Integrand size = 42, antiderivative size = 757 \[ \int \frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{x^2 \left (-b+a^2 x^2\right )^{3/2}} \, dx=-\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{x \sqrt {-b+a^2 x^2} \left (-b+2 a^2 x^2\right )+x \left (-2 a b x+2 a^3 x^3\right )}-\frac {a \arctan \left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{b}}\right )}{2 b^{15/8}}-\frac {\sqrt {2-\sqrt {2}} a \arctan \left (\frac {\left (\sqrt {\frac {2}{2-\sqrt {2}}} \sqrt [8]{b}-\frac {2 \sqrt [8]{b}}{\sqrt {2-\sqrt {2}}}\right ) \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{-\sqrt [4]{b}+\sqrt {a x+\sqrt {-b+a^2 x^2}}}\right )}{4 b^{15/8}}+\frac {a \arctan \left (\frac {\sqrt {2} \sqrt [8]{b} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{-\sqrt [4]{b}+\sqrt {a x+\sqrt {-b+a^2 x^2}}}\right )}{2 \sqrt {2} b^{15/8}}+\frac {\sqrt {2+\sqrt {2}} a \arctan \left (\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{b} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{-\sqrt [4]{b}+\sqrt {a x+\sqrt {-b+a^2 x^2}}}\right )}{4 b^{15/8}}-\frac {a \text {arctanh}\left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{b}}\right )}{2 b^{15/8}}-\frac {a \text {arctanh}\left (\frac {\frac {\sqrt [8]{b}}{\sqrt {2}}+\frac {\sqrt {a x+\sqrt {-b+a^2 x^2}}}{\sqrt {2} \sqrt [8]{b}}}{\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}\right )}{2 \sqrt {2} b^{15/8}}-\frac {\sqrt {2-\sqrt {2}} a \text {arctanh}\left (\frac {\frac {\sqrt [8]{b}}{\sqrt {2-\sqrt {2}}}+\frac {\sqrt {a x+\sqrt {-b+a^2 x^2}}}{\sqrt {2-\sqrt {2}} \sqrt [8]{b}}}{\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}\right )}{4 b^{15/8}}-\frac {\sqrt {2+\sqrt {2}} a \text {arctanh}\left (\frac {\frac {\sqrt [8]{b}}{\sqrt {2+\sqrt {2}}}+\frac {\sqrt {a x+\sqrt {-b+a^2 x^2}}}{\sqrt {2+\sqrt {2}} \sqrt [8]{b}}}{\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}\right )}{4 b^{15/8}} \]
-(a*x+(a^2*x^2-b)^(1/2))^(1/4)/(x*(a^2*x^2-b)^(1/2)*(2*a^2*x^2-b)+x*(2*a^3 *x^3-2*a*b*x))-1/2*a*arctan((a*x+(a^2*x^2-b)^(1/2))^(1/4)/b^(1/8))/b^(15/8 )-1/4*(2-2^(1/2))^(1/2)*a*arctan((2^(1/2)/(2-2^(1/2))^(1/2)*b^(1/8)-2*b^(1 /8)/(2-2^(1/2))^(1/2))*(a*x+(a^2*x^2-b)^(1/2))^(1/4)/(-b^(1/4)+(a*x+(a^2*x ^2-b)^(1/2))^(1/2)))/b^(15/8)+1/4*a*arctan(2^(1/2)*b^(1/8)*(a*x+(a^2*x^2-b )^(1/2))^(1/4)/(-b^(1/4)+(a*x+(a^2*x^2-b)^(1/2))^(1/2)))*2^(1/2)/b^(15/8)+ 1/4*(2+2^(1/2))^(1/2)*a*arctan((2+2^(1/2))^(1/2)*b^(1/8)*(a*x+(a^2*x^2-b)^ (1/2))^(1/4)/(-b^(1/4)+(a*x+(a^2*x^2-b)^(1/2))^(1/2)))/b^(15/8)-1/2*a*arct anh((a*x+(a^2*x^2-b)^(1/2))^(1/4)/b^(1/8))/b^(15/8)-1/4*a*arctanh((1/2*b^( 1/8)*2^(1/2)+1/2*(a*x+(a^2*x^2-b)^(1/2))^(1/2)*2^(1/2)/b^(1/8))/(a*x+(a^2* x^2-b)^(1/2))^(1/4))*2^(1/2)/b^(15/8)-1/4*(2-2^(1/2))^(1/2)*a*arctanh((b^( 1/8)/(2-2^(1/2))^(1/2)+(a*x+(a^2*x^2-b)^(1/2))^(1/2)/(2-2^(1/2))^(1/2)/b^( 1/8))/(a*x+(a^2*x^2-b)^(1/2))^(1/4))/b^(15/8)-1/4*(2+2^(1/2))^(1/2)*a*arct anh((b^(1/8)/(2+2^(1/2))^(1/2)+(a*x+(a^2*x^2-b)^(1/2))^(1/2)/(2+2^(1/2))^( 1/2)/b^(1/8))/(a*x+(a^2*x^2-b)^(1/2))^(1/4))/b^(15/8)
Time = 3.88 (sec) , antiderivative size = 679, normalized size of antiderivative = 0.90 \[ \int \frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{x^2 \left (-b+a^2 x^2\right )^{3/2}} \, dx=\frac {1}{4} \left (-\frac {4 \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{x \left (-2 a b x+2 a^3 x^3-\left (b-2 a^2 x^2\right ) \sqrt {-b+a^2 x^2}\right )}-\frac {2 a \arctan \left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{b}}\right )}{b^{15/8}}+\frac {\sqrt {2} a \arctan \left (\frac {\sqrt {2} \sqrt [8]{b} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{-\sqrt [4]{b}+\sqrt {a x+\sqrt {-b+a^2 x^2}}}\right )}{b^{15/8}}+\frac {\sqrt {2-\sqrt {2}} a \arctan \left (\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{b} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{-\sqrt [4]{b}+\sqrt {a x+\sqrt {-b+a^2 x^2}}}\right )}{b^{15/8}}+\frac {\sqrt {2+\sqrt {2}} a \arctan \left (\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{b} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{-\sqrt [4]{b}+\sqrt {a x+\sqrt {-b+a^2 x^2}}}\right )}{b^{15/8}}-\frac {2 a \text {arctanh}\left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{b}}\right )}{b^{15/8}}-\frac {\sqrt {2} a \text {arctanh}\left (\frac {\sqrt [4]{b}+\sqrt {a x+\sqrt {-b+a^2 x^2}}}{\sqrt {2} \sqrt [8]{b} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}\right )}{b^{15/8}}-\frac {\sqrt {2+\sqrt {2}} a \text {arctanh}\left (\frac {\sqrt {1-\frac {1}{\sqrt {2}}} \left (\sqrt [4]{b}+\sqrt {a x+\sqrt {-b+a^2 x^2}}\right )}{\sqrt [8]{b} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}\right )}{b^{15/8}}-\frac {\sqrt {2-\sqrt {2}} a \text {arctanh}\left (\frac {\sqrt {1+\frac {1}{\sqrt {2}}} \left (\sqrt [4]{b}+\sqrt {a x+\sqrt {-b+a^2 x^2}}\right )}{\sqrt [8]{b} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}\right )}{b^{15/8}}\right ) \]
