Integrand size = 24, antiderivative size = 633 \[ \int \frac {1}{\sqrt {x} \left (a+b x^2\right ) \left (c+d x^2\right )^3} \, dx=-\frac {d \sqrt {x}}{4 c (b c-a d) \left (c+d x^2\right )^2}-\frac {d (15 b c-7 a d) \sqrt {x}}{16 c^2 (b c-a d)^2 \left (c+d x^2\right )}-\frac {b^{11/4} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{3/4} (b c-a d)^3}+\frac {b^{11/4} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{3/4} (b c-a d)^3}+\frac {d^{3/4} \left (77 b^2 c^2-66 a b c d+21 a^2 d^2\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{11/4} (b c-a d)^3}-\frac {d^{3/4} \left (77 b^2 c^2-66 a b c d+21 a^2 d^2\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{11/4} (b c-a d)^3}-\frac {b^{11/4} \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{3/4} (b c-a d)^3}+\frac {b^{11/4} \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{3/4} (b c-a d)^3}+\frac {d^{3/4} \left (77 b^2 c^2-66 a b c d+21 a^2 d^2\right ) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{11/4} (b c-a d)^3}-\frac {d^{3/4} \left (77 b^2 c^2-66 a b c d+21 a^2 d^2\right ) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{11/4} (b c-a d)^3} \]
-1/2*b^(11/4)*arctan(1-b^(1/4)*2^(1/2)*x^(1/2)/a^(1/4))/a^(3/4)/(-a*d+b*c) ^3*2^(1/2)+1/2*b^(11/4)*arctan(1+b^(1/4)*2^(1/2)*x^(1/2)/a^(1/4))/a^(3/4)/ (-a*d+b*c)^3*2^(1/2)+1/64*d^(3/4)*(21*a^2*d^2-66*a*b*c*d+77*b^2*c^2)*arcta n(1-d^(1/4)*2^(1/2)*x^(1/2)/c^(1/4))/c^(11/4)/(-a*d+b*c)^3*2^(1/2)-1/64*d^ (3/4)*(21*a^2*d^2-66*a*b*c*d+77*b^2*c^2)*arctan(1+d^(1/4)*2^(1/2)*x^(1/2)/ c^(1/4))/c^(11/4)/(-a*d+b*c)^3*2^(1/2)-1/4*b^(11/4)*ln(a^(1/2)+x*b^(1/2)-a ^(1/4)*b^(1/4)*2^(1/2)*x^(1/2))/a^(3/4)/(-a*d+b*c)^3*2^(1/2)+1/4*b^(11/4)* ln(a^(1/2)+x*b^(1/2)+a^(1/4)*b^(1/4)*2^(1/2)*x^(1/2))/a^(3/4)/(-a*d+b*c)^3 *2^(1/2)+1/128*d^(3/4)*(21*a^2*d^2-66*a*b*c*d+77*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^(11/4)/(-a*d+b*c)^3*2^(1/2)-1/128* d^(3/4)*(21*a^2*d^2-66*a*b*c*d+77*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^(11/4)/(-a*d+b*c)^3*2^(1/2)-1/4*d*x^(1/2)/c/(-a*d +b*c)/(d*x^2+c)^2-1/16*d*(-7*a*d+15*b*c)*x^(1/2)/c^2/(-a*d+b*c)^2/(d*x^2+c )
Time = 1.25 (sec) , antiderivative size = 362, normalized size of antiderivative = 0.57 \[ \int \frac {1}{\sqrt {x} \left (a+b x^2\right ) \left (c+d x^2\right )^3} \, dx=\frac {1}{64} \left (\frac {4 d \sqrt {x} \left (a d \left (11 c+7 d x^2\right )-b c \left (19 c+15 d x^2\right )\right )}{c^2 (b c-a d)^2 \left (c+d x^2\right )^2}+\frac {32 \sqrt {2} b^{11/4} \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )}{a^{3/4} (-b c+a d)^3}+\frac {\sqrt {2} d^{3/4} \left (77 b^2 c^2-66 a b c d+21 a^2 d^2\right ) \arctan \left (\frac {\sqrt {c}-\sqrt {d} x}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}\right )}{c^{11/4} (b c-a d)^3}-\frac {32 \sqrt {2} b^{11/4} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{a^{3/4} (-b c+a d)^3}-\frac {\sqrt {2} d^{3/4} \left (77 b^2 c^2-66 a b c d+21 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}{\sqrt {c}+\sqrt {d} x}\right )}{c^{11/4} (b c-a d)^3}\right ) \]
