Integrand size = 28, antiderivative size = 387 \[ \int \frac {\sqrt {d x}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {(d x)^{3/2}}{10 a d \left (a+b x^2\right )^5}+\frac {17 (d x)^{3/2}}{160 a^2 d \left (a+b x^2\right )^4}+\frac {221 (d x)^{3/2}}{1920 a^3 d \left (a+b x^2\right )^3}+\frac {663 (d x)^{3/2}}{5120 a^4 d \left (a+b x^2\right )^2}+\frac {663 (d x)^{3/2}}{4096 a^5 d \left (a+b x^2\right )}-\frac {663 \sqrt {d} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} a^{21/4} b^{3/4}}+\frac {663 \sqrt {d} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} a^{21/4} b^{3/4}}+\frac {663 \sqrt {d} \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{16384 \sqrt {2} a^{21/4} b^{3/4}}-\frac {663 \sqrt {d} \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{16384 \sqrt {2} a^{21/4} b^{3/4}} \]
1/10*(d*x)^(3/2)/a/d/(b*x^2+a)^5+17/160*(d*x)^(3/2)/a^2/d/(b*x^2+a)^4+221/ 1920*(d*x)^(3/2)/a^3/d/(b*x^2+a)^3+663/5120*(d*x)^(3/2)/a^4/d/(b*x^2+a)^2+ 663/4096*(d*x)^(3/2)/a^5/d/(b*x^2+a)-663/16384*arctan(1-b^(1/4)*2^(1/2)*(d *x)^(1/2)/a^(1/4)/d^(1/2))*d^(1/2)/a^(21/4)/b^(3/4)*2^(1/2)+663/16384*arct an(1+b^(1/4)*2^(1/2)*(d*x)^(1/2)/a^(1/4)/d^(1/2))*d^(1/2)/a^(21/4)/b^(3/4) *2^(1/2)+663/32768*ln(a^(1/2)*d^(1/2)+x*b^(1/2)*d^(1/2)-a^(1/4)*b^(1/4)*2^ (1/2)*(d*x)^(1/2))*d^(1/2)/a^(21/4)/b^(3/4)*2^(1/2)-663/32768*ln(a^(1/2)*d ^(1/2)+x*b^(1/2)*d^(1/2)+a^(1/4)*b^(1/4)*2^(1/2)*(d*x)^(1/2))*d^(1/2)/a^(2 1/4)/b^(3/4)*2^(1/2)
Time = 0.39 (sec) , antiderivative size = 183, normalized size of antiderivative = 0.47 \[ \int \frac {\sqrt {d x}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {\sqrt {d x} \left (\frac {4 \sqrt [4]{a} x^{3/2} \left (37645 a^4+84320 a^3 b x^2+90610 a^2 b^2 x^4+47736 a b^3 x^6+9945 b^4 x^8\right )}{\left (a+b x^2\right )^5}-\frac {9945 \sqrt {2} \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )}{b^{3/4}}-\frac {9945 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{b^{3/4}}\right )}{245760 a^{21/4} \sqrt {x}} \]
(Sqrt[d*x]*((4*a^(1/4)*x^(3/2)*(37645*a^4 + 84320*a^3*b*x^2 + 90610*a^2*b^ 2*x^4 + 47736*a*b^3*x^6 + 9945*b^4*x^8))/(a + b*x^2)^5 - (9945*Sqrt[2]*Arc Tan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])])/b^(3/4) - (9 945*Sqrt[2]*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x )])/b^(3/4)))/(245760*a^(21/4)*Sqrt[x])
Time = 0.64 (sec) , antiderivative size = 437, normalized size of antiderivative = 1.13, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.607, Rules used = {1380, 27, 253, 253, 253, 253, 253, 266, 27, 826, 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.
