3.23.1 \(\int \frac {a (a b+a c-2 b c)-2 (a^2-b c) x+(2 a-b-c) x^2}{((-a+x) (-b+x) (-c+x))^{2/3} (a^2-b c d+(-2 a+b d+c d) x+(1-d) x^2)} \, dx\)
  3.23.2 \(\int \frac {1+x^4}{(-1+x^4) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx\)
  3.23.3 \(\int \frac {(-1+x^2)^2}{(1+x^2)^2 \sqrt {x^2+\sqrt {1+x^4}}} \, dx\)
  3.23.4 \(\int \frac {2 a b^2-b (2 a+b) x+x^3}{\sqrt [3]{x (-a+x) (-b+x)^2} (-a b^2+b (2 a+b) x-(a+2 b+d) x^2+x^3)} \, dx\)
  3.23.5 \(\int \frac {-d+c x}{x \sqrt [3]{-b+a x^3}} \, dx\)
  3.23.6 \(\int \frac {-b^4+c^2 x^2+a^4 x^4}{\sqrt {-b^4+a^4 x^4} (b^4+a^4 x^4)} \, dx\)
  3.23.7 \(\int \frac {x^2}{(1-x^4) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx\)
  3.23.8 \(\int \frac {x (-b+x) (-a^2 b+2 a^2 x+(-2 a+b) x^2)}{(x (-a+x) (-b+x))^{2/3} (-a^4+4 a^3 x+(-6 a^2+b^2 d) x^2+2 (2 a-b d) x^3+(-1+d) x^4)} \, dx\)
  3.23.9 \(\int \frac {(-1+x^3)^{2/3} (8-8 x^3+x^6)}{x^6 (-4+x^3) (-2+x^3)} \, dx\)
  3.23.10 \(\int \frac {(c+(a x+\sqrt {-b+a^2 x^2})^{3/4})^{4/3}}{\sqrt {-b+a^2 x^2}} \, dx\)
  3.23.11 \(\int \frac {1}{(-b+a x) \sqrt [3]{b^3+a^3 x^3}} \, dx\)
  3.23.12 \(\int \frac {\sqrt [3]{a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}}{\sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}} \, dx\)
  3.23.13 \(\int \frac {-b+x}{((-a+x) (-b+x)^2)^{2/3} (b-a d+(-1+d) x)} \, dx\)
  3.23.14 \(\int \frac {(b+a x)^2 (-2 a q+3 b p x^2+a p x^3)}{\sqrt {q+p x^3} (b^4 c+d q^2+4 a b^3 c x+6 a^2 b^2 c x^2+(4 a^3 b c+2 d p q) x^3+a^4 c x^4+d p^2 x^6)} \, dx\)
  3.23.15 \(\int \frac {-1+k^4 x^4}{\sqrt {(1-x) x (1-k^2 x)} (a+b x^2+a k^4 x^4)} \, dx\)
  3.23.16 \(\int \frac {x^5 (-4 a+3 x)}{\sqrt [3]{x^2 (-a+x)} (-a^2+2 a x-x^2+d x^8)} \, dx\)
  3.23.17 \(\int \frac {\sqrt {1-x}}{8 (1+x)^{7/2}} \, dx\)
  3.23.18 \(\int \frac {2 a b^2-b (2 a+b) x+x^3}{\sqrt [3]{x (-a+x) (-b+x)^2} (-a b^2 d+b (2 a+b) d x-(1+a d+2 b d) x^2+d x^3)} \, dx\)
  3.23.19 \(\int \frac {(-x+x^2) \sqrt [4]{-x^3+x^4}}{-1-x+x^2} \, dx\)
  3.23.20 \(\int \frac {1}{\sqrt {1+\sqrt {a x+\sqrt {-b+a^2 x^2}}}} \, dx\)
