Integrand size = 24, antiderivative size = 681 \[ \int \frac {1}{x^{5/2} \left (a+b x^2\right ) \left (c+d x^2\right )^3} \, dx=-\frac {32 b^2 c^2-133 a b c d+77 a^2 d^2}{48 a c^3 (b c-a d)^2 x^{3/2}}-\frac {d}{4 c (b c-a d) x^{3/2} \left (c+d x^2\right )^2}-\frac {d (19 b c-11 a d)}{16 c^2 (b c-a d)^2 x^{3/2} \left (c+d x^2\right )}+\frac {b^{15/4} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{7/4} (b c-a d)^3}-\frac {b^{15/4} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{7/4} (b c-a d)^3}-\frac {d^{7/4} \left (165 b^2 c^2-210 a b c d+77 a^2 d^2\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{15/4} (b c-a d)^3}+\frac {d^{7/4} \left (165 b^2 c^2-210 a b c d+77 a^2 d^2\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{15/4} (b c-a d)^3}+\frac {b^{15/4} \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{7/4} (b c-a d)^3}-\frac {b^{15/4} \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{7/4} (b c-a d)^3}-\frac {d^{7/4} \left (165 b^2 c^2-210 a b c d+77 a^2 d^2\right ) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{15/4} (b c-a d)^3}+\frac {d^{7/4} \left (165 b^2 c^2-210 a b c d+77 a^2 d^2\right ) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{15/4} (b c-a d)^3} \]
1/48*(-77*a^2*d^2+133*a*b*c*d-32*b^2*c^2)/a/c^3/(-a*d+b*c)^2/x^(3/2)-1/4*d /c/(-a*d+b*c)/x^(3/2)/(d*x^2+c)^2-1/16*d*(-11*a*d+19*b*c)/c^2/(-a*d+b*c)^2 /x^(3/2)/(d*x^2+c)+1/2*b^(15/4)*arctan(1-b^(1/4)*2^(1/2)*x^(1/2)/a^(1/4))/ a^(7/4)/(-a*d+b*c)^3*2^(1/2)-1/2*b^(15/4)*arctan(1+b^(1/4)*2^(1/2)*x^(1/2) /a^(1/4))/a^(7/4)/(-a*d+b*c)^3*2^(1/2)-1/64*d^(7/4)*(77*a^2*d^2-210*a*b*c* d+165*b^2*c^2)*arctan(1-d^(1/4)*2^(1/2)*x^(1/2)/c^(1/4))/c^(15/4)/(-a*d+b* c)^3*2^(1/2)+1/64*d^(7/4)*(77*a^2*d^2-210*a*b*c*d+165*b^2*c^2)*arctan(1+d^ (1/4)*2^(1/2)*x^(1/2)/c^(1/4))/c^(15/4)/(-a*d+b*c)^3*2^(1/2)+1/4*b^(15/4)* ln(a^(1/2)+x*b^(1/2)-a^(1/4)*b^(1/4)*2^(1/2)*x^(1/2))/a^(7/4)/(-a*d+b*c)^3 *2^(1/2)-1/4*b^(15/4)*ln(a^(1/2)+x*b^(1/2)+a^(1/4)*b^(1/4)*2^(1/2)*x^(1/2) )/a^(7/4)/(-a*d+b*c)^3*2^(1/2)-1/128*d^(7/4)*(77*a^2*d^2-210*a*b*c*d+165*b ^2*c^2)*ln(c^(1/2)+x*d^(1/2)-c^(1/4)*d^(1/4)*2^(1/2)*x^(1/2))/c^(15/4)/(-a *d+b*c)^3*2^(1/2)+1/128*d^(7/4)*(77*a^2*d^2-210*a*b*c*d+165*b^2*c^2)*ln(c^ (1/2)+x*d^(1/2)+c^(1/4)*d^(1/4)*2^(1/2)*x^(1/2))/c^(15/4)/(-a*d+b*c)^3*2^( 1/2)
Time = 0.97 (sec) , antiderivative size = 410, normalized size of antiderivative = 0.60 \[ \int \frac {1}{x^{5/2} \left (a+b x^2\right ) \left (c+d x^2\right )^3} \, dx=\frac {1}{192} \left (-\frac {4 \left (32 b^2 c^2 \left (c+d x^2\right )^2+a^2 d^2 \left (32 c^2+121 c d x^2+77 d^2 x^4\right )-a b c d \left (64 c^2+209 c d x^2+133 d^2 x^4\right )\right )}{a c^3 (b c-a d)^2 x^{3/2} \left (c+d x^2\right )^2}-\frac {96 \sqrt {2} b^{15/4} \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )}{a^{7/4} (-b c+a d)^3}-\frac {3 \sqrt {2} d^{7/4} \left (165 b^2 c^2-210 a b c d+77 a^2 d^2\right ) \arctan \left (\frac {\sqrt {c}-\sqrt {d} x}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}\right )}{c^{15/4} (b c-a d)^3}+\frac {96 \sqrt {2} b^{15/4} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{a^{7/4} (-b c+a d)^3}+\frac {3 \sqrt {2} d^{7/4} \left (165 b^2 c^2-210 a b c d+77 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}{\sqrt {c}+\sqrt {d} x}\right )}{c^{15/4} (b c-a d)^3}\right ) \]
