Optimal. Leaf size=981 \[ \frac{2}{11} \sqrt{a+\frac{b}{x}+\frac{c}{x^2}} \sqrt{d+e x} x^5+\frac{2 (a d+b e) \sqrt{a+\frac{b}{x}+\frac{c}{x^2}} (d+e x)^{7/2} x}{99 a e^4}-\frac{2 \left (29 a^2 d^2+8 b^2 e^2+a e (19 b d-18 c e)\right ) \sqrt{a+\frac{b}{x}+\frac{c}{x^2}} (d+e x)^{5/2} x}{693 a^2 e^4}+\frac{2 \left (233 a^3 d^3+4 a^2 e (18 b d-37 c e) d+48 b^3 e^3+a b e^2 (67 b d-157 c e)\right ) \sqrt{a+\frac{b}{x}+\frac{c}{x^2}} (d+e x)^{3/2} x}{3465 a^3 e^4}+\frac{\sqrt{2} \sqrt{b^2-4 a c} \left (128 a^5 d^5-4 a^4 e (14 b d-27 c e) d^3-a^3 e^2 \left (37 b^2 d^2-135 b c e d+156 c^2 e^2\right ) d+128 b^5 e^5-8 a b^3 e^4 (7 b d+87 c e)-a^2 b e^3 \left (37 b^2 d^2-258 b c e d-771 c^2 e^2\right )\right ) \sqrt{a+\frac{b}{x}+\frac{c}{x^2}} \sqrt{d+e x} \sqrt{-\frac{a \left (a x^2+b x+c\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 a x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 a d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right ) x}{3465 a^5 e^5 \sqrt{\frac{a (d+e x)}{2 a d-\left (b+\sqrt{b^2-4 a c}\right ) e}} \left (a x^2+b x+c\right )}-\frac{2 \sqrt{2} \sqrt{b^2-4 a c} \left (a d^2-e (b d-c e)\right ) \left (128 a^4 d^4+4 a^3 e (2 b d+3 c e) d^2-64 b^4 e^4-4 a b^2 e^3 (7 b d-69 c e)-3 a^2 e^2 \left (3 b^2 d^2-29 b c e d+50 c^2 e^2\right )\right ) \sqrt{a+\frac{b}{x}+\frac{c}{x^2}} \sqrt{\frac{a (d+e x)}{2 a d-\left (b+\sqrt{b^2-4 a c}\right ) e}} \sqrt{-\frac{a \left (a x^2+b x+c\right )}{b^2-4 a c}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 a x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right ),-\frac{2 \sqrt{b^2-4 a c} e}{2 a d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right ) x}{3465 a^5 e^5 \sqrt{d+e x} \left (a x^2+b x+c\right )}-\frac{2 \left (187 a^4 d^4-4 a^3 e (2 b d+3 c e) d^2+64 b^4 e^4+4 a b^2 e^3 (7 b d-69 c e)+3 a^2 e^2 \left (3 b^2 d^2-29 b c e d+50 c^2 e^2\right )\right ) \sqrt{a+\frac{b}{x}+\frac{c}{x^2}} \sqrt{d+e x} x}{3465 a^4 e^4} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 6.17238, antiderivative size = 981, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 7, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.241, Rules used = {1573, 918, 1653, 843, 718, 424, 419} \[ \frac{2}{11} \sqrt{a+\frac{b}{x}+\frac{c}{x^2}} \sqrt{d+e x} x^5+\frac{2 (a d+b e) \sqrt{a+\frac{b}{x}+\frac{c}{x^2}} (d+e x)^{7/2} x}{99 a e^4}-\frac{2 \left (29 a^2 d^2+8 b^2 e^2+a e (19 b d-18 c e)\right ) \sqrt{a+\frac{b}{x}+\frac{c}{x^2}} (d+e x)^{5/2} x}{693 a^2 e^4}+\frac{2 \left (233 a^3 d^3+4 a^2 e (18 b d-37 c e) d+48 b^3 e^3+a b e^2 (67 b d-157 c e)\right ) \sqrt{a+\frac{b}{x}+\frac{c}{x^2}} (d+e x)^{3/2} x}{3465 a^3 e^4}+\frac{\sqrt{2} \sqrt{b^2-4 a c} \left (128 a^5 d^5-4 a^4 e (14 b d-27 c e) d^3-a^3 e^2 \left (37 b^2 d^2-135 b c e d+156 c^2 e^2\right ) d+128 b^5 e^5-8 a b^3 e^4 (7 b d+87 c e)-a^2 b e^3 \left (37 b^2 d^2-258 b c e d-771 c^2 e^2\right )\right ) \sqrt{a+\frac{b}{x}+\frac{c}{x^2}} \sqrt{d+e