3.20.1 \(\int \frac {-1+x^3}{(1+x^3) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx\)
  3.20.2 \(\int \frac {5 x-4 (1+k) x^2+3 k x^3}{\sqrt [3]{(1-x) x (1-k x)} (-1+(1+k) x-k x^2+b x^5)} \, dx\)
  3.20.3 \(\int \frac {x^2 (8-7 (1+k) x+6 k x^2)}{\sqrt [3]{(1-x) x (1-k x)} (-1+(1+k) x-k x^2+b x^8)} \, dx\)
  3.20.4 \(\int \frac {x (-a b+x^2)}{(x^2 (-a+x) (-b+x))^{2/3} (a b-(a+b+d) x+x^2)} \, dx\)
  3.20.5 \(\int \frac {(b+x^3) (c+x^3)}{\sqrt [3]{a+x^3}} \, dx\)
  3.20.6 \(\int \frac {(-2 q+p x^3) \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{x^3 (a q+b x^2+a p x^3)} \, dx\)
  3.20.7 \(\int \frac {\sqrt [4]{x^2+x^6} (1+x^8)}{x^4 (-1+x^4)} \, dx\)
  3.20.8 \(\int \frac {\sqrt [4]{x^2+x^6} (1+x^8)}{x^4 (-1+x^4)} \, dx\)
  3.20.9 \(\int \frac {1}{x \sqrt [3]{2-3 x+x^2}} \, dx\)
  3.20.10 \(\int \frac {a b-x^2}{\sqrt [3]{x^2 (-a+x) (-b+x)} (a b-(a+b+d) x+x^2)} \, dx\)
  3.20.11 \(\int \frac {(-q+2 p x^3) \sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6} (b x^2+a (q+p x^3)^2)}{x^5} \, dx\)
  3.20.12 \(\int \frac {-d+c x}{x^7 \sqrt [3]{-b+a x^3}} \, dx\)
  3.20.13 \(\int \frac {1+x^2+x^4}{(1-x^4) \sqrt [4]{x^3+x^5}} \, dx\)
  3.20.14 \(\int \frac {\sqrt {1+x} \sqrt {1+\sqrt {1+x}}}{x \sqrt {1+\sqrt {1+\sqrt {1+x}}}} \, dx\)
  3.20.15 \(\int \frac {1+x}{(3+x) (1+2 x) \sqrt [3]{1+x^2}} \, dx\)
  3.20.16 \(\int \frac {x (5-4 (1+k) x+3 k x^2)}{\sqrt [3]{(1-x) x (1-k x)} (-b+(b+b k) x-b k x^2+x^5)} \, dx\)
  3.20.17 \(\int \frac {1+x^4}{(1-x^4) \sqrt [4]{x^3+x^5}} \, dx\)
  3.20.18 \(\int \frac {(1+x^3)^{2/3} (2+x^3+x^6)}{x^6 (-2+x^3)^2} \, dx\)
  3.20.19 \(\int \frac {1+x^8}{\sqrt [4]{x^2+x^6} (-1+x^8)} \, dx\)
  3.20.20 \(\int \frac {1+x^8}{\sqrt [4]{x^2+x^6} (-1+x^8)} \, dx\)
  3.20.21 \(\int \frac {x^2 (8-7 (1+k) x+6 k x^2)}{\sqrt [3]{(1-x) x (1-k x)} (-b+b (1+k) x-b k x^2+x^8)} \, dx\)
  3.20.22 \(\int \frac {1}{(1+x^2)^2 \sqrt {x+\sqrt {1+x^2}}} \, dx\)
  3.20.23 \(\int \frac {x (-a+x) (a b+(a-2 b) x)}{(x (-a+x) (-b+x)^2)^{3/4} (b^2 d+(a-2 b d) x+(-1+d) x^2)} \, dx\)
  3.20.24 \(\int \frac {(1+x^2) (1-3 x^2+x^4)}{x^2 \sqrt {\frac {-2+x+2 x^2}{-1+x+x^2}} (1-x-3 x^2+x^3+x^4)} \, dx\)
