Integrand size = 28, antiderivative size = 343 \[ \int \frac {(d x)^{27/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=-\frac {69615 a d^{13} \sqrt {d x}}{4096 b^7}+\frac {13923 d^{11} (d x)^{5/2}}{4096 b^6}-\frac {d (d x)^{25/2}}{10 b \left (a+b x^2\right )^5}-\frac {5 d^3 (d x)^{21/2}}{32 b^2 \left (a+b x^2\right )^4}-\frac {35 d^5 (d x)^{17/2}}{128 b^3 \left (a+b x^2\right )^3}-\frac {595 d^7 (d x)^{13/2}}{1024 b^4 \left (a+b x^2\right )^2}-\frac {7735 d^9 (d x)^{9/2}}{4096 b^5 \left (a+b x^2\right )}-\frac {69615 a^{5/4} d^{27/2} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} b^{29/4}}+\frac {69615 a^{5/4} d^{27/2} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} b^{29/4}}+\frac {69615 a^{5/4} d^{27/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}}{\sqrt {d} \left (\sqrt {a}+\sqrt {b} x\right )}\right )}{8192 \sqrt {2} b^{29/4}} \] Output:
-69615/4096*a*d^13*(d*x)^(1/2)/b^7+13923/4096*d^11*(d*x)^(5/2)/b^6-1/10*d* (d*x)^(25/2)/b/(b*x^2+a)^5-5/32*d^3*(d*x)^(21/2)/b^2/(b*x^2+a)^4-35/128*d^ 5*(d*x)^(17/2)/b^3/(b*x^2+a)^3-595/1024*d^7*(d*x)^(13/2)/b^4/(b*x^2+a)^2-7 735/4096*d^9*(d*x)^(9/2)/b^5/(b*x^2+a)-69615/16384*a^(5/4)*d^(27/2)*arctan (1-2^(1/2)*b^(1/4)*(d*x)^(1/2)/a^(1/4)/d^(1/2))*2^(1/2)/b^(29/4)+69615/163 84*a^(5/4)*d^(27/2)*arctan(1+2^(1/2)*b^(1/4)*(d*x)^(1/2)/a^(1/4)/d^(1/2))* 2^(1/2)/b^(29/4)+69615/16384*a^(5/4)*d^(27/2)*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^(29/4)
Time = 1.03 (sec) , antiderivative size = 227, normalized size of antiderivative = 0.66 \[ \int \frac {(d x)^{27/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {d^{13} \sqrt {d x} \left (4 \sqrt [4]{b} \sqrt {x} \left (-348075 a^6-1670760 a^5 b x^2-3171350 a^4 b^2 x^4-2951200 a^3 b^3 x^6-1317575 a^2 b^4 x^8-204800 a b^5 x^{10}+8192 b^6 x^{12}\right )+348075 \sqrt {2} a^{5/4} \left (a+b x^2\right )^5 \arctan \left (\frac {-\sqrt {a}+\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )+348075 \sqrt {2} a^{5/4} \left (a+b x^2\right )^5 \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )\right )}{81920 b^{29/4} \sqrt {x} \left (a+b x^2\right )^5} \] Input:
Integrate[(d*x)^(27/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]
Output:
(d^13*Sqrt[d*x]*(4*b^(1/4)*Sqrt[x]*(-348075*a^6 - 1670760*a^5*b*x^2 - 3171 350*a^4*b^2*x^4 - 2951200*a^3*b^3*x^6 - 1317575*a^2*b^4*x^8 - 204800*a*b^5 *x^10 + 8192*b^6*x^12) + 348075*Sqrt[2]*a^(5/4)*(a + b*x^2)^5*ArcTan[(-Sqr t[a] + Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])] + 348075*Sqrt[2]*a^(5 /4)*(a + b*x^2)^5*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqr t[b]*x)]))/(81920*b^(29/4)*Sqrt[x]*(a + b*x^2)^5)
Time = 1.20 (sec) , antiderivative size = 489, normalized size of antiderivative = 1.43, number of steps used = 20, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.679, Rules used = {1380, 27, 252, 252, 252, 252, 252, 262, 262, 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.