((-4*(32*b^2*c^2*(c + d*x^2)^2 + a^2*d^2*(32*c^2 + 121*c*d*x^2 + 77*d^2*x^ 4) - a*b*c*d*(64*c^2 + 209*c*d*x^2 + 133*d^2*x^4)))/(a*c^3*(b*c - a*d)^2*x ^(3/2)*(c + d*x^2)^2) - (96*Sqrt[2]*b^(15/4)*ArcTan[(Sqrt[a] - Sqrt[b]*x)/ (Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])])/(a^(7/4)*(-(b*c) + a*d)^3) - (3*Sqrt[2 ]*d^(7/4)*(165*b^2*c^2 - 210*a*b*c*d + 77*a^2*d^2)*ArcTan[(Sqrt[c] - Sqrt[ d]*x)/(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])])/(c^(15/4)*(b*c - a*d)^3) + (96*S qrt[2]*b^(15/4)*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[ b]*x)])/(a^(7/4)*(-(b*c) + a*d)^3) + (3*Sqrt[2]*d^(7/4)*(165*b^2*c^2 - 210 *a*b*c*d + 77*a^2*d^2)*ArcTanh[(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])/(Sqrt[c] + Sqrt[d]*x)])/(c^(15/4)*(b*c - a*d)^3))/192
Time = 1.03 (sec) , antiderivative size = 634, normalized size of antiderivative = 0.93, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {368, 972, 1049, 1053, 27, 1020, 755, 1476, 1082, 217, 1479, 25, 27, 1103}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {1}{x^{5/2} \left (a+b x^2\right ) \left (c+d x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 368 |
| \(\displaystyle 2 \int \frac {1}{x^2 \left (b x^2+a\right ) \left (d x^2+c\right )^3}d\sqrt {x}\) |
| \(\Big \downarrow \) 972 |
| \(\displaystyle 2 \left (\frac {\int \frac {-11 b d x^2+8 b c-11 a d}{x^2 \left (b x^2+a\right ) \left (d x^2+c\right )^2}d\sqrt {x}}{8 c (b c-a d)}-\frac {d}{8 c x^{3/2} \left (c+d x^2\right )^2 (b c-a d)}\right )\) |
| \(\Big \downarrow \) 1049 |
| \(\displaystyle 2 \left (\frac {\frac {\int \frac {32 b^2 c^2-133 a b d c+77 a^2 d^2-7 b d (19 b c-11 a d) x^2}{x^2 \left (b x^2+a\right ) \left (d x^2+c\right )}d\sqrt {x}}{4 c (b c-a d)}-\frac {d (19 b c-11 a d)}{4 c x^{3/2} \left (c+d x^2\right ) (b c-a d)}}{8 c (b c-a d)}-\frac {d}{8 c x^{3/2} \left (c+d x^2\right )^2 (b c-a d)}\right )\) |
| \(\Big \downarrow \) 1053 |
| \(\displaystyle 2 \left (\frac {\frac {-\frac {\int \frac {3 \left (32 b^3 c^3+32 a b^2 d c^2-133 a^2 b d^2 c+77 a^3 d^3+b d \left (32 b^2 c^2-133 a b d c+77 a^2 d^2\right ) x^2\right )}{\left (b x^2+a\right ) \left (d x^2+c\right )}d\sqrt {x}}{3 a c}-\frac {\frac {32 b^2 c}{a}+\frac {77 a d^2}{c}-133 b d}{3 x^{3/2}}}{4 c (b c-a d)}-\frac {d (19 b c-11 a d)}{4 c x^{3/2} \left (c+d x^2\right ) (b c-a d)}}{8 c (b c-a d)}-\frac {d}{8 c x^{3/2} \left (c+d x^2\right )^2 (b c-a d)}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \left (\frac {\frac {-\frac {\int \frac {32 b^3 c^3+32 a b^2 d c^2-133 a^2 b d^2 c+77 a^3 d^3+b d \left (32 b^2 c^2-133 a b d c+77 a^2 d^2\right ) x^2}{\left (b x^2+a\right ) \left (d x^2+c\right )}d\sqrt {x}}{a c}-\frac {\frac {32 b^2 c}{a}+\frac {77 a d^2}{c}-133 b d}{3 x^{3/2}}}{4 c (b c-a d)}-\frac {d (19 b c-11 a d)}{4 c x^{3/2} \left (c+d x^2\right ) (b c-a d)}}{8 c (b c-a d)}-\frac {d}{8 c x^{3/2} \left (c+d x^2\right )^2 (b c-a d)}\right )\) |
| \(\Big \downarrow \) 1020 |
| \(\displaystyle 2 \left (\frac {\frac {-\frac {\frac {32 b^4 c^3 \int \frac {1}{b x^2+a}d\sqrt {x}}{b c-a d}-\frac {a d^2 \left (77 a^2 d^2-210 a b c d+165 b^2 c^2\right ) \int \frac {1}{d x^2+c}d\sqrt {x}}{b c-a d}}{a c}-\frac {\frac {32 b^2 c}{a}+\frac {77 a d^2}{c}-133 b d}{3 x^{3/2}}}{4 c (b c-a d)}-\frac {d (19 b c-11 a d)}{4 c x^{3/2} \left (c+d x^2\right ) (b c-a d)}}{8 c (b c-a d)}-\frac {d}{8 c x^{3/2} \left (c+d x^2\right )^2 (b c-a d)}\right )\) |
| \(\Big \downarrow \) 755 |