  3.20.25 \(\int \frac {x^3 (5-4 (1+k) x+3 k x^2)}{((1-x) x (1-k x))^{2/3} (-1+(1+k) x-k x^2+b x^5)} \, dx\)
  3.20.26 \(\int \frac {\sqrt [3]{-1-x+5 x^2+2 x^3-10 x^4+2 x^5+7 x^6-5 x^7+x^8}}{x^2} \, dx\)
  3.20.27 \(\int \frac {(-b+a^2 x^2) \sqrt {a x^2+\sqrt {b+a^2 x^4}}}{\sqrt {b+a^2 x^4}} \, dx\)
  3.20.28 \(\int \frac {-i+\sqrt {k} x}{(i+\sqrt {k} x) \sqrt {(1-x^2) (1-k^2 x^2)}} \, dx\)
  3.20.29 \(\int \frac {i+\sqrt {k} x}{(-i+\sqrt {k} x) \sqrt {(1-x^2) (1-k^2 x^2)}} \, dx\)
  3.20.30 \(\int \frac {x^4}{\sqrt [4]{x^2+x^6} (-1+x^8)} \, dx\)
  3.20.31 \(\int \frac {x^4}{\sqrt [4]{x^2+x^6} (-1+x^8)} \, dx\)
  3.20.32 \(\int \frac {(-1+x^4) \sqrt [4]{x^2+x^6}}{1-x^4+x^8} \, dx\)
  3.20.33 \(\int \frac {(-1+x^4) \sqrt [4]{x^2+x^6}}{1-x^4+x^8} \, dx\)
  3.20.34 \(\int \frac {1-x^4+x^8}{\sqrt [4]{x^2+x^6} (-1+x^8)} \, dx\)
  3.20.35 \(\int \frac {1-x^4+x^8}{\sqrt [4]{x^2+x^6} (-1+x^8)} \, dx\)
  3.20.36 \(\int \frac {1}{(-b+a x) \sqrt [4]{-x^3+x^4}} \, dx\)
  3.20.37 \(\int \frac {-x+x^2}{\sqrt {(1-x) x (1-k^2 x)} (1-2 x+k^2 x^2)} \, dx\)
  3.20.38 \(\int \frac {1+x^2}{(-1+x^2) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx\)
  3.20.39 \(\int \frac {(-2 q+p x^3) \sqrt {q+p x^3}}{c x^4+b x^2 (q+p x^3)+a (q+p x^3)^2} \, dx\)
  3.20.40 \(\int \frac {(-1+x^4)^2}{(1+x^4)^2 \sqrt {x^2+\sqrt {1+x^4}}} \, dx\)
  3.20.41 \(\int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{(1+x^2)^2 \sqrt {1+x^4}} \, dx\)
  3.20.42 \(\int \frac {\sqrt {q+p x^5} (-2 q+3 p x^5)}{c x^4+b x^2 (q+p x^5)+a (q+p x^5)^2} \, dx\)
  3.20.43 \(\int \frac {1}{\sqrt [3]{1+x^2} (9+x^2)} \, dx\)
  3.20.44 \(\int \frac {-3-2 (1+k^2) x+(1+k^2) x^2+4 k^2 x^3+k^2 x^4}{((1-x^2) (1-k^2 x^2))^{2/3} (-1+d-(2+d) x-(1+d k^2) x^2+d k^2 x^3)} \, dx\)
  3.20.45 \(\int \frac {\sqrt [4]{x^3+x^5} (1+x^4+x^8)}{x^4 (-1+x^4)} \, dx\)
  3.20.46 \(\int \frac {1}{(b+a x^2) \sqrt [3]{x+x^3}} \, dx\)
  3.20.47 \(\int \frac {(-1+2 x^6) \sqrt [3]{x+x^7}}{(1-2 x^2+x^6) (1-x^2+x^6)} \, dx\)
  3.20.48 \(\int \frac {(-q+2 p x^3) \sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6} (b x^3+a (q+p x^3)^3)}{x^6} \, dx\)
  3.20.49 \(\int \frac {-1+(-1+2 k) x}{\sqrt [3]{(1-x) x (1-k x)} (1-(2+b) x+(1+b k) x^2)} \, dx\)