| \(\displaystyle \int \frac {(d x)^{27/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx\) |
| \(\Big \downarrow \) 1380 |
| \(\displaystyle b^6 \int \frac {(d x)^{27/2}}{b^6 \left (b x^2+a\right )^6}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \frac {(d x)^{27/2}}{\left (a+b x^2\right )^6}dx\) |
| \(\Big \downarrow \) 252 |
| \(\displaystyle \frac {5 d^2 \int \frac {(d x)^{23/2}}{\left (b x^2+a\right )^5}dx}{4 b}-\frac {d (d x)^{25/2}}{10 b \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 252 |
| \(\displaystyle \frac {5 d^2 \left (\frac {21 d^2 \int \frac {(d x)^{19/2}}{\left (b x^2+a\right )^4}dx}{16 b}-\frac {d (d x)^{21/2}}{8 b \left (a+b x^2\right )^4}\right )}{4 b}-\frac {d (d x)^{25/2}}{10 b \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 252 |
| \(\displaystyle \frac {5 d^2 \left (\frac {21 d^2 \left (\frac {17 d^2 \int \frac {(d x)^{15/2}}{\left (b x^2+a\right )^3}dx}{12 b}-\frac {d (d x)^{17/2}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{21/2}}{8 b \left (a+b x^2\right )^4}\right )}{4 b}-\frac {d (d x)^{25/2}}{10 b \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 252 |
| \(\displaystyle \frac {5 d^2 \left (\frac {21 d^2 \left (\frac {17 d^2 \left (\frac {13 d^2 \int \frac {(d x)^{11/2}}{\left (b x^2+a\right )^2}dx}{8 b}-\frac {d (d x)^{13/2}}{4 b \left (a+b x^2\right )^2}\right )}{12 b}-\frac {d (d x)^{17/2}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{21/2}}{8 b \left (a+b x^2\right )^4}\right )}{4 b}-\frac {d (d x)^{25/2}}{10 b \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 252 |
| \(\displaystyle \frac {5 d^2 \left (\frac {21 d^2 \left (\frac {17 d^2 \left (\frac {13 d^2 \left (\frac {9 d^2 \int \frac {(d x)^{7/2}}{b x^2+a}dx}{4 b}-\frac {d (d x)^{9/2}}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{13/2}}{4 b \left (a+b x^2\right )^2}\right )}{12 b}-\frac {d (d x)^{17/2}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{21/2}}{8 b \left (a+b x^2\right )^4}\right )}{4 b}-\frac {d (d x)^{25/2}}{10 b \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle \frac {5 d^2 \left (\frac {21 d^2 \left (\frac {17 d^2 \left (\frac {13 d^2 \left (\frac {9 d^2 \left (\frac {2 d (d x)^{5/2}}{5 b}-\frac {a d^2 \int \frac {(d x)^{3/2}}{b x^2+a}dx}{b}\right )}{4 b}-\frac {d (d x)^{9/2}}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{13/2}}{4 b \left (a+b x^2\right )^2}\right )}{12 b}-\frac {d (d x)^{17/2}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{21/2}}{8 b \left (a+b x^2\right )^4}\right )}{4 b}-\frac {d (d x)^{25/2}}{10 b \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle \frac {5 d^2 \left (\frac {21 d^2 \left (\frac {17 d^2 \left (\frac {13 d^2 \left (\frac {9 d^2 \left (\frac {2 d (d x)^{5/2}}{5 b}-\frac {a d^2 \left (\frac {2 d \sqrt {d x}}{b}-\frac {a d^2 \int \frac {1}{\sqrt {d x} \left (b x^2+a\right )}dx}{b}\right )}{b}\right )}{4 b}-\frac {d (d x)^{9/2}}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{13/2}}{4 b \left (a+b x^2\right )^2}\right )}{12 b}-\frac {d (d x)^{17/2}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{21/2}}{8 b \left (a+b x^2\right )^4}\right )}{4 b}-\frac {d (d x)^{25/2}}{10 b \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {5 d^2 \left (\frac {21 d^2 \left (\frac {17 d^2 \left (\frac {13 d^2 \left (\frac {9 d^2 \left (\frac {2 d (d x)^{5/2}}{5 b}-\frac {a d^2 \left (\frac {2 d \sqrt {d x}}{b}-\frac {2 a d \int \frac {1}{b x^2+a}d\sqrt {d x}}{b}\right )}{b}\right )}{4 b}-\frac {d (d x)^{9/2}}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{13/2}}{4 b \left (a+b x^2\right )^2}\right )}{12 b}-\frac {d (d x)^{17/2}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{21/2}}{8 b \left (a+b x^2\right )^4}\right )}{4 b}-\frac {d (d x)^{25/2}}{10 b \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 755 |
