Integrand size = 28, antiderivative size = 311 \[ \int \frac {(d x)^{15/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=-\frac {d (d x)^{13/2}}{10 b \left (a+b x^2\right )^5}-\frac {13 d^3 (d x)^{9/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac {39 d^5 (d x)^{5/2}}{640 b^3 \left (a+b x^2\right )^3}-\frac {39 d^7 \sqrt {d x}}{1024 b^4 \left (a+b x^2\right )^2}+\frac {39 d^7 \sqrt {d x}}{4096 a b^4 \left (a+b x^2\right )}-\frac {117 d^{15/2} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} a^{7/4} b^{17/4}}+\frac {117 d^{15/2} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} a^{7/4} b^{17/4}}+\frac {117 d^{15/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}}{\sqrt {d} \left (\sqrt {a}+\sqrt {b} x\right )}\right )}{8192 \sqrt {2} a^{7/4} b^{17/4}} \] Output:
-1/10*d*(d*x)^(13/2)/b/(b*x^2+a)^5-13/160*d^3*(d*x)^(9/2)/b^2/(b*x^2+a)^4- 39/640*d^5*(d*x)^(5/2)/b^3/(b*x^2+a)^3-39/1024*d^7*(d*x)^(1/2)/b^4/(b*x^2+ a)^2+39/4096*d^7*(d*x)^(1/2)/a/b^4/(b*x^2+a)-117/16384*d^(15/2)*arctan(1-2 ^(1/2)*b^(1/4)*(d*x)^(1/2)/a^(1/4)/d^(1/2))*2^(1/2)/a^(7/4)/b^(17/4)+117/1 6384*d^(15/2)*arctan(1+2^(1/2)*b^(1/4)*(d*x)^(1/2)/a^(1/4)/d^(1/2))*2^(1/2 )/a^(7/4)/b^(17/4)+117/16384*d^(15/2)*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^(7/4)/b^(17/4)
Time = 0.81 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.60 \[ \int \frac {(d x)^{15/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {d^7 \sqrt {d x} \left (-\frac {4 a^{3/4} \sqrt [4]{b} \left (585 a^4+2808 a^3 b x^2+5330 a^2 b^2 x^4+4960 a b^3 x^6-195 b^4 x^8\right )}{\left (a+b x^2\right )^5}-\frac {585 \sqrt {2} \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )}{\sqrt {x}}+\frac {585 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{\sqrt {x}}\right )}{81920 a^{7/4} b^{17/4}} \] Input:
Integrate[(d*x)^(15/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]
Output:
(d^7*Sqrt[d*x]*((-4*a^(3/4)*b^(1/4)*(585*a^4 + 2808*a^3*b*x^2 + 5330*a^2*b ^2*x^4 + 4960*a*b^3*x^6 - 195*b^4*x^8))/(a + b*x^2)^5 - (585*Sqrt[2]*ArcTa n[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])])/Sqrt[x] + (585 *Sqrt[2]*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)]) /Sqrt[x]))/(81920*a^(7/4)*b^(17/4))
Time = 1.12 (sec) , antiderivative size = 445, normalized size of antiderivative = 1.43, 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, 252, 252, 252, 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.
