Integrand size = 28, antiderivative size = 422 \[ \int \frac {1}{(d x)^{7/2} \left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=-\frac {13923}{4096 a^6 d (d x)^{5/2}}+\frac {69615 b}{4096 a^7 d^3 \sqrt {d x}}+\frac {1}{10 a d (d x)^{5/2} \left (a+b x^2\right )^5}+\frac {5}{32 a^2 d (d x)^{5/2} \left (a+b x^2\right )^4}+\frac {35}{128 a^3 d (d x)^{5/2} \left (a+b x^2\right )^3}+\frac {595}{1024 a^4 d (d x)^{5/2} \left (a+b x^2\right )^2}+\frac {7735}{4096 a^5 d (d x)^{5/2} \left (a+b x^2\right )}-\frac {69615 b^{5/4} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} a^{29/4} d^{7/2}}+\frac {69615 b^{5/4} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} a^{29/4} d^{7/2}}+\frac {69615 b^{5/4} \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{16384 \sqrt {2} a^{29/4} d^{7/2}}-\frac {69615 b^{5/4} \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{16384 \sqrt {2} a^{29/4} d^{7/2}} \]
-13923/4096/a^6/d/(d*x)^(5/2)+1/10/a/d/(d*x)^(5/2)/(b*x^2+a)^5+5/32/a^2/d/ (d*x)^(5/2)/(b*x^2+a)^4+35/128/a^3/d/(d*x)^(5/2)/(b*x^2+a)^3+595/1024/a^4/ d/(d*x)^(5/2)/(b*x^2+a)^2+7735/4096/a^5/d/(d*x)^(5/2)/(b*x^2+a)-69615/1638 4*b^(5/4)*arctan(1-b^(1/4)*2^(1/2)*(d*x)^(1/2)/a^(1/4)/d^(1/2))/a^(29/4)/d ^(7/2)*2^(1/2)+69615/16384*b^(5/4)*arctan(1+b^(1/4)*2^(1/2)*(d*x)^(1/2)/a^ (1/4)/d^(1/2))/a^(29/4)/d^(7/2)*2^(1/2)+69615/32768*b^(5/4)*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^(29/4)/d^(7/ 2)*2^(1/2)-69615/32768*b^(5/4)*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^(29/4)/d^(7/2)*2^(1/2)+69615/4096*b/a^7/d ^3/(d*x)^(1/2)
Time = 0.81 (sec) , antiderivative size = 211, normalized size of antiderivative = 0.50 \[ \int \frac {1}{(d x)^{7/2} \left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {\sqrt {d x} \left (\frac {4 \sqrt [4]{a} \left (-8192 a^6+204800 a^5 b x^2+1317575 a^4 b^2 x^4+2951200 a^3 b^3 x^6+3171350 a^2 b^4 x^8+1670760 a b^5 x^{10}+348075 b^6 x^{12}\right )}{\left (a+b x^2\right )^5}-348075 \sqrt {2} b^{5/4} x^{5/2} \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )-348075 \sqrt {2} b^{5/4} x^{5/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )\right )}{81920 a^{29/4} d^4 x^3} \]
(Sqrt[d*x]*((4*a^(1/4)*(-8192*a^6 + 204800*a^5*b*x^2 + 1317575*a^4*b^2*x^4 + 2951200*a^3*b^3*x^6 + 3171350*a^2*b^4*x^8 + 1670760*a*b^5*x^10 + 348075 *b^6*x^12))/(a + b*x^2)^5 - 348075*Sqrt[2]*b^(5/4)*x^(5/2)*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])] - 348075*Sqrt[2]*b^(5/4)* x^(5/2)*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)])) /(81920*a^(29/4)*d^4*x^3)
Time = 0.71 (sec) , antiderivative size = 488, normalized size of antiderivative = 1.16, number of steps used = 20, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.679, Rules used = {1380, 27, 253, 253, 253, 253, 253, 264, 264, 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.
