Integrand size = 30, antiderivative size = 496 \[ \int \frac {1}{(d x)^{5/2} \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx=\frac {1045}{1024 a^4 d (d x)^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {1}{8 a d (d x)^{3/2} \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {19}{96 a^2 d (d x)^{3/2} \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {95}{256 a^3 d (d x)^{3/2} \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {7315 \left (a+b x^2\right )}{3072 a^5 d (d x)^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {7315 b^{3/4} \left (a+b x^2\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{2048 \sqrt {2} a^{23/4} d^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {7315 b^{3/4} \left (a+b x^2\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{2048 \sqrt {2} a^{23/4} d^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {7315 b^{3/4} \left (a+b x^2\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}}{\sqrt {d} \left (\sqrt {a}+\sqrt {b} x\right )}\right )}{2048 \sqrt {2} a^{23/4} d^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}} \] Output:
1045/1024/a^4/d/(d*x)^(3/2)/((b*x^2+a)^2)^(1/2)+1/8/a/d/(d*x)^(3/2)/(b*x^2 +a)^3/((b*x^2+a)^2)^(1/2)+19/96/a^2/d/(d*x)^(3/2)/(b*x^2+a)^2/((b*x^2+a)^2 )^(1/2)+95/256/a^3/d/(d*x)^(3/2)/(b*x^2+a)/((b*x^2+a)^2)^(1/2)-7315/3072*( b*x^2+a)/a^5/d/(d*x)^(3/2)/((b*x^2+a)^2)^(1/2)+7315/4096*b^(3/4)*(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^(23/4)/d^ (5/2)/((b*x^2+a)^2)^(1/2)-7315/4096*b^(3/4)*(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^(23/4)/d^(5/2)/((b*x^2+a)^2)^( 1/2)-7315/4096*b^(3/4)*(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^(23/4)/d^(5/2)/((b*x^2+a)^2)^(1/ 2)
Time = 0.70 (sec) , antiderivative size = 213, normalized size of antiderivative = 0.43 \[ \int \frac {1}{(d x)^{5/2} \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx=\frac {x \left (a+b x^2\right ) \left (-4 a^{3/4} \left (2048 a^4+16967 a^3 b x^2+33345 a^2 b^2 x^4+26125 a b^3 x^6+7315 b^4 x^8\right )+21945 \sqrt {2} b^{3/4} x^{3/2} \left (a+b x^2\right )^4 \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )-21945 \sqrt {2} b^{3/4} x^{3/2} \left (a+b x^2\right )^4 \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )\right )}{12288 a^{23/4} (d x)^{5/2} \left (\left (a+b x^2\right )^2\right )^{5/2}} \] Input:
Integrate[1/((d*x)^(5/2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)),x]
Output:
(x*(a + b*x^2)*(-4*a^(3/4)*(2048*a^4 + 16967*a^3*b*x^2 + 33345*a^2*b^2*x^4 + 26125*a*b^3*x^6 + 7315*b^4*x^8) + 21945*Sqrt[2]*b^(3/4)*x^(3/2)*(a + b* x^2)^4*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])] - 2 1945*Sqrt[2]*b^(3/4)*x^(3/2)*(a + b*x^2)^4*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4 )*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)]))/(12288*a^(23/4)*(d*x)^(5/2)*((a + b*x^ 2)^2)^(5/2))
Time = 1.15 (sec) , antiderivative size = 461, normalized size of antiderivative = 0.93, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.567, Rules used = {1384, 27, 253, 253, 253, 253, 264, 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.
