Integrand size = 28, antiderivative size = 327 \[ \int \frac {1}{(d x)^{5/2} \left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=-\frac {33649}{12288 a^6 d (d x)^{3/2}}+\frac {1}{10 a d (d x)^{3/2} \left (a+b x^2\right )^5}+\frac {23}{160 a^2 d (d x)^{3/2} \left (a+b x^2\right )^4}+\frac {437}{1920 a^3 d (d x)^{3/2} \left (a+b x^2\right )^3}+\frac {437}{1024 a^4 d (d x)^{3/2} \left (a+b x^2\right )^2}+\frac {4807}{4096 a^5 d (d x)^{3/2} \left (a+b x^2\right )}+\frac {33649 b^{3/4} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} a^{27/4} d^{5/2}}-\frac {33649 b^{3/4} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} a^{27/4} d^{5/2}}-\frac {33649 b^{3/4} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}}{\sqrt {d} \left (\sqrt {a}+\sqrt {b} x\right )}\right )}{8192 \sqrt {2} a^{27/4} d^{5/2}} \] Output:
-33649/12288/a^6/d/(d*x)^(3/2)+1/10/a/d/(d*x)^(3/2)/(b*x^2+a)^5+23/160/a^2 /d/(d*x)^(3/2)/(b*x^2+a)^4+437/1920/a^3/d/(d*x)^(3/2)/(b*x^2+a)^3+437/1024 /a^4/d/(d*x)^(3/2)/(b*x^2+a)^2+4807/4096/a^5/d/(d*x)^(3/2)/(b*x^2+a)+33649 /16384*b^(3/4)*arctan(1-2^(1/2)*b^(1/4)*(d*x)^(1/2)/a^(1/4)/d^(1/2))*2^(1/ 2)/a^(27/4)/d^(5/2)-33649/16384*b^(3/4)*arctan(1+2^(1/2)*b^(1/4)*(d*x)^(1/ 2)/a^(1/4)/d^(1/2))*2^(1/2)/a^(27/4)/d^(5/2)-33649/16384*b^(3/4)*arctanh(2 ^(1/2)*a^(1/4)*b^(1/4)*(d*x)^(1/2)/d^(1/2)/(a^(1/2)+b^(1/2)*x))*2^(1/2)/a^ (27/4)/d^(5/2)
Time = 0.84 (sec) , antiderivative size = 195, normalized size of antiderivative = 0.60 \[ \int \frac {1}{(d x)^{5/2} \left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {x \left (-\frac {4 a^{3/4} \left (40960 a^5+437345 a^4 b x^2+1157176 a^3 b^2 x^4+1367810 a^2 b^3 x^6+769120 a b^4 x^8+168245 b^5 x^{10}\right )}{\left (a+b x^2\right )^5}+504735 \sqrt {2} b^{3/4} x^{3/2} \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )-504735 \sqrt {2} b^{3/4} x^{3/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )\right )}{245760 a^{27/4} (d x)^{5/2}} \] Input:
Integrate[1/((d*x)^(5/2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^3),x]
Output:
(x*((-4*a^(3/4)*(40960*a^5 + 437345*a^4*b*x^2 + 1157176*a^3*b^2*x^4 + 1367 810*a^2*b^3*x^6 + 769120*a*b^4*x^8 + 168245*b^5*x^10))/(a + b*x^2)^5 + 504 735*Sqrt[2]*b^(3/4)*x^(3/2)*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)* b^(1/4)*Sqrt[x])] - 504735*Sqrt[2]*b^(3/4)*x^(3/2)*ArcTanh[(Sqrt[2]*a^(1/4 )*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)]))/(245760*a^(27/4)*(d*x)^(5/2))
Time = 1.13 (sec) , antiderivative size = 465, normalized size of antiderivative = 1.42, number of steps used = 19, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.643, Rules used = {1380, 27, 253, 253, 253, 253, 253, 264, 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.