| \(\displaystyle \int \frac {1}{(d x)^{7/2} \left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx\) |
| \(\Big \downarrow \) 1380 |
| \(\displaystyle b^6 \int \frac {1}{b^6 (d x)^{7/2} \left (b x^2+a\right )^6}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \frac {1}{(d x)^{7/2} \left (a+b x^2\right )^6}dx\) |
| \(\Big \downarrow \) 253 |
| \(\displaystyle \frac {5 \int \frac {1}{(d x)^{7/2} \left (b x^2+a\right )^5}dx}{4 a}+\frac {1}{10 a d (d x)^{5/2} \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 253 |
| \(\displaystyle \frac {5 \left (\frac {21 \int \frac {1}{(d x)^{7/2} \left (b x^2+a\right )^4}dx}{16 a}+\frac {1}{8 a d (d x)^{5/2} \left (a+b x^2\right )^4}\right )}{4 a}+\frac {1}{10 a d (d x)^{5/2} \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 253 |
| \(\displaystyle \frac {5 \left (\frac {21 \left (\frac {17 \int \frac {1}{(d x)^{7/2} \left (b x^2+a\right )^3}dx}{12 a}+\frac {1}{6 a d (d x)^{5/2} \left (a+b x^2\right )^3}\right )}{16 a}+\frac {1}{8 a d (d x)^{5/2} \left (a+b x^2\right )^4}\right )}{4 a}+\frac {1}{10 a d (d x)^{5/2} \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 253 |
| \(\displaystyle \frac {5 \left (\frac {21 \left (\frac {17 \left (\frac {13 \int \frac {1}{(d x)^{7/2} \left (b x^2+a\right )^2}dx}{8 a}+\frac {1}{4 a d (d x)^{5/2} \left (a+b x^2\right )^2}\right )}{12 a}+\frac {1}{6 a d (d x)^{5/2} \left (a+b x^2\right )^3}\right )}{16 a}+\frac {1}{8 a d (d x)^{5/2} \left (a+b x^2\right )^4}\right )}{4 a}+\frac {1}{10 a d (d x)^{5/2} \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 253 |
| \(\displaystyle \frac {5 \left (\frac {21 \left (\frac {17 \left (\frac {13 \left (\frac {9 \int \frac {1}{(d x)^{7/2} \left (b x^2+a\right )}dx}{4 a}+\frac {1}{2 a d (d x)^{5/2} \left (a+b x^2\right )}\right )}{8 a}+\frac {1}{4 a d (d x)^{5/2} \left (a+b x^2\right )^2}\right )}{12 a}+\frac {1}{6 a d (d x)^{5/2} \left (a+b x^2\right )^3}\right )}{16 a}+\frac {1}{8 a d (d x)^{5/2} \left (a+b x^2\right )^4}\right )}{4 a}+\frac {1}{10 a d (d x)^{5/2} \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 264 |
| \(\displaystyle \frac {5 \left (\frac {21 \left (\frac {17 \left (\frac {13 \left (\frac {9 \left (-\frac {b \int \frac {1}{(d x)^{3/2} \left (b x^2+a\right )}dx}{a d^2}-\frac {2}{5 a d (d x)^{5/2}}\right )}{4 a}+\frac {1}{2 a d (d x)^{5/2} \left (a+b x^2\right )}\right )}{8 a}+\frac {1}{4 a d (d x)^{5/2} \left (a+b x^2\right )^2}\right )}{12 a}+\frac {1}{6 a d (d x)^{5/2} \left (a+b x^2\right )^3}\right )}{16 a}+\frac {1}{8 a d (d x)^{5/2} \left (a+b x^2\right )^4}\right )}{4 a}+\frac {1}{10 a d (d x)^{5/2} \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 264 |
| \(\displaystyle \frac {5 \left (\frac {21 \left (\frac {17 \left (\frac {13 \left (\frac {9 \left (-\frac {b \left (-\frac {b \int \frac {\sqrt {d x}}{b x^2+a}dx}{a d^2}-\frac {2}{a d \sqrt {d x}}\right )}{a d^2}-\frac {2}{5 a d (d x)^{5/2}}\right )}{4 a}+\frac {1}{2 a d (d x)^{5/2} \left (a+b x^2\right )}\right )}{8 a}+\frac {1}{4 a d (d x)^{5/2} \left (a+b x^2\right )^2}\right )}{12 a}+\frac {1}{6 a d (d x)^{5/2} \left (a+b x^2\right )^3}\right )}{16 a}+\frac {1}{8 a d (d x)^{5/2} \left (a+b x^2\right )^4}\right )}{4 a}+\frac {1}{10 a d (d x)^{5/2} \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {5 \left (\frac {21 \left (\frac {17 \left (\frac {13 \left (\frac {9 \left (-\frac {b \left (-\frac {2 b \int \frac {d^3 x}{b x^2 d^2+a d^2}d\sqrt {d x}}{a d^3}-\frac {2}{a d \sqrt {d x}}\right )}{a d^2}-\frac {2}{5 a d (d x)^{5/2}}\right )}{4 a}+\frac {1}{2 a d (d x)^{5/2} \left (a+b x^2\right )}\right )}{8 a}+\frac {1}{4 a d (d x)^{5/2} \left (a+b x^2\right )^2}\right )}{12 a}+\frac {1}{6 a d (d x)^{5/2} \left (a+b x^2\right )^3}\right )}{16 a}+\frac {1}{8 a d (d x)^{5/2} \left (a+b x^2\right )^4}\right )}{4 a}+\frac {1}{10 a d (d x)^{5/2} \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5 \left (\frac {21 \left (\frac {17 \left (\frac {13 \left (\frac {9 \left (-\frac {b \left (-\frac {2 b \int \frac {d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{a d}-\frac {2}{a d \sqrt {d x}}\right )}{a d^2}-\frac {2}{5 a d (d x)^{5/2}}\right )}{4 a}+\frac {1}{2 a d (d x)^{5/2} \left (a+b x^2\right )}\right )}{8 a}+\frac {1}{4 a d (d x)^{5/2} \left (a+b x^2\right )^2}\right )}{12 a}+\frac {1}{6 a d (d x)^{5/2} \left (a+b x^2\right )^3}\right )}{16 a}+\frac {1}{8 a d (d x)^{5/2} \left (a+b x^2\right )^4}\right )}{4 a}+\frac {1}{10 a d (d x)^{5/2} \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 826 |