  3.23.21 \(\int \frac {(-1+x) (-1+k x) (-2 x+(1+k) x^2)}{((1-x) x (1-k x))^{2/3} (b-2 b (1+k) x+(b+4 b k+b k^2) x^2-2 b k (1+k) x^3+(-1+b k^2) x^4)} \, dx\)
  3.23.22 \(\int \frac {(-1+2 (-2+k) x+3 k x^2) (-1+3 x-3 x^2+x^3)}{((1-x) x (1-k x))^{2/3} (-b+(1+5 b) x-(10 b+k) x^2+10 b x^3-5 b x^4+b x^5)} \, dx\)
  3.23.23 \(\int \frac {-b^{12}+a^{12} x^{12}}{\sqrt {-b^4+a^4 x^4} (b^{12}+a^{12} x^{12})} \, dx\)
  3.23.24 \(\int \frac {-a (a-5 b)-(3 a+5 b) x+4 x^2}{\sqrt [3]{(-a+x) (-b+x)} (b-a^5 d-(1-5 a^4 d) x-10 a^3 d x^2+10 a^2 d x^3-5 a d x^4+d x^5)} \, dx\)
  3.23.25 \(\int \frac {a b-(a+b) x+x^2}{((-a+x) (-b+x)^2)^{2/3} (a^2-b^2 d-2 (a-b d) x+(1-d) x^2)} \, dx\)
  3.23.26 \(\int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{(-1+x^4) \sqrt {1+x^4}} \, dx\)
  3.23.27 \(\int \frac {-1+x^6}{\sqrt [3]{-x^2+x^4} (1+x^6)} \, dx\)
  3.23.28 \(\int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{1+x} \, dx\)
  3.23.29 \(\int \frac {b+a x^2}{(b-a x^2) \sqrt [4]{b x^3+a x^5}} \, dx\)
  3.23.30 \(\int \frac {(4+x^3) (1+x^3+x^4)}{\sqrt [4]{1+x^3} (1+2 x^3+x^6+x^8)} \, dx\)
  3.23.31 \(\int \frac {(-b-a x+x^4) \sqrt [4]{b x^3+a x^4}}{-b+a x} \, dx\)
  3.23.32 \(\int \frac {(-4 a+b+3 x) (-b^3+3 b^2 x-3 b x^2+x^3)}{((-a+x) (-b+x)^2)^{2/3} (a+b^4 d-(1+4 b^3 d) x+6 b^2 d x^2-4 b d x^3+d x^4)} \, dx\)
  3.23.33 \(\int \frac {x^2 (-2+(1+k) x)}{\sqrt [3]{(1-x) x (1-k x)} (1-(2+2 k) x+(1+4 k+k^2) x^2-(2 k+2 k^2) x^3+(-b+k^2) x^4)} \, dx\)
  3.23.34 \(\int \frac {1}{c+\sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx\)
  3.23.35 \(\int \frac {(-7+x) \sqrt [3]{1+x-5 x^2+3 x^3}}{(-5+x) (-1+x)^3} \, dx\)
  3.23.36 \(\int \frac {b^2+a^2 x^2}{(-b^2+a^2 x^2)^3 \sqrt [3]{-b x^2+a x^3}} \, dx\)
  3.23.37 \(\int \frac {d+c x}{(-d+c x) \sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx\)
  3.23.38 \(\int \frac {x (-1+k x) (-1+(-1+2 k) x)}{\sqrt [3]{(1-x) x (1-k x)} (-1+4 x+(-6+b) x^2+(4-2 b k) x^3+(-1+b k^2) x^4)} \, dx\)
  3.23.39 \(\int \frac {1+(-2+3 k) x-(k+4 k^2) x^2+3 k^2 x^3}{\sqrt [3]{(1-x) x (1-k x)} (-b+(1+5 b k) x-(1+10 b k^2) x^2+10 b k^3 x^3-5 b k^4 x^4+b k^5 x^5)} \, dx\)
