Integrand size = 28, antiderivative size = 325 \[ \int \frac {(d x)^{25/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {33649 d^{11} (d x)^{3/2}}{12288 b^6}-\frac {d (d x)^{23/2}}{10 b \left (a+b x^2\right )^5}-\frac {23 d^3 (d x)^{19/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac {437 d^5 (d x)^{15/2}}{1920 b^3 \left (a+b x^2\right )^3}-\frac {437 d^7 (d x)^{11/2}}{1024 b^4 \left (a+b x^2\right )^2}-\frac {4807 d^9 (d x)^{7/2}}{4096 b^5 \left (a+b x^2\right )}+\frac {33649 a^{3/4} d^{25/2} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} b^{27/4}}-\frac {33649 a^{3/4} d^{25/2} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} b^{27/4}}+\frac {33649 a^{3/4} d^{25/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}}{\sqrt {d} \left (\sqrt {a}+\sqrt {b} x\right )}\right )}{8192 \sqrt {2} b^{27/4}} \] Output:
33649/12288*d^11*(d*x)^(3/2)/b^6-1/10*d*(d*x)^(23/2)/b/(b*x^2+a)^5-23/160* d^3*(d*x)^(19/2)/b^2/(b*x^2+a)^4-437/1920*d^5*(d*x)^(15/2)/b^3/(b*x^2+a)^3 -437/1024*d^7*(d*x)^(11/2)/b^4/(b*x^2+a)^2-4807/4096*d^9*(d*x)^(7/2)/b^5/( b*x^2+a)+33649/16384*a^(3/4)*d^(25/2)*arctan(1-2^(1/2)*b^(1/4)*(d*x)^(1/2) /a^(1/4)/d^(1/2))*2^(1/2)/b^(27/4)-33649/16384*a^(3/4)*d^(25/2)*arctan(1+2 ^(1/2)*b^(1/4)*(d*x)^(1/2)/a^(1/4)/d^(1/2))*2^(1/2)/b^(27/4)+33649/16384*a ^(3/4)*d^(25/2)*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)/b^(27/4)
Time = 1.04 (sec) , antiderivative size = 216, normalized size of antiderivative = 0.66 \[ \int \frac {(d x)^{25/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {d^{12} \sqrt {d x} \left (4 b^{3/4} x^{3/2} \left (168245 a^5+769120 a^4 b x^2+1367810 a^3 b^2 x^4+1157176 a^2 b^3 x^6+437345 a b^4 x^8+40960 b^5 x^{10}\right )-504735 \sqrt {2} a^{3/4} \left (a+b x^2\right )^5 \arctan \left (\frac {-\sqrt {a}+\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )+504735 \sqrt {2} a^{3/4} \left (a+b x^2\right )^5 \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )\right )}{245760 b^{27/4} \sqrt {x} \left (a+b x^2\right )^5} \] Input:
Integrate[(d*x)^(25/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]
Output:
(d^12*Sqrt[d*x]*(4*b^(3/4)*x^(3/2)*(168245*a^5 + 769120*a^4*b*x^2 + 136781 0*a^3*b^2*x^4 + 1157176*a^2*b^3*x^6 + 437345*a*b^4*x^8 + 40960*b^5*x^10) - 504735*Sqrt[2]*a^(3/4)*(a + b*x^2)^5*ArcTan[(-Sqrt[a] + Sqrt[b]*x)/(Sqrt[ 2]*a^(1/4)*b^(1/4)*Sqrt[x])] + 504735*Sqrt[2]*a^(3/4)*(a + b*x^2)^5*ArcTan h[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)]))/(245760*b^(27 /4)*Sqrt[x]*(a + b*x^2)^5)
Time = 1.16 (sec) , antiderivative size = 466, normalized size of antiderivative = 1.43, number of steps used = 19, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.643, Rules used = {1380, 27, 252, 252, 252, 252, 252, 262, 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.