x} \sqrt{-\frac{a \left (a x^2+b x+c\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 a x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 a d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right ) x}{3465 a^5 e^5 \sqrt{\frac{a (d+e x)}{2 a d-\left (b+\sqrt{b^2-4 a c}\right ) e}} \left (a x^2+b x+c\right )}-\frac{2 \sqrt{2} \sqrt{b^2-4 a c} \left (a d^2-e (b d-c e)\right ) \left (128 a^4 d^4+4 a^3 e (2 b d+3 c e) d^2-64 b^4 e^4-4 a b^2 e^3 (7 b d-69 c e)-3 a^2 e^2 \left (3 b^2 d^2-29 b c e d+50 c^2 e^2\right )\right ) \sqrt{a+\frac{b}{x}+\frac{c}{x^2}} \sqrt{\frac{a (d+e x)}{2 a d-\left (b+\sqrt{b^2-4 a c}\right ) e}} \sqrt{-\frac{a \left (a x^2+b x+c\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 a x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 a d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right ) x}{3465 a^5 e^5 \sqrt{d+e x} \left (a x^2+b x+c\right )}-\frac{2 \left (187 a^4 d^4-4 a^3 e (2 b d+3 c e) d^2+64 b^4 e^4+4 a b^2 e^3 (7 b d-69 c e)+3 a^2 e^2 \left (3 b^2 d^2-29 b c e d+50 c^2 e^2\right )\right ) \sqrt{a+\frac{b}{x}+\frac{c}{x^2}} \sqrt{d+e x} x}{3465 a^4 e^4} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1573
Rule 918
Rule 1653
Rule 843
Rule 718
Rule 424
Rule 419
Rubi steps
\begin{align*} \int \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x^4 \sqrt{d+e x} \, dx &=\frac{\left (\sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x\right ) \int x^3 \sqrt{d+e x} \sqrt{c+b x+a x^2} \, dx}{\sqrt{c+b x+a x^2}}\\ &=\frac{2}{11} \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x^5 \sqrt{d+e x}-\frac{\left (\sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x\right ) \int \frac{x^3 \left (-3 c d-2 (b d+c e) x-(a d+b e) x^2\right )}{\sqrt{d+e x} \sqrt{c+b x+a x^2}} \, dx}{11 \sqrt{c+b x+a x^2}}\\ &=\frac{2}{11} \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x^5 \sqrt{d+e x}+\frac{2 (a d+b e) \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x (d+e x)^{7/2}}{99 a e^4}-\frac{\left (2 \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x\right ) \int \frac{\frac{1}{2} d^3 e (a d+b e) (b d+7 c e)+\frac{1}{2} d^2 e (a d+b e) \left (2 a d^2+e (11 b d+21 c e)\right ) x+\frac{3}{2} d e^2 (a d+b e) \left (5 a d^2+e (9 b d+7 c e)\right ) x^2+\frac{1}{2} e^3 \left (33 a^2 d^3+2 a d e (29 b d-10 c e)+b e^2 (25 b d+7 c e)\right ) x^3+\frac{1}{2} e^4 \left (29 a^2 d^2+8 b^2 e^2+a e (19 b d-18 c e)\right ) x^4}{\sqrt{d+e x} \sqrt{c+b x+a x^2}} \, dx}{99 a e^5 \sqrt{c+b x+a x^2}}\\ &=\frac{2}{11} \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x^5 \sqrt{d+e x}-\frac{2 \left (29 a^2 d^2+8 b^2 e^2+a e (19 b d-18 c e)\right ) \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x (d+e x)^{5/2}}{693 a^2 e^4}+\frac{2 (a d+b e) \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x (d+e x)^{7/2}}{99 a e^4}-\frac{\left (4 \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x\right ) \int \frac{-\frac{1}{2} d^2 e^5 \left (4 b^2 e^2 (b d+5 c e)+a^2 d^2 (11 b d+48 c e)+a e \left (6 b^2 d^2+14 b c d e-45 c^2 e^2\right )\right )-\frac{1}{4} d e^5 \left (44 a^3 d^4+16 b^2 e^3 (4 b d+5 c e)+a^2 d^2 e (179 b d+107 c e)+a e^2 \left (91 b^2 d^2-101 b c d e-180 c^2 e^2\right )\right ) x-\frac{1}{2} e^6 \left (107 a^3 d^4+2 a^2 d^2 e (73 b d-50 c e)+4 b^2 e^3 (13 b d+5 c e)+a e^2 \left (73 b^2 d^2-143 b c d e-45 c^2 e^2\right )\right ) x^2-\frac{1}{4} e^7 \left (233 a^3 d^3+48 b^3 e^3+a b e^2 (67 b d-157 c e)+4 a^2 d e (18 b d-37 c e)\right ) x^3}{\sqrt{d+e x} \sqrt{c+b x+a x^2}} \, dx}{693 a^2 e^9 \sqrt{c+b x+a x^2}}\\ &=\frac{2}{11} \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x^5 \sqrt{d+e x}+\frac{2 \left (233 a^3 d^3+48 b^3 e^3+a b e^2 (67 b d-157 c e)+4 a^2 d e (18 b d-37 c e)\right ) \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x (d+e x)^{3/2}}{3465 a^3 e^4}-\frac{2 \left (29 a^2 d^2+8 b^2 e^2+a e (19 b d-18 c e)\right ) \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x (d+e x)^{5/2}}{693 a^2 e^4}+\frac{2 (a d+b e) \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x (d+e x)^{7/2}}{99 a e^4}-\frac{\left (8 \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x\right ) \int \frac{\frac{3}{8} d e^8 \left (16 b^3 e^3 (b d+3 c e)+a^3 d^3 (41 b d+73 c e)+a b e^2 \left (9 b^2 d^2-52 b c d e-157 c^2 e^2\right )+2 a^2 d e \left (2 b^2 d^2-12 b c d e+c^2 e^2\right )\right )+\frac{3}{8} e^8 \left (82 a^4 d^5+2 a^3 d^3 e (69 b d-22 c e)+16 b^3 e^4 (5 b d+3 c e)+a b e^3 \left (37 b^2 d^2-328 b c d e-157 c^2 e^2\right )+a^2 d e^2 \left (13 b^2 d^2-111 b c d e+152 c^2 e^2\right )\right ) x+\frac{3}{8} e^9 \left (187 a^4 d^4+64 b^4 e^4+4 a b^2 e^3 (7 b d-69 c e)-4 a^3 d^2 e (2 b d+3 c e)+3 a^2 e^2 \left (3 b^2 d^2-29 b c d e+50 c^2 e^2\right )\right ) x^2}{\sqrt{d+e x} \sqrt{c+b x+a x^2}} \, dx}{3465 a^3 e^{12} \sqrt{c+b x+a x^2}}\\ &=-\frac{2 \left (187 a^4 d^4+64 b^4 e^4+4 a b^2 e^3 (7 b d-69 c e)-4 a^3 d^2 e (2 b d+3 c e)+3 a^2 e^2 \left (3 b^2 d^2-29 b c d e+50 c^2 e^2\right )\right ) \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x \sqrt{d+e x}}{3465 a^4 e^4}+\frac{2}{11} \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x^5 \sqrt{d+e x}+\frac{2 \left (233 a^3 d^3+48 b^3 e^3+a b e^2 (67 b d-157 c e)+4 a^2 d e (18 b d-37 c e)\right ) \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x (d+e x)^{3/2}}{3465 a^3 e^4}-\frac{2 \left (29 a^2 d^2+8 b^2 e^2+a e (19 b d-18 c e)\right ) \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x (d+e x)^{5/2}}{693 a^2 e^4}+\frac{2 (a d+b e) \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x (d+e x)^{7/2}}{99 a e^4}-\frac{\left (16 \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x\right ) \int \frac{-\frac{3}{8} e^{10} \left (16 a^4 d^4 (2 b d-c e)+32 b^4 e^4 (b d+c e)-a^3 d^2 e \left (10 b^2 d^2-26 b c d e+9 c^2 e^2\right )-2 a b^2 e^3 \left (5 b^2 d^2+98 b c d e+69 