Optimal. Leaf size=3329 \[ \text {result too large to display} \]
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Rubi [A] time = 20.94, antiderivative size = 1806, normalized size of antiderivative = 0.54, number of steps used = 8, number of rules used = 5, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.122, Rules used = {1649, 823, 827, 1166, 208}
result too large to display
Antiderivative was successfully verified.
[In]
[Out]
Rule 208
Rule 823
Rule 827
Rule 1166
Rule 1649
Rubi steps
\begin {align*} \int \frac {(c_6+x c_7){}^2}{\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5}} \, dx &=(2 (c_1 c_2-c_0 c_3)) \operatorname {Subst}\left (\int \frac {x \left (-c_1 c_6+c_0 c_7+x^2 (c_3 c_6-c_2 c_7)\right ){}^2}{\left (c_1-x^2 c_3\right ){}^4 \sqrt {c_4+x c_5}} \, dx,x,\sqrt {\frac {c_0+x c_1}{c_2+x c_3}}\right )\\ &=\frac {(c_1 c_2-c_0 c_3){}^3 \left (c_4-\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5\right ) \sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5} c_7{}^2}{3 c_3{}^2 \left (c_1-\frac {(c_0+x c_1) c_3}{c_2+x c_3}\right ){}^3 \left (c_3 c_4{}^2-c_1 c_5{}^2\right )}+\frac {(c_1 c_2-c_0 c_3) \operatorname {Subst}\left (\int \frac {\frac {c_1 (c_1 c_2-c_0 c_3){}^2 c_4 c_5 c_7{}^2}{2 c_3{}^2}-\frac {6 x^3 c_1 \left (c_3 c_4{}^2-c_1 c_5{}^2\right ) (c_3 c_6-c_2 c_7){}^2}{c_3}-\frac {3 x c_1 \left (c_0 c_3{}^2 c_7 \left (8 c_3 c_4{}^2 c_6-\left (8 c_2 c_4{}^2-3 c_0 c_5{}^2\right ) c_7\right )+c_1{}^2 c_5{}^2 \left (4 c_3{}^2 c_6{}^2-c_2{}^2 c_7{}^2\right )-2 c_1 c_3 \left (2 c_3{}^2 c_4{}^2 c_6{}^2+4 c_0 c_3 c_5{}^2 c_6 c_7-c_2 \left (2 c_2 c_4{}^2+c_0 c_5{}^2\right ) c_7{}^2\right )\right )}{2 c_3{}^2}}{\left (c_1-x^2 c_3\right ){}^3 \sqrt {c_4+x c_5}} \, dx,x,\sqrt {\frac {c_0+x c_1}{c_2+x c_3}}\right )}{3 c_1 \left (c_3 c_4{}^2-c_1 c_5{}^2\right )}\\ &=\frac {(c_1 c_2-c_0 c_3){}^3 \left (c_4-\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5\right ) \sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5} c_7{}^2}{3 c_3{}^2 \left (c_1-\frac {(c_0+x c_1) c_3}{c_2+x c_3}\right ){}^3 \left (c_3 c_4{}^2-c_1 c_5{}^2\right )}-\frac {(c_1 c_2-c_0 c_3){}^2 \sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5} c_7 \left (\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5 \left (c_0 c_3{}^2 c_4{}^2 c_7-3 c_1{}^2 c_5{}^2 (8 c_3 c_6-5 c_2 c_7)+c_1 c_3 \left (24 c_3 c_4{}^2 c_6-\left (25 c_2 c_4{}^2-9 c_0 c_5{}^2\right ) c_7\right )\right )-2 c_1 c_4 \left (12 c_3{}^2 c_4{}^2 c_6+7 c_1 c_2 c_5{}^2 c_7-c_3 \left (12 c_1 c_5{}^2 c_6+\left (12 c_2 c_4{}^2-5 c_0 c_5{}^2\right ) c_7\right )\right )\right )}{24 c_1 c_3{}^2 \left (c_1-\frac {(c_0+x c_1) c_3}{c_2+x c_3}\right ){}^2 \left (c_3 c_4{}^2-c_1 