((-4*(a*x + Sqrt[-b + a^2*x^2])^(1/4))/(x*(-2*a*b*x + 2*a^3*x^3 - (b - 2*a ^2*x^2)*Sqrt[-b + a^2*x^2])) - (2*a*ArcTan[(a*x + Sqrt[-b + a^2*x^2])^(1/4 )/b^(1/8)])/b^(15/8) + (Sqrt[2]*a*ArcTan[(Sqrt[2]*b^(1/8)*(a*x + Sqrt[-b + a^2*x^2])^(1/4))/(-b^(1/4) + Sqrt[a*x + Sqrt[-b + a^2*x^2]])])/b^(15/8) + (Sqrt[2 - Sqrt[2]]*a*ArcTan[(Sqrt[2 - Sqrt[2]]*b^(1/8)*(a*x + Sqrt[-b + a ^2*x^2])^(1/4))/(-b^(1/4) + Sqrt[a*x + Sqrt[-b + a^2*x^2]])])/b^(15/8) + ( Sqrt[2 + Sqrt[2]]*a*ArcTan[(Sqrt[2 + Sqrt[2]]*b^(1/8)*(a*x + Sqrt[-b + a^2 *x^2])^(1/4))/(-b^(1/4) + Sqrt[a*x + Sqrt[-b + a^2*x^2]])])/b^(15/8) - (2* a*ArcTanh[(a*x + Sqrt[-b + a^2*x^2])^(1/4)/b^(1/8)])/b^(15/8) - (Sqrt[2]*a *ArcTanh[(b^(1/4) + Sqrt[a*x + Sqrt[-b + a^2*x^2]])/(Sqrt[2]*b^(1/8)*(a*x + Sqrt[-b + a^2*x^2])^(1/4))])/b^(15/8) - (Sqrt[2 + Sqrt[2]]*a*ArcTanh[(Sq rt[1 - 1/Sqrt[2]]*(b^(1/4) + Sqrt[a*x + Sqrt[-b + a^2*x^2]]))/(b^(1/8)*(a* x + Sqrt[-b + a^2*x^2])^(1/4))])/b^(15/8) - (Sqrt[2 - Sqrt[2]]*a*ArcTanh[( Sqrt[1 + 1/Sqrt[2]]*(b^(1/4) + Sqrt[a*x + Sqrt[-b + a^2*x^2]]))/(b^(1/8)*( a*x + Sqrt[-b + a^2*x^2])^(1/4))])/b^(15/8))/4
Time = 1.00 (sec) , antiderivative size = 863, normalized size of antiderivative = 1.14, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.405, Rules used = {2545, 335, 817, 851, 758, 758, 755, 756, 216, 219, 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.
| \(\displaystyle \int \frac {\sqrt [4]{\sqrt {a^2 x^2-b}+a x}}{x^2 \left (a^2 x^2-b\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 2545 |
| \(\displaystyle 16 a \int \frac {\left (a x+\sqrt {a^2 x^2-b}\right )^{13/4}}{\left (b-\left (a x+\sqrt {a^2 x^2-b}\right )^2\right )^2 \left (\left (a x+\sqrt {a^2 x^2-b}\right )^2+b\right )^2}d\left (a x+\sqrt {a^2 x^2-b}\right )\) |
| \(\Big \downarrow \) 335 |
| \(\displaystyle 16 a \int \frac {\left (a x+\sqrt {a^2 x^2-b}\right )^{13/4}}{\left (b^2-\left (a x+\sqrt {a^2 x^2-b}\right )^4\right )^2}d\left (a x+\sqrt {a^2 x^2-b}\right )\) |
| \(\Big \downarrow \) 817 |
| \(\displaystyle 16 a \left (\frac {\sqrt [4]{\sqrt {a^2 x^2-b}+a x}}{4 \left (b^2-\left (\sqrt {a^2 x^2-b}+a x\right )^4\right )}-\frac {1}{16} \int \frac {1}{\left (a x+\sqrt {a^2 x^2-b}\right )^{3/4} \left (b^2-\left (a x+\sqrt {a^2 x^2-b}\right )^4\right )}d\left (a x+\sqrt {a^2 x^2-b}\right )\right )\) |
| \(\Big \downarrow \) 851 |
| \(\displaystyle 16 a \left (\frac {\sqrt [4]{\sqrt {a^2 x^2-b}+a x}}{4 \left (b^2-\left (\sqrt {a^2 x^2-b}+a x\right )^4\right )}-\frac {1}{4} \int \frac {1}{b^2-\left (a x+\sqrt {a^2 x^2-b}\right )^4}d\sqrt [4]{a x+\sqrt {a^2 x^2-b}}\right )\) |
| \(\Big \downarrow \) 758 |
| \(\displaystyle 16 a \left (\frac {1}{4} \left (-\frac {\int \frac {1}{b-\left (a x+\sqrt {a^2 x^2-b}\right )^2}d\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{2 b}-\frac {\int \frac {1}{\left (a x+\sqrt {a^2 x^2-b}\right )^2+b}d\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{2 b}\right )+\frac {\sqrt [4]{\sqrt {a^2 x^2-b}+a x}}{4 \left (b^2-\left (\sqrt {a^2 x^2-b}+a x\right )^4\right )}\right )\) |
| \(\Big \downarrow \) 758 |
| \(\displaystyle 16 a \left (\frac {1}{4} \left (-\frac {-\frac {\int \frac {1}{-a x+\sqrt {-b}-\sqrt {a^2 x^2-b}}d\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{2 \sqrt {-b}}-\frac {\int \frac {1}{a x+\sqrt {-b}+\sqrt {a^2 x^2-b}}d\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{2 \sqrt {-b}}}{2 b}-\frac {\frac {\int \frac {1}{-a x+\sqrt {b}-\sqrt {a^2 x^2-b}}d\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{2 \sqrt {b}}+\frac {\int \frac {1}{a x+\sqrt {b}+\sqrt {a^2 x^2-b}}d\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{2 \sqrt {b}}}{2 b}\right )+\frac {\sqrt [4]{\sqrt {a^2 x^2-b}+a x}}{4 \left (b^2-\left (\sqrt {a^2 x^2-b}+a x\right )^4\right )}\right )\) |
| \(\Big \downarrow \) 755 |
