Integrand size = 28, antiderivative size = 387 \[ \int \frac {1}{\sqrt {d x} \left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {\sqrt {d x}}{10 a d \left (a+b x^2\right )^5}+\frac {19 \sqrt {d x}}{160 a^2 d \left (a+b x^2\right )^4}+\frac {19 \sqrt {d x}}{128 a^3 d \left (a+b x^2\right )^3}+\frac {209 \sqrt {d x}}{1024 a^4 d \left (a+b x^2\right )^2}+\frac {1463 \sqrt {d x}}{4096 a^5 d \left (a+b x^2\right )}-\frac {4389 \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} a^{23/4} \sqrt [4]{b} \sqrt {d}}+\frac {4389 \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} a^{23/4} \sqrt [4]{b} \sqrt {d}}-\frac {4389 \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{16384 \sqrt {2} a^{23/4} \sqrt [4]{b} \sqrt {d}}+\frac {4389 \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{16384 \sqrt {2} a^{23/4} \sqrt [4]{b} \sqrt {d}} \]
-4389/16384*arctan(1-b^(1/4)*2^(1/2)*(d*x)^(1/2)/a^(1/4)/d^(1/2))/a^(23/4) /b^(1/4)*2^(1/2)/d^(1/2)+4389/16384*arctan(1+b^(1/4)*2^(1/2)*(d*x)^(1/2)/a ^(1/4)/d^(1/2))/a^(23/4)/b^(1/4)*2^(1/2)/d^(1/2)-4389/32768*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))/a^(23/4)/b^(1/ 4)*2^(1/2)/d^(1/2)+4389/32768*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))/a^(23/4)/b^(1/4)*2^(1/2)/d^(1/2)+1/10*(d*x)^ (1/2)/a/d/(b*x^2+a)^5+19/160*(d*x)^(1/2)/a^2/d/(b*x^2+a)^4+19/128*(d*x)^(1 /2)/a^3/d/(b*x^2+a)^3+209/1024*(d*x)^(1/2)/a^4/d/(b*x^2+a)^2+1463/4096*(d* x)^(1/2)/a^5/d/(b*x^2+a)
Time = 0.37 (sec) , antiderivative size = 183, normalized size of antiderivative = 0.47 \[ \int \frac {1}{\sqrt {d x} \left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {\sqrt {x} \left (\frac {4 a^{3/4} \sqrt {x} \left (19015 a^4+50312 a^3 b x^2+59470 a^2 b^2 x^4+33440 a b^3 x^6+7315 b^4 x^8\right )}{\left (a+b x^2\right )^5}-\frac {21945 \sqrt {2} \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )}{\sqrt [4]{b}}+\frac {21945 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{\sqrt [4]{b}}\right )}{81920 a^{23/4} \sqrt {d x}} \]
(Sqrt[x]*((4*a^(3/4)*Sqrt[x]*(19015*a^4 + 50312*a^3*b*x^2 + 59470*a^2*b^2* x^4 + 33440*a*b^3*x^6 + 7315*b^4*x^8))/(a + b*x^2)^5 - (21945*Sqrt[2]*ArcT an[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])])/b^(1/4) + (21 945*Sqrt[2]*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x )])/b^(1/4)))/(81920*a^(23/4)*Sqrt[d*x])
Time = 0.65 (sec) , antiderivative size = 441, normalized size of antiderivative = 1.14, 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, 755, 27, 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.