c_5{}^2\right ){}^2}+\frac {(c_1 c_2-c_0 c_3) \operatorname {Subst}\left (\int \frac {-\frac {c_1 (c_1 c_2-c_0 c_3) c_4 c_5 c_7 \left (3 c_0 c_3{}^2 c_4{}^2 c_7+4 c_1{}^2 c_5{}^2 (3 c_3 c_6-c_2 c_7)-c_1 c_3 \left (12 c_3 c_4{}^2 c_6-\left (9 c_2 c_4{}^2-8 c_0 c_5{}^2\right ) c_7\right )\right )}{2 c_3{}^2}+\frac {x c_1 \left (5 c_0{}^2 c_3{}^3 c_4{}^2 c_5{}^2 c_7{}^2+3 c_1{}^3 c_5{}^4 \left (32 c_3{}^2 c_6{}^2-24 c_2 c_3 c_6 c_7+7 c_2{}^2 c_7{}^2\right )-c_1{}^2 c_3 c_5{}^2 \left (192 c_3{}^2 c_4{}^2 c_6{}^2-24 c_3 \left (11 c_2 c_4{}^2-5 c_0 c_5{}^2\right ) c_6 c_7+c_2 \left (67 c_2 c_4{}^2-30 c_0 c_5{}^2\right ) c_7{}^2\right )+c_1 c_3{}^2 \left (96 c_3{}^2 c_4{}^4 c_6{}^2-24 c_3 c_4{}^2 \left (8 c_2 c_4{}^2-5 c_0 c_5{}^2\right ) c_6 c_7+\left (96 c_2{}^2 c_4{}^4-130 c_0 c_2 c_4{}^2 c_5{}^2+45 c_0{}^2 c_5{}^4\right ) c_7{}^2\right )\right )}{4 c_3{}^2}}{\left (c_1-x^2 c_3\right ){}^2 \sqrt {c_4+x c_5}} \, dx,x,\sqrt {\frac {c_0+x c_1}{c_2+x c_3}}\right )}{12 c_1{}^2 \left (c_3 c_4{}^2-c_1 c_5{}^2\right ){}^2}\\ &=\frac {(c_1 c_2-c_0 c_3){}^3 \left (c_4-\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5\right ) \sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5} c_7{}^2}{3 c_3{}^2 \left (c_1-\frac {(c_0+x c_1) c_3}{c_2+x c_3}\right ){}^3 \left (c_3 c_4{}^2-c_1 c_5{}^2\right )}-\frac {(c_1 c_2-c_0 c_3){}^2 \sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5} c_7 \left (\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5 \left (c_0 c_3{}^2 c_4{}^2 c_7-3 c_1{}^2 c_5{}^2 (8 c_3 c_6-5 c_2 c_7)+c_1 c_3 \left (24 c_3 c_4{}^2 c_6-\left (25 c_2 c_4{}^2-9 c_0 c_5{}^2\right ) c_7\right )\right )-2 c_1 c_4 \left (12 c_3{}^2 c_4{}^2 c_6+7 c_1 c_2 c_5{}^2 c_7-c_3 \left (12 c_1 c_5{}^2 c_6+\left (12 c_2 c_4{}^2-5 c_0 c_5{}^2\right ) c_7\right )\right )\right )}{24 c_1 c_3{}^2 \left (c_1-\frac {(c_0+x c_1) c_3}{c_2+x c_3}\right ){}^2 \left (c_3 c_4{}^2-c_1 c_5{}^2\right ){}^2}+\frac {(c_2+x c_3) \sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5} \left (3 \sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5 \left (2 c_0{}^2 c_3{}^4 c_4{}^4 c_7{}^2-c_0 c_1 c_3{}^3 c_4{}^2 c_7 \left (8 c_3 c_4{}^2 c_6-\left (4 c_2 c_4{}^2-7 c_0 c_5{}^2\right ) c_7\right )-c_1{}^4 c_5{}^4 \left (32 c_3{}^2 c_6{}^2-24 c_2 c_3 c_6 c_7+7 c_2{}^2 c_7{}^2\right )+c_1{}^3 c_3 c_5{}^2 \left (64 c_3{}^2 c_4{}^2 c_6{}^2-8 c_3 \left (12 c_2 c_4{}^2-5 c_0 c_5{}^2\right ) c_6 c_7+5 c_2 \left (5 c_2 c_4{}^2-2 c_0 c_5{}^2\right ) c_7{}^2\right )-c_1{}^2 c_3{}^2 \left (32 c_3{}^2 c_4{}^4 c_6{}^2-8 c_3 c_4{}^2 \left (9 c_2 c_4{}^2-4 c_0 c_5{}^2\right ) c_6 c_7+\left (38 c_2{}^2 c_4{}^4-46 c_0 c_2 c_4{}^2 c_5{}^2+15 c_0{}^2 c_5{}^4\right ) c_7{}^2\right )\right )-c_1 