Integrand size = 28, antiderivative size = 325 \[ \int \frac {(d x)^{23/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {13923 d^{11} \sqrt {d x}}{4096 b^6}-\frac {d (d x)^{21/2}}{10 b \left (a+b x^2\right )^5}-\frac {21 d^3 (d x)^{17/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac {119 d^5 (d x)^{13/2}}{640 b^3 \left (a+b x^2\right )^3}-\frac {1547 d^7 (d x)^{9/2}}{5120 b^4 \left (a+b x^2\right )^2}-\frac {13923 d^9 (d x)^{5/2}}{20480 b^5 \left (a+b x^2\right )}+\frac {13923 \sqrt [4]{a} d^{23/2} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} b^{25/4}}-\frac {13923 \sqrt [4]{a} d^{23/2} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} b^{25/4}}-\frac {13923 \sqrt [4]{a} d^{23/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}}{\sqrt {d} \left (\sqrt {a}+\sqrt {b} x\right )}\right )}{8192 \sqrt {2} b^{25/4}} \] Output:
13923/4096*d^11*(d*x)^(1/2)/b^6-1/10*d*(d*x)^(21/2)/b/(b*x^2+a)^5-21/160*d ^3*(d*x)^(17/2)/b^2/(b*x^2+a)^4-119/640*d^5*(d*x)^(13/2)/b^3/(b*x^2+a)^3-1 547/5120*d^7*(d*x)^(9/2)/b^4/(b*x^2+a)^2-13923/20480*d^9*(d*x)^(5/2)/b^5/( b*x^2+a)+13923/16384*a^(1/4)*d^(23/2)*arctan(1-2^(1/2)*b^(1/4)*(d*x)^(1/2) /a^(1/4)/d^(1/2))*2^(1/2)/b^(25/4)-13923/16384*a^(1/4)*d^(23/2)*arctan(1+2 ^(1/2)*b^(1/4)*(d*x)^(1/2)/a^(1/4)/d^(1/2))*2^(1/2)/b^(25/4)-13923/16384*a ^(1/4)*d^(23/2)*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^(25/4)
Time = 0.96 (sec) , antiderivative size = 216, normalized size of antiderivative = 0.66 \[ \int \frac {(d x)^{23/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {d^{11} \sqrt {d x} \left (4 \sqrt [4]{b} \sqrt {x} \left (69615 a^5+334152 a^4 b x^2+634270 a^3 b^2 x^4+590240 a^2 b^3 x^6+263515 a b^4 x^8+40960 b^5 x^{10}\right )-69615 \sqrt {2} \sqrt [4]{a} \left (a+b x^2\right )^5 \arctan \left (\frac {-\sqrt {a}+\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )-69615 \sqrt {2} \sqrt [4]{a} \left (a+b x^2\right )^5 \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )\right )}{81920 b^{25/4} \sqrt {x} \left (a+b x^2\right )^5} \] Input:
Integrate[(d*x)^(23/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]
Output:
(d^11*Sqrt[d*x]*(4*b^(1/4)*Sqrt[x]*(69615*a^5 + 334152*a^4*b*x^2 + 634270* a^3*b^2*x^4 + 590240*a^2*b^3*x^6 + 263515*a*b^4*x^8 + 40960*b^5*x^10) - 69 615*Sqrt[2]*a^(1/4)*(a + b*x^2)^5*ArcTan[(-Sqrt[a] + Sqrt[b]*x)/(Sqrt[2]*a ^(1/4)*b^(1/4)*Sqrt[x])] - 69615*Sqrt[2]*a^(1/4)*(a + b*x^2)^5*ArcTanh[(Sq rt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)]))/(81920*b^(25/4)*Sq rt[x]*(a + b*x^2)^5)
Time = 1.15 (sec) , antiderivative size = 464, 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, 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.
