Integrand size = 28, antiderivative size = 327 \[ \int \frac {1}{(d x)^{3/2} \left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=-\frac {13923}{4096 a^6 d \sqrt {d x}}+\frac {1}{10 a d \sqrt {d x} \left (a+b x^2\right )^5}+\frac {21}{160 a^2 d \sqrt {d x} \left (a+b x^2\right )^4}+\frac {119}{640 a^3 d \sqrt {d x} \left (a+b x^2\right )^3}+\frac {1547}{5120 a^4 d \sqrt {d x} \left (a+b x^2\right )^2}+\frac {13923}{20480 a^5 d \sqrt {d x} \left (a+b x^2\right )}+\frac {13923 \sqrt [4]{b} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} a^{25/4} d^{3/2}}-\frac {13923 \sqrt [4]{b} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} a^{25/4} d^{3/2}}+\frac {13923 \sqrt [4]{b} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}}{\sqrt {d} \left (\sqrt {a}+\sqrt {b} x\right )}\right )}{8192 \sqrt {2} a^{25/4} d^{3/2}} \] Output:
-13923/4096/a^6/d/(d*x)^(1/2)+1/10/a/d/(d*x)^(1/2)/(b*x^2+a)^5+21/160/a^2/ d/(d*x)^(1/2)/(b*x^2+a)^4+119/640/a^3/d/(d*x)^(1/2)/(b*x^2+a)^3+1547/5120/ a^4/d/(d*x)^(1/2)/(b*x^2+a)^2+13923/20480/a^5/d/(d*x)^(1/2)/(b*x^2+a)+1392 3/16384*b^(1/4)*arctan(1-2^(1/2)*b^(1/4)*(d*x)^(1/2)/a^(1/4)/d^(1/2))*2^(1 /2)/a^(25/4)/d^(3/2)-13923/16384*b^(1/4)*arctan(1+2^(1/2)*b^(1/4)*(d*x)^(1 /2)/a^(1/4)/d^(1/2))*2^(1/2)/a^(25/4)/d^(3/2)+13923/16384*b^(1/4)*arctanh( 2^(1/2)*a^(1/4)*b^(1/4)*(d*x)^(1/2)/d^(1/2)/(a^(1/2)+b^(1/2)*x))*2^(1/2)/a ^(25/4)/d^(3/2)
Time = 0.87 (sec) , antiderivative size = 195, normalized size of antiderivative = 0.60 \[ \int \frac {1}{(d x)^{3/2} \left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {x \left (-\frac {4 \sqrt [4]{a} \left (40960 a^5+263515 a^4 b x^2+590240 a^3 b^2 x^4+634270 a^2 b^3 x^6+334152 a b^4 x^8+69615 b^5 x^{10}\right )}{\left (a+b x^2\right )^5}+69615 \sqrt {2} \sqrt [4]{b} \sqrt {x} \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )+69615 \sqrt {2} \sqrt [4]{b} \sqrt {x} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )\right )}{81920 a^{25/4} (d x)^{3/2}} \] Input:
Integrate[1/((d*x)^(3/2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^3),x]
Output:
(x*((-4*a^(1/4)*(40960*a^5 + 263515*a^4*b*x^2 + 590240*a^3*b^2*x^4 + 63427 0*a^2*b^3*x^6 + 334152*a*b^4*x^8 + 69615*b^5*x^10))/(a + b*x^2)^5 + 69615* Sqrt[2]*b^(1/4)*Sqrt[x]*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1 /4)*Sqrt[x])] + 69615*Sqrt[2]*b^(1/4)*Sqrt[x]*ArcTanh[(Sqrt[2]*a^(1/4)*b^( 1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)]))/(81920*a^(25/4)*(d*x)^(3/2))
Time = 1.16 (sec) , antiderivative size = 461, normalized size of antiderivative = 1.41, number of steps used = 19, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.643, Rules used = {1380, 27, 253, 253, 253, 253, 253, 264, 266, 27, 826, 1476, 1082, 217, 1479, 25, 27, 1103}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {1}{(d x)^{3/2} \left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx\) |
| \(\Big \downarrow \) 1380 |
| \(\displaystyle b^6 \int \frac {1}{b^6 (d x)^{3/2} \left (b x^2+a\right )^6}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \frac {1}{(d x)^{3/2} \left (a+b x^2\right )^6}dx\) |
| \(\Big \downarrow \) 253 |
| \(\displaystyle \frac {21 \int \frac {1}{(d x)^{3/2} \left (b x^2+a\right )^5}dx}{20 a}+\frac {1}{10 a d \sqrt {d x} \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 253 |
| \(\displaystyle \frac {21 \left (\frac {17 \int \frac {1}{(d x)^{3/2} \left (b x^2+a\right )^4}dx}{16 a}+\frac {1}{8 a d \sqrt {d x} \left (a+b x^2\right )^4}\right )}{20 a}+\frac {1}{10 a d \sqrt {d x} \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 253 |
| \(\displaystyle \frac {21 \left (\frac {17 \left (\frac {13 \int \frac {1}{(d x)^{3/2} \left (b x^2+a\right )^3}dx}{12 a}+\frac {1}{6 a d \sqrt {d x} \left (a+b x^2\right )^3}\right )}{16 a}+\frac {1}{8 a d \sqrt {d x} \left (a+b x^2\right )^4}\right )}{20 a}+\frac {1}{10 a d \sqrt {d x} \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 253 |