c_4 \left (c_0{}^2 c_3{}^3 c_4{}^2 c_5{}^2 c_7{}^2-c_1{}^3 c_5{}^4 \left (96 c_3{}^2 c_6{}^2-48 c_2 c_3 c_6 c_7+13 c_2{}^2 c_7{}^2\right )+c_1{}^2 c_3 c_5{}^2 \left (192 c_3{}^2 c_4{}^2 c_6{}^2-48 c_3 \left (5 c_2 c_4{}^2-3 c_0 c_5{}^2\right ) c_6 c_7+c_2 \left (49 c_2 c_4{}^2-22 c_0 c_5{}^2\right ) c_7{}^2\right )-c_1 c_3{}^2 \left (96 c_3{}^2 c_4{}^4 c_6{}^2-48 c_3 c_4{}^2 \left (4 c_2 c_4{}^2-3 c_0 c_5{}^2\right ) c_6 c_7+\left (96 c_2{}^2 c_4{}^4-142 c_0 c_2 c_4{}^2 c_5{}^2+61 c_0{}^2 c_5{}^4\right ) c_7{}^2\right )\right )\right )}{96 c_1{}^2 c_3{}^2 \left (c_3 c_4{}^2-c_1 c_5{}^2\right ){}^3}-\frac {(c_1 c_2-c_0 c_3) \operatorname {Subst}\left (\int \frac {-\frac {3 x c_1 c_5{}^2 \left (2 c_0{}^2 c_3{}^4 c_4{}^4 c_7{}^2-c_0 c_1 c_3{}^3 c_4{}^2 c_7 \left (8 c_3 c_4{}^2 c_6-\left (4 c_2 c_4{}^2-7 c_0 c_5{}^2\right ) c_7\right )-c_1{}^4 c_5{}^4 \left (32 c_3{}^2 c_6{}^2-24 c_2 c_3 c_6 c_7+7 c_2{}^2 c_7{}^2\right )+c_1{}^3 c_3 c_5{}^2 \left (64 c_3{}^2 c_4{}^2 c_6{}^2-8 c_3 \left (12 c_2 c_4{}^2-5 c_0 c_5{}^2\right ) c_6 c_7+5 c_2 \left (5 c_2 c_4{}^2-2 c_0 c_5{}^2\right ) c_7{}^2\right )-c_1{}^2 c_3{}^2 \left (32 c_3{}^2 c_4{}^4 c_6{}^2-8 c_3 c_4{}^2 \left (9 c_2 c_4{}^2-4 c_0 c_5{}^2\right ) c_6 c_7+\left (38 c_2{}^2 c_4{}^4-46 c_0 c_2 c_4{}^2 c_5{}^2+15 c_0{}^2 c_5{}^4\right ) c_7{}^2\right )\right )}{8 c_3}-\frac {3 c_1 c_4 c_5 \left (4 c_0{}^2 c_3{}^4 c_4{}^4 c_7{}^2-c_0 c_1 c_3{}^3 c_4{}^2 c_7 \left (16 c_3 c_4{}^2 c_6-\left (8 c_2 c_4{}^2-15 c_0 c_5{}^2\right ) c_7\right )+c_1{}^4 c_5{}^4 \left (32 c_3{}^2 c_6{}^2-c_2{}^2 c_7{}^2\right )-c_1{}^3 c_3 c_5{}^2 \left (64 c_3{}^2 c_4{}^2 c_6{}^2-16 c_3 \left (3 c_2 c_4{}^2-4 c_0 c_5{}^2\right ) c_6 c_7-c_2 \left (c_2 c_4{}^2+2 c_0 c_5{}^2\right ) c_7{}^2\right )+c_1{}^2 c_3{}^2 \left (32 c_3{}^2 c_4{}^4 c_6{}^2-16 c_3 c_4{}^2 \left (3 c_2 c_4{}^2-5 c_0 c_5{}^2\right ) c_6 c_7+\left (20 c_2{}^2 c_4{}^4-50 c_0 c_2 c_4{}^2 c_5{}^2+31 c_0{}^2 c_5{}^4\right ) c_7{}^2\right )\right )}{8 c_3}}{\left (c_1-x^2 c_3\right ) \sqrt {c_4+x c_5}} \, dx,x,\sqrt {\frac {c_0+x c_1}{c_2+x c_3}}\right )}{24 c_1{}^3 c_3 \left (c_3 c_4{}^2-c_1 c_5{}^2\right ){}^3}\\ &=\frac {(c_1 c_2-c_0 c_3){}^3 \left (c_4-\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5\right ) \sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5} c_7{}^2}{3 c_3{}^2 \left (c_1-\frac {(c_0+x c_1) c_3}{c_2+x c_3}\right ){}^3 \left (c_3 c_4{}^2-c_1 c_5{}^2\right )}-\frac {(c_1 c_2-c_0 c_3){}^2 \sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5} c_7 \left (\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5 \left (c_0 c_3{}^2 c_4{}^2 c_7-3 c_1{}^2 c_5{}^2 (8 c_3 c_6-5 c_2 c_7)+c_1 c_3 \left (24 c_3 c_4{}^2 c_6-\left (25 