Integrand size = 29, antiderivative size = 981 \[ \int \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x^4 \sqrt {d+e x} \, dx=-\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 (\arcsin \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-e (b d-c e)\right ) \left (128 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 {\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}} \operatorname {EllipticF}\left (\arcsin \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 )} \]
[Out]
Time = 3.62 (sec) , antiderivative size = 981, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {1587, 932, 1667, 857, 732, 435, 430} \[ \int \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x^4 \sqrt {d+e x} \, dx=\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 (\arcsin \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}} \operatorname {EllipticF}\left (\arcsin \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} \]
[In]
[Out]
Rule 430
Rule 435
Rule 732
Rule 857
Rule 932
Rule 1587
Rule 1667
Rubi steps \begin{align*} \text {integral}& = \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 ) \text {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 ) \text {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*}
Result contains complex when optimal does not.
Time = 35.47 (sec) , antiderivative size = 10904, normalized size of antiderivative = 11.12 \[ \int \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x^4 \sqrt {d+e x} \, dx=\text {Result too large to show} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(5003\) vs. \(2(899)=1798\).
Time = 2.70 (sec) , antiderivative size = 5004, normalized size of antiderivative = 5.10
method | result | size |
risch | \(\text {Expression too large to display}\) | \(5004\) |
default | \(\text {Expression too large to display}\) | \(11938\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.12 (sec) , antiderivative size = 920, normalized size of antiderivative = 0.94 \[ \int \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x^4 \sqrt {d+e x} \, dx=-\frac {2 \, {\left ({\left (128 \, a^{6} d^{6} - 120 \, a^{5} b d^{5} e - 3 \, {\left (11 \, a^{4} b^{2} - 68 \, a^{5} c\right )} d^{4} e^{2} - {\left (20 \, a^{3} b^{3} - 87 \, a^{4} b c\right )} d^{3} e^{3} - 3 \, {\left (11 \, a^{2} b^{4} - 53 \, a^{3} b^{2} c + 34 \, a^{4} c^{2}\right )} d^{2} e^{4} - 3 \, {\left (40 \, a b^{5} - 246 \, a^{2} b^{3} c + 329 \, a^{3} b c^{2}\right )} d e^{5} + {\left (128 \, b^{6} - 888 \, a b^{4} c + 1599 \, a^{2} b^{2} c^{2} - 450 \, a^{3} c^{3}\right )} e^{6}\right )} \sqrt {a e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (a^{2} d^{2} - a b d e + {\left (b^{2} - 3 \, a c\right )} e^{2}\right )}}{3 \, a^{2} e^{2}}, -\frac {4 \, {\left (2 \, a^{3} d^{3} - 3 \, a^{2} b d^{2} e - 3 \, {\left (a b^{2} - 6 \, a^{2} c\right )} d e^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} e^{3}\right )}}{27 \, a^{3} e^{3}}, \frac {3 \, a e x + a d + b e}{3 \, a e}\right ) + 3 \, {\left (128 \, a^{6} d^{5} e - 56 \, a^{5} b d^{4} e^{2} - {\left (37 \, a^{4} b^{2} - 108 \, a^{5} c\right )} d^{3} e^{3} - {\left (37 \, a^{3} b^{3} - 135 \, a^{4} b c\right )} d^{2} e^{4} - 2 \, {\left (28 \, a^{2} b^{4} - 129 \, a^{3} b^{2} c + 78 \, a^{4} c^{2}\right )} d e^{5} + {\left (128 \, a b^{5} - 696 \, a^{2} b^{3} c + 771 \, a^{3} b c^{2}\right )} e^{6}\right )} \sqrt {a e} {\rm weierstrassZeta}\left (\frac {4 \, {\left (a^{2} d^{2} - a b d e + {\left (b^{2} - 3 \, a c\right )} e^{2}\right )}}{3 \, a^{2} e^{2}}, -\frac {4 \, {\left (2 \, a^{3} d^{3} - 3 \, a^{2} b d^{2} e - 3 \, {\left (a b^{2} - 6 \, a^{2} c\right )} d e^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} e^{3}\right )}}{27 \, a^{3} e^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (a^{2} d^{2} - a b d e + {\left (b^{2} - 3 \, a c\right )} e^{2}\right )}}{3 \, a^{2} e^{2}}, -\frac {4 \, {\left (2 \, a^{3} d^{3} - 3 \, a^{2} b d^{2} e - 3 \, {\left (a b^{2} - 6 \, a^{2} c\right )} d e^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} e^{3}\right )}}{27 \, a^{3} e^{3}}, \frac {3 \, a e x + a d + b e}{3 \, a e}\right )\right ) - 3 \, {\left (315 \, a^{6} e^{6} x^{5} + 35 \, {\left (a^{6} d e^{5} + a^{5} b e^{6}\right )} x^{4} - 10 \, {\left (4 \, a^{6} d^{2} e^{4} - a^{5} b d e^{5} + {\left (4 \, a^{4} b^{2} - 9 \, a^{5} c\right )} e^{6}\right )} x^{3} + {\left (48 \, a^{6} d^{3} e^{3} - 13 \, a^{5} b d^{2} e^{4} - {\left (13 \, a^{4} b^{2} - 32 \, a^{5} c\right )} d e^{5} + {\left (48 \, a^{3} b^{3} - 157 \, a^{4} b c\right )} e^{6}\right )} x^{2} - 2 \, {\left (32 \, a^{6} d^{4} e^{2} - 10 \, a^{5} b d^{3} e^{3} - {\left (9 \, a^{4} b^{2} - 23 \, a^{5} c\right )} d^{2} e^{4} - 5 \, {\left (2 \, a^{3} b^{3} - 7 \, a^{4} b c\right )} d e^{5} + {\left (32 \, a^{2} b^{4} - 138 \, a^{3} b^{2} c + 75 \, a^{4} c^{2}\right )} e^{6}\right )} x\right )} \sqrt {e x + d} \sqrt {\frac {a x^{2} + b x + c}{x^{2}}}\right )}}{10395 \, a^{6} e^{6}} \]
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\[ \int \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x^4 \sqrt {d+e x} \, dx=\int x^{4} \sqrt {d + e x} \sqrt {a + \frac {b}{x} + \frac {c}{x^{2}}}\, dx \]
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\[ \int \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x^4 \sqrt {d+e x} \, dx=\int { \sqrt {e x + d} \sqrt {a + \frac {b}{x} + \frac {c}{x^{2}}} x^{4} \,d x } \]
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\[ \int \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x^4 \sqrt {d+e x} \, dx=\int { \sqrt {e x + d} \sqrt {a + \frac {b}{x} + \frac {c}{x^{2}}} x^{4} \,d x } \]
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Timed out. \[ \int \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x^4 \sqrt {d+e x} \, dx=\int x^4\,\sqrt {d+e\,x}\,\sqrt {a+\frac {b}{x}+\frac {c}{x^2}} \,d x \]
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