| \(\displaystyle \int \frac {1}{\sqrt {d x} \left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx\) |
| \(\Big \downarrow \) 1380 |
| \(\displaystyle b^6 \int \frac {1}{b^6 \sqrt {d x} \left (b x^2+a\right )^6}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \frac {1}{\sqrt {d x} \left (a+b x^2\right )^6}dx\) |
| \(\Big \downarrow \) 253 |
| \(\displaystyle \frac {19 \int \frac {1}{\sqrt {d x} \left (b x^2+a\right )^5}dx}{20 a}+\frac {\sqrt {d x}}{10 a d \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 253 |
| \(\displaystyle \frac {19 \left (\frac {15 \int \frac {1}{\sqrt {d x} \left (b x^2+a\right )^4}dx}{16 a}+\frac {\sqrt {d x}}{8 a d \left (a+b x^2\right )^4}\right )}{20 a}+\frac {\sqrt {d x}}{10 a d \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 253 |
| \(\displaystyle \frac {19 \left (\frac {15 \left (\frac {11 \int \frac {1}{\sqrt {d x} \left (b x^2+a\right )^3}dx}{12 a}+\frac {\sqrt {d x}}{6 a d \left (a+b x^2\right )^3}\right )}{16 a}+\frac {\sqrt {d x}}{8 a d \left (a+b x^2\right )^4}\right )}{20 a}+\frac {\sqrt {d x}}{10 a d \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 253 |
| \(\displaystyle \frac {19 \left (\frac {15 \left (\frac {11 \left (\frac {7 \int \frac {1}{\sqrt {d x} \left (b x^2+a\right )^2}dx}{8 a}+\frac {\sqrt {d x}}{4 a d \left (a+b x^2\right )^2}\right )}{12 a}+\frac {\sqrt {d x}}{6 a d \left (a+b x^2\right )^3}\right )}{16 a}+\frac {\sqrt {d x}}{8 a d \left (a+b x^2\right )^4}\right )}{20 a}+\frac {\sqrt {d x}}{10 a d \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 253 |
| \(\displaystyle \frac {19 \left (\frac {15 \left (\frac {11 \left (\frac {7 \left (\frac {3 \int \frac {1}{\sqrt {d x} \left (b x^2+a\right )}dx}{4 a}+\frac {\sqrt {d x}}{2 a d \left (a+b x^2\right )}\right )}{8 a}+\frac {\sqrt {d x}}{4 a d \left (a+b x^2\right )^2}\right )}{12 a}+\frac {\sqrt {d x}}{6 a d \left (a+b x^2\right )^3}\right )}{16 a}+\frac {\sqrt {d x}}{8 a d \left (a+b x^2\right )^4}\right )}{20 a}+\frac {\sqrt {d x}}{10 a d \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {19 \left (\frac {15 \left (\frac {11 \left (\frac {7 \left (\frac {3 \int \frac {1}{b x^2+a}d\sqrt {d x}}{2 a d}+\frac {\sqrt {d x}}{2 a d \left (a+b x^2\right )}\right )}{8 a}+\frac {\sqrt {d x}}{4 a d \left (a+b x^2\right )^2}\right )}{12 a}+\frac {\sqrt {d x}}{6 a d \left (a+b x^2\right )^3}\right )}{16 a}+\frac {\sqrt {d x}}{8 a d \left (a+b x^2\right )^4}\right )}{20 a}+\frac {\sqrt {d x}}{10 a d \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 755 |
| \(\displaystyle \frac {19 \left (\frac {15 \left (\frac {11 \left (\frac {7 \left (\frac {3 \left (\frac {\int \frac {d^2 \left (\sqrt {a} d-\sqrt {b} d x\right )}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {a} d}+\frac {\int \frac {d^2 \left (\sqrt {b} x d+\sqrt {a} d\right )}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {a} d}\right )}{2 a d}+\frac {\sqrt {d x}}{2 a d \left (a+b x^2\right )}\right )}{8 a}+\frac {\sqrt {d x}}{4 a d \left (a+b x^2\right )^2}\right )}{12 a}+\frac {\sqrt {d x}}{6 a d \left (a+b x^2\right )^3}\right )}{16 a}+\frac {\sqrt {d x}}{8 a d \left (a+b x^2\right )^4}\right )}{20 a}+\frac {\sqrt {d x}}{10 a d \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {19 \left (\frac {15 \left (\frac {11 \left (\frac {7 \left (\frac {3 \left (\frac {d \int \frac {\sqrt {a} d-\sqrt {b} d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {a}}+\frac {d \int \frac {\sqrt {b} x d+\sqrt {a} d}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {a}}\right )}{2 a d}+\frac {\sqrt {d x}}{2 a d \left (a+b x^2\right )}\right )}{8 a}+\frac {\sqrt {d x}}{4 a d \left (a+b x^2\right )^2}\right )}{12 a}+\frac {\sqrt {d x}}{6 a d \left (a+b x^2\right )^3}\right )}{16 a}+\frac {\sqrt {d x}}{8 a d \left (a+b x^2\right )^4}\right )}{20 a}+\frac {\sqrt {d x}}{10 a d \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {19 \left (\frac {15 \left (\frac {11 \left (\frac {7 \left (\frac {3 \left (\frac {d \int \frac {\sqrt {a} d-\sqrt {b} d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {a}}+\frac {d \left (\frac {\int \frac {1}{x d+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}}d\sqrt {d x}}{2 \sqrt {b}}+\frac {\int \frac {1}{x d+\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}}d\sqrt {d x}}{2 \sqrt {b}}\right )}{2 \sqrt {a}}\right )}{2 a d}+\frac {\sqrt {d x}}{2 a d \left (a+b x^2\right )}\right )}{8 a}+\frac {\sqrt {d x}}{4 a d \left (a+b x^2\right )^2}\right )}{12 a}+\frac {\sqrt {d x}}{6 a d \left (a+b x^2\right )^3}\right )}{16 a}+\frac {\sqrt {d x}}{8 a d \left (a+b x^2\right )^4}\right )}{20 a}+\frac {\sqrt {d x}}{10 a d \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {19 \left (\frac {15 \left (\frac {11 \left (\frac {7 \left (\frac {3 \left (\frac {d \int \frac {\sqrt {a} d-\sqrt {b} d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {a}}+\frac {d \left (\frac {\int \frac {1}{-d x-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\int \frac {1}{-d x-1}d\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}\right )}{2 a d}+\frac {\sqrt {d x}}{2 a d \left (a+b x^2\right )}\right )}{8 a}+\frac {\sqrt {d x}}{4 a d \left (a+b x^2\right )^2}\right )}{12 a}+\frac {\sqrt {d x}}{6 a d \left (a+b x^2\right )^3}\right )}{16 a}+\frac {\sqrt {d x}}{8 a d \left (a+b x^2\right )^4}\right )}{20 a}+\frac {\sqrt {d x}}{10 a d \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {19 \left (\frac {15 \left (\frac {11 \left (\frac {7 \left (\frac {3 \left (\frac {d \int \frac {\sqrt {a} d-\sqrt {b} d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {a}}+\frac {d \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}\right )}{2 a d}+\frac {\sqrt {d x}}{2 a d \left (a+b x^2\right )}\right )}{8 a}+\frac {\sqrt {d x}}{4 a d \left (a+b x^2\right )^2}\right )}{12 a}+\frac {\sqrt {d x}}{6 a d \left (a+b x^2\right )^3}\right )}{16 a}+\frac {\sqrt {d x}}{8 a d \left (a+b x^2\right )^4}\right )}{20 a}+\frac {\sqrt {d x}}{10 a d \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {19 \left (\frac {15 \left (\frac {11 \left (\frac {7 \left (\frac {3 \left (\frac {d \left (-\frac {\int -\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d}-2 \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{b} \left (x d+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt [4]{a} \sqrt {d}+\sqrt {2} \sqrt [4]{b} \sqrt {d x}\right )}{\sqrt [4]{b} \left (x d+\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}+\frac {d \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}\right )}{2 a d}+\frac {\sqrt {d x}}{2 a d \left (a+b x^2\right )}\right )}{8 a}+\frac {\sqrt {d x}}{4 a d \left (a+b x^2\right )^2}\right )}{12 a}+\frac {\sqrt {d x}}{6 a d \left (a+b x^2\right )^3}\right )}{16 a}+\frac {\sqrt {d x}}{8 a d \left (a+b x^2\right )^4}\right )}{20 a}+\frac {\sqrt {d x}}{10 a d \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {19 \left (\frac {15 \left (\frac {11 \left (\frac {7 \left (\frac {3 \left (\frac {d \left (\frac {\int \frac {\sqrt {2} \sqrt [4]{a} \sqrt {d}-2 \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{b} \left (x d+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}+\frac {\int \frac {\sqrt {2} \left (\sqrt [4]{a} \sqrt {d}+\sqrt {2} \sqrt [4]{b} \sqrt {d x}\right )}{\sqrt [4]{b} \left (x d+\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}+\frac {d \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}\right )}{2 a d}+\frac {\sqrt {d x}}{2 a d \left (a+b x^2\right )}\right )}{8 a}+\frac {\sqrt {d x}}{4 a d \left (a+b x^2\right )^2}\right )}{12 a}+\frac {\sqrt {d x}}{6 a d \left (a+b x^2\right )^3}\right )}{16 a}+\frac {\sqrt {d x}}{8 a d \left (a+b x^2\right )^4}\right )}{20 a}+\frac {\sqrt {d x}}{10 a d \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {19 \left (\frac {15 \left (\frac {11 \left (\frac {7 \left (\frac {3 \left (\frac {d \left (\frac {\int \frac {\sqrt {2} \sqrt [4]{a} \sqrt {d}-2 \sqrt [4]{b} \sqrt {d x}}{x d+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt {b} \sqrt {d}}+\frac {\int \frac {\sqrt [4]{a} \sqrt {d}+\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{x d+\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}}d\sqrt {d x}}{2 \sqrt [4]{a} \sqrt {b} \sqrt {d}}\right )}{2 \sqrt {a}}+\frac {d \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}\right )}{2 a d}+\frac {\sqrt {d x}}{2 a d \left (a+b x^2\right )}\right )}{8 a}+\frac {\sqrt {d x}}{4 a d \left (a+b x^2\right )^2}\right )}{12 a}+\frac {\sqrt {d x}}{6 a d \left (a+b x^2\right )^3}\right )}{16 a}+\frac {\sqrt {d x}}{8 a d \left (a+b x^2\right )^4}\right )}{20 a}+\frac {\sqrt {d x}}{10 a d \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {19 \left (\frac {15 \left (\frac {11 \left (\frac {7 \left (\frac {3 \left (\frac {d \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}+\frac {d \left (\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d} \sqrt {d x}+\sqrt {a} d+\sqrt {b} d x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d} \sqrt {d x}+\sqrt {a} d+\sqrt {b} d x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}\right )}{2 a d}+\frac {\sqrt {d x}}{2 a d \left (a+b x^2\right )}\right )}{8 a}+\frac {\sqrt {d x}}{4 a d \left (a+b x^2\right )^2}\right )}{12 a}+\frac {\sqrt {d x}}{6 a d \left (a+b x^2\right )^3}\right )}{16 a}+\frac {\sqrt {d x}}{8 a d \left (a+b x^2\right )^4}\right )}{20 a}+\frac {\sqrt {d x}}{10 a d \left (a+b x^2\right )^5}\) |