| \(\displaystyle \frac {21 \left (\frac {17 \left (\frac {13 \left (\frac {9 \int \frac {1}{(d x)^{3/2} \left (b x^2+a\right )^2}dx}{8 a}+\frac {1}{4 a d \sqrt {d x} \left (a+b x^2\right )^2}\right )}{12 a}+\frac {1}{6 a d \sqrt {d x} \left (a+b x^2\right )^3}\right )}{16 a}+\frac {1}{8 a d \sqrt {d x} \left (a+b x^2\right )^4}\right )}{20 a}+\frac {1}{10 a d \sqrt {d x} \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 253 |
| \(\displaystyle \frac {21 \left (\frac {17 \left (\frac {13 \left (\frac {9 \left (\frac {5 \int \frac {1}{(d x)^{3/2} \left (b x^2+a\right )}dx}{4 a}+\frac {1}{2 a d \sqrt {d x} \left (a+b x^2\right )}\right )}{8 a}+\frac {1}{4 a d \sqrt {d x} \left (a+b x^2\right )^2}\right )}{12 a}+\frac {1}{6 a d \sqrt {d x} \left (a+b x^2\right )^3}\right )}{16 a}+\frac {1}{8 a d \sqrt {d x} \left (a+b x^2\right )^4}\right )}{20 a}+\frac {1}{10 a d \sqrt {d x} \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 264 |
| \(\displaystyle \frac {21 \left (\frac {17 \left (\frac {13 \left (\frac {9 \left (\frac {5 \left (-\frac {b \int \frac {\sqrt {d x}}{b x^2+a}dx}{a d^2}-\frac {2}{a d \sqrt {d x}}\right )}{4 a}+\frac {1}{2 a d \sqrt {d x} \left (a+b x^2\right )}\right )}{8 a}+\frac {1}{4 a d \sqrt {d x} \left (a+b x^2\right )^2}\right )}{12 a}+\frac {1}{6 a d \sqrt {d x} \left (a+b x^2\right )^3}\right )}{16 a}+\frac {1}{8 a d \sqrt {d x} \left (a+b x^2\right )^4}\right )}{20 a}+\frac {1}{10 a d \sqrt {d x} \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {21 \left (\frac {17 \left (\frac {13 \left (\frac {9 \left (\frac {5 \left (-\frac {2 b \int \frac {d^3 x}{b x^2 d^2+a d^2}d\sqrt {d x}}{a d^3}-\frac {2}{a d \sqrt {d x}}\right )}{4 a}+\frac {1}{2 a d \sqrt {d x} \left (a+b x^2\right )}\right )}{8 a}+\frac {1}{4 a d \sqrt {d x} \left (a+b x^2\right )^2}\right )}{12 a}+\frac {1}{6 a d \sqrt {d x} \left (a+b x^2\right )^3}\right )}{16 a}+\frac {1}{8 a d \sqrt {d x} \left (a+b x^2\right )^4}\right )}{20 a}+\frac {1}{10 a d \sqrt {d x} \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {21 \left (\frac {17 \left (\frac {13 \left (\frac {9 \left (\frac {5 \left (-\frac {2 b \int \frac {d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{a d}-\frac {2}{a d \sqrt {d x}}\right )}{4 a}+\frac {1}{2 a d \sqrt {d x} \left (a+b x^2\right )}\right )}{8 a}+\frac {1}{4 a d \sqrt {d x} \left (a+b x^2\right )^2}\right )}{12 a}+\frac {1}{6 a d \sqrt {d x} \left (a+b x^2\right )^3}\right )}{16 a}+\frac {1}{8 a d \sqrt {d x} \left (a+b x^2\right )^4}\right )}{20 a}+\frac {1}{10 a d \sqrt {d x} \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 826 |
| \(\displaystyle \frac {21 \left (\frac {17 \left (\frac {13 \left (\frac {9 \left (\frac {5 \left (-\frac {2 b \left (\frac {\int \frac {\sqrt {b} x d+\sqrt {a} d}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {b}}-\frac {\int \frac {\sqrt {a} d-\sqrt {b} d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {b}}\right )}{a d}-\frac {2}{a d \sqrt {d x}}\right )}{4 a}+\frac {1}{2 a d \sqrt {d x} \left (a+b x^2\right )}\right )}{8 a}+\frac {1}{4 a d \sqrt {d x} \left (a+b x^2\right )^2}\right )}{12 a}+\frac {1}{6 a d \sqrt {d x} \left (a+b x^2\right )^3}\right )}{16 a}+\frac {1}{8 a d \sqrt {d x} \left (a+b x^2\right )^4}\right )}{20 a}+\frac {1}{10 a d \sqrt {d x} \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {21 \left (\frac {17 \left (\frac {13 \left (\frac {9 \left (\frac {5 \left (-\frac {2 b \left (\frac {\frac {\int \frac {1}{x d+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}}d\sqrt {d x}}{2 \sqrt {b}}+\frac {\int \frac {1}{x d+\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}}d\sqrt {d x}}{2 \sqrt {b}}}{2 \sqrt {b}}-\frac {\int \frac {\sqrt {a} d-\sqrt {b} d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {b}}\right )}{a d}-\frac {2}{a d \sqrt {d x}}\right )}{4 a}+\frac {1}{2 a d \sqrt {d x} \left (a+b x^2\right )}\right )}{8 a}+\frac {1}{4 a d \sqrt {d x} \left (a+b x^2\right )^2}\right )}{12 a}+\frac {1}{6 a d \sqrt {d x} \left (a+b x^2\right )^3}\right )}{16 a}+\frac {1}{8 a d \sqrt {d x} \left (a+b x^2\right )^4}\right )}{20 a}+\frac {1}{10 a d \sqrt {d x} \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {21 \left (\frac {17 \left (\frac {13 \left (\frac {9 \left (\frac {5 \left (-\frac {2 b \left (\frac {\frac {\int \frac {1}{-d x-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\int \frac {1}{-d x-1}d\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}}{2 \sqrt {b}}-\frac {\int \frac {\sqrt {a} d-\sqrt {b} d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {b}}\right )}{a d}-\frac {2}{a d \sqrt {d x}}\right )}{4 a}+\frac {1}{2 a d \sqrt {d x} \left (a+b x^2\right )}\right )}{8 a}+\frac {1}{4 a d \sqrt {d x} \left (a+b x^2\right )^2}\right )}{12 a}+\frac {1}{6 a d \sqrt {d x} \left (a+b x^2\right )^3}\right )}{16 a}+\frac {1}{8 a d \sqrt {d x} \left (a+b x^2\right )^4}\right )}{20 a}+\frac {1}{10 a d \sqrt {d x} \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {21 \left (\frac {17 \left (\frac {13 \left (\frac {9 \left (\frac {5 \left (-\frac {2 b \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}}{2 \sqrt {b}}-\frac {\int \frac {\sqrt {a} d-\sqrt {b} d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {b}}\right )}{a d}-\frac {2}{a d \sqrt {d x}}\right )}{4 a}+\frac {1}{2 a d \sqrt {d x} \left (a+b x^2\right )}\right )}{8 a}+\frac {1}{4 a d \sqrt {d x} \left (a+b x^2\right )^2}\right )}{12 a}+\frac {1}{6 a d \sqrt {d x} \left (a+b x^2\right )^3}\right )}{16 a}+\frac {1}{8 a d \sqrt {d x} \left (a+b x^2\right )^4}\right )}{20 a}+\frac {1}{10 a d \sqrt {d x} \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {21 \left (\frac {17 \left (\frac {13 \left (\frac {9 \left (\frac {5 \left (-\frac {2 b \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}}{2 \sqrt {b}}-\frac {-\frac {\int -\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d}-2 \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{b} \left (x d+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt [4]{a} \sqrt {d}+\sqrt {2} \sqrt [4]{b} \sqrt {d x}\right )}{\sqrt [4]{b} \left (x d+\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}}{2 \sqrt {b}}\right )}{a d}-\frac {2}{a d \sqrt {d x}}\right )}{4 a}+\frac {1}{2 a d \sqrt {d x} \left (a+b x^2\right )}\right )}{8 a}+\frac {1}{4 a d \sqrt {d x} \left (a+b x^2\right )^2}\right )}{12 a}+\frac {1}{6 a d \sqrt {d x} \left (a+b x^2\right )^3}\right )}{16 a}+\frac {1}{8 a d \sqrt {d x} \left (a+b x^2\right )^4}\right )}{20 a}+\frac {1}{10 a d \sqrt {d x} \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {21 \left (\frac {17 \left (\frac {13 \left (\frac {9 \left (\frac {5 \left (-\frac {2 b \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}}{2 \sqrt {b}}-\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{a} \sqrt {d}-2 \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{b} \left (x d+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}+\frac {\int \frac {\sqrt {2} \left (\sqrt [4]{a} \sqrt {d}+\sqrt {2} \sqrt [4]{b} \sqrt {d x}\right )}{\sqrt [4]{b} \left (x d+\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}}{2 \sqrt {b}}\right )}{a d}-\frac {2}{a d \sqrt {d x}}\right )}{4 a}+\frac {1}{2 a d \sqrt {d x} \left (a+b x^2\right )}\right )}{8 a}+\frac {1}{4 a d \sqrt {d x} \left (a+b x^2\right )^2}\right )}{12 a}+\frac {1}{6 a d \sqrt {d x} \left (a+b x^2\right )^3}\right )}{16 a}+\frac {1}{8 a d \sqrt {d x} \left (a+b x^2\right )^4}\right )}{20 a}+\frac {1}{10 a d \sqrt {d x} \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {21 \left (\frac {17 \left (\frac {13 \left (\frac {9 \left (\frac {5 \left (-\frac {2 b \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}}{2 \sqrt {b}}-\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{a} \sqrt {d}-2 \sqrt [4]{b} \sqrt {d x}}{x d+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt {b} \sqrt {d}}+\frac {\int \frac {\sqrt [4]{a} \sqrt {d}+\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{x d+\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}}d\sqrt {d x}}{2 \sqrt [4]{a} \sqrt {b} \sqrt {d}}}{2 \sqrt {b}}\right )}{a d}-\frac {2}{a d \sqrt {d x}}\right )}{4 a}+\frac {1}{2 a d \sqrt {d x} \left (a+b x^2\right )}\right )}{8 a}+\frac {1}{4 a d \sqrt {d x} \left (a+b x^2\right )^2}\right )}{12 a}+\frac {1}{6 a d \sqrt {d x} \left (a+b x^2\right )^3}\right )}{16 a}+\frac {1}{8 a d \sqrt {d x} \left (a+b x^2\right )^4}\right )}{20 a}+\frac {1}{10 a d \sqrt {d x} \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {21 \left (\frac {17 \left (\frac {13 \left (\frac {9 \left (\frac {5 \left (-\frac {2 b \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}}{2 \sqrt {b}}-\frac {\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d} \sqrt {d x}+\sqrt {a} d+\sqrt {b} d x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d} \sqrt {d x}+\sqrt {a} d+\sqrt {b} d x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}}{2 \sqrt {b}}\right )}{a d}-\frac {2}{a d \sqrt {d x}}\right )}{4 a}+\frac {1}{2 a d \sqrt {d x} \left (a+b x^2\right )}\right )}{8 a}+\frac {1}{4 a d \sqrt {d x} \left (a+b x^2\right )^2}\right )}{12 a}+\frac {1}{6 a d \sqrt {d x} \left (a+b x^2\right )^3}\right )}{16 a}+\frac {1}{8 a d \sqrt {d x} \left (a+b x^2\right )^4}\right )}{20 a}+\frac {1}{10 a d \sqrt {d x} \left (a+b x^2\right )^5}\) |
Input:
Int[1/((d*x)^(3/2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^3),x]
Output:
1/(10*a*d*Sqrt[d*x]*(a + b*x^2)^5) + (21*(1/(8*a*d*Sqrt[d*x]*(a + b*x^2)^4 ) + (17*(1/(6*a*d*Sqrt[d*x]*(a + b*x^2)^3) + (13*(1/(4*a*d*Sqrt[d*x]*(a + b*x^2)^2) + (9*(1/(2*a*d*Sqrt[d*x]*(a + b*x^2)) + (5*(-2/(a*d*Sqrt[d*x]) - (2*b*((-(ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])]/(Sqrt[ 2]*a^(1/4)*b^(1/4)*Sqrt[d])) + 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])))/(a*d)))/(4*a)))/(8*a)))/(12*a)))/(16*a)))/(20*a)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(c*x )^(m + 1))*((a + b*x^2)^(p + 1)/(2*a*c*(p + 1))), x] + Simp[(m + 2*p + 3)/( 2*a*(p + 1)) Int[(c*x)^m*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, m }, x] && LtQ[p, -1] && IntBinomialQ[a, b, c, 2, m, p, x]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^( m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + 2*p + 3)/(a*c ^2*(m + 1))) Int[(c*x)^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, p }, x] && LtQ[m, -1] && IntBinomialQ[a, b, c, 2, m, p, x]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{k = De nominator[m]}, Simp[k/c Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(2*k)/c^2)) ^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && FractionQ[m] && I ntBinomialQ[a, b, c, 2, m, p, x]
Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*s) Int[(r + s*x^2)/(a + b*x^ 4), x], x] - Simp[1/(2*s) Int[(r - s*x^2)/(a + b*x^4), x], x]] /; FreeQ[{ a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ, b]]))
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b )], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /; Fre eQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[(u_)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> S imp[1/c^p Int[u*(b/2 + c*x^n)^(2*p), x], x] /; FreeQ[{a, b, c, n, p}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 2*(d/e), 2]}, Simp[e/(2*c) Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ e/(2*c) Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ -2*(d/e), 2]}, Simp[e/(2*c*q) Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Simp[e/(2*c*q) Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
