Integrand size = 30, antiderivative size = 451 \[ \int \frac {(d x)^{3/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx=\frac {77 d \sqrt {d x}}{3072 a^3 b \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d \sqrt {d x}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {d \sqrt {d x}}{96 a b \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {11 d \sqrt {d x}}{768 a^2 b \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {77 d^{3/2} \left (a+b x^2\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{2048 \sqrt {2} a^{15/4} b^{5/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {77 d^{3/2} \left (a+b x^2\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{2048 \sqrt {2} a^{15/4} b^{5/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {77 d^{3/2} \left (a+b x^2\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}}{\sqrt {d} \left (\sqrt {a}+\sqrt {b} x\right )}\right )}{2048 \sqrt {2} a^{15/4} b^{5/4} \sqrt {a^2+2 a b x^2+b^2 x^4}} \] Output:
77/3072*d*(d*x)^(1/2)/a^3/b/((b*x^2+a)^2)^(1/2)-1/8*d*(d*x)^(1/2)/b/(b*x^2 +a)^3/((b*x^2+a)^2)^(1/2)+1/96*d*(d*x)^(1/2)/a/b/(b*x^2+a)^2/((b*x^2+a)^2) ^(1/2)+11/768*d*(d*x)^(1/2)/a^2/b/(b*x^2+a)/((b*x^2+a)^2)^(1/2)-77/4096*d^ (3/2)*(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^(15/4)/b^(5/4)/((b*x^2+a)^2)^(1/2)+77/4096*d^(3/2)*(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^(15/4)/b^(5/4)/(( b*x^2+a)^2)^(1/2)+77/4096*d^(3/2)*(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^(15/4)/b^(5/4)/((b*x^ 2+a)^2)^(1/2)
Time = 0.54 (sec) , antiderivative size = 201, normalized size of antiderivative = 0.45 \[ \int \frac {(d x)^{3/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx=\frac {(d x)^{3/2} \left (a+b x^2\right ) \left (4 a^{3/4} \sqrt [4]{b} \sqrt {x} \left (-231 a^3+351 a^2 b x^2+275 a b^2 x^4+77 b^3 x^6\right )-231 \sqrt {2} \left (a+b x^2\right )^4 \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )+231 \sqrt {2} \left (a+b x^2\right )^4 \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )\right )}{12288 a^{15/4} b^{5/4} x^{3/2} \left (\left (a+b x^2\right )^2\right )^{5/2}} \] Input:
Integrate[(d*x)^(3/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2),x]
Output:
((d*x)^(3/2)*(a + b*x^2)*(4*a^(3/4)*b^(1/4)*Sqrt[x]*(-231*a^3 + 351*a^2*b* x^2 + 275*a*b^2*x^4 + 77*b^3*x^6) - 231*Sqrt[2]*(a + b*x^2)^4*ArcTan[(Sqrt [a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])] + 231*Sqrt[2]*(a + b*x ^2)^4*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)]))/( 12288*a^(15/4)*b^(5/4)*x^(3/2)*((a + b*x^2)^2)^(5/2))
Time = 1.07 (sec) , antiderivative size = 438, normalized size of antiderivative = 0.97, 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, 755, 27, 1476, 1082, 217, 1479, 25, 27, 1103}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {(d x)^{3/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 1384 |
| \(\displaystyle \frac {b^5 \left (a+b x^2\right ) \int \frac {(d x)^{3/2}}{b^5 \left (b x^2+a\right )^5}dx}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\left (a+b x^2\right ) \int \frac {(d x)^{3/2}}{\left (b x^2+a\right )^5}dx}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\) |
| \(\Big \downarrow \) 252 |
| \(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {d^2 \int \frac {1}{\sqrt {d x} \left (b x^2+a\right )^4}dx}{16 b}-\frac {d \sqrt {d x}}{8 b \left (a+b x^2\right )^4}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\) |
| \(\Big \downarrow \) 253 |
