Integrand size = 30, antiderivative size = 451 \[ \int \frac {(d x)^{5/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx=\frac {45 d (d x)^{3/2}}{1024 a^3 b \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{3/2}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {d (d x)^{3/2}}{32 a b \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {9 d (d x)^{3/2}}{256 a^2 b \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {45 d^{5/2} \left (a+b x^2\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{2048 \sqrt {2} a^{13/4} b^{7/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {45 d^{5/2} \left (a+b x^2\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{2048 \sqrt {2} a^{13/4} b^{7/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {45 d^{5/2} \left (a+b x^2\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}}{\sqrt {d} \left (\sqrt {a}+\sqrt {b} x\right )}\right )}{2048 \sqrt {2} a^{13/4} b^{7/4} \sqrt {a^2+2 a b x^2+b^2 x^4}} \] Output:
45/1024*d*(d*x)^(3/2)/a^3/b/((b*x^2+a)^2)^(1/2)-1/8*d*(d*x)^(3/2)/b/(b*x^2 +a)^3/((b*x^2+a)^2)^(1/2)+1/32*d*(d*x)^(3/2)/a/b/(b*x^2+a)^2/((b*x^2+a)^2) ^(1/2)+9/256*d*(d*x)^(3/2)/a^2/b/(b*x^2+a)/((b*x^2+a)^2)^(1/2)-45/4096*d^( 5/2)*(b*x^2+a)*arctan(1-2^(1/2)*b^(1/4)*(d*x)^(1/2)/a^(1/4)/d^(1/2))*2^(1/ 2)/a^(13/4)/b^(7/4)/((b*x^2+a)^2)^(1/2)+45/4096*d^(5/2)*(b*x^2+a)*arctan(1 +2^(1/2)*b^(1/4)*(d*x)^(1/2)/a^(1/4)/d^(1/2))*2^(1/2)/a^(13/4)/b^(7/4)/((b *x^2+a)^2)^(1/2)-45/4096*d^(5/2)*(b*x^2+a)*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^(13/4)/b^(7/4)/((b*x^2 +a)^2)^(1/2)
Time = 0.57 (sec) , antiderivative size = 201, normalized size of antiderivative = 0.45 \[ \int \frac {(d x)^{5/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx=\frac {(d x)^{5/2} \left (a+b x^2\right ) \left (4 \sqrt [4]{a} b^{3/4} x^{3/2} \left (-15 a^3+239 a^2 b x^2+171 a b^2 x^4+45 b^3 x^6\right )-45 \sqrt {2} \left (a+b x^2\right )^4 \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )-45 \sqrt {2} \left (a+b x^2\right )^4 \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )\right )}{4096 a^{13/4} b^{7/4} x^{5/2} \left (\left (a+b x^2\right )^2\right )^{5/2}} \] Input:
Integrate[(d*x)^(5/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2),x]
Output:
((d*x)^(5/2)*(a + b*x^2)*(4*a^(1/4)*b^(3/4)*x^(3/2)*(-15*a^3 + 239*a^2*b*x ^2 + 171*a*b^2*x^4 + 45*b^3*x^6) - 45*Sqrt[2]*(a + b*x^2)^4*ArcTan[(Sqrt[a ] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])] - 45*Sqrt[2]*(a + b*x^2) ^4*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)]))/(409 6*a^(13/4)*b^(7/4)*x^(5/2)*((a + b*x^2)^2)^(5/2))
Time = 1.08 (sec) , antiderivative size = 434, normalized size of antiderivative = 0.96, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.533, Rules used = {1384, 27, 252, 253, 253, 253, 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.
