Integrand size = 28, antiderivative size = 388 \[ \int \frac {(d x)^{17/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=-\frac {d (d x)^{15/2}}{10 b \left (a+b x^2\right )^5}-\frac {3 d^3 (d x)^{11/2}}{32 b^2 \left (a+b x^2\right )^4}-\frac {11 d^5 (d x)^{7/2}}{128 b^3 \left (a+b x^2\right )^3}-\frac {77 d^7 (d x)^{3/2}}{1024 b^4 \left (a+b x^2\right )^2}+\frac {231 d^7 (d x)^{3/2}}{4096 a b^4 \left (a+b x^2\right )}-\frac {231 d^{17/2} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} a^{5/4} b^{19/4}}+\frac {231 d^{17/2} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} a^{5/4} b^{19/4}}+\frac {231 d^{17/2} \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{16384 \sqrt {2} a^{5/4} b^{19/4}}-\frac {231 d^{17/2} \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{16384 \sqrt {2} a^{5/4} b^{19/4}} \]
-1/10*d*(d*x)^(15/2)/b/(b*x^2+a)^5-3/32*d^3*(d*x)^(11/2)/b^2/(b*x^2+a)^4-1 1/128*d^5*(d*x)^(7/2)/b^3/(b*x^2+a)^3-77/1024*d^7*(d*x)^(3/2)/b^4/(b*x^2+a )^2+231/4096*d^7*(d*x)^(3/2)/a/b^4/(b*x^2+a)-231/16384*d^(17/2)*arctan(1-b ^(1/4)*2^(1/2)*(d*x)^(1/2)/a^(1/4)/d^(1/2))/a^(5/4)/b^(19/4)*2^(1/2)+231/1 6384*d^(17/2)*arctan(1+b^(1/4)*2^(1/2)*(d*x)^(1/2)/a^(1/4)/d^(1/2))/a^(5/4 )/b^(19/4)*2^(1/2)+231/32768*d^(17/2)*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^(5/4)/b^(19/4)*2^(1/2)-231/32768*d ^(17/2)*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^(5/4)/b^(19/4)*2^(1/2)
Time = 0.95 (sec) , antiderivative size = 205, normalized size of antiderivative = 0.53 \[ \int \frac {(d x)^{17/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {d^8 \sqrt {d x} \left (4 \sqrt [4]{a} b^{3/4} x^{3/2} \left (-385 a^4-1760 a^3 b x^2-3130 a^2 b^2 x^4-2648 a b^3 x^6+1155 b^4 x^8\right )+1155 \sqrt {2} \left (a+b x^2\right )^5 \arctan \left (\frac {-\sqrt {a}+\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )-1155 \sqrt {2} \left (a+b x^2\right )^5 \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )\right )}{81920 a^{5/4} b^{19/4} \sqrt {x} \left (a+b x^2\right )^5} \]
(d^8*Sqrt[d*x]*(4*a^(1/4)*b^(3/4)*x^(3/2)*(-385*a^4 - 1760*a^3*b*x^2 - 313 0*a^2*b^2*x^4 - 2648*a*b^3*x^6 + 1155*b^4*x^8) + 1155*Sqrt[2]*(a + b*x^2)^ 5*ArcTan[(-Sqrt[a] + Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])] - 1155* Sqrt[2]*(a + b*x^2)^5*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)]))/(81920*a^(5/4)*b^(19/4)*Sqrt[x]*(a + b*x^2)^5)
Time = 0.64 (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, 252, 252, 252, 252, 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.