| \(\displaystyle \int \frac {(d x)^{25/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx\) |
| \(\Big \downarrow \) 1380 |
| \(\displaystyle b^6 \int \frac {(d x)^{25/2}}{b^6 \left (b x^2+a\right )^6}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \frac {(d x)^{25/2}}{\left (a+b x^2\right )^6}dx\) |
| \(\Big \downarrow \) 252 |
| \(\displaystyle \frac {23 d^2 \int \frac {(d x)^{21/2}}{\left (b x^2+a\right )^5}dx}{20 b}-\frac {d (d x)^{23/2}}{10 b \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 252 |
| \(\displaystyle \frac {23 d^2 \left (\frac {19 d^2 \int \frac {(d x)^{17/2}}{\left (b x^2+a\right )^4}dx}{16 b}-\frac {d (d x)^{19/2}}{8 b \left (a+b x^2\right )^4}\right )}{20 b}-\frac {d (d x)^{23/2}}{10 b \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 252 |
| \(\displaystyle \frac {23 d^2 \left (\frac {19 d^2 \left (\frac {5 d^2 \int \frac {(d x)^{13/2}}{\left (b x^2+a\right )^3}dx}{4 b}-\frac {d (d x)^{15/2}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{19/2}}{8 b \left (a+b x^2\right )^4}\right )}{20 b}-\frac {d (d x)^{23/2}}{10 b \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 252 |
| \(\displaystyle \frac {23 d^2 \left (\frac {19 d^2 \left (\frac {5 d^2 \left (\frac {11 d^2 \int \frac {(d x)^{9/2}}{\left (b x^2+a\right )^2}dx}{8 b}-\frac {d (d x)^{11/2}}{4 b \left (a+b x^2\right )^2}\right )}{4 b}-\frac {d (d x)^{15/2}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{19/2}}{8 b \left (a+b x^2\right )^4}\right )}{20 b}-\frac {d (d x)^{23/2}}{10 b \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 252 |
| \(\displaystyle \frac {23 d^2 \left (\frac {19 d^2 \left (\frac {5 d^2 \left (\frac {11 d^2 \left (\frac {7 d^2 \int \frac {(d x)^{5/2}}{b x^2+a}dx}{4 b}-\frac {d (d x)^{7/2}}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{11/2}}{4 b \left (a+b x^2\right )^2}\right )}{4 b}-\frac {d (d x)^{15/2}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{19/2}}{8 b \left (a+b x^2\right )^4}\right )}{20 b}-\frac {d (d x)^{23/2}}{10 b \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle \frac {23 d^2 \left (\frac {19 d^2 \left (\frac {5 d^2 \left (\frac {11 d^2 \left (\frac {7 d^2 \left (\frac {2 d (d x)^{3/2}}{3 b}-\frac {a d^2 \int \frac {\sqrt {d x}}{b x^2+a}dx}{b}\right )}{4 b}-\frac {d (d x)^{7/2}}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{11/2}}{4 b \left (a+b x^2\right )^2}\right )}{4 b}-\frac {d (d x)^{15/2}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{19/2}}{8 b \left (a+b x^2\right )^4}\right )}{20 b}-\frac {d (d x)^{23/2}}{10 b \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {23 d^2 \left (\frac {19 d^2 \left (\frac {5 d^2 \left (\frac {11 d^2 \left (\frac {7 d^2 \left (\frac {2 d (d x)^{3/2}}{3 b}-\frac {2 a d \int \frac {d^3 x}{b x^2 d^2+a d^2}d\sqrt {d x}}{b}\right )}{4 b}-\frac {d (d x)^{7/2}}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{11/2}}{4 b \left (a+b x^2\right )^2}\right )}{4 b}-\frac {d (d x)^{15/2}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{19/2}}{8 b \left (a+b x^2\right )^4}\right )}{20 b}-\frac {d (d x)^{23/2}}{10 b \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {23 d^2 \left (\frac {19 d^2 \left (\frac {5 d^2 \left (\frac {11 d^2 \left (\frac {7 d^2 \left (\frac {2 d (d x)^{3/2}}{3 b}-\frac {2 a d^3 \int \frac {d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{b}\right )}{4 b}-\frac {d (d x)^{7/2}}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{11/2}}{4 b \left (a+b x^2\right )^2}\right )}{4 b}-\frac {d (d x)^{15/2}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{19/2}}{8 b \left (a+b x^2\right )^4}\right )}{20 b}-\frac {d (d x)^{23/2}}{10 b \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 826 |