| \(\displaystyle \int \frac {(d x)^{5/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 1384 |
| \(\displaystyle \frac {b^5 \left (a+b x^2\right ) \int \frac {(d x)^{5/2}}{b^5 \left (b x^2+a\right )^5}dx}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\left (a+b x^2\right ) \int \frac {(d x)^{5/2}}{\left (b x^2+a\right )^5}dx}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\) |
| \(\Big \downarrow \) 252 |
| \(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {3 d^2 \int \frac {\sqrt {d x}}{\left (b x^2+a\right )^4}dx}{16 b}-\frac {d (d x)^{3/2}}{8 b \left (a+b x^2\right )^4}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\) |
| \(\Big \downarrow \) 253 |
| \(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {3 d^2 \left (\frac {3 \int \frac {\sqrt {d x}}{\left (b x^2+a\right )^3}dx}{4 a}+\frac {(d x)^{3/2}}{6 a d \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{3/2}}{8 b \left (a+b x^2\right )^4}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\) |
| \(\Big \downarrow \) 253 |
| \(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {3 d^2 \left (\frac {3 \left (\frac {5 \int \frac {\sqrt {d x}}{\left (b x^2+a\right )^2}dx}{8 a}+\frac {(d x)^{3/2}}{4 a d \left (a+b x^2\right )^2}\right )}{4 a}+\frac {(d x)^{3/2}}{6 a d \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{3/2}}{8 b \left (a+b x^2\right )^4}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\) |
| \(\Big \downarrow \) 253 |
| \(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {3 d^2 \left (\frac {3 \left (\frac {5 \left (\frac {\int \frac {\sqrt {d x}}{b x^2+a}dx}{4 a}+\frac {(d x)^{3/2}}{2 a d \left (a+b x^2\right )}\right )}{8 a}+\frac {(d x)^{3/2}}{4 a d \left (a+b x^2\right )^2}\right )}{4 a}+\frac {(d x)^{3/2}}{6 a d \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{3/2}}{8 b \left (a+b x^2\right )^4}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {3 d^2 \left (\frac {3 \left (\frac {5 \left (\frac {\int \frac {d^3 x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 a d}+\frac {(d x)^{3/2}}{2 a d \left (a+b x^2\right )}\right )}{8 a}+\frac {(d x)^{3/2}}{4 a d \left (a+b x^2\right )^2}\right )}{4 a}+\frac {(d x)^{3/2}}{6 a d \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{3/2}}{8 b \left (a+b x^2\right )^4}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {3 d^2 \left (\frac {3 \left (\frac {5 \left (\frac {d \int \frac {d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 a}+\frac {(d x)^{3/2}}{2 a d \left (a+b x^2\right )}\right )}{8 a}+\frac {(d x)^{3/2}}{4 a d \left (a+b x^2\right )^2}\right )}{4 a}+\frac {(d x)^{3/2}}{6 a d \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{3/2}}{8 b \left (a+b x^2\right )^4}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\) |
| \(\Big \downarrow \) 826 |
| \(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {3 d^2 \left (\frac {3 \left (\frac {5 \left (\frac {d \left (\frac {\int \frac {\sqrt {b} x d+\sqrt {a} d}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {b}}-\frac {\int \frac {\sqrt {a} d-\sqrt {b} d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {b}}\right )}{2 a}+\frac {(d x)^{3/2}}{2 a d \left (a+b x^2\right )}\right )}{8 a}+\frac {(d x)^{3/2}}{4 a d \left (a+b x^2\right )^2}\right )}{4 a}+\frac {(d x)^{3/2}}{6 a d \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{3/2}}{8 b \left (a+b x^2\right )^4}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {3 d^2 \left (\frac {3 \left (\frac {5 \left (\frac {d \left (\frac {\frac {\int \frac {1}{x d+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}}d\sqrt {d x}}{2 \sqrt {b}}+\frac {\int \frac {1}{x d+\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}}d\sqrt {d x}}{2 \sqrt {b}}}{2 \sqrt {b}}-\frac {\int \frac {\sqrt {a} d-\sqrt {b} d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {b}}\right )}{2 a}+\frac {(d x)^{3/2}}{2 a d \left (a+b x^2\right )}\right )}{8 a}+\frac {(d x)^{3/2}}{4 a d \left (a+b x^2\right )^2}\right )}{4 a}+\frac {(d x)^{3/2}}{6 a d \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{3/2}}{8 b \left (a+b x^2\right )^4}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {3 d^2 \left (\frac {3 \left (\frac {5 \left (\frac {d \left (\frac {\frac {\int \frac {1}{-d x-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\int \frac {1}{-d x-1}d\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}}{2 \sqrt {b}}-\frac {\int \frac {\sqrt {a} d-\sqrt {b} d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {b}}\right )}{2 a}+\frac {(d x)^{3/2}}{2 a d \left (a+b x^2\right )}\right )}{8 a}+\frac {(d x)^{3/2}}{4 a d \left (a+b x^2\right )^2}\right )}{4 a}+\frac {(d x)^{3/2}}{6 a d \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{3/2}}{8 b \left (a+b x^2\right )^4}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {3 d^2 \left (\frac {3 \left (\frac {5 \left (\frac {d \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}}{2 \sqrt {b}}-\frac {\int \frac {\sqrt {a} d-\sqrt {b} d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {b}}\right )}{2 a}+\frac {(d x)^{3/2}}{2 a d \left (a+b x^2\right )}\right )}{8 a}+\frac {(d x)^{3/2}}{4 a d \left (a+b x^2\right )^2}\right )}{4 a}+\frac {(d x)^{3/2}}{6 a d \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{3/2}}{8 b \left (a+b x^2\right )^4}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {3 d^2 \left (\frac {3 \left (\frac {5 \left (\frac {d \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}}{2 \sqrt {b}}-\frac {-\frac {\int -\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d}-2 \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{b} \left (x d+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt [4]{a} \sqrt {d}+\sqrt {2} \sqrt [4]{b} \sqrt {d x}\right )}{\sqrt [4]{b} \left (x d+\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}}{2 \sqrt {b}}\right )}{2 a}+\frac {(d x)^{3/2}}{2 a d \left (a+b x^2\right )}\right )}{8 a}+\frac {(d x)^{3/2}}{4 a d \left (a+b x^2\right )^2}\right )}{4 a}+\frac {(d x)^{3/2}}{6 a d \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{3/2}}{8 b \left (a+b x^2\right )^4}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {3 d^2 \left (\frac {3 \left (\frac {5 \left (\frac {d \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}}{2 \sqrt {b}}-\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{a} \sqrt {d}-2 \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{b} \left (x d+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}+\frac {\int \frac {\sqrt {2} \left (\sqrt [4]{a} \sqrt {d}+\sqrt {2} \sqrt [4]{b} \sqrt {d x}\right )}{\sqrt [4]{b} \left (x d+\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}}{2 \sqrt {b}}\right )}{2 a}+\frac {(d x)^{3/2}}{2 a d \left (a+b x^2\right )}\right )}{8 a}+\frac {(d x)^{3/2}}{4 a d \left (a+b x^2\right )^2}\right )}{4 a}+\frac {(d x)^{3/2}}{6 a d \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{3/2}}{8 b \left (a+b x^2\right )^4}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {3 d^2 \left (\frac {3 \left (\frac {5 \left (\frac {d \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}}{2 \sqrt {b}}-\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{a} \sqrt {d}-2 \sqrt [4]{b} \sqrt {d x}}{x d+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt {b} \sqrt {d}}+\frac {\int \frac {\sqrt [4]{a} \sqrt {d}+\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{x d+\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}}d\sqrt {d x}}{2 \sqrt [4]{a} \sqrt {b} \sqrt {d}}}{2 \sqrt {b}}\right )}{2 a}+\frac {(d x)^{3/2}}{2 a d \left (a+b x^2\right )}\right )}{8 a}+\frac {(d x)^{3/2}}{4 a d \left (a+b x^2\right )^2}\right )}{4 a}+\frac {(d x)^{3/2}}{6 a d \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{3/2}}{8 b \left (a+b x^2\right )^4}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {3 d^2 \left (\frac {3 \left (\frac {5 \left (\frac {d \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}}{2 \sqrt {b}}-\frac {\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d} \sqrt {d x}+\sqrt {a} d+\sqrt {b} d x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d} \sqrt {d x}+\sqrt {a} d+\sqrt {b} d x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}}{2 \sqrt {b}}\right )}{2 a}+\frac {(d x)^{3/2}}{2 a d \left (a+b x^2\right )}\right )}{8 a}+\frac {(d x)^{3/2}}{4 a d \left (a+b x^2\right )^2}\right )}{4 a}+\frac {(d x)^{3/2}}{6 a d \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{3/2}}{8 b \left (a+b x^2\right )^4}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\) |
Input:
Int[(d*x)^(5/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2),x]
Output:
((a + b*x^2)*(-1/8*(d*(d*x)^(3/2))/(b*(a + b*x^2)^4) + (3*d^2*((d*x)^(3/2) /(6*a*d*(a + b*x^2)^3) + (3*((d*x)^(3/2)/(4*a*d*(a + b*x^2)^2) + (5*((d*x) ^(3/2)/(2*a*d*(a + b*x^2)) + (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[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 + S qrt[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])))/(2*a)))/(8*a)))/(4*a)))/(16*b)))/Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]
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[(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[(a + b*x^n + c*x^(2*n))^FracPart[p]/(c^IntPart[p]*(b/2 + c*x^n)^(2*Frac Part[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 - 1/2] && NeQ[u, x^(n - 1)] && NeQ[u, x^(2*n - 1)] && !(EqQ[p, 1/2] && EqQ[u, x^(-2*n - 1)])
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]
Leaf count of result is larger than twice the leaf count of optimal. \(1050\) vs. \(2(294)=588\).