| \(\displaystyle \int \frac {(d x)^{17/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx\) |
| \(\Big \downarrow \) 1380 |
| \(\displaystyle b^6 \int \frac {(d x)^{17/2}}{b^6 \left (b x^2+a\right )^6}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \frac {(d x)^{17/2}}{\left (a+b x^2\right )^6}dx\) |
| \(\Big \downarrow \) 252 |
| \(\displaystyle \frac {3 d^2 \int \frac {(d x)^{13/2}}{\left (b x^2+a\right )^5}dx}{4 b}-\frac {d (d x)^{15/2}}{10 b \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 252 |
| \(\displaystyle \frac {3 d^2 \left (\frac {11 d^2 \int \frac {(d x)^{9/2}}{\left (b x^2+a\right )^4}dx}{16 b}-\frac {d (d x)^{11/2}}{8 b \left (a+b x^2\right )^4}\right )}{4 b}-\frac {d (d x)^{15/2}}{10 b \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 252 |
| \(\displaystyle \frac {3 d^2 \left (\frac {11 d^2 \left (\frac {7 d^2 \int \frac {(d x)^{5/2}}{\left (b x^2+a\right )^3}dx}{12 b}-\frac {d (d x)^{7/2}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{11/2}}{8 b \left (a+b x^2\right )^4}\right )}{4 b}-\frac {d (d x)^{15/2}}{10 b \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 252 |
| \(\displaystyle \frac {3 d^2 \left (\frac {11 d^2 \left (\frac {7 d^2 \left (\frac {3 d^2 \int \frac {\sqrt {d x}}{\left (b x^2+a\right )^2}dx}{8 b}-\frac {d (d x)^{3/2}}{4 b \left (a+b x^2\right )^2}\right )}{12 b}-\frac {d (d x)^{7/2}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{11/2}}{8 b \left (a+b x^2\right )^4}\right )}{4 b}-\frac {d (d x)^{15/2}}{10 b \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 253 |
| \(\displaystyle \frac {3 d^2 \left (\frac {11 d^2 \left (\frac {7 d^2 \left (\frac {3 d^2 \left (\frac {\int \frac {\sqrt {d x}}{b x^2+a}dx}{4 a}+\frac {(d x)^{3/2}}{2 a d \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{3/2}}{4 b \left (a+b x^2\right )^2}\right )}{12 b}-\frac {d (d x)^{7/2}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{11/2}}{8 b \left (a+b x^2\right )^4}\right )}{4 b}-\frac {d (d x)^{15/2}}{10 b \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {3 d^2 \left (\frac {11 d^2 \left (\frac {7 d^2 \left (\frac {3 d^2 \left (\frac {\int \frac {d^3 x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 a d}+\frac {(d x)^{3/2}}{2 a d \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{3/2}}{4 b \left (a+b x^2\right )^2}\right )}{12 b}-\frac {d (d x)^{7/2}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{11/2}}{8 b \left (a+b x^2\right )^4}\right )}{4 b}-\frac {d (d x)^{15/2}}{10 b \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3 d^2 \left (\frac {11 d^2 \left (\frac {7 d^2 \left (\frac {3 d^2 \left (\frac {d \int \frac {d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 a}+\frac {(d x)^{3/2}}{2 a d \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{3/2}}{4 b \left (a+b x^2\right )^2}\right )}{12 b}-\frac {d (d x)^{7/2}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{11/2}}{8 b \left (a+b x^2\right )^4}\right )}{4 b}-\frac {d (d x)^{15/2}}{10 b \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 826 |