| \(\displaystyle \frac {23 d^2 \left (\frac {19 d^2 \left (\frac {5 d^2 \left (\frac {11 d^2 \left (\frac {7 d^2 \left (\frac {2 d (d x)^{3/2}}{3 b}-\frac {2 a d^3 \left (\frac {\int \frac {\sqrt {b} x d+\sqrt {a} d}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {b}}-\frac {\int \frac {\sqrt {a} d-\sqrt {b} d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {b}}\right )}{b}\right )}{4 b}-\frac {d (d x)^{7/2}}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{11/2}}{4 b \left (a+b x^2\right )^2}\right )}{4 b}-\frac {d (d x)^{15/2}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{19/2}}{8 b \left (a+b x^2\right )^4}\right )}{20 b}-\frac {d (d x)^{23/2}}{10 b \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {23 d^2 \left (\frac {19 d^2 \left (\frac {5 d^2 \left (\frac {11 d^2 \left (\frac {7 d^2 \left (\frac {2 d (d x)^{3/2}}{3 b}-\frac {2 a d^3 \left (\frac {\frac {\int \frac {1}{x d+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}}d\sqrt {d x}}{2 \sqrt {b}}+\frac {\int \frac {1}{x d+\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}}d\sqrt {d x}}{2 \sqrt {b}}}{2 \sqrt {b}}-\frac {\int \frac {\sqrt {a} d-\sqrt {b} d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {b}}\right )}{b}\right )}{4 b}-\frac {d (d x)^{7/2}}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{11/2}}{4 b \left (a+b x^2\right )^2}\right )}{4 b}-\frac {d (d x)^{15/2}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{19/2}}{8 b \left (a+b x^2\right )^4}\right )}{20 b}-\frac {d (d x)^{23/2}}{10 b \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {23 d^2 \left (\frac {19 d^2 \left (\frac {5 d^2 \left (\frac {11 d^2 \left (\frac {7 d^2 \left (\frac {2 d (d x)^{3/2}}{3 b}-\frac {2 a d^3 \left (\frac {\frac {\int \frac {1}{-d x-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\int \frac {1}{-d x-1}d\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}}{2 \sqrt {b}}-\frac {\int \frac {\sqrt {a} d-\sqrt {b} d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {b}}\right )}{b}\right )}{4 b}-\frac {d (d x)^{7/2}}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{11/2}}{4 b \left (a+b x^2\right )^2}\right )}{4 b}-\frac {d (d x)^{15/2}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{19/2}}{8 b \left (a+b x^2\right )^4}\right )}{20 b}-\frac {d (d x)^{23/2}}{10 b \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {23 d^2 \left (\frac {19 d^2 \left (\frac {5 d^2 \left (\frac {11 d^2 \left (\frac {7 d^2 \left (\frac {2 d (d x)^{3/2}}{3 b}-\frac {2 a d^3 \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}}{2 \sqrt {b}}-\frac {\int \frac {\sqrt {a} d-\sqrt {b} d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {b}}\right )}{b}\right )}{4 b}-\frac {d (d x)^{7/2}}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{11/2}}{4 b \left (a+b x^2\right )^2}\right )}{4 b}-\frac {d (d x)^{15/2}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{19/2}}{8 b \left (a+b x^2\right )^4}\right )}{20 b}-\frac {d (d x)^{23/2}}{10 b \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {23 d^2 \left (\frac {19 d^2 \left (\frac {5 d^2 \left (\frac {11 d^2 \left (\frac {7 d^2 \left (\frac {2 d (d x)^{3/2}}{3 b}-\frac {2 a d^3 \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}}{2 \sqrt {b}}-\frac {-\frac {\int -\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d}-2 \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{b} \left (x d+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt [4]{a} \sqrt {d}+\sqrt {2} \sqrt [4]{b} \sqrt {d x}\right )}{\sqrt [4]{b} \left (x d+\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}}{2 \sqrt {b}}\right )}{b}\right )}{4 