| \(\displaystyle \int \frac {1}{(d x)^{5/2} \left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx\) |
| \(\Big \downarrow \) 1380 |
| \(\displaystyle b^6 \int \frac {1}{b^6 (d x)^{5/2} \left (b x^2+a\right )^6}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \frac {1}{(d x)^{5/2} \left (a+b x^2\right )^6}dx\) |
| \(\Big \downarrow \) 253 |
| \(\displaystyle \frac {23 \int \frac {1}{(d x)^{5/2} \left (b x^2+a\right )^5}dx}{20 a}+\frac {1}{10 a d (d x)^{3/2} \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 253 |
| \(\displaystyle \frac {23 \left (\frac {19 \int \frac {1}{(d x)^{5/2} \left (b x^2+a\right )^4}dx}{16 a}+\frac {1}{8 a d (d x)^{3/2} \left (a+b x^2\right )^4}\right )}{20 a}+\frac {1}{10 a d (d x)^{3/2} \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 253 |
| \(\displaystyle \frac {23 \left (\frac {19 \left (\frac {5 \int \frac {1}{(d x)^{5/2} \left (b x^2+a\right )^3}dx}{4 a}+\frac {1}{6 a d (d x)^{3/2} \left (a+b x^2\right )^3}\right )}{16 a}+\frac {1}{8 a d (d x)^{3/2} \left (a+b x^2\right )^4}\right )}{20 a}+\frac {1}{10 a d (d x)^{3/2} \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 253 |
| \(\displaystyle \frac {23 \left (\frac {19 \left (\frac {5 \left (\frac {11 \int \frac {1}{(d x)^{5/2} \left (b x^2+a\right )^2}dx}{8 a}+\frac {1}{4 a d (d x)^{3/2} \left (a+b x^2\right )^2}\right )}{4 a}+\frac {1}{6 a d (d x)^{3/2} \left (a+b x^2\right )^3}\right )}{16 a}+\frac {1}{8 a d (d x)^{3/2} \left (a+b x^2\right )^4}\right )}{20 a}+\frac {1}{10 a d (d x)^{3/2} \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 253 |
| \(\displaystyle \frac {23 \left (\frac {19 \left (\frac {5 \left (\frac {11 \left (\frac {7 \int \frac {1}{(d x)^{5/2} \left (b x^2+a\right )}dx}{4 a}+\frac {1}{2 a d (d x)^{3/2} \left (a+b x^2\right )}\right )}{8 a}+\frac {1}{4 a d (d x)^{3/2} \left (a+b x^2\right )^2}\right )}{4 a}+\frac {1}{6 a d (d x)^{3/2} \left (a+b x^2\right )^3}\right )}{16 a}+\frac {1}{8 a d (d x)^{3/2} \left (a+b x^2\right )^4}\right )}{20 a}+\frac {1}{10 a d (d x)^{3/2} \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 264 |
| \(\displaystyle \frac {23 \left (\frac {19 \left (\frac {5 \left (\frac {11 \left (\frac {7 \left (-\frac {b \int \frac {1}{\sqrt {d x} \left (b x^2+a\right )}dx}{a d^2}-\frac {2}{3 a d (d x)^{3/2}}\right )}{4 a}+\frac {1}{2 a d (d x)^{3/2} \left (a+b x^2\right )}\right )}{8 a}+\frac {1}{4 a d (d x)^{3/2} \left (a+b x^2\right )^2}\right )}{4 a}+\frac {1}{6 a d (d x)^{3/2} \left (a+b x^2\right )^3}\right )}{16 a}+\frac {1}{8 a d (d x)^{3/2} \left (a+b x^2\right )^4}\right )}{20 a}+\frac {1}{10 a d (d x)^{3/2} \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {23 \left (\frac {19 \left (\frac {5 \left (\frac {11 \left (\frac {7 \left (-\frac {2 b \int \frac {1}{b x^2+a}d\sqrt {d x}}{a d^3}-\frac {2}{3 a d (d x)^{3/2}}\right )}{4 a}+\frac {1}{2 a d (d x)^{3/2} \left (a+b x^2\right )}\right )}{8 a}+\frac {1}{4 a d (d x)^{3/2} \left (a+b x^2\right )^2}\right )}{4 a}+\frac {1}{6 a d (d x)^{3/2} \left (a+b x^2\right )^3}\right )}{16 a}+\frac {1}{8 a d (d x)^{3/2} \left (a+b x^2\right )^4}\right )}{20 a}+\frac {1}{10 a d (d x)^{3/2} \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 755 |
