Integrand size = 28, antiderivative size = 312 \[ \int \frac {(d x)^{3/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=-\frac {d \sqrt {d x}}{10 b \left (a+b x^2\right )^5}+\frac {d \sqrt {d x}}{160 a b \left (a+b x^2\right )^4}+\frac {d \sqrt {d x}}{128 a^2 b \left (a+b x^2\right )^3}+\frac {11 d \sqrt {d x}}{1024 a^3 b \left (a+b x^2\right )^2}+\frac {77 d \sqrt {d x}}{4096 a^4 b \left (a+b x^2\right )}-\frac {231 d^{3/2} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} a^{19/4} b^{5/4}}+\frac {231 d^{3/2} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} a^{19/4} b^{5/4}}+\frac {231 d^{3/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}}{\sqrt {d} \left (\sqrt {a}+\sqrt {b} x\right )}\right )}{8192 \sqrt {2} a^{19/4} b^{5/4}} \] Output:
-1/10*d*(d*x)^(1/2)/b/(b*x^2+a)^5+1/160*d*(d*x)^(1/2)/a/b/(b*x^2+a)^4+1/12 8*d*(d*x)^(1/2)/a^2/b/(b*x^2+a)^3+11/1024*d*(d*x)^(1/2)/a^3/b/(b*x^2+a)^2+ 77/4096*d*(d*x)^(1/2)/a^4/b/(b*x^2+a)-231/16384*d^(3/2)*arctan(1-2^(1/2)*b ^(1/4)*(d*x)^(1/2)/a^(1/4)/d^(1/2))*2^(1/2)/a^(19/4)/b^(5/4)+231/16384*d^( 3/2)*arctan(1+2^(1/2)*b^(1/4)*(d*x)^(1/2)/a^(1/4)/d^(1/2))*2^(1/2)/a^(19/4 )/b^(5/4)+231/16384*d^(3/2)*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)/a^(19/4)/b^(5/4)
Time = 0.52 (sec) , antiderivative size = 183, normalized size of antiderivative = 0.59 \[ \int \frac {(d x)^{3/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {(d x)^{3/2} \left (\frac {4 a^{3/4} \sqrt [4]{b} \sqrt {x} \left (-1155 a^4+2648 a^3 b x^2+3130 a^2 b^2 x^4+1760 a b^3 x^6+385 b^4 x^8\right )}{\left (a+b x^2\right )^5}-1155 \sqrt {2} \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )+1155 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )\right )}{81920 a^{19/4} b^{5/4} x^{3/2}} \] Input:
Integrate[(d*x)^(3/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]
Output:
((d*x)^(3/2)*((4*a^(3/4)*b^(1/4)*Sqrt[x]*(-1155*a^4 + 2648*a^3*b*x^2 + 313 0*a^2*b^2*x^4 + 1760*a*b^3*x^6 + 385*b^4*x^8))/(a + b*x^2)^5 - 1155*Sqrt[2 ]*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])] + 1155*S qrt[2]*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)]))/ (81920*a^(19/4)*b^(5/4)*x^(3/2))
Time = 1.10 (sec) , antiderivative size = 442, normalized size of antiderivative = 1.42, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.607, Rules used = {1380, 27, 252, 253, 253, 253, 253, 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.