((4*d*Sqrt[x]*(a*d*(11*c + 7*d*x^2) - b*c*(19*c + 15*d*x^2)))/(c^2*(b*c - a*d)^2*(c + d*x^2)^2) + (32*Sqrt[2]*b^(11/4)*ArcTan[(Sqrt[a] - Sqrt[b]*x)/ (Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])])/(a^(3/4)*(-(b*c) + a*d)^3) + (Sqrt[2]* d^(3/4)*(77*b^2*c^2 - 66*a*b*c*d + 21*a^2*d^2)*ArcTan[(Sqrt[c] - Sqrt[d]*x )/(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])])/(c^(11/4)*(b*c - a*d)^3) - (32*Sqrt[ 2]*b^(11/4)*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x )])/(a^(3/4)*(-(b*c) + a*d)^3) - (Sqrt[2]*d^(3/4)*(77*b^2*c^2 - 66*a*b*c*d + 21*a^2*d^2)*ArcTanh[(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])/(Sqrt[c] + Sqrt[d ]*x)])/(c^(11/4)*(b*c - a*d)^3))/64
Time = 0.90 (sec) , antiderivative size = 590, normalized size of antiderivative = 0.93, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {368, 931, 1024, 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.
| \(\displaystyle \int \frac {1}{\sqrt {x} \left (a+b x^2\right ) \left (c+d x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 368 |
| \(\displaystyle 2 \int \frac {1}{\left (b x^2+a\right ) \left (d x^2+c\right )^3}d\sqrt {x}\) |
| \(\Big \downarrow \) 931 |
| \(\displaystyle 2 \left (\frac {\int \frac {-7 b d x^2+8 b c-7 a d}{\left (b x^2+a\right ) \left (d x^2+c\right )^2}d\sqrt {x}}{8 c (b c-a d)}-\frac {d \sqrt {x}}{8 c \left (c+d x^2\right )^2 (b c-a d)}\right )\) |
| \(\Big \downarrow \) 1024 |
| \(\displaystyle 2 \left (\frac {\frac {\int \frac {32 b^2 c^2-45 a b d c+21 a^2 d^2-3 b d (15 b c-7 a d) x^2}{\left (b x^2+a\right ) \left (d x^2+c\right )}d\sqrt {x}}{4 c (b c-a d)}-\frac {d \sqrt {x} (15 b c-7 a d)}{4 c \left (c+d x^2\right ) (b c-a d)}}{8 c (b c-a d)}-\frac {d \sqrt {x}}{8 c \left (c+d x^2\right )^2 (b c-a d)}\right )\) |
| \(\Big \downarrow \) 1020 |
| \(\displaystyle 2 \left (\frac {\frac {\frac {32 b^3 c^2 \int \frac {1}{b x^2+a}d\sqrt {x}}{b c-a d}-\frac {d \left (21 a^2 d^2-66 a b c d+77 b^2 c^2\right ) \int \frac {1}{d x^2+c}d\sqrt {x}}{b c-a d}}{4 c (b c-a d)}-\frac {d \sqrt {x} (15 b c-7 a d)}{4 c \left (c+d x^2\right ) (b c-a d)}}{8 c (b c-a d)}-\frac {d \sqrt {x}}{8 c \left (c+d x^2\right )^2 (b c-a d)}\right )\) |
| \(\Big \downarrow \) 755 |
| \(\displaystyle 2 \left (\frac {\frac {\frac {32 b^3 c^2 \left (\frac {\int \frac {\sqrt {a}-\sqrt {b} x}{b x^2+a}d\sqrt {x}}{2 \sqrt {a}}+\frac {\int \frac {\sqrt {b} x+\sqrt {a}}{b x^2+a}d\sqrt {x}}{2 \sqrt {a}}\right )}{b c-a d}-\frac {d \left (21 a^2 d^2-66 a b c d+77 b^2 c^2\right ) \left (\frac {\int \frac {\sqrt {c}-\sqrt {d} x}{d x^2+c}d\sqrt {x}}{2 \sqrt {c}}+\frac {\int \frac {\sqrt {d} x+\sqrt {c}}{d x^2+c}d\sqrt {x}}{2 \sqrt {c}}\right )}{b c-a d}}{4 c (b c-a d)}-\frac {d \sqrt {x} (15 b c-7 a d)}{4 c \left (c+d x^2\right ) (b c-a d)}}{8 c (b c-a d)}-\frac {d \sqrt {x}}{8 c \left (c+d x^2\right )^2 (b c-a d)}\right )\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle 2 \left (\frac {\frac {\frac {32 b^3 c^2 \left (\frac {\int \frac {\sqrt {a}-\sqrt {b} x}{b x^2+a}d\sqrt {x}}{2 \sqrt {a}}+\frac {\frac {\int \frac {1}{x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}d\sqrt {x}}{2 \sqrt {b}}+\frac {\int \frac {1}{x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}d\sqrt {x}}{2 \sqrt {b}}}{2 \sqrt {a}}\right )}{b c-a d}-\frac {d \left (21 a^2 d^2-66 a b c d+77 b^2 c^2\right ) \left (\frac {\int \frac {\sqrt {c}-\sqrt {d} x}{d x^2+c}d\sqrt {x}}{2 \sqrt {c}}+\frac {\frac {\int \frac {1}{x-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}}d\sqrt {x}}{2 \sqrt {d}}+\frac {\int \frac {1}{x+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}}d\sqrt {x}}{2 \sqrt {d}}}{2 \sqrt {c}}\right )}{b c-a d}}{4 c (b c-a d)}-\frac {d \sqrt {x} (15 b c-7 a d)}{4 c \left (c+d x^2\right ) (b c-a d)}}{8 c (b c-a d)}-\frac {d \sqrt {x}}{8 c \left (c+d x^2\right )^2 (b c-a d)}\right )\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle 2 \left (\frac {\frac {\frac {32 b^3 c^2 \left (\frac {\int \frac {\sqrt {a}-\sqrt {b} x}{b x^2+a}d\sqrt {x}}{2 \sqrt {a}}+\frac {\frac {\int \frac {1}{-x-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\int \frac {1}{-x-1}d\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{b c-a d}-\frac {d \left (21 a^2 d^2-66 a b c d+77 b^2 c^2\right ) \left (\frac {\int \frac {\sqrt {c}-\sqrt {d} x}{d x^2+c}d\sqrt {x}}{2 \sqrt {c}}+\frac {\frac {\int \frac {1}{-x-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\int \frac {1}{-x-1}d\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}\right )}{b c-a d}}{4 c (b c-a d)}-\frac {d \sqrt {x} (15 b c-7 a d)}{4 c \left (c+d x^2\right ) (b c-a d)}}{8 c (b c-a d)}-\frac {d \sqrt {x}}{8 c \left (c+d x^2\right )^2 (b c-a d)}\right )\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle 2 \left (\frac {\frac {\frac {32 b^3 c^2 \left (\frac {\int \frac {\sqrt {a}-\sqrt {b} x}{b x^2+a}d\sqrt {x}}{2 \sqrt {a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{b c-a d}-\frac {d \left (21 a^2 d^2-66 a b c d+77 b^2 c^2\right ) \left (\frac {\int \frac {\sqrt {c}-\sqrt {d} x}{d x^2+c}d\sqrt {x}}{2 \sqrt {c}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}\right )}{b c-a d}}{4 c (b c-a d)}-\frac {d \sqrt {x} (15 b c-7 a d)}{4 c \left (c+d x^2\right ) (b c-a d)}}{8 c (b c-a d)}-\frac {d \sqrt {x}}{8 c \left (c+d x^2\right )^2 (b c-a d)}\right )\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle 2 \left (\frac {\frac {\frac {32 b^3 c^2 \left (\frac {-\frac {\int -\frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{b} \left (x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{b} \sqrt {x}+\sqrt [4]{a}\right )}{\sqrt [4]{b} \left (x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{b c-a d}-\frac {d \left (21 a^2 d^2-66 a b c d+77 b^2 c^2\right ) \left (\frac {-\frac {\int -\frac {\sqrt {2} \sqrt [4]{c}-2 \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{d} \left (x-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{d} \sqrt {x}+\sqrt [4]{c}\right )}{\sqrt [4]{d} \left (x+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}\right )}{b c-a d}}{4 c (b c-a d)}-\frac {d \sqrt {x} (15 b c-7 a d)}{4 c \left (c+d x^2\right ) (b c-a d)}}{8 c (b c-a d)}-\frac {d \sqrt {x}}{8 c \left (c+d x^2\right )^2 (b c-a d)}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle 2 \left (\frac {\frac {\frac {32 b^3 c^2 \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{b} \left (x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{b} \sqrt {x}+\sqrt [4]{a}\right )}{\sqrt [4]{b} \left (x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{b c-a d}-\frac {d \left (21 a^2 d^2-66 a b c d+77 b^2 c^2\right ) \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{c}-2 \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{d} \left (x-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{d} \sqrt {x}+\sqrt [4]{c}\right )}{\sqrt [4]{d} \left (x+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}\right )}{b c-a d}}{4 c (b c-a d)}-\frac {d \sqrt {x} (15 b c-7 a d)}{4 c \left (c+d x^2\right ) (b c-a d)}}{8 c (b c-a d)}-\frac {d \sqrt {x}}{8 c \left (c+d x^2\right )^2 (b c-a d)}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \left (\frac {\frac {\frac {32 b^3 c^2 \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{b} \sqrt {x}}{x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt {b}}+\frac {\int \frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}+\sqrt [4]{a}}{x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}d\sqrt {x}}{2 \sqrt [4]{a} \sqrt {b}}}{2 \sqrt {a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{b c-a d}-\frac {d \left (21 a^2 d^2-66 a b c d+77 b^2 c^2\right ) \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{c}-2 \sqrt [4]{d} \sqrt {x}}{x-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{c} \sqrt {d}}+\frac {\int \frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}+\sqrt [4]{c}}{x+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}}d\sqrt {x}}{2 \sqrt [4]{c} \sqrt {d}}}{2 \sqrt {c}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}\right )}{b c-a d}}{4 c (b c-a d)}-\frac {d \sqrt {x} (15 b c-7 a d)}{4 c \left (c+d x^2\right ) (b c-a d)}}{8 c (b c-a d)}-\frac {d \sqrt {x}}{8 c \left (c+d x^2\right )^2 (b c-a d)}\right )\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle 2 \left (\frac {\frac {\frac {32 b^3 c^2 \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}+\frac {\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{b c-a d}-\frac {d \left (21 a^2 d^2-66 a b c d+77 b^2 c^2\right ) \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}+\frac {\frac {\log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{2 \sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{2 \sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}\right )}{b c-a d}}{4 c (b c-a d)}-\frac {d \sqrt {x} (15 b c-7 a d)}{4 c \left (c+d x^2\right ) (b c-a d)}}{8 c (b c-a d)}-\frac {d \sqrt {x}}{8 c \left (c+d x^2\right )^2 (b c-a d)}\right )\) |
2*(-1/8*(d*Sqrt[x])/(c*(b*c - a*d)*(c + d*x^2)^2) + (-1/4*(d*(15*b*c - 7*a *d)*Sqrt[x])/(c*(b*c - a*d)*(c + d*x^2)) + ((32*b^3*c^2*((-(ArcTan[1 - (Sq rt[2]*b^(1/4)*Sqrt[x])/a^(1/4)]/(Sqrt[2]*a^(1/4)*b^(1/4))) + ArcTan[1 + (S qrt[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) - (d*(77*b^2*c^ 2 - 66*a*b*c*d + 21*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))/(4*c*(b*c - a*d)))/(8*c*(b*c - a*d)))
3.5.84.3.1 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[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(-b)*x*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*n*(p + 1)*(b*c - a*d))), x] + Simp[1/(a*n*(p + 1)*(b*c - a*d)) Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[b*c + n*(p + 1)*(b*c - a*d) + d*b*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, n, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && !( !IntegerQ[p] && IntegerQ[q] && LtQ[q, -1]) && IntBinomialQ[a, b, c, d, 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[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f _.)*(x_)^(n_)), x_Symbol] :> Simp[(-(b*e - a*f))*x*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*n*(b*c - a*d)*(p + 1))), x] + Simp[1/(a*n*(b*c - a*d)*( p + 1)) Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(b*e - a*f) + e*n*(b *c - a*d)*(p + 1) + d*(b*e - a*f)*(n*(p + q + 2) + 1)*x^n, x], x], x] /; Fr eeQ[{a, b, c, d, e, f, n, q}, x] && LtQ[p, -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 = 2.74 (sec) , antiderivative size = 336, normalized size of antiderivative = 0.53