| \(\displaystyle \int \frac {(d x)^{23/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx\) |
| \(\Big \downarrow \) 1380 |
| \(\displaystyle b^6 \int \frac {(d x)^{23/2}}{b^6 \left (b x^2+a\right )^6}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \frac {(d x)^{23/2}}{\left (a+b x^2\right )^6}dx\) |
| \(\Big \downarrow \) 252 |
| \(\displaystyle \frac {21 d^2 \int \frac {(d x)^{19/2}}{\left (b x^2+a\right )^5}dx}{20 b}-\frac {d (d x)^{21/2}}{10 b \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 252 |
| \(\displaystyle \frac {21 d^2 \left (\frac {17 d^2 \int \frac {(d x)^{15/2}}{\left (b x^2+a\right )^4}dx}{16 b}-\frac {d (d x)^{17/2}}{8 b \left (a+b x^2\right )^4}\right )}{20 b}-\frac {d (d x)^{21/2}}{10 b \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 252 |
| \(\displaystyle \frac {21 d^2 \left (\frac {17 d^2 \left (\frac {13 d^2 \int \frac {(d x)^{11/2}}{\left (b x^2+a\right )^3}dx}{12 b}-\frac {d (d x)^{13/2}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{17/2}}{8 b \left (a+b x^2\right )^4}\right )}{20 b}-\frac {d (d x)^{21/2}}{10 b \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 252 |
| \(\displaystyle \frac {21 d^2 \left (\frac {17 d^2 \left (\frac {13 d^2 \left (\frac {9 d^2 \int \frac {(d x)^{7/2}}{\left (b x^2+a\right )^2}dx}{8 b}-\frac {d (d x)^{9/2}}{4 b \left (a+b x^2\right )^2}\right )}{12 b}-\frac {d (d x)^{13/2}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{17/2}}{8 b \left (a+b x^2\right )^4}\right )}{20 b}-\frac {d (d x)^{21/2}}{10 b \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 252 |
| \(\displaystyle \frac {21 d^2 \left (\frac {17 d^2 \left (\frac {13 d^2 \left (\frac {9 d^2 \left (\frac {5 d^2 \int \frac {(d x)^{3/2}}{b x^2+a}dx}{4 b}-\frac {d (d x)^{5/2}}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{9/2}}{4 b \left (a+b x^2\right )^2}\right )}{12 b}-\frac {d (d x)^{13/2}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{17/2}}{8 b \left (a+b x^2\right )^4}\right )}{20 b}-\frac {d (d x)^{21/2}}{10 b \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle \frac {21 d^2 \left (\frac {17 d^2 \left (\frac {13 d^2 \left (\frac {9 d^2 \left (\frac {5 d^2 \left (\frac {2 d \sqrt {d x}}{b}-\frac {a d^2 \int \frac {1}{\sqrt {d x} \left (b x^2+a\right )}dx}{b}\right )}{4 b}-\frac {d (d x)^{5/2}}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{9/2}}{4 b \left (a+b x^2\right )^2}\right )}{12 b}-\frac {d (d x)^{13/2}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{17/2}}{8 b \left (a+b x^2\right )^4}\right )}{20 b}-\frac {d (d x)^{21/2}}{10 b \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {21 d^2 \left (\frac {17 d^2 \left (\frac {13 d^2 \left (\frac {9 d^2 \left (\frac {5 d^2 \left (\frac {2 d \sqrt {d x}}{b}-\frac {2 a d \int \frac {1}{b x^2+a}d\sqrt {d x}}{b}\right )}{4 b}-\frac {d (d x)^{5/2}}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{9/2}}{4 b \left (a+b x^2\right )^2}\right )}{12 b}-\frac {d (d x)^{13/2}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{17/2}}{8 b \left (a+b x^2\right )^4}\right )}{20 b}-\frac {d (d x)^{21/2}}{10 b \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 755 |