Time = 0.14 (sec) , antiderivative size = 1051, normalized size of antiderivative = 2.33
Input:
int((d*x)^(5/2)/(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x,method=_RETURNVERBOSE)
Output:
1/8192*(45*2^(1/2)*ln(-((a*d^2/b)^(1/4)*(d*x)^(1/2)*2^(1/2)-d*x-(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)))*b^4*d^8* x^8+90*2^(1/2)*arctan((2^(1/2)*(d*x)^(1/2)+(a*d^2/b)^(1/4))/(a*d^2/b)^(1/4 ))*b^4*d^8*x^8+90*2^(1/2)*arctan((2^(1/2)*(d*x)^(1/2)-(a*d^2/b)^(1/4))/(a* d^2/b)^(1/4))*b^4*d^8*x^8+360*(d*x)^(15/2)*(a*d^2/b)^(1/4)*b^4+180*2^(1/2) *ln(-((a*d^2/b)^(1/4)*(d*x)^(1/2)*2^(1/2)-d*x-(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)))*a*b^3*d^8*x^6+360*2^(1/2)* arctan((2^(1/2)*(d*x)^(1/2)+(a*d^2/b)^(1/4))/(a*d^2/b)^(1/4))*a*b^3*d^8*x^ 6+360*2^(1/2)*arctan((2^(1/2)*(d*x)^(1/2)-(a*d^2/b)^(1/4))/(a*d^2/b)^(1/4) )*a*b^3*d^8*x^6+1368*(d*x)^(11/2)*(a*d^2/b)^(1/4)*a*b^3*d^2+270*2^(1/2)*ln (-((a*d^2/b)^(1/4)*(d*x)^(1/2)*2^(1/2)-d*x-(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)))*a^2*b^2*d^8*x^4+540*2^(1/2)*a rctan((2^(1/2)*(d*x)^(1/2)+(a*d^2/b)^(1/4))/(a*d^2/b)^(1/4))*a^2*b^2*d^8*x ^4+540*2^(1/2)*arctan((2^(1/2)*(d*x)^(1/2)-(a*d^2/b)^(1/4))/(a*d^2/b)^(1/4 ))*a^2*b^2*d^8*x^4+1912*(d*x)^(7/2)*(a*d^2/b)^(1/4)*a^2*b^2*d^4+180*2^(1/2 )*ln(-((a*d^2/b)^(1/4)*(d*x)^(1/2)*2^(1/2)-d*x-(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)))*a^3*b*d^8*x^2+360*2^(1/2) *arctan((2^(1/2)*(d*x)^(1/2)+(a*d^2/b)^(1/4))/(a*d^2/b)^(1/4))*a^3*b*d^8*x ^2+360*2^(1/2)*arctan((2^(1/2)*(d*x)^(1/2)-(a*d^2/b)^(1/4))/(a*d^2/b)^(1/4 ))*a^3*b*d^8*x^2-120*(d*x)^(3/2)*(a*d^2/b)^(1/4)*a^3*b*d^6+45*2^(1/2)*l...
Result contains complex when optimal does not.
