\(\int \frac {\sqrt [6]{a x+\sqrt {-b+a^2 x^2}}}{x^3 \sqrt {-b+a^2 x^2}} \, dx\) [3118]

Optimal result

Integrand size = 42, antiderivative size = 678 \[ \int \frac {\sqrt [6]{a x+\sqrt {-b+a^2 x^2}}}{x^3 \sqrt {-b+a^2 x^2}} \, dx=-\frac {5}{24 x^2 \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/6}}+\frac {7 \left (a x+\sqrt {-b+a^2 x^2}\right )^{7/6}}{24 b x^2}-\frac {35 \sqrt {2+\sqrt {3}} a^2 \arctan \left (\frac {\left (\sqrt {\frac {3}{2}} \sqrt [12]{b}-\frac {\sqrt [12]{b}}{\sqrt {2}}\right ) \sqrt [6]{a x+\sqrt {-b+a^2 x^2}}}{-\sqrt [6]{b}+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}\right )}{72 b^{17/12}}-\frac {35 \sqrt {2-\sqrt {3}} a^2 \arctan \left (\frac {\left (\sqrt {\frac {3}{2}} \sqrt [12]{b}+\frac {\sqrt [12]{b}}{\sqrt {2}}\right ) \sqrt [6]{a x+\sqrt {-b+a^2 x^2}}}{-\sqrt [6]{b}+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}\right )}{72 b^{17/12}}+\frac {35 a^2 \arctan \left (\frac {\sqrt {2} \sqrt [12]{b} \sqrt [6]{a x+\sqrt {-b+a^2 x^2}}}{-\sqrt [6]{b}+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}\right )}{36 \sqrt {2} b^{17/12}}+\frac {35 a^2 \text {arctanh}\left (\frac {\frac {\sqrt [12]{b}}{\sqrt {2}}+\frac {\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt {2} \sqrt [12]{b}}}{\sqrt [6]{a x+\sqrt {-b+a^2 x^2}}}\right )}{36 \sqrt {2} b^{17/12}}-\frac {35 \sqrt {2+\sqrt {3}} a^2 \text {arctanh}\left (\frac {\frac {\sqrt {2} \sqrt [12]{b}}{-1+\sqrt {3}}+\frac {\sqrt {2} \sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}{\left (-1+\sqrt {3}\right ) \sqrt [12]{b}}}{\sqrt [6]{a x+\sqrt {-b+a^2 x^2}}}\right )}{72 b^{17/12}}-\frac {35 \sqrt {2-\sqrt {3}} a^2 \text {arctanh}\left (\frac {\frac {\sqrt {2} \sqrt [12]{b}}{1+\sqrt {3}}+\frac {\sqrt {2} \sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}{\left (1+\sqrt {3}\right ) \sqrt [12]{b}}}{\sqrt [6]{a x+\sqrt {-b+a^2 x^2}}}\right )}{72 b^{17/12}} \] Output:

-5/24/x^2/(a*x+(a^2*x^2-b)^(1/2))^(5/6)+7/24*(a*x+(a^2*x^2-b)^(1/2))^(7/6) 
/b/x^2-35/72*(1/2*6^(1/2)+1/2*2^(1/2))*a^2*arctan((1/2*6^(1/2)*b^(1/12)-1/ 
2*b^(1/12)*2^(1/2))*(a*x+(a^2*x^2-b)^(1/2))^(1/6)/(-b^(1/6)+(a*x+(a^2*x^2- 
b)^(1/2))^(1/3)))/b^(17/12)-35/72*(1/2*6^(1/2)-1/2*2^(1/2))*a^2*arctan((1/ 
2*6^(1/2)*b^(1/12)+1/2*b^(1/12)*2^(1/2))*(a*x+(a^2*x^2-b)^(1/2))^(1/6)/(-b 
^(1/6)+(a*x+(a^2*x^2-b)^(1/2))^(1/3)))/b^(17/12)+35/72*a^2*arctan(2^(1/2)* 
b^(1/12)*(a*x+(a^2*x^2-b)^(1/2))^(1/6)/(-b^(1/6)+(a*x+(a^2*x^2-b)^(1/2))^( 
1/3)))*2^(1/2)/b^(17/12)+35/72*a^2*arctanh((1/2*b^(1/12)*2^(1/2)+1/2*(a*x+ 
(a^2*x^2-b)^(1/2))^(1/3)*2^(1/2)/b^(1/12))/(a*x+(a^2*x^2-b)^(1/2))^(1/6))* 
2^(1/2)/b^(17/12)-35/72*(1/2*6^(1/2)+1/2*2^(1/2))*a^2*arctanh((2^(1/2)*b^( 
1/12)/(3^(1/2)-1)+2^(1/2)*(a*x+(a^2*x^2-b)^(1/2))^(1/3)/(3^(1/2)-1)/b^(1/1 
2))/(a*x+(a^2*x^2-b)^(1/2))^(1/6))/b^(17/12)-35/72*(1/2*6^(1/2)-1/2*2^(1/2 
))*a^2*arctanh((2^(1/2)*b^(1/12)/(1+3^(1/2))+2^(1/2)*(a*x+(a^2*x^2-b)^(1/2 
))^(1/3)/(1+3^(1/2))/b^(1/12))/(a*x+(a^2*x^2-b)^(1/2))^(1/6))/b^(17/12)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 2.53 (sec) , antiderivative size = 606, normalized size of antiderivative = 0.89 \[ \int \frac {\sqrt [6]{a x+\sqrt {-b+a^2 x^2}}}{x^3 \sqrt {-b+a^2 x^2}} \, dx=\frac {1}{72} \left (-\frac {15}{x^2 \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/6}}+\frac {21 \left (a x+\sqrt {-b+a^2 x^2}\right )^{7/6}}{b x^2}+\frac {35 \sqrt {2+\sqrt {3}} a^2 \arctan \left (\frac {\sqrt {2-\sqrt {3}} \sqrt [12]{b} \sqrt [6]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [6]{b}-\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}\right )}{b^{17/12}}+\frac {35 \sqrt {2-\sqrt {3}} a^2 \arctan \left (\frac {\sqrt {2+\sqrt {3}} \sqrt [12]{b} \sqrt [6]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [6]{b}-\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}\right )}{b^{17/12}}+\frac {35 \sqrt {2} a^2 \arctan \left (\frac {\sqrt {2} \sqrt [12]{b} \sqrt [6]{a x+\sqrt {-b+a^2 x^2}}}{-\sqrt [6]{b}+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}\right )}{b^{17/12}}+\frac {35 \sqrt {2} a^2 \text {arctanh}\left (\frac {\sqrt [6]{b}+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt {2} \sqrt [12]{b} \sqrt [6]{a x+\sqrt {-b+a^2 x^2}}}\right )}{b^{17/12}}-\frac {35 \sqrt {2-\sqrt {3}} a^2 \text {arctanh}\left (\frac {\sqrt {2-\sqrt {3}} \left (\sqrt [6]{b}+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}\right )}{\sqrt [12]{b} \sqrt [6]{a x+\sqrt {-b+a^2 x^2}}}\right )}{b^{17/12}}-\frac {35 \sqrt {2+\sqrt {3}} a^2 \text {arctanh}\left (\frac {\sqrt {2+\sqrt {3}} \left (\sqrt [6]{b}+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}\right )}{\sqrt [12]{b} \sqrt [6]{a x+\sqrt {-b+a^2 x^2}}}\right )}{b^{17/12}}\right ) \] Input:

Integrate[(a*x + Sqrt[-b + a^2*x^2])^(1/6)/(x^3*Sqrt[-b + a^2*x^2]),x]
 

Output:

(-15/(x^2*(a*x + Sqrt[-b + a^2*x^2])^(5/6)) + (21*(a*x + Sqrt[-b + a^2*x^2 
])^(7/6))/(b*x^2) + (35*Sqrt[2 + Sqrt[3]]*a^2*ArcTan[(Sqrt[2 - Sqrt[3]]*b^ 
(1/12)*(a*x + Sqrt[-b + a^2*x^2])^(1/6))/(b^(1/6) - (a*x + Sqrt[-b + a^2*x 
^2])^(1/3))])/b^(17/12) + (35*Sqrt[2 - Sqrt[3]]*a^2*ArcTan[(Sqrt[2 + Sqrt[ 
3]]*b^(1/12)*(a*x + Sqrt[-b + a^2*x^2])^(1/6))/(b^(1/6) - (a*x + Sqrt[-b + 
 a^2*x^2])^(1/3))])/b^(17/12) + (35*Sqrt[2]*a^2*ArcTan[(Sqrt[2]*b^(1/12)*( 
a*x + Sqrt[-b + a^2*x^2])^(1/6))/(-b^(1/6) + (a*x + Sqrt[-b + a^2*x^2])^(1 
/3))])/b^(17/12) + (35*Sqrt[2]*a^2*ArcTanh[(b^(1/6) + (a*x + Sqrt[-b + a^2 
*x^2])^(1/3))/(Sqrt[2]*b^(1/12)*(a*x + Sqrt[-b + a^2*x^2])^(1/6))])/b^(17/ 
12) - (35*Sqrt[2 - Sqrt[3]]*a^2*ArcTanh[(Sqrt[2 - Sqrt[3]]*(b^(1/6) + (a*x 
 + Sqrt[-b + a^2*x^2])^(1/3)))/(b^(1/12)*(a*x + Sqrt[-b + a^2*x^2])^(1/6)) 
])/b^(17/12) - (35*Sqrt[2 + Sqrt[3]]*a^2*ArcTanh[(Sqrt[2 + Sqrt[3]]*(b^(1/ 
6) + (a*x + Sqrt[-b + a^2*x^2])^(1/3)))/(b^(1/12)*(a*x + Sqrt[-b + a^2*x^2 
])^(1/6))])/b^(17/12))/72
 

Rubi [A] (verified)

Time = 1.09 (sec) , antiderivative size = 749, normalized size of antiderivative = 1.10, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {2545, 252, 253, 266, 830, 753, 27, 216, 754, 27, 219, 1142, 25, 1082, 217, 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.

\(\displaystyle \int \frac {\sqrt [6]{\sqrt {a^2 x^2-b}+a x}}{x^3 \sqrt {a^2 x^2-b}} \, dx\)

\(\Big \downarrow \) 2545

\(\displaystyle 8 a^2 \int \frac {\left (a x+\sqrt {a^2 x^2-b}\right )^{13/6}}{\left (\left (a x+\sqrt {a^2 x^2-b}\right )^2+b\right )^3}d\left (a x+\sqrt {a^2 x^2-b}\right )\)

\(\Big \downarrow \) 252

\(\displaystyle 8 a^2 \left (\frac {7}{24} \int \frac {\sqrt [6]{a x+\sqrt {a^2 x^2-b}}}{\left (\left (a x+\sqrt {a^2 x^2-b}\right )^2+b\right )^2}d\left (a x+\sqrt {a^2 x^2-b}\right )-\frac {\left (\sqrt {a^2 x^2-b}+a x\right )^{7/6}}{4 \left (\left (\sqrt {a^2 x^2-b}+a x\right )^2+b\right )^2}\right )\)

\(\Big \downarrow \) 253

\(\displaystyle 8 a^2 \left (\frac {7}{24} \left (\frac {5 \int \frac {\sqrt [6]{a x+\sqrt {a^2 x^2-b}}}{\left (a x+\sqrt {a^2 x^2-b}\right )^2+b}d\left (a x+\sqrt {a^2 x^2-b}\right )}{12 b}+\frac {\left (\sqrt {a^2 x^2-b}+a x\right )^{7/6}}{2 b \left (\left (\sqrt {a^2 x^2-b}+a x\right )^2+b\right )}\right )-\frac {\left (\sqrt {a^2 x^2-b}+a x\right )^{7/6}}{4 \left (\left (\sqrt {a^2 x^2-b}+a x\right )^2+b\right )^2}\right )\)