b}-\frac {d (d x)^{7/2}}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{11/2}}{4 b \left (a+b x^2\right )^2}\right )}{4 b}-\frac {d (d x)^{15/2}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{19/2}}{8 b \left (a+b x^2\right )^4}\right )}{20 b}-\frac {d (d x)^{23/2}}{10 b \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {23 d^2 \left (\frac {19 d^2 \left (\frac {5 d^2 \left (\frac {11 d^2 \left (\frac {7 d^2 \left (\frac {2 d (d x)^{3/2}}{3 b}-\frac {2 a d^3 \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}}{2 \sqrt {b}}-\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{a} \sqrt {d}-2 \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{b} \left (x d+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}+\frac {\int \frac {\sqrt {2} \left (\sqrt [4]{a} \sqrt {d}+\sqrt {2} \sqrt [4]{b} \sqrt {d x}\right )}{\sqrt [4]{b} \left (x d+\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}}{2 \sqrt {b}}\right )}{b}\right )}{4 b}-\frac {d (d x)^{7/2}}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{11/2}}{4 b \left (a+b x^2\right )^2}\right )}{4 b}-\frac {d (d x)^{15/2}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{19/2}}{8 b \left (a+b x^2\right )^4}\right )}{20 b}-\frac {d (d x)^{23/2}}{10 b \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {23 d^2 \left (\frac {19 d^2 \left (\frac {5 d^2 \left (\frac {11 d^2 \left (\frac {7 d^2 \left (\frac {2 d (d x)^{3/2}}{3 b}-\frac {2 a d^3 \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}}{2 \sqrt {b}}-\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{a} \sqrt {d}-2 \sqrt [4]{b} \sqrt {d x}}{x d+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt {b} \sqrt {d}}+\frac {\int \frac {\sqrt [4]{a} \sqrt {d}+\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{x d+\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}}d\sqrt {d x}}{2 \sqrt [4]{a} \sqrt {b} \sqrt {d}}}{2 \sqrt {b}}\right )}{b}\right )}{4 b}-\frac {d (d x)^{7/2}}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{11/2}}{4 b \left (a+b x^2\right )^2}\right )}{4 b}-\frac {d (d x)^{15/2}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{19/2}}{8 b \left (a+b x^2\right )^4}\right )}{20 b}-\frac {d (d x)^{23/2}}{10 b \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {23 d^2 \left (\frac {19 d^2 \left (\frac {5 d^2 \left (\frac {11 d^2 \left (\frac {7 d^2 \left (\frac {2 d (d x)^{3/2}}{3 b}-\frac {2 a d^3 \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}}{2 \sqrt {b}}-\frac {\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d} \sqrt {d x}+\sqrt {a} d+\sqrt {b} d x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d} \sqrt {d x}+\sqrt {a} d+\sqrt {b} d x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}}{2 \sqrt {b}}\right )}{b}\right )}{4 b}-\frac {d (d x)^{7/2}}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{11/2}}{4 b \left (a+b x^2\right )^2}\right )}{4 b}-\frac {d (d x)^{15/2}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{19/2}}{8 b \left (a+b x^2\right )^4}\right )}{20 b}-\frac {d (d x)^{23/2}}{10 b \left (a+b x^2\right )^5}\) |
Input:
Int[(d*x)^(25/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]
Output:
-1/10*(d*(d*x)^(23/2))/(b*(a + b*x^2)^5) + (23*d^2*(-1/8*(d*(d*x)^(19/2))/ (b*(a + b*x^2)^4) + (19*d^2*(-1/6*(d*(d*x)^(15/2))/(b*(a + b*x^2)^3) + (5* d^2*(-1/4*(d*(d*x)^(11/2))/(b*(a + b*x^2)^2) + (11*d^2*(-1/2*(d*(d*x)^(7/2 ))/(b*(a + b*x^2)) + (7*d^2*((2*d*(d*x)^(3/2))/(3*b) - (2*a*d^3*((-(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])))/b))/(4*b) ))/(8*b)))/(4*b)))/(16*b)))/(20*b)