Time = 0.09 (sec) , antiderivative size = 498, normalized size of antiderivative = 1.10 \[ \int \frac {(d x)^{5/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx=\frac {45 \, {\left (a^{3} b^{5} x^{8} + 4 \, a^{4} b^{4} x^{6} + 6 \, a^{5} b^{3} x^{4} + 4 \, a^{6} b^{2} x^{2} + a^{7} b\right )} \left (-\frac {d^{10}}{a^{13} b^{7}}\right )^{\frac {1}{4}} \log \left (91125 \, a^{10} b^{5} \left (-\frac {d^{10}}{a^{13} b^{7}}\right )^{\frac {3}{4}} + 91125 \, \sqrt {d x} d^{7}\right ) - 45 \, {\left (i \, a^{3} b^{5} x^{8} + 4 i \, a^{4} b^{4} x^{6} + 6 i \, a^{5} b^{3} x^{4} + 4 i \, a^{6} b^{2} x^{2} + i \, a^{7} b\right )} \left (-\frac {d^{10}}{a^{13} b^{7}}\right )^{\frac {1}{4}} \log \left (91125 i \, a^{10} b^{5} \left (-\frac {d^{10}}{a^{13} b^{7}}\right )^{\frac {3}{4}} + 91125 \, \sqrt {d x} d^{7}\right ) - 45 \, {\left (-i \, a^{3} b^{5} x^{8} - 4 i \, a^{4} b^{4} x^{6} - 6 i \, a^{5} b^{3} x^{4} - 4 i \, a^{6} b^{2} x^{2} - i \, a^{7} b\right )} \left (-\frac {d^{10}}{a^{13} b^{7}}\right )^{\frac {1}{4}} \log \left (-91125 i \, a^{10} b^{5} \left (-\frac {d^{10}}{a^{13} b^{7}}\right )^{\frac {3}{4}} + 91125 \, \sqrt {d x} d^{7}\right ) - 45 \, {\left (a^{3} b^{5} x^{8} + 4 \, a^{4} b^{4} x^{6} + 6 \, a^{5} b^{3} x^{4} + 4 \, a^{6} b^{2} x^{2} + a^{7} b\right )} \left (-\frac {d^{10}}{a^{13} b^{7}}\right )^{\frac {1}{4}} \log \left (-91125 \, a^{10} b^{5} \left (-\frac {d^{10}}{a^{13} b^{7}}\right )^{\frac {3}{4}} + 91125 \, \sqrt {d x} d^{7}\right ) + 4 \, {\left (45 \, b^{3} d^{2} x^{7} + 171 \, a b^{2} d^{2} x^{5} + 239 \, a^{2} b d^{2} x^{3} - 15 \, a^{3} d^{2} x\right )} \sqrt {d x}}{4096 \, {\left (a^{3} b^{5} x^{8} + 4 \, a^{4} b^{4} x^{6} + 6 \, a^{5} b^{3} x^{4} + 4 \, a^{6} b^{2} x^{2} + a^{7} b\right )}} \] Input:
integrate((d*x)^(5/2)/(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x, algorithm="fricas")
Output:
1/4096*(45*(a^3*b^5*x^8 + 4*a^4*b^4*x^6 + 6*a^5*b^3*x^4 + 4*a^6*b^2*x^2 + a^7*b)*(-d^10/(a^13*b^7))^(1/4)*log(91125*a^10*b^5*(-d^10/(a^13*b^7))^(3/4 ) + 91125*sqrt(d*x)*d^7) - 45*(I*a^3*b^5*x^8 + 4*I*a^4*b^4*x^6 + 6*I*a^5*b ^3*x^4 + 4*I*a^6*b^2*x^2 + I*a^7*b)*(-d^10/(a^13*b^7))^(1/4)*log(91125*I*a ^10*b^5*(-d^10/(a^13*b^7))^(3/4) + 91125*sqrt(d*x)*d^7) - 45*(-I*a^3*b^5*x ^8 - 4*I*a^4*b^4*x^6 - 6*I*a^5*b^3*x^4 - 4*I*a^6*b^2*x^2 - I*a^7*b)*(-d^10 /(a^13*b^7))^(1/4)*log(-91125*I*a^10*b^5*(-d^10/(a^13*b^7))^(3/4) + 91125* sqrt(d*x)*d^7) - 45*(a^3*b^5*x^8 + 4*a^4*b^4*x^6 + 6*a^5*b^3*x^4 + 4*a^6*b ^2*x^2 + a^7*b)*(-d^10/(a^13*b^7))^(1/4)*log(-91125*a^10*b^5*(-d^10/(a^13* b^7))^(3/4) + 91125*sqrt(d*x)*d^7) + 4*(45*b^3*d^2*x^7 + 171*a*b^2*d^2*x^5 + 239*a^2*b*d^2*x^3 - 15*a^3*d^2*x)*sqrt(d*x))/(a^3*b^5*x^8 + 4*a^4*b^4*x ^6 + 6*a^5*b^3*x^4 + 4*a^6*b^2*x^2 + a^7*b)
\[ \int \frac {(d x)^{5/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx=\int \frac {\left (d x\right )^{\frac {5}{2}}}{\left (\left (a + b x^{2}\right )^{2}\right )^{\frac {5}{2}}}\, dx \] Input:
integrate((d*x)**(5/2)/(b**2*x**4+2*a*b*x**2+a**2)**(5/2),x)
Output:
Integral((d*x)**(5/2)/((a + b*x**2)**2)**(5/2), x)