c_2 c_4{}^2-9 c_0 c_5{}^2\right ) c_7\right )\right )-2 c_1 c_4 \left (12 c_3{}^2 c_4{}^2 c_6+7 c_1 c_2 c_5{}^2 c_7-c_3 \left (12 c_1 c_5{}^2 c_6+\left (12 c_2 c_4{}^2-5 c_0 c_5{}^2\right ) c_7\right )\right )\right )}{24 c_1 c_3{}^2 \left (c_1-\frac {(c_0+x c_1) c_3}{c_2+x c_3}\right ){}^2 \left (c_3 c_4{}^2-c_1 c_5{}^2\right ){}^2}+\frac {(c_2+x c_3) \sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5} \left (3 \sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5 \left (2 c_0{}^2 c_3{}^4 c_4{}^4 c_7{}^2-c_0 c_1 c_3{}^3 c_4{}^2 c_7 \left (8 c_3 c_4{}^2 c_6-\left (4 c_2 c_4{}^2-7 c_0 c_5{}^2\right ) c_7\right )-c_1{}^4 c_5{}^4 \left (32 c_3{}^2 c_6{}^2-24 c_2 c_3 c_6 c_7+7 c_2{}^2 c_7{}^2\right )+c_1{}^3 c_3 c_5{}^2 \left (64 c_3{}^2 c_4{}^2 c_6{}^2-8 c_3 \left (12 c_2 c_4{}^2-5 c_0 c_5{}^2\right ) c_6 c_7+5 c_2 \left (5 c_2 c_4{}^2-2 c_0 c_5{}^2\right ) c_7{}^2\right )-c_1{}^2 c_3{}^2 \left (32 c_3{}^2 c_4{}^4 c_6{}^2-8 c_3 c_4{}^2 \left (9 c_2 c_4{}^2-4 c_0 c_5{}^2\right ) c_6 c_7+\left (38 c_2{}^2 c_4{}^4-46 c_0 c_2 c_4{}^2 c_5{}^2+15 c_0{}^2 c_5{}^4\right ) c_7{}^2\right )\right )-c_1 c_4 \left (c_0{}^2 c_3{}^3 c_4{}^2 c_5{}^2 c_7{}^2-c_1{}^3 c_5{}^4 \left (96 c_3{}^2 c_6{}^2-48 c_2 c_3 c_6 c_7+13 c_2{}^2 c_7{}^2\right )+c_1{}^2 c_3 c_5{}^2 \left (192 c_3{}^2 c_4{}^2 c_6{}^2-48 c_3 \left (5 c_2 c_4{}^2-3 c_0 c_5{}^2\right ) c_6 c_7+c_2 \left (49 c_2 c_4{}^2-22 c_0 c_5{}^2\right ) c_7{}^2\right )-c_1 c_3{}^2 \left (96 c_3{}^2 c_4{}^4 c_6{}^2-48 c_3 c_4{}^2 \left (4 c_2 c_4{}^2-3 c_0 c_5{}^2\right ) c_6 c_7+\left (96 c_2{}^2 c_4{}^4-142 c_0 c_2 c_4{}^2 c_5{}^2+61 c_0{}^2 c_5{}^4\right ) c_7{}^2\right )\right )\right )}{96 c_1{}^2 c_3{}^2 \left (c_3 c_4{}^2-c_1 c_5{}^2\right ){}^3}-\frac {(c_1 c_2-c_0 c_3) \operatorname {Subst}\left (\int \frac {-\frac {3 x^2 c_1 c_5{}^2 \left (2 c_0{}^2 c_3{}^4 c_4{}^4 c_7{}^2-c_0 c_1 c_3{}^3 c_4{}^2 c_7 \left (8 c_3 c_4{}^2 c_6-\left (4 c_2 c_4{}^2-7 c_0 c_5{}^2\right ) c_7\right )-c_1{}^4 c_5{}^4 \left (32 c_3{}^2 c_6{}^2-24 c_2 c_3 c_6 c_7+7 c_2{}^2 c_7{}^2\right )+c_1{}^3 c_3 c_5{}^2 \left (64 c_3{}^2 c_4{}^2 c_6{}^2-8 c_3 \left (12 c_2 c_4{}^2-5 c_0 c_5{}^2\right ) c_6 c_7+5 c_2 \left (5 c_2 c_4{}^2-2 c_0 c_5{}^2\right ) c_7{}^2\right )-c_1{}^2 c_3{}^2 \left (32 c_3{}^2 c_4{}^4 c_6{}^2-8 c_3 c_4{}^2 \left (9 c_2 c_4{}^2-4 c_0 c_5{}^2\right ) c_6 c_7+\left (38 c_2{}^2 c_4{}^4-46 c_0 c_2 c_4{}^2 c_5{}^2+15 c_0{}^2 c_5{}^4\right ) c_7{}^2\right )\right )}{8 c_3}+\frac {3 c_1 c_4 c_5{}^2 \left (2 c_0{}^2 c_3{}^4 c_4{}^4 c_7{}^2-c_0 c_1 c_3{}^3 c_4{}^2 c_7 \left (8 c_3 c_4{}^2 c_6-\left (4 c_2 c_4{}^2-7 c_0 c_5{}^2\right ) c_7\right )-c_1{}^4 c_5{}^4 \left (32 c_3{}^2 c_6{}^2-24 c_2 c_3 c_6 c_7+7 