| method | result | size |
| derivativedivides | \(-\frac {b^{3} \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 \left (a d -b c \right )^{3} a}+\frac {2 d \left (\frac {\frac {d \left (7 a^{2} d^{2}-22 a b c d +15 b^{2} c^{2}\right ) x^{\frac {5}{2}}}{32 c^{2}}+\frac {\left (11 a^{2} d^{2}-30 a b c d +19 b^{2} c^{2}\right ) \sqrt {x}}{32 c}}{\left (d \,x^{2}+c \right )^{2}}+\frac {\left (21 a^{2} d^{2}-66 a b c d +77 b^{2} c^{2}\right ) \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{256 c^{3}}\right )}{\left (a d -b c \right )^{3}}\) | \(336\) |
| default | \(-\frac {b^{3} \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 \left (a d -b c \right )^{3} a}+\frac {2 d \left (\frac {\frac {d \left (7 a^{2} d^{2}-22 a b c d +15 b^{2} c^{2}\right ) x^{\frac {5}{2}}}{32 c^{2}}+\frac {\left (11 a^{2} d^{2}-30 a b c d +19 b^{2} c^{2}\right ) \sqrt {x}}{32 c}}{\left (d \,x^{2}+c \right )^{2}}+\frac {\left (21 a^{2} d^{2}-66 a b c d +77 b^{2} c^{2}\right ) \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{256 c^{3}}\right )}{\left (a d -b c \right )^{3}}\) | \(336\) |
-1/4*b^3/(a*d-b*c)^3*(a/b)^(1/4)/a*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*d/( a*d-b*c)^3*((1/32*d*(7*a^2*d^2-22*a*b*c*d+15*b^2*c^2)/c^2*x^(5/2)+1/32*(11 *a^2*d^2-30*a*b*c*d+19*b^2*c^2)/c*x^(1/2))/(d*x^2+c)^2+1/256*(21*a^2*d^2-6 6*a*b*c*d+77*b^2*c^2)/c^3*(c/d)^(1/4)*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)))
Result contains complex when optimal does not.
Time = 150.89 (sec) , antiderivative size = 5099, normalized size of antiderivative = 8.06 \[ \int \frac {1}{\sqrt {x} \left (a+b x^2\right ) \left (c+d x^2\right )^3} \, dx=\text {Too large to display} \]
Timed out. \[ \int \frac {1}{\sqrt {x} \left (a+b x^2\right ) \left (c+d x^2\right )^3} \, dx=\text {Timed out} \]
Time = 0.28 (sec) , antiderivative size = 675, normalized size of antiderivative = 1.07 \[ \int \frac {1}{\sqrt {x} \left (a+b x^2\right ) \left (c+d x^2\right )^3} \, dx=-\frac {{\left (15 \, b c d^{2} - 7 \, a d^{3}\right )} x^{\frac {5}{2}} + {\left (19 \, b c^{2} d - 11 \, a c d^{2}\right )} \sqrt {x}}{16 \, {\left (b^{2} c^{6} - 2 \, a b c^{5} d + a^{2} c^{4} d^{2} + {\left (b^{2} c^{4} d^{2} - 2 \, a b c^{3} d^{3} + a^{2} c^{2} d^{4}\right )} x^{4} + 2 \, {\left (b^{2} c^{5} d - 2 \, a b c^{4} d^{2} + a^{2} c^{3} d^{3}\right )} x^{2}\right )}} + \frac {\frac {2 \, \sqrt {2} b^{3} \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^{3} \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 {11}{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 {11}{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 (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )}} - \frac {\frac {2 \, \sqrt {2} {\left (77 \, b^{2} c^{2} d - 66 \, a b c d^{2} + 21 \, a^{2} d^{3}\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 (77 \, b^{2} c^{2} d - 66 \, a b c d^{2} + 21 \, a^{2} d^{3}\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 (77 \, b^{2} c^{2} d - 66 \, a b c d^{2} + 21 \, a^{2} d^{3}\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 (77 \, b^{2} c^{2} d - 66 \, a b c d^{2} + 21 \, a^{2} d^{3}\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^{5} - 3 \, a b^{2} c^{4} d + 3 \, a^{2} b c^{3} d^{2} - a^{3} c^{2} d^{3}\right )}} \]