| \(\displaystyle \frac {5 \left (\frac {21 \left (\frac {17 \left (\frac {13 \left (\frac {9 \left (-\frac {b \left (-\frac {2 b \left (\frac {\int \frac {\sqrt {b} x d+\sqrt {a} d}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {b}}-\frac {\int \frac {\sqrt {a} d-\sqrt {b} d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {b}}\right )}{a d}-\frac {2}{a d \sqrt {d x}}\right )}{a d^2}-\frac {2}{5 a d (d x)^{5/2}}\right )}{4 a}+\frac {1}{2 a d (d x)^{5/2} \left (a+b x^2\right )}\right )}{8 a}+\frac {1}{4 a d (d x)^{5/2} \left (a+b x^2\right )^2}\right )}{12 a}+\frac {1}{6 a d (d x)^{5/2} \left (a+b x^2\right )^3}\right )}{16 a}+\frac {1}{8 a d (d x)^{5/2} \left (a+b x^2\right )^4}\right )}{4 a}+\frac {1}{10 a d (d x)^{5/2} \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {5 \left (\frac {21 \left (\frac {17 \left (\frac {13 \left (\frac {9 \left (-\frac {b \left (-\frac {2 b \left (\frac {\frac {\int \frac {1}{x d+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}}d\sqrt {d x}}{2 \sqrt {b}}+\frac {\int \frac {1}{x d+\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}}d\sqrt {d x}}{2 \sqrt {b}}}{2 \sqrt {b}}-\frac {\int \frac {\sqrt {a} d-\sqrt {b} d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {b}}\right )}{a d}-\frac {2}{a d \sqrt {d x}}\right )}{a d^2}-\frac {2}{5 a d (d x)^{5/2}}\right )}{4 a}+\frac {1}{2 a d (d x)^{5/2} \left (a+b x^2\right )}\right )}{8 a}+\frac {1}{4 a d (d x)^{5/2} \left (a+b x^2\right )^2}\right )}{12 a}+\frac {1}{6 a d (d x)^{5/2} \left (a+b x^2\right )^3}\right )}{16 a}+\frac {1}{8 a d (d x)^{5/2} \left (a+b x^2\right )^4}\right )}{4 a}+\frac {1}{10 a d (d x)^{5/2} \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {5 \left (\frac {21 \left (\frac {17 \left (\frac {13 \left (\frac {9 \left (-\frac {b \left (-\frac {2 b \left (\frac {\frac {\int \frac {1}{-d x-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\int \frac {1}{-d x-1}d\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}}{2 \sqrt {b}}-\frac {\int \frac {\sqrt {a} d-\sqrt {b} d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {b}}\right )}{a d}-\frac {2}{a d \sqrt {d x}}\right )}{a d^2}-\frac {2}{5 a d (d x)^{5/2}}\right )}{4 a}+\frac {1}{2 a d (d x)^{5/2} \left (a+b x^2\right )}\right )}{8 a}+\frac {1}{4 a d (d x)^{5/2} \left (a+b x^2\right )^2}\right )}{12 a}+\frac {1}{6 a d (d x)^{5/2} \left (a+b x^2\right )^3}\right )}{16 a}+\frac {1}{8 a d (d x)^{5/2} \left (a+b x^2\right )^4}\right )}{4 a}+\frac {1}{10 a d (d x)^{5/2} \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {5 \left (\frac {21 \left (\frac {17 \left (\frac {13 \left (\frac {9 \left (-\frac {b \left (-\frac {2 b \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}}{2 \sqrt {b}}-\frac {\int \frac {\sqrt {a} d-\sqrt {b} d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {b}}\right )}{a d}-\frac {2}{a d \sqrt {d x}}\right )}{a d^2}-\frac {2}{5 a d (d x)^{5/2}}\right )}{4 a}+\frac {1}{2 a d (d x)^{5/2} \left (a+b x^2\right )}\right )}{8 a}+\frac {1}{4 a d (d x)^{5/2} \left (a+b x^2\right )^2}\right )}{12 a}+\frac {1}{6 a d (d x)^{5/2} \left (a+b x^2\right )^3}\right )}{16 a}+\frac {1}{8 a d (d x)^{5/2} \left (a+b x^2\right )^4}\right )}{4 a}+\frac {1}{10 a d (d x)^{5/2} \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {5 \left (\frac {21 \left (\frac {17 \left (\frac {13 \left (\frac {9 \left (-\frac {b \left (-\frac {2 b \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}}{2 \sqrt {b}}-\frac {-\frac {\int -\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d}-2 \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{b} \left (x d+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt [4]{a} \sqrt {d}+\sqrt {2} \sqrt [4]{b} \sqrt {d x}\right )}{\sqrt [4]{b} \left (x d+\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}}{2 \sqrt {b}}\right )}{a d}-\frac {2}{a d \sqrt {d x}}\right )}{a d^2}-\frac {2}{5 a d (d x)^{5/2}}\right )}{4 a}+\frac {1}{2 a d (d x)^{5/2} \left (a+b x^2\right )}\right )}{8 a}+\frac {1}{4 a d (d x)^{5/2} \left (a+b x^2\right )^2}\right )}{12 a}+\frac {1}{6 a d (d x)^{5/2} \left (a+b x^2\right )^3}\right )}{16 a}+\frac {1}{8 a d (d x)^{5/2} \left (a+b x^2\right )^4}\right )}{4 a}+\frac {1}{10 a d (d x)^{5/2} \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {5 \left (\frac {21 \left (\frac {17 \left (\frac {13 \left (\frac {9 \left (-\frac {b \left (-\frac {2 b \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}}{2 \sqrt {b}}-\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{a} \sqrt {d}-2 \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{b} \left (x d+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}+\frac {\int \frac {\sqrt {2} \left (\sqrt [4]{a} \sqrt {d}+\sqrt {2} \sqrt [4]{b} \sqrt {d x}\right )}{\sqrt [4]{b} \left (x d+\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}}{2 \sqrt {b}}\right )}{a d}-\frac {2}{a d \sqrt {d x}}\right )}{a d^2}-\frac {2}{5 a d (d x)^{5/2}}\right )}{4 a}+\frac {1}{2 a d (d x)^{5/2} \left (a+b x^2\right )}\right )}{8 a}+\frac {1}{4 a d (d x)^{5/2} \left (a+b x^2\right )^2}\right )}{12 a}+\frac {1}{6 a d (d x)^{5/2} \left (a+b x^2\right )^3}\right )}{16 a}+\frac {1}{8 a d (d x)^{5/2} \left (a+b x^2\right )^4}\right )}{4 a}+\frac {1}{10 a d (d x)^{5/2} \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5 \left (\frac {21 \left (\frac {17 \left (\frac {13 \left (\frac {9 \left (-\frac {b \left (-\frac {2 b \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}}{2 \sqrt {b}}-\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{a} \sqrt {d}-2 \sqrt [4]{b} \sqrt {d x}}{x d+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt {b} \sqrt {d}}+\frac {\int \frac {\sqrt [4]{a} \sqrt {d}+\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{x d+\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}}d\sqrt {d x}}{2 \sqrt [4]{a} \sqrt {b} \sqrt {d}}}{2 \sqrt {b}}\right )}{a d}-\frac {2}{a d \sqrt {d x}}\right )}{a d^2}-\frac {2}{5 a d (d x)^{5/2}}\right )}{4 a}+\frac {1}{2 a d (d x)^{5/2} \left (a+b x^2\right )}\right )}{8 a}+\frac {1}{4 a d (d x)^{5/2} \left (a+b x^2\right )^2}\right )}{12 a}+\frac {1}{6 a d (d x)^{5/2} \left (a+b x^2\right )^3}\right )}{16 a}+\frac {1}{8 a d (d x)^{5/2} \left (a+b x^2\right )^4}\right )}{4 a}+\frac {1}{10 a d (d x)^{5/2} \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {5 \left (\frac {21 \left (\frac {17 \left (\frac {13 \left (\frac {9 \left (-\frac {b \left (-\frac {2 b \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}}{2 \sqrt {b}}-\frac {\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d} \sqrt {d x}+\sqrt {a} d+\sqrt {b} d x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d} \sqrt {d x}+\sqrt {a} d+\sqrt {b} d x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}}{2 \sqrt {b}}\right )}{a d}-\frac {2}{a d \sqrt {d x}}\right )}{a d^2}-\frac {2}{5 a d (d x)^{5/2}}\right )}{4 a}+\frac {1}{2 a d (d x)^{5/2} \left (a+b x^2\right )}\right )}{8 a}+\frac {1}{4 a d (d x)^{5/2} \left (a+b x^2\right )^2}\right )}{12 a}+\frac {1}{6 a d (d x)^{5/2} \left (a+b x^2\right )^3}\right )}{16 a}+\frac {1}{8 a d (d x)^{5/2} \left (a+b x^2\right )^4}\right )}{4 a}+\frac {1}{10 a d (d x)^{5/2} \left (a+b x^2\right )^5}\) |
1/(10*a*d*(d*x)^(5/2)*(a + b*x^2)^5) + (5*(1/(8*a*d*(d*x)^(5/2)*(a + b*x^2 )^4) + (21*(1/(6*a*d*(d*x)^(5/2)*(a + b*x^2)^3) + (17*(1/(4*a*d*(d*x)^(5/2 )*(a + b*x^2)^2) + (13*(1/(2*a*d*(d*x)^(5/2)*(a + b*x^2)) + (9*(-2/(5*a*d* (d*x)^(5/2)) - (b*(-2/(a*d*Sqrt[d*x]) - (2*b*((-(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])) + ArcT an[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*Sq rt[2]*a^(1/4)*b^(1/4)*Sqrt[d]))/(2*Sqrt[b])))/(a*d)))/(a*d^2)))/(4*a)))/(8 *a)))/(12*a)))/(16*a)))/(4*a)