Sqrt[d*x]/(10*a*d*(a + b*x^2)^5) + (19*(Sqrt[d*x]/(8*a*d*(a + b*x^2)^4) + (15*(Sqrt[d*x]/(6*a*d*(a + b*x^2)^3) + (11*(Sqrt[d*x]/(4*a*d*(a + b*x^2)^2 ) + (7*(Sqrt[d*x]/(2*a*d*(a + b*x^2)) + (3*((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])) + Arc Tan[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[a]) + (d*(-1/2*Log[Sqrt[a]*d + Sqrt[b]*d*x - Sqrt [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[a])))/(2*a*d)))/(8*a)))/(12* a)))/(16*a)))/(20*a)
3.8.24.3.1 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[((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_) + (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 = 0.40 (sec) , antiderivative size = 237, normalized size of antiderivative = 0.61
| method | result | size |
| pseudoelliptic | \(\frac {\left (58520 a \,x^{8} b^{4}+267520 a^{2} x^{6} b^{3}+475760 a^{3} x^{4} b^{2}+402496 x^{2} a^{4} b +152120 a^{5}\right ) \sqrt {d x}+21945 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (b \,x^{2}+a \right )^{5} \left (\ln \left (\frac {d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}-\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}+\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )\right )}{163840 d \,a^{6} \left (b \,x^{2}+a \right )^{5}}\) | \(237\) |
| derivativedivides | \(2 d^{11} \left (\frac {\frac {3803 \sqrt {d x}}{8192 a \,d^{2}}+\frac {6289 b \left (d x \right )^{\frac {5}{2}}}{5120 a^{2} d^{4}}+\frac {5947 b^{2} \left (d x \right )^{\frac {9}{2}}}{4096 a^{3} d^{6}}+\frac {209 b^{3} \left (d x \right )^{\frac {13}{2}}}{256 a^{4} d^{8}}+\frac {1463 b^{4} \left (d x \right )^{\frac {17}{2}}}{8192 a^{5} d^{10}}}{\left (b \,d^{2} x^{2}+a \,d^{2}\right )^{5}}+\frac {4389 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{65536 a^{6} d^{12}}\right )\) | \(241\) |
| default | \(2 d^{11} \left (\frac {\frac {3803 \sqrt {d x}}{8192 a \,d^{2}}+\frac {6289 b \left (d x \right )^{\frac {5}{2}}}{5120 a^{2} d^{4}}+\frac {5947 b^{2} \left (d x \right )^{\frac {9}{2}}}{4096 a^{3} d^{6}}+\frac {209 b^{3} \left (d x \right )^{\frac {13}{2}}}{256 a^{4} d^{8}}+\frac {1463 b^{4} \left (d x \right )^{\frac {17}{2}}}{8192 a^{5} d^{10}}}{\left (b \,d^{2} x^{2}+a \,d^{2}\right )^{5}}+\frac {4389 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{65536 a^{6} d^{12}}\right )\) | \(241\) |
1/163840*((58520*a*b^4*x^8+267520*a^2*b^3*x^6+475760*a^3*b^2*x^4+402496*a^ 4*b*x^2+152120*a^5)*(d*x)^(1/2)+21945*(a*d^2/b)^(1/4)*2^(1/2)*(b*x^2+a)^5* (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)*(d*x)^(1/ 2)-(a*d^2/b)^(1/4))/(a*d^2/b)^(1/4))+2*arctan((2^(1/2)*(d*x)^(1/2)+(a*d^2/ b)^(1/4))/(a*d^2/b)^(1/4))))/d/a^6/(b*x^2+a)^5
Result contains complex when optimal does not.