-1/16*((15*b*c*d^2 - 7*a*d^3)*x^(5/2) + (19*b*c^2*d - 11*a*c*d^2)*sqrt(x)) /(b^2*c^6 - 2*a*b*c^5*d + a^2*c^4*d^2 + (b^2*c^4*d^2 - 2*a*b*c^3*d^3 + a^2 *c^2*d^4)*x^4 + 2*(b^2*c^5*d - 2*a*b*c^4*d^2 + a^2*c^3*d^3)*x^2) + 1/4*(2* sqrt(2)*b^3*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))) + 2*sqrt(2)*b^3* arctan(-1/2*sqrt(2)*(sqrt(2)*a^(1/4)*b^(1/4) - 2*sqrt(b)*sqrt(x))/sqrt(sqr t(a)*sqrt(b)))/(sqrt(a)*sqrt(sqrt(a)*sqrt(b))) + sqrt(2)*b^(11/4)*log(sqrt (2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x + sqrt(a))/a^(3/4) - sqrt(2)*b^(11 /4)*log(-sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x + sqrt(a))/a^(3/4))/( b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3) - 1/128*(2*sqrt(2)*(77* b^2*c^2*d - 66*a*b*c*d^2 + 21*a^2*d^3)*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)*(77*b^2*c^2*d - 66*a*b*c*d^2 + 21*a^2*d^3)*arctan(- 1/2*sqrt(2)*(sqrt(2)*c^(1/4)*d^(1/4) - 2*sqrt(d)*sqrt(x))/sqrt(sqrt(c)*sqr t(d)))/(sqrt(c)*sqrt(sqrt(c)*sqrt(d))) + sqrt(2)*(77*b^2*c^2*d - 66*a*b*c* d^2 + 21*a^2*d^3)*log(sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x + sqrt(c ))/(c^(3/4)*d^(1/4)) - sqrt(2)*(77*b^2*c^2*d - 66*a*b*c*d^2 + 21*a^2*d^3)* 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^5 - 3*a*b^2*c^4*d + 3*a^2*b*c^3*d^2 - a^3*c^2*d^3)
Time = 0.44 (sec) , antiderivative size = 960, normalized size of antiderivative = 1.52 \[ \int \frac {1}{\sqrt {x} \left (a+b x^2\right ) \left (c+d x^2\right )^3} \, dx=\text {Too large to display} \]
(a*b^3)^(1/4)*b^2*arctan(1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) + 2*sqrt(x))/(a/ b)^(1/4))/(sqrt(2)*a*b^3*c^3 - 3*sqrt(2)*a^2*b^2*c^2*d + 3*sqrt(2)*a^3*b*c *d^2 - sqrt(2)*a^4*d^3) + (a*b^3)^(1/4)*b^2*arctan(-1/2*sqrt(2)*(sqrt(2)*( a/b)^(1/4) - 2*sqrt(x))/(a/b)^(1/4))/(sqrt(2)*a*b^3*c^3 - 3*sqrt(2)*a^2*b^ 2*c^2*d + 3*sqrt(2)*a^3*b*c*d^2 - sqrt(2)*a^4*d^3) + 1/2*(a*b^3)^(1/4)*b^2 *log(sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(sqrt(2)*a*b^3*c^3 - 3*s qrt(2)*a^2*b^2*c^2*d + 3*sqrt(2)*a^3*b*c*d^2 - sqrt(2)*a^4*d^3) - 1/2*(a*b ^3)^(1/4)*b^2*log(-sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(sqrt(2)*a *b^3*c^3 - 3*sqrt(2)*a^2*b^2*c^2*d + 3*sqrt(2)*a^3*b*c*d^2 - sqrt(2)*a^4*d ^3) - 1/32*(77*(c*d^3)^(1/4)*b^2*c^2 - 66*(c*d^3)^(1/4)*a*b*c*d + 21*(c*d^ 3)^(1/4)*a^2*d^2)*arctan(1/2*sqrt(2)*(sqrt(2)*(c/d)^(1/4) + 2*sqrt(x))/(c/ d)^(1/4))/(sqrt(2)*b^3*c^6 - 3*sqrt(2)*a*b^2*c^5*d + 3*sqrt(2)*a^2*b*c^4*d ^2 - sqrt(2)*a^3*c^3*d^3) - 1/32*(77*(c*d^3)^(1/4)*b^2*c^2 - 66*(c*d^3)^(1 /4)*a*b*c*d + 21*(c*d^3)^(1/4)*a^2*d^2)*arctan(-1/2*sqrt(2)*(sqrt(2)*(c/d) ^(1/4) - 2*sqrt(x))/(c/d)^(1/4))/(sqrt(2)*b^3*c^6 - 3*sqrt(2)*a*b^2*c^5*d + 3*sqrt(2)*a^2*b*c^4*d^2 - sqrt(2)*a^3*c^3*d^3) - 1/64*(77*(c*d^3)^(1/4)* b^2*c^2 - 66*(c*d^3)^(1/4)*a*b*c*d + 21*(c*d^3)^(1/4)*a^2*d^2)*log(sqrt(2) *sqrt(x)*(c/d)^(1/4) + x + sqrt(c/d))/(sqrt(2)*b^3*c^6 - 3*sqrt(2)*a*b^2*c ^5*d + 3*sqrt(2)*a^2*b*c^4*d^2 - sqrt(2)*a^3*c^3*d^3) + 1/64*(77*(c*d^3)^( 1/4)*b^2*c^2 - 66*(c*d^3)^(1/4)*a*b*c*d + 21*(c*d^3)^(1/4)*a^2*d^2)*log...