| \(\displaystyle 2 \left (\frac {\frac {-\frac {\frac {32 b^4 c^3 \left (\frac {\int \frac {\sqrt {a}-\sqrt {b} x}{b x^2+a}d\sqrt {x}}{2 \sqrt {a}}+\frac {\int \frac {\sqrt {b} x+\sqrt {a}}{b x^2+a}d\sqrt {x}}{2 \sqrt {a}}\right )}{b c-a d}-\frac {a d^2 \left (77 a^2 d^2-210 a b c d+165 b^2 c^2\right ) \left (\frac {\int \frac {\sqrt {c}-\sqrt {d} x}{d x^2+c}d\sqrt {x}}{2 \sqrt {c}}+\frac {\int \frac {\sqrt {d} x+\sqrt {c}}{d x^2+c}d\sqrt {x}}{2 \sqrt {c}}\right )}{b c-a d}}{a c}-\frac {\frac {32 b^2 c}{a}+\frac {77 a d^2}{c}-133 b d}{3 x^{3/2}}}{4 c (b c-a d)}-\frac {d (19 b c-11 a d)}{4 c x^{3/2} \left (c+d x^2\right ) (b c-a d)}}{8 c (b c-a d)}-\frac {d}{8 c x^{3/2} \left (c+d x^2\right )^2 (b c-a d)}\right )\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle 2 \left (\frac {\frac {-\frac {\frac {32 b^4 c^3 \left (\frac {\int \frac {\sqrt {a}-\sqrt {b} x}{b x^2+a}d\sqrt {x}}{2 \sqrt {a}}+\frac {\frac {\int \frac {1}{x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}d\sqrt {x}}{2 \sqrt {b}}+\frac {\int \frac {1}{x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}d\sqrt {x}}{2 \sqrt {b}}}{2 \sqrt {a}}\right )}{b c-a d}-\frac {a d^2 \left (77 a^2 d^2-210 a b c d+165 b^2 c^2\right ) \left (\frac {\int \frac {\sqrt {c}-\sqrt {d} x}{d x^2+c}d\sqrt {x}}{2 \sqrt {c}}+\frac {\frac {\int \frac {1}{x-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}}d\sqrt {x}}{2 \sqrt {d}}+\frac {\int \frac {1}{x+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}}d\sqrt {x}}{2 \sqrt {d}}}{2 \sqrt {c}}\right )}{b c-a d}}{a c}-\frac {\frac {32 b^2 c}{a}+\frac {77 a d^2}{c}-133 b d}{3 x^{3/2}}}{4 c (b c-a d)}-\frac {d (19 b c-11 a d)}{4 c x^{3/2} \left (c+d x^2\right ) (b c-a d)}}{8 c (b c-a d)}-\frac {d}{8 c x^{3/2} \left (c+d x^2\right )^2 (b c-a d)}\right )\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle 2 \left (\frac {\frac {-\frac {\frac {32 b^4 c^3 \left (\frac {\int \frac {\sqrt {a}-\sqrt {b} x}{b x^2+a}d\sqrt {x}}{2 \sqrt {a}}+\frac {\frac {\int \frac {1}{-x-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\int \frac {1}{-x-1}d\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{b c-a d}-\frac {a d^2 \left (77 a^2 d^2-210 a b c d+165 b^2 c^2\right ) \left (\frac {\int \frac {\sqrt {c}-\sqrt {d} x}{d x^2+c}d\sqrt {x}}{2 \sqrt {c}}+\frac {\frac {\int \frac {1}{-x-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\int \frac {1}{-x-1}d\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}\right )}{b c-a d}}{a c}-\frac {\frac {32 b^2 c}{a}+\frac {77 a d^2}{c}-133 b d}{3 x^{3/2}}}{4 c (b c-a d)}-\frac {d (19 b c-11 a d)}{4 c x^{3/2} \left (c+d x^2\right ) (b c-a d)}}{8 c (b c-a d)}-\frac {d}{8 c x^{3/2} \left (c+d x^2\right )^2 (b c-a d)}\right )\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle 2 \left (\frac {\frac {-\frac {\frac {32 b^4 c^3 \left (\frac {\int \frac {\sqrt {a}-\sqrt {b} x}{b x^2+a}d\sqrt {x}}{2 \sqrt {a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{b c-a d}-\frac {a d^2 \left (77 a^2 d^2-210 a b c d+165 b^2 c^2\right ) \left (\frac {\int \frac {\sqrt {c}-\sqrt {d} x}{d x^2+c}d\sqrt {x}}{2 \sqrt {c}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}\right )}{b c-a d}}{a c}-\frac {\frac {32 b^2 c}{a}+\frac {77 a d^2}{c}-133 b d}{3 x^{3/2}}}{4 c (b c-a d)}-\frac {d (19 b c-11 a d)}{4 c x^{3/2} \left (c+d x^2\right ) (b c-a d)}}{8 c (b c-a d)}-\frac {d}{8 c x^{3/2} \left (c+d x^2\right )^2 (b c-a d)}\right )\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle 2 \left (\frac {\frac {-\frac {\frac {32 c b^2}{a}-133 d b+\frac {77 a d^2}{c}}{3 x^{3/2}}-\frac {\frac {32 b^4 c^3 \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}+\frac {-\frac {\int -\frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{b} \left (x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{b} \sqrt {x}+\sqrt [4]{a}\right )}{\sqrt [4]{b} \left (x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{b