| \(\displaystyle \int \frac {1}{(d x)^{5/2} \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 1384 |
| \(\displaystyle \frac {b^5 \left (a+b x^2\right ) \int \frac {1}{b^5 (d x)^{5/2} \left (b x^2+a\right )^5}dx}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\left (a+b x^2\right ) \int \frac {1}{(d x)^{5/2} \left (b x^2+a\right )^5}dx}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\) |
| \(\Big \downarrow \) 253 |
| \(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {19 \int \frac {1}{(d x)^{5/2} \left (b x^2+a\right )^4}dx}{16 a}+\frac {1}{8 a d (d x)^{3/2} \left (a+b x^2\right )^4}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\) |
| \(\Big \downarrow \) 253 |
| \(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {19 \left (\frac {5 \int \frac {1}{(d x)^{5/2} \left (b x^2+a\right )^3}dx}{4 a}+\frac {1}{6 a d (d x)^{3/2} \left (a+b x^2\right )^3}\right )}{16 a}+\frac {1}{8 a d (d x)^{3/2} \left (a+b x^2\right )^4}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\) |
| \(\Big \downarrow \) 253 |
| \(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {19 \left (\frac {5 \left (\frac {11 \int \frac {1}{(d x)^{5/2} \left (b x^2+a\right )^2}dx}{8 a}+\frac {1}{4 a d (d x)^{3/2} \left (a+b x^2\right )^2}\right )}{4 a}+\frac {1}{6 a d (d x)^{3/2} \left (a+b x^2\right )^3}\right )}{16 a}+\frac {1}{8 a d (d x)^{3/2} \left (a+b x^2\right )^4}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\) |
| \(\Big \downarrow \) 253 |
| \(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {19 \left (\frac {5 \left (\frac {11 \left (\frac {7 \int \frac {1}{(d x)^{5/2} \left (b x^2+a\right )}dx}{4 a}+\frac {1}{2 a d (d x)^{3/2} \left (a+b x^2\right )}\right )}{8 a}+\frac {1}{4 a d (d x)^{3/2} \left (a+b x^2\right )^2}\right )}{4 a}+\frac {1}{6 a d (d x)^{3/2} \left (a+b x^2\right )^3}\right )}{16 a}+\frac {1}{8 a d (d x)^{3/2} \left (a+b x^2\right )^4}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\) |
| \(\Big \downarrow \) 264 |
| \(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {19 \left (\frac {5 \left (\frac {11 \left (\frac {7 \left (-\frac {b \int \frac {1}{\sqrt {d x} \left (b x^2+a\right )}dx}{a d^2}-\frac {2}{3 a d (d x)^{3/2}}\right )}{4 a}+\frac {1}{2 a d (d x)^{3/2} \left (a+b x^2\right )}\right )}{8 a}+\frac {1}{4 a d (d x)^{3/2} \left (a+b x^2\right )^2}\right )}{4 a}+\frac {1}{6 a d (d x)^{3/2} \left (a+b x^2\right )^3}\right )}{16 a}+\frac {1}{8 a d (d x)^{3/2} \left (a+b x^2\right )^4}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {19 \left (\frac {5 \left (\frac {11 \left (\frac {7 \left (-\frac {2 b \int \frac {1}{b x^2+a}d\sqrt {d x}}{a d^3}-\frac {2}{3 a d (d x)^{3/2}}\right )}{4 a}+\frac {1}{2 a d (d x)^{3/2} \left (a+b x^2\right )}\right )}{8 a}+\frac {1}{4 a d (d x)^{3/2} \left (a+b x^2\right )^2}\right )}{4 a}+\frac {1}{6 a d (d x)^{3/2} \left (a+b x^2\right )^3}\right )}{16 a}+\frac {1}{8 a d (d x)^{3/2} \left (a+b x^2\right )^4}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\) |