3.8.27.3.1 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] :> Simp[(c*x)^( m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + 2*p + 3)/(a*c ^2*(m + 1))) Int[(c*x)^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, p }, x] && LtQ[m, -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 = 0.58 (sec) , antiderivative size = 261, normalized size of antiderivative = 0.62
| method | result | size |
| risch | \(-\frac {2 \left (-30 b \,x^{2}+a \right )}{5 a^{7} \sqrt {d x}\, x^{2} d^{3}}+\frac {b^{2} \left (\frac {\frac {34139 a^{4} d^{8} \left (d x \right )^{\frac {3}{2}}}{4096}+\frac {3597 a^{3} b \,d^{6} \left (d x \right )^{\frac {7}{2}}}{128}+\frac {75471 a^{2} d^{4} b^{2} \left (d x \right )^{\frac {11}{2}}}{2048}+\frac {56269 a \,b^{3} d^{2} \left (d x \right )^{\frac {15}{2}}}{2560}+\frac {20463 b^{4} \left (d x \right )^{\frac {19}{2}}}{4096}}{\left (b \,d^{2} x^{2}+a \,d^{2}\right )^{5}}+\frac {69615 \sqrt {2}\, \left (\ln \left (\frac {d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{32768 b \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{a^{7} d^{3}}\) | \(261\) |
| derivativedivides | \(2 d^{11} \left (-\frac {1}{5 a^{6} d^{12} \left (d x \right )^{\frac {5}{2}}}+\frac {6 b}{a^{7} d^{14} \sqrt {d x}}+\frac {b^{2} \left (\frac {\frac {34139 a^{4} d^{8} \left (d x \right )^{\frac {3}{2}}}{8192}+\frac {3597 a^{3} b \,d^{6} \left (d x \right )^{\frac {7}{2}}}{256}+\frac {75471 a^{2} d^{4} b^{2} \left (d x \right )^{\frac {11}{2}}}{4096}+\frac {56269 a \,b^{3} d^{2} \left (d x \right )^{\frac {15}{2}}}{5120}+\frac {20463 b^{4} \left (d x \right )^{\frac {19}{2}}}{8192}}{\left (b \,d^{2} x^{2}+a \,d^{2}\right )^{5}}+\frac {69615 \sqrt {2}\, \left (\ln \left (\frac {d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{65536 b \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{a^{7} d^{14}}\right )\) | \(268\) |
| default | \(2 d^{11} \left (-\frac {1}{5 a^{6} d^{12} \left (d x \right )^{\frac {5}{2}}}+\frac {6 b}{a^{7} d^{14} \sqrt {d x}}+\frac {b^{2} \left (\frac {\frac {34139 a^{4} d^{8} \left (d x \right )^{\frac {3}{2}}}{8192}+\frac {3597 a^{3} b \,d^{6} \left (d x \right )^{\frac {7}{2}}}{256}+\frac {75471 a^{2} d^{4} b^{2} \left (d x \right )^{\frac {11}{2}}}{4096}+\frac {56269 a \,b^{3} d^{2} \left (d x \right )^{\frac {15}{2}}}{5120}+\frac {20463 b^{4} \left (d x \right )^{\frac {19}{2}}}{8192}}{\left (b \,d^{2} x^{2}+a \,d^{2}\right )^{5}}+\frac {69615 \sqrt {2}\, \left (\ln \left (\frac {d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{65536 b \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{a^{7} d^{14}}\right )\) | \(268\) |
| pseudoelliptic | \(-\frac {2 \left (-\frac {348075 b \,x^{2} \sqrt {2}\, \left (b \,x^{2}+a \right )^{5} \left (2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}-\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}+\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )+\ln \left (\frac {d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )\right ) \sqrt {d x}}{65536}+\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \left (-\frac {348075}{8192} b^{6} x^{12}+a^{6}-\frac {1317575}{8192} a^{4} b^{2} x^{4}-25 a^{5} b \,x^{2}-\frac {208845}{1024} a \,b^{5} x^{10}-\frac {1585675}{4096} a^{2} b^{4} x^{8}-\frac {92225}{256} a^{3} b^{3} x^{6}\right )\right )}{5 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, d^{3} a^{7} x^{2} \left (b \,x^{2}+a \right )^{5}}\) | \(276\) |
-2/5*(-30*b*x^2+a)/a^7/(d*x)^(1/2)/x^2/d^3+b^2/a^7*(2*(34139/8192*a^4*d^8* (d*x)^(3/2)+3597/256*a^3*b*d^6*(d*x)^(7/2)+75471/4096*a^2*d^4*b^2*(d*x)^(1 1/2)+56269/5120*a*b^3*d^2*(d*x)^(15/2)+20463/8192*b^4*(d*x)^(19/2))/(b*d^2 *x^2+a*d^2)^5+69615/32768/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)))/d^3
Result contains complex when optimal does not.