| \(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {d^2 \left (\frac {11 \int \frac {1}{\sqrt {d x} \left (b x^2+a\right )^3}dx}{12 a}+\frac {\sqrt {d x}}{6 a d \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d \sqrt {d x}}{8 b \left (a+b x^2\right )^4}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\) |
| \(\Big \downarrow \) 253 |
| \(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {d^2 \left (\frac {11 \left (\frac {7 \int \frac {1}{\sqrt {d x} \left (b x^2+a\right )^2}dx}{8 a}+\frac {\sqrt {d x}}{4 a d \left (a+b x^2\right )^2}\right )}{12 a}+\frac {\sqrt {d x}}{6 a d \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d \sqrt {d x}}{8 b \left (a+b x^2\right )^4}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\) |
| \(\Big \downarrow \) 253 |
| \(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {d^2 \left (\frac {11 \left (\frac {7 \left (\frac {3 \int \frac {1}{\sqrt {d x} \left (b x^2+a\right )}dx}{4 a}+\frac {\sqrt {d x}}{2 a d \left (a+b x^2\right )}\right )}{8 a}+\frac {\sqrt {d x}}{4 a d \left (a+b x^2\right )^2}\right )}{12 a}+\frac {\sqrt {d x}}{6 a d \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d \sqrt {d x}}{8 b \left (a+b x^2\right )^4}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {d^2 \left (\frac {11 \left (\frac {7 \left (\frac {3 \int \frac {1}{b x^2+a}d\sqrt {d x}}{2 a d}+\frac {\sqrt {d x}}{2 a d \left (a+b x^2\right )}\right )}{8 a}+\frac {\sqrt {d x}}{4 a d \left (a+b x^2\right )^2}\right )}{12 a}+\frac {\sqrt {d x}}{6 a d \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d \sqrt {d x}}{8 b \left (a+b x^2\right )^4}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\) |
| \(\Big \downarrow \) 755 |
| \(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {d^2 \left (\frac {11 \left (\frac {7 \left (\frac {3 \left (\frac {\int \frac {d^2 \left (\sqrt {a} d-\sqrt {b} d x\right )}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {a} d}+\frac {\int \frac {d^2 \left (\sqrt {b} x d+\sqrt {a} d\right )}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {a} d}\right )}{2 a d}+\frac {\sqrt {d x}}{2 a d \left (a+b x^2\right )}\right )}{8 a}+\frac {\sqrt {d x}}{4 a d \left (a+b x^2\right )^2}\right )}{12 a}+\frac {\sqrt {d x}}{6 a d \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d \sqrt {d x}}{8 b \left (a+b x^2\right )^4}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {d^2 \left (\frac {11 \left (\frac {7 \left (\frac {3 \left (\frac {d \int \frac {\sqrt {a} d-\sqrt {b} d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {a}}+\frac {d \int \frac {\sqrt {b} x d+\sqrt {a} d}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {a}}\right )}{2 a d}+\frac {\sqrt {d x}}{2 a d \left (a+b x^2\right )}\right )}{8 a}+\frac {\sqrt {d x}}{4 a d \left (a+b x^2\right )^2}\right )}{12 a}+\frac {\sqrt {d x}}{6 a d \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d \sqrt {d x}}{8 b \left (a+b x^2\right )^4}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {d^2 \left (\frac {11 \left (\frac {7 \left (\frac {3 \left (\frac {d \int \frac {\sqrt {a} d-\sqrt {b} d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {a}}+\frac {d \left (\frac {\int \frac {1}{x d+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}}d\sqrt {d x}}{2 \sqrt {b}}+\frac {\int \frac {1}{x d+\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}}d\sqrt {d x}}{2 \sqrt {b}}\right )}{2 \sqrt {a}}\right )}{2 a d}+\frac {\sqrt {d x}}{2 a d \left (a+b x^2\right )}\right )}{8 a}+\frac {\sqrt {d x}}{4 a d \left (a+b x^2\right )^2}\right )}{12 a}+\frac {\sqrt {d x}}{6 a d \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d \sqrt {d x}}{8 b \left (a+b x^2\right )^4}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {d^2 \left (\frac {11 \left (\frac {7 \left (\frac {3 \left (\frac {d \int \frac {\sqrt {a} d-\sqrt {b} d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {a}}+\frac {d \left (\frac {\int \frac {1}{-d x-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\int \frac {1}{-d x-1}d\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}\right )}{2 a d}+\frac {\sqrt {d x}}{2 a d \left (a+b x^2\right )}\right )}{8 a}+\frac {\sqrt {d x}}{4 a d \left (a+b x^2\right )^2}\right )}{12 a}+\frac {\sqrt {d x}}{6 a d \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d \sqrt {d x}}{8 b \left (a+b x^2\right )^4}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {d^2 \left (\frac {11 \left (\frac {7 \left (\frac {3 \left (\frac {d \int \frac {\sqrt {a} d-\sqrt {b} d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {a}}+\frac {d \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}\right )}{2 a d}+\frac {\sqrt {d x}}{2 a d \left (a+b x^2\right )}\right )}{8 a}+\frac {\sqrt {d x}}{4 a d \left (a+b x^2\right )^2}\right )}{12 a}+\frac {\sqrt {d x}}{6 a d \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d \sqrt {d x}}{8 b \left (a+b x^2\right )^4}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {d^2 \left (\frac {11 \left (\frac {7 \left (\frac {3 \left (\frac {d \left (-\frac {\int -\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d}-2 \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{b} \left (x d+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt [4]{a} \sqrt {d}+\sqrt {2} \sqrt [4]{b} \sqrt {d x}\right )}{\sqrt [4]{b} \left (x d+\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}+\frac {d \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}\right )}{2 a d}+\frac {\sqrt {d x}}{2 a d \left (a+b x^2\right )}\right )}{8 a}+\frac {\sqrt {d x}}{4 a d \left (a+b x^2\right )^2}\right )}{12 a}+\frac {\sqrt {d x}}{6 a d \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d \sqrt {d x}}{8 b \left (a+b x^2\right )^4}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {d^2 \left (\frac {11 \left (\frac {7 \left (\frac {3 \left (\frac {d \left (\frac {\int \frac {\sqrt {2} \sqrt [4]{a} \sqrt {d}-2 \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{b} \left (x d+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}+\frac {\int \frac {\sqrt {2} \left (\sqrt [4]{a} \sqrt {d}+\sqrt {2} \sqrt [4]{b} \sqrt {d x}\right )}{\sqrt [4]{b} \left (x d+\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}+\frac {d \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}\right )}{2 a d}+\frac {\sqrt {d x}}{2 a d \left (a+b x^2\right )}\right )}{8 a}+\frac {\sqrt {d x}}{4 a d \left (a+b x^2\right )^2}\right )}{12 a}+\frac {\sqrt {d x}}{6 a d \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d \sqrt {d x}}{8 b \left (a+b x^2\right )^4}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {d^2 \left (\frac {11 \left (\frac {7 \left (\frac {3 \left (\frac {d \left (\frac {\int \frac {\sqrt {2} \sqrt [4]{a} \sqrt {d}-2 \sqrt [4]{b} \sqrt {d x}}{x d+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt {b} \sqrt {d}}+\frac {\int \frac {\sqrt [4]{a} \sqrt {d}+\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{x d+\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}}d\sqrt {d x}}{2 \sqrt [4]{a} \sqrt {b} \sqrt {d}}\right )}{2 \sqrt {a}}+\frac {d \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}\right )}{2 a d}+\frac {\sqrt {d x}}{2 a d \left (a+b x^2\right )}\right )}{8 a}+\frac {\sqrt {d x}}{4 a d \left (a+b x^2\right )^2}\right )}{12 a}+\frac {\sqrt {d x}}{6 a d \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d \sqrt {d x}}{8 b \left (a+b x^2\right )^4}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {d^2 \left (\frac {11 \left (\frac {7 \left (\frac {3 \left (\frac {d \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}+\frac {d \left (\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d} \sqrt {d x}+\sqrt {a} d+\sqrt {b} d x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d} \sqrt {d x}+\sqrt {a} d+\sqrt {b} d x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}\right )}{2 a d}+\frac {\sqrt {d x}}{2 a d \left (a+b x^2\right )}\right )}{8 a}+\frac {\sqrt {d x}}{4 a d \left (a+b x^2\right )^2}\right )}{12 a}+\frac {\sqrt {d x}}{6 a d \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d \sqrt {d x}}{8 b \left (a+b x^2\right )^4}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\) |
Input:
Int[(d*x)^(3/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2),x]
Output:
((a + b*x^2)*(-1/8*(d*Sqrt[d*x])/(b*(a + b*x^2)^4) + (d^2*(Sqrt[d*x]/(6*a* d*(a + b*x^2)^3) + (11*(Sqrt[d*x]/(4*a*d*(a + b*x^2)^2) + (7*(Sqrt[d*x]/(2 *a*d*(a + b*x^2)) + (3*((d*(-(ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1 /4)*Sqrt[d])]/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d])) + ArcTan[1 + (Sqrt[2]*b^( 1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])]/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d])))/(2* Sqrt[a]) + (d*(-1/2*Log[Sqrt[a]*d + Sqrt[b]*d*x - Sqrt[2]*a^(1/4)*b^(1/4)* Sqrt[d]*Sqrt[d*x]]/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d]) + Log[Sqrt[a]*d + Sqr t[b]*d*x + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d]*Sqrt[d*x]]/(2*Sqrt[2]*a^(1/4)*b ^(1/4)*Sqrt[d])))/(2*Sqrt[a])))/(2*a*d)))/(8*a)))/(12*a)))/(16*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[((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]
Leaf count of result is larger than twice the leaf count of optimal. \(1135\) vs. \(2(294)=588\).
Time = 0.16 (sec) , antiderivative size = 1136, normalized size of antiderivative = 2.52
Input:
int((d*x)^(3/2)/(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x,method=_RETURNVERBOSE)
Output:
1/24576*(231*(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)))*b^4*d^6*x^8+462*(a*d^2/b)^(1/4)*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^6*x^8+462*(a*d^2/b)^(1/4)*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^6*x ^8+924*(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) ))*a*b^3*d^6*x^6+1848*(a*d^2/b)^(1/4)*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^6*x^6+1848*(a*d^2/b)^(1/4)*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^6 *x^6+616*(d*x)^(13/2)*a*b^3+1386*(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)))*a^2*b^2*d^6*x^4+2772*(a*d^2/b)^(1/4)*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^6* x^4+2772*(a*d^2/b)^(1/4)*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^6*x^4+2200*(d*x)^(9/2)*a^2*b^2*d^2+924*(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)))*a^3*b*d^ 6*x^2+1848*(a*d^2/b)^(1/4)*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^6*x^2+1848*(a*d^2/b)^(1/4)*2^(1/2)*arcta...
Result contains complex when optimal does not.
