Integrand size = 28, antiderivative size = 293 \[ \int \frac {1}{(d x)^{7/2} \left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=-\frac {663}{320 a^4 d (d x)^{5/2}}+\frac {663 b}{64 a^5 d^3 \sqrt {d x}}+\frac {1}{6 a d (d x)^{5/2} \left (a+b x^2\right )^3}+\frac {17}{48 a^2 d (d x)^{5/2} \left (a+b x^2\right )^2}+\frac {221}{192 a^3 d (d x)^{5/2} \left (a+b x^2\right )}-\frac {663 b^{5/4} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{128 \sqrt {2} a^{21/4} d^{7/2}}+\frac {663 b^{5/4} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{128 \sqrt {2} a^{21/4} d^{7/2}}-\frac {663 b^{5/4} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}}{\sqrt {d} \left (\sqrt {a}+\sqrt {b} x\right )}\right )}{128 \sqrt {2} a^{21/4} d^{7/2}} \] Output:
-663/320/a^4/d/(d*x)^(5/2)+663/64*b/a^5/d^3/(d*x)^(1/2)+1/6/a/d/(d*x)^(5/2 )/(b*x^2+a)^3+17/48/a^2/d/(d*x)^(5/2)/(b*x^2+a)^2+221/192/a^3/d/(d*x)^(5/2 )/(b*x^2+a)-663/256*b^(5/4)*arctan(1-2^(1/2)*b^(1/4)*(d*x)^(1/2)/a^(1/4)/d ^(1/2))*2^(1/2)/a^(21/4)/d^(7/2)+663/256*b^(5/4)*arctan(1+2^(1/2)*b^(1/4)* (d*x)^(1/2)/a^(1/4)/d^(1/2))*2^(1/2)/a^(21/4)/d^(7/2)-663/256*b^(5/4)*arct anh(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^(21/4)/d^(7/2)
Time = 0.56 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.65 \[ \int \frac {1}{(d x)^{7/2} \left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=\frac {\sqrt {d x} \left (\frac {4 \sqrt [4]{a} \left (-384 a^4+6528 a^3 b x^2+24973 a^2 b^2 x^4+27846 a b^3 x^6+9945 b^4 x^8\right )}{\left (a+b x^2\right )^3}-9945 \sqrt {2} b^{5/4} x^{5/2} \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )-9945 \sqrt {2} b^{5/4} x^{5/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )\right )}{3840 a^{21/4} d^4 x^3} \] Input:
Integrate[1/((d*x)^(7/2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^2),x]
Output:
(Sqrt[d*x]*((4*a^(1/4)*(-384*a^4 + 6528*a^3*b*x^2 + 24973*a^2*b^2*x^4 + 27 846*a*b^3*x^6 + 9945*b^4*x^8))/(a + b*x^2)^3 - 9945*Sqrt[2]*b^(5/4)*x^(5/2 )*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])] - 9945*S qrt[2]*b^(5/4)*x^(5/2)*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)]))/(3840*a^(21/4)*d^4*x^3)
Time = 1.06 (sec) , antiderivative size = 420, 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, 253, 253, 253, 264, 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.