  3.20.50 \(\int \frac {-3+2 (1+k^2) x+(1+k^2) x^2-4 k^2 x^3+k^2 x^4}{((1-x^2) (1-k^2 x^2))^{2/3} (1-d-(2+d) x+(1+d k^2) x^2+d k^2 x^3)} \, dx\)
  3.20.51 \(\int \frac {b-3 a x^3+3 x^6}{x^6 (-b+2 a x^3) \sqrt [4]{-b x+a x^4}} \, dx\)
  3.20.52 \(\int \frac {(-2 q+p x^3) \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6} (b x^6+a (q+p x^3)^3)}{x^{11}} \, dx\)
  3.20.53 \(\int \frac {x^2}{\sqrt {\frac {b+a x}{d+c x}}} \, dx\)
  3.20.54 \(\int \frac {b+2 a x}{(-b+a x) (2 b+a x) \sqrt [4]{-1+b x+a x^2}} \, dx\)
  3.20.55 \(\int \frac {-1+a k x+k x^2}{(1+k x^2) \sqrt {(1-x^2) (1-k^2 x^2)}} \, dx\)
  3.20.56 \(\int \frac {(-b+x^3) (b+x^3)}{\sqrt [3]{a x^2+x^3}} \, dx\)
  3.20.57 \(\int \frac {x}{(1-x^2) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx\)
  3.20.58 \(\int \frac {1}{\sqrt [4]{-1+x^4} (-1-x^4+x^8)} \, dx\)
  3.20.59 \(\int \frac {-1+2 x^4}{\sqrt [4]{-1+x^4} (-1-x^4+x^8)} \, dx\)
  3.20.60 \(\int \frac {1}{\sqrt [4]{1+x^4} (-1+x^4+x^8)} \, dx\)
  3.20.61 \(\int \frac {-2+x^4}{\sqrt [4]{1+x^4} (-1+x^4+x^8)} \, dx\)
  3.20.62 \(\int \frac {\sqrt [4]{b x^3+a x^4} (-d+c x^8)}{x^8} \, dx\)
  3.20.63 \(\int \frac {x^4 (2+x^5)}{\sqrt {1+x^5} (-1-x^5+a x^{10})} \, dx\)
  3.20.64 \(\int \frac {x^4 (-2+x^5)}{\sqrt {-1+x^5} (1-x^5+a x^{10})} \, dx\)
  3.20.65 \(\int \frac {x^4 (3+x^5)}{\sqrt {1+x^5} (-1+a-(1+2 a) x^5+a x^{10})} \, dx\)
  3.20.66 \(\int \frac {1}{\sqrt {-b+a^2 x^2} (c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}})^{2/3}} \, dx\)
  3.20.67 \(\int \frac {1}{\sqrt {-b+a^2 x^2} \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}} \, dx\)
  3.20.68 \(\int \frac {1}{\sqrt [3]{1-3 x^2} (-3+x^2)} \, dx\)
  3.20.69 \(\int \frac {-a+x}{\sqrt [3]{x^2 (-a+x)} (a^2 d-2 a d x+(-1+d) x^2)} \, dx\)
  3.20.70 \(\int \frac {b-a x^4+2 x^8}{\sqrt [4]{-b+a x^4} (b+3 a x^4)} \, dx\)
  3.20.71 \(\int \frac {d+c x^2}{\sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx\)
  3.20.72 \(\int \frac {\sqrt [4]{b x^3+a x^4} (-c+d x^8)}{x^4} \, dx\)
  3.20.73 \(\int \frac {(b^2+a x^2)^{5/2}}{\sqrt {b+\sqrt {b^2+a x^2}}} \, dx\)
  3.20.74 \(\int \sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}} \, dx\)