| \(\displaystyle \int \frac {\sqrt {d x}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx\) |
| \(\Big \downarrow \) 1380 |
| \(\displaystyle b^6 \int \frac {\sqrt {d x}}{b^6 \left (b x^2+a\right )^6}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \frac {\sqrt {d x}}{\left (a+b x^2\right )^6}dx\) |
| \(\Big \downarrow \) 253 |
| \(\displaystyle \frac {17 \int \frac {\sqrt {d x}}{\left (b x^2+a\right )^5}dx}{20 a}+\frac {(d x)^{3/2}}{10 a d \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 253 |
| \(\displaystyle \frac {17 \left (\frac {13 \int \frac {\sqrt {d x}}{\left (b x^2+a\right )^4}dx}{16 a}+\frac {(d x)^{3/2}}{8 a d \left (a+b x^2\right )^4}\right )}{20 a}+\frac {(d x)^{3/2}}{10 a d \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 253 |
| \(\displaystyle \frac {17 \left (\frac {13 \left (\frac {3 \int \frac {\sqrt {d x}}{\left (b x^2+a\right )^3}dx}{4 a}+\frac {(d x)^{3/2}}{6 a d \left (a+b x^2\right )^3}\right )}{16 a}+\frac {(d x)^{3/2}}{8 a d \left (a+b x^2\right )^4}\right )}{20 a}+\frac {(d x)^{3/2}}{10 a d \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 253 |
| \(\displaystyle \frac {17 \left (\frac {13 \left (\frac {3 \left (\frac {5 \int \frac {\sqrt {d x}}{\left (b x^2+a\right )^2}dx}{8 a}+\frac {(d x)^{3/2}}{4 a d \left (a+b x^2\right )^2}\right )}{4 a}+\frac {(d x)^{3/2}}{6 a d \left (a+b x^2\right )^3}\right )}{16 a}+\frac {(d x)^{3/2}}{8 a d \left (a+b x^2\right )^4}\right )}{20 a}+\frac {(d x)^{3/2}}{10 a d \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 253 |
| \(\displaystyle \frac {17 \left (\frac {13 \left (\frac {3 \left (\frac {5 \left (\frac {\int \frac {\sqrt {d x}}{b x^2+a}dx}{4 a}+\frac {(d x)^{3/2}}{2 a d \left (a+b x^2\right )}\right )}{8 a}+\frac {(d x)^{3/2}}{4 a d \left (a+b x^2\right )^2}\right )}{4 a}+\frac {(d x)^{3/2}}{6 a d \left (a+b x^2\right )^3}\right )}{16 a}+\frac {(d x)^{3/2}}{8 a d \left (a+b x^2\right )^4}\right )}{20 a}+\frac {(d x)^{3/2}}{10 a d \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {17 \left (\frac {13 \left (\frac {3 \left (\frac {5 \left (\frac {\int \frac {d^3 x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 a d}+\frac {(d x)^{3/2}}{2 a d \left (a+b x^2\right )}\right )}{8 a}+\frac {(d x)^{3/2}}{4 a d \left (a+b x^2\right )^2}\right )}{4 a}+\frac {(d x)^{3/2}}{6 a d \left (a+b x^2\right )^3}\right )}{16 a}+\frac {(d x)^{3/2}}{8 a d \left (a+b x^2\right )^4}\right )}{20 a}+\frac {(d x)^{3/2}}{10 a d \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {17 \left (\frac {13 \left (\frac {3 \left (\frac {5 \left (\frac {d \int \frac {d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 a}+\frac {(d x)^{3/2}}{2 a d \left (a+b x^2\right )}\right )}{8 a}+\frac {(d x)^{3/2}}{4 a d \left (a+b x^2\right )^2}\right )}{4 a}+\frac {(d x)^{3/2}}{6 a d \left (a+b x^2\right )^3}\right )}{16 a}+\frac {(d x)^{3/2}}{8 a d \left (a+b x^2\right )^4}\right )}{20 a}+\frac {(d x)^{3/2}}{10 a d \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 826 |