| \(\Big \downarrow \) 755 |
| \(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {19 \left (\frac {5 \left (\frac {11 \left (\frac {7 \left (-\frac {2 b \left (\frac {\int \frac {d^2 \left (\sqrt {a} d-\sqrt {b} d x\right )}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {a} d}+\frac {\int \frac {d^2 \left (\sqrt {b} x d+\sqrt {a} d\right )}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {a} d}\right )}{a d^3}-\frac {2}{3 a d (d x)^{3/2}}\right )}{4 a}+\frac {1}{2 a d (d x)^{3/2} \left (a+b x^2\right )}\right )}{8 a}+\frac {1}{4 a d (d x)^{3/2} \left (a+b x^2\right )^2}\right )}{4 a}+\frac {1}{6 a d (d x)^{3/2} \left (a+b x^2\right )^3}\right )}{16 a}+\frac {1}{8 a d (d x)^{3/2} \left (a+b x^2\right )^4}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {19 \left (\frac {5 \left (\frac {11 \left (\frac {7 \left (-\frac {2 b \left (\frac {d \int \frac {\sqrt {a} d-\sqrt {b} d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {a}}+\frac {d \int \frac {\sqrt {b} x d+\sqrt {a} d}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {a}}\right )}{a d^3}-\frac {2}{3 a d (d x)^{3/2}}\right )}{4 a}+\frac {1}{2 a d (d x)^{3/2} \left (a+b x^2\right )}\right )}{8 a}+\frac {1}{4 a d (d x)^{3/2} \left (a+b x^2\right )^2}\right )}{4 a}+\frac {1}{6 a d (d x)^{3/2} \left (a+b x^2\right )^3}\right )}{16 a}+\frac {1}{8 a d (d x)^{3/2} \left (a+b x^2\right )^4}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {19 \left (\frac {5 \left (\frac {11 \left (\frac {7 \left (-\frac {2 b \left (\frac {d \int \frac {\sqrt {a} d-\sqrt {b} d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {a}}+\frac {d \left (\frac {\int \frac {1}{x d+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}}d\sqrt {d x}}{2 \sqrt {b}}+\frac {\int \frac {1}{x d+\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}}d\sqrt {d x}}{2 \sqrt {b}}\right )}{2 \sqrt {a}}\right )}{a d^3}-\frac {2}{3 a d (d x)^{3/2}}\right )}{4 a}+\frac {1}{2 a d (d x)^{3/2} \left (a+b x^2\right )}\right )}{8 a}+\frac {1}{4 a d (d x)^{3/2} \left (a+b x^2\right )^2}\right )}{4 a}+\frac {1}{6 a d (d x)^{3/2} \left (a+b x^2\right )^3}\right )}{16 a}+\frac {1}{8 a d (d x)^{3/2} \left (a+b x^2\right )^4}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {19 \left (\frac {5 \left (\frac {11 \left (\frac {7 \left (-\frac {2 b \left (\frac {d \int \frac {\sqrt {a} d-\sqrt {b} d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {a}}+\frac {d \left (\frac {\int \frac {1}{-d x-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\int \frac {1}{-d x-1}d\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}\right )}{a d^3}-\frac {2}{3 a d (d x)^{3/2}}\right )}{4 a}+\frac {1}{2 a d (d x)^{3/2} \left (a+b x^2\right )}\right )}{8 a}+\frac {1}{4 a d (d x)^{3/2} \left (a+b x^2\right )^2}\right )}{4 a}+\frac {1}{6 a d (d x)^{3/2} \left (a+b x^2\right )^3}\right )}{16 a}+\frac {1}{8 a d (d x)^{3/2} \left (a+b x^2\right )^4}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {19 \left (\frac {5 \left (\frac {11 \left (\frac {7 \left (-\frac {2 b \left (\frac {d \int \frac {\sqrt {a} d-\sqrt {b} d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {a}}+\frac {d \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}\right )}{a d^3}-\frac {2}{3 a d (d x)^{3/2}}\right )}{4 a}+\frac {1}{2 a d (d x)^{3/2} \left (a+b x^2\right )}\right )}{8 a}+\frac {1}{4 a d (d x)^{3/2} \left (a+b x^2\right )^2}\right )}{4 a}+\frac {1}{6 a d (d x)^{3/2} \left (a+b x^2\right )^3}\right )}{16 a}+\frac {1}{8 a d (d x)^{3/2} \left (a+b x^2\right )^4}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {19 \left (\frac {5 \left (\frac {11 \left (\frac {7 \left (-\frac {2 b \left (\frac {d \left (-\frac {\int -\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d}-2 \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{b} \left (x d+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt [4]{a} \sqrt {d}+\sqrt {2} \sqrt [4]{b} \sqrt {d x}\right )}{\sqrt [4]{b} \left (x d+\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}+\frac {d \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}\right )}{a d^3}-\frac {2}{3 a d (d x)^{3/2}}\right )}{4 a}+\frac {1}{2 a d (d x)^{3/2} \left (a+b x^2\right )}\right )}{8 a}+\frac {1}{4 a d (d x)^{3/2} \left (a+b x^2\right )^2}\right )}{4 a}+\frac {1}{6 a d (d x)^{3/2} \left (a+b x^2\right )^3}\right )}{16 a}+\frac {1}{8 a d (d x)^{3/2} \left (a+b x^2\right )^4}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {19 \left (\frac {5 \left (\frac {11 \left (\frac {7 \left (-\frac {2 b \left (\frac {d \left (\frac {\int \frac {\sqrt {2} \sqrt [4]{a} \sqrt {d}-2 \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{b} \left (x d+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}+\frac {\int \frac {\sqrt {2} \left (\sqrt [4]{a} \sqrt {d}+\sqrt {2} \sqrt [4]{b} \sqrt {d x}\right )}{\sqrt [4]{b} \left (x d+\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}+\frac {d \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}\right )}{a d^3}-\frac {2}{3 a d (d x)^{3/2}}\right )}{4 a}+\frac {1}{2 a d (d x)^{3/2} \left (a+b x^2\right )}\right )}{8 a}+\frac {1}{4 a d (d x)^{3/2} \left (a+b x^2\right )^2}\right )}{4 a}+\frac {1}{6 a d (d x)^{3/2} \left (a+b x^2\right )^3}\right )}{16 a}+\frac {1}{8 a d (d x)^{3/2} \left (a+b x^2\right )^4}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {19 \left (\frac {5 \left (\frac {11 \left (\frac {7 \left (-\frac {2 b \left (\frac {d \left (\frac {\int \frac {\sqrt {2} \sqrt [4]{a} \sqrt {d}-2 \sqrt [4]{b} \sqrt {d x}}{x d+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt {b} \sqrt {d}}+\frac {\int \frac {\sqrt [4]{a} \sqrt {d}+\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{x d+\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}}d\sqrt {d x}}{2 \sqrt [4]{a} \sqrt {b} \sqrt {d}}\right )}{2 \sqrt {a}}+\frac {d \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}\right )}{a d^3}-\frac {2}{3 a d (d x)^{3/2}}\right )}{4 a}+\frac {1}{2 a d (d x)^{3/2} \left (a+b x^2\right )}\right )}{8 a}+\frac {1}{4 a d (d x)^{3/2} \left (a+b x^2\right )^2}\right )}{4 a}+\frac {1}{6 a d (d x)^{3/2} \left (a+b x^2\right )^3}\right )}{16 a}+\frac {1}{8 a d (d x)^{3/2} \left (a+b x^2\right )^4}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {19 \left (\frac {5 \left (\frac {11 \left (\frac {7 \left (-\frac {2 b \left (\frac {d \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}+\frac {d \left (\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d} \sqrt {d x}+\sqrt {a} d+\sqrt {b} d x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d} \sqrt {d x}+\sqrt {a} d+\sqrt {b} d x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}\right )}{a d^3}-\frac {2}{3 a d (d x)^{3/2}}\right )}{4 a}+\frac {1}{2 a d (d x)^{3/2} \left (a+b x^2\right )}\right )}{8 a}+\frac {1}{4 a d (d x)^{3/2} \left (a+b x^2\right )^2}\right )}{4 a}+\frac {1}{6 a d (d x)^{3/2} \left (a+b x^2\right )^3}\right )}{16 a}+\frac {1}{8 a d (d x)^{3/2} \left (a+b x^2\right )^4}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\) |
Input:
Int[1/((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*a*d*(d*x)^(3/2)*(a + b*x^2)^4) + (19*(1/(6*a*d*(d*x)^(3 /2)*(a + b*x^2)^3) + (5*(1/(4*a*d*(d*x)^(3/2)*(a + b*x^2)^2) + (11*(1/(2*a *d*(d*x)^(3/2)*(a + b*x^2)) + (7*(-2/(3*a*d*(d*x)^(3/2)) - (2*b*((d*(-(Arc Tan[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])))/( a*d^3)))/(4*a)))/(8*a)))/(4*a)))/(16*a)))/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*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[((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[(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]
Time = 0.21 (sec) , antiderivative size = 278, normalized size of antiderivative = 0.56
| method | result | size |
| risch | \(-\frac {2 \sqrt {\left (b \,x^{2}+a \right )^{2}}}{3 a^{5} x \sqrt {d x}\, d^{2} \left (b \,x^{2}+a \right )}-\frac {2 b \left (\frac {\frac {2925 a^{3} d^{6} \sqrt {d x}}{2048}+\frac {7019 b \,d^{4} a^{2} \left (d x \right )^{\frac {5}{2}}}{2048}+\frac {17933 a \,d^{2} b^{2} \left (d x \right )^{\frac {9}{2}}}{6144}+\frac {5267 b^{3} \left (d x \right )^{\frac {13}{2}}}{6144}}{\left (b \,d^{2} x^{2}+a \,d^{2}\right )^{4}}+\frac {7315 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{16384 a \,d^{2}}\right ) \sqrt {\left (b \,x^{2}+a \right )^{2}}}{a^{5} d \left (b \,x^{2}+a \right )}\) | \(278\) |
| default | \(\text {Expression too large to display}\) | \(1183\) |
Input:
int(1/(d*x)^(5/2)/(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x,method=_RETURNVERBOSE)
Output:
-2/3/a^5/x/(d*x)^(1/2)/d^2*((b*x^2+a)^2)^(1/2)/(b*x^2+a)-2/a^5*b/d*((2925/ 2048*a^3*d^6*(d*x)^(1/2)+7019/2048*b*d^4*a^2*(d*x)^(5/2)+17933/6144*a*d^2* b^2*(d*x)^(9/2)+5267/6144*b^3*(d*x)^(13/2))/(b*d^2*x^2+a*d^2)^4+7315/16384 *(a*d^2/b)^(1/4)/a/d^2*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)))*((b*x^2+a)^2)^(1/2)/(b*x^2+a)