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*(c*x) ^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ (b*(m + 2*p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b , c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && 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 = 14.73 (sec) , antiderivative size = 250, normalized size of antiderivative = 0.77
| method | result | size |
| derivativedivides | \(2 d^{11} \left (\frac {\left (d x \right )^{\frac {3}{2}}}{3 b^{6}}-\frac {a \,d^{2} \left (\frac {-\frac {25457 a^{4} d^{8} \left (d x \right )^{\frac {3}{2}}}{24576}-\frac {3527 a^{3} d^{6} b \left (d x \right )^{\frac {7}{2}}}{768}-\frac {95821 a^{2} d^{4} b^{2} \left (d x \right )^{\frac {11}{2}}}{12288}-\frac {31149 a \,d^{2} b^{3} \left (d x \right )^{\frac {15}{2}}}{5120}-\frac {15503 b^{4} \left (d x \right )^{\frac {19}{2}}}{8192}}{\left (b \,d^{2} x^{2}+a \,d^{2}\right )^{5}}+\frac {33649 \sqrt {2}\, \left (\ln \left (\frac {d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{65536 b \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{b^{6}}\right )\) | \(250\) |
| default | \(2 d^{11} \left (\frac {\left (d x \right )^{\frac {3}{2}}}{3 b^{6}}-\frac {a \,d^{2} \left (\frac {-\frac {25457 a^{4} d^{8} \left (d x \right )^{\frac {3}{2}}}{24576}-\frac {3527 a^{3} d^{6} b \left (d x \right )^{\frac {7}{2}}}{768}-\frac {95821 a^{2} d^{4} b^{2} \left (d x \right )^{\frac {11}{2}}}{12288}-\frac {31149 a \,d^{2} b^{3} \left (d x \right )^{\frac {15}{2}}}{5120}-\frac {15503 b^{4} \left (d x \right )^{\frac {19}{2}}}{8192}}{\left (b \,d^{2} x^{2}+a \,d^{2}\right )^{5}}+\frac {33649 \sqrt {2}\, \left (\ln \left (\frac {d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{65536 b \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{b^{6}}\right )\) | \(250\) |
| risch | \(\frac {2 x^{2} d^{13}}{3 b^{6} \sqrt {d x}}-\frac {a \left (\frac {-\frac {25457 a^{4} d^{8} \left (d x \right )^{\frac {3}{2}}}{12288}-\frac {3527 a^{3} d^{6} b \left (d x \right )^{\frac {7}{2}}}{384}-\frac {95821 a^{2} d^{4} b^{2} \left (d x \right )^{\frac {11}{2}}}{6144}-\frac {31149 a \,d^{2} b^{3} \left (d x \right )^{\frac {15}{2}}}{2560}-\frac {15503 b^{4} \left (d x \right )^{\frac {19}{2}}}{4096}}{\left (b \,d^{2} x^{2}+a \,d^{2}\right )^{5}}+\frac {33649 \sqrt {2}\, \left (\ln \left (\frac {d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{32768 b \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right ) d^{13}}{b^{6}}\) | \(252\) |
| pseudoelliptic | \(\frac {33649 \left (8 \left (\frac {8192}{33649} x^{10} b^{5}+\frac {3803}{1463} a \,x^{8} b^{4}+\frac {2648}{385} a^{2} x^{6} b^{3}+\frac {626}{77} a^{3} x^{4} b^{2}+\frac {32}{7} x^{2} a^{4} b +a^{5}\right ) \sqrt {d x}\, b x \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}-3 \sqrt {2}\, a d \left (b \,x^{2}+a \right )^{5} \left (\ln \left (\frac {d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}-\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}+\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )\right )\right ) d^{12}}{98304 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} b^{7} \left (b \,x^{2}+a \right )^{5}}\) | \(258\) |
Input:
int((d*x)^(25/2)/(b^2*x^4+2*a*b*x^2+a^2)^3,x,method=_RETURNVERBOSE)
Output:
2*d^11*(1/3*(d*x)^(3/2)/b^6-a*d^2/b^6*((-25457/24576*a^4*d^8*(d*x)^(3/2)-3 527/768*a^3*d^6*b*(d*x)^(7/2)-95821/12288*a^2*d^4*b^2*(d*x)^(11/2)-31149/5 120*a*d^2*b^3*(d*x)^(15/2)-15503/8192*b^4*(d*x)^(19/2))/(b*d^2*x^2+a*d^2)^ 5+33649/65536/b/(a*d^2/b)^(1/4)*2^(1/2)*(ln((d*x-(a*d^2/b)^(1/4)*(d*x)^(1/ 2)*2^(1/2)+(a*d^2/b)^(1/2))/(d*x+(a*d^2/b)^(1/4)*(d*x)^(1/2)*2^(1/2)+(a*d^ 2/b)^(1/2)))+2*arctan(2^(1/2)/(a*d^2/b)^(1/4)*(d*x)^(1/2)+1)+2*arctan(2^(1 /2)/(a*d^2/b)^(1/4)*(d*x)^(1/2)-1))))
Result contains complex when optimal does not.