  3.23.40 \(\int \frac {(d+c x)^2}{\sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx\)
  3.23.41 \(\int \frac {1-k x}{(1+(-2+k) x) ((1-x) x (1-k x))^{2/3}} \, dx\)
  3.23.42 \(\int \sqrt {a x+\sqrt {b+a^2 x^2}} \sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}} \, dx\)
  3.23.43 \(\int x^4 \sqrt {b+a^2 x^4} \sqrt {a x^2+\sqrt {b+a^2 x^4}} \, dx\)
  3.23.44 \(\int \frac {x^2}{(b+a x^2) \sqrt [3]{x+x^3}} \, dx\)
  3.23.45 \(\int \frac {1+x^6}{\sqrt [3]{x^2+x^4} (-1+x^6)} \, dx\)
  3.23.46 \(\int \frac {1}{(-b x+a^2 x^2)^{5/2} (a x^2+x \sqrt {-b x+a^2 x^2})^{3/2}} \, dx\)
  3.23.47 \(\int \frac {(a-2 b+x) (-b+x)}{\sqrt [3]{(-a+x) (-b+x)} (a^4-b^2 d-2 (2 a^3-b d) x+(6 a^2-d) x^2-4 a x^3+x^4)} \, dx\)
  3.23.48 \(\int \frac {(-1+x) x (1+(-2+k) x)}{\sqrt [3]{(1-x) x (1-k x)} (1-4 k x+(-b+6 k^2) x^2+(2 b-4 k^3) x^3+(-b+k^4) x^4)} \, dx\)
  3.23.49 \(\int \frac {(-1+x^3)^{2/3} (4+x^6)}{x^6 (-4+x^6)} \, dx\)
  3.23.50 \(\int \frac {\sqrt [6]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt {-b+a^2 x^2}} \, dx\)
  3.23.51 \(\int \frac {(-b+x) (a b-2 b x+x^2)}{(x (-a+x) (-b+x)^2)^{2/3} (b d-(a+d) x+x^2)} \, dx\)
  3.23.52 \(\int \frac {-d+c x^4}{x \sqrt [3]{-b+a x^3}} \, dx\)
  3.23.53 \(\int \frac {a b-2 b x+x^2}{\sqrt [3]{x (-a+x) (-b+x)^2} (b-(1+a d) x+d x^2)} \, dx\)
  3.23.54 \(\int \frac {-a b^2+(4 a-b) b x-3 a x^2+x^3}{\sqrt [3]{x (-a+x) (-b+x)^2} (-a^2+(2 a+b^2 d) x-(1+2 b d) x^2+d x^3)} \, dx\)
  3.23.55 \(\int \frac {(-4 a+b+3 x) (-b^3+3 b^2 x-3 b x^2+x^3)}{((-a+x) (-b+x)^2)^{2/3} (b^4+a d-(4 b^3+d) x+6 b^2 x^2-4 b x^3+x^4)} \, dx\)
  3.23.56 \(\int \frac {\sqrt [4]{-b x^3+a x^4}}{x (-d+c x)} \, dx\)
  3.23.57 \(\int \frac {(-a+x) (-b+x) (-2 a b x+(a+b) x^2)}{(x (-a+x) (-b+x))^{2/3} (a^2 b^2 d-2 a b (a+b) d x+(a^2+4 a b+b^2) d x^2-2 (a+b) d x^3+(-1+d) x^4)} \, dx\)
  3.23.58 \(\int \frac {1+x^3+x^6}{\sqrt [4]{x^3+x^5} (1-x^6)} \, dx\)
  3.23.59 \(\int \frac {-a-b x+(b+a k^2) x^2}{\sqrt {(1-x) x (1-k^2 x)} (1-2 x+k^2 x^2)} \, dx\)
  3.23.60 \(\int \frac {1-2 x^4+x^8}{\sqrt [4]{-1+x^4} (1-2 x^4+2 x^8)} \, dx\)