Time = 0.09 (sec) , antiderivative size = 473, normalized size of antiderivative = 1.05 \[ \int \frac {(d x)^{3/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx=\frac {231 \, {\left (a^{3} b^{5} x^{8} + 4 \, a^{4} b^{4} x^{6} + 6 \, a^{5} b^{3} x^{4} + 4 \, a^{6} b^{2} x^{2} + a^{7} b\right )} \left (-\frac {d^{6}}{a^{15} b^{5}}\right )^{\frac {1}{4}} \log \left (77 \, a^{4} b \left (-\frac {d^{6}}{a^{15} b^{5}}\right )^{\frac {1}{4}} + 77 \, \sqrt {d x} d\right ) - 231 \, {\left (-i \, a^{3} b^{5} x^{8} - 4 i \, a^{4} b^{4} x^{6} - 6 i \, a^{5} b^{3} x^{4} - 4 i \, a^{6} b^{2} x^{2} - i \, a^{7} b\right )} \left (-\frac {d^{6}}{a^{15} b^{5}}\right )^{\frac {1}{4}} \log \left (77 i \, a^{4} b \left (-\frac {d^{6}}{a^{15} b^{5}}\right )^{\frac {1}{4}} + 77 \, \sqrt {d x} d\right ) - 231 \, {\left (i \, a^{3} b^{5} x^{8} + 4 i \, a^{4} b^{4} x^{6} + 6 i \, a^{5} b^{3} x^{4} + 4 i \, a^{6} b^{2} x^{2} + i \, a^{7} b\right )} \left (-\frac {d^{6}}{a^{15} b^{5}}\right )^{\frac {1}{4}} \log \left (-77 i \, a^{4} b \left (-\frac {d^{6}}{a^{15} b^{5}}\right )^{\frac {1}{4}} + 77 \, \sqrt {d x} d\right ) - 231 \, {\left (a^{3} b^{5} x^{8} + 4 \, a^{4} b^{4} x^{6} + 6 \, a^{5} b^{3} x^{4} + 4 \, a^{6} b^{2} x^{2} + a^{7} b\right )} \left (-\frac {d^{6}}{a^{15} b^{5}}\right )^{\frac {1}{4}} \log \left (-77 \, a^{4} b \left (-\frac {d^{6}}{a^{15} b^{5}}\right )^{\frac {1}{4}} + 77 \, \sqrt {d x} d\right ) + 4 \, {\left (77 \, b^{3} d x^{6} + 275 \, a b^{2} d x^{4} + 351 \, a^{2} b d x^{2} - 231 \, a^{3} d\right )} \sqrt {d x}}{12288 \, {\left (a^{3} b^{5} x^{8} + 4 \, a^{4} b^{4} x^{6} + 6 \, a^{5} b^{3} x^{4} + 4 \, a^{6} b^{2} x^{2} + a^{7} b\right )}} \] Input:
integrate((d*x)^(3/2)/(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x, algorithm="fricas")
Output:
1/12288*(231*(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^6/(a^15*b^5))^(1/4)*log(77*a^4*b*(-d^6/(a^15*b^5))^(1/4) + 77 *sqrt(d*x)*d) - 231*(-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^6/(a^15*b^5))^(1/4)*log(77*I*a^4*b*(-d^6/(a ^15*b^5))^(1/4) + 77*sqrt(d*x)*d) - 231*(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^6/(a^15*b^5))^(1/4)*log( -77*I*a^4*b*(-d^6/(a^15*b^5))^(1/4) + 77*sqrt(d*x)*d) - 231*(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^6/(a^15*b^5))^ (1/4)*log(-77*a^4*b*(-d^6/(a^15*b^5))^(1/4) + 77*sqrt(d*x)*d) + 4*(77*b^3* d*x^6 + 275*a*b^2*d*x^4 + 351*a^2*b*d*x^2 - 231*a^3*d)*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)
\[ \int \frac {(d x)^{3/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx=\int \frac {\left (d x\right )^{\frac {3}{2}}}{\left (\left (a + b x^{2}\right )^{2}\right )^{\frac {5}{2}}}\, dx \] Input:
integrate((d*x)**(3/2)/(b**2*x**4+2*a*b*x**2+a**2)**(5/2),x)
Output:
Integral((d*x)**(3/2)/((a + b*x**2)**2)**(5/2), x)