| \(\displaystyle \frac {21 d^2 \left (\frac {17 d^2 \left (\frac {13 d^2 \left (\frac {9 d^2 \left (\frac {5 d^2 \left (\frac {2 d \sqrt {d x}}{b}-\frac {2 a d \left (\frac {\int \frac {d^2 \left (\sqrt {a} d-\sqrt {b} d x\right )}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {a} d}+\frac {\int \frac {d^2 \left (\sqrt {b} x d+\sqrt {a} d\right )}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {a} d}\right )}{b}\right )}{4 b}-\frac {d (d x)^{5/2}}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{9/2}}{4 b \left (a+b x^2\right )^2}\right )}{12 b}-\frac {d (d x)^{13/2}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{17/2}}{8 b \left (a+b x^2\right )^4}\right )}{20 b}-\frac {d (d x)^{21/2}}{10 b \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {21 d^2 \left (\frac {17 d^2 \left (\frac {13 d^2 \left (\frac {9 d^2 \left (\frac {5 d^2 \left (\frac {2 d \sqrt {d x}}{b}-\frac {2 a d \left (\frac {d \int \frac {\sqrt {a} d-\sqrt {b} d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {a}}+\frac {d \int \frac {\sqrt {b} x d+\sqrt {a} d}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {a}}\right )}{b}\right )}{4 b}-\frac {d (d x)^{5/2}}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{9/2}}{4 b \left (a+b x^2\right )^2}\right )}{12 b}-\frac {d (d x)^{13/2}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{17/2}}{8 b \left (a+b x^2\right )^4}\right )}{20 b}-\frac {d (d x)^{21/2}}{10 b \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {21 d^2 \left (\frac {17 d^2 \left (\frac {13 d^2 \left (\frac {9 d^2 \left (\frac {5 d^2 \left (\frac {2 d \sqrt {d x}}{b}-\frac {2 a d \left (\frac {d \int \frac {\sqrt {a} d-\sqrt {b} d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {a}}+\frac {d \left (\frac {\int \frac {1}{x d+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}}d\sqrt {d x}}{2 \sqrt {b}}+\frac {\int \frac {1}{x d+\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}}d\sqrt {d x}}{2 \sqrt {b}}\right )}{2 \sqrt {a}}\right )}{b}\right )}{4 b}-\frac {d (d x)^{5/2}}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{9/2}}{4 b \left (a+b x^2\right )^2}\right )}{12 b}-\frac {d (d x)^{13/2}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{17/2}}{8 b \left (a+b x^2\right )^4}\right )}{20 b}-\frac {d (d x)^{21/2}}{10 b \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {21 d^2 \left (\frac {17 d^2 \left (\frac {13 d^2 \left (\frac {9 d^2 \left (\frac {5 d^2 \left (\frac {2 d \sqrt {d x}}{b}-\frac {2 a d \left (\frac {d \int \frac {\sqrt {a} d-\sqrt {b} d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {a}}+\frac {d \left (\frac {\int \frac {1}{-d x-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\int \frac {1}{-d x-1}d\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}\right )}{b}\right )}{4 b}-\frac {d (d x)^{5/2}}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{9/2}}{4 b \left (a+b x^2\right )^2}\right )}{12 b}-\frac {d (d x)^{13/2}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{17/2}}{8 b \left (a+b x^2\right )^4}\right )}{20 b}-\frac {d (d x)^{21/2}}{10 b \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {21 d^2 \left (\frac {17 d^2 \left (\frac {13 d^2 \left (\frac {9 d^2 \left (\frac {5 d^2 \left (\frac {2 d \sqrt {d x}}{b}-\frac {2 a d \left (\frac {d \int \frac {\sqrt {a} d-\sqrt {b} d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {a}}+\frac {d \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}\right )}{b}\right )}{4 b}-\frac {d (d x)^{5/2}}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{9/2}}{4 b \left (a+b x^2\right )^2}\right )}{12 b}-\frac {d (d x)^{13/2}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{17/2}}{8 b \left (a+b x^2\right )^4}\right )}{20 b}-\frac {d (d x)^{21/2}}{10 b \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {21 d^2 \left (\frac {17 d^2 \left (\frac {13 d^2 \left (\frac {9 d^2 \left (\frac {5 d^2 \left (\frac {2 d \sqrt {d x}}{b}-\frac {2 a d \left (\frac {d \left (-\frac {\int -\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d}-2 \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{b} \left (x d+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt [4]{a} \sqrt {d}+\sqrt {2} \sqrt [4]{b} \sqrt {d x}\right )}{\sqrt [4]{b} \left (x d+\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}+\frac {d \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}\right )}{b}\right )}{4 b}-\frac {d (d x)^{5/2}}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{9/2}}{4 b \left (a+b x^2\right )^2}\right )}{12 b}-\frac {d (d x)^{13/2}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{17/2}}{8 b \left (a+b x^2\right )^4}\right )}{20 b}-\frac {d (d x)^{21/2}}{10 b \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {21 d^2 \left (\frac {17 d^2 \left (\frac {13 d^2 \left (\frac {9 d^2 \left (\frac {5 d^2 \left (\frac {2 d \sqrt {d x}}{b}-\frac {2 a d \left (\frac {d \left (\frac {\int \frac {\sqrt {2} \sqrt [4]{a} \sqrt {d}-2 \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{b} \left (x d+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}+\frac {\int \frac {\sqrt {2} \left (\sqrt [4]{a} \sqrt {d}+\sqrt {2} \sqrt [4]{b} \sqrt {d x}\right )}{\sqrt [4]{b} \left (x d+\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}+\frac {d \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}\right )}{b}\right )}{4 b}-\frac {d (d x)^{5/2}}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{9/2}}{4 b \left (a+b x^2\right )^2}\right )}{12 b}-\frac {d (d x)^{13/2}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{17/2}}{8 b \left (a+b x^2\right )^4}\right )}{20 b}-\frac {d (d x)^{21/2}}{10 b \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {21 d^2 \left (\frac {17 d^2 \left (\frac {13 d^2 \left (\frac {9 d^2 \left (\frac {5 d^2 \left (\frac {2 d \sqrt {d x}}{b}-\frac {2 a d \left (\frac {d \left (\frac {\int \frac {\sqrt {2} \sqrt [4]{a} \sqrt {d}-2 \sqrt [4]{b} \sqrt {d x}}{x d+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt {b} \sqrt {d}}+\frac {\int \frac {\sqrt [4]{a} \sqrt {d}+\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{x d+\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}}d\sqrt {d x}}{2 \sqrt [4]{a} \sqrt {b} \sqrt {d}}\right )}{2 \sqrt {a}}+\frac {d \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}\right )}{b}\right )}{4 b}-\frac {d (d x)^{5/2}}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{9/2}}{4 b \left (a+b x^2\right )^2}\right )}{12 b}-\frac {d (d x)^{13/2}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{17/2}}{8 b \left (a+b x^2\right )^4}\right )}{20 b}-\frac {d (d x)^{21/2}}{10 b \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {21 d^2 \left (\frac {17 d^2 \left (\frac {13 d^2 \left (\frac {9 d^2 \left (\frac {5 d^2 \left (\frac {2 d \sqrt {d x}}{b}-\frac {2 a d \left (\frac {d \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}+\frac {d \left (\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d} \sqrt {d x}+\sqrt {a} d+\sqrt {b} d x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d} \sqrt {d x}+\sqrt {a} d+\sqrt {b} d x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}\right )}{b}\right )}{4 b}-\frac {d (d x)^{5/2}}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{9/2}}{4 b \left (a+b x^2\right )^2}\right )}{12 b}-\frac {d (d x)^{13/2}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{17/2}}{8 b \left (a+b x^2\right )^4}\right )}{20 b}-\frac {d (d x)^{21/2}}{10 b \left (a+b x^2\right )^5}\) |