Time = 0.28 (sec) , antiderivative size = 537, normalized size of antiderivative = 1.39 \[ \int \frac {1}{\sqrt {d x} \left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {21945 \, {\left (a^{5} b^{5} d x^{10} + 5 \, a^{6} b^{4} d x^{8} + 10 \, a^{7} b^{3} d x^{6} + 10 \, a^{8} b^{2} d x^{4} + 5 \, a^{9} b d x^{2} + a^{10} d\right )} \left (-\frac {1}{a^{23} b d^{2}}\right )^{\frac {1}{4}} \log \left (a^{6} d \left (-\frac {1}{a^{23} b d^{2}}\right )^{\frac {1}{4}} + \sqrt {d x}\right ) - 21945 \, {\left (-i \, a^{5} b^{5} d x^{10} - 5 i \, a^{6} b^{4} d x^{8} - 10 i \, a^{7} b^{3} d x^{6} - 10 i \, a^{8} b^{2} d x^{4} - 5 i \, a^{9} b d x^{2} - i \, a^{10} d\right )} \left (-\frac {1}{a^{23} b d^{2}}\right )^{\frac {1}{4}} \log \left (i \, a^{6} d \left (-\frac {1}{a^{23} b d^{2}}\right )^{\frac {1}{4}} + \sqrt {d x}\right ) - 21945 \, {\left (i \, a^{5} b^{5} d x^{10} + 5 i \, a^{6} b^{4} d x^{8} + 10 i \, a^{7} b^{3} d x^{6} + 10 i \, a^{8} b^{2} d x^{4} + 5 i \, a^{9} b d x^{2} + i \, a^{10} d\right )} \left (-\frac {1}{a^{23} b d^{2}}\right )^{\frac {1}{4}} \log \left (-i \, a^{6} d \left (-\frac {1}{a^{23} b d^{2}}\right )^{\frac {1}{4}} + \sqrt {d x}\right ) - 21945 \, {\left (a^{5} b^{5} d x^{10} + 5 \, a^{6} b^{4} d x^{8} + 10 \, a^{7} b^{3} d x^{6} + 10 \, a^{8} b^{2} d x^{4} + 5 \, a^{9} b d x^{2} + a^{10} d\right )} \left (-\frac {1}{a^{23} b d^{2}}\right )^{\frac {1}{4}} \log \left (-a^{6} d \left (-\frac {1}{a^{23} b d^{2}}\right )^{\frac {1}{4}} + \sqrt {d x}\right ) + 4 \, {\left (7315 \, b^{4} x^{8} + 33440 \, a b^{3} x^{6} + 59470 \, a^{2} b^{2} x^{4} + 50312 \, a^{3} b x^{2} + 19015 \, a^{4}\right )} \sqrt {d x}}{81920 \, {\left (a^{5} b^{5} d x^{10} + 5 \, a^{6} b^{4} d x^{8} + 10 \, a^{7} b^{3} d x^{6} + 10 \, a^{8} b^{2} d x^{4} + 5 \, a^{9} b d x^{2} + a^{10} d\right )}} \]
1/81920*(21945*(a^5*b^5*d*x^10 + 5*a^6*b^4*d*x^8 + 10*a^7*b^3*d*x^6 + 10*a ^8*b^2*d*x^4 + 5*a^9*b*d*x^2 + a^10*d)*(-1/(a^23*b*d^2))^(1/4)*log(a^6*d*( -1/(a^23*b*d^2))^(1/4) + sqrt(d*x)) - 21945*(-I*a^5*b^5*d*x^10 - 5*I*a^6*b ^4*d*x^8 - 10*I*a^7*b^3*d*x^6 - 10*I*a^8*b^2*d*x^4 - 5*I*a^9*b*d*x^2 - I*a ^10*d)*(-1/(a^23*b*d^2))^(1/4)*log(I*a^6*d*(-1/(a^23*b*d^2))^(1/4) + sqrt( d*x)) - 21945*(I*a^5*b^5*d*x^10 + 5*I*a^6*b^4*d*x^8 + 10*I*a^7*b^3*d*x^6 + 10*I*a^8*b^2*d*x^4 + 5*I*a^9*b*d*x^2 + I*a^10*d)*(-1/(a^23*b*d^2))^(1/4)* log(-I*a^6*d*(-1/(a^23*b*d^2))^(1/4) + sqrt(d*x)) - 21945*(a^5*b^5*d*x^10 + 5*a^6*b^4*d*x^8 + 10*a^7*b^3*d*x^6 + 10*a^8*b^2*d*x^4 + 5*a^9*b*d*x^2 + a^10*d)*(-1/(a^23*b*d^2))^(1/4)*log(-a^6*d*(-1/(a^23*b*d^2))^(1/4) + sqrt( d*x)) + 4*(7315*b^4*x^8 + 33440*a*b^3*x^6 + 59470*a^2*b^2*x^4 + 50312*a^3* b*x^2 + 19015*a^4)*sqrt(d*x))/(a^5*b^5*d*x^10 + 5*a^6*b^4*d*x^8 + 10*a^7*b ^3*d*x^6 + 10*a^8*b^2*d*x^4 + 5*a^9*b*d*x^2 + a^10*d)