| \(\displaystyle \int \frac {1}{(d x)^{7/2} \left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx\) |
| \(\Big \downarrow \) 1380 |
| \(\displaystyle b^4 \int \frac {1}{b^4 (d x)^{7/2} \left (b x^2+a\right )^4}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \frac {1}{(d x)^{7/2} \left (a+b x^2\right )^4}dx\) |
| \(\Big \downarrow \) 253 |
| \(\displaystyle \frac {17 \int \frac {1}{(d x)^{7/2} \left (b x^2+a\right )^3}dx}{12 a}+\frac {1}{6 a d (d x)^{5/2} \left (a+b x^2\right )^3}\) |
| \(\Big \downarrow \) 253 |
| \(\displaystyle \frac {17 \left (\frac {13 \int \frac {1}{(d x)^{7/2} \left (b x^2+a\right )^2}dx}{8 a}+\frac {1}{4 a d (d x)^{5/2} \left (a+b x^2\right )^2}\right )}{12 a}+\frac {1}{6 a d (d x)^{5/2} \left (a+b x^2\right )^3}\) |
| \(\Big \downarrow \) 253 |
| \(\displaystyle \frac {17 \left (\frac {13 \left (\frac {9 \int \frac {1}{(d x)^{7/2} \left (b x^2+a\right )}dx}{4 a}+\frac {1}{2 a d (d x)^{5/2} \left (a+b x^2\right )}\right )}{8 a}+\frac {1}{4 a d (d x)^{5/2} \left (a+b x^2\right )^2}\right )}{12 a}+\frac {1}{6 a d (d x)^{5/2} \left (a+b x^2\right )^3}\) |
| \(\Big \downarrow \) 264 |
| \(\displaystyle \frac {17 \left (\frac {13 \left (\frac {9 \left (-\frac {b \int \frac {1}{(d x)^{3/2} \left (b x^2+a\right )}dx}{a d^2}-\frac {2}{5 a d (d x)^{5/2}}\right )}{4 a}+\frac {1}{2 a d (d x)^{5/2} \left (a+b x^2\right )}\right )}{8 a}+\frac {1}{4 a d (d x)^{5/2} \left (a+b x^2\right )^2}\right )}{12 a}+\frac {1}{6 a d (d x)^{5/2} \left (a+b x^2\right )^3}\) |
| \(\Big \downarrow \) 264 |
| \(\displaystyle \frac {17 \left (\frac {13 \left (\frac {9 \left (-\frac {b \left (-\frac {b \int \frac {\sqrt {d x}}{b x^2+a}dx}{a d^2}-\frac {2}{a d \sqrt {d x}}\right )}{a d^2}-\frac {2}{5 a d (d x)^{5/2}}\right )}{4 a}+\frac {1}{2 a d (d x)^{5/2} \left (a+b x^2\right )}\right )}{8 a}+\frac {1}{4 a d (d x)^{5/2} \left (a+b x^2\right )^2}\right )}{12 a}+\frac {1}{6 a d (d x)^{5/2} \left (a+b x^2\right )^3}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {17 \left (\frac {13 \left (\frac {9 \left (-\frac {b \left (-\frac {2 b \int \frac {d^3 x}{b x^2 d^2+a d^2}d\sqrt {d x}}{a d^3}-\frac {2}{a d \sqrt {d x}}\right )}{a d^2}-\frac {2}{5 a d (d x)^{5/2}}\right )}{4 a}+\frac {1}{2 a d (d x)^{5/2} \left (a+b x^2\right )}\right )}{8 a}+\frac {1}{4 a d (d x)^{5/2} \left (a+b x^2\right )^2}\right )}{12 a}+\frac {1}{6 a d (d x)^{5/2} \left (a+b x^2\right )^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {17 \left (\frac {13 \left (\frac {9 \left (-\frac {b \left (-\frac {2 b \int \frac {d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{a d}-\frac {2}{a d \sqrt {d x}}\right )}{a d^2}-\frac {2}{5 a d (d x)^{5/2}}\right )}{4 a}+\frac {1}{2 a d (d x)^{5/2} \left (a+b x^2\right )}\right )}{8 a}+\frac {1}{4 a d (d x)^{5/2} \left (a+b x^2\right )^2}\right )}{12 a}+\frac {1}{6 a d (d x)^{5/2} \left (a+b x^2\right )^3}\) |
| \(\Big \downarrow \) 826 |