| \(\displaystyle 16 a \left (\frac {1}{4} \left (-\frac {-\frac {\int \frac {1}{-a x+\sqrt {-b}-\sqrt {a^2 x^2-b}}d\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{2 \sqrt {-b}}-\frac {\frac {\int \frac {\sqrt [4]{-b}-\sqrt {a x+\sqrt {a^2 x^2-b}}}{a x+\sqrt {-b}+\sqrt {a^2 x^2-b}}d\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{2 \sqrt [4]{-b}}+\frac {\int \frac {\sqrt {a x+\sqrt {a^2 x^2-b}}+\sqrt [4]{-b}}{a x+\sqrt {-b}+\sqrt {a^2 x^2-b}}d\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{2 \sqrt [4]{-b}}}{2 \sqrt {-b}}}{2 b}-\frac {\frac {\int \frac {1}{-a x+\sqrt {b}-\sqrt {a^2 x^2-b}}d\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{2 \sqrt {b}}+\frac {\frac {\int \frac {\sqrt [4]{b}-\sqrt {a x+\sqrt {a^2 x^2-b}}}{a x+\sqrt {b}+\sqrt {a^2 x^2-b}}d\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{2 \sqrt [4]{b}}+\frac {\int \frac {\sqrt {a x+\sqrt {a^2 x^2-b}}+\sqrt [4]{b}}{a x+\sqrt {b}+\sqrt {a^2 x^2-b}}d\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{2 \sqrt [4]{b}}}{2 \sqrt {b}}}{2 b}\right )+\frac {\sqrt [4]{\sqrt {a^2 x^2-b}+a x}}{4 \left (b^2-\left (\sqrt {a^2 x^2-b}+a x\right )^4\right )}\right )\) |
| \(\Big \downarrow \) 756 |
| \(\displaystyle 16 a \left (\frac {1}{4} \left (-\frac {-\frac {\frac {\int \frac {1}{\sqrt [4]{-b}-\sqrt {a x+\sqrt {a^2 x^2-b}}}d\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{2 \sqrt [4]{-b}}+\frac {\int \frac {1}{\sqrt {a x+\sqrt {a^2 x^2-b}}+\sqrt [4]{-b}}d\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{2 \sqrt [4]{-b}}}{2 \sqrt {-b}}-\frac {\frac {\int \frac {\sqrt [4]{-b}-\sqrt {a x+\sqrt {a^2 x^2-b}}}{a x+\sqrt {-b}+\sqrt {a^2 x^2-b}}d\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{2 \sqrt [4]{-b}}+\frac {\int \frac {\sqrt {a x+\sqrt {a^2 x^2-b}}+\sqrt [4]{-b}}{a x+\sqrt {-b}+\sqrt {a^2 x^2-b}}d\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{2 \sqrt [4]{-b}}}{2 \sqrt {-b}}}{2 b}-\frac {\frac {\frac {\int \frac {1}{\sqrt [4]{b}-\sqrt {a x+\sqrt {a^2 x^2-b}}}d\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{2 \sqrt [4]{b}}+\frac {\int \frac {1}{\sqrt {a x+\sqrt {a^2 x^2-b}}+\sqrt [4]{b}}d\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{2 \sqrt [4]{b}}}{2 \sqrt {b}}+\frac {\frac {\int \frac {\sqrt [4]{b}-\sqrt {a x+\sqrt {a^2 x^2-b}}}{a x+\sqrt {b}+\sqrt {a^2 x^2-b}}d\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{2 \sqrt [4]{b}}+\frac {\int \frac {\sqrt {a x+\sqrt {a^2 x^2-b}}+\sqrt [4]{b}}{a x+\sqrt {b}+\sqrt {a^2 x^2-b}}d\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{2 \sqrt [4]{b}}}{2 \sqrt {b}}}{2 b}\right )+\frac {\sqrt [4]{\sqrt {a^2 x^2-b}+a x}}{4 \left (b^2-\left (\sqrt {a^2 x^2-b}+a x\right )^4\right )}\right )\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle 16 a \left (\frac {1}{4} \left (-\frac {\frac {\frac {\int \frac {1}{\sqrt [4]{b}-\sqrt {a x+\sqrt {a^2 x^2-b}}}d\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{2 \sqrt [4]{b}}+\frac {\arctan \left (\frac {\sqrt [4]{\sqrt {a^2 x^2-b}+a x}}{\sqrt [8]{b}}\right )}{2 b^{3/8}}}{2 \sqrt {b}}+\frac {\frac {\int \frac {\sqrt [4]{b}-\sqrt {a x+\sqrt {a^2 x^2-b}}}{a x+\sqrt {b}+\sqrt {a^2 x^2-b}}d\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{2 \sqrt [4]{b}}+\frac {\int \frac {\sqrt {a x+\sqrt {a^2 x^2-b}}+\sqrt [4]{b}}{a x+\sqrt {b}+\sqrt {a^2 x^2-b}}d\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{2 \sqrt [4]{b}}}{2 \sqrt {b}}}{2 b}-\frac {-\frac {\frac {\int \frac {1}{\sqrt [4]{-b}-\sqrt {a x+\sqrt {a^2 x^2-b}}}d\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{2 \sqrt [4]{-b}}+\frac {\arctan \left (\frac {\sqrt [4]{\sqrt {a^2 x^2-b}+a x}}{\sqrt [8]{-b}}\right )}{2 (-b)^{3/8}}}{2 \sqrt {-b}}-\frac {\frac {\int \frac {\sqrt [4]{-b}-\sqrt {a x+\sqrt {a^2 x^2-b}}}{a x+\sqrt {-b}+\sqrt {a^2 x^2-b}}d\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{2 \sqrt [4]{-b}}+\frac {\int \frac {\sqrt {a x+\sqrt {a^2 x^2-b}}+\sqrt [4]{-b}}{a x+\sqrt {-b}+\sqrt {a^2 x^2-b}}d\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{2 \sqrt [4]{-b}}}{2 \sqrt {-b}}}{2 b}\right )+\frac {\sqrt [4]{\sqrt {a^2 x^2-b}+a x}}{4 \left (b^2-\left (\sqrt {a^2 x^2-b}+a x\right )^4\right )}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle 16 a \left (\frac {1}{4} \left (-\frac {\frac {\frac {\int \frac {\sqrt [4]{b}-\sqrt {a x+\sqrt {a^2 x^2-b}}}{a x+\sqrt {b}+\sqrt {a^2 x^2-b}}d\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{2 \sqrt [4]{b}}+\frac {\int \frac {\sqrt {a x+\sqrt {a^2 x^2-b}}+\sqrt [4]{b}}{a x+\sqrt {b}+\sqrt {a^2 x^2-b}}d\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{2 \sqrt [4]{b}}}{2 \sqrt {b}}+\frac {\frac {\arctan \left (\frac {\sqrt [4]{\sqrt {a^2 x^2-b}+a x}}{\sqrt [8]{b}}\right )}{2 b^{3/8}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{\sqrt {a^2 x^2-b}+a x}}{\sqrt [8]{b}}\right )}{2 b^{3/8}}}{2 \sqrt {b}}}{2 b}-\frac {-\frac {\frac {\int \frac {\sqrt [4]{-b}-\sqrt {a x+\sqrt {a^2 x^2-b}}}{a x+\sqrt {-b}+\sqrt {a^2 x^2-b}}d\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{2 \sqrt [4]{-b}}+\frac {\int \frac {\sqrt {a x+\sqrt {a^2 x^2-b}}+\sqrt [4]{-b}}{a x+\sqrt {-b}+\sqrt {a^2 x^2-b}}d\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{2 \sqrt [4]{-b}}}{2 \sqrt {-b}}-\frac {\frac {\arctan \left (\frac {\sqrt [4]{\sqrt {a^2 x^2-b}+a x}}{\sqrt [8]{-b}}\right )}{2 (-b)^{3/8}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{\sqrt {a^2 x^2-b}+a x}}{\sqrt [8]{-b}}\right )}{2 (-b)^{3/8}}}{2 \sqrt {-b}}}{2 b}\right )+\frac {\sqrt [4]{\sqrt {a^2 x^2-b}+a x}}{4 \left (b^2-\left (\sqrt {a^2 x^2-b}+a x\right )^4\right )}\right )\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle 16 a \left (\frac {1}{4} \left (-\frac {-\frac {\frac {\arctan \left (\frac {\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [8]{-b}}\right )}{2 (-b)^{3/8}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [8]{-b}}\right )}{2 (-b)^{3/8}}}{2 \sqrt {-b}}-\frac {\frac {\int \frac {\sqrt [4]{-b}-\sqrt {a x+\sqrt {a^2 x^2-b}}}{a x+\sqrt {-b}+\sqrt {a^2 x^2-b}}d\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{2 \sqrt [4]{-b}}+\frac {\frac {1}{2} \int \frac {1}{\sqrt {a x+\sqrt {a^2 x^2-b}}-\sqrt {2} \sqrt [8]{-b} \sqrt [4]{a x+\sqrt {a^2 x^2-b}}+\sqrt [4]{-b}}d\sqrt [4]{a x+\sqrt {a^2 x^2-b}}+\frac {1}{2} \int \frac {1}{\sqrt {a x+\sqrt {a^2 x^2-b}}+\sqrt {2} \sqrt [8]{-b} \sqrt [4]{a x+\sqrt {a^2 x^2-b}}+\sqrt [4]{-b}}d\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{2 \sqrt [4]{-b}}}{2 \sqrt {-b}}}{2 b}-\frac {\frac {\frac {\arctan \left (\frac {\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [8]{b}}\right )}{2 b^{3/8}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [8]{b}}\right )}{2 b^{3/8}}}{2 \sqrt {b}}+\frac {\frac {\int \frac {\sqrt [4]{b}-\sqrt {a x+\sqrt {a^2 x^2-b}}}{a x+\sqrt {b}+\sqrt {a^2 x^2-b}}d\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{2 \sqrt [4]{b}}+\frac {\frac {1}{2} \int \frac {1}{\sqrt {a x+\sqrt {a^2 x^2-b}}-\sqrt {2} \sqrt [8]{b} \sqrt [4]{a x+\sqrt {a^2 x^2-b}}+\sqrt [4]{b}}d\sqrt [4]{a x+\sqrt {a^2 x^2-b}}+\frac {1}{2} \int \frac {1}{\sqrt {a x+\sqrt {a^2 x^2-b}}+\sqrt {2} \sqrt [8]{b} \sqrt [4]{a x+\sqrt {a^2 x^2-b}}+\sqrt [4]{b}}d\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{2 \sqrt [4]{b}}}{2 \sqrt {b}}}{2 b}\right )+\frac {\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{4 \left (b^2-\left (a x+\sqrt {a^2 x^2-b}\right )^4\right )}\right )\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle 16 a \left (\frac {1}{4} \left (-\frac {-\frac {\frac {\arctan \left (\frac {\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [8]{-b}}\right )}{2 (-b)^{3/8}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [8]{-b}}\right )}{2 (-b)^{3/8}}}{2 \sqrt {-b}}-\frac {\frac {\frac {\int \frac {1}{-\sqrt {a x+\sqrt {a^2 x^2-b}}-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [8]{-b}}\right )}{\sqrt {2} \sqrt [8]{-b}}-\frac {\int \frac {1}{-\sqrt {a x+\sqrt {a^2 x^2-b}}-1}d\left (\frac {\sqrt {2} \sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [8]{-b}}+1\right )}{\sqrt {2} \sqrt [8]{-b}}}{2 \sqrt [4]{-b}}+\frac {\int \frac {\sqrt [4]{-b}-\sqrt {a x+\sqrt {a^2 x^2-b}}}{a x+\sqrt {-b}+\sqrt {a^2 x^2-b}}d\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{2 \sqrt [4]{-b}}}{2 \sqrt {-b}}}{2 b}-\frac {\frac {\frac {\arctan \left (\frac {\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [8]{b}}\right )}{2 b^{3/8}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [8]{b}}\right )}{2 b^{3/8}}}{2 \sqrt {b}}+\frac {\frac {\frac {\int \frac {1}{-\sqrt {a x+\sqrt {a^2 x^2-b}}-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [8]{b}}\right )}{\sqrt {2} \sqrt [8]{b}}-\frac {\int \frac {1}{-\sqrt {a x+\sqrt {a^2 x^2-b}}-1}d\left (\frac {\sqrt {2} \sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [8]{b}}+1\right )}{\sqrt {2} \sqrt [8]{b}}}{2 \sqrt [4]{b}}+\frac {\int \frac {\sqrt [4]{b}-\sqrt {a x+\sqrt {a^2 x^2-b}}}{a x+\sqrt {b}+\sqrt {a^2 x^2-b}}d\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{2 \sqrt [4]{b}}}{2 \sqrt {b}}}{2 b}\right )+\frac {\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{4 \left (b^2-\left (a x+\sqrt {a^2 x^2-b}\right )^4\right )}\right )\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle 16 a \left (\frac {1}{4} \left (-\frac {\frac {\frac {\int \frac {\sqrt [4]{b}-\sqrt {a x+\sqrt {a^2 x^2-b}}}{a x+\sqrt {b}+\sqrt {a^2 x^2-b}}d\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{2 \sqrt [4]{b}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{\sqrt {a^2 x^2-b}+a x}}{\sqrt [8]{b}}+1\right )}{\sqrt {2} \sqrt [8]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{\sqrt {a^2 x^2-b}+a x}}{\sqrt [8]{b}}\right )}{\sqrt {2} \sqrt [8]{b}}}{2 \sqrt [4]{b}}}{2 \sqrt {b}}+\frac {\frac {\arctan \left (\frac {\sqrt [4]{\sqrt {a^2 x^2-b}+a x}}{\sqrt [8]{b}}\right )}{2 b^{3/8}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{\sqrt {a^2 x^2-b}+a x}}{\sqrt [8]{b}}\right )}{2 b^{3/8}}}{2 \sqrt {b}}}{2 b}-\frac {-\frac {\frac {\int \frac {\sqrt [4]{-b}-\sqrt {a x+\sqrt {a^2 x^2-b}}}{a x+\sqrt {-b}+\sqrt {a^2 x^2-b}}d\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{2 \sqrt [4]{-b}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{\sqrt {a^2 x^2-b}+a x}}{\sqrt [8]{-b}}+1\right )}{\sqrt {2} \sqrt [8]{-b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{\sqrt {a^2 x^2-b}+a x}}{\sqrt [8]{-b}}\right )}{\sqrt {2} \sqrt [8]{-b}}}{2 \sqrt [4]{-b}}}{2 \sqrt {-b}}-\frac {\frac {\arctan \left (\frac {\sqrt [4]{\sqrt {a^2 x^2-b}+a x}}{\sqrt [8]{-b}}\right )}{2 (-b)^{3/8}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{\sqrt {a^2 x^2-b}+a x}}{\sqrt [8]{-b}}\right )}{2 (-b)^{3/8}}}{2 \sqrt {-b}}}{2 b}\right )+\frac {\sqrt [4]{\sqrt {a^2 x^2-b}+a x}}{4 \left (b^2-\left (\sqrt {a^2 x^2-b}+a x\right )^4\right )}\right )\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle 16 a \left (\frac {1}{4} \left (-\frac {-\frac {\frac {\arctan \left (\frac {\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [8]{-b}}\right )}{2 (-b)^{3/8}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [8]{-b}}\right )}{2 (-b)^{3/8}}}{2 \sqrt {-b}}-\frac {\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [8]{-b}}+1\right )}{\sqrt {2} \sqrt [8]{-b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [8]{-b}}\right )}{\sqrt {2} \sqrt [8]{-b}}}{2 \sqrt [4]{-b}}+\frac {-\frac {\int -\frac {\sqrt {2} \sqrt [8]{-b}-2 \sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{\sqrt {a x+\sqrt {a^2 x^2-b}}-\sqrt {2} \sqrt [8]{-b} \sqrt [4]{a x+\sqrt {a^2 x^2-b}}+\sqrt [4]{-b}}d\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{2 \sqrt {2} \sqrt [8]{-b}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{a x+\sqrt {a^2 x^2-b}}+\sqrt [8]{-b}\right )}{\sqrt {a x+\sqrt {a^2 x^2-b}}+\sqrt {2} \sqrt [8]{-b} \sqrt [4]{a x+\sqrt {a^2 x^2-b}}+\sqrt [4]{-b}}d\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{2 \sqrt {2} \sqrt [8]{-b}}}{2 \sqrt [4]{-b}}}{2 \sqrt {-b}}}{2 b}-\frac {\frac {\frac {\arctan \left (\frac {\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [8]{b}}\right )}{2 b^{3/8}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [8]{b}}\right )}{2 b^{3/8}}}{2 \sqrt {b}}+\frac {\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [8]{b}}+1\right )}{\sqrt {2} \sqrt [8]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [8]{b}}\right )}{\sqrt {2} \sqrt [8]{b}}}{2 \sqrt [4]{b}}+\frac {-\frac {\int -\frac {\sqrt {2} \sqrt [8]{b}-2 \sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{\sqrt {a x+\sqrt {a^2 x^2-b}}-\sqrt {2} \sqrt [8]{b} \sqrt [4]{a x+\sqrt {a^2 x^2-b}}+\sqrt [4]{b}}d\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{2 \sqrt {2} \sqrt [8]{b}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{a x+\sqrt {a^2 x^2-b}}+\sqrt [8]{b}\right )}{\sqrt {a x+\sqrt {a^2 x^2-b}}+\sqrt {2} \sqrt [8]{b} \sqrt [4]{a x+\sqrt {a^2 x^2-b}}+\sqrt [4]{b}}d\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{2 \sqrt {2} \sqrt [8]{b}}}{2 \sqrt [4]{b}}}{2 \sqrt {b}}}{2 b}\right )+\frac {\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{4 \left (b^2-\left (a x+\sqrt {a^2 x^2-b}\right )^4\right )}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle 16 a \left (\frac {1}{4} \left (-\frac {-\frac {\frac {\arctan \left (\frac {\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [8]{-b}}\right )}{2 (-b)^{3/8}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [8]{-b}}\right )}{2 (-b)^{3/8}}}{2 \sqrt {-b}}-\frac {\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [8]{-b}}+1\right )}{\sqrt {2} \sqrt [8]{-b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [8]{-b}}\right )}{\sqrt {2} \sqrt [8]{-b}}}{2 \sqrt [4]{-b}}+\frac {\frac {\int \frac {\sqrt {2} \sqrt [8]{-b}-2 \sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{\sqrt {a x+\sqrt {a^2 x^2-b}}-\sqrt {2} \sqrt [8]{-b} \sqrt [4]{a x+\sqrt {a^2 x^2-b}}+\sqrt [4]{-b}}d\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{2 \sqrt {2} \sqrt [8]{-b}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{a