| \(\displaystyle \frac {3 d^2 \left (\frac {11 d^2 \left (\frac {7 d^2 \left (\frac {3 d^2 \left (\frac {d \left (\frac {\int \frac {\sqrt {b} x d+\sqrt {a} d}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {b}}-\frac {\int \frac {\sqrt {a} d-\sqrt {b} d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {b}}\right )}{2 a}+\frac {(d x)^{3/2}}{2 a d \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{3/2}}{4 b \left (a+b x^2\right )^2}\right )}{12 b}-\frac {d (d x)^{7/2}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{11/2}}{8 b \left (a+b x^2\right )^4}\right )}{4 b}-\frac {d (d x)^{15/2}}{10 b \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {3 d^2 \left (\frac {11 d^2 \left (\frac {7 d^2 \left (\frac {3 d^2 \left (\frac {d \left (\frac {\frac {\int \frac {1}{x d+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}}d\sqrt {d x}}{2 \sqrt {b}}+\frac {\int \frac {1}{x d+\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}}d\sqrt {d x}}{2 \sqrt {b}}}{2 \sqrt {b}}-\frac {\int \frac {\sqrt {a} d-\sqrt {b} d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {b}}\right )}{2 a}+\frac {(d x)^{3/2}}{2 a d \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{3/2}}{4 b \left (a+b x^2\right )^2}\right )}{12 b}-\frac {d (d x)^{7/2}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{11/2}}{8 b \left (a+b x^2\right )^4}\right )}{4 b}-\frac {d (d x)^{15/2}}{10 b \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {3 d^2 \left (\frac {11 d^2 \left (\frac {7 d^2 \left (\frac {3 d^2 \left (\frac {d \left (\frac {\frac {\int \frac {1}{-d x-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\int \frac {1}{-d x-1}d\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}}{2 \sqrt {b}}-\frac {\int \frac {\sqrt {a} d-\sqrt {b} d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {b}}\right )}{2 a}+\frac {(d x)^{3/2}}{2 a d \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{3/2}}{4 b \left (a+b x^2\right )^2}\right )}{12 b}-\frac {d (d x)^{7/2}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{11/2}}{8 b \left (a+b x^2\right )^4}\right )}{4 b}-\frac {d (d x)^{15/2}}{10 b \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {3 d^2 \left (\frac {11 d^2 \left (\frac {7 d^2 \left (\frac {3 d^2 \left (\frac {d \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}}{2 \sqrt {b}}-\frac {\int \frac {\sqrt {a} d-\sqrt {b} d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {b}}\right )}{2 a}+\frac {(d x)^{3/2}}{2 a d \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{3/2}}{4 b \left (a+b x^2\right )^2}\right )}{12 b}-\frac {d (d x)^{7/2}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{11/2}}{8 b \left (a+b x^2\right )^4}\right )}{4 b}-\frac {d (d x)^{15/2}}{10 b \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {3 d^2 \left (\frac {11 d^2 \left (\frac {7 d^2 \left (\frac {3 d^2 \left (\frac {d \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}}{2 \sqrt {b}}-\frac {-\frac {\int -\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d}-2 \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{b} \left (x d+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt [4]{a} \sqrt {d}+\sqrt {2} \sqrt [4]{b} \sqrt {d x}\right )}{\sqrt [4]{b} \left (x d+\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}}{2 \sqrt {b}}\right )}{2 a}+\frac {(d x)^{3/2}}{2 a d \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{3/2}}{4 b \left (a+b x^2\right )^2}\right )}{12 b}-\frac {d (d x)^{7/2}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{11/2}}{8 b \left (a+b x^2\right )^4}\right )}{4 b}-\frac {d (d x)^{15/2}}{10 b \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {3 d^2 \left (\frac {11 d^2 \left (\frac {7 d^2 \left (\frac {3 d^2 \left (\frac {d \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}}{2 \sqrt {b}}-\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{a} \sqrt {d}-2 \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{b} \left (x d+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}+\frac {\int \frac {\sqrt {2} \left (\sqrt [4]{a} \sqrt {d}+\sqrt {2} \sqrt [4]{b} \sqrt {d x}\right )}{\sqrt [4]{b} \left (x d+\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}}{2 \sqrt {b}}\right )}{2 a}+\frac {(d x)^{3/2}}{2 a d \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{3/2}}{4 b \left (a+b x^2\right )^2}\right )}{12 b}-\frac {d (d x)^{7/2}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{11/2}}{8 b \left (a+b x^2\right )^4}\right )}{4 b}-\frac {d (d x)^{15/2}}{10 b \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3 d^2 \left (\frac {11 d^2 \left (\frac {7 d^2 \left (\frac {3 d^2 \left (\frac {d \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}}{2 \sqrt {b}}-\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{a} \sqrt {d}-2 \sqrt [4]{b} \sqrt {d x}}{x d+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt {b} \sqrt {d}}+\frac {\int \frac {\sqrt [4]{a} \sqrt {d}+\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{x d+\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}}d\sqrt {d x}}{2 \sqrt [4]{a} \sqrt {b} \sqrt {d}}}{2 \sqrt {b}}\right )}{2 a}+\frac {(d x)^{3/2}}{2 a d \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{3/2}}{4 b \left (a+b x^2\right )^2}\right )}{12 b}-\frac {d (d x)^{7/2}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{11/2}}{8 b \left (a+b x^2\right )^4}\right )}{4 b}-\frac {d (d x)^{15/2}}{10 b \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {3 d^2 \left (\frac {11 d^2 \left (\frac {7 d^2 \left (\frac {3 d^2 \left (\frac {d \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}}{2 \sqrt {b}}-\frac {\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d} \sqrt {d x}+\sqrt {a} d+\sqrt {b} d x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d} \sqrt {d x}+\sqrt {a} d+\sqrt {b} d x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}}{2 \sqrt {b}}\right )}{2 a}+\frac {(d x)^{3/2}}{2 a d \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{3/2}}{4 b \left (a+b x^2\right )^2}\right )}{12 b}-\frac {d (d x)^{7/2}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{11/2}}{8 b \left (a+b x^2\right )^4}\right )}{4 b}-\frac {d (d x)^{15/2}}{10 b \left (a+b x^2\right )^5}\) |
-1/10*(d*(d*x)^(15/2))/(b*(a + b*x^2)^5) + (3*d^2*(-1/8*(d*(d*x)^(11/2))/( b*(a + b*x^2)^4) + (11*d^2*(-1/6*(d*(d*x)^(7/2))/(b*(a + b*x^2)^3) + (7*d^ 2*(-1/4*(d*(d*x)^(3/2))/(b*(a + b*x^2)^2) + (3*d^2*((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 - 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[b])))/(2*a)))/(8*b)))/(12*b)))/(16*b)))/(4*b)
3.8.15.3.1 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*(c*x )^(m - 1)*((a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] - Simp[c^2*((m - 1)/(2*b* (p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c }, x] && LtQ[p, -1] && GtQ[m, 1] && !ILtQ[(m + 2*p + 3)/2, 0] && IntBinomi alQ[a, b, c, 2, m, p, x]
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 = 20.19 (sec) , antiderivative size = 238, normalized size of antiderivative = 0.61
| method | result | size |