| \(\displaystyle \frac {17 \left (\frac {13 \left (\frac {3 \left (\frac {5 \left (\frac {d \left (\frac {\int \frac {\sqrt {b} x d+\sqrt {a} d}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {b}}-\frac {\int \frac {\sqrt {a} d-\sqrt {b} d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {b}}\right )}{2 a}+\frac {(d x)^{3/2}}{2 a d \left (a+b x^2\right )}\right )}{8 a}+\frac {(d x)^{3/2}}{4 a d \left (a+b x^2\right )^2}\right )}{4 a}+\frac {(d x)^{3/2}}{6 a d \left (a+b x^2\right )^3}\right )}{16 a}+\frac {(d x)^{3/2}}{8 a d \left (a+b x^2\right )^4}\right )}{20 a}+\frac {(d x)^{3/2}}{10 a d \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {17 \left (\frac {13 \left (\frac {3 \left (\frac {5 \left (\frac {d \left (\frac {\frac {\int \frac {1}{x d+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}}d\sqrt {d x}}{2 \sqrt {b}}+\frac {\int \frac {1}{x d+\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}}d\sqrt {d x}}{2 \sqrt {b}}}{2 \sqrt {b}}-\frac {\int \frac {\sqrt {a} d-\sqrt {b} d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {b}}\right )}{2 a}+\frac {(d x)^{3/2}}{2 a d \left (a+b x^2\right )}\right )}{8 a}+\frac {(d x)^{3/2}}{4 a d \left (a+b x^2\right )^2}\right )}{4 a}+\frac {(d x)^{3/2}}{6 a d \left (a+b x^2\right )^3}\right )}{16 a}+\frac {(d x)^{3/2}}{8 a d \left (a+b x^2\right )^4}\right )}{20 a}+\frac {(d x)^{3/2}}{10 a d \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {17 \left (\frac {13 \left (\frac {3 \left (\frac {5 \left (\frac {d \left (\frac {\frac {\int \frac {1}{-d x-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\int \frac {1}{-d x-1}d\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}}{2 \sqrt {b}}-\frac {\int \frac {\sqrt {a} d-\sqrt {b} d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {b}}\right )}{2 a}+\frac {(d x)^{3/2}}{2 a d \left (a+b x^2\right )}\right )}{8 a}+\frac {(d x)^{3/2}}{4 a d \left (a+b x^2\right )^2}\right )}{4 a}+\frac {(d x)^{3/2}}{6 a d \left (a+b x^2\right )^3}\right )}{16 a}+\frac {(d x)^{3/2}}{8 a d \left (a+b x^2\right )^4}\right )}{20 a}+\frac {(d x)^{3/2}}{10 a d \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {17 \left (\frac {13 \left (\frac {3 \left (\frac {5 \left (\frac {d \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}}{2 \sqrt {b}}-\frac {\int \frac {\sqrt {a} d-\sqrt {b} d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {b}}\right )}{2 a}+\frac {(d x)^{3/2}}{2 a d \left (a+b x^2\right )}\right )}{8 a}+\frac {(d x)^{3/2}}{4 a d \left (a+b x^2\right )^2}\right )}{4 a}+\frac {(d x)^{3/2}}{6 a d \left (a+b x^2\right )^3}\right )}{16 a}+\frac {(d x)^{3/2}}{8 a d \left (a+b x^2\right )^4}\right )}{20 a}+\frac {(d x)^{3/2}}{10 a d \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {17 \left (\frac {13 \left (\frac {3 \left (\frac {5 \left (\frac {d \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}}{2 \sqrt {b}}-\frac {-\frac {\int -\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d}-2 \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{b} \left (x d+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt [4]{a} \sqrt {d}+\sqrt {2} \sqrt [4]{b} \sqrt {d x}\right )}{\sqrt [4]{b} \left (x d+\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}}{2 \sqrt {b}}\right )}{2 a}+\frac {(d x)^{3/2}}{2 a d \left (a+b x^2\right )}\right )}{8 a}+\frac {(d x)^{3/2}}{4 a d \left (a+b x^2\right )^2}\right )}{4 a}+\frac {(d x)^{3/2}}{6 a d \left (a+b x^2\right )^3}\right )}{16 a}+\frac {(d x)^{3/2}}{8 a d \left (a+b x^2\right )^4}\right )}{20 a}+\frac {(d x)^{3/2}}{10 a d \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {17 \left (\frac {13 \left (\frac {3 \left (\frac {5 \left (\frac {d \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}}{2 \sqrt {b}}-\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{a} \sqrt {d}-2 \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{b} \left (x d+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}+\frac {\int \frac {\sqrt {2} \left (\sqrt [4]{a} \sqrt {d}+\sqrt {2} \sqrt [4]{b} \sqrt {d x}\right )}{\sqrt [4]{b} \left (x d+\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}}{2 \sqrt {b}}\right )}{2 a}+\frac {(d x)^{3/2}}{2 a d \left (a+b x^2\right )}\right )}{8 a}+\frac {(d x)^{3/2}}{4 a d \left (a+b x^2\right )^2}\right )}{4 a}+\frac {(d x)^{3/2}}{6 a d \left (a+b x^2\right )^3}\right )}{16 a}+\frac {(d x)^{3/2}}{8 a d \left (a+b x^2\right )^4}\right )}{20 a}+\frac {(d x)^{3/2}}{10 a d \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {17 \left (\frac {13 \left (\frac {3 \left (\frac {5 \left (\frac {d \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}}{2 \sqrt {b}}-\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{a} \sqrt {d}-2 \sqrt [4]{b} \sqrt {d x}}{x d+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt {b} \sqrt {d}}+\frac {\int \frac {\sqrt [4]{a} \sqrt {d}+\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{x d+\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}}d\sqrt {d x}}{2 \sqrt [4]{a} \sqrt {b} \sqrt {d}}}{2 \sqrt {b}}\right )}{2 a}+\frac {(d x)^{3/2}}{2 a d \left (a+b x^2\right )}\right )}{8 a}+\frac {(d x)^{3/2}}{4 a d \left (a+b x^2\right )^2}\right )}{4 a}+\frac {(d x)^{3/2}}{6 a d \left (a+b x^2\right )^3}\right )}{16 a}+\frac {(d x)^{3/2}}{8 a d \left (a+b x^2\right )^4}\right )}{20 a}+\frac {(d x)^{3/2}}{10 a d \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {17 \left (\frac {13 \left (\frac {3 \left (\frac {5 \left (\frac {d \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}}{2 \sqrt {b}}-\frac {\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d} \sqrt {d x}+\sqrt {a} d+\sqrt {b} d x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d} \sqrt {d x}+\sqrt {a} d+\sqrt {b} d x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}}{2 \sqrt {b}}\right )}{2 a}+\frac {(d x)^{3/2}}{2 a d \left (a+b x^2\right )}\right )}{8 a}+\frac {(d x)^{3/2}}{4 a d \left (a+b x^2\right )^2}\right )}{4 a}+\frac {(d x)^{3/2}}{6 a d \left (a+b x^2\right )^3}\right )}{16 a}+\frac {(d x)^{3/2}}{8 a d \left (a+b x^2\right )^4}\right )}{20 a}+\frac {(d x)^{3/2}}{10 a d \left (a+b x^2\right )^5}\) |
(d*x)^(3/2)/(10*a*d*(a + b*x^2)^5) + (17*((d*x)^(3/2)/(8*a*d*(a + b*x^2)^4 ) + (13*((d*x)^(3/2)/(6*a*d*(a + b*x^2)^3) + (3*((d*x)^(3/2)/(4*a*d*(a + b *x^2)^2) + (5*((d*x)^(3/2)/(2*a*d*(a + b*x^2)) + (d*((-(ArcTan[1 - (Sqrt[2 ]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])]/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d])) + ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])]/(Sqrt[2]*a^(1 /4)*b^(1/4)*Sqrt[d]))/(2*Sqrt[b]) - (-1/2*Log[Sqrt[a]*d + Sqrt[b]*d*x - Sq rt[2]*a^(1/4)*b^(1/4)*Sqrt[d]*Sqrt[d*x]]/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d]) + Log[Sqrt[a]*d + Sqrt[b]*d*x + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d]*Sqrt[d*x] ]/(2*Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d]))/(2*Sqrt[b])))/(2*a)))/(8*a)))/(4*a) ))/(16*a)))/(20*a)