Result contains complex when optimal does not.
Time = 0.09 (sec) , antiderivative size = 563, normalized size of antiderivative = 1.14 \[ \int \frac {1}{(d x)^{5/2} \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx=-\frac {21945 \, {\left (a^{5} b^{4} d^{3} x^{10} + 4 \, a^{6} b^{3} d^{3} x^{8} + 6 \, a^{7} b^{2} d^{3} x^{6} + 4 \, a^{8} b d^{3} x^{4} + a^{9} d^{3} x^{2}\right )} \left (-\frac {b^{3}}{a^{23} d^{10}}\right )^{\frac {1}{4}} \log \left (7315 \, a^{6} d^{3} \left (-\frac {b^{3}}{a^{23} d^{10}}\right )^{\frac {1}{4}} + 7315 \, \sqrt {d x} b\right ) + 21945 \, {\left (i \, a^{5} b^{4} d^{3} x^{10} + 4 i \, a^{6} b^{3} d^{3} x^{8} + 6 i \, a^{7} b^{2} d^{3} x^{6} + 4 i \, a^{8} b d^{3} x^{4} + i \, a^{9} d^{3} x^{2}\right )} \left (-\frac {b^{3}}{a^{23} d^{10}}\right )^{\frac {1}{4}} \log \left (7315 i \, a^{6} d^{3} \left (-\frac {b^{3}}{a^{23} d^{10}}\right )^{\frac {1}{4}} + 7315 \, \sqrt {d x} b\right ) + 21945 \, {\left (-i \, a^{5} b^{4} d^{3} x^{10} - 4 i \, a^{6} b^{3} d^{3} x^{8} - 6 i \, a^{7} b^{2} d^{3} x^{6} - 4 i \, a^{8} b d^{3} x^{4} - i \, a^{9} d^{3} x^{2}\right )} \left (-\frac {b^{3}}{a^{23} d^{10}}\right )^{\frac {1}{4}} \log \left (-7315 i \, a^{6} d^{3} \left (-\frac {b^{3}}{a^{23} d^{10}}\right )^{\frac {1}{4}} + 7315 \, \sqrt {d x} b\right ) - 21945 \, {\left (a^{5} b^{4} d^{3} x^{10} + 4 \, a^{6} b^{3} d^{3} x^{8} + 6 \, a^{7} b^{2} d^{3} x^{6} + 4 \, a^{8} b d^{3} x^{4} + a^{9} d^{3} x^{2}\right )} \left (-\frac {b^{3}}{a^{23} d^{10}}\right )^{\frac {1}{4}} \log \left (-7315 \, a^{6} d^{3} \left (-\frac {b^{3}}{a^{23} d^{10}}\right )^{\frac {1}{4}} + 7315 \, \sqrt {d x} b\right ) + 4 \, {\left (7315 \, b^{4} x^{8} + 26125 \, a b^{3} x^{6} + 33345 \, a^{2} b^{2} x^{4} + 16967 \, a^{3} b x^{2} + 2048 \, a^{4}\right )} \sqrt {d x}}{12288 \, {\left (a^{5} b^{4} d^{3} x^{10} + 4 \, a^{6} b^{3} d^{3} x^{8} + 6 \, a^{7} b^{2} d^{3} x^{6} + 4 \, a^{8} b d^{3} x^{4} + a^{9} d^{3} x^{2}\right )}} \] Input:
integrate(1/(d*x)^(5/2)/(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x, algorithm="fricas ")
Output:
-1/12288*(21945*(a^5*b^4*d^3*x^10 + 4*a^6*b^3*d^3*x^8 + 6*a^7*b^2*d^3*x^6 + 4*a^8*b*d^3*x^4 + a^9*d^3*x^2)*(-b^3/(a^23*d^10))^(1/4)*log(7315*a^6*d^3 *(-b^3/(a^23*d^10))^(1/4) + 7315*sqrt(d*x)*b) + 21945*(I*a^5*b^4*d^3*x^10 + 4*I*a^6*b^3*d^3*x^8 + 6*I*a^7*b^2*d^3*x^6 + 4*I*a^8*b*d^3*x^4 + I*a^9*d^ 3*x^2)*(-b^3/(a^23*d^10))^(1/4)*log(7315*I*a^6*d^3*(-b^3/(a^23*d^10))^(1/4 ) + 7315*sqrt(d*x)*b) + 21945*(-I*a^5*b^4*d^3*x^10 - 4*I*a^6*b^3*d^3*x^8 - 6*I*a^7*b^2*d^3*x^6 - 4*I*a^8*b*d^3*x^4 - I*a^9*d^3*x^2)*(-b^3/(a^23*d^10 ))^(1/4)*log(-7315*I*a^6*d^3*(-b^3/(a^23*d^10))^(1/4) + 7315*sqrt(d*x)*b) - 21945*(a^5*b^4*d^3*x^10 + 4*a^6*b^3*d^3*x^8 + 6*a^7*b^2*d^3*x^6 + 4*a^8* b*d^3*x^4 + a^9*d^3*x^2)*(-b^3/(a^23*d^10))^(1/4)*log(-7315*a^6*d^3*(-b^3/ (a^23*d^10))^(1/4) + 7315*sqrt(d*x)*b) + 4*(7315*b^4*x^8 + 26125*a*b^3*x^6 + 33345*a^2*b^2*x^4 + 16967*a^3*b*x^2 + 2048*a^4)*sqrt(d*x))/(a^5*b^4*d^3 *x^10 + 4*a^6*b^3*d^3*x^8 + 6*a^7*b^2*d^3*x^6 + 4*a^8*b*d^3*x^4 + a^9*d^3* x^2)