| derivativedivides | \(2 d^{11} \left (\frac {-\frac {77 d^{6} a^{3} \left (d x \right )^{\frac {3}{2}}}{8192 b^{4}}-\frac {11 d^{4} a^{2} \left (d x \right )^{\frac {7}{2}}}{256 b^{3}}-\frac {313 d^{2} a \left (d x \right )^{\frac {11}{2}}}{4096 b^{2}}-\frac {331 \left (d x \right )^{\frac {15}{2}}}{5120 b}+\frac {231 \left (d x \right )^{\frac {19}{2}}}{8192 a \,d^{2}}}{\left (b \,d^{2} x^{2}+a \,d^{2}\right )^{5}}+\frac {231 \sqrt {2}\, \left (\ln \left (\frac {d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{65536 a \,d^{2} b^{5} \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )\) | \(238\) |
| default | \(2 d^{11} \left (\frac {-\frac {77 d^{6} a^{3} \left (d x \right )^{\frac {3}{2}}}{8192 b^{4}}-\frac {11 d^{4} a^{2} \left (d x \right )^{\frac {7}{2}}}{256 b^{3}}-\frac {313 d^{2} a \left (d x \right )^{\frac {11}{2}}}{4096 b^{2}}-\frac {331 \left (d x \right )^{\frac {15}{2}}}{5120 b}+\frac {231 \left (d x \right )^{\frac {19}{2}}}{8192 a \,d^{2}}}{\left (b \,d^{2} x^{2}+a \,d^{2}\right )^{5}}+\frac {231 \sqrt {2}\, \left (\ln \left (\frac {d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{65536 a \,d^{2} b^{5} \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )\) | \(238\) |
| pseudoelliptic | \(-\frac {77 d^{8} \left (8 x b \left (-3 b^{4} x^{8}+\frac {2648}{385} a \,b^{3} x^{6}+\frac {626}{77} a^{2} b^{2} x^{4}+\frac {32}{7} a^{3} b \,x^{2}+a^{4}\right ) \sqrt {d x}\, \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}-3 \sqrt {2}\, d \left (b \,x^{2}+a \right )^{5} \left (2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}-\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}+\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )+\ln \left (\frac {d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )\right )\right )}{32768 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} b^{5} \left (b \,x^{2}+a \right )^{5} a}\) | \(249\) |
2*d^11*((-77/8192/b^4*d^6*a^3*(d*x)^(3/2)-11/256/b^3*d^4*a^2*(d*x)^(7/2)-3 13/4096/b^2*d^2*a*(d*x)^(11/2)-331/5120/b*(d*x)^(15/2)+231/8192/a/d^2*(d*x )^(19/2))/(b*d^2*x^2+a*d^2)^5+231/65536/a/d^2/b^5/(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 = 0.28 (sec) , antiderivative size = 567, normalized size of antiderivative = 1.46 \[ \int \frac {(d x)^{17/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {1155 \, {\left (a b^{9} x^{10} + 5 \, a^{2} b^{8} x^{8} + 10 \, a^{3} b^{7} x^{6} + 10 \, a^{4} b^{6} x^{4} + 5 \, a^{5} b^{5} x^{2} + a^{6} b^{4}\right )} \left (-\frac {d^{34}}{a^{5} b^{19}}\right )^{\frac {1}{4}} \log \left (12326391 \, \sqrt {d x} d^{25} + 12326391 \, \left (-\frac {d^{34}}{a^{5} b^{19}}\right )^{\frac {3}{4}} a^{4} b^{14}\right ) - 1155 \, {\left (i \, a b^{9} x^{10} + 5 i \, a^{2} b^{8} x^{8} + 10 i \, a^{3} b^{7} x^{6} + 10 i \, a^{4} b^{6} x^{4} + 5 i \, a^{5} b^{5} x^{2} + i \, a^{6} b^{4}\right )} \left (-\frac {d^{34}}{a^{5} b^{19}}\right )^{\frac {1}{4}} \log \left (12326391 \, \sqrt {d x} d^{25} + 12326391 i \, \left (-\frac {d^{34}}{a^{5} b^{19}}\right )^{\frac {3}{4}} a^{4} b^{14}\right ) - 1155 \, {\left (-i \, a b^{9} x^{10} - 5 i \, a^{2} b^{8} x^{8} - 10 i \, a^{3} b^{7} x^{6} - 10 i \, a^{4} b^{6} x^{4} - 5 i \, a^{5} b^{5} x^{2} - i \, a^{6} b^{4}\right )} \left (-\frac {d^{34}}{a^{5} b^{19}}\right )^{\frac {1}{4}} \log \left (12326391 \, \sqrt {d x} d^{25} - 12326391 