c-a d}-\frac {a d^2 \left (165 b^2 c^2-210 a b d c+77 a^2 d^2\right ) \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}+\frac {-\frac {\int -\frac {\sqrt {2} \sqrt [4]{c}-2 \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{d} \left (x-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{d} \sqrt {x}+\sqrt [4]{c}\right )}{\sqrt [4]{d} \left (x+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}\right )}{b c-a d}}{a c}}{4 c (b c-a d)}-\frac {d (19 b c-11 a d)}{4 c (b c-a d) x^{3/2} \left (d x^2+c\right )}}{8 c (b c-a d)}-\frac {d}{8 c (b c-a d) x^{3/2} \left (d x^2+c\right )^2}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle 2 \left (\frac {\frac {-\frac {\frac {32 b^4 c^3 \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{b} \left (x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{b} \sqrt {x}+\sqrt [4]{a}\right )}{\sqrt [4]{b} \left (x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{b c-a d}-\frac {a d^2 \left (77 a^2 d^2-210 a b c d+165 b^2 c^2\right ) \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{c}-2 \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{d} \left (x-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{d} \sqrt {x}+\sqrt [4]{c}\right )}{\sqrt [4]{d} \left (x+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}\right )}{b c-a d}}{a c}-\frac {\frac {32 b^2 c}{a}+\frac {77 a d^2}{c}-133 b d}{3 x^{3/2}}}{4 c (b c-a d)}-\frac {d (19 b c-11 a d)}{4 c x^{3/2} \left (c+d x^2\right ) (b c-a d)}}{8 c (b c-a d)}-\frac {d}{8 c x^{3/2} \left (c+d x^2\right )^2 (b c-a d)}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \left (\frac {\frac {-\frac {\frac {32 b^4 c^3 \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{b} \sqrt {x}}{x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt {b}}+\frac {\int \frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}+\sqrt [4]{a}}{x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}d\sqrt {x}}{2 \sqrt [4]{a} \sqrt {b}}}{2 \sqrt {a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{b c-a d}-\frac {a d^2 \left (77 a^2 d^2-210 a b c d+165 b^2 c^2\right ) \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{c}-2 \sqrt [4]{d} \sqrt {x}}{x-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{c} \sqrt {d}}+\frac {\int \frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}+\sqrt [4]{c}}{x+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}}d\sqrt {x}}{2 \sqrt [4]{c} \sqrt {d}}}{2 \sqrt {c}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}\right )}{b c-a d}}{a c}-\frac {\frac {32 b^2 c}{a}+\frac {77 a d^2}{c}-133 b d}{3 x^{3/2}}}{4 c (b c-a d)}-\frac {d (19 b c-11 a d)}{4 c x^{3/2} \left (c+d x^2\right ) (b c-a d)}}{8 c (b c-a d)}-\frac {d}{8 c x^{3/2} \left (c+d x^2\right )^2 (b c-a d)}\right )\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle 2 \left (\frac {\frac {-\frac {\frac {32 b^4 c^3 \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}+\frac {\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{b c-a d}-\frac {a d^2 \left (77 a^2 d^2-210 a b c d+165 b^2 c^2\right ) \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}+\frac {\frac {\log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{2 \sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{2 \sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}\right )}{b c-a d}}{a c}-\frac {\frac {32 b^2 c}{a}+\frac {77 a d^2}{c}-133 b d}{3 x^{3/2}}}{4 c (b c-a d)}-\frac {d (19 b c-11 a d)}{4 c x^{3/2} \left (c+d x^2\right ) (b c-a d)}}{8 c (b c-a d)}-\frac {d}{8 c x^{3/2} \left (c+d x^2\right )^2 (b c-a d)}\right )\) |