| \(\displaystyle \frac {5 d^2 \left (\frac {21 d^2 \left (\frac {17 d^2 \left (\frac {13 d^2 \left (\frac {9 d^2 \left (\frac {2 d (d x)^{5/2}}{5 b}-\frac {a d^2 \left (\frac {2 d \sqrt {d x}}{b}-\frac {2 a d \left (\frac {\int \frac {d^2 \left (\sqrt {a} d-\sqrt {b} d x\right )}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {a} d}+\frac {\int \frac {d^2 \left (\sqrt {b} x d+\sqrt {a} d\right )}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {a} d}\right )}{b}\right )}{b}\right )}{4 b}-\frac {d (d x)^{9/2}}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{13/2}}{4 b \left (a+b x^2\right )^2}\right )}{12 b}-\frac {d (d x)^{17/2}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{21/2}}{8 b \left (a+b x^2\right )^4}\right )}{4 b}-\frac {d (d x)^{25/2}}{10 b \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5 d^2 \left (\frac {21 d^2 \left (\frac {17 d^2 \left (\frac {13 d^2 \left (\frac {9 d^2 \left (\frac {2 d (d x)^{5/2}}{5 b}-\frac {a d^2 \left (\frac {2 d \sqrt {d x}}{b}-\frac {2 a d \left (\frac {d \int \frac {\sqrt {a} d-\sqrt {b} d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {a}}+\frac {d \int \frac {\sqrt {b} x d+\sqrt {a} d}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {a}}\right )}{b}\right )}{b}\right )}{4 b}-\frac {d (d x)^{9/2}}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{13/2}}{4 b \left (a+b x^2\right )^2}\right )}{12 b}-\frac {d (d x)^{17/2}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{21/2}}{8 b \left (a+b x^2\right )^4}\right )}{4 b}-\frac {d (d x)^{25/2}}{10 b \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {5 d^2 \left (\frac {21 d^2 \left (\frac {17 d^2 \left (\frac {13 d^2 \left (\frac {9 d^2 \left (\frac {2 d (d x)^{5/2}}{5 b}-\frac {a d^2 \left (\frac {2 d \sqrt {d x}}{b}-\frac {2 a d \left (\frac {d \int \frac {\sqrt {a} d-\sqrt {b} d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {a}}+\frac {d \left (\frac {\int \frac {1}{x d+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}}d\sqrt {d x}}{2 \sqrt {b}}+\frac {\int \frac {1}{x d+\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}}d\sqrt {d x}}{2 \sqrt {b}}\right )}{2 \sqrt {a}}\right )}{b}\right )}{b}\right )}{4 b}-\frac {d (d x)^{9/2}}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{13/2}}{4 b \left (a+b x^2\right )^2}\right )}{12 b}-\frac {d (d x)^{17/2}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{21/2}}{8 b \left (a+b x^2\right )^4}\right )}{4 b}-\frac {d (d x)^{25/2}}{10 b \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {5 d^2 \left (\frac {21 d^2 \left (\frac {17 d^2 \left (\frac {13 d^2 \left (\frac {9 d^2 \left (\frac {2 d (d x)^{5/2}}{5 b}-\frac {a d^2 \left (\frac {2 d \sqrt {d x}}{b}-\frac {2 a d \left (\frac {d \int \frac {\sqrt {a} d-\sqrt {b} d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {a}}+\frac {d \left (\frac {\int \frac {1}{-d x-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\int \frac {1}{-d x-1}d\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}\right )}{b}\right )}{b}\right )}{4 b}-\frac {d (d x)^{9/2}}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{13/2}}{4 b \left (a+b x^2\right )^2}\right )}{12 b}-\frac {d (d x)^{17/2}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{21/2}}{8 b \left (a+b x^2\right )^4}\right )}{4 b}-\frac {d (d x)^{25/2}}{10 b \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {5 d^2 \left (\frac {21 d^2 \left (\frac {17 d^2 \left (\frac {13 d^2 \left (\frac {9 d^2 \left (\frac {2 d (d x)^{5/2}}{5 b}-\frac {a d^2 \left (\frac {2 d \sqrt {d x}}{b}-\frac {2 a d \left (\frac {d \int \frac {\sqrt {a} d-\sqrt {b} d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {a}}+\frac {d \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}\right )}{b}\right )}{b}\right )}{4 b}-\frac {d (d x)^{9/2}}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{13/2}}{4 b \left (a+b x^2\right )^2}\right )}{12 b}-\frac {d (d x)^{17/2}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{21/2}}{8 b \left (a+b x^2\right )^4}\right )}{4 b}-\frac {d (d x)^{25/2}}{10 b \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {5 d^2 \left (\frac {21 d^2 \left (\frac {17 d^2 \left (\frac {13 d^2 \left (\frac {9 d^2 \left (\frac {2 d (d x)^{5/2}}{5 b}-\frac {a d^2 \left (\frac {2 d \sqrt {d x}}{b}-\frac {2 a d \left (\frac {d \left (-\frac {\int -\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d}-2 \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{b} \left (x d+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt [4]{a} \sqrt {d}+\sqrt {2} \sqrt [4]{b} \sqrt {d x}\right )}{\sqrt [4]{b} \left (x d+\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}+\frac {d \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}\right )}{b}\right )}{b}\right )}{4 b}-\frac {d (d x)^{9/2}}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{13/2}}{4 b \left (a+b x^2\right )^2}\right )}{12 b}-\frac {d (d x)^{17/2}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{21/2}}{8 b \left (a+b x^2\right )^4}\right )}{4 b}-\frac {d (d x)^{25/2}}{10 b \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {5 d^2 \left (\frac {21 d^2 \left (\frac {17 d^2 \left (\frac {13 d^2 \left (\frac {9 d^2 \left (\frac {2 d (d x)^{5/2}}{5 b}-\frac {a d^2 \left (\frac {2 d \sqrt {d x}}{b}-\frac {2 a d \left (\frac {d \left (\frac {\int \frac {\sqrt {2} \sqrt [4]{a} \sqrt {d}-2 \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{b} \left (x d+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}+\frac {\int \frac {\sqrt {2} \left (\sqrt [4]{a} \sqrt {d}+\sqrt {2} \sqrt [4]{b} \sqrt {d x}\right )}{\sqrt [4]{b} \left (x d+\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}+\frac {d \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}\right )}{b}\right )}{b}\right )}{4 b}-\frac {d (d x)^{9/2}}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{13/2}}{4 b \left (a+b x^2\right )^2}\right )}{12 b}-\frac {d (d x)^{17/2}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{21/2}}{8 b \left (a+b x^2\right )^4}\right )}{4 b}-\frac {d (d x)^{25/2}}{10 b \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5 d^2 \left (\frac {21 d^2 \left (\frac {17 d^2 \left (\frac {13 d^2 \left (\frac {9 d^2 \left (\frac {2 d (d x)^{5/2}}{5 b}-\frac {a d^2 \left (\frac {2 d \sqrt {d x}}{b}-\frac {2 a d \left (\frac {d \left (\frac {\int \frac {\sqrt {2} \sqrt [4]{a} \sqrt {d}-2 \sqrt [4]{b} \sqrt {d x}}{x d+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt {b} \sqrt {d}}+\frac {\int \frac {\sqrt [4]{a} \sqrt {d}+\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{x d+\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}}d\sqrt {d x}}{2 \sqrt [4]{a} \sqrt {b} \sqrt {d}}\right )}{2 \sqrt {a}}+\frac {d \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}\right )}{b}\right )}{b}\right )}{4 b}-\frac {d (d x)^{9/2}}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{13/2}}{4 b \left (a+b x^2\right )^2}\right )}{12 b}-\frac {d (d x)^{17/2}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{21/2}}{8 b \left (a+b x^2\right )^4}\right )}{4 b}-\frac {d (d x)^{25/2}}{10 b \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {5 d^2 \left (\frac {21 d^2 \left (\frac {17 d^2 \left (\frac {13 d^2 \left (\frac {9 d^2 \left (\frac {2 d (d x)^{5/2}}{5 b}-\frac {a d^2 \left (\frac {2 d \sqrt {d x}}{b}-\frac {2 a d \left (\frac {d \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}+\frac {d \left (\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d} \sqrt {d x}+\sqrt {a} d+\sqrt {b} d x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d} \sqrt {d x}+\sqrt {a} d+\sqrt {b} d x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}\right )}{b}\right )}{b}\right )}{4 b}-\frac {d (d x)^{9/2}}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{13/2}}{4 b \left (a+b x^2\right )^2}\right )}{12 b}-\frac {d (d x)^{17/2}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{21/2}}{8 b \left (a+b x^2\right )^4}\right )}{4 b}-\frac {d (d x)^{25/2}}{10 b \left (a+b x^2\right )^5}\) |
Input:
Int[(d*x)^(27/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]
Output:
-1/10*(d*(d*x)^(25/2))/(b*(a + b*x^2)^5) + (5*d^2*(-1/8*(d*(d*x)^(21/2))/( b*(a + b*x^2)^4) + (21*d^2*(-1/6*(d*(d*x)^(17/2))/(b*(a + b*x^2)^3) + (17* d^2*(-1/4*(d*(d*x)^(13/2))/(b*(a + b*x^2)^2) + (13*d^2*(-1/2*(d*(d*x)^(9/2 ))/(b*(a + b*x^2)) + (9*d^2*((2*d*(d*x)^(5/2))/(5*b) - (a*d^2*((2*d*Sqrt[d *x])/b - (2*a*d*((d*(-(ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqr t[d])]/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d])) + ArcTan[1 + (Sqrt[2]*b^(1/4)*Sq rt[d*x])/(a^(1/4)*Sqrt[d])]/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d])))/(2*Sqrt[a] ) + (d*(-1/2*Log[Sqrt[a]*d + Sqrt[b]*d*x - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d] *Sqrt[d*x]]/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d]) + Log[Sqrt[a]*d + Sqrt[b]*d* x + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d]*Sqrt[d*x]]/(2*Sqrt[2]*a^(1/4)*b^(1/4)* Sqrt[d])))/(2*Sqrt[a])))/b))/b))/(4*b)))/(8*b)))/(12*b)))/(16*b)))/(4*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[((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 = 14.83 (sec) , antiderivative size = 256, normalized size of antiderivative = 0.75
| method | result | size |
| pseudoelliptic | \(\frac {d^{13} \left (\left (65536 b^{6} x^{12}-1638400 a \,b^{5} x^{10}-10540600 a^{2} b^{4} x^{8}-23609600 a^{3} b^{3} x^{6}-25370800 a^{4} b^{2} x^{4}-13366080 a^{5} b \,x^{2}-2784600 a^{6}\right ) \sqrt {d x}+348075 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} a \sqrt {2}\, \left (b \,x^{2}+a \right )^{5} \left (\ln \left (\frac {d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )-2 \arctan \left (\frac {-\sqrt {2}\, \sqrt {d x}+\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}+\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )\right )\right )}{163840 b^{7} \left (b \,x^{2}+a \right )^{5}}\) | \(256\) |
| risch | \(-\frac {2 \left (-b \,x^{2}+30 a \right ) x \,d^{14}}{5 b^{7} \sqrt {d x}}+\frac {2 a^{2} d^{15} \left (\frac {-\frac {20463 a^{4} d^{8} \sqrt {d x}}{8192}-\frac {56269 a^{3} d^{6} b \left (d x \right )^{\frac {5}{2}}}{5120}-\frac {75471 a^{2} d^{4} b^{2} \left (d x \right )^{\frac {9}{2}}}{4096}-\frac {3597 a \,d^{2} b^{3} \left (d x \right )^{\frac {13}{2}}}{256}-\frac {34139 b^{4} \left (d x \right )^{\frac {17}{2}}}{8192}}{\left (b \,d^{2} x^{2}+a \,d^{2}\right )^{5}}+\frac {69615 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{65536 a \,d^{2}}\right )}{b^{7}}\) | \(264\) |
| derivativedivides | \(2 d^{11} \left (-\frac {-\frac {b \left (d x \right )^{\frac {5}{2}}}{5}+6 \sqrt {d x}\, a \,d^{2}}{b^{7}}+\frac {a^{2} d^{4} \left (\frac {-\frac {20463 a^{4} d^{8} \sqrt {d x}}{8192}-\frac {56269 a^{3} d^{6} b \left (d x \right )^{\frac {5}{2}}}{5120}-\frac {75471 a^{2} d^{4} b^{2} \left (d x \right )^{\frac {9}{2}}}{4096}-\frac {3597 a \,d^{2} b^{3} \left (d x \right )^{\frac {13}{2}}}{256}-\frac {34139 b^{4} \left (d x \right )^{\frac {17}{2}}}{8192}}{\left (b \,d^{2} x^{2}+a \,d^{2}\right )^{5}}+\frac {69615 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{65536 a \,d^{2}}\right )}{b^{7}}\right )\) | \(269\) |
| default | \(2 d^{11} \left (-\frac {-\frac {b \left (d x \right )^{\frac {5}{2}}}{5}+6 \sqrt {d x}\, a \,d^{2}}{b^{7}}+\frac {a^{2} d^{4} \left (\frac {-\frac {20463 a^{4} d^{8} \sqrt {d x}}{8192}-\frac {56269 a^{3} d^{6} b \left (d x \right )^{\frac {5}{2}}}{5120}-\frac {75471 a^{2} d^{4} b^{2} \left (d x \right )^{\frac {9}{2}}}{4096}-\frac {3597 a \,d^{2} b^{3} \left (d x \right )^{\frac {13}{2}}}{256}-\frac {34139 b^{4} \left (d x \right )^{\frac {17}{2}}}{8192}}{\left (b \,d^{2} x^{2}+a \,d^{2}\right )^{5}}+\frac {69615 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{65536 a \,d^{2}}\right )}{b^{7}}\right )\) | \(269\) |
Input:
int((d*x)^(27/2)/(b^2*x^4+2*a*b*x^2+a^2)^3,x,method=_RETURNVERBOSE)
Output:
1/163840*d^13*((65536*b^6*x^12-1638400*a*b^5*x^10-10540600*a^2*b^4*x^8-236 09600*a^3*b^3*x^6-25370800*a^4*b^2*x^4-13366080*a^5*b*x^2-2784600*a^6)*(d* x)^(1/2)+348075*(a*d^2/b)^(1/4)*a*2^(1/2)*(b*x^2+a)^5*(ln((d*x+(a*d^2/b)^( 1/4)*(d*x)^(1/2)*2^(1/2)+(a*d^2/b)^(1/2))/(d*x-(a*d^2/b)^(1/4)*(d*x)^(1/2) *2^(1/2)+(a*d^2/b)^(1/2)))-2*arctan((-2^(1/2)*(d*x)^(1/2)+(a*d^2/b)^(1/4)) /(a*d^2/b)^(1/4))+2*arctan((2^(1/2)*(d*x)^(1/2)+(a*d^2/b)^(1/4))/(a*d^2/b) ^(1/4))))/b^7/(b*x^2+a)^5
Result contains complex when optimal does not.