3.8.23.3.1 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[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(c*x )^(m + 1))*((a + b*x^2)^(p + 1)/(2*a*c*(p + 1))), x] + Simp[(m + 2*p + 3)/( 2*a*(p + 1)) Int[(c*x)^m*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, m }, x] && LtQ[p, -1] && IntBinomialQ[a, b, c, 2, m, p, x]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{k = De nominator[m]}, Simp[k/c Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(2*k)/c^2)) ^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && FractionQ[m] && I ntBinomialQ[a, b, c, 2, m, p, x]
Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*s) Int[(r + s*x^2)/(a + b*x^ 4), x], x] - Simp[1/(2*s) 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_) + (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[(u_)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> S imp[1/c^p Int[u*(b/2 + c*x^n)^(2*p), x], x] /; FreeQ[{a, b, c, n, p}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]
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 = 0.42 (sec) , antiderivative size = 244, normalized size of antiderivative = 0.63
| method | result | size |
| derivativedivides | \(2 d^{11} \left (\frac {\frac {7529 \left (d x \right )^{\frac {3}{2}}}{24576 a \,d^{2}}+\frac {527 b \left (d x \right )^{\frac {7}{2}}}{768 a^{2} d^{4}}+\frac {9061 b^{2} \left (d x \right )^{\frac {11}{2}}}{12288 a^{3} d^{6}}+\frac {1989 b^{3} \left (d x \right )^{\frac {15}{2}}}{5120 a^{4} d^{8}}+\frac {663 b^{4} \left (d x \right )^{\frac {19}{2}}}{8192 a^{5} d^{10}}}{\left (b \,d^{2} x^{2}+a \,d^{2}\right )^{5}}+\frac {663 \sqrt {2}\, \left (\ln \left (\frac {d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{65536 a^{5} d^{10} b \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )\) | \(244\) |
| default | \(2 d^{11} \left (\frac {\frac {7529 \left (d x \right )^{\frac {3}{2}}}{24576 a \,d^{2}}+\frac {527 b \left (d x \right )^{\frac {7}{2}}}{768 a^{2} d^{4}}+\frac {9061 b^{2} \left (d x \right )^{\frac {11}{2}}}{12288 a^{3} d^{6}}+\frac {1989 b^{3} \left (d x \right )^{\frac {15}{2}}}{5120 a^{4} d^{8}}+\frac {663 b^{4} \left (d x \right )^{\frac {19}{2}}}{8192 a^{5} d^{10}}}{\left (b \,d^{2} x^{2}+a \,d^{2}\right )^{5}}+\frac {663 \sqrt {2}\, \left (\ln \left (\frac {d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{65536 a^{5} d^{10} b \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )\) | \(244\) |
| pseudoelliptic | \(\frac {60232 \left (\frac {1989}{7529} b^{4} x^{8}+\frac {47736}{37645} a \,b^{3} x^{6}+\frac {18122}{7529} a^{2} b^{2} x^{4}+\frac {16864}{7529} a^{3} b \,x^{2}+a^{4}\right ) x b \sqrt {d x}\, \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}+1989 \sqrt {2}\, d \left (b \,x^{2}+a \right )^{5} \left (2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}-\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}+\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )+\ln \left (\frac {d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )\right )}{98304 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} b \,a^{5} \left (b \,x^{2}+a \right )^{5}}\) | \(246\) |
2*d^11*((7529/24576/a/d^2*(d*x)^(3/2)+527/768/a^2/d^4*b*(d*x)^(7/2)+9061/1 2288/a^3/d^6*b^2*(d*x)^(11/2)+1989/5120/a^4/d^8*b^3*(d*x)^(15/2)+663/8192/ a^5/d^10*b^4*(d*x)^(19/2))/(b*d^2*x^2+a*d^2)^5+663/65536/a^5/d^10/b/(a*d^2 /b)^(1/4)*2^(1/2)*(ln((d*x-(a*d^2/b)^(1/4)*(d*x)^(1/2)*2^(1/2)+(a*d^2/b)^( 1/2))/(d*x+(a*d^2/b)^(1/4)*(d*x)^(1/2)*2^(1/2)+(a*d^2/b)^(1/2)))+2*arctan( 2^(1/2)/(a*d^2/b)^(1/4)*(d*x)^(1/2)+1)+2*arctan(2^(1/2)/(a*d^2/b)^(1/4)*(d *x)^(1/2)-1)))
Result contains complex when optimal does not.