i \, \left (-\frac {d^{34}}{a^{5} b^{19}}\right )^{\frac {3}{4}} a^{4} b^{14}\right ) - 1155 \, {\left (a b^{9} x^{10} + 5 \, a^{2} b^{8} x^{8} + 10 \, a^{3} b^{7} x^{6} + 10 \, a^{4} b^{6} x^{4} + 5 \, a^{5} b^{5} x^{2} + a^{6} b^{4}\right )} \left (-\frac {d^{34}}{a^{5} b^{19}}\right )^{\frac {1}{4}} \log \left (12326391 \, \sqrt {d x} d^{25} - 12326391 \, \left (-\frac {d^{34}}{a^{5} b^{19}}\right )^{\frac {3}{4}} a^{4} b^{14}\right ) + 4 \, {\left (1155 \, b^{4} d^{8} x^{9} - 2648 \, a b^{3} d^{8} x^{7} - 3130 \, a^{2} b^{2} d^{8} x^{5} - 1760 \, a^{3} b d^{8} x^{3} - 385 \, a^{4} d^{8} x\right )} \sqrt {d x}}{81920 \, {\left (a b^{9} x^{10} + 5 \, a^{2} b^{8} x^{8} + 10 \, a^{3} b^{7} x^{6} + 10 \, a^{4} b^{6} x^{4} + 5 \, a^{5} b^{5} x^{2} + a^{6} b^{4}\right )}} \]
1/81920*(1155*(a*b^9*x^10 + 5*a^2*b^8*x^8 + 10*a^3*b^7*x^6 + 10*a^4*b^6*x^ 4 + 5*a^5*b^5*x^2 + a^6*b^4)*(-d^34/(a^5*b^19))^(1/4)*log(12326391*sqrt(d* x)*d^25 + 12326391*(-d^34/(a^5*b^19))^(3/4)*a^4*b^14) - 1155*(I*a*b^9*x^10 + 5*I*a^2*b^8*x^8 + 10*I*a^3*b^7*x^6 + 10*I*a^4*b^6*x^4 + 5*I*a^5*b^5*x^2 + I*a^6*b^4)*(-d^34/(a^5*b^19))^(1/4)*log(12326391*sqrt(d*x)*d^25 + 12326 391*I*(-d^34/(a^5*b^19))^(3/4)*a^4*b^14) - 1155*(-I*a*b^9*x^10 - 5*I*a^2*b ^8*x^8 - 10*I*a^3*b^7*x^6 - 10*I*a^4*b^6*x^4 - 5*I*a^5*b^5*x^2 - I*a^6*b^4 )*(-d^34/(a^5*b^19))^(1/4)*log(12326391*sqrt(d*x)*d^25 - 12326391*I*(-d^34 /(a^5*b^19))^(3/4)*a^4*b^14) - 1155*(a*b^9*x^10 + 5*a^2*b^8*x^8 + 10*a^3*b ^7*x^6 + 10*a^4*b^6*x^4 + 5*a^5*b^5*x^2 + a^6*b^4)*(-d^34/(a^5*b^19))^(1/4 )*log(12326391*sqrt(d*x)*d^25 - 12326391*(-d^34/(a^5*b^19))^(3/4)*a^4*b^14 ) + 4*(1155*b^4*d^8*x^9 - 2648*a*b^3*d^8*x^7 - 3130*a^2*b^2*d^8*x^5 - 1760 *a^3*b*d^8*x^3 - 385*a^4*d^8*x)*sqrt(d*x))/(a*b^9*x^10 + 5*a^2*b^8*x^8 + 1 0*a^3*b^7*x^6 + 10*a^4*b^6*x^4 + 5*a^5*b^5*x^2 + a^6*b^4)
Timed out. \[ \int \frac {(d x)^{17/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\text {Timed out} \]
Time = 0.29 (sec) , antiderivative size = 383, normalized size of antiderivative = 0.99 \[ \int \frac {(d x)^{17/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {\frac {1155 \, d^{10} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {d x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} d}}\right )}{\sqrt {\sqrt {a} \sqrt {b} d} \sqrt {b}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {d x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} d}}\right )}{\sqrt {\sqrt {a} \sqrt {b} d} \sqrt {b}} - \frac {\sqrt {2} \log \left (\sqrt {b} d x + \sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} \sqrt {d x} b^{\frac {1}{4}} + \sqrt {a} d\right )}{\left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (\sqrt {b} d x - \sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} \sqrt {d x} b^{\frac {1}{4}} + \sqrt {a} d\right )}{\left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {3}{4}}}\right )}}{a b^{4}} + \frac {8 \, {\left (1155 \, \left (d x\right )^{\frac {19}{2}} b^{4} d^{10} - 2648 \, \left (d x\right )^{\frac {15}{2}} a b^{3} d^{12} - 3130 \, \left (d x\right )^{\frac {11}{2}} a^{2} b^{2} d^{14} - 1760 \, \left (d x\right )^{\frac {7}{2}} a^{3} b d^{16} - 385 \, \left (d x\right )^{\frac {3}{2}} a^{4} d^{18}\right )}}{a b^{9} d^{10} x^{10} + 5 \, a^{2} b^{8} d^{10} x^{8} + 10 \, a^{3} b^{7} d^{10} x^{6} + 10 \, a^{4} b^{6} d^{10} x^{4} + 5 \, a^{5} b^{5} d^{10} x^{2} + a^{6} b^{4} d^{10}}}{163840 \, d} \]