Time = 0.17 (sec) , antiderivative size = 582, normalized size of antiderivative = 1.29 \[ \int \frac {(d x)^{5/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx=\frac {135 \, b^{3} d^{\frac {5}{2}} x^{\frac {15}{2}} + 657 \, a b^{2} d^{\frac {5}{2}} x^{\frac {11}{2}} + 173 \, a^{2} b d^{\frac {5}{2}} x^{\frac {7}{2}} + 35 \, a^{3} d^{\frac {5}{2}} x^{\frac {3}{2}}}{3072 \, {\left (a^{3} b^{5} x^{8} + 4 \, a^{4} b^{4} x^{6} + 6 \, a^{5} b^{3} x^{4} + 4 \, a^{6} b^{2} x^{2} + a^{7} b\right )}} - \frac {{\left (9 \, b^{4} d^{\frac {5}{2}} x^{5} - 38 \, a b^{3} d^{\frac {5}{2}} x^{3} - 15 \, a^{2} b^{2} d^{\frac {5}{2}} x\right )} x^{\frac {9}{2}} + 2 \, {\left (11 \, a b^{3} d^{\frac {5}{2}} x^{5} - 42 \, a^{2} b^{2} d^{\frac {5}{2}} x^{3} - 21 \, a^{3} b d^{\frac {5}{2}} x\right )} x^{\frac {5}{2}} + {\left (45 \, a^{2} b^{2} d^{\frac {5}{2}} x^{5} + 18 \, a^{3} b d^{\frac {5}{2}} x^{3} + 5 \, a^{4} d^{\frac {5}{2}} x\right )} \sqrt {x}}{192 \, {\left (a^{5} b^{4} x^{6} + 3 \, a^{6} b^{3} x^{4} + 3 \, a^{7} b^{2} x^{2} + a^{8} b + {\left (a^{2} b^{7} x^{6} + 3 \, a^{3} b^{6} x^{4} + 3 \, a^{4} b^{5} x^{2} + a^{5} b^{4}\right )} x^{6} + 3 \, {\left (a^{3} b^{6} x^{6} + 3 \, a^{4} b^{5} x^{4} + 3 \, a^{5} b^{4} x^{2} + a^{6} b^{3}\right )} x^{4} + 3 \, {\left (a^{4} b^{5} x^{6} + 3 \, a^{5} b^{4} x^{4} + 3 \, a^{6} b^{3} x^{2} + a^{7} b^{2}\right )} x^{2}\right )}} + \frac {45 \, d^{\frac {5}{2}} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} - \frac {\sqrt {2} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}}\right )}}{8192 \, a^{3} b} \] Input:
integrate((d*x)^(5/2)/(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x, algorithm="maxima")
Output:
1/3072*(135*b^3*d^(5/2)*x^(15/2) + 657*a*b^2*d^(5/2)*x^(11/2) + 173*a^2*b* d^(5/2)*x^(7/2) + 35*a^3*d^(5/2)*x^(3/2))/(a^3*b^5*x^8 + 4*a^4*b^4*x^6 + 6 *a^5*b^3*x^4 + 4*a^6*b^2*x^2 + a^7*b) - 1/192*((9*b^4*d^(5/2)*x^5 - 38*a*b ^3*d^(5/2)*x^3 - 15*a^2*b^2*d^(5/2)*x)*x^(9/2) + 2*(11*a*b^3*d^(5/2)*x^5 - 42*a^2*b^2*d^(5/2)*x^3 - 21*a^3*b*d^(5/2)*x)*x^(5/2) + (45*a^2*b^2*d^(5/2 )*x^5 + 18*a^3*b*d^(5/2)*x^3 + 5*a^4*d^(5/2)*x)*sqrt(x))/(a^5*b^4*x^6 + 3* a^6*b^3*x^4 + 3*a^7*b^2*x^2 + a^8*b + (a^2*b^7*x^6 + 3*a^3*b^6*x^4 + 3*a^4 *b^5*x^2 + a^5*b^4)*x^6 + 3*(a^3*b^6*x^6 + 3*a^4*b^5*x^4 + 3*a^5*b^4*x^2 + a^6*b^3)*x^4 + 3*(a^4*b^5*x^6 + 3*a^5*b^4*x^4 + 3*a^6*b^3*x^2 + a^7*b^2)* x^2) + 45/8192*d^(5/2)*(2*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2)*a^(1/4)*b^(1 /4) + 2*sqrt(b)*sqrt(x))/sqrt(sqrt(a)*sqrt(b)))/(sqrt(sqrt(a)*sqrt(b))*sqr t(b)) + 2*sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2)*a^(1/4)*b^(1/4) - 2*sqrt(b) *sqrt(x))/sqrt(sqrt(a)*sqrt(b)))/(sqrt(sqrt(a)*sqrt(b))*sqrt(b)) - sqrt(2) *log(sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x + sqrt(a))/(a^(1/4)*b^(3/ 4)) + sqrt(2)*log(-sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x + sqrt(a))/ (a^(1/4)*b^(3/4)))/(a^3*b)