| \(\displaystyle \frac {17 \left (\frac {13 \left (\frac {9 \left (-\frac {b \left (-\frac {2 b \left (\frac {\int \frac {\sqrt {b} x d+\sqrt {a} d}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {b}}-\frac {\int \frac {\sqrt {a} d-\sqrt {b} d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {b}}\right )}{a d}-\frac {2}{a d \sqrt {d x}}\right )}{a d^2}-\frac {2}{5 a d (d x)^{5/2}}\right )}{4 a}+\frac {1}{2 a d (d x)^{5/2} \left (a+b x^2\right )}\right )}{8 a}+\frac {1}{4 a d (d x)^{5/2} \left (a+b x^2\right )^2}\right )}{12 a}+\frac {1}{6 a d (d x)^{5/2} \left (a+b x^2\right )^3}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {17 \left (\frac {13 \left (\frac {9 \left (-\frac {b \left (-\frac {2 b \left (\frac {\frac {\int \frac {1}{x d+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}}d\sqrt {d x}}{2 \sqrt {b}}+\frac {\int \frac {1}{x d+\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}}d\sqrt {d x}}{2 \sqrt {b}}}{2 \sqrt {b}}-\frac {\int \frac {\sqrt {a} d-\sqrt {b} d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {b}}\right )}{a d}-\frac {2}{a d \sqrt {d x}}\right )}{a d^2}-\frac {2}{5 a d (d x)^{5/2}}\right )}{4 a}+\frac {1}{2 a d (d x)^{5/2} \left (a+b x^2\right )}\right )}{8 a}+\frac {1}{4 a d (d x)^{5/2} \left (a+b x^2\right )^2}\right )}{12 a}+\frac {1}{6 a d (d x)^{5/2} \left (a+b x^2\right )^3}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {17 \left (\frac {13 \left (\frac {9 \left (-\frac {b \left (-\frac {2 b \left (\frac {\frac {\int \frac {1}{-d x-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\int \frac {1}{-d x-1}d\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}}{2 \sqrt {b}}-\frac {\int \frac {\sqrt {a} d-\sqrt {b} d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {b}}\right )}{a d}-\frac {2}{a d \sqrt {d x}}\right )}{a d^2}-\frac {2}{5 a d (d x)^{5/2}}\right )}{4 a}+\frac {1}{2 a d (d x)^{5/2} \left (a+b x^2\right )}\right )}{8 a}+\frac {1}{4 a d (d x)^{5/2} \left (a+b x^2\right )^2}\right )}{12 a}+\frac {1}{6 a d (d x)^{5/2} \left (a+b x^2\right )^3}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {17 \left (\frac {13 \left (\frac {9 \left (-\frac {b \left (-\frac {2 b \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}}{2 \sqrt {b}}-\frac {\int \frac {\sqrt {a} d-\sqrt {b} d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {b}}\right )}{a d}-\frac {2}{a d \sqrt {d x}}\right )}{a d^2}-\frac {2}{5 a d (d x)^{5/2}}\right )}{4 a}+\frac {1}{2 a d (d x)^{5/2} \left (a+b x^2\right )}\right )}{8 a}+\frac {1}{4 a d (d x)^{5/2} \left (a+b x^2\right )^2}\right )}{12 a}+\frac {1}{6 a d (d x)^{5/2} \left (a+b x^2\right )^3}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {17 \left (\frac {13 \left (\frac {9 \left (-\frac {b \left (-\frac {2 b \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}}{2 \sqrt {b}}-\frac {-\frac {\int -\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d}-2 \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{b} \left (x d+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt [4]{a} \sqrt {d}+\sqrt {2} \sqrt [4]{b} \sqrt {d x}\right )}{\sqrt [4]{b} \left (x d+\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}}{2 \sqrt {b}}\right )}{a d}-\frac {2}{a d \sqrt {d x}}\right )}{a d^2}-\frac {2}{5 a d (d x)^{5/2}}\right )}{4 a}+\frac {1}{2 a d (d x)^{5/2} \left (a+b x^2\right )}\right )}{8 a}+\frac {1}{4 a d (d x)^{5/2} \left (a+b x^2\right )^2}\right )}{12 a}+\frac {1}{6 a d (d x)^{5/2} \left (a+b x^2\right )^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {17 \left (\frac {13 \left (\frac {9 \left (-\frac {b \left (-\frac {2 b \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}}{2 \sqrt {b}}-\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{a} \sqrt {d}-2 \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{b} \left (x d+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}+\frac {\int \frac {\sqrt {2} \left (\sqrt [4]{a} \sqrt {d}+\sqrt {2} \sqrt [4]{b} \sqrt {d x}\right )}{\sqrt [4]{b} \left (x d+\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}}{2 \sqrt {b}}\right )}{a d}-\frac {2}{a d \sqrt {d x}}\right )}{a d^2}-\frac {2}{5 a d (d x)^{5/2}}\right )}{4 a}+\frac {1}{2 a d (d x)^{5/2} \left (a+b x^2\right )}\right )}{8 a}+\frac {1}{4 a d (d x)^{5/2} \left (a+b x^2\right )^2}\right )}{12 a}+\frac {1}{6 a d (d x)^{5/2} \left (a+b x^2\right )^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {17 \left (\frac {13 \left (\frac {9 \left (-\frac {b \left (-\frac {2 b \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}}{2 \sqrt {b}}-\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{a} \sqrt {d}-2 \sqrt [4]{b} \sqrt {d x}}{x d+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt {b} \sqrt {d}}+\frac {\int \frac {\sqrt [4]{a} \sqrt {d}+\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{x d+\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}}d\sqrt {d x}}{2 \sqrt [4]{a} \sqrt {b} \sqrt {d}}}{2 \sqrt {b}}\right )}{a d}-\frac {2}{a d \sqrt {d x}}\right )}{a d^2}-\frac {2}{5 a d (d x)^{5/2}}\right )}{4 a}+\frac {1}{2 a d (d x)^{5/2} \left (a+b x^2\right )}\right )}{8 a}+\frac {1}{4 a d (d x)^{5/2} \left (a+b x^2\right )^2}\right )}{12 a}+\frac {1}{6 a d (d x)^{5/2} \left (a+b x^2\right )^3}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {17 \left (\frac {13 \left (\frac {9 \left (-\frac {b \left (-\frac {2 b \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}}{2 \sqrt {b}}-\frac {\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d} \sqrt {d x}+\sqrt {a} d+\sqrt {b} d x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d} \sqrt {d x}+\sqrt {a} d+\sqrt {b} d x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}}{2 \sqrt {b}}\right )}{a d}-\frac {2}{a d \sqrt {d x}}\right )}{a d^2}-\frac {2}{5 a d (d x)^{5/2}}\right )}{4 a}+\frac {1}{2 a d (d x)^{5/2} \left (a+b x^2\right )}\right )}{8 a}+\frac {1}{4 a d (d x)^{5/2} \left (a+b x^2\right )^2}\right )}{12 a}+\frac {1}{6 a d (d x)^{5/2} \left (a+b x^2\right )^3}\) |
Input:
Int[1/((d*x)^(7/2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^2),x]
Output:
1/(6*a*d*(d*x)^(5/2)*(a + b*x^2)^3) + (17*(1/(4*a*d*(d*x)^(5/2)*(a + b*x^2 )^2) + (13*(1/(2*a*d*(d*x)^(5/2)*(a + b*x^2)) + (9*(-2/(5*a*d*(d*x)^(5/2)) - (b*(-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)))/(a*d^2)))/(4*a)))/(8*a)))/(12*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 = 0.65 (sec) , antiderivative size = 229, normalized size of antiderivative = 0.78
| method | result | size |
| risch | \(-\frac {2 \left (-20 b \,x^{2}+a \right )}{5 a^{5} \sqrt {d x}\, x^{2} d^{3}}+\frac {b^{2} \left (\frac {\frac {151 b^{2} \left (d x \right )^{\frac {11}{2}}}{64}+\frac {173 a b \,d^{2} \left (d x \right )^{\frac {7}{2}}}{32}+\frac {617 a^{2} d^{4} \left (d x \right )^{\frac {3}{2}}}{192}}{\left (b \,d^{2} x^{2}+a \,d^{2}\right )^{3}}+\frac {663 \sqrt {2}\, \left (\ln \left (\frac {d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{512 b \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{a^{5} d^{3}}\) | \(229\) |
| derivativedivides | \(2 d^{7} \left (-\frac {1}{5 d^{8} a^{4} \left (d x \right )^{\frac {5}{2}}}+\frac {4 b}{d^{10} a^{5} \sqrt {d x}}+\frac {b^{2} \left (\frac {\frac {151 b^{2} \left (d x \right )^{\frac {11}{2}}}{128}+\frac {173 a b \,d^{2} \left (d x \right )^{\frac {7}{2}}}{64}+\frac {617 a^{2} d^{4} \left (d x \right )^{\frac {3}{2}}}{384}}{\left (b \,d^{2} x^{2}+a \,d^{2}\right )^{3}}+\frac {663 \sqrt {2}\, \left (\ln \left (\frac {d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{1024 b \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{d^{10} a^{5}}\right )\) | \(236\) |
| default | \(2 d^{7} \left (-\frac {1}{5 d^{8} a^{4} \left (d x \right )^{\frac {5}{2}}}+\frac {4 b}{d^{10} a^{5} \sqrt {d x}}+\frac {b^{2} \left (\frac {\frac {151 b^{2} \left (d x \right )^{\frac {11}{2}}}{128}+\frac {173 a b \,d^{2} \left (d x \right )^{\frac {7}{2}}}{64}+\frac {617 a^{2} d^{4} \left (d x \right )^{\frac {3}{2}}}{384}}{\left (b \,d^{2} x^{2}+a \,d^{2}\right )^{3}}+\frac {663 \sqrt {2}\, \left (\ln \left (\frac {d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{1024 b \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{d^{10} a^{5}}\right )\) | \(236\) |
| pseudoelliptic | \(-\frac {2 \left (-\frac {3315 b \,x^{2} \sqrt {2}\, \left (b \,x^{2}+a \right )^{3} \left (\ln \left (\frac {d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}-\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}+\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )\right ) \sqrt {d x}}{1024}+\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \left (-\frac {3315}{128} b^{4} x^{8}-\frac {4641}{64} a \,b^{3} x^{6}-\frac {24973}{384} a^{2} b^{2} x^{4}-17 a^{3} b \,x^{2}+a^{4}\right )\right )}{5 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, d^{3} a^{5} x^{2} \left (b \,x^{2}+a \right )^{3}}\) | \(254\) |
Input:
int(1/(d*x)^(7/2)/(b^2*x^4+2*a*b*x^2+a^2)^2,x,method=_RETURNVERBOSE)
Output:
-2/5*(-20*b*x^2+a)/a^5/(d*x)^(1/2)/x^2/d^3+b^2/a^5*(2*(151/128*b^2*(d*x)^( 11/2)+173/64*a*b*d^2*(d*x)^(7/2)+617/384*a^2*d^4*(d*x)^(3/2))/(b*d^2*x^2+a *d^2)^3+663/512/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^3
Result contains complex when optimal does not.