\(\Big \downarrow \) 266

\(\displaystyle 8 a^2 \left (\frac {7}{24} \left (\frac {5 \int \frac {a x+\sqrt {a^2 x^2-b}}{\left (a x+\sqrt {a^2 x^2-b}\right )^2+b}d\sqrt [6]{a x+\sqrt {a^2 x^2-b}}}{2 b}+\frac {\left (\sqrt {a^2 x^2-b}+a x\right )^{7/6}}{2 b \left (\left (\sqrt {a^2 x^2-b}+a x\right )^2+b\right )}\right )-\frac {\left (\sqrt {a^2 x^2-b}+a x\right )^{7/6}}{4 \left (\left (\sqrt {a^2 x^2-b}+a x\right )^2+b\right )^2}\right )\)

\(\Big \downarrow \) 830

\(\displaystyle 8 a^2 \left (\frac {7}{24} \left (\frac {5 \left (\frac {1}{2} \int \frac {1}{a x+\sqrt {-b}+\sqrt {a^2 x^2-b}}d\sqrt [6]{a x+\sqrt {a^2 x^2-b}}-\frac {1}{2} \int \frac {1}{-a x+\sqrt {-b}-\sqrt {a^2 x^2-b}}d\sqrt [6]{a x+\sqrt {a^2 x^2-b}}\right )}{2 b}+\frac {\left (\sqrt {a^2 x^2-b}+a x\right )^{7/6}}{2 b \left (\left (\sqrt {a^2 x^2-b}+a x\right )^2+b\right )}\right )-\frac {\left (\sqrt {a^2 x^2-b}+a x\right )^{7/6}}{4 \left (\left (\sqrt {a^2 x^2-b}+a x\right )^2+b\right )^2}\right )\)