Time = 0.75 (sec) , antiderivative size = 249, normalized size of antiderivative = 0.76
| method | result | size |
| risch | \(-\frac {2}{a^{6} d \sqrt {d x}}-\frac {b \left (\frac {\frac {11743 a^{4} d^{8} \left (d x \right )^{\frac {3}{2}}}{4096}+\frac {1129 a^{3} d^{6} b \left (d x \right )^{\frac {7}{2}}}{128}+\frac {22467 a^{2} d^{4} b^{2} \left (d x \right )^{\frac {11}{2}}}{2048}+\frac {16169 a \,d^{2} b^{3} \left (d x \right )^{\frac {15}{2}}}{2560}+\frac {5731 b^{4} \left (d x \right )^{\frac {19}{2}}}{4096}}{\left (b \,d^{2} x^{2}+a \,d^{2}\right )^{5}}+\frac {13923 \sqrt {2}\, \left (\ln \left (\frac {d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{32768 b \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{a^{6} d}\) | \(249\) |
| derivativedivides | \(2 d^{11} \left (-\frac {b \left (\frac {\frac {11743 a^{4} d^{8} \left (d x \right )^{\frac {3}{2}}}{8192}+\frac {1129 a^{3} d^{6} b \left (d x \right )^{\frac {7}{2}}}{256}+\frac {22467 a^{2} d^{4} b^{2} \left (d x \right )^{\frac {11}{2}}}{4096}+\frac {16169 a \,d^{2} b^{3} \left (d x \right )^{\frac {15}{2}}}{5120}+\frac {5731 b^{4} \left (d x \right )^{\frac {19}{2}}}{8192}}{\left (b \,d^{2} x^{2}+a \,d^{2}\right )^{5}}+\frac {13923 \sqrt {2}\, \left (\ln \left (\frac {d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{65536 b \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{a^{6} d^{12}}-\frac {1}{a^{6} d^{12} \sqrt {d x}}\right )\) | \(253\) |
| default | \(2 d^{11} \left (-\frac {b \left (\frac {\frac {11743 a^{4} d^{8} \left (d x \right )^{\frac {3}{2}}}{8192}+\frac {1129 a^{3} d^{6} b \left (d x \right )^{\frac {7}{2}}}{256}+\frac {22467 a^{2} d^{4} b^{2} \left (d x \right )^{\frac {11}{2}}}{4096}+\frac {16169 a \,d^{2} b^{3} \left (d x \right )^{\frac {15}{2}}}{5120}+\frac {5731 b^{4} \left (d x \right )^{\frac {19}{2}}}{8192}}{\left (b \,d^{2} x^{2}+a \,d^{2}\right )^{5}}+\frac {13923 \sqrt {2}\, \left (\ln \left (\frac {d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{65536 b \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{a^{6} d^{12}}-\frac {1}{a^{6} d^{12} \sqrt {d x}}\right )\) | \(253\) |
| pseudoelliptic | \(-\frac {13923 \left (\frac {\sqrt {2}\, \left (b \,x^{2}+a \right )^{5} \left (\ln \left (\frac {d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}-\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}+\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )\right ) \sqrt {d x}}{2}+\frac {32768 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \left (\frac {13923}{8192} x^{10} b^{5}+\frac {41769}{5120} a \,x^{8} b^{4}+\frac {63427}{4096} a^{2} x^{6} b^{3}+\frac {3689}{256} a^{3} x^{4} b^{2}+\frac {52703}{8192} x^{2} a^{4} b +a^{5}\right )}{13923}\right )}{16384 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, d \,a^{6} \left (b \,x^{2}+a \right )^{5}}\) | \(259\) |
Input:
int(1/(d*x)^(3/2)/(b^2*x^4+2*a*b*x^2+a^2)^3,x,method=_RETURNVERBOSE)
Output:
-2/a^6/d/(d*x)^(1/2)-1/a^6*b*(2*(11743/8192*a^4*d^8*(d*x)^(3/2)+1129/256*a ^3*d^6*b*(d*x)^(7/2)+22467/4096*a^2*d^4*b^2*(d*x)^(11/2)+16169/5120*a*d^2* b^3*(d*x)^(15/2)+5731/8192*b^4*(d*x)^(19/2))/(b*d^2*x^2+a*d^2)^5+13923/327 68/b/(a*d^2/b)^(1/4)*2^(1/2)*(ln((d*x-(a*d^2/b)^(1/4)*(d*x)^(1/2)*2^(1/2)+ (a*d^2/b)^(1/2))/(d*x+(a*d^2/b)^(1/4)*(d*x)^(1/2)*2^(1/2)+(a*d^2/b)^(1/2)) )+2*arctan(2^(1/2)/(a*d^2/b)^(1/4)*(d*x)^(1/2)+1)+2*arctan(2^(1/2)/(a*d^2/ b)^(1/4)*(d*x)^(1/2)-1)))/d
Result contains complex when optimal does not.
