d}{e}}\right )}{\sqrt{a} \left (c+b x+a x^2\right )}\\ \end{align*}
Mathematica [C] time = 10.6806, size = 1258, normalized size = 1.32 \[ \frac{x \sqrt{a+\frac{c+b x}{x^2}} \left (\frac{4 (a d+b e) \sqrt{\frac{a d^2+e (c e-b d)}{-2 a d+b e+\sqrt{\left (b^2-4 a c\right ) e^2}}} (c+x (b+a x)) e^2}{(d+e x)^2}+\frac{6 i \sqrt{2} a c \sqrt{\frac{-2 c e^2+2 a d x e+b (d-e x) e+\sqrt{\left (b^2-4 a c\right ) e^2} (d+e x)}{\left (2 a d-b e+\sqrt{\left (b^2-4 a c\right ) e^2}\right ) (d+e x)}} \sqrt{\frac{2 c e^2-2 a d x e+b (e x-d) e+\sqrt{\left (b^2-4 a c\right ) e^2} (d+e x)}{\left (-2 a d+b e+\sqrt{\left (b^2-4 a c\right ) e^2}\right ) (d+e x)}} \Pi \left (\frac{d \left (2 a d-b e-\sqrt{\left (b^2-4 a c\right ) e^2}\right )}{2 \left (a d^2+e (c e-b d)\right )};i \sinh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{a d^2-b e d+c e^2}{-2 a d+b e+\sqrt{\left (b^2-4 a c\right ) e^2}}}}{\sqrt{d+e x}}\right )|-\frac{-2 a d+b e+\sqrt{\left (b^2-4 a c\right ) e^2}}{2 a d-b e+\sqrt{\left (b^2-4 a c\right ) e^2}}\right ) e^2}{\sqrt{d+e x}}-\frac{i \sqrt{2} (a d+b e) \left (2 a d-b e+\sqrt{\left (b^2-4 a c\right ) e^2}\right ) \sqrt{\frac{-2 c e^2+2 a d x e+b (d-e x) e+\sqrt{\left (b^2-4 a c\right ) e^2} (d+e x)}{\left (2 a d-b e+\sqrt{\left (b^2-4 a c\right ) e^2}\right ) (d+e x)}} \sqrt{\frac{2 c e^2-2 a d x e+b (e x-d) e+\sqrt{\left (b^2-4 a c\right ) e^2} (d+e x)}{\left (-2 a d+b e+\sqrt{\left (b^2-4 a c\right ) e^2}\right ) (d+e x)}} E\left (i \sinh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{a d^2-b e d+c e^2}{-2 a d+b e+\sqrt{\left (b^2-4 a c\right ) e^2}}}}{\sqrt{d+e x}}\right )|-\frac{-2 a d+b e+\sqrt{\left (b^2-4 a c\right ) e^2}}{2 a d-b e+\sqrt{\left (b^2-4 a c\right ) e^2}}\right )}{\sqrt{d+e x}}+\frac{i \sqrt{2} \left (b e \left (\sqrt{\left (b^2-4 a c\right ) e^2}-b e\right )+a \left (-2 c e^2+3 b d e+d \sqrt{\left (b^2-4 a c\right ) e^2}\right )\right ) \sqrt{\frac{-2 c e^2+2 a d x e+b (d-e x) e+\sqrt{\left (b^2-4 a c\right ) e^2} (d+e x)}{\left (2 a d-b e+\sqrt{\left (b^2-4 a c\right ) e^2}\right ) (d+e x)}} \sqrt{\frac{2 c e^2-2 a d x e+b (e x-d) e+\sqrt{\left (b^2-4 a c\right ) e^2} (d+e x)}{\left (-2 a d+b e+\sqrt{\left (b^2-4 a c\right ) e^2}\right ) (d+e x)}} \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{a d^2-b e d+c e^2}{-2 a d+b e+\sqrt{\left (b^2-4 a c\right ) e^2}}}}{\sqrt{d+e x}}\right ),-\frac{-2 a d+b e+\sqrt{\left (b^2-4 a c\right ) e^2}}{2 a d-b e+\sqrt{\left (b^2-4 a c\right ) e^2}}\right )}{\sqrt{d+e x}}\right ) (d+e x)^{3/2}}{6 a e^2 \sqrt{\frac{a d^2+e (c e-b d)}{-2 a d+b e+\sqrt{\left (b^2-4 a c\right ) e^2}}} (c+x (b+a x))}+\frac{2}{3} x \sqrt{a+\frac{c+b x}{x^2}} \sqrt{d+e x} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.048, size = 3023, normalized size = 3.2 \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 \sqrt{e x + d} \sqrt{a + \frac{b}{x} + \frac{c}{x^{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(-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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