Time = 0.12 (sec) , antiderivative size = 566, normalized size of antiderivative = 1.74 \[ \int \frac {(d x)^{25/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=-\frac {504735 \, \left (-\frac {a^{3} d^{50}}{b^{27}}\right )^{\frac {1}{4}} {\left (b^{11} x^{10} + 5 \, a b^{10} x^{8} + 10 \, a^{2} b^{9} x^{6} + 10 \, a^{3} b^{8} x^{4} + 5 \, a^{4} b^{7} x^{2} + a^{5} b^{6}\right )} \log \left (38099255258449 \, \sqrt {d x} a^{2} d^{37} + 38099255258449 \, \left (-\frac {a^{3} d^{50}}{b^{27}}\right )^{\frac {3}{4}} b^{20}\right ) + 504735 \, \left (-\frac {a^{3} d^{50}}{b^{27}}\right )^{\frac {1}{4}} {\left (-i \, b^{11} x^{10} - 5 i \, a b^{10} x^{8} - 10 i \, a^{2} b^{9} x^{6} - 10 i \, a^{3} b^{8} x^{4} - 5 i \, a^{4} b^{7} x^{2} - i \, a^{5} b^{6}\right )} \log \left (38099255258449 \, \sqrt {d x} a^{2} d^{37} + 38099255258449 i \, \left (-\frac {a^{3} d^{50}}{b^{27}}\right )^{\frac {3}{4}} b^{20}\right ) + 504735 \, \left (-\frac {a^{3} d^{50}}{b^{27}}\right )^{\frac {1}{4}} {\left (i \, b^{11} x^{10} + 5 i \, a b^{10} x^{8} + 10 i \, a^{2} b^{9} x^{6} + 10 i \, a^{3} b^{8} x^{4} + 5 i \, a^{4} b^{7} x^{2} + i \, a^{5} b^{6}\right )} \log \left (38099255258449 \, \sqrt {d x} a^{2} d^{37} - 38099255258449 i \, \left (-\frac {a^{3} d^{50}}{b^{27}}\right )^{\frac {3}{4}} b^{20}\right ) - 504735 \, \left (-\frac {a^{3} d^{50}}{b^{27}}\right )^{\frac {1}{4}} {\left (b^{11} x^{10} + 5 \, a b^{10} x^{8} + 10 \, a^{2} b^{9} x^{6} + 10 \, a^{3} b^{8} x^{4} + 5 \, a^{4} b^{7} x^{2} + a^{5} b^{6}\right )} \log \left (38099255258449 \, \sqrt {d x} a^{2} d^{37} - 38099255258449 \, \left (-\frac {a^{3} d^{50}}{b^{27}}\right )^{\frac {3}{4}} b^{20}\right ) - 4 \, {\left (40960 \, b^{5} d^{12} x^{11} + 437345 \, a b^{4} d^{12} x^{9} + 1157176 \, a^{2} b^{3} d^{12} x^{7} + 1367810 \, a^{3} b^{2} d^{12} x^{5} + 769120 \, a^{4} b d^{12} x^{3} + 168245 \, a^{5} d^{12} x\right )} \sqrt {d x}}{245760 \, {\left (b^{11} x^{10} + 5 \, a b^{10} x^{8} + 10 \, a^{2} b^{9} x^{6} + 10 \, a^{3} b^{8} x^{4} + 5 \, a^{4} b^{7} x^{2} + a^{5} b^{6}\right )}} \] Input:
integrate((d*x)^(25/2)/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="fricas")
Output:
-1/245760*(504735*(-a^3*d^50/b^27)^(1/4)*(b^11*x^10 + 5*a*b^10*x^8 + 10*a^ 2*b^9*x^6 + 10*a^3*b^8*x^4 + 5*a^4*b^7*x^2 + a^5*b^6)*log(38099255258449*s qrt(d*x)*a^2*d^37 + 38099255258449*(-a^3*d^50/b^27)^(3/4)*b^20) + 504735*( -a^3*d^50/b^27)^(1/4)*(-I*b^11*x^10 - 5*I*a*b^10*x^8 - 10*I*a^2*b^9*x^6 - 10*I*a^3*b^8*x^4 - 5*I*a^4*b^7*x^2 - I*a^5*b^6)*log(38099255258449*sqrt(d* x)*a^2*d^37 + 38099255258449*I*(-a^3*d^50/b^27)^(3/4)*b^20) + 504735*(-a^3 *d^50/b^27)^(1/4)*(I*b^11*x^10 + 5*I*a*b^10*x^8 + 10*I*a^2*b^9*x^6 + 10*I* a^3*b^8*x^4 + 5*I*a^4*b^7*x^2 + I*a^5*b^6)*log(38099255258449*sqrt(d*x)*a^ 2*d^37 - 38099255258449*I*(-a^3*d^50/b^27)^(3/4)*b^20) - 504735*(-a^3*d^50 /b^27)^(1/4)*(b^11*x^10 + 5*a*b^10*x^8 + 10*a^2*b^9*x^6 + 10*a^3*b^8*x^4 + 5*a^4*b^7*x^2 + a^5*b^6)*log(38099255258449*sqrt(d*x)*a^2*d^37 - 38099255 258449*(-a^3*d^50/b^27)^(3/4)*b^20) - 4*(40960*b^5*d^12*x^11 + 437345*a*b^ 4*d^12*x^9 + 1157176*a^2*b^3*d^12*x^7 + 1367810*a^3*b^2*d^12*x^5 + 769120* a^4*b*d^12*x^3 + 168245*a^5*d^12*x)*sqrt(d*x))/(b^11*x^10 + 5*a*b^10*x^8 + 10*a^2*b^9*x^6 + 10*a^3*b^8*x^4 + 5*a^4*b^7*x^2 + a^5*b^6)
Timed out. \[ \int \frac {(d x)^{25/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\text {Timed out} \] Input:
integrate((d*x)**(25/2)/(b**2*x**4+2*a*b*x**2+a**2)**3,x)