Time = 0.23 (sec) , antiderivative size = 618, normalized size of antiderivative = 1.89 \[ \int \frac {1}{(d x)^{3/2} \left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=-\frac {69615 \, {\left (a^{6} b^{5} d^{2} x^{11} + 5 \, a^{7} b^{4} d^{2} x^{9} + 10 \, a^{8} b^{3} d^{2} x^{7} + 10 \, a^{9} b^{2} d^{2} x^{5} + 5 \, a^{10} b d^{2} x^{3} + a^{11} d^{2} x\right )} \left (-\frac {b}{a^{25} d^{6}}\right )^{\frac {1}{4}} \log \left (2698972561467 \, a^{19} d^{5} \left (-\frac {b}{a^{25} d^{6}}\right )^{\frac {3}{4}} + 2698972561467 \, \sqrt {d x} b\right ) + 69615 \, {\left (-i \, a^{6} b^{5} d^{2} x^{11} - 5 i \, a^{7} b^{4} d^{2} x^{9} - 10 i \, a^{8} b^{3} d^{2} x^{7} - 10 i \, a^{9} b^{2} d^{2} x^{5} - 5 i \, a^{10} b d^{2} x^{3} - i \, a^{11} d^{2} x\right )} \left (-\frac {b}{a^{25} d^{6}}\right )^{\frac {1}{4}} \log \left (2698972561467 i \, a^{19} d^{5} \left (-\frac {b}{a^{25} d^{6}}\right )^{\frac {3}{4}} + 2698972561467 \, \sqrt {d x} b\right ) + 69615 \, {\left (i \, a^{6} b^{5} d^{2} x^{11} + 5 i \, a^{7} b^{4} d^{2} x^{9} + 10 i \, a^{8} b^{3} d^{2} x^{7} + 10 i \, a^{9} b^{2} d^{2} x^{5} + 5 i \, a^{10} b d^{2} x^{3} + i \, a^{11} d^{2} x\right )} \left (-\frac {b}{a^{25} d^{6}}\right )^{\frac {1}{4}} \log \left (-2698972561467 i \, a^{19} d^{5} \left (-\frac {b}{a^{25} d^{6}}\right )^{\frac {3}{4}} + 2698972561467 \, \sqrt {d x} b\right ) - 69615 \, {\left (a^{6} b^{5} d^{2} x^{11} + 5 \, a^{7} b^{4} d^{2} x^{9} + 10 \, a^{8} b^{3} d^{2} x^{7} + 10 \, a^{9} b^{2} d^{2} x^{5} + 5 \, a^{10} b d^{2} x^{3} + a^{11} d^{2} x\right )} \left (-\frac {b}{a^{25} d^{6}}\right )^{\frac {1}{4}} \log \left (-2698972561467 \, a^{19} d^{5} \left (-\frac {b}{a^{25} d^{6}}\right )^{\frac {3}{4}} + 2698972561467 \, \sqrt {d x} b\right ) + 4 \, {\left (69615 \, b^{5} x^{10} + 334152 \, a b^{4} x^{8} + 634270 \, a^{2} b^{3} x^{6} + 590240 \, a^{3} b^{2} x^{4} + 263515 \, a^{4} b x^{2} + 40960 \, a^{5}\right )} \sqrt {d x}}{81920 \, {\left (a^{6} b^{5} d^{2} x^{11} + 5 \, a^{7} b^{4} d^{2} x^{9} + 10 \, a^{8} b^{3} d^{2} x^{7} + 10 \, a^{9} b^{2} d^{2} x^{5} + 5 \, a^{10} b d^{2} x^{3} + a^{11} d^{2} x\right )}} \] Input:
integrate(1/(d*x)^(3/2)/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="fricas")
Output:
-1/81920*(69615*(a^6*b^5*d^2*x^11 + 5*a^7*b^4*d^2*x^9 + 10*a^8*b^3*d^2*x^7 + 10*a^9*b^2*d^2*x^5 + 5*a^10*b*d^2*x^3 + a^11*d^2*x)*(-b/(a^25*d^6))^(1/ 4)*log(2698972561467*a^19*d^5*(-b/(a^25*d^6))^(3/4) + 2698972561467*sqrt(d *x)*b) + 69615*(-I*a^6*b^5*d^2*x^11 - 5*I*a^7*b^4*d^2*x^9 - 10*I*a^8*b^3*d ^2*x^7 - 10*I*a^9*b^2*d^2*x^5 - 5*I*a^10*b*d^2*x^3 - I*a^11*d^2*x)*(-b/(a^ 25*d^6))^(1/4)*log(2698972561467*I*a^19*d^5*(-b/(a^25*d^6))^(3/4) + 269897 2561467*sqrt(d*x)*b) + 69615*(I*a^6*b^5*d^2*x^11 + 5*I*a^7*b^4*d^2*x^9 + 1 0*I*a^8*b^3*d^2*x^7 + 10*I*a^9*b^2*d^2*x^5 + 5*I*a^10*b*d^2*x^3 + I*a^11*d ^2*x)*(-b/(a^25*d^6))^(1/4)*log(-2698972561467*I*a^19*d^5*(-b/(a^25*d^6))^ (3/4) + 2698972561467*sqrt(d*x)*b) - 69615*(a^6*b^5*d^2*x^11 + 5*a^7*b^4*d ^2*x^9 + 10*a^8*b^3*d^2*x^7 + 10*a^9*b^2*d^2*x^5 + 5*a^10*b*d^2*x^3 + a^11 *d^2*x)*(-b/(a^25*d^6))^(1/4)*log(-2698972561467*a^19*d^5*(-b/(a^25*d^6))^ (3/4) + 2698972561467*sqrt(d*x)*b) + 4*(69615*b^5*x^10 + 334152*a*b^4*x^8 + 634270*a^2*b^3*x^6 + 590240*a^3*b^2*x^4 + 263515*a^4*b*x^2 + 40960*a^5)* sqrt(d*x))/(a^6*b^5*d^2*x^11 + 5*a^7*b^4*d^2*x^9 + 10*a^8*b^3*d^2*x^7 + 10 *a^9*b^2*d^2*x^5 + 5*a^10*b*d^2*x^3 + a^11*d^2*x)
\[ \int \frac {1}{(d x)^{3/2} \left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\int \frac {1}{\left (d x\right )^{\frac {3}{2}} \left (a + b x^{2}\right )^{6}}\, dx \] Input:
integrate(1/(d*x)**(3/2)/(b**2*x**4+2*a*b*x**2+a**2)**3,x)
Output:
Integral(1/((d*x)**(3/2)*(a + b*x**2)**6), x)
Time = 0.13 (sec) , antiderivative size = 388, normalized size of antiderivative = 1.19 \[ \int \frac {1}{(d x)^{3/2} \left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=-\frac {\frac {8 \, {\left (69615 \, b^{5} d^{10} x^{10} + 334152 \, a b^{4} d^{10} x^{8} + 634270 \, a^{2} b^{3} d^{10} x^{6} + 590240 \, a^{3} b^{2} d^{10} x^{4} + 263515 \, a^{4} b d^{10} x^{2} + 40960 \, a^{5} d^{10}\right )}}{\left (d x\right )^{\frac {21}{2}} a^{6} b^{5} + 5 \, \left (d x\right )^{\frac {17}{2}} a^{7} b^{4} d^{2} + 10 \, \left (d x\right )^{\frac {13}{2}} a^{8} b^{3} d^{4} + 10 \, \left (d x\right )^{\frac {9}{2}} a^{9} b^{2} d^{6} + 5 \, \left (d x\right )^{\frac {5}{2}} a^{10} b d^{8} + \sqrt {d x} a^{11} d^{10}} + \frac {69615 \, b {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {d x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} d}}\right )}{\sqrt {\sqrt {a} \sqrt {b} d} \sqrt {b}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {d x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} d}}\right )}{\sqrt {\sqrt {a} \sqrt {b} d} \sqrt {b}} - \frac {\sqrt {2} \log \left (\sqrt {b} d x + \sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} \sqrt {d x} b^{\frac {1}{4}} + \sqrt {a} d\right )}{\left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (\sqrt {b} d x - \sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} \sqrt {d x} b^{\frac {1}{4}} + \sqrt {a} d\right )}{\left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {3}{4}}}\right )}}{a^{6}}}{163840 \, d} \] Input:
integrate(1/(d*x)^(3/2)/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="maxima")
Output:
-1/163840*(8*(69615*b^5*d^10*x^10 + 334152*a*b^4*d^10*x^8 + 634270*a^2*b^3 *d^10*x^6 + 590240*a^3*b^2*d^10*x^4 + 263515*a^4*b*d^10*x^2 + 40960*a^5*d^ 10)/((d*x)^(21/2)*a^6*b^5 + 5*(d*x)^(17/2)*a^7*b^4*d^2 + 10*(d*x)^(13/2)*a ^8*b^3*d^4 + 10*(d*x)^(9/2)*a^9*b^2*d^6 + 5*(d*x)^(5/2)*a^10*b*d^8 + sqrt( d*x)*a^11*d^10) + 69615*b*(2*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2)*(a*d^2)^( 1/4)*b^(1/4) + 2*sqrt(d*x)*sqrt(b))/sqrt(sqrt(a)*sqrt(b)*d))/(sqrt(sqrt(a) *sqrt(b)*d)*sqrt(b)) + 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)*sq rt(b)*d)*sqrt(b)) - sqrt(2)*log(sqrt(b)*d*x + sqrt(2)*(a*d^2)^(1/4)*sqrt(d *x)*b^(1/4) + sqrt(a)*d)/((a*d^2)^(1/4)*b^(3/4)) + sqrt(2)*log(sqrt(b)*d*x - sqrt(2)*(a*d^2)^(1/4)*sqrt(d*x)*b^(1/4) + sqrt(a)*d)/((a*d^2)^(1/4)*b^( 3/4)))/a^6)/d
Time = 0.13 (sec) , antiderivative size = 365, normalized size of antiderivative = 1.12 \[ \int \frac {1}{(d x)^{3/2} \left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=-\frac {\frac {327680}{\sqrt {d x} a^{6}} + \frac {139230 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{a^{7} b^{2} d^{2}} + \frac {139230 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{a^{7} b^{2} d^{2}} - \frac {69615 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \log \left (d x + \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{a^{7} b^{2} d^{2}} + \frac {69615 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \log \left (d x - \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{a^{7} b^{2} d^{2}} + \frac {8 \, {\left (28655 \, \sqrt {d x} b^{5} d^{9} x^{9} + 129352 \, \sqrt {d x} a b^{4} d^{9} x^{7} + 224670 \, \sqrt {d x} a^{2} b^{3} d^{9} x^{5} + 180640 \, \sqrt {d x} a^{3} b^{2} d^{9} x^{3} + 58715 \, \sqrt {d x} a^{4} b d^{9} x\right )}}{{\left (b d^{2} x^{2} + a d^{2}\right )}^{5} a^{6}}}{163840 \, d} \] Input:
integrate(1/(d*x)^(3/2)/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="giac")
Output:
-1/163840*(327680/(sqrt(d*x)*a^6) + 139230*sqrt(2)*(a*b^3*d^2)^(3/4)*arcta n(1/2*sqrt(2)*(sqrt(2)*(a*d^2/b)^(1/4) + 2*sqrt(d*x))/(a*d^2/b)^(1/4))/(a^ 7*b^2*d^2) + 139230*sqrt(2)*(a*b^3*d^2)^(3/4)*arctan(-1/2*sqrt(2)*(sqrt(2) *(a*d^2/b)^(1/4) - 2*sqrt(d*x))/(a*d^2/b)^(1/4))/(a^7*b^2*d^2) - 69615*sqr t(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))/(a^7*b^2*d^2) + 69615*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))/(a^7*b^2*d^2) + 8*(28655*sqrt (d*x)*b^5*d^9*x^9 + 129352*sqrt(d*x)*a*b^4*d^9*x^7 + 224670*sqrt(d*x)*a^2* b^3*d^9*x^5 + 180640*sqrt(d*x)*a^3*b^2*d^9*x^3 + 58715*sqrt(d*x)*a^4*b*d^9 *x)/((b*d^2*x^2 + a*d^2)^5*a^6))/d
Time = 0.22 (sec) , antiderivative size = 226, normalized size of antiderivative = 0.69 \[ \int \frac {1}{(d x)^{3/2} \left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {13923\,{\left (-b\right )}^{1/4}\,\mathrm {atanh}\left (\frac {{\left (-b\right )}^{1/4}\,\sqrt {d\,x}}{a^{1/4}\,\sqrt {d}}\right )}{8192\,a^{25/4}\,d^{3/2}}-\frac {13923\,{\left (-b\right )}^{1/4}\,\mathrm {atan}\left (\frac {{\left (-b\right )}^{1/4}\,\sqrt {d\,x}}{a^{1/4}\,\sqrt {d}}\right )}{8192\,a^{25/4}\,d^{3/2}}-\frac {\frac {2\,d^9}{a}+\frac {52703\,b\,d^9\,x^2}{4096\,a^2}+\frac {3689\,b^2\,d^9\,x^4}{128\,a^3}+\frac {63427\,b^3\,d^9\,x^6}{2048\,a^4}+\frac {41769\,b^4\,d^9\,x^8}{2560\,a^5}+\frac {13923\,b^5\,d^9\,x^{10}}{4096\,a^6}}{b^5\,{\left (d\,x\right )}^{21/2}+a^5\,d^{10}\,\sqrt {d\,x}+10\,a^3\,b^2\,d^6\,{\left (d\,x\right )}^{9/2}+10\,a^2\,b^3\,d^4\,{\left (d\,x\right )}^{13/2}+5\,a^4\,b\,d^8\,{\left (d\,x\right )}^{5/2}+5\,a\,b^4\,d^2\,{\left (d\,x\right )}^{17/2}} \] Input:
int(1/((d*x)^(3/2)*(a^2 + b^2*x^4 + 2*a*b*x^2)^3),x)
Output:
(13923*(-b)^(1/4)*atanh(((-b)^(1/4)*(d*x)^(1/2))/(a^(1/4)*d^(1/2))))/(8192 *a^(25/4)*d^(3/2)) - (13923*(-b)^(1/4)*atan(((-b)^(1/4)*(d*x)^(1/2))/(a^(1 /4)*d^(1/2))))/(8192*a^(25/4)*d^(3/2)) - ((2*d^9)/a + (52703*b*d^9*x^2)/(4 096*a^2) + (3689*b^2*d^9*x^4)/(128*a^3) + (63427*b^3*d^9*x^6)/(2048*a^4) + (41769*b^4*d^9*x^8)/(2560*a^5) + (13923*b^5*d^9*x^10)/(4096*a^6))/(b^5*(d *x)^(21/2) + a^5*d^10*(d*x)^(1/2) + 10*a^3*b^2*d^6*(d*x)^(9/2) + 10*a^2*b^ 3*d^4*(d*x)^(13/2) + 5*a^4*b*d^8*(d*x)^(5/2) + 5*a*b^4*d^2*(d*x)^(17/2))
Time = 0.19 (sec) , antiderivative size = 1042, normalized size of antiderivative = 3.19 \[ \int \frac {1}{(d x)^{3/2} \left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx =\text {Too large to display} \] Input:
int(1/(d*x)^(3/2)/(b^2*x^4+2*a*b*x^2+a^2)^3,x)
Output:
(sqrt(d)*(139230*sqrt(x)*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 + 696150*s qrt(x)*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 + 1392300*sqrt(x)*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 + 1392300*sqrt(x)*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 + 696150*sqrt(x)*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 + 139230*sqrt(x)*b**(1/4)*a**(3/4)*sqrt(2)*ata n((b**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt( 2)))*b**5*x**10 - 139230*sqrt(x)*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 - 696150*sqrt(x)*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 - 1392300*sqr t(x)*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 - 1392300*sqrt(x)*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 - 696150*sqrt(x)*b**(1/4)*a **(3/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) + 2*sqrt(x)*sqrt(b))/(b...