| \(\displaystyle \int \frac {(d x)^{3/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx\) |
| \(\Big \downarrow \) 1380 |
| \(\displaystyle b^6 \int \frac {(d x)^{3/2}}{b^6 \left (b x^2+a\right )^6}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \frac {(d x)^{3/2}}{\left (a+b x^2\right )^6}dx\) |
| \(\Big \downarrow \) 252 |
| \(\displaystyle \frac {d^2 \int \frac {1}{\sqrt {d x} \left (b x^2+a\right )^5}dx}{20 b}-\frac {d \sqrt {d x}}{10 b \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 253 |
| \(\displaystyle \frac {d^2 \left (\frac {15 \int \frac {1}{\sqrt {d x} \left (b x^2+a\right )^4}dx}{16 a}+\frac {\sqrt {d x}}{8 a d \left (a+b x^2\right )^4}\right )}{20 b}-\frac {d \sqrt {d x}}{10 b \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 253 |
| \(\displaystyle \frac {d^2 \left (\frac {15 \left (\frac {11 \int \frac {1}{\sqrt {d x} \left (b x^2+a\right )^3}dx}{12 a}+\frac {\sqrt {d x}}{6 a d \left (a+b x^2\right )^3}\right )}{16 a}+\frac {\sqrt {d x}}{8 a d \left (a+b x^2\right )^4}\right )}{20 b}-\frac {d \sqrt {d x}}{10 b \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 253 |
| \(\displaystyle \frac {d^2 \left (\frac {15 \left (\frac {11 \left (\frac {7 \int \frac {1}{\sqrt {d x} \left (b x^2+a\right )^2}dx}{8 a}+\frac {\sqrt {d x}}{4 a d \left (a+b x^2\right )^2}\right )}{12 a}+\frac {\sqrt {d x}}{6 a d \left (a+b x^2\right )^3}\right )}{16 a}+\frac {\sqrt {d x}}{8 a d \left (a+b x^2\right )^4}\right )}{20 b}-\frac {d \sqrt {d x}}{10 b \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 253 |
| \(\displaystyle \frac {d^2 \left (\frac {15 \left (\frac {11 \left (\frac {7 \left (\frac {3 \int \frac {1}{\sqrt {d x} \left (b x^2+a\right )}dx}{4 a}+\frac {\sqrt {d x}}{2 a d \left (a+b x^2\right )}\right )}{8 a}+\frac {\sqrt {d x}}{4 a d \left (a+b x^2\right )^2}\right )}{12 a}+\frac {\sqrt {d x}}{6 a d \left (a+b x^2\right )^3}\right )}{16 a}+\frac {\sqrt {d x}}{8 a d \left (a+b x^2\right )^4}\right )}{20 b}-\frac {d \sqrt {d x}}{10 b \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {d^2 \left (\frac {15 \left (\frac {11 \left (\frac {7 \left (\frac {3 \int \frac {1}{b x^2+a}d\sqrt {d x}}{2 a d}+\frac {\sqrt {d x}}{2 a d \left (a+b x^2\right )}\right )}{8 a}+\frac {\sqrt {d x}}{4 a d \left (a+b x^2\right )^2}\right )}{12 a}+\frac {\sqrt {d x}}{6 a d \left (a+b x^2\right )^3}\right )}{16 a}+\frac {\sqrt {d x}}{8 a d \left (a+b x^2\right )^4}\right )}{20 b}-\frac {d \sqrt {d x}}{10 b \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 755 |
| \(\displaystyle \frac {d^2 \left (\frac {15 \left (\frac {11 \left (\frac {7 \left (\frac {3 \left (\frac {\int \frac {d^2 \left (\sqrt {a} d-\sqrt {b} d x\right )}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {a} d}+\frac {\int \frac {d^2 \left (\sqrt {b} x d+\sqrt {a} d\right )}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {a} d}\right )}{2 a d}+\frac {\sqrt {d x}}{2 a d \left (a+b x^2\right )}\right )}{8 a}+\frac {\sqrt {d x}}{4 a d \left (a+b x^2\right )^2}\right )}{12 a}+\frac {\sqrt {d x}}{6 a d \left (a+b x^2\right )^3}\right )}{16 a}+\frac {\sqrt {d x}}{8 a d \left (a+b x^2\right )^4}\right )}{20 b}-\frac {d \sqrt {d x}}{10 b \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {d^2 \left (\frac {15 \left (\frac {11 \left (\frac {7 \left (\frac {3 \left (\frac {d \int \frac {\sqrt {a} d-\sqrt {b} d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {a}}+\frac {d \int \frac {\sqrt {b} x d+\sqrt {a} d}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {a}}\right )}{2 a d}+\frac {\sqrt {d x}}{2 a d \left (a+b x^2\right )}\right )}{8 a}+\frac {\sqrt {d x}}{4 a d \left (a+b x^2\right )^2}\right )}{12 a}+\frac {\sqrt {d x}}{6 a d \left (a+b x^2\right )^3}\right )}{16 a}+\frac {\sqrt {d x}}{8 a d \left (a+b x^2\right )^4}\right )}{20 b}-\frac {d \sqrt {d x}}{10 b \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {d^2 \left (\frac {15 \left (\frac {11 \left (\frac {7 \left (\frac {3 \left (\frac {d \int \frac {\sqrt {a} d-\sqrt {b} d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {a}}+\frac {d \left (\frac {\int \frac {1}{x d+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}}d\sqrt {d x}}{2 \sqrt {b}}+\frac {\int \frac {1}{x d+\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}}d\sqrt {d x}}{2 \sqrt {b}}\right )}{2 \sqrt {a}}\right )}{2 a d}+\frac {\sqrt {d x}}{2 a d \left (a+b x^2\right )}\right )}{8 a}+\frac {\sqrt {d x}}{4 a d \left (a+b x^2\right )^2}\right )}{12 a}+\frac {\sqrt {d x}}{6 a d \left (a+b x^2\right )^3}\right )}{16 a}+\frac {\sqrt {d x}}{8 a d \left (a+b x^2\right )^4}\right )}{20 b}-\frac {d \sqrt {d x}}{10 b \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {d^2 \left (\frac {15 \left (\frac {11 \left (\frac {7 \left (\frac {3 \left (\frac {d \int \frac {\sqrt {a} d-\sqrt {b} d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {a}}+\frac {d \left (\frac {\int \frac {1}{-d x-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\int \frac {1}{-d x-1}d\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}\right )}{2 a d}+\frac {\sqrt {d x}}{2 a d \left (a+b x^2\right )}\right )}{8 a}+\frac {\sqrt {d x}}{4 a d \left (a+b x^2\right )^2}\right )}{12 a}+\frac {\sqrt {d x}}{6 a d \left (a+b x^2\right )^3}\right )}{16 a}+\frac {\sqrt {d x}}{8 a d \left (a+b x^2\right )^4}\right )}{20 b}-\frac {d \sqrt {d x}}{10 b \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {d^2 \left (\frac {15 \left (\frac {11 \left (\frac {7 \left (\frac {3 \left (\frac {d \int \frac {\sqrt {a} d-\sqrt {b} d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {a}}+\frac {d \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}\right )}{2 a d}+\frac {\sqrt {d x}}{2 a d \left (a+b x^2\right )}\right )}{8 a}+\frac {\sqrt {d x}}{4 a d \left (a+b x^2\right )^2}\right )}{12 a}+\frac {\sqrt {d x}}{6 a d \left (a+b x^2\right )^3}\right )}{16 a}+\frac {\sqrt {d x}}{8 a d \left (a+b x^2\right )^4}\right )}{20 b}-\frac {d \sqrt {d x}}{10 b \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {d^2 \left (\frac {15 \left (\frac {11 \left (\frac {7 \left (\frac {3 \left (\frac {d \left (-\frac {\int -\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d}-2 \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{b} \left (x d+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt [4]{a} \sqrt {d}+\sqrt {2} \sqrt [4]{b} \sqrt {d x}\right )}{\sqrt [4]{b} \left (x d+\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}+\frac {d \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}\right )}{2 a d}+\frac {\sqrt {d x}}{2 a d \left (a+b x^2\right )}\right )}{8 a}+\frac {\sqrt {d x}}{4 a d \left (a+b x^2\right )^2}\right )}{12 a}+\frac {\sqrt {d x}}{6 a d \left (a+b x^2\right )^3}\right )}{16 a}+\frac {\sqrt {d x}}{8 a d \left (a+b x^2\right )^4}\right )}{20 b}-\frac {d \sqrt {d x}}{10 b \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {d^2 \left (\frac {15 \left (\frac {11 \left (\frac {7 \left (\frac {3 \left (\frac {d \left (\frac {\int \frac {\sqrt {2} \sqrt [4]{a} \sqrt {d}-2 \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{b} \left (x d+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}+\frac {\int \frac {\sqrt {2} \left (\sqrt [4]{a} \sqrt {d}+\sqrt {2} \sqrt [4]{b} \sqrt {d x}\right )}{\sqrt [4]{b} \left (x d+\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}+\frac {d \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}\right )}{2 a d}+\frac {\sqrt {d x}}{2 a d \left (a+b x^2\right )}\right )}{8 a}+\frac {\sqrt {d x}}{4 a d \left (a+b x^2\right )^2}\right )}{12 a}+\frac {\sqrt {d x}}{6 a d \left (a+b x^2\right )^3}\right )}{16 a}+\frac {\sqrt {d x}}{8 a d \left (a+b x^2\right )^4}\right )}{20 b}-\frac {d \sqrt {d x}}{10 b \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {d^2 \left (\frac {15 \left (\frac {11 \left (\frac {7 \left (\frac {3 \left (\frac {d \left (\frac {\int \frac {\sqrt {2} \sqrt [4]{a} \sqrt {d}-2 \sqrt [4]{b} \sqrt {d x}}{x d+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt {b} \sqrt {d}}+\frac {\int \frac {\sqrt [4]{a} \sqrt {d}+\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{x d+\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}}d\sqrt {d x}}{2 \sqrt [4]{a} \sqrt {b} \sqrt {d}}\right )}{2 \sqrt {a}}+\frac {d \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}\right )}{2 a d}+\frac {\sqrt {d x}}{2 a d \left (a+b x^2\right )}\right )}{8 a}+\frac {\sqrt {d x}}{4 a d \left (a+b x^2\right )^2}\right )}{12 a}+\frac {\sqrt {d x}}{6 a d \left (a+b x^2\right )^3}\right )}{16 a}+\frac {\sqrt {d x}}{8 a d \left (a+b x^2\right )^4}\right )}{20 b}-\frac {d \sqrt {d x}}{10 b \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {d^2 \left (\frac {15 \left (\frac {11 \left (\frac {7 \left (\frac {3 \left (\frac {d \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}+\frac {d \left (\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d} \sqrt {d x}+\sqrt {a} d+\sqrt {b} d x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d} \sqrt {d x}+\sqrt {a} d+\sqrt {b} d x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}\right )}{2 a d}+\frac {\sqrt {d x}}{2 a d \left (a+b x^2\right )}\right )}{8 a}+\frac {\sqrt {d x}}{4 a d \left (a+b x^2\right )^2}\right )}{12 a}+\frac {\sqrt {d x}}{6 a d \left (a+b x^2\right )^3}\right )}{16 a}+\frac {\sqrt {d x}}{8 a d \left (a+b x^2\right )^4}\right )}{20 b}-\frac {d \sqrt {d x}}{10 b \left (a+b x^2\right )^5}\) |
Input:
Int[(d*x)^(3/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]
Output:
-1/10*(d*Sqrt[d*x])/(b*(a + b*x^2)^5) + (d^2*(Sqrt[d*x]/(8*a*d*(a + b*x^2) ^4) + (15*(Sqrt[d*x]/(6*a*d*(a + b*x^2)^3) + (11*(Sqrt[d*x]/(4*a*d*(a + b* x^2)^2) + (7*(Sqrt[d*x]/(2*a*d*(a + b*x^2)) + (3*((d*(-(ArcTan[1 - (Sqrt[2 ]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])]/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d])) + ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])]/(Sqrt[2]*a^(1 /4)*b^(1/4)*Sqrt[d])))/(2*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])))/(2*a*d)))/(8*a)) )/(12*a)))/(16*a)))/(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*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] :> With[{k = De nominator[m]}, Simp[k/c Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(2*k)/c^2)) ^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && FractionQ[m] && I ntBinomialQ[a, b, c, 2, m, p, x]
Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2] ], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*r) Int[(r - s*x^2)/(a + b*x^4) , x], x] + Simp[1/(2*r) Int[(r + s*x^2)/(a + b*x^4), x], x]] /; FreeQ[{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] & & AtomQ[SplitProduct[SumBaseQ, b]]))
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b )], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /; Fre eQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[(u_)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> S imp[1/c^p Int[u*(b/2 + c*x^n)^(2*p), x], x] /; FreeQ[{a, b, c, n, p}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 2*(d/e), 2]}, Simp[e/(2*c) Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ e/(2*c) Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ -2*(d/e), 2]}, Simp[e/(2*c*q) Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Simp[e/(2*c*q) Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
Time = 0.59 (sec) , antiderivative size = 238, normalized size of antiderivative = 0.76
| method | result | size |
| derivativedivides | \(2 d^{11} \left (\frac {-\frac {231 \sqrt {d x}}{8192 b}+\frac {331 \left (d x \right )^{\frac {5}{2}}}{5120 a \,d^{2}}+\frac {313 b \left (d x \right )^{\frac {9}{2}}}{4096 a^{2} d^{4}}+\frac {11 b^{2} \left (d x \right )^{\frac {13}{2}}}{256 a^{3} d^{6}}+\frac {77 b^{3} \left (d x \right )^{\frac {17}{2}}}{8192 a^{4} d^{8}}}{\left (b \,d^{2} x^{2}+a \,d^{2}\right )^{5}}+\frac {231 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{65536 a^{5} d^{10} b}\right )\) | \(238\) |
| default | \(2 d^{11} \left (\frac {-\frac {231 \sqrt {d x}}{8192 b}+\frac {331 \left (d x \right )^{\frac {5}{2}}}{5120 a \,d^{2}}+\frac {313 b \left (d x \right )^{\frac {9}{2}}}{4096 a^{2} d^{4}}+\frac {11 b^{2} \left (d x \right )^{\frac {13}{2}}}{256 a^{3} d^{6}}+\frac {77 b^{3} \left (d x \right )^{\frac {17}{2}}}{8192 a^{4} d^{8}}}{\left (b \,d^{2} x^{2}+a \,d^{2}\right )^{5}}+\frac {231 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{65536 a^{5} d^{10} b}\right )\) | \(238\) |
| pseudoelliptic | \(\frac {d \left (\left (3080 a \,x^{8} b^{4}+14080 a^{2} x^{6} b^{3}+25040 a^{3} x^{4} b^{2}+21184 x^{2} a^{4} b -9240 a^{5}\right ) \sqrt {d x}+1155 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (b \,x^{2}+a \right )^{5} \left (\ln \left (\frac {d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}-\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}+\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )\right )\right )}{163840 b \,a^{5} \left (b \,x^{2}+a \right )^{5}}\) | \(238\) |
Input:
int((d*x)^(3/2)/(b^2*x^4+2*a*b*x^2+a^2)^3,x,method=_RETURNVERBOSE)
Output:
2*d^11*((-231/8192/b*(d*x)^(1/2)+331/5120/a/d^2*(d*x)^(5/2)+313/4096/a^2/d ^4*b*(d*x)^(9/2)+11/256/a^3/d^6*b^2*(d*x)^(13/2)+77/8192/a^4/d^8*b^3*(d*x) ^(17/2))/(b*d^2*x^2+a*d^2)^5+231/65536/a^5/d^10/b*(a*d^2/b)^(1/4)*2^(1/2)* (ln((d*x+(a*d^2/b)^(1/4)*(d*x)^(1/2)*2^(1/2)+(a*d^2/b)^(1/2))/(d*x-(a*d^2/ b)^(1/4)*(d*x)^(1/2)*2^(1/2)+(a*d^2/b)^(1/2)))+2*arctan(2^(1/2)/(a*d^2/b)^ (1/4)*(d*x)^(1/2)+1)+2*arctan(2^(1/2)/(a*d^2/b)^(1/4)*(d*x)^(1/2)-1)))