Time = 9.34 (sec) , antiderivative size = 36997, normalized size of antiderivative = 58.45 \[ \int \frac {1}{\sqrt {x} \left (a+b x^2\right ) \left (c+d x^2\right )^3} \, dx=\text {Too large to display} \]
atan(((-b^11/(16*a^15*d^12 + 16*a^3*b^12*c^12 - 192*a^4*b^11*c^11*d + 1056 *a^5*b^10*c^10*d^2 - 3520*a^6*b^9*c^9*d^3 + 7920*a^7*b^8*c^8*d^4 - 12672*a ^8*b^7*c^7*d^5 + 14784*a^9*b^6*c^6*d^6 - 12672*a^10*b^5*c^5*d^7 + 7920*a^1 1*b^4*c^4*d^8 - 3520*a^12*b^3*c^3*d^9 + 1056*a^13*b^2*c^2*d^10 - 192*a^14* b*c*d^11))^(1/4)*((((194481*a^8*b^8*d^14)/2048 + 1232*b^16*c^8*d^6 - (3479 2593*a*b^15*c^7*d^7)/2048 - (2250423*a^7*b^9*c*d^13)/2048 + (86420247*a^2* b^14*c^6*d^8)/2048 - (106888869*a^3*b^13*c^5*d^9)/2048 + (80271027*a^4*b^1 2*c^4*d^10)/2048 - (38915667*a^5*b^11*c^3*d^11)/2048 + (12127941*a^6*b^10* c^2*d^12)/2048)/(b^8*c^16 + a^8*c^8*d^8 - 8*a^7*b*c^9*d^7 + 28*a^2*b^6*c^1 4*d^2 - 56*a^3*b^5*c^13*d^3 + 70*a^4*b^4*c^12*d^4 - 56*a^5*b^3*c^11*d^5 + 28*a^6*b^2*c^10*d^6 - 8*a*b^7*c^15*d) + ((x^(1/2)*(16777216*b^23*c^23*d^4 - 201326592*a*b^22*c^22*d^5 + 1107296256*a^2*b^21*c^21*d^6 - 3593846784*a^ 3*b^20*c^20*d^7 + 6972506112*a^4*b^19*c^19*d^8 - 4753588224*a^5*b^18*c^18* d^9 - 18397265920*a^6*b^17*c^17*d^10 + 80192667648*a^7*b^16*c^16*d^11 - 18 1503787008*a^8*b^15*c^15*d^12 + 289980416000*a^9*b^14*c^14*d^13 - 35225862 1440*a^10*b^13*c^13*d^14 + 334222688256*a^11*b^12*c^12*d^15 - 249961119744 *a^12*b^11*c^11*d^16 + 147248775168*a^13*b^10*c^10*d^17 - 67718086656*a^14 *b^9*c^9*d^18 + 23871029248*a^15*b^8*c^8*d^19 - 6245842944*a^16*b^7*c^7*d^ 20 + 1146224640*a^17*b^6*c^6*d^21 - 132120576*a^18*b^5*c^5*d^22 + 7225344* a^19*b^4*c^4*d^23))/(4096*(b^12*c^20 + a^12*c^8*d^12 - 12*a^11*b*c^9*d^...