Time = 0.29 (sec) , antiderivative size = 663, normalized size of antiderivative = 1.57 \[ \int \frac {1}{(d x)^{7/2} \left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {348075 \, {\left (a^{7} b^{5} d^{4} x^{13} + 5 \, a^{8} b^{4} d^{4} x^{11} + 10 \, a^{9} b^{3} d^{4} x^{9} + 10 \, a^{10} b^{2} d^{4} x^{7} + 5 \, a^{11} b d^{4} x^{5} + a^{12} d^{4} x^{3}\right )} \left (-\frac {b^{5}}{a^{29} d^{14}}\right )^{\frac {1}{4}} \log \left (337371570183375 \, a^{22} d^{11} \left (-\frac {b^{5}}{a^{29} d^{14}}\right )^{\frac {3}{4}} + 337371570183375 \, \sqrt {d x} b^{4}\right ) - 348075 \, {\left (i \, a^{7} b^{5} d^{4} x^{13} + 5 i \, a^{8} b^{4} d^{4} x^{11} + 10 i \, a^{9} b^{3} d^{4} x^{9} + 10 i \, a^{10} b^{2} d^{4} x^{7} + 5 i \, a^{11} b d^{4} x^{5} + i \, a^{12} d^{4} x^{3}\right )} \left (-\frac {b^{5}}{a^{29} d^{14}}\right )^{\frac {1}{4}} \log \left (337371570183375 i \, a^{22} d^{11} \left (-\frac {b^{5}}{a^{29} d^{14}}\right )^{\frac {3}{4}} + 337371570183375 \, \sqrt {d x} b^{4}\right ) - 348075 \, {\left (-i \, a^{7} b^{5} d^{4} x^{13} - 5 i \, a^{8} b^{4} d^{4} x^{11} - 10 i \, a^{9} b^{3} d^{4} x^{9} - 10 i \, a^{10} b^{2} d^{4} x^{7} - 5 i \, a^{11} b d^{4} x^{5} - i \, a^{12} d^{4} x^{3}\right )} \left (-\frac {b^{5}}{a^{29} d^{14}}\right )^{\frac {1}{4}} \log \left (-337371570183375 i \, a^{22} d^{11} \left (-\frac {b^{5}}{a^{29} d^{14}}\right )^{\frac {3}{4}} + 337371570183375 \, \sqrt {d x} b^{4}\right ) - 348075 \, {\left (a^{7} b^{5} d^{4} x^{13} + 5 \, a^{8} b^{4} d^{4} x^{11} + 10 \, a^{9} b^{3} d^{4} x^{9} + 10 \, a^{10} b^{2} d^{4} x^{7} + 5 \, a^{11} b d^{4} x^{5} + a^{12} d^{4} x^{3}\right )} \left (-\frac {b^{5}}{a^{29} d^{14}}\right )^{\frac {1}{4}} \log \left (-337371570183375 \, a^{22} d^{11} \left (-\frac {b^{5}}{a^{29} d^{14}}\right )^{\frac {3}{4}} + 337371570183375 \, \sqrt {d x} b^{4}\right ) + 4 \, {\left (348075 \, b^{6} x^{12} + 1670760 \, a b^{5} x^{10} + 3171350 \, a^{2} b^{4} x^{8} + 2951200 \, a^{3} b^{3} x^{6} + 1317575 \, a^{4} b^{2} x^{4} + 204800 \, a^{5} b x^{2} - 8192 \, a^{6}\right )} \sqrt {d x}}{81920 \, {\left (a^{7} b^{5} d^{4} x^{13} + 5 \, a^{8} b^{4} d^{4} x^{11} + 10 \, a^{9} b^{3} d^{4} x^{9} + 10 \, a^{10} b^{2} d^{4} x^{7} + 5 \, a^{11} b d^{4} x^{5} + a^{12} d^{4} x^{3}\right )}} \]
1/81920*(348075*(a^7*b^5*d^4*x^13 + 5*a^8*b^4*d^4*x^11 + 10*a^9*b^3*d^4*x^ 9 + 10*a^10*b^2*d^4*x^7 + 5*a^11*b*d^4*x^5 + a^12*d^4*x^3)*(-b^5/(a^29*d^1 4))^(1/4)*log(337371570183375*a^22*d^11*(-b^5/(a^29*d^14))^(3/4) + 3373715 70183375*sqrt(d*x)*b^4) - 348075*(I*a^7*b^5*d^4*x^13 + 5*I*a^8*b^4*d^4*x^1 1 + 10*I*a^9*b^3*d^4*x^9 + 10*I*a^10*b^2*d^4*x^7 + 5*I*a^11*b*d^4*x^5 + I* a^12*d^4*x^3)*(-b^5/(a^29*d^14))^(1/4)*log(337371570183375*I*a^22*d^11*(-b ^5/(a^29*d^14))^(3/4) + 337371570183375*sqrt(d*x)*b^4) - 348075*(-I*a^7*b^ 5*d^4*x^13 - 5*I*a^8*b^4*d^4*x^11 - 10*I*a^9*b^3*d^4*x^9 - 10*I*a^10*b^2*d ^4*x^7 - 5*I*a^11*b*d^4*x^5 - I*a^12*d^4*x^3)*(-b^5/(a^29*d^14))^(1/4)*log (-337371570183375*I*a^22*d^11*(-b^5/(a^29*d^14))^(3/4) + 337371570183375*s qrt(d*x)*b^4) - 348075*(a^7*b^5*d^4*x^13 + 5*a^8*b^4*d^4*x^11 + 10*a^9*b^3 *d^4*x^9 + 10*a^10*b^2*d^4*x^7 + 5*a^11*b*d^4*x^5 + a^12*d^4*x^3)*(-b^5/(a ^29*d^14))^(1/4)*log(-337371570183375*a^22*d^11*(-b^5/(a^29*d^14))^(3/4) + 337371570183375*sqrt(d*x)*b^4) + 4*(348075*b^6*x^12 + 1670760*a*b^5*x^10 + 3171350*a^2*b^4*x^8 + 2951200*a^3*b^3*x^6 + 1317575*a^4*b^2*x^4 + 204800 *a^5*b*x^2 - 8192*a^6)*sqrt(d*x))/(a^7*b^5*d^4*x^13 + 5*a^8*b^4*d^4*x^11 + 10*a^9*b^3*d^4*x^9 + 10*a^10*b^2*d^4*x^7 + 5*a^11*b*d^4*x^5 + a^12*d^4*x^ 3)
\[ \int \frac {1}{(d x)^{7/2} \left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\int \frac {1}{\left (d x\right )^{\frac {7}{2}} \left (a + b x^{2}\right )^{6}}\, dx \]
Time = 0.42 (sec) , antiderivative size = 410, normalized size of antiderivative = 0.97 \[ \int \frac {1}{(d x)^{7/2} \left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {\frac {8 \, {\left (348075 \, b^{6} d^{12} x^{12} + 1670760 \, a b^{5} d^{12} x^{10} + 3171350 \, a^{2} b^{4} d^{12} x^{8} + 2951200 \, a^{3} b^{3} d^{12} x^{6} + 1317575 \, a^{4} b^{2} d^{12} x^{4} + 204800 \, a^{5} b d^{12} x^{2} - 8192 \, a^{6} d^{12}\right )}}{\left (d x\right )^{\frac {25}{2}} a^{7} b^{5} d^{2} + 5 \, \left (d x\right )^{\frac {21}{2}} a^{8} b^{4} d^{4} + 10 \, \left (d x\right )^{\frac {17}{2}} a^{9} b^{3} d^{6} + 10 \, \left (d x\right )^{\frac {13}{2}} a^{10} b^{2} d^{8} + 5 \, \left (d x\right )^{\frac {9}{2}} a^{11} b d^{10} + \left (d x\right )^{\frac {5}{2}} a^{12} d^{12}} + \frac {348075 \, b^{2} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {d x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} d}}\right )}{\sqrt {\sqrt {a} \sqrt {b} d} \sqrt {b}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {d x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} d}}\right )}{\sqrt {\sqrt {a} \sqrt {b} d} \sqrt {b}} - \frac {\sqrt {2} \log \left (\sqrt {b} d x + \sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} \sqrt {d x} b^{\frac {1}{4}} + \sqrt {a} d\right )}{\left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (\sqrt {b} d x - \sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} \sqrt {d x} b^{\frac {1}{4}} + \sqrt {a} d\right )}{\left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {3}{4}}}\right )}}{a^{7} d^{2}}}{163840 \, d} \]
1/163840*(8*(348075*b^6*d^12*x^12 + 1670760*a*b^5*d^12*x^10 + 3171350*a^2* b^4*d^12*x^8 + 2951200*a^3*b^3*d^12*x^6 + 1317575*a^4*b^2*d^12*x^4 + 20480 0*a^5*b*d^12*x^2 - 8192*a^6*d^12)/((d*x)^(25/2)*a^7*b^5*d^2 + 5*(d*x)^(21/ 2)*a^8*b^4*d^4 + 10*(d*x)^(17/2)*a^9*b^3*d^6 + 10*(d*x)^(13/2)*a^10*b^2*d^ 8 + 5*(d*x)^(9/2)*a^11*b*d^10 + (d*x)^(5/2)*a^12*d^12) + 348075*b^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^7*d^2))/d