Input:
Int[(d*x)^(23/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]
Output:
-1/10*(d*(d*x)^(21/2))/(b*(a + b*x^2)^5) + (21*d^2*(-1/8*(d*(d*x)^(17/2))/ (b*(a + b*x^2)^4) + (17*d^2*(-1/6*(d*(d*x)^(13/2))/(b*(a + b*x^2)^3) + (13 *d^2*(-1/4*(d*(d*x)^(9/2))/(b*(a + b*x^2)^2) + (9*d^2*(-1/2*(d*(d*x)^(5/2) )/(b*(a + b*x^2)) + (5*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)*Sqrt[d])]/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqr t[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[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))/(4*b)) )/(8*b)))/(12*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[((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 = 14.77 (sec) , antiderivative size = 245, normalized size of antiderivative = 0.75
| method | result | size |
| pseudoelliptic | \(-\frac {d^{11} \left (\left (-327680 x^{10} b^{5}-2108120 a \,x^{8} b^{4}-4721920 a^{2} x^{6} b^{3}-5074160 a^{3} x^{4} b^{2}-2673216 x^{2} a^{4} b -556920 a^{5}\right ) \sqrt {d x}+69615 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (b \,x^{2}+a \right )^{5} \left (\ln \left (\frac {d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}-\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}+\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )\right )\right )}{163840 b^{6} \left (b \,x^{2}+a \right )^{5}}\) | \(245\) |
| derivativedivides | \(2 d^{11} \left (\frac {\sqrt {d x}}{b^{6}}-\frac {a \,d^{2} \left (\frac {-\frac {5731 a^{4} d^{8} \sqrt {d x}}{8192}-\frac {16169 a^{3} d^{6} b \left (d x \right )^{\frac {5}{2}}}{5120}-\frac {22467 a^{2} d^{4} b^{2} \left (d x \right )^{\frac {9}{2}}}{4096}-\frac {1129 a \,d^{2} b^{3} \left (d x \right )^{\frac {13}{2}}}{256}-\frac {11743 b^{4} \left (d x \right )^{\frac {17}{2}}}{8192}}{\left (b \,d^{2} x^{2}+a \,d^{2}\right )^{5}}+\frac {13923 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{65536 a \,d^{2}}\right )}{b^{6}}\right )\) | \(252\) |
| default | \(2 d^{11} \left (\frac {\sqrt {d x}}{b^{6}}-\frac {a \,d^{2} \left (\frac {-\frac {5731 a^{4} d^{8} \sqrt {d x}}{8192}-\frac {16169 a^{3} d^{6} b \left (d x \right )^{\frac {5}{2}}}{5120}-\frac {22467 a^{2} d^{4} b^{2} \left (d x \right )^{\frac {9}{2}}}{4096}-\frac {1129 a \,d^{2} b^{3} \left (d x \right )^{\frac {13}{2}}}{256}-\frac {11743 b^{4} \left (d x \right )^{\frac {17}{2}}}{8192}}{\left (b \,d^{2} x^{2}+a \,d^{2}\right )^{5}}+\frac {13923 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{65536 a \,d^{2}}\right )}{b^{6}}\right )\) | \(252\) |
| risch | \(\frac {2 x \,d^{12}}{b^{6} \sqrt {d x}}-\frac {2 a \,d^{13} \left (\frac {-\frac {5731 a^{4} d^{8} \sqrt {d x}}{8192}-\frac {16169 a^{3} d^{6} b \left (d x \right )^{\frac {5}{2}}}{5120}-\frac {22467 a^{2} d^{4} b^{2} \left (d x \right )^{\frac {9}{2}}}{4096}-\frac {1129 a \,d^{2} b^{3} \left (d x \right )^{\frac {13}{2}}}{256}-\frac {11743 b^{4} \left (d x \right )^{\frac {17}{2}}}{8192}}{\left (b \,d^{2} x^{2}+a \,d^{2}\right )^{5}}+\frac {13923 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{65536 a \,d^{2}}\right )}{b^{6}}\) | \(252\) |
Input:
int((d*x)^(23/2)/(b^2*x^4+2*a*b*x^2+a^2)^3,x,method=_RETURNVERBOSE)
Output:
-1/163840*d^11*((-327680*b^5*x^10-2108120*a*b^4*x^8-4721920*a^2*b^3*x^6-50 74160*a^3*b^2*x^4-2673216*a^4*b*x^2-556920*a^5)*(d*x)^(1/2)+69615*(a*d^2/b )^(1/4)*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^6/(b*x^2+a)^5
Result contains complex when optimal does not.