| \(\displaystyle \int \frac {(d x)^{15/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx\) |
| \(\Big \downarrow \) 1380 |
| \(\displaystyle b^6 \int \frac {(d x)^{15/2}}{b^6 \left (b x^2+a\right )^6}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \frac {(d x)^{15/2}}{\left (a+b x^2\right )^6}dx\) |
| \(\Big \downarrow \) 252 |
| \(\displaystyle \frac {13 d^2 \int \frac {(d x)^{11/2}}{\left (b x^2+a\right )^5}dx}{20 b}-\frac {d (d x)^{13/2}}{10 b \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 252 |
| \(\displaystyle \frac {13 d^2 \left (\frac {9 d^2 \int \frac {(d x)^{7/2}}{\left (b x^2+a\right )^4}dx}{16 b}-\frac {d (d x)^{9/2}}{8 b \left (a+b x^2\right )^4}\right )}{20 b}-\frac {d (d x)^{13/2}}{10 b \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 252 |
| \(\displaystyle \frac {13 d^2 \left (\frac {9 d^2 \left (\frac {5 d^2 \int \frac {(d x)^{3/2}}{\left (b x^2+a\right )^3}dx}{12 b}-\frac {d (d x)^{5/2}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{9/2}}{8 b \left (a+b x^2\right )^4}\right )}{20 b}-\frac {d (d x)^{13/2}}{10 b \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 252 |
| \(\displaystyle \frac {13 d^2 \left (\frac {9 d^2 \left (\frac {5 d^2 \left (\frac {d^2 \int \frac {1}{\sqrt {d x} \left (b x^2+a\right )^2}dx}{8 b}-\frac {d \sqrt {d x}}{4 b \left (a+b x^2\right )^2}\right )}{12 b}-\frac {d (d x)^{5/2}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{9/2}}{8 b \left (a+b x^2\right )^4}\right )}{20 b}-\frac {d (d x)^{13/2}}{10 b \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 253 |
| \(\displaystyle \frac {13 d^2 \left (\frac {9 d^2 \left (\frac {5 d^2 \left (\frac {d^2 \left (\frac {3 \int \frac {1}{\sqrt {d x} \left (b x^2+a\right )}dx}{4 a}+\frac {\sqrt {d x}}{2 a d \left (a+b x^2\right )}\right )}{8 b}-\frac {d \sqrt {d x}}{4 b \left (a+b x^2\right )^2}\right )}{12 b}-\frac {d (d x)^{5/2}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{9/2}}{8 b \left (a+b x^2\right )^4}\right )}{20 b}-\frac {d (d x)^{13/2}}{10 b \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {13 d^2 \left (\frac {9 d^2 \left (\frac {5 d^2 \left (\frac {d^2 \left (\frac {3 \int \frac {1}{b x^2+a}d\sqrt {d x}}{2 a d}+\frac {\sqrt {d x}}{2 a d \left (a+b x^2\right )}\right )}{8 b}-\frac {d \sqrt {d x}}{4 b \left (a+b x^2\right )^2}\right )}{12 b}-\frac {d (d x)^{5/2}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{9/2}}{8 b \left (a+b x^2\right )^4}\right )}{20 b}-\frac {d (d x)^{13/2}}{10 b \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 755 |
| \(\displaystyle \frac {13 d^2 \left (\frac {9 d^2 \left (\frac {5 d^2 \left (\frac {d^2 \left (\frac {3 \left (\frac {\int \frac {d^2 \left (\sqrt {a} d-\sqrt {b} d x\right )}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {a} d}+\frac {\int \frac {d^2 \left (\sqrt {b} x d+\sqrt {a} d\right )}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {a} d}\right )}{2 a d}+\frac {\sqrt {d x}}{2 a d \left (a+b x^2\right )}\right )}{8 b}-\frac {d \sqrt {d x}}{4 b \left (a+b x^2\right )^2}\right )}{12 b}-\frac {d (d x)^{5/2}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{9/2}}{8 