2*(-1/8*d/(c*(b*c - a*d)*x^(3/2)*(c + d*x^2)^2) + (-1/4*(d*(19*b*c - 11*a* d))/(c*(b*c - a*d)*x^(3/2)*(c + d*x^2)) + (-1/3*((32*b^2*c)/a - 133*b*d + (77*a*d^2)/c)/x^(3/2) - ((32*b^4*c^3*((-(ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[ x])/a^(1/4)]/(Sqrt[2]*a^(1/4)*b^(1/4))) + ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt [x])/a^(1/4)]/(Sqrt[2]*a^(1/4)*b^(1/4)))/(2*Sqrt[a]) + (-1/2*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x]/(Sqrt[2]*a^(1/4)*b^(1/4)) + Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x]/(2*Sqrt[2]*a^(1 /4)*b^(1/4)))/(2*Sqrt[a])))/(b*c - a*d) - (a*d^2*(165*b^2*c^2 - 210*a*b*c* d + 77*a^2*d^2)*((-(ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)]/(Sqrt[2] *c^(1/4)*d^(1/4))) + ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)]/(Sqrt[2 ]*c^(1/4)*d^(1/4)))/(2*Sqrt[c]) + (-1/2*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1 /4)*Sqrt[x] + Sqrt[d]*x]/(Sqrt[2]*c^(1/4)*d^(1/4)) + Log[Sqrt[c] + Sqrt[2] *c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x]/(2*Sqrt[2]*c^(1/4)*d^(1/4)))/(2*Sqrt [c])))/(b*c - a*d))/(a*c))/(4*c*(b*c - a*d)))/(8*c*(b*c - a*d)))
3.5.86.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_) , x_Symbol] :> With[{k = Denominator[m]}, Simp[k/e Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(k*2)/e^2))^p*(c + d*(x^(k*2)/e^2))^q, x], x, (e*x)^(1/k)], x]] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b*c - a*d, 0] && FractionQ[m ] && IntegerQ[p]
Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2] ], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*r) Int[(r - s*x^2)/(a + b*x^4) , x], x] + Simp[1/(2*r) Int[(r + s*x^2)/(a + b*x^4), x], x]] /; FreeQ[{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] & & AtomQ[SplitProduct[SumBaseQ, b]]))
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_ ))^(q_), x_Symbol] :> Simp[(-b)*(e*x)^(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x ^n)^(q + 1)/(a*e*n*(b*c - a*d)*(p + 1))), x] + Simp[1/(a*n*(b*c - a*d)*(p + 1)) Int[(e*x)^m*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*b*(m + 1) + n*( b*c - a*d)*(p + 1) + d*b*(m + n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{ a, b, c, d, e, m, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[p, -1] & & IntBinomialQ[a, b, c, d, e, m, n, p, q, x]
Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*((c_) + (d_.)*(x_)^( n_))), x_Symbol] :> Simp[(b*e - a*f)/(b*c - a*d) Int[1/(a + b*x^n), x], x ] - Simp[(d*e - c*f)/(b*c - a*d) Int[1/(c + d*x^n), x], x] /; FreeQ[{a, b , c, d, e, f, n}, x]
Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_ ))^(q_)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> Simp[(-(b*e - a*f))*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*g*n*(b*c - a*d)*(p + 1))) , x] + Simp[1/(a*n*(b*c - a*d)*(p + 1)) Int[(g*x)^m*(a + b*x^n)^(p + 1)*( c + d*x^n)^q*Simp[c*(b*e - a*f)*(m + 1) + e*n*(b*c - a*d)*(p + 1) + d*(b*e - a*f)*(m + n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m, q}, x] && IGtQ[n, 0] && LtQ[p, -1]
Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_ ))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> Simp[e*(g*x)^(m + 1)*(a + b *x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*c*g*(m + 1))), x] + Simp[1/(a*c*g^n*( m + 1)) Int[(g*x)^(m + n)*(a + b*x^n)^p*(c + d*x^n)^q*Simp[a*f*c*(m + 1) - e*(b*c + a*d)*(m + n + 1) - e*n*(b*c*p + a*d*q) - b*e*d*(m + n*(p + q + 2 ) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] && IGtQ[n, 0] && LtQ[m, -1]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b )], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /; Fre eQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 2*(d/e), 2]}, Simp[e/(2*c) Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ e/(2*c) Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ -2*(d/e), 2]}, Simp[e/(2*c*q) Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Simp[e/(2*c*q) Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