| \(\displaystyle \frac {23 \left (\frac {19 \left (\frac {5 \left (\frac {11 \left (\frac {7 \left (-\frac {2 b \left (\frac {\int \frac {d^2 \left (\sqrt {a} d-\sqrt {b} d x\right )}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {a} d}+\frac {\int \frac {d^2 \left (\sqrt {b} x d+\sqrt {a} d\right )}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {a} d}\right )}{a d^3}-\frac {2}{3 a d (d x)^{3/2}}\right )}{4 a}+\frac {1}{2 a d (d x)^{3/2} \left (a+b x^2\right )}\right )}{8 a}+\frac {1}{4 a d (d x)^{3/2} \left (a+b x^2\right )^2}\right )}{4 a}+\frac {1}{6 a d (d x)^{3/2} \left (a+b x^2\right )^3}\right )}{16 a}+\frac {1}{8 a d (d x)^{3/2} \left (a+b x^2\right )^4}\right )}{20 a}+\frac {1}{10 a d (d x)^{3/2} \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {23 \left (\frac {19 \left (\frac {5 \left (\frac {11 \left (\frac {7 \left (-\frac {2 b \left (\frac {d \int \frac {\sqrt {a} d-\sqrt {b} d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {a}}+\frac {d \int \frac {\sqrt {b} x d+\sqrt {a} d}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {a}}\right )}{a d^3}-\frac {2}{3 a d (d x)^{3/2}}\right )}{4 a}+\frac {1}{2 a d (d x)^{3/2} \left (a+b x^2\right )}\right )}{8 a}+\frac {1}{4 a d (d x)^{3/2} \left (a+b x^2\right )^2}\right )}{4 a}+\frac {1}{6 a d (d x)^{3/2} \left (a+b x^2\right )^3}\right )}{16 a}+\frac {1}{8 a d (d x)^{3/2} \left (a+b x^2\right )^4}\right )}{20 a}+\frac {1}{10 a d (d x)^{3/2} \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {23 \left (\frac {19 \left (\frac {5 \left (\frac {11 \left (\frac {7 \left (-\frac {2 b \left (\frac {d \int \frac {\sqrt {a} d-\sqrt {b} d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {a}}+\frac {d \left (\frac {\int \frac {1}{x d+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}}d\sqrt {d x}}{2 \sqrt {b}}+\frac {\int \frac {1}{x d+\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}}d\sqrt {d x}}{2 \sqrt {b}}\right )}{2 \sqrt {a}}\right )}{a d^3}-\frac {2}{3 a d (d x)^{3/2}}\right )}{4 a}+\frac {1}{2 a d (d x)^{3/2} \left (a+b x^2\right )}\right )}{8 a}+\frac {1}{4 a d (d x)^{3/2} \left (a+b x^2\right )^2}\right )}{4 a}+\frac {1}{6 a d (d x)^{3/2} \left (a+b x^2\right )^3}\right )}{16 a}+\frac {1}{8 a d (d x)^{3/2} \left (a+b x^2\right )^4}\right )}{20 a}+\frac {1}{10 a d (d x)^{3/2} \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {23 \left (\frac {19 \left (\frac {5 \left (\frac {11 \left (\frac {7 \left (-\frac {2 b \left (\frac {d \int \frac {\sqrt {a} d-\sqrt {b} d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {a}}+\frac {d \left (\frac {\int \frac {1}{-d x-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\int \frac {1}{-d x-1}d\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}\right )}{a d^3}-\frac {2}{3 a d (d x)^{3/2}}\right )}{4 a}+\frac {1}{2 a d (d x)^{3/2} \left (a+b x^2\right )}\right )}{8 a}+\frac {1}{4 a d (d x)^{3/2} \left (a+b x^2\right )^2}\right )}{4 a}+\frac {1}{6 a d (d x)^{3/2} \left (a+b x^2\right )^3}\right )}{16 a}+\frac {1}{8 a d (d x)^{3/2} \left (a+b x^2\right )^4}\right )}{20 a}+\frac {1}{10 a d (d x)^{3/2} \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {23 \left (\frac {19 \left (\frac {5 \left (\frac {11 \left (\frac {7 \left (-\frac {2 b \left (\frac {d \int \frac {\sqrt {a} d-\sqrt {b} d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {a}}+\frac {d \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}\right )}{a d^3}-\frac {2}{3 a d (d x)^{3/2}}\right )}{4 a}+\frac {1}{2 a d (d x)^{3/2} \left (a+b x^2\right )}\right )}{8 a}+\frac {1}{4 a d (d x)^{3/2} \left (a+b x^2\right )^2}\right )}{4 a}+\frac {1}{6 a d (d x)^{3/2} \left (a+b x^2\right )^3}\right )}{16 a}+\frac {1}{8 a d (d x)^{3/2} \left (a+b x^2\right )^4}\right )}{20 a}+\frac {1}{10 a d (d x)^{3/2} \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {23 \left (\frac {19 \left (\frac {5 \left (\frac {11 \left (\frac {7 \left (-\frac {2 b \left (\frac {d \left (-\frac {\int -\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d}-2 \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{b} \left (x