  3.20.75 \(\int \frac {1+x^6}{\sqrt [3]{x+x^5} (-1+x^6)} \, dx\)
  3.20.76 \(\int \frac {b^8+a^8 x^8}{\sqrt {b^4+a^4 x^4} (-b^8+a^8 x^8)} \, dx\)
  3.20.77 \(\int \frac {\sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}{x \sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}} \, dx\)
  3.20.78 \(\int \sqrt {1+\sqrt {x+\sqrt {1+x^2}}} \, dx\)
  3.20.79 \(\int \frac {2+3 x}{\sqrt [3]{4+3 x^2} (-12+52 x+9 x^2)} \, dx\)
  3.20.80 \(\int (c+b x+a x^2)^{5/2} \, dx\)
  3.20.81 \(\int \frac {1-x^4+x^8}{x^2 (-1+x^4)^{3/4} (-1-x^4+x^8)} \, dx\)
  3.20.82 \(\int \frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\sqrt {x+\sqrt {1+x^2}}} \, dx\)
  3.20.83 \(\int \frac {1}{x \sqrt [3]{4-6 x+3 x^2}} \, dx\)
  3.20.84 \(\int \frac {x^2 \sqrt [4]{b x^3+a x^4}}{-b+a x} \, dx\)
  3.20.85 \(\int \frac {-1+x^6}{\sqrt [3]{x+x^5} (1+x^6)} \, dx\)
  3.20.86 \(\int \frac {\sqrt [4]{-b x^3+a x^4} (-d+c x^8)}{x^4} \, dx\)
  3.20.87 \(\int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{x^6} \, dx\)
  3.20.88 \(\int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{x^6 \sqrt {b^2+a x^2}} \, dx\)
  3.20.89 \(\int \sqrt {d+\sqrt {c+\sqrt {b+a x}}} \, dx\)
  3.20.90 \(\int \frac {1}{(-b+a x^2) \sqrt [3]{-x+x^3}} \, dx\)
  3.20.91 \(\int \frac {b+a x^4}{(-b+a x^4) \sqrt [4]{b^2+c x^4+a^2 x^8}} \, dx\)
  3.20.92 \(\int \frac {2 x^4-x^9}{\sqrt {-1+x^5} (a-a x^5+x^{10})} \, dx\)
  3.20.93 \(\int \frac {1+x^2}{(-1+x^2) \sqrt {x+\sqrt {1+x^2}}} \, dx\)
  3.20.94 \(\int \frac {(-q+p x^2) (a q+b x+a p x^2) \sqrt {q^2+p^2 x^4}}{x^3 (c q+d x+c p x^2)} \, dx\)
  3.20.95 \(\int \frac {x (3+7 x^4)}{\sqrt [3]{1+x^4} (-4+x^3+x^7)} \, dx\)
  3.20.96 \(\int \frac {\sqrt {-b x+a^2 x^2}}{\sqrt {a x^2+x \sqrt {-b x+a^2 x^2}}} \, dx\)
  3.20.97 \(\int \frac {a-2 b+x}{\sqrt [3]{(-a+x) (-b+x)} (a^2+b d-(2 a+d) x+x^2)} \, dx\)
  3.20.98 \(\int \frac {-a (a-2 b)-2 b x+x^2}{((-a+x) (-b+x))^{2/3} (b+a^2 d-(1+2 a d) x+d x^2)} \, dx\)
  3.20.99 \(\int \frac {-1+(2-k) x}{\sqrt [3]{(1-x) x (1-k x)} (1-(b+2 k) x+(b+k^2) x^2)} \, dx\)
  3.20.100 \(\int \frac {-3 k-2 (1+k^2) x+k (1+k^2) x^2+4 k^2 x^3+k^3 x^4}{((1-x^2) (1-k^2 x^2))^{2/3} (-1+d-(2+d) k x-(d+k^2) x^2+d k x^3)} \, dx\)