Time = 2.82 (sec) , antiderivative size = 348, normalized size of antiderivative = 0.51
| method | result | size |
| derivativedivides | \(\frac {b^{4} \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 a^{2} \left (a d -b c \right )^{3}}-\frac {2}{3 a \,c^{3} x^{\frac {3}{2}}}-\frac {2 d^{2} \left (\frac {\left (\frac {15}{32} a^{2} d^{3}-\frac {19}{16} a b c \,d^{2}+\frac {23}{32} b^{2} c^{2} d \right ) x^{\frac {5}{2}}+\frac {c \left (19 a^{2} d^{2}-46 a b c d +27 b^{2} c^{2}\right ) \sqrt {x}}{32}}{\left (d \,x^{2}+c \right )^{2}}+\frac {\left (77 a^{2} d^{2}-210 a b c d +165 b^{2} c^{2}\right ) \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{256 c}\right )}{\left (a d -b c \right )^{3} c^{3}}\) | \(348\) |
| default | \(\frac {b^{4} \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 a^{2} \left (a d -b c \right )^{3}}-\frac {2}{3 a \,c^{3} x^{\frac {3}{2}}}-\frac {2 d^{2} \left (\frac {\left (\frac {15}{32} a^{2} d^{3}-\frac {19}{16} a b c \,d^{2}+\frac {23}{32} b^{2} c^{2} d \right ) x^{\frac {5}{2}}+\frac {c \left (19 a^{2} d^{2}-46 a b c d +27 b^{2} c^{2}\right ) \sqrt {x}}{32}}{\left (d \,x^{2}+c \right )^{2}}+\frac {\left (77 a^{2} d^{2}-210 a b c d +165 b^{2} c^{2}\right ) \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{256 c}\right )}{\left (a d -b c \right )^{3} c^{3}}\) | \(348\) |
| risch | \(-\frac {2}{3 a \,c^{3} x^{\frac {3}{2}}}-\frac {-\frac {c^{3} b^{4} \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 \left (a d -b c \right )^{3} a}+\frac {2 a \,d^{2} \left (\frac {\left (\frac {15}{32} a^{2} d^{3}-\frac {19}{16} a b c \,d^{2}+\frac {23}{32} b^{2} c^{2} d \right ) x^{\frac {5}{2}}+\frac {c \left (19 a^{2} d^{2}-46 a b c d +27 b^{2} c^{2}\right ) \sqrt {x}}{32}}{\left (d \,x^{2}+c \right )^{2}}+\frac {\left (77 a^{2} d^{2}-210 a b c d +165 b^{2} c^{2}\right ) \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{256 c}\right )}{\left (a d -b c \right )^{3}}}{a \,c^{3}}\) | \(358\) |
1/4/a^2*b^4/(a*d-b*c)^3*(a/b)^(1/4)*2^(1/2)*(ln((x+(a/b)^(1/4)*x^(1/2)*2^( 1/2)+(a/b)^(1/2))/(x-(a/b)^(1/4)*x^(1/2)*2^(1/2)+(a/b)^(1/2)))+2*arctan(2^ (1/2)/(a/b)^(1/4)*x^(1/2)+1)+2*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)-1))-2/3/ a/c^3/x^(3/2)-2*d^2/(a*d-b*c)^3/c^3*(((15/32*a^2*d^3-19/16*a*b*c*d^2+23/32 *b^2*c^2*d)*x^(5/2)+1/32*c*(19*a^2*d^2-46*a*b*c*d+27*b^2*c^2)*x^(1/2))/(d* x^2+c)^2+1/256*(77*a^2*d^2-210*a*b*c*d+165*b^2*c^2)*(c/d)^(1/4)/c*2^(1/2)* (ln((x+(c/d)^(1/4)*x^(1/2)*2^(1/2)+(c/d)^(1/2))/(x-(c/d)^(1/4)*x^(1/2)*2^( 1/2)+(c/d)^(1/2)))+2*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)+1)+2*arctan(2^(1/2 )/(c/d)^(1/4)*x^(1/2)-1)))
Timed out. \[ \int \frac {1}{x^{5/2} \left (a+b x^2\right ) \left (c+d x^2\right )^3} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {1}{x^{5/2} \left (a+b x^2\right ) \left (c+d x^2\right )^3} \, dx=\text {Timed out} \]