Time = 0.17 (sec) , antiderivative size = 571, normalized size of antiderivative = 1.66 \[ \int \frac {(d x)^{27/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {348075 \, \left (-\frac {a^{5} d^{54}}{b^{29}}\right )^{\frac {1}{4}} {\left (b^{12} x^{10} + 5 \, a b^{11} x^{8} + 10 \, a^{2} b^{10} x^{6} + 10 \, a^{3} b^{9} x^{4} + 5 \, a^{4} b^{8} x^{2} + a^{5} b^{7}\right )} \log \left (69615 \, \sqrt {d x} a d^{13} + 69615 \, \left (-\frac {a^{5} d^{54}}{b^{29}}\right )^{\frac {1}{4}} b^{7}\right ) - 348075 \, \left (-\frac {a^{5} d^{54}}{b^{29}}\right )^{\frac {1}{4}} {\left (-i \, b^{12} x^{10} - 5 i \, a b^{11} x^{8} - 10 i \, a^{2} b^{10} x^{6} - 10 i \, a^{3} b^{9} x^{4} - 5 i \, a^{4} b^{8} x^{2} - i \, a^{5} b^{7}\right )} \log \left (69615 \, \sqrt {d x} a d^{13} + 69615 i \, \left (-\frac {a^{5} d^{54}}{b^{29}}\right )^{\frac {1}{4}} b^{7}\right ) - 348075 \, \left (-\frac {a^{5} d^{54}}{b^{29}}\right )^{\frac {1}{4}} {\left (i \, b^{12} x^{10} + 5 i \, a b^{11} x^{8} + 10 i \, a^{2} b^{10} x^{6} + 10 i \, a^{3} b^{9} x^{4} + 5 i \, a^{4} b^{8} x^{2} + i \, a^{5} b^{7}\right )} \log \left (69615 \, \sqrt {d x} a d^{13} - 69615 i \, \left (-\frac {a^{5} d^{54}}{b^{29}}\right )^{\frac {1}{4}} b^{7}\right ) - 348075 \, \left (-\frac {a^{5} d^{54}}{b^{29}}\right )^{\frac {1}{4}} {\left (b^{12} x^{10} + 5 \, a b^{11} x^{8} + 10 \, a^{2} b^{10} x^{6} + 10 \, a^{3} b^{9} x^{4} + 5 \, a^{4} b^{8} x^{2} + a^{5} b^{7}\right )} \log \left (69615 \, \sqrt {d x} a d^{13} - 69615 \, \left (-\frac {a^{5} d^{54}}{b^{29}}\right )^{\frac {1}{4}} b^{7}\right ) + 4 \, {\left (8192 \, b^{6} d^{13} x^{12} - 204800 \, a b^{5} d^{13} x^{10} - 1317575 \, a^{2} b^{4} d^{13} x^{8} - 2951200 \, a^{3} b^{3} d^{13} x^{6} - 3171350 \, a^{4} b^{2} d^{13} x^{4} - 1670760 \, a^{5} b d^{13} x^{2} - 348075 \, a^{6} d^{13}\right )} \sqrt {d x}}{81920 \, {\left (b^{12} x^{10} + 5 \, a b^{11} x^{8} + 10 \, a^{2} b^{10} x^{6} + 10 \, a^{3} b^{9} x^{4} + 5 \, a^{4} b^{8} x^{2} + a^{5} b^{7}\right )}} \] Input:
integrate((d*x)^(27/2)/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="fricas")
Output:
1/81920*(348075*(-a^5*d^54/b^29)^(1/4)*(b^12*x^10 + 5*a*b^11*x^8 + 10*a^2* b^10*x^6 + 10*a^3*b^9*x^4 + 5*a^4*b^8*x^2 + a^5*b^7)*log(69615*sqrt(d*x)*a *d^13 + 69615*(-a^5*d^54/b^29)^(1/4)*b^7) - 348075*(-a^5*d^54/b^29)^(1/4)* (-I*b^12*x^10 - 5*I*a*b^11*x^8 - 10*I*a^2*b^10*x^6 - 10*I*a^3*b^9*x^4 - 5* I*a^4*b^8*x^2 - I*a^5*b^7)*log(69615*sqrt(d*x)*a*d^13 + 69615*I*(-a^5*d^54 /b^29)^(1/4)*b^7) - 348075*(-a^5*d^54/b^29)^(1/4)*(I*b^12*x^10 + 5*I*a*b^1 1*x^8 + 10*I*a^2*b^10*x^6 + 10*I*a^3*b^9*x^4 + 5*I*a^4*b^8*x^2 + I*a^5*b^7 )*log(69615*sqrt(d*x)*a*d^13 - 69615*I*(-a^5*d^54/b^29)^(1/4)*b^7) - 34807 5*(-a^5*d^54/b^29)^(1/4)*(b^12*x^10 + 5*a*b^11*x^8 + 10*a^2*b^10*x^6 + 10* a^3*b^9*x^4 + 5*a^4*b^8*x^2 + a^5*b^7)*log(69615*sqrt(d*x)*a*d^13 - 69615* (-a^5*d^54/b^29)^(1/4)*b^7) + 4*(8192*b^6*d^13*x^12 - 204800*a*b^5*d^13*x^ 10 - 1317575*a^2*b^4*d^13*x^8 - 2951200*a^3*b^3*d^13*x^6 - 3171350*a^4*b^2 *d^13*x^4 - 1670760*a^5*b*d^13*x^2 - 348075*a^6*d^13)*sqrt(d*x))/(b^12*x^1 0 + 5*a*b^11*x^8 + 10*a^2*b^10*x^6 + 10*a^3*b^9*x^4 + 5*a^4*b^8*x^2 + a^5* b^7)
Timed out. \[ \int \frac {(d x)^{27/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\text {Timed out} \] Input:
integrate((d*x)**(27/2)/(b**2*x**4+2*a*b*x**2+a**2)**3,x)
Output:
Timed out