Output:
Timed out
Time = 0.12 (sec) , antiderivative size = 394, normalized size of antiderivative = 1.21 \[ \int \frac {(d x)^{25/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=-\frac {\frac {504735 \, a d^{14} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {d x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} d}}\right )}{\sqrt {\sqrt {a} \sqrt {b} d} \sqrt {b}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {d x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} d}}\right )}{\sqrt {\sqrt {a} \sqrt {b} d} \sqrt {b}} - \frac {\sqrt {2} \log \left (\sqrt {b} d x + \sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} \sqrt {d x} b^{\frac {1}{4}} + \sqrt {a} d\right )}{\left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (\sqrt {b} d x - \sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} \sqrt {d x} b^{\frac {1}{4}} + \sqrt {a} d\right )}{\left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {3}{4}}}\right )}}{b^{6}} - \frac {327680 \, \left (d x\right )^{\frac {3}{2}} d^{12}}{b^{6}} - \frac {8 \, {\left (232545 \, \left (d x\right )^{\frac {19}{2}} a b^{4} d^{14} + 747576 \, \left (d x\right )^{\frac {15}{2}} a^{2} b^{3} d^{16} + 958210 \, \left (d x\right )^{\frac {11}{2}} a^{3} b^{2} d^{18} + 564320 \, \left (d x\right )^{\frac {7}{2}} a^{4} b d^{20} + 127285 \, \left (d x\right )^{\frac {3}{2}} a^{5} d^{22}\right )}}{b^{11} d^{10} x^{10} + 5 \, a b^{10} d^{10} x^{8} + 10 \, a^{2} b^{9} d^{10} x^{6} + 10 \, a^{3} b^{8} d^{10} x^{4} + 5 \, a^{4} b^{7} d^{10} x^{2} + a^{5} b^{6} d^{10}}}{491520 \, d} \] Input:
integrate((d*x)^(25/2)/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="maxima")
Output:
-1/491520*(504735*a*d^14*(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)*sqr t(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)))/b^6 - 327680*(d*x)^(3/2)*d^12/b^6 - 8*(232545*(d*x)^(19/2)*a*b^4*d^1 4 + 747576*(d*x)^(15/2)*a^2*b^3*d^16 + 958210*(d*x)^(11/2)*a^3*b^2*d^18 + 564320*(d*x)^(7/2)*a^4*b*d^20 + 127285*(d*x)^(3/2)*a^5*d^22)/(b^11*d^10*x^ 10 + 5*a*b^10*d^10*x^8 + 10*a^2*b^9*d^10*x^6 + 10*a^3*b^8*d^10*x^4 + 5*a^4 *b^7*d^10*x^2 + a^5*b^6*d^10))/d
Time = 0.15 (sec) , antiderivative size = 354, normalized size of antiderivative = 1.09 \[ \int \frac {(d x)^{25/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {1}{491520} \, d^{12} {\left (\frac {327680 \, \sqrt {d x} x}{b^{6}} - \frac {1009470 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{b^{9} d} - \frac {1009470 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{b^{9} d} + \frac {504735 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \log \left (d x + \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{b^{9} d} - \frac {504735 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \log \left (d x - \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{b^{9} d} + \frac {8 \, {\left (232545 \, \sqrt {d x} a b^{4} d^{10} x^{9} + 747576 \, \sqrt {d x} a^{2} b^{3} d^{10} x^{7} + 958210 \, \sqrt {d x} a^{3} b^{2} d^{10} x^{5} + 564320 \, \sqrt {d x} a^{4} b d^{10} x^{3} + 127285 \, \sqrt {d x} a^{5} d^{10} x\right )}}{{\left (b d^{2} x^{2} + a d^{2}\right )}^{5} b^{6}}\right )} \] Input:
integrate((d*x)^(25/2)/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="giac")
Output:
1/491520*d^12*(327680*sqrt(d*x)*x/b^6 - 1009470*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) )/(b^9*d) - 1009470*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))/(b^9*d) + 504735*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))/(b^9*d) - 504735*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))/(b^9*d) + 8*(232545*sqrt(d*x)*a*b^4*d^1 0*x^9 + 747576*sqrt(d*x)*a^2*b^3*d^10*x^7 + 958210*sqrt(d*x)*a^3*b^2*d^10* x^5 + 564320*sqrt(d*x)*a^4*b*d^10*x^3 + 127285*sqrt(d*x)*a^5*d^10*x)/((b*d ^2*x^2 + a*d^2)^5*b^6))
Time = 18.63 (sec) , antiderivative size = 231, normalized size of antiderivative = 0.71 \[ \int \frac {(d x)^{25/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {\frac {25457\,a^5\,d^{21}\,{\left (d\,x\right )}^{3/2}}{12288}+\frac {95821\,a^3\,b^2\,d^{17}\,{\left (d\,x\right )}^{11/2}}{6144}+\frac {31149\,a^2\,b^3\,d^{15}\,{\left (d\,x\right )}^{15/2}}{2560}+\frac {3527\,a^4\,b\,d^{19}\,{\left (d\,x\right )}^{7/2}}{384}+\frac {15503\,a\,b^4\,d^{13}\,{\left (d\,x\right )}^{19/2}}{4096}}{a^5\,b^6\,d^{10}+5\,a^4\,b^7\,d^{10}\,x^2+10\,a^3\,b^8\,d^{10}\,x^4+10\,a^2\,b^9\,d^{10}\,x^6+5\,a\,b^{10}\,d^{10}\,x^8+b^{11}\,d^{10}\,x^{10}}+\frac {2\,d^{11}\,{\left (d\,x\right )}^{3/2}}{3\,b^6}+\frac {33649\,{\left (-a\right )}^{3/4}\,d^{25/2}\,\mathrm {atan}\left (\frac {b^{1/4}\,\sqrt {d\,x}}{{\left (-a\right )}^{1/4}\,\sqrt {d}}\right )}{8192\,b^{27/4}}+\frac {{\left (-a\right )}^{3/4}\,d^{25/2}\,\mathrm {atan}\left (\frac {b^{1/4}\,\sqrt {d\,x}\,1{}\mathrm {i}}{{\left (-a\right )}^{1/4}\,\sqrt {d}}\right )\,33649{}\mathrm {i}}{8192\,b^{27/4}} \] Input:
int((d*x)^(25/2)/(a^2 + b^2*x^4 + 2*a*b*x^2)^3,x)
Output:
((25457*a^5*d^21*(d*x)^(3/2))/12288 + (95821*a^3*b^2*d^17*(d*x)^(11/2))/61 44 + (31149*a^2*b^3*d^15*(d*x)^(15/2))/2560 + (3527*a^4*b*d^19*(d*x)^(7/2) )/384 + (15503*a*b^4*d^13*(d*x)^(19/2))/4096)/(a^5*b^6*d^10 + b^11*d^10*x^ 10 + 5*a*b^10*d^10*x^8 + 5*a^4*b^7*d^10*x^2 + 10*a^3*b^8*d^10*x^4 + 10*a^2 *b^9*d^10*x^6) + (2*d^11*(d*x)^(3/2))/(3*b^6) + (33649*(-a)^(3/4)*d^(25/2) *atan((b^(1/4)*(d*x)^(1/2))/((-a)^(1/4)*d^(1/2))))/(8192*b^(27/4)) + ((-a) ^(3/4)*d^(25/2)*atan((b^(1/4)*(d*x)^(1/2)*1i)/((-a)^(1/4)*d^(1/2)))*33649i )/(8192*b^(27/4))
Time = 0.19 (sec) , antiderivative size = 1003, normalized size of antiderivative = 3.09 \[ \int \frac {(d x)^{25/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx =\text {Too large to display} \] Input:
int((d*x)^(25/2)/(b^2*x^4+2*a*b*x^2+a^2)^3,x)
Output:
(sqrt(d)*d**12*(1009470*b**(1/4)*a**(3/4)*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 + 5047350*b **(1/4)*a**(3/4)*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**2 + 10094700*b**(1/4)*a**(3/4)* 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**4 + 10094700*b**(1/4)*a**(3/4)*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**6 + 5047350*b**(1/4)*a**(3/4)*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**8 + 1009470*b**(1/4)*a**(3/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) - 2*sqr t(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*b**5*x**10 - 1009470*b**(1/4)*a **(3/4)*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 - 5047350*b**(1/4)*a**(3/4)*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**2 - 10094700*b**(1/4)*a**(3/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sq rt(2) + 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*a**3*b**2*x**4 - 1 0094700*b**(1/4)*a**(3/4)*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**6 - 5047350*b**(1/4 )*a**(3/4)*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**8 - 1009470*b**(1/4)*a**(3/4)*sqrt...