c^2 e^2\right )-3 a^2 e^2 \left (3 b^3 d^3-13 b^2 c d^2 e-89 b c^2 d e^2-25 c^3 e^3\right )\right )-\frac{3}{16} e^{10} \left (128 a^5 d^5+128 b^5 e^5-4 a^4 d^3 e (14 b d-27 c e)-8 a b^3 e^4 (7 b d+87 c e)-a^2 b e^3 \left (37 b^2 d^2-258 b c d e-771 c^2 e^2\right )-a^3 d e^2 \left (37 b^2 d^2-135 b c d e+156 c^2 e^2\right )\right ) x}{\sqrt{d+e x} \sqrt{c+b x+a x^2}} \, dx}{10395 a^4 e^{14} \sqrt{c+b x+a x^2}}\\ &=-\frac{2 \left (187 a^4 d^4+64 b^4 e^4+4 a b^2 e^3 (7 b d-69 c e)-4 a^3 d^2 e (2 b d+3 c e)+3 a^2 e^2 \left (3 b^2 d^2-29 b c d e+50 c^2 e^2\right )\right ) \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x \sqrt{d+e x}}{3465 a^4 e^4}+\frac{2}{11} \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x^5 \sqrt{d+e x}+\frac{2 \left (233 a^3 d^3+48 b^3 e^3+a b e^2 (67 b d-157 c e)+4 a^2 d e (18 b d-37 c e)\right ) \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x (d+e x)^{3/2}}{3465 a^3 e^4}-\frac{2 \left (29 a^2 d^2+8 b^2 e^2+a e (19 b d-18 c e)\right ) \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x (d+e x)^{5/2}}{693 a^2 e^4}+\frac{2 (a d+b e) \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x (d+e x)^{7/2}}{99 a e^4}+\frac{\left (\left (128 a^5 d^5+128 b^5 e^5-4 a^4 d^3 e (14 b d-27 c e)-8 a b^3 e^4 (7 b d+87 c e)-a^2 b e^3 \left (37 b^2 d^2-258 b c d e-771 c^2 e^2\right )-a^3 d e^2 \left (37 b^2 d^2-135 b c d e+156 c^2 e^2\right )\right ) \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}{3465 a^4 e^5 \sqrt{c+b x+a x^2}}-\frac{\left (16 \left (\frac{3}{16} d e^{10} \left (128 a^5 d^5+128 b^5 e^5-4 a^4 d^3 e (14 b d-27 c e)-8 a b^3 e^4 (7 b d+87 c e)-a^2 b e^3 \left (37 b^2 d^2-258 b c d e-771 c^2 e^2\right )-a^3 d e^2 \left (37 b^2 d^2-135 b c d e+156 c^2 e^2\right )\right )-\frac{3}{8} e^{11} \left (16 a^4 d^4 (2 b d-c e)+32 b^4 e^4 (b d+c e)-a^3 d^2 e \left (10 b^2 d^2-26 b c d e+9 c^2 e^2\right )-2 a b^2 e^3 \left (5 b^2 d^2+98 b c d e+69 c^2 e^2\right )-3 a^2 e^2 \left (3 b^3 d^3-13 b^2 c d^2 e-89 b c^2 d e^2-25 c^3 e^3\right )\right )\right ) \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}{10395 a^4 e^{15} \sqrt{c+b x+a x^2}}\\ &=-\frac{2 \left (187 a^4 d^4+64 b^4 e^4+4 a b^2 e^3 (7 b d-69 c e)-4 a^3 d^2 e (2 b d+3 c e)+3 a^2 e^2 \left (3 b^2 d^2-29 b c d e+50 c^2 e^2\right )\right ) \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x \sqrt{d+e x}}{3465 a^4 e^4}+\frac{2}{11} \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x^5 \sqrt{d+e x}+\frac{2 \left (233 a^3 d^3+48 b^3 e^3+a b e^2 (67 b d-157 c e)+4 a^2 d e (18 b d-37 c e)\right ) \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x (d+e x)^{3/2}}{3465 a^3 e^4}-\frac{2 \left (29 a^2 d^2+8 b^2 e^2+a e (19 b d-18 c e)\right ) \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x (d+e x)^{5/2}}{693 a^2 e^4}+\frac{2 (a d+b e) \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x (d+e x)^{7/2}}{99 a e^4}+\frac{\left (\sqrt{2} \sqrt{b^2-4 a