Time = 0.13 (sec) , antiderivative size = 421, normalized size of antiderivative = 1.23 \[ \int \frac {(d x)^{27/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=-\frac {\frac {8 \, {\left (170695 \, \left (d x\right )^{\frac {17}{2}} a^{2} b^{4} d^{16} + 575520 \, \left (d x\right )^{\frac {13}{2}} a^{3} b^{3} d^{18} + 754710 \, \left (d x\right )^{\frac {9}{2}} a^{4} b^{2} d^{20} + 450152 \, \left (d x\right )^{\frac {5}{2}} a^{5} b d^{22} + 102315 \, \sqrt {d x} a^{6} d^{24}\right )}}{b^{12} d^{10} x^{10} + 5 \, a b^{11} d^{10} x^{8} + 10 \, a^{2} b^{10} d^{10} x^{6} + 10 \, a^{3} b^{9} d^{10} x^{4} + 5 \, a^{4} b^{8} d^{10} x^{2} + a^{5} b^{7} d^{10}} - \frac {348075 \, {\left (\frac {\sqrt {2} d^{16} \log \left (\sqrt {b} d x + \sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} \sqrt {d x} b^{\frac {1}{4}} + \sqrt {a} d\right )}{\left (a d^{2}\right )^{\frac {3}{4}} b^{\frac {1}{4}}} - \frac {\sqrt {2} d^{16} \log \left (\sqrt {b} d x - \sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} \sqrt {d x} b^{\frac {1}{4}} + \sqrt {a} d\right )}{\left (a d^{2}\right )^{\frac {3}{4}} b^{\frac {1}{4}}} + \frac {2 \, \sqrt {2} d^{15} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {d x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} d}}\right )}{\sqrt {\sqrt {a} \sqrt {b} d} \sqrt {a}} + \frac {2 \, \sqrt {2} d^{15} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {d x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} d}}\right )}{\sqrt {\sqrt {a} \sqrt {b} d} \sqrt {a}}\right )} a^{2}}{b^{7}} - \frac {65536 \, {\left (\left (d x\right )^{\frac {5}{2}} b d^{12} - 30 \, \sqrt {d x} a d^{14}\right )}}{b^{7}}}{163840 \, d} \] Input:
integrate((d*x)^(27/2)/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="maxima")
Output:
-1/163840*(8*(170695*(d*x)^(17/2)*a^2*b^4*d^16 + 575520*(d*x)^(13/2)*a^3*b ^3*d^18 + 754710*(d*x)^(9/2)*a^4*b^2*d^20 + 450152*(d*x)^(5/2)*a^5*b*d^22 + 102315*sqrt(d*x)*a^6*d^24)/(b^12*d^10*x^10 + 5*a*b^11*d^10*x^8 + 10*a^2* b^10*d^10*x^6 + 10*a^3*b^9*d^10*x^4 + 5*a^4*b^8*d^10*x^2 + a^5*b^7*d^10) - 348075*(sqrt(2)*d^16*log(sqrt(b)*d*x + sqrt(2)*(a*d^2)^(1/4)*sqrt(d*x)*b^ (1/4) + sqrt(a)*d)/((a*d^2)^(3/4)*b^(1/4)) - sqrt(2)*d^16*log(sqrt(b)*d*x - sqrt(2)*(a*d^2)^(1/4)*sqrt(d*x)*b^(1/4) + sqrt(a)*d)/((a*d^2)^(3/4)*b^(1 /4)) + 2*sqrt(2)*d^15*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)*sqr t(a)) + 2*sqrt(2)*d^15*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)*s qrt(a)))*a^2/b^7 - 65536*((d*x)^(5/2)*b*d^12 - 30*sqrt(d*x)*a*d^14)/b^7)/d
Time = 0.15 (sec) , antiderivative size = 383, normalized size of antiderivative = 1.12 \[ \int \frac {(d x)^{27/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {\frac {696150 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} a d^{14} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{b^{8}} + \frac {696150 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} a d^{14} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{b^{8}} + \frac {348075 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} a d^{14} \log \left (d x + \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{b^{8}} - \frac {348075 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} a d^{14} \log \left (d x - \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{b^{8}} - \frac {8 \, {\left (170695 \, \sqrt {d x} a^{2} b^{4} d^{24} x^{8} + 575520 \, \sqrt {d x} a^{3} b^{3} d^{24} x^{6} + 754710 \, \sqrt {d x} a^{4} b^{2} d^{24} x^{4} + 450152 \, \sqrt {d x} a^{5} b d^{24} x^{2} + 102315 \, \sqrt {d x} a^{6} d^{24}\right )}}{{\left (b d^{2} x^{2} + a d^{2}\right )}^{5} b^{7}} + \frac {65536 \, {\left (\sqrt {d x} b^{24} d^{14} x^{2} - 30 \, \sqrt {d x} a b^{23} d^{14}\right )}}{b^{30}}}{163840 \, d} \] Input:
integrate((d*x)^(27/2)/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="giac")
Output:
1/163840*(696150*sqrt(2)*(a*b^3*d^2)^(1/4)*a*d^14*arctan(1/2*sqrt(2)*(sqrt (2)*(a*d^2/b)^(1/4) + 2*sqrt(d*x))/(a*d^2/b)^(1/4))/b^8 + 696150*sqrt(2)*( a*b^3*d^2)^(1/4)*a*d^14*arctan(-1/2*sqrt(2)*(sqrt(2)*(a*d^2/b)^(1/4) - 2*s qrt(d*x))/(a*d^2/b)^(1/4))/b^8 + 348075*sqrt(2)*(a*b^3*d^2)^(1/4)*a*d^14*l og(d*x + sqrt(2)*(a*d^2/b)^(1/4)*sqrt(d*x) + sqrt(a*d^2/b))/b^8 - 348075*s qrt(2)*(a*b^3*d^2)^(1/4)*a*d^14*log(d*x - sqrt(2)*(a*d^2/b)^(1/4)*sqrt(d*x ) + sqrt(a*d^2/b))/b^8 - 8*(170695*sqrt(d*x)*a^2*b^4*d^24*x^8 + 575520*sqr t(d*x)*a^3*b^3*d^24*x^6 + 754710*sqrt(d*x)*a^4*b^2*d^24*x^4 + 450152*sqrt( d*x)*a^5*b*d^24*x^2 + 102315*sqrt(d*x)*a^6*d^24)/((b*d^2*x^2 + a*d^2)^5*b^ 7) + 65536*(sqrt(d*x)*b^24*d^14*x^2 - 30*sqrt(d*x)*a*b^23*d^14)/b^30)/d
Time = 18.67 (sec) , antiderivative size = 248, normalized size of antiderivative = 0.72 \[ \int \frac {(d x)^{27/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {2\,d^{11}\,{\left (d\,x\right )}^{5/2}}{5\,b^6}-\frac {\frac {20463\,a^6\,d^{23}\,\sqrt {d\,x}}{4096}+\frac {75471\,a^4\,b^2\,d^{19}\,{\left (d\,x\right )}^{9/2}}{2048}+\frac {3597\,a^3\,b^3\,d^{17}\,{\left (d\,x\right )}^{13/2}}{128}+\frac {34139\,a^2\,b^4\,d^{15}\,{\left (d\,x\right )}^{17/2}}{4096}+\frac {56269\,a^5\,b\,d^{21}\,{\left (d\,x\right )}^{5/2}}{2560}}{a^5\,b^7\,d^{10}+5\,a^4\,b^8\,d^{10}\,x^2+10\,a^3\,b^9\,d^{10}\,x^4+10\,a^2\,b^{10}\,d^{10}\,x^6+5\,a\,b^{11}\,d^{10}\,x^8+b^{12}\,d^{10}\,x^{10}}-\frac {69615\,{\left (-a\right )}^{5/4}\,d^{27/2}\,\mathrm {atan}\left (\frac {b^{1/4}\,\sqrt {d\,x}}{{\left (-a\right )}^{1/4}\,\sqrt {d}}\right )}{8192\,b^{29/4}}-\frac {12\,a\,d^{13}\,\sqrt {d\,x}}{b^7}+\frac {{\left (-a\right )}^{5/4}\,d^{27/2}\,\mathrm {atan}\left (\frac {b^{1/4}\,\sqrt {d\,x}\,1{}\mathrm {i}}{{\left (-a\right )}^{1/4}\,\sqrt {d}}\right )\,69615{}\mathrm {i}}{8192\,b^{29/4}} \] Input:
int((d*x)^(27/2)/(a^2 + b^2*x^4 + 2*a*b*x^2)^3,x)
Output:
(2*d^11*(d*x)^(5/2))/(5*b^6) - ((20463*a^6*d^23*(d*x)^(1/2))/4096 + (75471 *a^4*b^2*d^19*(d*x)^(9/2))/2048 + (3597*a^3*b^3*d^17*(d*x)^(13/2))/128 + ( 34139*a^2*b^4*d^15*(d*x)^(17/2))/4096 + (56269*a^5*b*d^21*(d*x)^(5/2))/256 0)/(a^5*b^7*d^10 + b^12*d^10*x^10 + 5*a*b^11*d^10*x^8 + 5*a^4*b^8*d^10*x^2 + 10*a^3*b^9*d^10*x^4 + 10*a^2*b^10*d^10*x^6) - (69615*(-a)^(5/4)*d^(27/2 )*atan((b^(1/4)*(d*x)^(1/2))/((-a)^(1/4)*d^(1/2))))/(8192*b^(29/4)) + ((-a )^(5/4)*d^(27/2)*atan((b^(1/4)*(d*x)^(1/2)*1i)/((-a)^(1/4)*d^(1/2)))*69615 i)/(8192*b^(29/4)) - (12*a*d^13*(d*x)^(1/2))/b^7
Time = 0.19 (sec) , antiderivative size = 1015, normalized size of antiderivative = 2.96 \[ \int \frac {(d x)^{27/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx =\text {Too large to display} \] Input:
int((d*x)^(27/2)/(b^2*x^4+2*a*b*x^2+a^2)^3,x)
Output:
(sqrt(d)*d**13*( - 696150*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**6 - 3480750 *b**(3/4)*a**(1/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(x)*sqr t(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*a**5*b*x**2 - 6961500*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**2*x**4 - 6961500*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**3*x**6 - 3480750*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**4*x** 8 - 696150*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*b**5*x**10 + 696150*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**6 + 3480750*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*b*x**2 + 6961500*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**2*x**4 + 6961500*b**(3/4)*a**(1/4)*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)))*a**3*b**3*x**6 + 3480750*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**4*x**8 + 696150*b**(3/4)*a**(1/4)*s...