Time = 0.27 (sec) , antiderivative size = 526, normalized size of antiderivative = 1.36 \[ \int \frac {\sqrt {d x}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {9945 \, {\left (a^{5} b^{5} x^{10} + 5 \, a^{6} b^{4} x^{8} + 10 \, a^{7} b^{3} x^{6} + 10 \, a^{8} b^{2} x^{4} + 5 \, a^{9} b x^{2} + a^{10}\right )} \left (-\frac {d^{2}}{a^{21} b^{3}}\right )^{\frac {1}{4}} \log \left (291434247 \, a^{16} b^{2} \left (-\frac {d^{2}}{a^{21} b^{3}}\right )^{\frac {3}{4}} + 291434247 \, \sqrt {d x} d\right ) - 9945 \, {\left (i \, a^{5} b^{5} x^{10} + 5 i \, a^{6} b^{4} x^{8} + 10 i \, a^{7} b^{3} x^{6} + 10 i \, a^{8} b^{2} x^{4} + 5 i \, a^{9} b x^{2} + i \, a^{10}\right )} \left (-\frac {d^{2}}{a^{21} b^{3}}\right )^{\frac {1}{4}} \log \left (291434247 i \, a^{16} b^{2} \left (-\frac {d^{2}}{a^{21} b^{3}}\right )^{\frac {3}{4}} + 291434247 \, \sqrt {d x} d\right ) - 9945 \, {\left (-i \, a^{5} b^{5} x^{10} - 5 i \, a^{6} b^{4} x^{8} - 10 i \, a^{7} b^{3} x^{6} - 10 i \, a^{8} b^{2} x^{4} - 5 i \, a^{9} b x^{2} - i \, a^{10}\right )} \left (-\frac {d^{2}}{a^{21} b^{3}}\right )^{\frac {1}{4}} \log \left (-291434247 i \, a^{16} b^{2} \left (-\frac {d^{2}}{a^{21} b^{3}}\right )^{\frac {3}{4}} + 291434247 \, \sqrt {d x} d\right ) - 9945 \, {\left (a^{5} b^{5} x^{10} + 5 \, a^{6} b^{4} x^{8} + 10 \, a^{7} b^{3} x^{6} + 10 \, a^{8} b^{2} x^{4} + 5 \, a^{9} b x^{2} + a^{10}\right )} \left (-\frac {d^{2}}{a^{21} b^{3}}\right )^{\frac {1}{4}} \log \left (-291434247 \, a^{16} b^{2} \left (-\frac {d^{2}}{a^{21} b^{3}}\right )^{\frac {3}{4}} + 291434247 \, \sqrt {d x} d\right ) + 4 \, {\left (9945 \, b^{4} x^{9} + 47736 \, a b^{3} x^{7} + 90610 \, a^{2} b^{2} x^{5} + 84320 \, a^{3} b x^{3} + 37645 \, a^{4} x\right )} \sqrt {d x}}{245760 \, {\left (a^{5} b^{5} x^{10} + 5 \, a^{6} b^{4} x^{8} + 10 \, a^{7} b^{3} x^{6} + 10 \, a^{8} b^{2} x^{4} + 5 \, a^{9} b x^{2} + a^{10}\right )}} \]
1/245760*(9945*(a^5*b^5*x^10 + 5*a^6*b^4*x^8 + 10*a^7*b^3*x^6 + 10*a^8*b^2 *x^4 + 5*a^9*b*x^2 + a^10)*(-d^2/(a^21*b^3))^(1/4)*log(291434247*a^16*b^2* (-d^2/(a^21*b^3))^(3/4) + 291434247*sqrt(d*x)*d) - 9945*(I*a^5*b^5*x^10 + 5*I*a^6*b^4*x^8 + 10*I*a^7*b^3*x^6 + 10*I*a^8*b^2*x^4 + 5*I*a^9*b*x^2 + I* a^10)*(-d^2/(a^21*b^3))^(1/4)*log(291434247*I*a^16*b^2*(-d^2/(a^21*b^3))^( 3/4) + 291434247*sqrt(d*x)*d) - 9945*(-I*a^5*b^5*x^10 - 5*I*a^6*b^4*x^8 - 10*I*a^7*b^3*x^6 - 10*I*a^8*b^2*x^4 - 5*I*a^9*b*x^2 - I*a^10)*(-d^2/(a^21* b^3))^(1/4)*log(-291434247*I*a^16*b^2*(-d^2/(a^21*b^3))^(3/4) + 291434247* sqrt(d*x)*d) - 9945*(a^5*b^5*x^10 + 5*a^6*b^4*x^8 + 10*a^7*b^3*x^6 + 10*a^ 8*b^2*x^4 + 5*a^9*b*x^2 + a^10)*(-d^2/(a^21*b^3))^(1/4)*log(-291434247*a^1 6*b^2*(-d^2/(a^21*b^3))^(3/4) + 291434247*sqrt(d*x)*d) + 4*(9945*b^4*x^9 + 47736*a*b^3*x^7 + 90610*a^2*b^2*x^5 + 84320*a^3*b*x^3 + 37645*a^4*x)*sqrt (d*x))/(a^5*b^5*x^10 + 5*a^6*b^4*x^8 + 10*a^7*b^3*x^6 + 10*a^8*b^2*x^4 + 5 *a^9*b*x^2 + a^10)