b \left (a+b x^2\right )^4}\right )}{20 b}-\frac {d (d x)^{13/2}}{10 b \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {13 d^2 \left (\frac {9 d^2 \left (\frac {5 d^2 \left (\frac {d^2 \left (\frac {3 \left (\frac {d \int \frac {\sqrt {a} d-\sqrt {b} d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {a}}+\frac {d \int \frac {\sqrt {b} x d+\sqrt {a} d}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {a}}\right )}{2 a d}+\frac {\sqrt {d x}}{2 a d \left (a+b x^2\right )}\right )}{8 b}-\frac {d \sqrt {d x}}{4 b \left (a+b x^2\right )^2}\right )}{12 b}-\frac {d (d x)^{5/2}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{9/2}}{8 b \left (a+b x^2\right )^4}\right )}{20 b}-\frac {d (d x)^{13/2}}{10 b \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {13 d^2 \left (\frac {9 d^2 \left (\frac {5 d^2 \left (\frac {d^2 \left (\frac {3 \left (\frac {d \int \frac {\sqrt {a} d-\sqrt {b} d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {a}}+\frac {d \left (\frac {\int \frac {1}{x d+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}}d\sqrt {d x}}{2 \sqrt {b}}+\frac {\int \frac {1}{x d+\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}}d\sqrt {d x}}{2 \sqrt {b}}\right )}{2 \sqrt {a}}\right )}{2 a d}+\frac {\sqrt {d x}}{2 a d \left (a+b x^2\right )}\right )}{8 b}-\frac {d \sqrt {d x}}{4 b \left (a+b x^2\right )^2}\right )}{12 b}-\frac {d (d x)^{5/2}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{9/2}}{8 b \left (a+b x^2\right )^4}\right )}{20 b}-\frac {d (d x)^{13/2}}{10 b \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {13 d^2 \left (\frac {9 d^2 \left (\frac {5 d^2 \left (\frac {d^2 \left (\frac {3 \left (\frac {d \int \frac {\sqrt {a} d-\sqrt {b} d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {a}}+\frac {d \left (\frac {\int \frac {1}{-d x-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\int \frac {1}{-d x-1}d\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}\right )}{2 a d}+\frac {\sqrt {d x}}{2 a d \left (a+b x^2\right )}\right )}{8 b}-\frac {d \sqrt {d x}}{4 b \left (a+b x^2\right )^2}\right )}{12 b}-\frac {d (d x)^{5/2}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{9/2}}{8 b \left (a+b x^2\right )^4}\right )}{20 b}-\frac {d (d x)^{13/2}}{10 b \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {13 d^2 \left (\frac {9 d^2 \left (\frac {5 d^2 \left (\frac {d^2 \left (\frac {3 \left (\frac {d \int \frac {\sqrt {a} d-\sqrt {b} d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {a}}+\frac {d \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}\right )}{2 a d}+\frac {\sqrt {d x}}{2 a d \left (a+b x^2\right )}\right )}{8 b}-\frac {d \sqrt {d x}}{4 b \left (a+b x^2\right )^2}\right )}{12 b}-\frac {d (d x)^{5/2}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{9/2}}{8 b \left (a+b x^2\right )^4}\right )}{20 b}-\frac {d (d x)^{13/2}}{10 b \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {13 d^2 \left (\frac {9 d^2 \left (\frac {5 d^2 \left (\frac {d^2 \left (\frac {3 \left (\frac {d \left (-\frac {\int -\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d}-2 \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{b} \left (x d+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt [4]{a} \sqrt {d}+\sqrt {2} \sqrt [4]{b} \sqrt {d x}\right )}{\sqrt [4]{b} \left (x d+\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}+\frac {d \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}\right )}{2 a d}+\frac {\sqrt {d x}}{2 a d \left (a+b x^2\right )}\right )}{8 b}-\frac {d \sqrt {d x}}{4 b \left (a+b x^2\right )^2}\right )}{12 b}-\frac {d (d x)^{5/2}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{9/2}}{8 b \left (a+b x^2\right )^4}\right )}{20 b}-\frac {d (d x)^{13/2}}{10 b \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {13 d^2 \left (\frac {9 d^2 \left (\frac {5 d^2 \left (\frac {d^2 \left (\frac {3 \left (\frac {d \left (\frac {\int \frac {\sqrt {2} \sqrt [4]{a} \sqrt {d}-2 \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{b} \left (x d+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}+\frac {\int \frac {\sqrt {2} \left (\sqrt [4]{a} \sqrt {d}+\sqrt {2} \sqrt [4]{b} \sqrt {d x}\right )}{\sqrt [4]{b} \left (x d+\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}+\frac {d \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}\right )}{2 a d}+\frac {\sqrt {d x}}{2 a d \left (a+b x^2\right )}\right )}{8 b}-\frac {d \sqrt {d x}}{4 b \left (a+b x^2\right )^2}\right )}{12 b}-\frac {d (d x)^{5/2}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{9/2}}{8 b \left (a+b x^2\right )^4}\right )}{20 b}-\frac {d (d x)^{13/2}}{10 b \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {13 d^2 \left (\frac {9 d^2 \left (\frac {5 d^2 \left (\frac {d^2 \left (\frac {3 \left (\frac {d \left (\frac {\int \frac {\sqrt {2} \sqrt [4]{a} \sqrt {d}-2 \sqrt [4]{b} \sqrt {d x}}{x d+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt {b} \sqrt {d}}+\frac {\int \frac {\sqrt [4]{a} \sqrt {d}+\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{x d+\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}}d\sqrt {d x}}{2 \sqrt [4]{a} \sqrt {b} \sqrt {d}}\right )}{2 \sqrt {a}}+\frac {d \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}\right )}{2 a d}+\frac {\sqrt {d x}}{2 a d \left (a+b x^2\right )}\right )}{8 b}-\frac {d \sqrt {d x}}{4 b \left (a+b x^2\right )^2}\right )}{12 b}-\frac {d (d x)^{5/2}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{9/2}}{8 b \left (a+b x^2\right )^4}\right )}{20 b}-\frac {d (d x)^{13/2}}{10 b \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {13 d^2 \left (\frac {9 d^2 \left (\frac {5 d^2 \left (\frac {d^2 \left (\frac {3 \left (\frac {d \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}+\frac {d \left (\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d} \sqrt {d x}+\sqrt {a} d+\sqrt {b} d x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d} \sqrt {d x}+\sqrt {a} d+\sqrt {b} d x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}\right )}{2 a d}+\frac {\sqrt {d x}}{2 a d \left (a+b x^2\right )}\right )}{8 b}-\frac {d \sqrt {d x}}{4 b \left (a+b x^2\right )^2}\right )}{12 b}-\frac {d (d x)^{5/2}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{9/2}}{8 b \left (a+b x^2\right )^4}\right )}{20 b}-\frac {d (d x)^{13/2}}{10 b \left (a+b x^2\right )^5}\) |