d+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt [4]{a} \sqrt {d}+\sqrt {2} \sqrt [4]{b} \sqrt {d x}\right )}{\sqrt [4]{b} \left (x d+\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}+\frac {d \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}\right )}{a d^3}-\frac {2}{3 a d (d x)^{3/2}}\right )}{4 a}+\frac {1}{2 a d (d x)^{3/2} \left (a+b x^2\right )}\right )}{8 a}+\frac {1}{4 a d (d x)^{3/2} \left (a+b x^2\right )^2}\right )}{4 a}+\frac {1}{6 a d (d x)^{3/2} \left (a+b x^2\right )^3}\right )}{16 a}+\frac {1}{8 a d (d x)^{3/2} \left (a+b x^2\right )^4}\right )}{20 a}+\frac {1}{10 a d (d x)^{3/2} \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {23 \left (\frac {19 \left (\frac {5 \left (\frac {11 \left (\frac {7 \left (-\frac {2 b \left (\frac {d \left (\frac {\int \frac {\sqrt {2} \sqrt [4]{a} \sqrt {d}-2 \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{b} \left (x d+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}+\frac {\int \frac {\sqrt {2} \left (\sqrt [4]{a} \sqrt {d}+\sqrt {2} \sqrt [4]{b} \sqrt {d x}\right )}{\sqrt [4]{b} \left (x d+\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}+\frac {d \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}\right )}{a d^3}-\frac {2}{3 a d (d x)^{3/2}}\right )}{4 a}+\frac {1}{2 a d (d x)^{3/2} \left (a+b x^2\right )}\right )}{8 a}+\frac {1}{4 a d (d x)^{3/2} \left (a+b x^2\right )^2}\right )}{4 a}+\frac {1}{6 a d (d x)^{3/2} \left (a+b x^2\right )^3}\right )}{16 a}+\frac {1}{8 a d (d x)^{3/2} \left (a+b x^2\right )^4}\right )}{20 a}+\frac {1}{10 a d (d x)^{3/2} \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {23 \left (\frac {19 \left (\frac {5 \left (\frac {11 \left (\frac {7 \left (-\frac {2 b \left (\frac {d \left (\frac {\int \frac {\sqrt {2} \sqrt [4]{a} \sqrt {d}-2 \sqrt [4]{b} \sqrt {d x}}{x d+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt {b} \sqrt {d}}+\frac {\int \frac {\sqrt [4]{a} \sqrt {d}+\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{x d+\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}}d\sqrt {d x}}{2 \sqrt [4]{a} \sqrt {b} \sqrt {d}}\right )}{2 \sqrt {a}}+\frac {d \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}\right )}{a d^3}-\frac {2}{3 a d (d x)^{3/2}}\right )}{4 a}+\frac {1}{2 a d (d x)^{3/2} \left (a+b x^2\right )}\right )}{8 a}+\frac {1}{4 a d (d x)^{3/2} \left (a+b x^2\right )^2}\right )}{4 a}+\frac {1}{6 a d (d x)^{3/2} \left (a+b x^2\right )^3}\right )}{16 a}+\frac {1}{8 a d (d x)^{3/2} \left (a+b x^2\right )^4}\right )}{20 a}+\frac {1}{10 a d (d x)^{3/2} \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {23 \left (\frac {19 \left (\frac {5 \left (\frac {11 \left (\frac {7 \left (-\frac {2 b \left (\frac {d \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}+\frac {d \left (\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d} \sqrt {d x}+\sqrt {a} d+\sqrt {b} d x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d} \sqrt {d x}+\sqrt {a} d+\sqrt {b} d x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}\right )}{a d^3}-\frac {2}{3 a d (d x)^{3/2}}\right )}{4 a}+\frac {1}{2 a d (d x)^{3/2} \left (a+b x^2\right )}\right )}{8 a}+\frac {1}{4 a d (d x)^{3/2} \left (a+b x^2\right )^2}\right )}{4 a}+\frac {1}{6 a d (d x)^{3/2} \left (a+b x^2\right )^3}\right )}{16 a}+\frac {1}{8 a d (d x)^{3/2} \left (a+b x^2\right )^4}\right )}{20 a}+\frac {1}{10 a d (d x)^{3/2} \left (a+b x^2\right )^5}\) |
Input:
Int[1/((d*x)^(5/2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^3),x]
Output:
1/(10*a*d*(d*x)^(3/2)*(a + b*x^2)^5) + (23*(1/(8*a*d*(d*x)^(3/2)*(a + b*x^ 2)^4) + (19*(1/(6*a*d*(d*x)^(3/2)*(a + b*x^2)^3) + (5*(1/(4*a*d*(d*x)^(3/2 )*(a + b*x^2)^2) + (11*(1/(2*a*d*(d*x)^(3/2)*(a + b*x^2)) + (7*(-2/(3*a*d* (d*x)^(3/2)) - (2*b*((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*Sqr t[a]) + (d*(-1/2*Log[Sqrt[a]*d + Sqrt[b]*d*x - Sqrt[2]*a^(1/4)*b^(1/4)*Sqr t[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])))/(a*d^3)))/(4*a)))/(8*a)))/(4*a)))/(16*a)))/(2 0*a)
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[((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 = 0.84 (sec) , antiderivative size = 254, normalized size of antiderivative = 0.78
| method | result | size |
| risch | \(-\frac {2}{3 a^{6} x \sqrt {d x}\, d^{2}}-\frac {2 b \left (\frac {\frac {15503 a^{4} d^{8} \sqrt {d x}}{8192}+\frac {31149 a^{3} d^{6} b \left (d x \right )^{\frac {5}{2}}}{5120}+\frac {95821 a^{2} d^{4} b^{2} \left (d x \right )^{\frac {9}{2}}}{12288}+\frac {3527 a \,d^{2} b^{3} \left (d x \right )^{\frac {13}{2}}}{768}+\frac {25457 b^{4} \left (d x \right )^{\frac {17}{2}}}{24576}}{\left (b \,d^{2} x^{2}+a \,d^{2}\right )^{5}}+\frac {33649 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{65536 a \,d^{2}}\right )}{a^{6} d}\) | \(254\) |
| pseudoelliptic | \(-\frac {33649 \left (\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} b x \sqrt {2}\, \left (b \,x^{2}+a \right )^{5} \left (\ln \left (\frac {d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}-\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}+\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )\right ) \sqrt {d x}+\frac {65536 d a \left (\frac {33649}{8192} x^{10} b^{5}+\frac {4807}{256} a \,x^{8} b^{4}+\frac {136781}{4096} a^{2} x^{6} b^{3}+\frac {144647}{5120} a^{3} x^{4} b^{2}+\frac {87469}{8192} x^{2} a^{4} b +a^{5}\right )}{100947}\right )}{32768 \sqrt {d x}\, d^{3} a^{7} x \left (b \,x^{2}+a \right )^{5}}\) | \(255\) |
| derivativedivides | \(2 d^{11} \left (-\frac {b \left (\frac {\frac {15503 a^{4} d^{8} \sqrt {d x}}{8192}+\frac {31149 a^{3} d^{6} b \left (d x \right )^{\frac {5}{2}}}{5120}+\frac {95821 a^{2} d^{4} b^{2} \left (d x \right )^{\frac {9}{2}}}{12288}+\frac {3527 a \,d^{2} b^{3} \left (d x \right )^{\frac {13}{2}}}{768}+\frac {25457 b^{4} \left (d x \right )^{\frac {17}{2}}}{24576}}{\left (b \,d^{2} x^{2}+a \,d^{2}\right )^{5}}+\frac {33649 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{65536 a \,d^{2}}\right )}{a^{6} d^{12}}-\frac {1}{3 a^{6} d^{12} \left (d x \right )^{\frac {3}{2}}}\right )\) | \(256\) |
| default | \(2 d^{11} \left (-\frac {b \left (\frac {\frac {15503 a^{4} d^{8} \sqrt {d x}}{8192}+\frac {31149 a^{3} d^{6} b \left (d x \right )^{\frac {5}{2}}}{5120}+\frac {95821 a^{2} d^{4} b^{2} \left (d x \right )^{\frac {9}{2}}}{12288}+\frac {3527 a \,d^{2} b^{3} \left (d x \right )^{\frac {13}{2}}}{768}+\frac {25457 b^{4} \left (d x \right )^{\frac {17}{2}}}{24576}}{\left (b \,d^{2} x^{2}+a \,d^{2}\right )^{5}}+\frac {33649 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{65536 a \,d^{2}}\right )}{a^{6} d^{12}}-\frac {1}{3 a^{6} d^{12} \left (d x \right )^{\frac {3}{2}}}\right )\) | \(256\) |
Input:
int(1/(d*x)^(5/2)/(b^2*x^4+2*a*b*x^2+a^2)^3,x,method=_RETURNVERBOSE)
Output:
-2/3/a^6/x/(d*x)^(1/2)/d^2-2/a^6*b/d*((15503/8192*a^4*d^8*(d*x)^(1/2)+3114 9/5120*a^3*d^6*b*(d*x)^(5/2)+95821/12288*a^2*d^4*b^2*(d*x)^(9/2)+3527/768* a*d^2*b^3*(d*x)^(13/2)+25457/24576*b^4*(d*x)^(17/2))/(b*d^2*x^2+a*d^2)^5+3 3649/65536*(a*d^2/b)^(1/4)/a/d^2*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 = 0.25 (sec) , antiderivative size = 644, normalized size of antiderivative = 1.97 \[ \int \frac {1}{(d x)^{5/2} \left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=-\frac {504735 \, {\left (a^{6} b^{5} d^{3} x^{12} + 5 \, a^{7} b^{4} d^{3} x^{10} + 10 \, a^{8} b^{3} d^{3} x^{8} + 10 \, a^{9} b^{2} d^{3} x^{6} + 5 \, a^{10} b d^{3} x^{4} + a^{11} d^{3} x^{2}\right )} \left (-\frac {b^{3}}{a^{27} d^{10}}\right )^{\frac {1}{4}} \log \left (33649 \, a^{7} d^{3} \left (-\frac {b^{3}}{a^{27} d^{10}}\right )^{\frac {1}{4}} + 33649 \, \sqrt {d x} b\right ) + 504735 \, {\left (i \, a^{6} b^{5} d^{3} x^{12} + 5 i \, a^{7} b^{4} d^{3} x^{10} + 10 i \, a^{8} b^{3} d^{3} x^{8} + 10 i \, a^{9} b^{2} d^{3} x^{6} + 5 i \, a^{10} b d^{3} x^{4} + i \, a^{11} d^{3} x^{2}\right )} \left (-\frac {b^{3}}{a^{27} d^{10}}\right )^{\frac {1}{4}} \log \left (33649 i \, a^{7} d^{3} \left (-\frac {b^{3}}{a^{27} d^{10}}\right )^{\frac {1}{4}} + 33649 \, \sqrt {d x} b\right ) + 504735 \, {\left (-i \, a^{6} b^{5} d^{3} x^{12} - 5 i \, a^{7} b^{4} d^{3} x^{10} - 10 i \, a^{8} b^{3} d^{3} x^{8} - 10 i \, a^{9} b^{2} d^{3} x^{6} - 5 i \, a^{10} b d^{3} x^{4} - i \, a^{11} d^{3} x^{2}\right )} \left (-\frac {b^{3}}{a^{27} d^{10}}\right )^{\frac {1}{4}} \log \left (-33649 i \, a^{7} d^{3} \left (-\frac {b^{3}}{a^{27} d^{10}}\right )^{\frac {1}{4}} + 33649 \, \sqrt {d x} b\right ) - 504735 \, {\left (a^{6} b^{5} d^{3} x^{12} + 5 \, a^{7} b^{4} d^{3} x^{10} + 10 \, a^{8} b^{3} d^{3} x^{8} + 10 \, a^{9} b^{2} d^{3} x^{6} + 5 \, a^{10} b d^{3} x^{4} + a^{11} d^{3} x^{2}\right )} \left (-\frac {b^{3}}{a^{27} d^{10}}\right )^{\frac {1}{4}} \log \left (-33649 \, a^{7} d^{3} \left (-\frac {b^{3}}{a^{27} d^{10}}\right )^{\frac {1}{4}} + 33649 \, \sqrt {d x} b\right ) + 4 \, {\left (168245 \, b^{5} x^{10} + 769120 \, a b^{4} x^{8} + 1367810 \, a^{2} b^{3} x^{6} + 1157176 \, a^{3} b^{2} x^{4} + 437345 \, a^{4} b x^{2} + 40960 \, a^{5}\right )} \sqrt {d x}}{245760 \, {\left (a^{6} b^{5} d^{3} x^{12} + 5 \, a^{7} b^{4} d^{3} x^{10} + 10 \, a^{8} b^{3} d^{3} x^{8} + 10 \, a^{9} b^{2} d^{3} x^{6} + 5 \, a^{10} b d^{3} x^{4} + a^{11} d^{3} x^{2}\right )}} \] Input:
integrate(1/(d*x)^(5/2)/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="fricas")
Output:
-1/245760*(504735*(a^6*b^5*d^3*x^12 + 5*a^7*b^4*d^3*x^10 + 10*a^8*b^3*d^3* x^8 + 10*a^9*b^2*d^3*x^6 + 5*a^10*b*d^3*x^4 + a^11*d^3*x^2)*(-b^3/(a^27*d^ 10))^(1/4)*log(33649*a^7*d^3*(-b^3/(a^27*d^10))^(1/4) + 33649*sqrt(d*x)*b) + 504735*(I*a^6*b^5*d^3*x^12 + 5*I*a^7*b^4*d^3*x^10 + 10*I*a^8*b^3*d^3*x^ 8 + 10*I*a^9*b^2*d^3*x^6 + 5*I*a^10*b*d^3*x^4 + I*a^11*d^3*x^2)*(-b^3/(a^2 7*d^10))^(1/4)*log(33649*I*a^7*d^3*(-b^3/(a^27*d^10))^(1/4) + 33649*sqrt(d *x)*b) + 504735*(-I*a^6*b^5*d^3*x^12 - 5*I*a^7*b^4*d^3*x^10 - 10*I*a^8*b^3 *d^3*x^8 - 10*I*a^9*b^2*d^3*x^6 - 5*I*a^10*b*d^3*x^4 - I*a^11*d^3*x^2)*(-b ^3/(a^27*d^10))^(1/4)*log(-33649*I*a^7*d^3*(-b^3/(a^27*d^10))^(1/4) + 3364 9*sqrt(d*x)*b) - 504735*(a^6*b^5*d^3*x^12 + 5*a^7*b^4*d^3*x^10 + 10*a^8*b^ 3*d^3*x^8 + 10*a^9*b^2*d^3*x^6 + 5*a^10*b*d^3*x^4 + a^11*d^3*x^2)*(-b^3/(a ^27*d^10))^(1/4)*log(-33649*a^7*d^3*(-b^3/(a^27*d^10))^(1/4) + 33649*sqrt( d*x)*b) + 4*(168245*b^5*x^10 + 769120*a*b^4*x^8 + 1367810*a^2*b^3*x^6 + 11 57176*a^3*b^2*x^4 + 437345*a^4*b*x^2 + 40960*a^5)*sqrt(d*x))/(a^6*b^5*d^3* x^12 + 5*a^7*b^4*d^3*x^10 + 10*a^8*b^3*d^3*x^8 + 10*a^9*b^2*d^3*x^6 + 5*a^ 10*b*d^3*x^4 + a^11*d^3*x^2)
\[ \int \frac {1}{(d x)^{5/2} \left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\int \frac {1}{\left (d x\right )^{\frac {5}{2}} \left (a + b x^{2}\right )^{6}}\, dx \] Input:
integrate(1/(d*x)**(5/2)/(b**2*x**4+2*a*b*x**2+a**2)**3,x)
Output:
Integral(1/((d*x)**(5/2)*(a + b*x**2)**6), x)