Time = 0.16 (sec) , antiderivative size = 586, normalized size of antiderivative = 1.30 \[ \int \frac {(d x)^{3/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx=-\frac {385 \, b^{3} d^{\frac {3}{2}} x^{\frac {13}{2}} + 495 \, a b^{2} d^{\frac {3}{2}} x^{\frac {9}{2}} + 315 \, a^{2} b d^{\frac {3}{2}} x^{\frac {5}{2}} + 77 \, a^{3} d^{\frac {3}{2}} \sqrt {x}}{1024 \, {\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 (77 \, b^{4} d^{\frac {3}{2}} x^{5} + 66 \, a b^{3} d^{\frac {3}{2}} x^{3} + 21 \, a^{2} b^{2} d^{\frac {3}{2}} x\right )} x^{\frac {11}{2}} + 2 \, {\left (99 \, a b^{3} d^{\frac {3}{2}} x^{5} + 102 \, a^{2} b^{2} d^{\frac {3}{2}} x^{3} + 35 \, a^{3} b d^{\frac {3}{2}} x\right )} x^{\frac {7}{2}} + {\left (153 \, a^{2} b^{2} d^{\frac {3}{2}} x^{5} + 202 \, a^{3} b d^{\frac {3}{2}} x^{3} + 81 \, a^{4} d^{\frac {3}{2}} x\right )} x^{\frac {3}{2}}}{192 \, {\left (a^{6} b^{3} x^{6} + 3 \, a^{7} b^{2} x^{4} + 3 \, a^{8} b x^{2} + a^{9} + {\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^{6} + 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^{4} + 3 \, {\left (a^{5} b^{4} x^{6} + 3 \, a^{6} b^{3} x^{4} + 3 \, a^{7} b^{2} x^{2} + a^{8} b\right )} x^{2}\right )}} + \frac {77 \, d {\left (\frac {2 \, \sqrt {2} \sqrt {d} \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 {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {2 \, \sqrt {2} \sqrt {d} \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 {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {\sqrt {2} \sqrt {d} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {1}{4}}} - \frac {\sqrt {2} \sqrt {d} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {1}{4}}}\right )}}{8192 \, a^{3} b} \] Input:
integrate((d*x)^(3/2)/(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x, algorithm="maxima")
Output:
-1/1024*(385*b^3*d^(3/2)*x^(13/2) + 495*a*b^2*d^(3/2)*x^(9/2) + 315*a^2*b* d^(3/2)*x^(5/2) + 77*a^3*d^(3/2)*sqrt(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) + 1/192*((77*b^4*d^(3/2)*x^5 + 66*a* b^3*d^(3/2)*x^3 + 21*a^2*b^2*d^(3/2)*x)*x^(11/2) + 2*(99*a*b^3*d^(3/2)*x^5 + 102*a^2*b^2*d^(3/2)*x^3 + 35*a^3*b*d^(3/2)*x)*x^(7/2) + (153*a^2*b^2*d^ (3/2)*x^5 + 202*a^3*b*d^(3/2)*x^3 + 81*a^4*d^(3/2)*x)*x^(3/2))/(a^6*b^3*x^ 6 + 3*a^7*b^2*x^4 + 3*a^8*b*x^2 + a^9 + (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^6 + 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^4 + 3*(a^5*b^4*x^6 + 3*a^6*b^3*x^4 + 3*a^7*b^2*x^2 + a^8*b)* x^2) + 77/8192*d*(2*sqrt(2)*sqrt(d)*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)*sq rt(b))) + 2*sqrt(2)*sqrt(d)*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)*sqrt(d)*log(sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x + sqrt( a))/(a^(3/4)*b^(1/4)) - sqrt(2)*sqrt(d)*log(-sqrt(2)*a^(1/4)*b^(1/4)*sqrt( x) + sqrt(b)*x + sqrt(a))/(a^(3/4)*b^(1/4)))/(a^3*b)
Time = 0.19 (sec) , antiderivative size = 385, normalized size of antiderivative = 0.85 \[ \int \frac {(d x)^{3/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx=\frac {\frac {462 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} d^{2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{a^{4} b^{2} \mathrm {sgn}\left (b x^{2} + a\right )} + \frac {462 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} d^{2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{a^{4} b^{2} \mathrm {sgn}\left (b x^{2} + a\right )} + \frac {231 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} d^{2} \log \left (d x + \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{a^{4} b^{2} \mathrm {sgn}\left (b x^{2} + a\right )} - \frac {231 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} d^{2} \log \left (d x - \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{a^{4} b^{2} \mathrm {sgn}\left (b x^{2} + a\right )} + \frac {8 \, {\left (77 \, \sqrt {d x} b^{3} d^{10} x^{6} + 275 \, \sqrt {d x} a b^{2} d^{10} x^{4} + 351 \, \sqrt {d x} a^{2} b d^{10} x^{2} - 231 \, \sqrt {d x} a^{3} d^{10}\right )}}{{\left (b d^{2} x^{2} + a d^{2}\right )}^{4} a^{3} b \mathrm {sgn}\left (b x^{2} + a\right )}}{24576 \, d} \] Input:
integrate((d*x)^(3/2)/(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x, algorithm="giac")
Output:
1/24576*(462*sqrt(2)*(a*b^3*d^2)^(1/4)*d^2*arctan(1/2*sqrt(2)*(sqrt(2)*(a* d^2/b)^(1/4) + 2*sqrt(d*x))/(a*d^2/b)^(1/4))/(a^4*b^2*sgn(b*x^2 + a)) + 46 2*sqrt(2)*(a*b^3*d^2)^(1/4)*d^2*arctan(-1/2*sqrt(2)*(sqrt(2)*(a*d^2/b)^(1/ 4) - 2*sqrt(d*x))/(a*d^2/b)^(1/4))/(a^4*b^2*sgn(b*x^2 + a)) + 231*sqrt(2)* (a*b^3*d^2)^(1/4)*d^2*log(d*x + sqrt(2)*(a*d^2/b)^(1/4)*sqrt(d*x) + sqrt(a *d^2/b))/(a^4*b^2*sgn(b*x^2 + a)) - 231*sqrt(2)*(a*b^3*d^2)^(1/4)*d^2*log( d*x - sqrt(2)*(a*d^2/b)^(1/4)*sqrt(d*x) + sqrt(a*d^2/b))/(a^4*b^2*sgn(b*x^ 2 + a)) + 8*(77*sqrt(d*x)*b^3*d^10*x^6 + 275*sqrt(d*x)*a*b^2*d^10*x^4 + 35 1*sqrt(d*x)*a^2*b*d^10*x^2 - 231*sqrt(d*x)*a^3*d^10)/((b*d^2*x^2 + a*d^2)^ 4*a^3*b*sgn(b*x^2 + a)))/d
Timed out. \[ \int \frac {(d x)^{3/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx=\int \frac {{\left (d\,x\right )}^{3/2}}{{\left (a^2+2\,a\,b\,x^2+b^2\,x^4\right )}^{5/2}} \,d x \] Input:
int((d*x)^(3/2)/(a^2 + b^2*x^4 + 2*a*b*x^2)^(5/2),x)
Output:
int((d*x)^(3/2)/(a^2 + b^2*x^4 + 2*a*b*x^2)^(5/2), x)
Time = 0.18 (sec) , antiderivative size = 822, normalized size of antiderivative = 1.82 \[ \int \frac {(d x)^{3/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx =\text {Too large to display} \] Input:
int((d*x)^(3/2)/(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x)
Output:
(sqrt(d)*d*( - 462*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 - 1848*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**2 - 2772*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**4 - 1848*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**6 - 462*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**8 + 462*b**(3/4)*a**(1/4)*sq rt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) + 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1 /4)*sqrt(2)))*a**4 + 1848*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**2 + 2772*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**4 + 1848*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**6 + 462*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**8 - 231*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 - 924*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**3*b*x**2 -...