x+\sqrt {a^2 x^2-b}}+\sqrt [8]{-b}\right )}{\sqrt {a x+\sqrt {a^2 x^2-b}}+\sqrt {2} \sqrt [8]{-b} \sqrt [4]{a x+\sqrt {a^2 x^2-b}}+\sqrt [4]{-b}}d\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{2 \sqrt {2} \sqrt [8]{-b}}}{2 \sqrt [4]{-b}}}{2 \sqrt {-b}}}{2 b}-\frac {\frac {\frac {\arctan \left (\frac {\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [8]{b}}\right )}{2 b^{3/8}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [8]{b}}\right )}{2 b^{3/8}}}{2 \sqrt {b}}+\frac {\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [8]{b}}+1\right )}{\sqrt {2} \sqrt [8]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [8]{b}}\right )}{\sqrt {2} \sqrt [8]{b}}}{2 \sqrt [4]{b}}+\frac {\frac {\int \frac {\sqrt {2} \sqrt [8]{b}-2 \sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{\sqrt {a x+\sqrt {a^2 x^2-b}}-\sqrt {2} \sqrt [8]{b} \sqrt [4]{a x+\sqrt {a^2 x^2-b}}+\sqrt [4]{b}}d\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{2 \sqrt {2} \sqrt [8]{b}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{a x+\sqrt {a^2 x^2-b}}+\sqrt [8]{b}\right )}{\sqrt {a x+\sqrt {a^2 x^2-b}}+\sqrt {2} \sqrt [8]{b} \sqrt [4]{a x+\sqrt {a^2 x^2-b}}+\sqrt [4]{b}}d\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{2 \sqrt {2} \sqrt [8]{b}}}{2 \sqrt [4]{b}}}{2 \sqrt {b}}}{2 b}\right )+\frac {\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{4 \left (b^2-\left (a x+\sqrt {a^2 x^2-b}\right )^4\right )}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 16 a \left (\frac {1}{4} \left (-\frac {-\frac {\frac {\arctan \left (\frac {\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [8]{-b}}\right )}{2 (-b)^{3/8}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [8]{-b}}\right )}{2 (-b)^{3/8}}}{2 \sqrt {-b}}-\frac {\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [8]{-b}}+1\right )}{\sqrt {2} \sqrt [8]{-b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [8]{-b}}\right )}{\sqrt {2} \sqrt [8]{-b}}}{2 \sqrt [4]{-b}}+\frac {\frac {\int \frac {\sqrt {2} \sqrt [8]{-b}-2 \sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{\sqrt {a x+\sqrt {a^2 x^2-b}}-\sqrt {2} \sqrt [8]{-b} \sqrt [4]{a x+\sqrt {a^2 x^2-b}}+\sqrt [4]{-b}}d\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{2 \sqrt {2} \sqrt [8]{-b}}+\frac {\int \frac {\sqrt {2} \sqrt [4]{a x+\sqrt {a^2 x^2-b}}+\sqrt [8]{-b}}{\sqrt {a x+\sqrt {a^2 x^2-b}}+\sqrt {2} \sqrt [8]{-b} \sqrt [4]{a x+\sqrt {a^2 x^2-b}}+\sqrt [4]{-b}}d\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{2 \sqrt [8]{-b}}}{2 \sqrt [4]{-b}}}{2 \sqrt {-b}}}{2 b}-\frac {\frac {\frac {\arctan \left (\frac {\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [8]{b}}\right )}{2 b^{3/8}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [8]{b}}\right )}{2 b^{3/8}}}{2 \sqrt {b}}+\frac {\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [8]{b}}+1\right )}{\sqrt {2} \sqrt [8]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [8]{b}}\right )}{\sqrt {2} \sqrt [8]{b}}}{2 \sqrt [4]{b}}+\frac {\frac {\int \frac {\sqrt {2} \sqrt [8]{b}-2 \sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{\sqrt {a x+\sqrt {a^2 x^2-b}}-\sqrt {2} \sqrt [8]{b} \sqrt [4]{a x+\sqrt {a^2 x^2-b}}+\sqrt [4]{b}}d\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{2 \sqrt {2} \sqrt [8]{b}}+\frac {\int \frac {\sqrt {2} \sqrt [4]{a x+\sqrt {a^2 x^2-b}}+\sqrt [8]{b}}{\sqrt {a x+\sqrt {a^2 x^2-b}}+\sqrt {2} \sqrt [8]{b} \sqrt [4]{a x+\sqrt {a^2 x^2-b}}+\sqrt [4]{b}}d\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{2 \sqrt [8]{b}}}{2 \sqrt [4]{b}}}{2 \sqrt {b}}}{2 b}\right )+\frac {\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{4 \left (b^2-\left (a x+\sqrt {a^2 x^2-b}\right )^4\right )}\right )\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle 16 a \left (\frac {1}{4} \left (-\frac {-\frac {\frac {\arctan \left (\frac {\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [8]{-b}}\right )}{2 (-b)^{3/8}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [8]{-b}}\right )}{2 (-b)^{3/8}}}{2 \sqrt {-b}}-\frac {\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [8]{-b}}+1\right )}{\sqrt {2} \sqrt [8]{-b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [8]{-b}}\right )}{\sqrt {2} \sqrt [8]{-b}}}{2 \sqrt [4]{-b}}+\frac {\frac {\log \left (\sqrt {a x+\sqrt {a^2 x^2-b}}+\sqrt {2} \sqrt [8]{-b} \sqrt [4]{a x+\sqrt {a^2 x^2-b}}+\sqrt [4]{-b}\right )}{2 \sqrt {2} \sqrt [8]{-b}}-\frac {\log \left (\sqrt {a x+\sqrt {a^2 x^2-b}}-\sqrt {2} \sqrt [8]{-b} \sqrt [4]{a x+\sqrt {a^2 x^2-b}}+\sqrt [4]{-b}\right )}{2 \sqrt {2} \sqrt [8]{-b}}}{2 \sqrt [4]{-b}}}{2 \sqrt {-b}}}{2 b}-\frac {\frac {\frac {\arctan \left (\frac {\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [8]{b}}\right )}{2 b^{3/8}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [8]{b}}\right )}{2 b^{3/8}}}{2 \sqrt {b}}+\frac {\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [8]{b}}+1\right )}{\sqrt {2} \sqrt [8]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [8]{b}}\right )}{\sqrt {2} \sqrt [8]{b}}}{2 \sqrt [4]{b}}+\frac {\frac {\log \left (\sqrt {a x+\sqrt {a^2 x^2-b}}+\sqrt {2} \sqrt [8]{b} \sqrt [4]{a x+\sqrt {a^2 x^2-b}}+\sqrt [4]{b}\right )}{2 \sqrt {2} \sqrt [8]{b}}-\frac {\log \left (\sqrt {a x+\sqrt {a^2 x^2-b}}-\sqrt {2} \sqrt [8]{b} \sqrt [4]{a x+\sqrt {a^2 x^2-b}}+\sqrt [4]{b}\right )}{2 \sqrt {2} \sqrt [8]{b}}}{2 \sqrt [4]{b}}}{2 \sqrt {b}}}{2 b}\right )+\frac {\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{4 \left (b^2-\left (a x+\sqrt {a^2 x^2-b}\right )^4\right )}\right )\) |
16*a*((a*x + Sqrt[-b + a^2*x^2])^(1/4)/(4*(b^2 - (a*x + Sqrt[-b + a^2*x^2] )^4)) + (-1/2*(-1/2*(ArcTan[(a*x + Sqrt[-b + a^2*x^2])^(1/4)/(-b)^(1/8)]/( 2*(-b)^(3/8)) + ArcTanh[(a*x + Sqrt[-b + a^2*x^2])^(1/4)/(-b)^(1/8)]/(2*(- b)^(3/8)))/Sqrt[-b] - ((-(ArcTan[1 - (Sqrt[2]*(a*x + Sqrt[-b + a^2*x^2])^( 1/4))/(-b)^(1/8)]/(Sqrt[2]*(-b)^(1/8))) + ArcTan[1 + (Sqrt[2]*(a*x + Sqrt[ -b + a^2*x^2])^(1/4))/(-b)^(1/8)]/(Sqrt[2]*(-b)^(1/8)))/(2*(-b)^(1/4)) + ( -1/2*Log[(-b)^(1/4) - Sqrt[2]*(-b)^(1/8)*(a*x + Sqrt[-b + a^2*x^2])^(1/4) + Sqrt[a*x + Sqrt[-b + a^2*x^2]]]/(Sqrt[2]*(-b)^(1/8)) + Log[(-b)^(1/4) + Sqrt[2]*(-b)^(1/8)*(a*x + Sqrt[-b + a^2*x^2])^(1/4) + Sqrt[a*x + Sqrt[-b + a^2*x^2]]]/(2*Sqrt[2]*(-b)^(1/8)))/(2*(-b)^(1/4)))/(2*Sqrt[-b]))/b - ((Ar cTan[(a*x + Sqrt[-b + a^2*x^2])^(1/4)/b^(1/8)]/(2*b^(3/8)) + ArcTanh[(a*x + Sqrt[-b + a^2*x^2])^(1/4)/b^(1/8)]/(2*b^(3/8)))/(2*Sqrt[b]) + ((-(ArcTan [1 - (Sqrt[2]*(a*x + Sqrt[-b + a^2*x^2])^(1/4))/b^(1/8)]/(Sqrt[2]*b^(1/8)) ) + ArcTan[1 + (Sqrt[2]*(a*x + Sqrt[-b + a^2*x^2])^(1/4))/b^(1/8)]/(Sqrt[2 ]*b^(1/8)))/(2*b^(1/4)) + (-1/2*Log[b^(1/4) - Sqrt[2]*b^(1/8)*(a*x + Sqrt[ -b + a^2*x^2])^(1/4) + Sqrt[a*x + Sqrt[-b + a^2*x^2]]]/(Sqrt[2]*b^(1/8)) + Log[b^(1/4) + Sqrt[2]*b^(1/8)*(a*x + Sqrt[-b + a^2*x^2])^(1/4) + Sqrt[a*x + Sqrt[-b + a^2*x^2]]]/(2*Sqrt[2]*b^(1/8)))/(2*b^(1/4)))/(2*Sqrt[b]))/(2* b))/4)
3.32.29.3.1 Defintions of rubi rules used
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[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
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[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(p _.), x_Symbol] :> Int[(e*x)^m*(a*c + b*d*x^4)^p, x] /; FreeQ[{a, b, c, d, e , m, p}, x] && EqQ[b*c + a*d, 0] && (IntegerQ[p] || (GtQ[a, 0] && GtQ[c, 0] ))
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_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2 ]], s = Denominator[Rt[-a/b, 2]]}, Simp[r/(2*a) Int[1/(r - s*x^2), x], x] + Simp[r/(2*a) Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] && !GtQ[a /b, 0]
Int[((a_) + (b_.)*(x_)^(n_))^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b , 2]], s = Denominator[Rt[-a/b, 2]]}, Simp[r/(2*a) Int[1/(r - s*x^(n/2)), x], x] + Simp[r/(2*a) Int[1/(r + s*x^(n/2)), x], x]] /; FreeQ[{a, b}, x] && IGtQ[n/4, 1] && !GtQ[a/b, 0]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^( n - 1)*(c*x)^(m - n + 1)*((a + b*x^n)^(p + 1)/(b*n*(p + 1))), x] - Simp[c^n *((m - n + 1)/(b*n*(p + 1))) Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x], x ] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] && ! ILtQ[(m + n*(p + 1) + 1)/n, 0] && IntBinomialQ[a, b, c, n, m, p, x]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k/c Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/c^ n))^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]
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[((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]
Int[(x_)^(p_.)*((g_) + (i_.)*(x_)^2)^(m_.)*((e_.)*(x_) + (f_.)*Sqrt[(a_) + (c_.)*(x_)^2])^(n_.), x_Symbol] :> Simp[(1/(2^(2*m + p + 1)*e^(p + 1)*f^(2* m)))*(i/c)^m Subst[Int[x^(n - 2*m - p - 2)*((-a)*f^2 + x^2)^p*(a*f^2 + x^ 2)^(2*m + 1), x], x, e*x + f*Sqrt[a + c*x^2]], x] /; FreeQ[{a, c, e, f, g, i, n}, x] && EqQ[e^2 - c*f^2, 0] && EqQ[c*g - a*i, 0] && IntegersQ[p, 2*m] && (IntegerQ[m] || GtQ[i/c, 0])
\[\int \frac {\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{4}}}{x^{2} \left (a^{2} x^{2}-b \right )^{\frac {3}{2}}}d x\]
Result contains complex when optimal does not.