\[ \int \frac {1}{\sqrt {d x} \left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\int \frac {1}{\sqrt {d x} \left (a + b x^{2}\right )^{6}}\, dx \]
Time = 0.29 (sec) , antiderivative size = 382, normalized size of antiderivative = 0.99 \[ \int \frac {1}{\sqrt {d x} \left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {\frac {8 \, {\left (7315 \, \left (d x\right )^{\frac {17}{2}} b^{4} d^{2} + 33440 \, \left (d x\right )^{\frac {13}{2}} a b^{3} d^{4} + 59470 \, \left (d x\right )^{\frac {9}{2}} a^{2} b^{2} d^{6} + 50312 \, \left (d x\right )^{\frac {5}{2}} a^{3} b d^{8} + 19015 \, \sqrt {d x} a^{4} d^{10}\right )}}{a^{5} b^{5} d^{10} x^{10} + 5 \, a^{6} b^{4} d^{10} x^{8} + 10 \, a^{7} b^{3} d^{10} x^{6} + 10 \, a^{8} b^{2} d^{10} x^{4} + 5 \, a^{9} b d^{10} x^{2} + a^{10} d^{10}} + \frac {21945 \, {\left (\frac {\sqrt {2} d^{2} \log \left (\sqrt {b} d x + \sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} \sqrt {d x} b^{\frac {1}{4}} + \sqrt {a} d\right )}{\left (a d^{2}\right )^{\frac {3}{4}} b^{\frac {1}{4}}} - \frac {\sqrt {2} d^{2} \log \left (\sqrt {b} d x - \sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} \sqrt {d x} b^{\frac {1}{4}} + \sqrt {a} d\right )}{\left (a d^{2}\right )^{\frac {3}{4}} b^{\frac {1}{4}}} + \frac {2 \, \sqrt {2} d \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {d x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} d}}\right )}{\sqrt {\sqrt {a} \sqrt {b} d} \sqrt {a}} + \frac {2 \, \sqrt {2} d \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {d x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} d}}\right )}{\sqrt {\sqrt {a} \sqrt {b} d} \sqrt {a}}\right )}}{a^{5}}}{163840 \, d} \]
1/163840*(8*(7315*(d*x)^(17/2)*b^4*d^2 + 33440*(d*x)^(13/2)*a*b^3*d^4 + 59 470*(d*x)^(9/2)*a^2*b^2*d^6 + 50312*(d*x)^(5/2)*a^3*b*d^8 + 19015*sqrt(d*x )*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) + 21945*(sqrt(2)*d^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)^(3/4)*b^(1/4)) - sqrt(2)*d^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)^(3/4)*b^(1/4)) + 2*sqrt(2)*d*arc tan(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(a)) + 2*sqrt(2)*d*arcta n(-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(a)))/a^5)/d