Time = 0.29 (sec) , antiderivative size = 755, normalized size of antiderivative = 1.11 \[ \int \frac {1}{x^{5/2} \left (a+b x^2\right ) \left (c+d x^2\right )^3} \, dx=-\frac {32 \, b^{2} c^{4} - 64 \, a b c^{3} d + 32 \, a^{2} c^{2} d^{2} + {\left (32 \, b^{2} c^{2} d^{2} - 133 \, a b c d^{3} + 77 \, a^{2} d^{4}\right )} x^{4} + {\left (64 \, b^{2} c^{3} d - 209 \, a b c^{2} d^{2} + 121 \, a^{2} c d^{3}\right )} x^{2}}{48 \, {\left ({\left (a b^{2} c^{5} d^{2} - 2 \, a^{2} b c^{4} d^{3} + a^{3} c^{3} d^{4}\right )} x^{\frac {11}{2}} + 2 \, {\left (a b^{2} c^{6} d - 2 \, a^{2} b c^{5} d^{2} + a^{3} c^{4} d^{3}\right )} x^{\frac {7}{2}} + {\left (a b^{2} c^{7} - 2 \, a^{2} b c^{6} d + a^{3} c^{5} d^{2}\right )} x^{\frac {3}{2}}\right )}} - \frac {\frac {2 \, \sqrt {2} b^{4} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {2 \, \sqrt {2} b^{4} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {\sqrt {2} b^{\frac {15}{4}} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {3}{4}}} - \frac {\sqrt {2} b^{\frac {15}{4}} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {3}{4}}}}{4 \, {\left (a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2} - a^{4} d^{3}\right )}} + \frac {\frac {2 \, \sqrt {2} {\left (165 \, b^{2} c^{2} d^{2} - 210 \, a b c d^{3} + 77 \, a^{2} d^{4}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} + 2 \, \sqrt {d} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {c} \sqrt {\sqrt {c} \sqrt {d}}} + \frac {2 \, \sqrt {2} {\left (165 \, b^{2} c^{2} d^{2} - 210 \, a b c d^{3} + 77 \, a^{2} d^{4}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} - 2 \, \sqrt {d} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {c} \sqrt {\sqrt {c} \sqrt {d}}} + \frac {\sqrt {2} {\left (165 \, b^{2} c^{2} d^{2} - 210 \, a b c d^{3} + 77 \, a^{2} d^{4}\right )} \log \left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {3}{4}} d^{\frac {1}{4}}} - \frac {\sqrt {2} {\left (165 \, b^{2} c^{2} d^{2} - 210 \, a b c d^{3} + 77 \, a^{2} d^{4}\right )} \log \left (-\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {3}{4}} d^{\frac {1}{4}}}}{128 \, {\left (b^{3} c^{6} - 3 \, a b^{2} c^{5} d + 3 \, a^{2} b c^{4} d^{2} - a^{3} c^{3} d^{3}\right )}} \]
-1/48*(32*b^2*c^4 - 64*a*b*c^3*d + 32*a^2*c^2*d^2 + (32*b^2*c^2*d^2 - 133* a*b*c*d^3 + 77*a^2*d^4)*x^4 + (64*b^2*c^3*d - 209*a*b*c^2*d^2 + 121*a^2*c* d^3)*x^2)/((a*b^2*c^5*d^2 - 2*a^2*b*c^4*d^3 + a^3*c^3*d^4)*x^(11/2) + 2*(a *b^2*c^6*d - 2*a^2*b*c^5*d^2 + a^3*c^4*d^3)*x^(7/2) + (a*b^2*c^7 - 2*a^2*b *c^6*d + a^3*c^5*d^2)*x^(3/2)) - 1/4*(2*sqrt(2)*b^4*arctan(1/2*sqrt(2)*(sq rt(2)*a^(1/4)*b^(1/4) + 2*sqrt(b)*sqrt(x))/sqrt(sqrt(a)*sqrt(b)))/(sqrt(a) *sqrt(sqrt(a)*sqrt(b))) + 2*sqrt(2)*b^4*arctan(-1/2*sqrt(2)*(sqrt(2)*a^(1/ 4)*b^(1/4) - 2*sqrt(b)*sqrt(x))/sqrt(sqrt(a)*sqrt(b)))/(sqrt(a)*sqrt(sqrt( a)*sqrt(b))) + sqrt(2)*b^(15/4)*log(sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt (b)*x + sqrt(a))/a^(3/4) - sqrt(2)*b^(15/4)*log(-sqrt(2)*a^(1/4)*b^(1/4)*s qrt(x) + sqrt(b)*x + sqrt(a))/a^(3/4))/(a*b^3*c^3 - 3*a^2*b^2*c^2*d + 3*a^ 3*b*c*d^2 - a^4*d^3) + 1/128*(2*sqrt(2)*(165*b^2*c^2*d^2 - 210*a*b*c*d^3 + 77*a^2*d^4)*arctan(1/2*sqrt(2)*(sqrt(2)*c^(1/4)*d^(1/4) + 2*sqrt(d)*sqrt( x))/sqrt(sqrt(c)*sqrt(d)))/(sqrt(c)*sqrt(sqrt(c)*sqrt(d))) + 2*sqrt(2)*(16 5*b^2*c^2*d^2 - 210*a*b*c*d^3 + 77*a^2*d^4)*arctan(-1/2*sqrt(2)*(sqrt(2)*c ^(1/4)*d^(1/4) - 2*sqrt(d)*sqrt(x))/sqrt(sqrt(c)*sqrt(d)))/(sqrt(c)*sqrt(s qrt(c)*sqrt(d))) + sqrt(2)*(165*b^2*c^2*d^2 - 210*a*b*c*d^3 + 77*a^2*d^4)* log(sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x + sqrt(c))/(c^(3/4)*d^(1/4 )) - sqrt(2)*(165*b^2*c^2*d^2 - 210*a*b*c*d^3 + 77*a^2*d^4)*log(-sqrt(2)*c ^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x + sqrt(c))/(c^(3/4)*d^(1/4)))/(b^3*c...