1/163840*(1155*d^10*(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)) + 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 - sqr t(2)*(a*d^2)^(1/4)*sqrt(d*x)*b^(1/4) + sqrt(a)*d)/((a*d^2)^(1/4)*b^(3/4))) /(a*b^4) + 8*(1155*(d*x)^(19/2)*b^4*d^10 - 2648*(d*x)^(15/2)*a*b^3*d^12 - 3130*(d*x)^(11/2)*a^2*b^2*d^14 - 1760*(d*x)^(7/2)*a^3*b*d^16 - 385*(d*x)^( 3/2)*a^4*d^18)/(a*b^9*d^10*x^10 + 5*a^2*b^8*d^10*x^8 + 10*a^3*b^7*d^10*x^6 + 10*a^4*b^6*d^10*x^4 + 5*a^5*b^5*d^10*x^2 + a^6*b^4*d^10))/d
Time = 0.28 (sec) , antiderivative size = 355, normalized size of antiderivative = 0.91 \[ \int \frac {(d x)^{17/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {1}{163840} \, d^{8} {\left (\frac {2310 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{a^{2} b^{7} d} + \frac {2310 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{a^{2} b^{7} d} - \frac {1155 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \log \left (d x + \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{a^{2} b^{7} d} + \frac {1155 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \log \left (d x - \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{a^{2} b^{7} d} + \frac {8 \, {\left (1155 \, \sqrt {d x} b^{4} d^{10} x^{9} - 2648 \, \sqrt {d x} a b^{3} d^{10} x^{7} - 3130 \, \sqrt {d x} a^{2} b^{2} d^{10} x^{5} - 1760 \, \sqrt {d x} a^{3} b d^{10} x^{3} - 385 \, \sqrt {d x} a^{4} d^{10} x\right )}}{{\left (b d^{2} x^{2} + a d^{2}\right )}^{5} a b^{4}}\right )} \]
1/163840*d^8*(2310*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^2*b^7*d) + 2310*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^2*b^7*d) - 1155*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^2*b^7*d) + 1155*sq rt(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^2*b^7*d) + 8*(1155*sqrt(d*x)*b^4*d^10*x^9 - 2648*sqrt(d*x)*a *b^3*d^10*x^7 - 3130*sqrt(d*x)*a^2*b^2*d^10*x^5 - 1760*sqrt(d*x)*a^3*b*d^1 0*x^3 - 385*sqrt(d*x)*a^4*d^10*x)/((b*d^2*x^2 + a*d^2)^5*a*b^4))
Time = 14.28 (sec) , antiderivative size = 210, normalized size of antiderivative = 0.54 \[ \int \frac {(d x)^{17/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {231\,d^{17/2}\,\mathrm {atanh}\left (\frac {b^{1/4}\,\sqrt {d\,x}}{{\left (-a\right )}^{1/4}\,\sqrt {d}}\right )}{8192\,{\left (-a\right )}^{5/4}\,b^{19/4}}-\frac {231\,d^{17/2}\,\mathrm {atan}\left (\frac {b^{1/4}\,\sqrt {d\,x}}{{\left (-a\right )}^{1/4}\,\sqrt {d}}\right )}{8192\,{\left (-a\right )}^{5/4}\,b^{19/4}}-\frac {\frac {331\,d^{11}\,{\left (d\,x\right )}^{15/2}}{2560\,b}-\frac {231\,d^9\,{\left (d\,x\right )}^{19/2}}{4096\,a}+\frac {11\,a^2\,d^{15}\,{\left (d\,x\right )}^{7/2}}{128\,b^3}+\frac {77\,a^3\,d^{17}\,{\left (d\,x\right )}^{3/2}}{4096\,b^4}+\frac {313\,a\,d^{13}\,{\left (d\,x\right )}^{11/2}}{2048\,b^2}}{a^5\,d^{10}+5\,a^4\,b\,d^{10}\,x^2+10\,a^3\,b^2\,d^{10}\,x^4+10\,a^2\,b^3\,d^{10}\,x^6+5\,a\,b^4\,d^{10}\,x^8+b^5\,d^{10}\,x^{10}} \]
(231*d^(17/2)*atanh((b^(1/4)*(d*x)^(1/2))/((-a)^(1/4)*d^(1/2))))/(8192*(-a )^(5/4)*b^(19/4)) - (231*d^(17/2)*atan((b^(1/4)*(d*x)^(1/2))/((-a)^(1/4)*d ^(1/2))))/(8192*(-a)^(5/4)*b^(19/4)) - ((331*d^11*(d*x)^(15/2))/(2560*b) - (231*d^9*(d*x)^(19/2))/(4096*a) + (11*a^2*d^15*(d*x)^(7/2))/(128*b^3) + ( 77*a^3*d^17*(d*x)^(3/2))/(4096*b^4) + (313*a*d^13*(d*x)^(11/2))/(2048*b^2) )/(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)