Time = 0.13 (sec) , antiderivative size = 386, normalized size of antiderivative = 0.86 \[ \int \frac {(d x)^{5/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx=\frac {1}{8192} \, d^{2} {\left (\frac {90 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{a^{4} b^{4} d \mathrm {sgn}\left (b x^{2} + a\right )} + \frac {90 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{a^{4} b^{4} d \mathrm {sgn}\left (b x^{2} + a\right )} - \frac {45 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \log \left (d x + \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{a^{4} b^{4} d \mathrm {sgn}\left (b x^{2} + a\right )} + \frac {45 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \log \left (d x - \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{a^{4} b^{4} d \mathrm {sgn}\left (b x^{2} + a\right )} + \frac {8 \, {\left (45 \, \sqrt {d x} b^{3} d^{8} x^{7} + 171 \, \sqrt {d x} a b^{2} d^{8} x^{5} + 239 \, \sqrt {d x} a^{2} b d^{8} x^{3} - 15 \, \sqrt {d x} a^{3} d^{8} x\right )}}{{\left (b d^{2} x^{2} + a d^{2}\right )}^{4} a^{3} b \mathrm {sgn}\left (b x^{2} + a\right )}\right )} \] Input:
integrate((d*x)^(5/2)/(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x, algorithm="giac")
Output:
1/8192*d^2*(90*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^4*b^4*d*sgn(b*x^2 + a)) + 90 *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^4*b^4*d*sgn(b*x^2 + a)) - 45*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^4*b^4*d*sgn(b*x^2 + a)) + 45*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^4*b^4*d*sgn(b*x^2 + a)) + 8*(45*sqrt(d*x)*b^3*d^8*x^7 + 171*sqrt(d*x)*a*b^2*d^8*x^5 + 239*sqrt(d*x) *a^2*b*d^8*x^3 - 15*sqrt(d*x)*a^3*d^8*x)/((b*d^2*x^2 + a*d^2)^4*a^3*b*sgn( b*x^2 + a)))
Timed out. \[ \int \frac {(d x)^{5/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx=\int \frac {{\left (d\,x\right )}^{5/2}}{{\left (a^2+2\,a\,b\,x^2+b^2\,x^4\right )}^{5/2}} \,d x \] Input:
int((d*x)^(5/2)/(a^2 + b^2*x^4 + 2*a*b*x^2)^(5/2),x)
Output:
int((d*x)^(5/2)/(a^2 + b^2*x^4 + 2*a*b*x^2)^(5/2), x)
Time = 0.18 (sec) , antiderivative size = 825, normalized size of antiderivative = 1.83 \[ \int \frac {(d x)^{5/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx =\text {Too large to display} \] Input:
int((d*x)^(5/2)/(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x)
Output:
(sqrt(d)*d**2*( - 90*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)))*a**4 - 360*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 - 540*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 - 360*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 - 9 0*b**(1/4)*a**(3/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(x)*sq rt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*b**4*x**8 + 90*b**(1/4)*a**(3/4)*sqrt( 2)*atan((b**(1/4)*a**(1/4)*sqrt(2) + 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4) *sqrt(2)))*a**4 + 360*b**(1/4)*a**(3/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sq rt(2) + 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*a**3*b*x**2 + 540* 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 + 360*b**(1/4)*a**(3/4)*s qrt(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 + 90*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 + 45*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**4 + 180*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 + 270*b**...