Input:
Int[(d*x)^(15/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]
Output:
-1/10*(d*(d*x)^(13/2))/(b*(a + b*x^2)^5) + (13*d^2*(-1/8*(d*(d*x)^(9/2))/( b*(a + b*x^2)^4) + (9*d^2*(-1/6*(d*(d*x)^(5/2))/(b*(a + b*x^2)^3) + (5*d^2 *(-1/4*(d*Sqrt[d*x])/(b*(a + b*x^2)^2) + (d^2*(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*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*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 = 14.64 (sec) , antiderivative size = 238, normalized size of antiderivative = 0.77
| method | result | size |
| derivativedivides | \(2 d^{11} \left (\frac {-\frac {117 a^{3} d^{6} \sqrt {d x}}{8192 b^{4}}-\frac {351 a^{2} d^{4} \left (d x \right )^{\frac {5}{2}}}{5120 b^{3}}-\frac {533 a \,d^{2} \left (d x \right )^{\frac {9}{2}}}{4096 b^{2}}-\frac {31 \left (d x \right )^{\frac {13}{2}}}{256 b}+\frac {39 \left (d x \right )^{\frac {17}{2}}}{8192 a \,d^{2}}}{\left (b \,d^{2} x^{2}+a \,d^{2}\right )^{5}}+\frac {117 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{65536 a^{2} d^{4} b^{4}}\right )\) | \(238\) |
| default | \(2 d^{11} \left (\frac {-\frac {117 a^{3} d^{6} \sqrt {d x}}{8192 b^{4}}-\frac {351 a^{2} d^{4} \left (d x \right )^{\frac {5}{2}}}{5120 b^{3}}-\frac {533 a \,d^{2} \left (d x \right )^{\frac {9}{2}}}{4096 b^{2}}-\frac {31 \left (d x \right )^{\frac {13}{2}}}{256 b}+\frac {39 \left (d x \right )^{\frac {17}{2}}}{8192 a \,d^{2}}}{\left (b \,d^{2} x^{2}+a \,d^{2}\right )^{5}}+\frac {117 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{65536 a^{2} d^{4} b^{4}}\right )\) | \(238\) |
| pseudoelliptic | \(-\frac {\left (\left (-1560 a \,x^{8} b^{4}+39680 a^{2} x^{6} b^{3}+42640 a^{3} x^{4} b^{2}+22464 x^{2} a^{4} b +4680 a^{5}\right ) \sqrt {d x}+585 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (b \,x^{2}+a \right )^{5} \left (2 \arctan \left (\frac {-\sqrt {2}\, \sqrt {d x}+\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )-2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}+\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )-\ln \left (\frac {d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )\right )\right ) d^{7}}{163840 b^{4} a^{2} \left (b \,x^{2}+a \right )^{5}}\) | \(241\) |
Input:
int((d*x)^(15/2)/(b^2*x^4+2*a*b*x^2+a^2)^3,x,method=_RETURNVERBOSE)
Output:
2*d^11*((-117/8192*a^3*d^6/b^4*(d*x)^(1/2)-351/5120*a^2*d^4/b^3*(d*x)^(5/2 )-533/4096/b^2*a*d^2*(d*x)^(9/2)-31/256/b*(d*x)^(13/2)+39/8192/a/d^2*(d*x) ^(17/2))/(b*d^2*x^2+a*d^2)^5+117/65536/a^2/d^4/b^4*(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 = 0.17 (sec) , antiderivative size = 566, normalized size of antiderivative = 1.82 \[ \int \frac {(d x)^{15/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {585 \, {\left (a b^{9} x^{10} + 5 \, a^{2} b^{8} x^{8} + 10 \, a^{3} b^{7} x^{6} + 10 \, a^{4} b^{6} x^{4} + 5 \, a^{5} b^{5} x^{2} + a^{6} b^{4}\right )} \left (-\frac {d^{30}}{a^{7} b^{17}}\right )^{\frac {1}{4}} \log \left (117 \, \sqrt {d x} d^{7} + 117 \, \left (-\frac {d^{30}}{a^{7} b^{17}}\right )^{\frac {1}{4}} a^{2} b^{4}\right ) - 585 \, {\left (-i \, a b^{9} x^{10} - 5 i \, a^{2} b^{8} x^{8} - 10 i \, a^{3} b^{7} x^{6} - 10 i \, a^{4} b^{6} x^{4} - 5 i \, a^{5} b^{5} x^{2} - i \, a^{6} b^{4}\right )} \left (-\frac {d^{30}}{a^{7} b^{17}}\right )^{\frac {1}{4}} \log \left (117 \, \sqrt {d x} d^{7} + 117 i \, \left (-\frac {d^{30}}{a^{7} b^{17}}\right )^{\frac {1}{4}} a^{2} b^{4}\right ) - 585 \, {\left (i \, a b^{9} x^{10} + 5 i \, a^{2} b^{8} x^{8} + 10 i \, a^{3} b^{7} x^{6} + 10 i \, a^{4} b^{6} x^{4} + 5 i \, a^{5} b^{5} x^{2} + i \, a^{6} b^{4}\right )} \left (-\frac {d^{30}}{a^{7} b^{17}}\right )^{\frac {1}{4}} \log \left (117 \, \sqrt {d x} d^{7} - 117 i \, \left (-\frac {d^{30}}{a^{7} b^{17}}\right )^{\frac {1}{4}} a^{2} b^{4}\right ) - 585 \, {\left (a b^{9} x^{10} + 5 \, a^{2} b^{8} x^{8} + 10 \, a^{3} b^{7} x^{6} + 10 \, a^{4} b^{6} x^{4} + 5 \, a^{5} b^{5} x^{2} + a^{6} b^{4}\right )} \left (-\frac {d^{30}}{a^{7} b^{17}}\right )^{\frac {1}{4}} \log \left (117 \, \sqrt {d x} d^{7} - 117 \, \left (-\frac {d^{30}}{a^{7} b^{17}}\right )^{\frac {1}{4}} a^{2} b^{4}\right ) + 4 \, {\left (195 \, b^{4} d^{7} x^{8} - 4960 \, a b^{3} d^{7} x^{6} - 5330 \, a^{2} b^{2} d^{7} x^{4} - 2808 \, a^{3} b d^{7} x^{2} - 585 \, a^{4} d^{7}\right )} \sqrt {d x}}{81920 \, {\left (a b^{9} x^{10} + 5 \, a^{2} b^{8} x^{8} + 10 \, a^{3} b^{7} x^{6} + 10 \, a^{4} b^{6} x^{4} + 5 \, a^{5} b^{5} x^{2} + a^{6} b^{4}\right )}} \] Input:
integrate((d*x)^(15/2)/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="fricas")
Output:
1/81920*(585*(a*b^9*x^10 + 5*a^2*b^8*x^8 + 10*a^3*b^7*x^6 + 10*a^4*b^6*x^4 + 5*a^5*b^5*x^2 + a^6*b^4)*(-d^30/(a^7*b^17))^(1/4)*log(117*sqrt(d*x)*d^7 + 117*(-d^30/(a^7*b^17))^(1/4)*a^2*b^4) - 585*(-I*a*b^9*x^10 - 5*I*a^2*b^ 8*x^8 - 10*I*a^3*b^7*x^6 - 10*I*a^4*b^6*x^4 - 5*I*a^5*b^5*x^2 - I*a^6*b^4) *(-d^30/(a^7*b^17))^(1/4)*log(117*sqrt(d*x)*d^7 + 117*I*(-d^30/(a^7*b^17)) ^(1/4)*a^2*b^4) - 585*(I*a*b^9*x^10 + 5*I*a^2*b^8*x^8 + 10*I*a^3*b^7*x^6 + 10*I*a^4*b^6*x^4 + 5*I*a^5*b^5*x^2 + I*a^6*b^4)*(-d^30/(a^7*b^17))^(1/4)* log(117*sqrt(d*x)*d^7 - 117*I*(-d^30/(a^7*b^17))^(1/4)*a^2*b^4) - 585*(a*b ^9*x^10 + 5*a^2*b^8*x^8 + 10*a^3*b^7*x^6 + 10*a^4*b^6*x^4 + 5*a^5*b^5*x^2 + a^6*b^4)*(-d^30/(a^7*b^17))^(1/4)*log(117*sqrt(d*x)*d^7 - 117*(-d^30/(a^ 7*b^17))^(1/4)*a^2*b^4) + 4*(195*b^4*d^7*x^8 - 4960*a*b^3*d^7*x^6 - 5330*a ^2*b^2*d^7*x^4 - 2808*a^3*b*d^7*x^2 - 585*a^4*d^7)*sqrt(d*x))/(a*b^9*x^10 + 5*a^2*b^8*x^8 + 10*a^3*b^7*x^6 + 10*a^4*b^6*x^4 + 5*a^5*b^5*x^2 + a^6*b^ 4)
Timed out. \[ \int \frac {(d x)^{15/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\text {Timed out} \] Input:
integrate((d*x)**(15/2)/(b**2*x**4+2*a*b*x**2+a**2)**3,x)
Output:
Timed out