c} \left (128 a^5 d^5+128 b^5 e^5-4 a^4 d^3 e (14 b d-27 c e)-8 a b^3 e^4 (7 b d+87 c e)-a^2 b e^3 \left (37 b^2 d^2-258 b c d e-771 c^2 e^2\right )-a^3 d e^2 \left (37 b^2 d^2-135 b c d e+156 c^2 e^2\right )\right ) \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 )}{3465 a^5 e^5 \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 (32 \sqrt{2} \sqrt{b^2-4 a c} \left (\frac{3}{16} d e^{10} \left (128 a^5 d^5+128 b^5 e^5-4 a^4 d^3 e (14 b d-27 c e)-8 a b^3 e^4 (7 b d+87 c e)-a^2 b e^3 \left (37 b^2 d^2-258 b c d e-771 c^2 e^2\right )-a^3 d e^2 \left (37 b^2 d^2-135 b c d e+156 c^2 e^2\right )\right )-\frac{3}{8} e^{11} \left (16 a^4 d^4 (2 b d-c e)+32 b^4 e^4 (b d+c e)-a^3 d^2 e \left (10 b^2 d^2-26 b c d e+9 c^2 e^2\right )-2 a b^2 e^3 \left (5 b^2 d^2+98 b c d e+69 c^2 e^2\right )-3 a^2 e^2 \left (3 b^3 d^3-13 b^2 c d^2 e-89 b c^2 d e^2-25 c^3 e^3\right )\right )\right ) \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 )}{10395 a^5 e^{15} \sqrt{d+e x} \left (c+b x+a x^2\right )}\\ &=-\frac{2 \left (187 a^4 d^4+64 b^4 e^4+4 a b^2 e^3 (7 b d-69 c e)-4 a^3 d^2 e (2 b d+3 c e)+3 a^2 e^2 \left (3 b^2 d^2-29 b c d e+50 c^2 e^2\right )\right ) \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x \sqrt{d+e x}}{3465 a^4 e^4}+\frac{2}{11} \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x^5 \sqrt{d+e x}+\frac{2 \left (233 a^3 d^3+48 b^3 e^3+a b e^2 (67 b d-157 c e)+4 a^2 d e (18 b d-37 c e)\right ) \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x (d+e x)^{3/2}}{3465 a^3 e^4}-\frac{2 \left (29 a^2 d^2+8 b^2 e^2+a e (19 b d-18 c e)\right ) \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x (d+e x)^{5/2}}{693 a^2 e^4}+\frac{2 (a d+b e) \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x (d+e x)^{7/2}}{99 a e^4}+\frac{\sqrt{2} \sqrt{b^2-4 a c} \left (128 a^5 d^5+128 b^5 e^5-4 a^4 d^3 e (14 b d-27 c e)-8 a b^3 e^4 (7 b d+87 c e)-a^2 b e^3 \left (37 b^2 d^2-258 b c d e-771 c^2 e^2\right )-a^3 d e^2 \left (37 b^2 d^2-135 b c d e+156 c^2 e^2\right )\right ) \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 )}{3465 a^5 e^5 \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} \left (a d^2-b d e+c e^2\right ) \left (128 a^4 d^4+8 a^3 b d^3 e-9 a^2 b^2 d^2 e^2+12 a^3 c d^2 e^2-28 a b^3 d e^3+87 a^2 b c d e^3-64 b^4 e^4+276 a b^2 c e^4-150 a^2 c^2 e^4\right ) \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 )}{3465 a^5 e^5 \sqrt{d+e x} \left (c+b x+a x^2\right )}\\ \end{align*}
Mathematica [C] time = 14.3281, size = 10904, normalized size = 11.12 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.148, size = 11938, normalized size = 12.2 \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}}} x^{4}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\sqrt{e x + d} x^{4} \sqrt{\frac{a x^{2} + b x + c}{x^{2}}}, x\right ) \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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