  3.23.61 \(\int \sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}} \, dx\)
  3.23.62 \(\int \frac {x^3}{\sqrt [3]{x^2 (-a+x)} (-a^4+4 a^3 x-6 a^2 x^2+4 a x^3+(-1+d) x^4)} \, dx\)
  3.23.63 \(\int \frac {\sqrt {c+\sqrt {a x^2+x \sqrt {-b+a^2 x^2}}}}{\sqrt {-b+a^2 x^2}} \, dx\)
  3.23.64 \(\int \frac {-a (a b+a c-2 b c)+2 (a^2-b c) x+(-2 a+b+c) x^2}{((-a+x) (-b+x) (-c+x))^{2/3} (-b c+a^2 d+(b+c-2 a d) x+(-1+d) x^2)} \, dx\)
  3.23.65 \(\int \frac {x^2 (-2 a b+(a+b) x)}{\sqrt [3]{x (-a+x) (-b+x)} (-a^2 b^2+2 a b (a+b) x-(a^2+4 a b+b^2) x^2+2 (a+b) x^3+(-1+d) x^4)} \, dx\)
  3.23.66 \(\int \frac {1+x^6}{\sqrt [4]{x^3+x^5} (1-x^6)} \, dx\)
  3.23.67 \(\int \frac {b^{16}+a^{16} x^{16}}{\sqrt {-b^4+a^4 x^4} (-b^{16}+a^{16} x^{16})} \, dx\)
  3.23.68 \(\int \frac {b^8+x^4+a^8 x^8}{\sqrt {-b^4+a^4 x^4} (-b^8+a^8 x^8)} \, dx\)
  3.23.69 \(\int \frac {-b+x}{((-a+x) (-b+x)^2)^{2/3} (a-b d+(-1+d) x)} \, dx\)
  3.23.70 \(\int \frac {\sqrt {-b+x^2} (c+x^4) \sqrt {x+\sqrt {-b+x^2}}}{x} \, dx\)
  3.23.71 \(\int \frac {x^8}{\sqrt {-b^4+a^4 x^4} (-b^{16}+a^{16} x^{16})} \, dx\)
  3.23.72 \(\int \frac {x^2}{(-b+a x^2) \sqrt [3]{-x+x^3}} \, dx\)
  3.23.73 \(\int \frac {(1+x^4) \sqrt {x+\sqrt {1+x^2}}}{-1+x^4} \, dx\)
  3.23.74 \(\int \frac {(-a+x) (-b+x)}{((-a+x) (-b+x)^2)^{2/3} (-b^2+a^2 d+2 (b-a d) x+(-1+d) x^2)} \, dx\)
  3.23.75 \(\int \frac {a b-(a+b) x+x^2}{((-a+x) (-b+x)^2)^{2/3} (-b^2+a^2 d+2 (b-a d) x+(-1+d) x^2)} \, dx\)
  3.23.76 \(\int \frac {a b-2 b x+x^2}{\sqrt [3]{x (-a+x) (-b+x)^2} (b d-(a+d) x+x^2)} \, dx\)
  3.23.77 \(\int \frac {-a b^2+(4 a-b) b x-3 a x^2+x^3}{\sqrt [3]{x (-a+x) (-b+x)^2} (-a^2 d+(b^2+2 a d) x-(2 b+d) x^2+x^3)} \, dx\)
  3.23.78 \(\int \frac {-d+c x^7}{x \sqrt [3]{-b+a x^3}} \, dx\)
  3.23.79 \(\int \frac {\sqrt {c+\sqrt {b+a x}}}{x-\sqrt {b+a x}} \, dx\)
  3.23.80 \(\int \frac {1+x}{(1+2 x) \sqrt [3]{27+27 x+36 x^2+28 x^3+9 x^4+x^5}} \, dx\)
  3.23.81 \(\int \frac {x (-b+x) (a b+(-2 a+b) x)}{\sqrt [3]{x (-a+x) (-b+x)} (-a^4+4 a^3 x+(-6 a^2+b^2 d) x^2+2 (2 a-b d) x^3+(-1+d) x^4)} \, dx\)