Result contains complex when optimal does not.
Time = 0.12 (sec) , antiderivative size = 540, normalized size of antiderivative = 1.73 \[ \int \frac {(d x)^{3/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {1155 \, {\left (a^{4} b^{6} x^{10} + 5 \, a^{5} b^{5} x^{8} + 10 \, a^{6} b^{4} x^{6} + 10 \, a^{7} b^{3} x^{4} + 5 \, a^{8} b^{2} x^{2} + a^{9} b\right )} \left (-\frac {d^{6}}{a^{19} b^{5}}\right )^{\frac {1}{4}} \log \left (231 \, a^{5} b \left (-\frac {d^{6}}{a^{19} b^{5}}\right )^{\frac {1}{4}} + 231 \, \sqrt {d x} d\right ) - 1155 \, {\left (-i \, a^{4} b^{6} x^{10} - 5 i \, a^{5} b^{5} x^{8} - 10 i \, a^{6} b^{4} x^{6} - 10 i \, a^{7} b^{3} x^{4} - 5 i \, a^{8} b^{2} x^{2} - i \, a^{9} b\right )} \left (-\frac {d^{6}}{a^{19} b^{5}}\right )^{\frac {1}{4}} \log \left (231 i \, a^{5} b \left (-\frac {d^{6}}{a^{19} b^{5}}\right )^{\frac {1}{4}} + 231 \, \sqrt {d x} d\right ) - 1155 \, {\left (i \, a^{4} b^{6} x^{10} + 5 i \, a^{5} b^{5} x^{8} + 10 i \, a^{6} b^{4} x^{6} + 10 i \, a^{7} b^{3} x^{4} + 5 i \, a^{8} b^{2} x^{2} + i \, a^{9} b\right )} \left (-\frac {d^{6}}{a^{19} b^{5}}\right )^{\frac {1}{4}} \log \left (-231 i \, a^{5} b \left (-\frac {d^{6}}{a^{19} b^{5}}\right )^{\frac {1}{4}} + 231 \, \sqrt {d x} d\right ) - 1155 \, {\left (a^{4} b^{6} x^{10} + 5 \, a^{5} b^{5} x^{8} + 10 \, a^{6} b^{4} x^{6} + 10 \, a^{7} b^{3} x^{4} + 5 \, a^{8} b^{2} x^{2} + a^{9} b\right )} \left (-\frac {d^{6}}{a^{19} b^{5}}\right )^{\frac {1}{4}} \log \left (-231 \, a^{5} b \left (-\frac {d^{6}}{a^{19} b^{5}}\right )^{\frac {1}{4}} + 231 \, \sqrt {d x} d\right ) + 4 \, {\left (385 \, b^{4} d x^{8} + 1760 \, a b^{3} d x^{6} + 3130 \, a^{2} b^{2} d x^{4} + 2648 \, a^{3} b d x^{2} - 1155 \, a^{4} d\right )} \sqrt {d x}}{81920 \, {\left (a^{4} b^{6} x^{10} + 5 \, a^{5} b^{5} x^{8} + 10 \, a^{6} b^{4} x^{6} + 10 \, a^{7} b^{3} x^{4} + 5 \, a^{8} b^{2} x^{2} + a^{9} b\right )}} \] Input:
integrate((d*x)^(3/2)/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="fricas")
Output:
1/81920*(1155*(a^4*b^6*x^10 + 5*a^5*b^5*x^8 + 10*a^6*b^4*x^6 + 10*a^7*b^3* x^4 + 5*a^8*b^2*x^2 + a^9*b)*(-d^6/(a^19*b^5))^(1/4)*log(231*a^5*b*(-d^6/( a^19*b^5))^(1/4) + 231*sqrt(d*x)*d) - 1155*(-I*a^4*b^6*x^10 - 5*I*a^5*b^5* x^8 - 10*I*a^6*b^4*x^6 - 10*I*a^7*b^3*x^4 - 5*I*a^8*b^2*x^2 - I*a^9*b)*(-d ^6/(a^19*b^5))^(1/4)*log(231*I*a^5*b*(-d^6/(a^19*b^5))^(1/4) + 231*sqrt(d* x)*d) - 1155*(I*a^4*b^6*x^10 + 5*I*a^5*b^5*x^8 + 10*I*a^6*b^4*x^6 + 10*I*a ^7*b^3*x^4 + 5*I*a^8*b^2*x^2 + I*a^9*b)*(-d^6/(a^19*b^5))^(1/4)*log(-231*I *a^5*b*(-d^6/(a^19*b^5))^(1/4) + 231*sqrt(d*x)*d) - 1155*(a^4*b^6*x^10 + 5 *a^5*b^5*x^8 + 10*a^6*b^4*x^6 + 10*a^7*b^3*x^4 + 5*a^8*b^2*x^2 + a^9*b)*(- d^6/(a^19*b^5))^(1/4)*log(-231*a^5*b*(-d^6/(a^19*b^5))^(1/4) + 231*sqrt(d* x)*d) + 4*(385*b^4*d*x^8 + 1760*a*b^3*d*x^6 + 3130*a^2*b^2*d*x^4 + 2648*a^ 3*b*d*x^2 - 1155*a^4*d)*sqrt(d*x))/(a^4*b^6*x^10 + 5*a^5*b^5*x^8 + 10*a^6* b^4*x^6 + 10*a^7*b^3*x^4 + 5*a^8*b^2*x^2 + a^9*b)
\[ \int \frac {(d x)^{3/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\int \frac {\left (d x\right )^{\frac {3}{2}}}{\left (a + b x^{2}\right )^{6}}\, dx \] Input:
integrate((d*x)**(3/2)/(b**2*x**4+2*a*b*x**2+a**2)**3,x)
Output:
Integral((d*x)**(3/2)/(a + b*x**2)**6, x)