Time = 0.19 (sec) , antiderivative size = 501, normalized size of antiderivative = 1.71 \[ \int \frac {1}{(d x)^{7/2} \left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=\frac {9945 \, {\left (a^{5} b^{3} d^{4} x^{9} + 3 \, a^{6} b^{2} d^{4} x^{7} + 3 \, a^{7} b d^{4} x^{5} + a^{8} d^{4} x^{3}\right )} \left (-\frac {b^{5}}{a^{21} d^{14}}\right )^{\frac {1}{4}} \log \left (291434247 \, a^{16} d^{11} \left (-\frac {b^{5}}{a^{21} d^{14}}\right )^{\frac {3}{4}} + 291434247 \, \sqrt {d x} b^{4}\right ) - 9945 \, {\left (i \, a^{5} b^{3} d^{4} x^{9} + 3 i \, a^{6} b^{2} d^{4} x^{7} + 3 i \, a^{7} b d^{4} x^{5} + i \, a^{8} d^{4} x^{3}\right )} \left (-\frac {b^{5}}{a^{21} d^{14}}\right )^{\frac {1}{4}} \log \left (291434247 i \, a^{16} d^{11} \left (-\frac {b^{5}}{a^{21} d^{14}}\right )^{\frac {3}{4}} + 291434247 \, \sqrt {d x} b^{4}\right ) - 9945 \, {\left (-i \, a^{5} b^{3} d^{4} x^{9} - 3 i \, a^{6} b^{2} d^{4} x^{7} - 3 i \, a^{7} b d^{4} x^{5} - i \, a^{8} d^{4} x^{3}\right )} \left (-\frac {b^{5}}{a^{21} d^{14}}\right )^{\frac {1}{4}} \log \left (-291434247 i \, a^{16} d^{11} \left (-\frac {b^{5}}{a^{21} d^{14}}\right )^{\frac {3}{4}} + 291434247 \, \sqrt {d x} b^{4}\right ) - 9945 \, {\left (a^{5} b^{3} d^{4} x^{9} + 3 \, a^{6} b^{2} d^{4} x^{7} + 3 \, a^{7} b d^{4} x^{5} + a^{8} d^{4} x^{3}\right )} \left (-\frac {b^{5}}{a^{21} d^{14}}\right )^{\frac {1}{4}} \log \left (-291434247 \, a^{16} d^{11} \left (-\frac {b^{5}}{a^{21} d^{14}}\right )^{\frac {3}{4}} + 291434247 \, \sqrt {d x} b^{4}\right ) + 4 \, {\left (9945 \, b^{4} x^{8} + 27846 \, a b^{3} x^{6} + 24973 \, a^{2} b^{2} x^{4} + 6528 \, a^{3} b x^{2} - 384 \, a^{4}\right )} \sqrt {d x}}{3840 \, {\left (a^{5} b^{3} d^{4} x^{9} + 3 \, a^{6} b^{2} d^{4} x^{7} + 3 \, a^{7} b d^{4} x^{5} + a^{8} d^{4} x^{3}\right )}} \] Input:
integrate(1/(d*x)^(7/2)/(b^2*x^4+2*a*b*x^2+a^2)^2,x, algorithm="fricas")
Output:
1/3840*(9945*(a^5*b^3*d^4*x^9 + 3*a^6*b^2*d^4*x^7 + 3*a^7*b*d^4*x^5 + a^8* d^4*x^3)*(-b^5/(a^21*d^14))^(1/4)*log(291434247*a^16*d^11*(-b^5/(a^21*d^14 ))^(3/4) + 291434247*sqrt(d*x)*b^4) - 9945*(I*a^5*b^3*d^4*x^9 + 3*I*a^6*b^ 2*d^4*x^7 + 3*I*a^7*b*d^4*x^5 + I*a^8*d^4*x^3)*(-b^5/(a^21*d^14))^(1/4)*lo g(291434247*I*a^16*d^11*(-b^5/(a^21*d^14))^(3/4) + 291434247*sqrt(d*x)*b^4 ) - 9945*(-I*a^5*b^3*d^4*x^9 - 3*I*a^6*b^2*d^4*x^7 - 3*I*a^7*b*d^4*x^5 - I *a^8*d^4*x^3)*(-b^5/(a^21*d^14))^(1/4)*log(-291434247*I*a^16*d^11*(-b^5/(a ^21*d^14))^(3/4) + 291434247*sqrt(d*x)*b^4) - 9945*(a^5*b^3*d^4*x^9 + 3*a^ 6*b^2*d^4*x^7 + 3*a^7*b*d^4*x^5 + a^8*d^4*x^3)*(-b^5/(a^21*d^14))^(1/4)*lo g(-291434247*a^16*d^11*(-b^5/(a^21*d^14))^(3/4) + 291434247*sqrt(d*x)*b^4) + 4*(9945*b^4*x^8 + 27846*a*b^3*x^6 + 24973*a^2*b^2*x^4 + 6528*a^3*b*x^2 - 384*a^4)*sqrt(d*x))/(a^5*b^3*d^4*x^9 + 3*a^6*b^2*d^4*x^7 + 3*a^7*b*d^4*x ^5 + a^8*d^4*x^3)
\[ \int \frac {1}{(d x)^{7/2} \left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=\int \frac {1}{\left (d x\right )^{\frac {7}{2}} \left (a + b x^{2}\right )^{4}}\, dx \] Input:
integrate(1/(d*x)**(7/2)/(b**2*x**4+2*a*b*x**2+a**2)**2,x)
Output:
Integral(1/((d*x)**(7/2)*(a + b*x**2)**4), x)