  3.23.82 \(\int \frac {-a b-a c+2 b c+(2 a-b-c) x}{\sqrt [3]{(-a+x) (-b+x) (-c+x)} (-b c+a^2 d+(b+c-2 a d) x+(-1+d) x^2)} \, dx\)
  3.23.83 \(\int \frac {x (-a b+x^2)}{\sqrt [3]{x^2 (-a+x) (-b+x)} (a^2 b^2-2 a b (a+b) x+(a^2+4 a b+b^2-d) x^2-2 (a+b) x^3+x^4)} \, dx\)
  3.23.84 \(\int \frac {-b^6+a^6 x^6}{\sqrt {-b^2 x+a^2 x^3} (b^6+a^6 x^6)} \, dx\)
  3.23.85 \(\int \frac {\sqrt {1+x^4} \sqrt {x^2+\sqrt {1+x^4}}}{-1+x^4} \, dx\)
  3.23.86 \(\int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{(1+x)^2 \sqrt {1+x^4}} \, dx\)
  3.23.87 \(\int \frac {(-b+x) (a b-2 b x+x^2)}{\sqrt [3]{x (-a+x) (-b+x)^2} (-b^2 d+2 b d x-(-a^2+d) x^2-2 a x^3+x^4)} \, dx\)
  3.23.88 \(\int \frac {b^8+a^8 x^8}{\sqrt {-b^4+a^4 x^4} (-b^8+a^8 x^8)} \, dx\)
  3.23.89 \(\int \frac {1}{\sqrt {-1+x^2} (\sqrt {x}+\sqrt {-1+x^2})^2} \, dx\)
  3.23.90 \(\int \frac {b+a x^2}{(d+c x^2) \sqrt [3]{x+x^3}} \, dx\)
  3.23.91 \(\int \frac {x^3 (4 a b-3 (a+b) x+2 x^2)}{(x^2 (-a+x) (-b+x))^{2/3} (-a b+(a+b) x-x^2+d x^4)} \, dx\)
  3.23.92 \(\int (1+x^2)^{3/2} \sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}} \, dx\)
  3.23.93 \(\int \frac {(a-5 b+4 x) (-a^3+3 a^2 x-3 a x^2+x^3)}{((-a+x) (-b+x))^{2/3} (b-a^5 d-(1-5 a^4 d) x-10 a^3 d x^2+10 a^2 d x^3-5 a d x^4+d x^5)} \, dx\)
  3.23.94 \(\int \frac {(2-2 x+2 x^2-3 x^3+3 x^4) \sqrt [3]{-x-x^3-x^4+x^6}}{(1+x) (-1+2 x-2 x^2+x^3) (-1-x^3+x^5)} \, dx\)
  3.23.95 \(\int \frac {1}{(-b+a x) \sqrt [3]{-b^2 x^2+a^3 x^3}} \, dx\)
  3.23.96 \(\int \frac {(-b+x) (-4 a+b+3 x)}{\sqrt [3]{(-a+x) (-b+x)^2} (b^4+a d-(4 b^3+d) x+6 b^2 x^2-4 b x^3+x^4)} \, dx\)
  3.23.97 \(\int \frac {x^4}{\sqrt {-b^4+a^4 x^4} (-b^8+a^8 x^8)} \, dx\)
  3.23.98 \(\int \frac {1}{\sqrt {-b+a^2 x^2} \sqrt {a x+\sqrt {-b+a^2 x^2}} \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}} \, dx\)
  3.23.99 \(\int \frac {x^3 (-b+x) (2 a b-3 a x+x^2)}{\sqrt [3]{x^2 (-a+x) (-b+x)} (-a^4+4 a^3 x-6 a^2 x^2+4 a x^3+(-1+b^2 d) x^4-2 b d x^5+d x^6)} \, dx\)
  3.23.100 \(\int \frac {(1+x^3)^{2/3} (-1+x^6)}{x^6 (-1-2 x^3+2 x^6)} \, dx\)