Time = 0.21 (sec) , antiderivative size = 537, normalized size of antiderivative = 1.65 \[ \int \frac {(d x)^{23/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=-\frac {69615 \, \left (-\frac {a d^{46}}{b^{25}}\right )^{\frac {1}{4}} {\left (b^{11} x^{10} + 5 \, a b^{10} x^{8} + 10 \, a^{2} b^{9} x^{6} + 10 \, a^{3} b^{8} x^{4} + 5 \, a^{4} b^{7} x^{2} + a^{5} b^{6}\right )} \log \left (13923 \, \sqrt {d x} d^{11} + 13923 \, \left (-\frac {a d^{46}}{b^{25}}\right )^{\frac {1}{4}} b^{6}\right ) + 69615 \, \left (-\frac {a d^{46}}{b^{25}}\right )^{\frac {1}{4}} {\left (i \, b^{11} x^{10} + 5 i \, a b^{10} x^{8} + 10 i \, a^{2} b^{9} x^{6} + 10 i \, a^{3} b^{8} x^{4} + 5 i \, a^{4} b^{7} x^{2} + i \, a^{5} b^{6}\right )} \log \left (13923 \, \sqrt {d x} d^{11} + 13923 i \, \left (-\frac {a d^{46}}{b^{25}}\right )^{\frac {1}{4}} b^{6}\right ) + 69615 \, \left (-\frac {a d^{46}}{b^{25}}\right )^{\frac {1}{4}} {\left (-i \, b^{11} x^{10} - 5 i \, a b^{10} x^{8} - 10 i \, a^{2} b^{9} x^{6} - 10 i \, a^{3} b^{8} x^{4} - 5 i \, a^{4} b^{7} x^{2} - i \, a^{5} b^{6}\right )} \log \left (13923 \, \sqrt {d x} d^{11} - 13923 i \, \left (-\frac {a d^{46}}{b^{25}}\right )^{\frac {1}{4}} b^{6}\right ) - 69615 \, \left (-\frac {a d^{46}}{b^{25}}\right )^{\frac {1}{4}} {\left (b^{11} x^{10} + 5 \, a b^{10} x^{8} + 10 \, a^{2} b^{9} x^{6} + 10 \, a^{3} b^{8} x^{4} + 5 \, a^{4} b^{7} x^{2} + a^{5} b^{6}\right )} \log \left (13923 \, \sqrt {d x} d^{11} - 13923 \, \left (-\frac {a d^{46}}{b^{25}}\right )^{\frac {1}{4}} b^{6}\right ) - 4 \, {\left (40960 \, b^{5} d^{11} x^{10} + 263515 \, a b^{4} d^{11} x^{8} + 590240 \, a^{2} b^{3} d^{11} x^{6} + 634270 \, a^{3} b^{2} d^{11} x^{4} + 334152 \, a^{4} b d^{11} x^{2} + 69615 \, a^{5} d^{11}\right )} \sqrt {d x}}{81920 \, {\left (b^{11} x^{10} + 5 \, a b^{10} x^{8} + 10 \, a^{2} b^{9} x^{6} + 10 \, a^{3} b^{8} x^{4} + 5 \, a^{4} b^{7} x^{2} + a^{5} b^{6}\right )}} \] Input:
integrate((d*x)^(23/2)/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="fricas")
Output:
-1/81920*(69615*(-a*d^46/b^25)^(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(13923*sqrt(d*x)*d^11 + 13923*(-a*d^46/b^25)^(1/4)*b^6) + 69615*(-a*d^46/b^25)^(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(13923*sqrt(d*x)*d^11 + 13923*I*(-a*d^46/b^25)^(1/4)*b^6 ) + 69615*(-a*d^46/b^25)^(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(13923*sqrt(d* x)*d^11 - 13923*I*(-a*d^46/b^25)^(1/4)*b^6) - 69615*(-a*d^46/b^25)^(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(13923*sqrt(d*x)*d^11 - 13923*(-a*d^46/b^25)^(1/4)*b^6) - 4 *(40960*b^5*d^11*x^10 + 263515*a*b^4*d^11*x^8 + 590240*a^2*b^3*d^11*x^6 + 634270*a^3*b^2*d^11*x^4 + 334152*a^4*b*d^11*x^2 + 69615*a^5*d^11)*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. \[ \int \frac {(d x)^{23/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\text {Timed out} \] Input:
integrate((d*x)**(23/2)/(b**2*x**4+2*a*b*x**2+a**2)**3,x)
Output:
Timed out