\[ \int \frac {\sqrt {d x}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\int \frac {\sqrt {d x}}{\left (a + b x^{2}\right )^{6}}\, dx \]
Time = 0.30 (sec) , antiderivative size = 377, normalized size of antiderivative = 0.97 \[ \int \frac {\sqrt {d x}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {\frac {8 \, {\left (9945 \, \left (d x\right )^{\frac {19}{2}} b^{4} d^{2} + 47736 \, \left (d x\right )^{\frac {15}{2}} a b^{3} d^{4} + 90610 \, \left (d x\right )^{\frac {11}{2}} a^{2} b^{2} d^{6} + 84320 \, \left (d x\right )^{\frac {7}{2}} a^{3} b d^{8} + 37645 \, \left (d x\right )^{\frac {3}{2}} a^{4} d^{10}\right )}}{a^{5} b^{5} d^{10} x^{10} + 5 \, a^{6} b^{4} d^{10} x^{8} + 10 \, a^{7} b^{3} d^{10} x^{6} + 10 \, a^{8} b^{2} d^{10} x^{4} + 5 \, a^{9} b d^{10} x^{2} + a^{10} d^{10}} + \frac {9945 \, d^{2} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {d x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} d}}\right )}{\sqrt {\sqrt {a} \sqrt {b} d} \sqrt {b}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {d x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} d}}\right )}{\sqrt {\sqrt {a} \sqrt {b} d} \sqrt {b}} - \frac {\sqrt {2} \log \left (\sqrt {b} d x + \sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} \sqrt {d x} b^{\frac {1}{4}} + \sqrt {a} d\right )}{\left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (\sqrt {b} d x - \sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} \sqrt {d x} b^{\frac {1}{4}} + \sqrt {a} d\right )}{\left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {3}{4}}}\right )}}{a^{5}}}{491520 \, d} \]
1/491520*(8*(9945*(d*x)^(19/2)*b^4*d^2 + 47736*(d*x)^(15/2)*a*b^3*d^4 + 90 610*(d*x)^(11/2)*a^2*b^2*d^6 + 84320*(d*x)^(7/2)*a^3*b*d^8 + 37645*(d*x)^( 3/2)*a^4*d^10)/(a^5*b^5*d^10*x^10 + 5*a^6*b^4*d^10*x^8 + 10*a^7*b^3*d^10*x ^6 + 10*a^8*b^2*d^10*x^4 + 5*a^9*b*d^10*x^2 + a^10*d^10) + 9945*d^2*(2*sqr t(2)*arctan(1/2*sqrt(2)*(sqrt(2)*(a*d^2)^(1/4)*b^(1/4) + 2*sqrt(d*x)*sqrt( b))/sqrt(sqrt(a)*sqrt(b)*d))/(sqrt(sqrt(a)*sqrt(b)*d)*sqrt(b)) + 2*sqrt(2) *arctan(-1/2*sqrt(2)*(sqrt(2)*(a*d^2)^(1/4)*b^(1/4) - 2*sqrt(d*x)*sqrt(b)) /sqrt(sqrt(a)*sqrt(b)*d))/(sqrt(sqrt(a)*sqrt(b)*d)*sqrt(b)) - sqrt(2)*log( sqrt(b)*d*x + sqrt(2)*(a*d^2)^(1/4)*sqrt(d*x)*b^(1/4) + sqrt(a)*d)/((a*d^2 )^(1/4)*b^(3/4)) + sqrt(2)*log(sqrt(b)*d*x - sqrt(2)*(a*d^2)^(1/4)*sqrt(d* x)*b^(1/4) + sqrt(a)*d)/((a*d^2)^(1/4)*b^(3/4)))/a^5)/d