Time = 0.46 (sec) , antiderivative size = 995, normalized size of antiderivative = 1.46 \[ \int \frac {1}{x^{5/2} \left (a+b x^2\right ) \left (c+d x^2\right )^3} \, dx=\text {Too large to display} \]
-(a*b^3)^(1/4)*b^3*arctan(1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) + 2*sqrt(x))/(a /b)^(1/4))/(sqrt(2)*a^2*b^3*c^3 - 3*sqrt(2)*a^3*b^2*c^2*d + 3*sqrt(2)*a^4* b*c*d^2 - sqrt(2)*a^5*d^3) - (a*b^3)^(1/4)*b^3*arctan(-1/2*sqrt(2)*(sqrt(2 )*(a/b)^(1/4) - 2*sqrt(x))/(a/b)^(1/4))/(sqrt(2)*a^2*b^3*c^3 - 3*sqrt(2)*a ^3*b^2*c^2*d + 3*sqrt(2)*a^4*b*c*d^2 - sqrt(2)*a^5*d^3) - 1/2*(a*b^3)^(1/4 )*b^3*log(sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(sqrt(2)*a^2*b^3*c^ 3 - 3*sqrt(2)*a^3*b^2*c^2*d + 3*sqrt(2)*a^4*b*c*d^2 - sqrt(2)*a^5*d^3) + 1 /2*(a*b^3)^(1/4)*b^3*log(-sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(sq rt(2)*a^2*b^3*c^3 - 3*sqrt(2)*a^3*b^2*c^2*d + 3*sqrt(2)*a^4*b*c*d^2 - sqrt (2)*a^5*d^3) + 1/32*(165*(c*d^3)^(1/4)*b^2*c^2*d - 210*(c*d^3)^(1/4)*a*b*c *d^2 + 77*(c*d^3)^(1/4)*a^2*d^3)*arctan(1/2*sqrt(2)*(sqrt(2)*(c/d)^(1/4) + 2*sqrt(x))/(c/d)^(1/4))/(sqrt(2)*b^3*c^7 - 3*sqrt(2)*a*b^2*c^6*d + 3*sqrt (2)*a^2*b*c^5*d^2 - sqrt(2)*a^3*c^4*d^3) + 1/32*(165*(c*d^3)^(1/4)*b^2*c^2 *d - 210*(c*d^3)^(1/4)*a*b*c*d^2 + 77*(c*d^3)^(1/4)*a^2*d^3)*arctan(-1/2*s qrt(2)*(sqrt(2)*(c/d)^(1/4) - 2*sqrt(x))/(c/d)^(1/4))/(sqrt(2)*b^3*c^7 - 3 *sqrt(2)*a*b^2*c^6*d + 3*sqrt(2)*a^2*b*c^5*d^2 - sqrt(2)*a^3*c^4*d^3) + 1/ 64*(165*(c*d^3)^(1/4)*b^2*c^2*d - 210*(c*d^3)^(1/4)*a*b*c*d^2 + 77*(c*d^3) ^(1/4)*a^2*d^3)*log(sqrt(2)*sqrt(x)*(c/d)^(1/4) + x + sqrt(c/d))/(sqrt(2)* b^3*c^7 - 3*sqrt(2)*a*b^2*c^6*d + 3*sqrt(2)*a^2*b*c^5*d^2 - sqrt(2)*a^3*c^ 4*d^3) - 1/64*(165*(c*d^3)^(1/4)*b^2*c^2*d - 210*(c*d^3)^(1/4)*a*b*c*d^...
Time = 17.27 (sec) , antiderivative size = 44524, normalized size of antiderivative = 65.38 \[ \int \frac {1}{x^{5/2} \left (a+b x^2\right ) \left (c+d x^2\right )^3} \, dx=\text {Too large to display} \]
atan(((-(35153041*a^8*d^15 + 741200625*b^8*c^8*d^7 - 3773385000*a*b^7*c^7* d^8 + 8587309500*a^2*b^6*c^6*d^9 - 11394999000*a^3*b^5*c^5*d^10 + 96367981 50*a^4*b^4*c^4*d^11 - 5317666200*a^5*b^3*c^3*d^12 + 1870125180*a^6*b^2*c^2 *d^13 - 383487720*a^7*b*c*d^14)/(16777216*b^12*c^27 + 16777216*a^12*c^15*d ^12 - 201326592*a^11*b*c^16*d^11 + 1107296256*a^2*b^10*c^25*d^2 - 36909875 20*a^3*b^9*c^24*d^3 + 8304721920*a^4*b^8*c^23*d^4 - 13287555072*a^5*b^7*c^ 22*d^5 + 15502147584*a^6*b^6*c^21*d^6 - 13287555072*a^7*b^5*c^20*d^7 + 830 4721920*a^8*b^4*c^19*d^8 - 3690987520*a^9*b^3*c^18*d^9 + 1107296256*a^10*b ^2*c^17*d^10 - 201326592*a*b^11*c^26*d))^(1/4)*((-(35153041*a^8*d^15 + 741 200625*b^8*c^8*d^7 - 3773385000*a*b^7*c^7*d^8 + 8587309500*a^2*b^6*c^6*d^9 - 11394999000*a^3*b^5*c^5*d^10 + 9636798150*a^4*b^4*c^4*d^11 - 5317666200 *a^5*b^3*c^3*d^12 + 1870125180*a^6*b^2*c^2*d^13 - 383487720*a^7*b*c*d^14)/ (16777216*b^12*c^27 + 16777216*a^12*c^15*d^12 - 201326592*a^11*b*c^16*d^11 + 1107296256*a^2*b^10*c^25*d^2 - 3690987520*a^3*b^9*c^24*d^3 + 8304721920 *a^4*b^8*c^23*d^4 - 13287555072*a^5*b^7*c^22*d^5 + 15502147584*a^6*b^6*c^2 1*d^6 - 13287555072*a^7*b^5*c^20*d^7 + 8304721920*a^8*b^4*c^19*d^8 - 36909 87520*a^9*b^3*c^18*d^9 + 1107296256*a^10*b^2*c^17*d^10 - 201326592*a*b^11* c^26*d))^(1/4)*((-(35153041*a^8*d^15 + 741200625*b^8*c^8*d^7 - 3773385000* a*b^7*c^7*d^8 + 8587309500*a^2*b^6*c^6*d^9 - 11394999000*a^3*b^5*c^5*d^10 + 9636798150*a^4*b^4*c^4*d^11 - 5317666200*a^5*b^3*c^3*d^12 + 187012518...