Time = 0.12 (sec) , antiderivative size = 395, normalized size of antiderivative = 1.21 \[ \int \frac {1}{(d x)^{5/2} \left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=-\frac {\frac {8 \, {\left (168245 \, b^{5} d^{10} x^{10} + 769120 \, a b^{4} d^{10} x^{8} + 1367810 \, a^{2} b^{3} d^{10} x^{6} + 1157176 \, a^{3} b^{2} d^{10} x^{4} + 437345 \, a^{4} b d^{10} x^{2} + 40960 \, a^{5} d^{10}\right )}}{\left (d x\right )^{\frac {23}{2}} a^{6} b^{5} + 5 \, \left (d x\right )^{\frac {19}{2}} a^{7} b^{4} d^{2} + 10 \, \left (d x\right )^{\frac {15}{2}} a^{8} b^{3} d^{4} + 10 \, \left (d x\right )^{\frac {11}{2}} a^{9} b^{2} d^{6} + 5 \, \left (d x\right )^{\frac {7}{2}} a^{10} b d^{8} + \left (d x\right )^{\frac {3}{2}} a^{11} d^{10}} + \frac {504735 \, {\left (\frac {\sqrt {2} b^{\frac {3}{4}} \log \left (\sqrt {b} d x + \sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} \sqrt {d x} b^{\frac {1}{4}} + \sqrt {a} d\right )}{\left (a d^{2}\right )^{\frac {3}{4}}} - \frac {\sqrt {2} b^{\frac {3}{4}} \log \left (\sqrt {b} d x - \sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} \sqrt {d x} b^{\frac {1}{4}} + \sqrt {a} d\right )}{\left (a d^{2}\right )^{\frac {3}{4}}} + \frac {2 \, \sqrt {2} b \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {d x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} d}}\right )}{\sqrt {\sqrt {a} \sqrt {b} d} \sqrt {a} d} + \frac {2 \, \sqrt {2} b \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {d x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} d}}\right )}{\sqrt {\sqrt {a} \sqrt {b} d} \sqrt {a} d}\right )}}{a^{6}}}{491520 \, d} \] Input:
integrate(1/(d*x)^(5/2)/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="maxima")
Output:
-1/491520*(8*(168245*b^5*d^10*x^10 + 769120*a*b^4*d^10*x^8 + 1367810*a^2*b ^3*d^10*x^6 + 1157176*a^3*b^2*d^10*x^4 + 437345*a^4*b*d^10*x^2 + 40960*a^5 *d^10)/((d*x)^(23/2)*a^6*b^5 + 5*(d*x)^(19/2)*a^7*b^4*d^2 + 10*(d*x)^(15/2 )*a^8*b^3*d^4 + 10*(d*x)^(11/2)*a^9*b^2*d^6 + 5*(d*x)^(7/2)*a^10*b*d^8 + ( d*x)^(3/2)*a^11*d^10) + 504735*(sqrt(2)*b^(3/4)*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) - sqrt(2)*b^(3/ 4)*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) + 2*sqrt(2)*b*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(a)*d) + 2*sqrt(2)*b*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(a)*d))/a^6)/d
Time = 0.16 (sec) , antiderivative size = 356, normalized size of antiderivative = 1.09 \[ \int \frac {1}{(d x)^{5/2} \left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=-\frac {33649 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{16384 \, a^{7} d^{3}} - \frac {33649 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{16384 \, a^{7} d^{3}} - \frac {33649 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} \log \left (d x + \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{32768 \, a^{7} d^{3}} + \frac {33649 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} \log \left (d x - \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{32768 \, a^{7} d^{3}} - \frac {2}{3 \, \sqrt {d x} a^{6} d^{2} x} - \frac {127285 \, \sqrt {d x} b^{5} d^{8} x^{8} + 564320 \, \sqrt {d x} a b^{4} d^{8} x^{6} + 958210 \, \sqrt {d x} a^{2} b^{3} d^{8} x^{4} + 747576 \, \sqrt {d x} a^{3} b^{2} d^{8} x^{2} + 232545 \, \sqrt {d x} a^{4} b d^{8}}{61440 \, {\left (b d^{2} x^{2} + a d^{2}\right )}^{5} a^{6} d} \] Input:
integrate(1/(d*x)^(5/2)/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="giac")
Output:
-33649/16384*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^7*d^3) - 33649/16384*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^7*d^3) - 33649/32768*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^7*d^3) + 33649 /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^7*d^3) - 2/3/(sqrt(d*x)*a^6*d^2*x) - 1/61440*(12728 5*sqrt(d*x)*b^5*d^8*x^8 + 564320*sqrt(d*x)*a*b^4*d^8*x^6 + 958210*sqrt(d*x )*a^2*b^3*d^8*x^4 + 747576*sqrt(d*x)*a^3*b^2*d^8*x^2 + 232545*sqrt(d*x)*a^ 4*b*d^8)/((b*d^2*x^2 + a*d^2)^5*a^6*d)