Time = 0.31 (sec) , antiderivative size = 340, normalized size of antiderivative = 0.88 \[ \int \frac {\sqrt {d x}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {\frac {19890 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{a^{6} b^{3}} + \frac {19890 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{a^{6} b^{3}} - \frac {9945 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \log \left (d x + \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{a^{6} b^{3}} + \frac {9945 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \log \left (d x - \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{a^{6} b^{3}} + \frac {8 \, {\left (9945 \, \sqrt {d x} b^{4} d^{11} x^{9} + 47736 \, \sqrt {d x} a b^{3} d^{11} x^{7} + 90610 \, \sqrt {d x} a^{2} b^{2} d^{11} x^{5} + 84320 \, \sqrt {d x} a^{3} b d^{11} x^{3} + 37645 \, \sqrt {d x} a^{4} d^{11} x\right )}}{{\left (b d^{2} x^{2} + a d^{2}\right )}^{5} a^{5}}}{491520 \, d} \]
1/491520*(19890*sqrt(2)*(a*b^3*d^2)^(3/4)*arctan(1/2*sqrt(2)*(sqrt(2)*(a*d ^2/b)^(1/4) + 2*sqrt(d*x))/(a*d^2/b)^(1/4))/(a^6*b^3) + 19890*sqrt(2)*(a*b ^3*d^2)^(3/4)*arctan(-1/2*sqrt(2)*(sqrt(2)*(a*d^2/b)^(1/4) - 2*sqrt(d*x))/ (a*d^2/b)^(1/4))/(a^6*b^3) - 9945*sqrt(2)*(a*b^3*d^2)^(3/4)*log(d*x + sqrt (2)*(a*d^2/b)^(1/4)*sqrt(d*x) + sqrt(a*d^2/b))/(a^6*b^3) + 9945*sqrt(2)*(a *b^3*d^2)^(3/4)*log(d*x - sqrt(2)*(a*d^2/b)^(1/4)*sqrt(d*x) + sqrt(a*d^2/b ))/(a^6*b^3) + 8*(9945*sqrt(d*x)*b^4*d^11*x^9 + 47736*sqrt(d*x)*a*b^3*d^11 *x^7 + 90610*sqrt(d*x)*a^2*b^2*d^11*x^5 + 84320*sqrt(d*x)*a^3*b*d^11*x^3 + 37645*sqrt(d*x)*a^4*d^11*x)/((b*d^2*x^2 + a*d^2)^5*a^5))/d
Time = 14.01 (sec) , antiderivative size = 210, normalized size of antiderivative = 0.54 \[ \int \frac {\sqrt {d x}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {\frac {7529\,d^9\,{\left (d\,x\right )}^{3/2}}{12288\,a}+\frac {9061\,b^2\,d^5\,{\left (d\,x\right )}^{11/2}}{6144\,a^3}+\frac {1989\,b^3\,d^3\,{\left (d\,x\right )}^{15/2}}{2560\,a^4}+\frac {527\,b\,d^7\,{\left (d\,x\right )}^{7/2}}{384\,a^2}+\frac {663\,b^4\,d\,{\left (d\,x\right )}^{19/2}}{4096\,a^5}}{a^5\,d^{10}+5\,a^4\,b\,d^{10}\,x^2+10\,a^3\,b^2\,d^{10}\,x^4+10\,a^2\,b^3\,d^{10}\,x^6+5\,a\,b^4\,d^{10}\,x^8+b^5\,d^{10}\,x^{10}}-\frac {663\,\sqrt {d}\,\mathrm {atan}\left (\frac {b^{1/4}\,\sqrt {d\,x}}{{\left (-a\right )}^{1/4}\,\sqrt {d}}\right )}{8192\,{\left (-a\right )}^{21/4}\,b^{3/4}}+\frac {663\,\sqrt {d}\,\mathrm {atanh}\left (\frac {b^{1/4}\,\sqrt {d\,x}}{{\left (-a\right )}^{1/4}\,\sqrt {d}}\right )}{8192\,{\left (-a\right )}^{21/4}\,b^{3/4}} \]
((7529*d^9*(d*x)^(3/2))/(12288*a) + (9061*b^2*d^5*(d*x)^(11/2))/(6144*a^3) + (1989*b^3*d^3*(d*x)^(15/2))/(2560*a^4) + (527*b*d^7*(d*x)^(7/2))/(384*a ^2) + (663*b^4*d*(d*x)^(19/2))/(4096*a^5))/(a^5*d^10 + b^5*d^10*x^10 + 5*a ^4*b*d^10*x^2 + 5*a*b^4*d^10*x^8 + 10*a^3*b^2*d^10*x^4 + 10*a^2*b^3*d^10*x ^6) - (663*d^(1/2)*atan((b^(1/4)*(d*x)^(1/2))/((-a)^(1/4)*d^(1/2))))/(8192 *(-a)^(21/4)*b^(3/4)) + (663*d^(1/2)*atanh((b^(1/4)*(d*x)^(1/2))/((-a)^(1/ 4)*d^(1/2))))/(8192*(-a)^(21/4)*b^(3/4))