Time = 0.29 (sec) , antiderivative size = 362, normalized size of antiderivative = 0.86 \[ \int \frac {1}{(d x)^{7/2} \left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {69615 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{16384 \, a^{8} b d^{5}} + \frac {69615 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{16384 \, a^{8} b d^{5}} - \frac {69615 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \log \left (d x + \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{32768 \, a^{8} b d^{5}} + \frac {69615 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \log \left (d x - \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{32768 \, a^{8} b d^{5}} + \frac {348075 \, b^{6} d^{12} x^{12} + 1670760 \, a b^{5} d^{12} x^{10} + 3171350 \, a^{2} b^{4} d^{12} x^{8} + 2951200 \, a^{3} b^{3} d^{12} x^{6} + 1317575 \, a^{4} b^{2} d^{12} x^{4} + 204800 \, a^{5} b d^{12} x^{2} - 8192 \, a^{6} d^{12}}{20480 \, {\left (\sqrt {d x} b d^{2} x^{2} + \sqrt {d x} a d^{2}\right )}^{5} a^{7} d^{3}} \]
69615/16384*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^8*b*d^5) + 69615/16384*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^8*b*d^5) - 69615/32768*sqrt(2)*(a*b^3*d^2)^(3/4)*l og(d*x + sqrt(2)*(a*d^2/b)^(1/4)*sqrt(d*x) + sqrt(a*d^2/b))/(a^8*b*d^5) + 69615/32768*sqrt(2)*(a*b^3*d^2)^(3/4)*log(d*x - sqrt(2)*(a*d^2/b)^(1/4)*sq rt(d*x) + sqrt(a*d^2/b))/(a^8*b*d^5) + 1/20480*(348075*b^6*d^12*x^12 + 167 0760*a*b^5*d^12*x^10 + 3171350*a^2*b^4*d^12*x^8 + 2951200*a^3*b^3*d^12*x^6 + 1317575*a^4*b^2*d^12*x^4 + 204800*a^5*b*d^12*x^2 - 8192*a^6*d^12)/((sqr t(d*x)*b*d^2*x^2 + sqrt(d*x)*a*d^2)^5*a^7*d^3)
Time = 13.43 (sec) , antiderivative size = 239, normalized size of antiderivative = 0.57 \[ \int \frac {1}{(d x)^{7/2} \left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {\frac {10\,b\,d^9\,x^2}{a^2}-\frac {2\,d^9}{5\,a}+\frac {263515\,b^2\,d^9\,x^4}{4096\,a^3}+\frac {18445\,b^3\,d^9\,x^6}{128\,a^4}+\frac {317135\,b^4\,d^9\,x^8}{2048\,a^5}+\frac {41769\,b^5\,d^9\,x^{10}}{512\,a^6}+\frac {69615\,b^6\,d^9\,x^{12}}{4096\,a^7}}{b^5\,{\left (d\,x\right )}^{25/2}+a^5\,d^{10}\,{\left (d\,x\right )}^{5/2}+10\,a^3\,b^2\,d^6\,{\left (d\,x\right )}^{13/2}+10\,a^2\,b^3\,d^4\,{\left (d\,x\right )}^{17/2}+5\,a^4\,b\,d^8\,{\left (d\,x\right )}^{9/2}+5\,a\,b^4\,d^2\,{\left (d\,x\right )}^{21/2}}-\frac {69615\,{\left (-b\right )}^{5/4}\,\mathrm {atan}\left (\frac {{\left (-b\right )}^{1/4}\,\sqrt {d\,x}}{a^{1/4}\,\sqrt {d}}\right )}{8192\,a^{29/4}\,d^{7/2}}+\frac {69615\,{\left (-b\right )}^{5/4}\,\mathrm {atanh}\left (\frac {{\left (-b\right )}^{1/4}\,\sqrt {d\,x}}{a^{1/4}\,\sqrt {d}}\right )}{8192\,a^{29/4}\,d^{7/2}} \]
((10*b*d^9*x^2)/a^2 - (2*d^9)/(5*a) + (263515*b^2*d^9*x^4)/(4096*a^3) + (1 8445*b^3*d^9*x^6)/(128*a^4) + (317135*b^4*d^9*x^8)/(2048*a^5) + (41769*b^5 *d^9*x^10)/(512*a^6) + (69615*b^6*d^9*x^12)/(4096*a^7))/(b^5*(d*x)^(25/2) + a^5*d^10*(d*x)^(5/2) + 10*a^3*b^2*d^6*(d*x)^(13/2) + 10*a^2*b^3*d^4*(d*x )^(17/2) + 5*a^4*b*d^8*(d*x)^(9/2) + 5*a*b^4*d^2*(d*x)^(21/2)) - (69615*(- b)^(5/4)*atan(((-b)^(1/4)*(d*x)^(1/2))/(a^(1/4)*d^(1/2))))/(8192*a^(29/4)* d^(7/2)) + (69615*(-b)^(5/4)*atanh(((-b)^(1/4)*(d*x)^(1/2))/(a^(1/4)*d^(1/ 2))))/(8192*a^(29/4)*d^(7/2))