Time = 0.13 (sec) , antiderivative size = 392, normalized size of antiderivative = 1.26 \[ \int \frac {(d x)^{15/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {\frac {8 \, {\left (195 \, \left (d x\right )^{\frac {17}{2}} b^{4} d^{10} - 4960 \, \left (d x\right )^{\frac {13}{2}} a b^{3} d^{12} - 5330 \, \left (d x\right )^{\frac {9}{2}} a^{2} b^{2} d^{14} - 2808 \, \left (d x\right )^{\frac {5}{2}} a^{3} b d^{16} - 585 \, \sqrt {d x} a^{4} d^{18}\right )}}{a b^{9} d^{10} x^{10} + 5 \, a^{2} b^{8} d^{10} x^{8} + 10 \, a^{3} b^{7} d^{10} x^{6} + 10 \, a^{4} b^{6} d^{10} x^{4} + 5 \, a^{5} b^{5} d^{10} x^{2} + a^{6} b^{4} d^{10}} + \frac {585 \, {\left (\frac {\sqrt {2} d^{10} \log \left (\sqrt {b} d x + \sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} \sqrt {d x} b^{\frac {1}{4}} + \sqrt {a} d\right )}{\left (a d^{2}\right )^{\frac {3}{4}} b^{\frac {1}{4}}} - \frac {\sqrt {2} d^{10} \log \left (\sqrt {b} d x - \sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} \sqrt {d x} b^{\frac {1}{4}} + \sqrt {a} d\right )}{\left (a d^{2}\right )^{\frac {3}{4}} b^{\frac {1}{4}}} + \frac {2 \, \sqrt {2} d^{9} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {d x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} d}}\right )}{\sqrt {\sqrt {a} \sqrt {b} d} \sqrt {a}} + \frac {2 \, \sqrt {2} d^{9} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {d x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} d}}\right )}{\sqrt {\sqrt {a} \sqrt {b} d} \sqrt {a}}\right )}}{a b^{4}}}{163840 \, d} \] Input:
integrate((d*x)^(15/2)/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="maxima")
Output:
1/163840*(8*(195*(d*x)^(17/2)*b^4*d^10 - 4960*(d*x)^(13/2)*a*b^3*d^12 - 53 30*(d*x)^(9/2)*a^2*b^2*d^14 - 2808*(d*x)^(5/2)*a^3*b*d^16 - 585*sqrt(d*x)* a^4*d^18)/(a*b^9*d^10*x^10 + 5*a^2*b^8*d^10*x^8 + 10*a^3*b^7*d^10*x^6 + 10 *a^4*b^6*d^10*x^4 + 5*a^5*b^5*d^10*x^2 + a^6*b^4*d^10) + 585*(sqrt(2)*d^10 *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^10*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^9* arctan(1/2*sqrt(2)*(sqrt(2)*(a*d^2)^(1/4)*b^(1/4) + 2*sqrt(d*x)*sqrt(b))/s qrt(sqrt(a)*sqrt(b)*d))/(sqrt(sqrt(a)*sqrt(b)*d)*sqrt(a)) + 2*sqrt(2)*d^9* 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^4))/d
Time = 0.12 (sec) , antiderivative size = 354, normalized size of antiderivative = 1.14 \[ \int \frac {(d x)^{15/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {\frac {1170 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} d^{8} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{a^{2} b^{5}} + \frac {1170 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} d^{8} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{a^{2} b^{5}} + \frac {585 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} d^{8} \log \left (d x + \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{a^{2} b^{5}} - \frac {585 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} d^{8} \log \left (d x - \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{a^{2} b^{5}} + \frac {8 \, {\left (195 \, \sqrt {d x} b^{4} d^{18} x^{8} - 4960 \, \sqrt {d x} a b^{3} d^{18} x^{6} - 5330 \, \sqrt {d x} a^{2} b^{2} d^{18} x^{4} - 2808 \, \sqrt {d x} a^{3} b d^{18} x^{2} - 585 \, \sqrt {d x} a^{4} d^{18}\right )}}{{\left (b d^{2} x^{2} + a d^{2}\right )}^{5} a b^{4}}}{163840 \, d} \] Input:
integrate((d*x)^(15/2)/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="giac")
Output:
1/163840*(1170*sqrt(2)*(a*b^3*d^2)^(1/4)*d^8*arctan(1/2*sqrt(2)*(sqrt(2)*( a*d^2/b)^(1/4) + 2*sqrt(d*x))/(a*d^2/b)^(1/4))/(a^2*b^5) + 1170*sqrt(2)*(a *b^3*d^2)^(1/4)*d^8*arctan(-1/2*sqrt(2)*(sqrt(2)*(a*d^2/b)^(1/4) - 2*sqrt( d*x))/(a*d^2/b)^(1/4))/(a^2*b^5) + 585*sqrt(2)*(a*b^3*d^2)^(1/4)*d^8*log(d *x + sqrt(2)*(a*d^2/b)^(1/4)*sqrt(d*x) + sqrt(a*d^2/b))/(a^2*b^5) - 585*sq rt(2)*(a*b^3*d^2)^(1/4)*d^8*log(d*x - sqrt(2)*(a*d^2/b)^(1/4)*sqrt(d*x) + sqrt(a*d^2/b))/(a^2*b^5) + 8*(195*sqrt(d*x)*b^4*d^18*x^8 - 4960*sqrt(d*x)* a*b^3*d^18*x^6 - 5330*sqrt(d*x)*a^2*b^2*d^18*x^4 - 2808*sqrt(d*x)*a^3*b*d^ 18*x^2 - 585*sqrt(d*x)*a^4*d^18)/((b*d^2*x^2 + a*d^2)^5*a*b^4))/d
Time = 0.13 (sec) , antiderivative size = 210, normalized size of antiderivative = 0.68 \[ \int \frac {(d x)^{15/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {117\,d^{15/2}\,\mathrm {atan}\left (\frac {b^{1/4}\,\sqrt {d\,x}}{{\left (-a\right )}^{1/4}\,\sqrt {d}}\right )}{8192\,{\left (-a\right )}^{7/4}\,b^{17/4}}-\frac {\frac {31\,d^{11}\,{\left (d\,x\right )}^{13/2}}{128\,b}-\frac {39\,d^9\,{\left (d\,x\right )}^{17/2}}{4096\,a}+\frac {351\,a^2\,d^{15}\,{\left (d\,x\right )}^{5/2}}{2560\,b^3}+\frac {117\,a^3\,d^{17}\,\sqrt {d\,x}}{4096\,b^4}+\frac {533\,a\,d^{13}\,{\left (d\,x\right )}^{9/2}}{2048\,b^2}}{a^5\,d^{10}+5\,a^4\,b\,d^{10}\,x^2+10\,a^3\,b^2\,d^{10}\,x^4+10\,a^2\,b^3\,d^{10}\,x^6+5\,a\,b^4\,d^{10}\,x^8+b^5\,d^{10}\,x^{10}}+\frac {117\,d^{15/2}\,\mathrm {atanh}\left (\frac {b^{1/4}\,\sqrt {d\,x}}{{\left (-a\right )}^{1/4}\,\sqrt {d}}\right )}{8192\,{\left (-a\right )}^{7/4}\,b^{17/4}} \] Input:
int((d*x)^(15/2)/(a^2 + b^2*x^4 + 2*a*b*x^2)^3,x)
Output:
(117*d^(15/2)*atan((b^(1/4)*(d*x)^(1/2))/((-a)^(1/4)*d^(1/2))))/(8192*(-a) ^(7/4)*b^(17/4)) - ((31*d^11*(d*x)^(13/2))/(128*b) - (39*d^9*(d*x)^(17/2)) /(4096*a) + (351*a^2*d^15*(d*x)^(5/2))/(2560*b^3) + (117*a^3*d^17*(d*x)^(1 /2))/(4096*b^4) + (533*a*d^13*(d*x)^(9/2))/(2048*b^2))/(a^5*d^10 + b^5*d^1 0*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) + (117*d^(15/2)*atanh((b^(1/4)*(d*x)^(1/2))/((-a)^(1/4)*d^ (1/2))))/(8192*(-a)^(7/4)*b^(17/4))
Time = 0.19 (sec) , antiderivative size = 995, normalized size of antiderivative = 3.20 \[ \int \frac {(d x)^{15/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx =\text {Too large to display} \] Input:
int((d*x)^(15/2)/(b^2*x^4+2*a*b*x^2+a^2)^3,x)
Output:
(sqrt(d)*d**7*( - 1170*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 - 5850*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 - 11700*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)*s qrt(2)))*a**3*b**2*x**4 - 11700*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 - 5850*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 - 1170*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 + 1170*b**(3/4)*a**(1/4)*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)))*a**5 + 5850*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 + 11700*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 + 11700*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 + 5850*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 + 1170*b**(3/4)*a**(1/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2...