Time = 17.83 (sec) , antiderivative size = 226, normalized size of antiderivative = 0.69 \[ \int \frac {1}{(d x)^{5/2} \left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {33649\,{\left (-b\right )}^{3/4}\,\mathrm {atan}\left (\frac {{\left (-b\right )}^{1/4}\,\sqrt {d\,x}}{a^{1/4}\,\sqrt {d}}\right )}{8192\,a^{27/4}\,d^{5/2}}-\frac {\frac {2\,d^9}{3\,a}+\frac {87469\,b\,d^9\,x^2}{12288\,a^2}+\frac {144647\,b^2\,d^9\,x^4}{7680\,a^3}+\frac {136781\,b^3\,d^9\,x^6}{6144\,a^4}+\frac {4807\,b^4\,d^9\,x^8}{384\,a^5}+\frac {33649\,b^5\,d^9\,x^{10}}{12288\,a^6}}{b^5\,{\left (d\,x\right )}^{23/2}+a^5\,d^{10}\,{\left (d\,x\right )}^{3/2}+10\,a^3\,b^2\,d^6\,{\left (d\,x\right )}^{11/2}+10\,a^2\,b^3\,d^4\,{\left (d\,x\right )}^{15/2}+5\,a^4\,b\,d^8\,{\left (d\,x\right )}^{7/2}+5\,a\,b^4\,d^2\,{\left (d\,x\right )}^{19/2}}+\frac {33649\,{\left (-b\right )}^{3/4}\,\mathrm {atanh}\left (\frac {{\left (-b\right )}^{1/4}\,\sqrt {d\,x}}{a^{1/4}\,\sqrt {d}}\right )}{8192\,a^{27/4}\,d^{5/2}} \] Input:
int(1/((d*x)^(5/2)*(a^2 + b^2*x^4 + 2*a*b*x^2)^3),x)
Output:
(33649*(-b)^(3/4)*atan(((-b)^(1/4)*(d*x)^(1/2))/(a^(1/4)*d^(1/2))))/(8192* a^(27/4)*d^(5/2)) - ((2*d^9)/(3*a) + (87469*b*d^9*x^2)/(12288*a^2) + (1446 47*b^2*d^9*x^4)/(7680*a^3) + (136781*b^3*d^9*x^6)/(6144*a^4) + (4807*b^4*d ^9*x^8)/(384*a^5) + (33649*b^5*d^9*x^10)/(12288*a^6))/(b^5*(d*x)^(23/2) + a^5*d^10*(d*x)^(3/2) + 10*a^3*b^2*d^6*(d*x)^(11/2) + 10*a^2*b^3*d^4*(d*x)^ (15/2) + 5*a^4*b*d^8*(d*x)^(7/2) + 5*a*b^4*d^2*(d*x)^(19/2)) + (33649*(-b) ^(3/4)*atanh(((-b)^(1/4)*(d*x)^(1/2))/(a^(1/4)*d^(1/2))))/(8192*a^(27/4)*d ^(5/2))
Time = 0.19 (sec) , antiderivative size = 1049, normalized size of antiderivative = 3.21 \[ \int \frac {1}{(d x)^{5/2} \left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx =\text {Too large to display} \] Input:
int(1/(d*x)^(5/2)/(b^2*x^4+2*a*b*x^2+a^2)^3,x)
Output:
(sqrt(d)*(1009470*sqrt(x)*b**(3/4)*a**(1/4)*sqrt(2)*atan((b**(1/4)*a**(1/4 )*sqrt(2) - 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*a**5*x + 50473 50*sqrt(x)*b**(3/4)*a**(1/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) - 2*s qrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*a**4*b*x**3 + 10094700*sqrt(x )*b**(3/4)*a**(1/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(x)*sq rt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*a**3*b**2*x**5 + 10094700*sqrt(x)*b**( 3/4)*a**(1/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(x)*sqrt(b)) /(b**(1/4)*a**(1/4)*sqrt(2)))*a**2*b**3*x**7 + 5047350*sqrt(x)*b**(3/4)*a* *(1/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(x)*sqrt(b))/(b**(1 /4)*a**(1/4)*sqrt(2)))*a*b**4*x**9 + 1009470*sqrt(x)*b**(3/4)*a**(1/4)*sqr t(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/ 4)*sqrt(2)))*b**5*x**11 - 1009470*sqrt(x)*b**(3/4)*a**(1/4)*sqrt(2)*atan(( b**(1/4)*a**(1/4)*sqrt(2) + 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)) )*a**5*x - 5047350*sqrt(x)*b**(3/4)*a**(1/4)*sqrt(2)*atan((b**(1/4)*a**(1/ 4)*sqrt(2) + 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*a**4*b*x**3 - 10094700*sqrt(x)*b**(3/4)*a**(1/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2 ) + 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*a**3*b**2*x**5 - 10094 700*sqrt(x)*b**(3/4)*a**(1/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) + 2* sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*a**2*b**3*x**7 - 5047350*sqr t(x)*b**(3/4)*a**(1/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) + 2*sqrt...