Time = 0.14 (sec) , antiderivative size = 350, normalized size of antiderivative = 1.19 \[ \int \frac {1}{(d x)^{7/2} \left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=\frac {\frac {8 \, {\left (9945 \, b^{4} d^{8} x^{8} + 27846 \, a b^{3} d^{8} x^{6} + 24973 \, a^{2} b^{2} d^{8} x^{4} + 6528 \, a^{3} b d^{8} x^{2} - 384 \, a^{4} d^{8}\right )}}{\left (d x\right )^{\frac {17}{2}} a^{5} b^{3} d^{2} + 3 \, \left (d x\right )^{\frac {13}{2}} a^{6} b^{2} d^{4} + 3 \, \left (d x\right )^{\frac {9}{2}} a^{7} b d^{6} + \left (d x\right )^{\frac {5}{2}} a^{8} d^{8}} + \frac {9945 \, b^{2} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {d x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} d}}\right )}{\sqrt {\sqrt {a} \sqrt {b} d} \sqrt {b}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {d x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} d}}\right )}{\sqrt {\sqrt {a} \sqrt {b} d} \sqrt {b}} - \frac {\sqrt {2} \log \left (\sqrt {b} d x + \sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} \sqrt {d x} b^{\frac {1}{4}} + \sqrt {a} d\right )}{\left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (\sqrt {b} d x - \sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} \sqrt {d x} b^{\frac {1}{4}} + \sqrt {a} d\right )}{\left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {3}{4}}}\right )}}{a^{5} d^{2}}}{7680 \, d} \] Input:
integrate(1/(d*x)^(7/2)/(b^2*x^4+2*a*b*x^2+a^2)^2,x, algorithm="maxima")
Output:
1/7680*(8*(9945*b^4*d^8*x^8 + 27846*a*b^3*d^8*x^6 + 24973*a^2*b^2*d^8*x^4 + 6528*a^3*b*d^8*x^2 - 384*a^4*d^8)/((d*x)^(17/2)*a^5*b^3*d^2 + 3*(d*x)^(1 3/2)*a^6*b^2*d^4 + 3*(d*x)^(9/2)*a^7*b*d^6 + (d*x)^(5/2)*a^8*d^8) + 9945*b ^2*(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)*sqrt(b)*d)*sqrt(b)) - sqr t(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^5*d^2))/d
Time = 0.19 (sec) , antiderivative size = 349, normalized size of antiderivative = 1.19 \[ \int \frac {1}{(d x)^{7/2} \left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=\frac {663 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{256 \, a^{6} b d^{5}} + \frac {663 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{256 \, a^{6} b d^{5}} - \frac {663 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \log \left (d x + \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{512 \, a^{6} b d^{5}} + \frac {663 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \log \left (d x - \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{512 \, a^{6} b d^{5}} + \frac {453 \, \sqrt {d x} b^{4} d^{5} x^{5} + 1038 \, \sqrt {d x} a b^{3} d^{5} x^{3} + 617 \, \sqrt {d x} a^{2} b^{2} d^{5} x}{192 \, {\left (b d^{2} x^{2} + a d^{2}\right )}^{3} a^{5} d^{3}} + \frac {2 \, {\left (20 \, b d^{2} x^{2} - a d^{2}\right )}}{5 \, \sqrt {d x} a^{5} d^{5} x^{2}} \] Input:
integrate(1/(d*x)^(7/2)/(b^2*x^4+2*a*b*x^2+a^2)^2,x, algorithm="giac")
Output:
663/256*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^6*b*d^5) + 663/256*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^6*b*d^5) - 663/512*sqrt(2)*(a*b^3*d^2)^(3/4)*log(d*x + sqr t(2)*(a*d^2/b)^(1/4)*sqrt(d*x) + sqrt(a*d^2/b))/(a^6*b*d^5) + 663/512*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^6*b*d^5) + 1/192*(453*sqrt(d*x)*b^4*d^5*x^5 + 1038*sqrt(d*x)*a *b^3*d^5*x^3 + 617*sqrt(d*x)*a^2*b^2*d^5*x)/((b*d^2*x^2 + a*d^2)^3*a^5*d^3 ) + 2/5*(20*b*d^2*x^2 - a*d^2)/(sqrt(d*x)*a^5*d^5*x^2)
Time = 18.67 (sec) , antiderivative size = 179, normalized size of antiderivative = 0.61 \[ \int \frac {1}{(d x)^{7/2} \left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=\frac {\frac {34\,b\,d^5\,x^2}{5\,a^2}-\frac {2\,d^5}{5\,a}+\frac {24973\,b^2\,d^5\,x^4}{960\,a^3}+\frac {4641\,b^3\,d^5\,x^6}{160\,a^4}+\frac {663\,b^4\,d^5\,x^8}{64\,a^5}}{b^3\,{\left (d\,x\right )}^{17/2}+a^3\,d^6\,{\left (d\,x\right )}^{5/2}+3\,a^2\,b\,d^4\,{\left (d\,x\right )}^{9/2}+3\,a\,b^2\,d^2\,{\left (d\,x\right )}^{13/2}}-\frac {663\,{\left (-b\right )}^{5/4}\,\mathrm {atan}\left (\frac {{\left (-b\right )}^{1/4}\,\sqrt {d\,x}}{a^{1/4}\,\sqrt {d}}\right )}{128\,a^{21/4}\,d^{7/2}}+\frac {663\,{\left (-b\right )}^{5/4}\,\mathrm {atanh}\left (\frac {{\left (-b\right )}^{1/4}\,\sqrt {d\,x}}{a^{1/4}\,\sqrt {d}}\right )}{128\,a^{21/4}\,d^{7/2}} \] Input:
int(1/((d*x)^(7/2)*(a^2 + b^2*x^4 + 2*a*b*x^2)^2),x)
Output:
((34*b*d^5*x^2)/(5*a^2) - (2*d^5)/(5*a) + (24973*b^2*d^5*x^4)/(960*a^3) + (4641*b^3*d^5*x^6)/(160*a^4) + (663*b^4*d^5*x^8)/(64*a^5))/(b^3*(d*x)^(17/ 2) + a^3*d^6*(d*x)^(5/2) + 3*a^2*b*d^4*(d*x)^(9/2) + 3*a*b^2*d^2*(d*x)^(13 /2)) - (663*(-b)^(5/4)*atan(((-b)^(1/4)*(d*x)^(1/2))/(a^(1/4)*d^(1/2))))/( 128*a^(21/4)*d^(7/2)) + (663*(-b)^(5/4)*atanh(((-b)^(1/4)*(d*x)^(1/2))/(a^ (1/4)*d^(1/2))))/(128*a^(21/4)*d^(7/2))
Time = 0.18 (sec) , antiderivative size = 714, normalized size of antiderivative = 2.44 \[ \int \frac {1}{(d x)^{7/2} \left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx =\text {Too large to display} \] Input:
int(1/(d*x)^(7/2)/(b^2*x^4+2*a*b*x^2+a^2)^2,x)
Output:
(sqrt(d)*( - 19890*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*x**2 - 59670*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**2*x**4 - 59670*sq rt(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**3*x**6 - 19890*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)))*b**4*x**8 + 19890*sqrt(x)*b**(1/4)*a**(3/4)*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**3*b*x**2 + 59670*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**2*x**4 + 59670*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**3* x**6 + 19890*sqrt(x)*b**(1/4)*a**(3/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqr t(2) + 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*b**4*x**8 + 9945*sq rt(x)*b**(1/4)*a**(3/4)*sqrt(2)*log( - sqrt(x)*b**(1/4)*a**(1/4)*sqrt(2) + sqrt(a) + sqrt(b)*x)*a**3*b*x**2 + 29835*sqrt(x)*b**(1/4)*a**(3/4)*sqrt(2 )*log( - sqrt(x)*b**(1/4)*a**(1/4)*sqrt(2) + sqrt(a) + sqrt(b)*x)*a**2*b** 2*x**4 + 29835*sqrt(x)*b**(1/4)*a**(3/4)*sqrt(2)*log( - sqrt(x)*b**(1/4)*a **(1/4)*sqrt(2) + sqrt(a) + sqrt(b)*x)*a*b**3*x**6 + 9945*sqrt(x)*b**(1...