Time = 0.13 (sec) , antiderivative size = 403, normalized size of antiderivative = 1.24 \[ \int \frac {(d x)^{23/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {\frac {327680 \, \sqrt {d x} d^{12}}{b^{6}} + \frac {8 \, {\left (58715 \, \left (d x\right )^{\frac {17}{2}} a b^{4} d^{14} + 180640 \, \left (d x\right )^{\frac {13}{2}} a^{2} b^{3} d^{16} + 224670 \, \left (d x\right )^{\frac {9}{2}} a^{3} b^{2} d^{18} + 129352 \, \left (d x\right )^{\frac {5}{2}} a^{4} b d^{20} + 28655 \, \sqrt {d x} a^{5} d^{22}\right )}}{b^{11} d^{10} x^{10} + 5 \, a b^{10} d^{10} x^{8} + 10 \, a^{2} b^{9} d^{10} x^{6} + 10 \, a^{3} b^{8} d^{10} x^{4} + 5 \, a^{4} b^{7} d^{10} x^{2} + a^{5} b^{6} d^{10}} - \frac {69615 \, {\left (\frac {\sqrt {2} d^{14} \log \left (\sqrt {b} d x + \sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} \sqrt {d x} b^{\frac {1}{4}} + \sqrt {a} d\right )}{\left (a d^{2}\right )^{\frac {3}{4}} b^{\frac {1}{4}}} - \frac {\sqrt {2} d^{14} \log \left (\sqrt {b} d x - \sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} \sqrt {d x} b^{\frac {1}{4}} + \sqrt {a} d\right )}{\left (a d^{2}\right )^{\frac {3}{4}} b^{\frac {1}{4}}} + \frac {2 \, \sqrt {2} d^{13} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {d x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} d}}\right )}{\sqrt {\sqrt {a} \sqrt {b} d} \sqrt {a}} + \frac {2 \, \sqrt {2} d^{13} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {d x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} d}}\right )}{\sqrt {\sqrt {a} \sqrt {b} d} \sqrt {a}}\right )} a}{b^{6}}}{163840 \, d} \] Input:
integrate((d*x)^(23/2)/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="maxima")
Output:
1/163840*(327680*sqrt(d*x)*d^12/b^6 + 8*(58715*(d*x)^(17/2)*a*b^4*d^14 + 1 80640*(d*x)^(13/2)*a^2*b^3*d^16 + 224670*(d*x)^(9/2)*a^3*b^2*d^18 + 129352 *(d*x)^(5/2)*a^4*b*d^20 + 28655*sqrt(d*x)*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) - 69615*(sqrt(2)*d^14*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^14*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^13*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(s qrt(a)*sqrt(b)*d)*sqrt(a)) + 2*sqrt(2)*d^13*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)))*a/b^6)/d
Time = 0.14 (sec) , antiderivative size = 355, normalized size of antiderivative = 1.09 \[ \int \frac {(d x)^{23/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=-\frac {\frac {139230 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} d^{12} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{b^{7}} + \frac {139230 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} d^{12} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{b^{7}} + \frac {69615 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} d^{12} \log \left (d x + \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{b^{7}} - \frac {69615 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} d^{12} \log \left (d x - \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{b^{7}} - \frac {327680 \, \sqrt {d x} d^{12}}{b^{6}} - \frac {8 \, {\left (58715 \, \sqrt {d x} a b^{4} d^{22} x^{8} + 180640 \, \sqrt {d x} a^{2} b^{3} d^{22} x^{6} + 224670 \, \sqrt {d x} a^{3} b^{2} d^{22} x^{4} + 129352 \, \sqrt {d x} a^{4} b d^{22} x^{2} + 28655 \, \sqrt {d x} a^{5} d^{22}\right )}}{{\left (b d^{2} x^{2} + a d^{2}\right )}^{5} b^{6}}}{163840 \, d} \] Input:
integrate((d*x)^(23/2)/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="giac")
Output:
-1/163840*(139230*sqrt(2)*(a*b^3*d^2)^(1/4)*d^12*arctan(1/2*sqrt(2)*(sqrt( 2)*(a*d^2/b)^(1/4) + 2*sqrt(d*x))/(a*d^2/b)^(1/4))/b^7 + 139230*sqrt(2)*(a *b^3*d^2)^(1/4)*d^12*arctan(-1/2*sqrt(2)*(sqrt(2)*(a*d^2/b)^(1/4) - 2*sqrt (d*x))/(a*d^2/b)^(1/4))/b^7 + 69615*sqrt(2)*(a*b^3*d^2)^(1/4)*d^12*log(d*x + sqrt(2)*(a*d^2/b)^(1/4)*sqrt(d*x) + sqrt(a*d^2/b))/b^7 - 69615*sqrt(2)* (a*b^3*d^2)^(1/4)*d^12*log(d*x - sqrt(2)*(a*d^2/b)^(1/4)*sqrt(d*x) + sqrt( a*d^2/b))/b^7 - 327680*sqrt(d*x)*d^12/b^6 - 8*(58715*sqrt(d*x)*a*b^4*d^22* x^8 + 180640*sqrt(d*x)*a^2*b^3*d^22*x^6 + 224670*sqrt(d*x)*a^3*b^2*d^22*x^ 4 + 129352*sqrt(d*x)*a^4*b*d^22*x^2 + 28655*sqrt(d*x)*a^5*d^22)/((b*d^2*x^ 2 + a*d^2)^5*b^6))/d
Time = 18.49 (sec) , antiderivative size = 231, normalized size of antiderivative = 0.71 \[ \int \frac {(d x)^{23/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {\frac {5731\,a^5\,d^{21}\,\sqrt {d\,x}}{4096}+\frac {22467\,a^3\,b^2\,d^{17}\,{\left (d\,x\right )}^{9/2}}{2048}+\frac {1129\,a^2\,b^3\,d^{15}\,{\left (d\,x\right )}^{13/2}}{128}+\frac {16169\,a^4\,b\,d^{19}\,{\left (d\,x\right )}^{5/2}}{2560}+\frac {11743\,a\,b^4\,d^{13}\,{\left (d\,x\right )}^{17/2}}{4096}}{a^5\,b^6\,d^{10}+5\,a^4\,b^7\,d^{10}\,x^2+10\,a^3\,b^8\,d^{10}\,x^4+10\,a^2\,b^9\,d^{10}\,x^6+5\,a\,b^{10}\,d^{10}\,x^8+b^{11}\,d^{10}\,x^{10}}+\frac {2\,d^{11}\,\sqrt {d\,x}}{b^6}-\frac {13923\,{\left (-a\right )}^{1/4}\,d^{23/2}\,\mathrm {atan}\left (\frac {b^{1/4}\,\sqrt {d\,x}}{{\left (-a\right )}^{1/4}\,\sqrt {d}}\right )}{8192\,b^{25/4}}+\frac {{\left (-a\right )}^{1/4}\,d^{23/2}\,\mathrm {atan}\left (\frac {b^{1/4}\,\sqrt {d\,x}\,1{}\mathrm {i}}{{\left (-a\right )}^{1/4}\,\sqrt {d}}\right )\,13923{}\mathrm {i}}{8192\,b^{25/4}} \] Input:
int((d*x)^(23/2)/(a^2 + b^2*x^4 + 2*a*b*x^2)^3,x)
Output:
((5731*a^5*d^21*(d*x)^(1/2))/4096 + (22467*a^3*b^2*d^17*(d*x)^(9/2))/2048 + (1129*a^2*b^3*d^15*(d*x)^(13/2))/128 + (16169*a^4*b*d^19*(d*x)^(5/2))/25 60 + (11743*a*b^4*d^13*(d*x)^(17/2))/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)^(1/2))/b^6 - (13923*(-a)^(1/4)*d^(23/2)*atan(( b^(1/4)*(d*x)^(1/2))/((-a)^(1/4)*d^(1/2))))/(8192*b^(25/4)) + ((-a)^(1/4)* d^(23/2)*atan((b^(1/4)*(d*x)^(1/2)*1i)/((-a)^(1/4)*d^(1/2)))*13923i)/(8192 *b^(25/4))
Time = 0.19 (sec) , antiderivative size = 1002, normalized size of antiderivative = 3.08 \[ \int \frac {(d x)^{23/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx =\text {Too large to display} \] Input:
int((d*x)^(23/2)/(b^2*x^4+2*a*b*x^2+a^2)^3,x)
Output:
(sqrt(d)*d**11*(139230*b**(3/4)*a**(1/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*s qrt(2) - 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*a**5 + 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**4*b*x**2 + 1392300*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)))*a**3*b**2*x**4 + 1392300*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**6 + 696150*b**(3/4)*a**(1/4)*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*b**4*x**8 + 1392 30*b**(3/4)*a**(1/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(x)*s qrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*b**5*x**10 - 139230*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 - 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**4*b*x* *2 - 1392300*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**4 - 1392300*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**6 - 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*b**4*x**8 - 139230*b**(3/4)*a**(1/4)*sqrt(2)*atan((b*...