\[ \int \frac {1}{(d x)^{5/2} \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx=\int \frac {1}{\left (d x\right )^{\frac {5}{2}} \left (\left (a + b x^{2}\right )^{2}\right )^{\frac {5}{2}}}\, dx \] Input:
integrate(1/(d*x)**(5/2)/(b**2*x**4+2*a*b*x**2+a**2)**(5/2),x)
Output:
Integral(1/((d*x)**(5/2)*((a + b*x**2)**2)**(5/2)), x)
\[ \int \frac {1}{(d x)^{5/2} \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx=\int { \frac {1}{{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{\frac {5}{2}} \left (d x\right )^{\frac {5}{2}}} \,d x } \] Input:
integrate(1/(d*x)^(5/2)/(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x, algorithm="maxima ")
Output:
-4*b*integrate(1/((a^5*b*d^(5/2)*x^2 + a^6*d^(5/2))*sqrt(x)), x) - 1/3072* (13795*b^4*x^(13/2) + 34285*a*b^3*x^(9/2) + 29649*a^2*b^2*x^(5/2) + 8775*a ^3*b*sqrt(x))/(a^5*b^4*d^(5/2)*x^8 + 4*a^6*b^3*d^(5/2)*x^6 + 6*a^7*b^2*d^( 5/2)*x^4 + 4*a^8*b*d^(5/2)*x^2 + a^9*d^(5/2)) + 1/192*((533*b^6*x^5 + 882* a*b^5*x^3 + 381*a^2*b^4*x)*x^(11/2) + 2*(603*a*b^5*x^5 + 1014*a^2*b^4*x^3 + 443*a^3*b^3*x)*x^(7/2) + (705*a^2*b^4*x^5 + 1210*a^3*b^3*x^3 + 537*a^4*b ^2*x)*x^(3/2))/(a^8*b^3*d^(5/2)*x^6 + 3*a^9*b^2*d^(5/2)*x^4 + 3*a^10*b*d^( 5/2)*x^2 + a^11*d^(5/2) + (a^5*b^6*d^(5/2)*x^6 + 3*a^6*b^5*d^(5/2)*x^4 + 3 *a^7*b^4*d^(5/2)*x^2 + a^8*b^3*d^(5/2))*x^6 + 3*(a^6*b^5*d^(5/2)*x^6 + 3*a ^7*b^4*d^(5/2)*x^4 + 3*a^8*b^3*d^(5/2)*x^2 + a^9*b^2*d^(5/2))*x^4 + 3*(a^7 *b^4*d^(5/2)*x^6 + 3*a^8*b^3*d^(5/2)*x^4 + 3*a^9*b^2*d^(5/2)*x^2 + a^10*b* d^(5/2))*x^2) + 2925/8192*(2*sqrt(2)*b*arctan(1/2*sqrt(2)*(sqrt(2)*a^(1/4) *b^(1/4) + 2*sqrt(b)*sqrt(x))/sqrt(sqrt(a)*sqrt(b)))/(sqrt(a)*sqrt(sqrt(a) *sqrt(b))) + 2*sqrt(2)*b*arctan(-1/2*sqrt(2)*(sqrt(2)*a^(1/4)*b^(1/4) - 2* sqrt(b)*sqrt(x))/sqrt(sqrt(a)*sqrt(b)))/(sqrt(a)*sqrt(sqrt(a)*sqrt(b))) + sqrt(2)*b^(3/4)*log(sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x + sqrt(a)) /a^(3/4) - sqrt(2)*b^(3/4)*log(-sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)* x + sqrt(a))/a^(3/4))/(a^5*d^(5/2)) + integrate(1/((a^4*b*d^(5/2)*x^2 + a^ 5*d^(5/2))*x^(5/2)), x)
Time = 0.12 (sec) , antiderivative size = 397, normalized size of antiderivative = 0.80 \[ \int \frac {1}{(d x)^{5/2} \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx=-\frac {7315 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{4096 \, a^{6} d^{3} \mathrm {sgn}\left (b x^{2} + a\right )} - \frac {7315 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{4096 \, a^{6} d^{3} \mathrm {sgn}\left (b x^{2} + a\right )} - \frac {7315 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} \log \left (d x + \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{8192 \, a^{6} d^{3} \mathrm {sgn}\left (b x^{2} + a\right )} + \frac {7315 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} \log \left (d x - \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{8192 \, a^{6} d^{3} \mathrm {sgn}\left (b x^{2} + a\right )} - \frac {2}{3 \, \sqrt {d x} a^{5} d^{2} x \mathrm {sgn}\left (b x^{2} + a\right )} - \frac {5267 \, \sqrt {d x} b^{4} d^{6} x^{6} + 17933 \, \sqrt {d x} a b^{3} d^{6} x^{4} + 21057 \, \sqrt {d x} a^{2} b^{2} d^{6} x^{2} + 8775 \, \sqrt {d x} a^{3} b d^{6}}{3072 \, {\left (b d^{2} x^{2} + a d^{2}\right )}^{4} a^{5} d \mathrm {sgn}\left (b x^{2} + a\right )} \] Input:
integrate(1/(d*x)^(5/2)/(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x, algorithm="giac")
Output:
-7315/4096*sqrt(2)*(a*b^3*d^2)^(1/4)*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*d^3*sgn(b*x^2 + a)) - 7315/409 6*sqrt(2)*(a*b^3*d^2)^(1/4)*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*d^3*sgn(b*x^2 + a)) - 7315/8192*sqrt(2 )*(a*b^3*d^2)^(1/4)*log(d*x + sqrt(2)*(a*d^2/b)^(1/4)*sqrt(d*x) + sqrt(a*d ^2/b))/(a^6*d^3*sgn(b*x^2 + a)) + 7315/8192*sqrt(2)*(a*b^3*d^2)^(1/4)*log( d*x - sqrt(2)*(a*d^2/b)^(1/4)*sqrt(d*x) + sqrt(a*d^2/b))/(a^6*d^3*sgn(b*x^ 2 + a)) - 2/3/(sqrt(d*x)*a^5*d^2*x*sgn(b*x^2 + a)) - 1/3072*(5267*sqrt(d*x )*b^4*d^6*x^6 + 17933*sqrt(d*x)*a*b^3*d^6*x^4 + 21057*sqrt(d*x)*a^2*b^2*d^ 6*x^2 + 8775*sqrt(d*x)*a^3*b*d^6)/((b*d^2*x^2 + a*d^2)^4*a^5*d*sgn(b*x^2 + a))
Timed out. \[ \int \frac {1}{(d x)^{5/2} \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx=\int \frac {1}{{\left (d\,x\right )}^{5/2}\,{\left (a^2+2\,a\,b\,x^2+b^2\,x^4\right )}^{5/2}} \,d x \] Input:
int(1/((d*x)^(5/2)*(a^2 + b^2*x^4 + 2*a*b*x^2)^(5/2)),x)
Output:
int(1/((d*x)^(5/2)*(a^2 + b^2*x^4 + 2*a*b*x^2)^(5/2)), x)
Time = 0.18 (sec) , antiderivative size = 872, normalized size of antiderivative = 1.76 \[ \int \frac {1}{(d x)^{5/2} \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx =\text {Too large to display} \] Input:
int(1/(d*x)^(5/2)/(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x)
Output:
(sqrt(d)*(43890*sqrt(x)*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*x + 175560* sqrt(x)*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*x**3 + 263340*sqrt(x)*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**2*x**5 + 175560*sqrt(x)*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**3*x**7 + 43890*sqrt(x)*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**4*x**9 - 43890*sqrt(x)*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*x - 175560*sqrt(x)*b**(3/4)*a**(1/4)*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**3*b*x**3 - 26334 0*sqrt(x)*b**(3/4)*a**(1/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) + 2*sq rt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*a**2*b**2*x**5 - 175560*sqrt(x )*b**(3/4)*a**(1/4)*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)))*a*b**3*x**7 - 43890*sqrt(x)*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**4*x**9 + 21945*sqrt(x)*b**(3/4)*a**(1/4)*sqrt(2 )*log( - sqrt(x)*b**(1/4)*a**(1/4)*sqrt(2) + sqrt(a) + sqrt(b)*x)*a**4*...