Time = 0.13 (sec) , antiderivative size = 392, normalized size of antiderivative = 1.26 \[ \int \frac {(d x)^{3/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {\frac {8 \, {\left (385 \, \left (d x\right )^{\frac {17}{2}} b^{4} d^{4} + 1760 \, \left (d x\right )^{\frac {13}{2}} a b^{3} d^{6} + 3130 \, \left (d x\right )^{\frac {9}{2}} a^{2} b^{2} d^{8} + 2648 \, \left (d x\right )^{\frac {5}{2}} a^{3} b d^{10} - 1155 \, \sqrt {d x} a^{4} d^{12}\right )}}{a^{4} b^{6} d^{10} x^{10} + 5 \, a^{5} b^{5} d^{10} x^{8} + 10 \, a^{6} b^{4} d^{10} x^{6} + 10 \, a^{7} b^{3} d^{10} x^{4} + 5 \, a^{8} b^{2} d^{10} x^{2} + a^{9} b d^{10}} + \frac {1155 \, {\left (\frac {\sqrt {2} d^{4} \log \left (\sqrt {b} d x + \sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} \sqrt {d x} b^{\frac {1}{4}} + \sqrt {a} d\right )}{\left (a d^{2}\right )^{\frac {3}{4}} b^{\frac {1}{4}}} - \frac {\sqrt {2} d^{4} \log \left (\sqrt {b} d x - \sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} \sqrt {d x} b^{\frac {1}{4}} + \sqrt {a} d\right )}{\left (a d^{2}\right )^{\frac {3}{4}} b^{\frac {1}{4}}} + \frac {2 \, \sqrt {2} d^{3} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {d x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} d}}\right )}{\sqrt {\sqrt {a} \sqrt {b} d} \sqrt {a}} + \frac {2 \, \sqrt {2} d^{3} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {d x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} d}}\right )}{\sqrt {\sqrt {a} \sqrt {b} d} \sqrt {a}}\right )}}{a^{4} b}}{163840 \, d} \] Input:
integrate((d*x)^(3/2)/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="maxima")
Output:
1/163840*(8*(385*(d*x)^(17/2)*b^4*d^4 + 1760*(d*x)^(13/2)*a*b^3*d^6 + 3130 *(d*x)^(9/2)*a^2*b^2*d^8 + 2648*(d*x)^(5/2)*a^3*b*d^10 - 1155*sqrt(d*x)*a^ 4*d^12)/(a^4*b^6*d^10*x^10 + 5*a^5*b^5*d^10*x^8 + 10*a^6*b^4*d^10*x^6 + 10 *a^7*b^3*d^10*x^4 + 5*a^8*b^2*d^10*x^2 + a^9*b*d^10) + 1155*(sqrt(2)*d^4*l og(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^4*log(sqrt(b)*d*x - sqrt(2)*(a*d^2)^(1/4)* sqrt(d*x)*b^(1/4) + sqrt(a)*d)/((a*d^2)^(3/4)*b^(1/4)) + 2*sqrt(2)*d^3*arc tan(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)) + 2*sqrt(2)*d^3*arc tan(-1/2*sqrt(2)*(sqrt(2)*(a*d^2)^(1/4)*b^(1/4) - 2*sqrt(d*x)*sqrt(b))/sqr t(sqrt(a)*sqrt(b)*d))/(sqrt(sqrt(a)*sqrt(b)*d)*sqrt(a)))/(a^4*b))/d
Time = 0.16 (sec) , antiderivative size = 354, normalized size of antiderivative = 1.13 \[ \int \frac {(d x)^{3/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {\frac {2310 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} d^{2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{a^{5} b^{2}} + \frac {2310 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} d^{2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{a^{5} b^{2}} + \frac {1155 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} d^{2} \log \left (d x + \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{a^{5} b^{2}} - \frac {1155 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} d^{2} \log \left (d x - \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{a^{5} b^{2}} + \frac {8 \, {\left (385 \, \sqrt {d x} b^{4} d^{12} x^{8} + 1760 \, \sqrt {d x} a b^{3} d^{12} x^{6} + 3130 \, \sqrt {d x} a^{2} b^{2} d^{12} x^{4} + 2648 \, \sqrt {d x} a^{3} b d^{12} x^{2} - 1155 \, \sqrt {d x} a^{4} d^{12}\right )}}{{\left (b d^{2} x^{2} + a d^{2}\right )}^{5} a^{4} b}}{163840 \, d} \] Input:
integrate((d*x)^(3/2)/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="giac")
Output:
1/163840*(2310*sqrt(2)*(a*b^3*d^2)^(1/4)*d^2*arctan(1/2*sqrt(2)*(sqrt(2)*( a*d^2/b)^(1/4) + 2*sqrt(d*x))/(a*d^2/b)^(1/4))/(a^5*b^2) + 2310*sqrt(2)*(a *b^3*d^2)^(1/4)*d^2*arctan(-1/2*sqrt(2)*(sqrt(2)*(a*d^2/b)^(1/4) - 2*sqrt( d*x))/(a*d^2/b)^(1/4))/(a^5*b^2) + 1155*sqrt(2)*(a*b^3*d^2)^(1/4)*d^2*log( d*x + sqrt(2)*(a*d^2/b)^(1/4)*sqrt(d*x) + sqrt(a*d^2/b))/(a^5*b^2) - 1155* sqrt(2)*(a*b^3*d^2)^(1/4)*d^2*log(d*x - sqrt(2)*(a*d^2/b)^(1/4)*sqrt(d*x) + sqrt(a*d^2/b))/(a^5*b^2) + 8*(385*sqrt(d*x)*b^4*d^12*x^8 + 1760*sqrt(d*x )*a*b^3*d^12*x^6 + 3130*sqrt(d*x)*a^2*b^2*d^12*x^4 + 2648*sqrt(d*x)*a^3*b* d^12*x^2 - 1155*sqrt(d*x)*a^4*d^12)/((b*d^2*x^2 + a*d^2)^5*a^4*b))/d
Time = 0.14 (sec) , antiderivative size = 209, normalized size of antiderivative = 0.67 \[ \int \frac {(d x)^{3/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {\frac {331\,d^9\,{\left (d\,x\right )}^{5/2}}{2560\,a}-\frac {231\,d^{11}\,\sqrt {d\,x}}{4096\,b}+\frac {11\,b^2\,d^5\,{\left (d\,x\right )}^{13/2}}{128\,a^3}+\frac {77\,b^3\,d^3\,{\left (d\,x\right )}^{17/2}}{4096\,a^4}+\frac {313\,b\,d^7\,{\left (d\,x\right )}^{9/2}}{2048\,a^2}}{a^5\,d^{10}+5\,a^4\,b\,d^{10}\,x^2+10\,a^3\,b^2\,d^{10}\,x^4+10\,a^2\,b^3\,d^{10}\,x^6+5\,a\,b^4\,d^{10}\,x^8+b^5\,d^{10}\,x^{10}}-\frac {231\,d^{3/2}\,\mathrm {atan}\left (\frac {b^{1/4}\,\sqrt {d\,x}}{{\left (-a\right )}^{1/4}\,\sqrt {d}}\right )}{8192\,{\left (-a\right )}^{19/4}\,b^{5/4}}-\frac {231\,d^{3/2}\,\mathrm {atanh}\left (\frac {b^{1/4}\,\sqrt {d\,x}}{{\left (-a\right )}^{1/4}\,\sqrt {d}}\right )}{8192\,{\left (-a\right )}^{19/4}\,b^{5/4}} \] Input:
int((d*x)^(3/2)/(a^2 + b^2*x^4 + 2*a*b*x^2)^3,x)
Output:
((331*d^9*(d*x)^(5/2))/(2560*a) - (231*d^11*(d*x)^(1/2))/(4096*b) + (11*b^ 2*d^5*(d*x)^(13/2))/(128*a^3) + (77*b^3*d^3*(d*x)^(17/2))/(4096*a^4) + (31 3*b*d^7*(d*x)^(9/2))/(2048*a^2))/(a^5*d^10 + b^5*d^10*x^10 + 5*a^4*b*d^10* x^2 + 5*a*b^4*d^10*x^8 + 10*a^3*b^2*d^10*x^4 + 10*a^2*b^3*d^10*x^6) - (231 *d^(3/2)*atan((b^(1/4)*(d*x)^(1/2))/((-a)^(1/4)*d^(1/2))))/(8192*(-a)^(19/ 4)*b^(5/4)) - (231*d^(3/2)*atanh((b^(1/4)*(d*x)^(1/2))/((-a)^(1/4)*d^(1/2) )))/(8192*(-a)^(19/4)*b^(5/4))
Time = 0.19 (sec) , antiderivative size = 993, normalized size of antiderivative = 3.18 \[ \int \frac {(d x)^{3/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx =\text {Too large to display} \] Input:
int((d*x)^(3/2)/(b^2*x^4+2*a*b*x^2+a^2)^3,x)
Output:
(sqrt(d)*d*( - 2310*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 - 11550*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 - 23100*b**(3/4)*a**(1/4)*sqrt(2)*a tan((b**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqr t(2)))*a**3*b**2*x**4 - 23100*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 - 11550*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 - 2310*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)))*b**5*x**10 + 2310*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 + 11550*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 + 23100*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 + 23100*b**(3/4)*a**(1/4)*sq rt(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 + 11550*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 + 2310*b**(3/4)*a**(1/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(...