c_2{}^2 c_7{}^2\right )+c_1{}^3 c_3 c_5{}^2 \left (64 c_3{}^2 c_4{}^2 c_6{}^2-8 c_3 \left (12 c_2 c_4{}^2-5 c_0 c_5{}^2\right ) c_6 c_7+5 c_2 \left (5 c_2 c_4{}^2-2 c_0 c_5{}^2\right ) c_7{}^2\right )-c_1{}^2 c_3{}^2 \left (32 c_3{}^2 c_4{}^4 c_6{}^2-8 c_3 c_4{}^2 \left (9 c_2 c_4{}^2-4 c_0 c_5{}^2\right ) c_6 c_7+\left (38 c_2{}^2 c_4{}^4-46 c_0 c_2 c_4{}^2 c_5{}^2+15 c_0{}^2 c_5{}^4\right ) c_7{}^2\right )\right )}{8 c_3}-\frac {3 c_1 c_4 c_5{}^2 \left (4 c_0{}^2 c_3{}^4 c_4{}^4 c_7{}^2-c_0 c_1 c_3{}^3 c_4{}^2 c_7 \left (16 c_3 c_4{}^2 c_6-\left (8 c_2 c_4{}^2-15 c_0 c_5{}^2\right ) c_7\right )+c_1{}^4 c_5{}^4 \left (32 c_3{}^2 c_6{}^2-c_2{}^2 c_7{}^2\right )-c_1{}^3 c_3 c_5{}^2 \left (64 c_3{}^2 c_4{}^2 c_6{}^2-16 c_3 \left (3 c_2 c_4{}^2-4 c_0 c_5{}^2\right ) c_6 c_7-c_2 \left (c_2 c_4{}^2+2 c_0 c_5{}^2\right ) c_7{}^2\right )+c_1{}^2 c_3{}^2 \left (32 c_3{}^2 c_4{}^4 c_6{}^2-16 c_3 c_4{}^2 \left (3 c_2 c_4{}^2-5 c_0 c_5{}^2\right ) c_6 c_7+\left (20 c_2{}^2 c_4{}^4-50 c_0 c_2 c_4{}^2 c_5{}^2+31 c_0{}^2 c_5{}^4\right ) c_7{}^2\right )\right )}{8 c_3}}{-x^4 c_3+2 x^2 c_3 c_4-c_3 c_4{}^2+c_1 c_5{}^2} \, dx,x,\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5}\right )}{12 c_1{}^3 c_3 \left (c_3 c_4{}^2-c_1 c_5{}^2\right ){}^3}\\ &=\frac {(c_1 c_2-c_0 c_3){}^3 \left (c_4-\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5\right ) \sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5} c_7{}^2}{3 c_3{}^2 \left (c_1-\frac {(c_0+x c_1) c_3}{c_2+x c_3}\right ){}^3 \left (c_3 c_4{}^2-c_1 c_5{}^2\right )}-\frac {(c_1 c_2-c_0 c_3){}^2 \sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5} c_7 \left (\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5 \left (c_0 c_3{}^2 c_4{}^2 c_7-3 c_1{}^2 c_5{}^2 (8 c_3 c_6-5 c_2 c_7)+c_1 c_3 \left (24 c_3 c_4{}^2 c_6-\left (25 c_2 c_4{}^2-9 c_0 c_5{}^2\right ) c_7\right )\right )-2 c_1 c_4 \left (12 c_3{}^2 c_4{}^2 c_6+7 c_1 c_2 c_5{}^2 c_7-c_3 \left (12 c_1 c_5{}^2 c_6+\left (12 c_2 c_4{}^2-5 c_0 c_5{}^2\right ) c_7\right )\right )\right )}{24 c_1 c_3{}^2 \left (c_1-\frac {(c_0+x c_1) c_3}{c_2+x c_3}\right ){}^2 \left (c_3 c_4{}^2-c_1 c_5{}^2\right ){}^2}+\frac {(c_2+x c_3) \sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5} \left (3 \sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5 \left (2 c_0{}^2 c_3{}^4 c_4{}^4 c_7{}^2-c_0 c_1 c_3{}^3 c_4{}^2 c_7 \left (8 c_3 c_4{}^2 c_6-\left (4 c_2 c_4{}^2-7 c_0 c_5{}^2\right ) c_7\right )-c_1{}^4 c_5{}^4 \left (32 c_3{}^2 c_6{}^2-24 c_2 c_3 c_6 c_7+7 c_2{}^2 c_7{}^2\right )+c_1{}^3 c_3 c_5{}^2 \left (64 c_3{}^2 c_4{}^2 c_6{}^2-8 c_3 \left (12 c_2 c_4{}^2-5 c_0 c_5{}^2\right ) c_6 c_7+5 c_2 \left (5 c_2 c_4{}^2-2 c_0 c_5{}^2\right ) c_7{}^2\right )-c_1{}^2 c_3{}^2 \left (32 c_3{}^2 c_4{}^4 c_6{}^2-8 c_3 c_4{}^2 \left (9 c_2 c_4{}^2-4 c_0 c_5{}^2\right ) c_6 c_7+\left (38 c_2{}^2 c_4{}^4-46 c_0 c_2 c_4{}^2 c_5{}^2+15 c_0{}^2 c_5{}^4\right ) c_7{}^2\right )\right )-c_1 c_4 \left (c_0{}^2 c_3{}^3 c_4{}^2 c_5{}^2 c_7{}^2-c_1{}^3 c_5{}^4 \left (96 c_3{}^2 c_6{}^2-48 c_2 c_3 c_6 c_7+13 c_2{}^2 c_7{}^2\right )+c_1{}^2 c_3 c_5{}^2 \left (192 c_3{}^2 c_4{}^2 c_6{}^2-48 c_3 \left (5 c_2 c_4{}^2-3 c_0 c_5{}^2\right ) c_6 c_7+c_2 \left (49 c_2 c_4{}^2-22 c_0 c_5{}^2\right ) c_7{}^2\right )-c_1 c_3{}^2 \left (96 c_3{}^2 c_4{}^4 c_6{}^2-48 c_3 c_4{}^2 \left (4 c_2 c_4{}^2-3 c_0 c_5{}^2\right ) c_6 c_7+\left (96 c_2{}^2 c_4{}^4-142 c_0 c_2 c_4{}^2 c_5{}^2+61 c_0{}^2 c_5{}^4\right ) c_7{}^2\right )\right )\right )}{96 c_1{}^2 c_3{}^2 \left (c_3 c_4{}^2-c_1 c_5{}^2\right ){}^3}-\frac {\left ((c_1 c_2-c_0 c_3) c_5 \left (4 c_0{}^2 c_3{}^3 c_4{}^2 c_7{}^2-14 c_0{}^2 \sqrt {c_1} c_3{}^{5/2} c_4 c_5 c_7{}^2+28 c_0 c_1{}^{3/2} c_3{}^{3/2} c_4 c_5 c_7 (2 c_3 c_6-c_2 c_7)-c_0 c_1 c_3{}^2 c_7 \left (16 c_3 c_4{}^2 c_6-\left (8 c_2 c_4{}^2+15 c_0 c_5{}^2\right ) c_7\right )+c_1{}^3 c_5{}^2 \left (32 c_3{}^2 c_6{}^2-24 c_2 c_3 c_6 c_7+7 c_2{}^2 c_7{}^2\right )-2 c_1{}^{5/2} \sqrt {c_3} c_4 c_5 \left (32 c_3{}^2 c_6{}^2-36 c_2 c_3 c_6 c_7+11 c_2{}^2 c_7{}^2\right )+2 c_1{}^2 c_3 \left (16 c_3{}^2 c_4{}^2 c_6{}^2-4 c_3 \left (6 c_2 c_4{}^2+5 c_0 c_5{}^2\right ) c_6 c_7+5 c_2 \left (2 c_2 c_4{}^2+c_0 c_5{}^2\right ) c_7{}^2\right )\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-x^2 c_3+c_3 c_4-\sqrt {c_1} \sqrt {c_3} c_5} \, dx,x,\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5}\right )}{64 c_1{}^{5/2} c_3{}^2 \left (\sqrt {c_3} c_4-\sqrt {c_1} c_5\right ){}^3}+\frac {\left ((c_1 c_2-c_0 c_3) c_5 \left (4 c_0{}^2 c_3{}^3 c_4{}^2 c_7{}^2+14 c_0{}^2 \sqrt {c_1} c_3{}^{5/2} c_4 c_5 c_7{}^2-28 c_0 c_1{}^{3/2} c_3{}^{3/2} c_4 c_5 c_7 (2 c_3 c_6-c_2 c_7)-c_0 c_1 c_3{}^2 c_7 \left (16 c_3 c_4{}^2 c_6-\left (8 c_2 c_4{}^2+15 c_0 c_5{}^2\right ) c_7\right )+c_1{}^3 c_5{}^2 \left (32 c_3{}^2 c_6{}^2-24 c_2 c_3 c_6 c_7+7 c_2{}^2 c_7{}^2\right )+2 c_1{}^{5/2} \sqrt {c_3} c_4 c_5 \left (32 c_3{}^2 c_6{}^2-36 c_2 c_3 c_6 c_7+11 c_2{}^2 c_7{}^2\right )+2 c_1{}^2 c_3 \left (16 c_3{}^2 c_4{}^2 c_6{}^2-4 c_3 \left (6 c_2 c_4{}^2+5 c_0 c_5{}^2\right ) c_6 c_7+5 c_2 \left (2 c_2 c_4{}^2+c_0 c_5{}^2\right ) c_7{}^2\right )\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-x^2 c_3+c_3 c_4+\sqrt {c_1} \sqrt {c_3} c_5} \, dx,x,\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5}\right )}{64 c_1{}^{5/2} c_3{}^2 \left (\sqrt {c_3} c_4+\sqrt {c_1} c_5\right ){}^3}\\ &=\frac {(c_1 c_2-c_0 c_3){}^3 \left (c_4-\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5\right ) \sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5} c_7{}^2}{3 c_3{}^2 \left (c_1-\frac {(c_0+x c_1) c_3}{c_2+x c_3}\right ){}^3 \left (c_3 c_4{}^2-c_1 c_5{}^2\right )}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{c_3} \sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5}}{\sqrt {\sqrt {c_3} c_4-\sqrt {c_1} c_5}}\right ) (c_1 c_2-c_0 c_3) c_5 \left (4 c_0{}^2 c_3{}^3 c_4{}^2 c_7{}^2-14 c_0{}^2 \sqrt {c_1} c_3{}^{5/2} c_4 c_5 c_7{}^2+28 c_0 c_1{}^{3/2} c_3{}^{3/2} c_4 c_5 c_7 (2 c_3 c_6-c_2 c_7)-c_0 c_1 c_3{}^2 c_7 \left (16 c_3 c_4{}^2 c_6-\left (8 c_2 c_4{}^2+15 c_0 c_5{}^2\right ) c_7\right )+c_1{}^3 c_5{}^2 \left (32 c_3{}^2 c_6{}^2-24 c_2 c_3 c_6 c_7+7 c_2{}^2 c_7{}^2\right )-2 c_1{}^{5/2} \sqrt {c_3} c_4 c_5 \left (32 c_3{}^2 c_6{}^2-36 c_2 c_3 c_6 c_7+11 c_2{}^2 c_7{}^2\right )+2 c_1{}^2 c_3 \left (16 c_3{}^2 c_4{}^2 c_6{}^2-4 c_3 \left (6 c_2 c_4{}^2+5 c_0 c_5{}^2\right ) c_6 c_7+5 c_2 \left (2 c_2 c_4{}^2+c_0 c_5{}^2\right ) c_7{}^2\right )\right )}{64 c_1{}^{5/2} c_3{}^{11/4} \left (\sqrt {c_3} c_4-\sqrt {c_1} c_5\right ){}^{7/2}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{c_3} \sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5}}{\sqrt {\sqrt {c_3} c_4+\sqrt {c_1} c_5}}\right ) (c_1 c_2-c_0 c_3) c_5 \left (4 c_0{}^2 c_3{}^3 c_4{}^2 c_7{}^2+14 c_0{}^2 \sqrt {c_1} c_3{}^{5/2} c_4 c_5 c_7{}^2-28 c_0 c_1{}^{3/2} c_3{}^{3/2} c_4 c_5 c_7 (2 c_3 c_6-c_2 c_7)-c_0 c_1 c_3{}^2 c_7 \left (16 c_3 c_4{}^2 c_6-\left (8 c_2 c_4{}^2+15 c_0 c_5{}^2\right ) c_7\right )+c_1{}^3 c_5{}^2 \left (32 c_3{}^2 c_6{}^2-24 c_2 c_3 c_6 c_7+7 c_2{}^2 c_7{}^2\right )+2 c_1{}^{5/2} \sqrt {c_3} c_4 c_5 \left (32 c_3{}^2 c_6{}^2-36 c_2 c_3 c_6 c_7+11 c_2{}^2 c_7{}^2\right )+2 c_1{}^2 c_3 \left (16 c_3{}^2 c_4{}^2 c_6{}^2-4 c_3 \left (6 c_2 c_4{}^2+5 c_0 c_5{}^2\right ) c_6 c_7+5 c_2 \left (2 c_2 c_4{}^2+c_0 c_5{}^2\right ) c_7{}^2\right )\right )}{64 c_1{}^{5/2} c_3{}^{11/4} \left (\sqrt {c_3} c_4+\sqrt {c_1} c_5\right ){}^{7/2}}-\frac {(c_1 c_2-c_0 c_3){}^2 \sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5} c_7 \left (\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5 \left (c_0 c_3{}^2 c_4{}^2 c_7-3 c_1{}^2 c_5{}^2 (8 c_3 c_6-5 c_2 c_7)+c_1 c_3 \left (24 c_3 c_4{}^2 c_6-\left (25 c_2 c_4{}^2-9 c_0 c_5{}^2\right ) c_7\right )\right )-2 c_1 c_4 \left (12 c_3{}^2 c_4{}^2 c_6+7 c_1 c_2 c_5{}^2 c_7-c_3 \left (12 c_1 c_5{}^2 c_6+\left (12 c_2 c_4{}^2-5 c_0 c_5{}^2\right ) c_7\right )\right )\right )}{24 c_1 c_3{}^2 \left (c_1-\frac {(c_0+x c_1) c_3}{c_2+x c_3}\right ){}^2 \left (c_3 c_4{}^2-c_1 c_5{}^2\right ){}^2}+\frac {(c_2+x c_3) \sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5} \left (3 \sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5 \left (2 c_0{}^2 c_3{}^4 c_4{}^4 c_7{}^2-c_0 c_1 c_3{}^3 c_4{}^2 c_7 \left (8 c_3 c_4{}^2 c_6-\left (4 c_2 c_4{}^2-7 c_0 c_5{}^2\right ) c_7\right )-c_1{}^4 c_5{}^4 \left (32 c_3{}^2 c_6{}^2-24 c_2 c_3 c_6 c_7+7 c_2{}^2 c_7{}^2\right )+c_1{}^3 c_3 c_5{}^2 \left (64 c_3{}^2 c_4{}^2 c_6{}^2-8 c_3 \left (12 c_2 c_4{}^2-5 c_0 c_5{}^2\right ) c_6 c_7+5 c_2 \left (5 c_2 c_4{}^2-2 c_0 c_5{}^2\right ) c_7{}^2\right )-c_1{}^2 c_3{}^2 \left (32 c_3{}^2 c_4{}^4 c_6{}^2-8 c_3 c_4{}^2 \left (9 c_2 c_4{}^2-4 c_0 c_5{}^2\right ) c_6 c_7+\left (38 c_2{}^2 c_4{}^4-46 c_0 c_2 c_4{}^2 c_5{}^2+15 c_0{}^2 c_5{}^4\right ) c_7{}^2\right )\right )-c_1 c_4 \left (c_0{}^2 c_3{}^3 c_4{}^2 c_5{}^2 c_7{}^2-c_1{}^3 c_5{}^4 \left (96 c_3{}^2 c_6{}^2-48 c_2 c_3 c_6 c_7+13 c_2{}^2 c_7{}^2\right )+c_1{}^2 c_3 c_5{}^2 \left (192 c_3{}^2 c_4{}^2 c_6{}^2-48 c_3 \left (5 c_2 c_4{}^2-3 c_0 c_5{}^2\right ) c_6 c_7+c_2 \left (49 c_2 c_4{}^2-22 c_0 c_5{}^2\right ) c_7{}^2\right )-c_1 c_3{}^2 \left (96 c_3{}^2 c_4{}^4 c_6{}^2-48 c_3 c_4{}^2 \left (4 c_2 c_4{}^2-3 c_0 c_5{}^2\right ) c_6 c_7+\left (96 c_2{}^2 c_4{}^4-142 c_0 c_2 c_4{}^2 c_5{}^2+61 c_0{}^2 c_5{}^4\right ) c_7{}^2\right )\right )\right )}{96 c_1{}^2 c_3{}^2 \left (c_3 c_4{}^2-c_1 c_5{}^2\right ){}^3}\\ \end {align*}
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Mathematica [B] time = 18.38, size = 8468, normalized size = 2.54 \begin {gather*} \text {Result too large to show} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [B] time = 20.10, size = 9401, normalized size = 2.82 \begin {gather*} \text {Result too large to show} \end {gather*}
Antiderivative was successfully verified.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.08, size = 0, normalized size = 0.00 \[\int \frac {\left (\textit {\_C7} x +\textit {\_C6} \right )^{2}}{\sqrt {\textit {\_C4} +\sqrt {\frac {\textit {\_C1} x +\textit {\_C0}}{\textit {\_C3} x +\textit {\_C2}}}\, \textit {\_C5}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (\_{C_{7}} x + \_{C_{6}}\right )}^{2}}{\sqrt {\_{C_{5}} \sqrt {\frac {\_{C_{1}} x + \_{C_{0}}}{\_{C_{3}} x + \_{C_{2}}}} + \_{C_{4}}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (_{\mathrm {C6}}+_{\mathrm {C7}}\,x\right )}^2}{\sqrt {_{\mathrm {C4}}+_{\mathrm {C5}}\,\sqrt {\frac {_{\mathrm {C0}}+_{\mathrm {C1}}\,x}{_{\mathrm {C2}}+_{\mathrm {C3}}\,x}}}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (_C6 + _C7 x\right )^{2}}{\sqrt {_C4 + _C5 \sqrt {\frac {_C0}{_C2 + _C3 x} + \frac {_C1 x}{_C2 + _C3 x}}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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