Time = 0.30 (sec) , antiderivative size = 346, normalized size of antiderivative = 0.89 \[ \int \frac {1}{\sqrt {d x} \left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {4389 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{16384 \, a^{6} b d} + \frac {4389 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{16384 \, a^{6} b d} + \frac {4389 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} \log \left (d x + \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{32768 \, a^{6} b d} - \frac {4389 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} \log \left (d x - \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{32768 \, a^{6} b d} + \frac {7315 \, \sqrt {d x} b^{4} d^{9} x^{8} + 33440 \, \sqrt {d x} a b^{3} d^{9} x^{6} + 59470 \, \sqrt {d x} a^{2} b^{2} d^{9} x^{4} + 50312 \, \sqrt {d x} a^{3} b d^{9} x^{2} + 19015 \, \sqrt {d x} a^{4} d^{9}}{20480 \, {\left (b d^{2} x^{2} + a d^{2}\right )}^{5} a^{5}} \]
4389/16384*sqrt(2)*(a*b^3*d^2)^(1/4)*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*d) + 4389/16384*sqrt(2)*(a*b ^3*d^2)^(1/4)*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*d) + 4389/32768*sqrt(2)*(a*b^3*d^2)^(1/4)*log(d*x + sqrt(2)*(a*d^2/b)^(1/4)*sqrt(d*x) + sqrt(a*d^2/b))/(a^6*b*d) - 4389/3276 8*sqrt(2)*(a*b^3*d^2)^(1/4)*log(d*x - sqrt(2)*(a*d^2/b)^(1/4)*sqrt(d*x) + sqrt(a*d^2/b))/(a^6*b*d) + 1/20480*(7315*sqrt(d*x)*b^4*d^9*x^8 + 33440*sqr t(d*x)*a*b^3*d^9*x^6 + 59470*sqrt(d*x)*a^2*b^2*d^9*x^4 + 50312*sqrt(d*x)*a ^3*b*d^9*x^2 + 19015*sqrt(d*x)*a^4*d^9)/((b*d^2*x^2 + a*d^2)^5*a^5)
Time = 14.19 (sec) , antiderivative size = 210, normalized size of antiderivative = 0.54 \[ \int \frac {1}{\sqrt {d x} \left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {\frac {3803\,d^9\,\sqrt {d\,x}}{4096\,a}+\frac {5947\,b^2\,d^5\,{\left (d\,x\right )}^{9/2}}{2048\,a^3}+\frac {209\,b^3\,d^3\,{\left (d\,x\right )}^{13/2}}{128\,a^4}+\frac {6289\,b\,d^7\,{\left (d\,x\right )}^{5/2}}{2560\,a^2}+\frac {1463\,b^4\,d\,{\left (d\,x\right )}^{17/2}}{4096\,a^5}}{a^5\,d^{10}+5\,a^4\,b\,d^{10}\,x^2+10\,a^3\,b^2\,d^{10}\,x^4+10\,a^2\,b^3\,d^{10}\,x^6+5\,a\,b^4\,d^{10}\,x^8+b^5\,d^{10}\,x^{10}}+\frac {4389\,\mathrm {atan}\left (\frac {b^{1/4}\,\sqrt {d\,x}}{{\left (-a\right )}^{1/4}\,\sqrt {d}}\right )}{8192\,{\left (-a\right )}^{23/4}\,b^{1/4}\,\sqrt {d}}+\frac {4389\,\mathrm {atanh}\left (\frac {b^{1/4}\,\sqrt {d\,x}}{{\left (-a\right )}^{1/4}\,\sqrt {d}}\right )}{8192\,{\left (-a\right )}^{23/4}\,b^{1/4}\,\sqrt {d}} \]
((3803*d^9*(d*x)^(1/2))/(4096*a) + (5947*b^2*d^5*(d*x)^(9/2))/(2048*a^3) + (209*b^3*d^3*(d*x)^(13/2))/(128*a^4) + (6289*b*d^7*(d*x)^(5/2))/(2560*a^2 ) + (1463*b^4*d*(d*x)^(17/2))/(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) + (4389*atan((b^(1/4)*(d*x)^(1/2))/((-a)^(1/4)*d^(1/2))))/(8192*(-a)^(2 3/4)*b^(1/4)*d^(1/2)) + (4389*atanh((b^(1/4)*(d*x)^(1/2))/((-a)^(1/4)*d^(1 /2))))/(8192*(-a)^(23/4)*b^(1/4)*d^(1/2))