Time = 0.32 (sec) , antiderivative size = 1190, normalized size of antiderivative = 1.57 \[ \int \frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{x^2 \left (-b+a^2 x^2\right )^{3/2}} \, dx=\text {Too large to display} \]
1/8*(sqrt(2)*(-(I + 1)*a^2*b^2*x^3 + (I + 1)*b^3*x)*(a^8/b^15)^(1/8)*log(( 1/2*I + 1/2)*sqrt(2)*b^2*(a^8/b^15)^(1/8) + (a*x + sqrt(a^2*x^2 - b))^(1/4 )*a) + sqrt(2)*((I - 1)*a^2*b^2*x^3 - (I - 1)*b^3*x)*(a^8/b^15)^(1/8)*log( -(1/2*I - 1/2)*sqrt(2)*b^2*(a^8/b^15)^(1/8) + (a*x + sqrt(a^2*x^2 - b))^(1 /4)*a) + sqrt(2)*(-(I - 1)*a^2*b^2*x^3 + (I - 1)*b^3*x)*(a^8/b^15)^(1/8)*l og((1/2*I - 1/2)*sqrt(2)*b^2*(a^8/b^15)^(1/8) + (a*x + sqrt(a^2*x^2 - b))^ (1/4)*a) + sqrt(2)*((I + 1)*a^2*b^2*x^3 - (I + 1)*b^3*x)*(a^8/b^15)^(1/8)* log(-(1/2*I + 1/2)*sqrt(2)*b^2*(a^8/b^15)^(1/8) + (a*x + sqrt(a^2*x^2 - b) )^(1/4)*a) + sqrt(2)*(-(I + 1)*a^2*b^2*x^3 + (I + 1)*b^3*x)*(-a^8/b^15)^(1 /8)*log((1/2*I + 1/2)*sqrt(2)*b^2*(-a^8/b^15)^(1/8) + (a*x + sqrt(a^2*x^2 - b))^(1/4)*a) + sqrt(2)*((I - 1)*a^2*b^2*x^3 - (I - 1)*b^3*x)*(-a^8/b^15) ^(1/8)*log(-(1/2*I - 1/2)*sqrt(2)*b^2*(-a^8/b^15)^(1/8) + (a*x + sqrt(a^2* x^2 - b))^(1/4)*a) + sqrt(2)*(-(I - 1)*a^2*b^2*x^3 + (I - 1)*b^3*x)*(-a^8/ b^15)^(1/8)*log((1/2*I - 1/2)*sqrt(2)*b^2*(-a^8/b^15)^(1/8) + (a*x + sqrt( a^2*x^2 - b))^(1/4)*a) + sqrt(2)*((I + 1)*a^2*b^2*x^3 - (I + 1)*b^3*x)*(-a ^8/b^15)^(1/8)*log(-(1/2*I + 1/2)*sqrt(2)*b^2*(-a^8/b^15)^(1/8) + (a*x + s qrt(a^2*x^2 - b))^(1/4)*a) - 2*(a^2*b^2*x^3 - b^3*x)*(a^8/b^15)^(1/8)*log( b^2*(a^8/b^15)^(1/8) + (a*x + sqrt(a^2*x^2 - b))^(1/4)*a) - 2*(I*a^2*b^2*x ^3 - I*b^3*x)*(a^8/b^15)^(1/8)*log(I*b^2*(a^8/b^15)^(1/8) + (a*x + sqrt(a^ 2*x^2 - b))^(1/4)*a) - 2*(-I*a^2*b^2*x^3 + I*b^3*x)*(a^8/b^15)^(1/8)*lo...
\[ \int \frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{x^2 \left (-b+a^2 x^2\right )^{3/2}} \, dx=\int \frac {\sqrt [4]{a x + \sqrt {a^{2} x^{2} - b}}}{x^{2} \left (a^{2} x^{2} - b\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{x^2 \left (-b+a^2 x^2\right )^{3/2}} \, dx=\int { \frac {{\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}}{{\left (a^{2} x^{2} - b\right )}^{\frac {3}{2}} x^{2}} \,d x } \]
Timed out. \[ \int \frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{x^2 \left (-b+a^2 x^2\right )^{3/2}} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{x^2 \left (-b+a^2 x^2\right )^{3/2}} \, dx=\int \frac {{\left (a\,x+\sqrt {a^2\,x^2-b}\right )}^{1/4}}{x^2\,{\left (a^2\,x^2-b\right )}^{3/2}} \,d x \]