\(\Big \downarrow \) 753

\(\displaystyle 8 a^2 \left (\frac {7}{24} \left (\frac {5 \left (\frac {1}{2} \left (\frac {\int \frac {1}{\sqrt [3]{a x+\sqrt {a^2 x^2-b}}+\sqrt [6]{-b}}d\sqrt [6]{a x+\sqrt {a^2 x^2-b}}}{3 \sqrt [3]{-b}}+\frac {\int \frac {2 \sqrt [12]{-b}-\sqrt {3} \sqrt [6]{a x+\sqrt {a^2 x^2-b}}}{2 \left (\sqrt [3]{a x+\sqrt {a^2 x^2-b}}-\sqrt {3} \sqrt [12]{-b} \sqrt [6]{a x+\sqrt {a^2 x^2-b}}+\sqrt [6]{-b}\right )}d\sqrt [6]{a x+\sqrt {a^2 x^2-b}}}{3 (-b)^{5/12}}+\frac {\int \frac {\sqrt {3} \sqrt [6]{a x+\sqrt {a^2 x^2-b}}+2 \sqrt [12]{-b}}{2 \left (\sqrt [3]{a x+\sqrt {a^2 x^2-b}}+\sqrt {3} \sqrt [12]{-b} \sqrt [6]{a x+\sqrt {a^2 x^2-b}}+\sqrt [6]{-b}\right )}d\sqrt [6]{a x+\sqrt {a^2 x^2-b}}}{3 (-b)^{5/12}}\right )-\frac {1}{2} \int \frac {1}{-a x+\sqrt {-b}-\sqrt {a^2 x^2-b}}d\sqrt [6]{a x+\sqrt {a^2 x^2-b}}\right )}{2 b}+\frac {\left (\sqrt {a^2 x^2-b}+a x\right )^{7/6}}{2 b \left (\left (\sqrt {a^2 x^2-b}+a x\right )^2+b\right )}\right )-\frac {\left (\sqrt {a^2 x^2-b}+a x\right )^{7/6}}{4 \left (\left (\sqrt {a^2 x^2-b}+a x\right )^2+b\right )^2}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle 8 a^2 \left (\frac {7}{24} \left (\frac {5 \left (\frac {1}{2} \left (\frac {\int \frac {1}{\sqrt [3]{a x+\sqrt {a^2 x^2-b}}+\sqrt [6]{-b}}d\sqrt [6]{a x+\sqrt {a^2 x^2-b}}}{3 \sqrt [3]{-b}}+\frac {\int \frac {2 \sqrt [12]{-b}-\sqrt {3} \sqrt [6]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [3]{a x+\sqrt {a^2 x^2-b}}-\sqrt {3} \sqrt [12]{-b} \sqrt [6]{a x+\sqrt {a^2 x^2-b}}+\sqrt [6]{-b}}d\sqrt [6]{a x+\sqrt {a^2 x^2-b}}}{6 (-b)^{5/12}}+\frac {\int \frac {\sqrt {3} \sqrt [6]{a x+\sqrt {a^2 x^2-b}}+2 \sqrt [12]{-b}}{\sqrt [3]{a x+\sqrt {a^2 x^2-b}}+\sqrt {3} \sqrt [12]{-b} \sqrt [6]{a x+\sqrt {a^2 x^2-b}}+\sqrt [6]{-b}}d\sqrt [6]{a x+\sqrt {a^2 x^2-b}}}{6 (-b)^{5/12}}\right )-\frac {1}{2} \int \frac {1}{-a x+\sqrt {-b}-\sqrt {a^2 x^2-b}}d\sqrt [6]{a x+\sqrt {a^2 x^2-b}}\right )}{2 b}+\frac {\left (\sqrt {a^2 x^2-b}+a x\right )^{7/6}}{2 b \left (\left (\sqrt {a^2 x^2-b}+a x\right )^2+b\right )}\right )-\frac {\left (\sqrt {a^2 x^2-b}+a x\right )^{7/6}}{4 \left (\left (\sqrt {a^2 x^2-b}+a x\right )^2+b\right )^2}\right )\)

\(\Big \downarrow \) 216

\(\displaystyle 8 a^2 \left (\frac {7}{24} \left (\frac {5 \left (\frac {1}{2} \left (\frac {\int \frac {2 \sqrt [12]{-b}-\sqrt {3} \sqrt [6]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [3]{a x+\sqrt {a^2 x^2-b}}-\sqrt {3} \sqrt [12]{-b} \sqrt [6]{a x+\sqrt {a^2 x^2-b}}+\sqrt [6]{-b}}d\sqrt [6]{a x+\sqrt {a^2 x^2-b}}}{6 (-b)^{5/12}}+\frac {\int \frac {\sqrt {3} \sqrt [6]{a x+\sqrt {a^2 x^2-b}}+2 \sqrt [12]{-b}}{\sqrt [3]{a x+\sqrt {a^2 x^2-b}}+\sqrt {3} \sqrt [12]{-b} \sqrt [6]{a x+\sqrt {a^2 x^2-b}}+\sqrt [6]{-b}}d\sqrt [6]{a x+\sqrt {a^2 x^2-b}}}{6 (-b)^{5/12}}+\frac {\arctan \left (\frac {\sqrt [6]{\sqrt {a^2 x^2-b}+a x}}{\sqrt [12]{-b}}\right )}{3 (-b)^{5/12}}\right )-\frac {1}{2} \int \frac {1}{-a x+\sqrt {-b}-\sqrt {a^2 x^2-b}}d\sqrt [6]{a x+\sqrt {a^2 x^2-b}}\right )}{2 b}+\frac {\left (\sqrt {a^2 x^2-b}+a x\right )^{7/6}}{2 b \left (\left (\sqrt {a^2 x^2-b}+a x\right )^2+b\right )}\right )-\frac {\left (\sqrt {a^2 x^2-b}+a x\right )^{7/6}}{4 \left (\left (\sqrt {a^2 x^2-b}+a x\right )^2+b\right )^2}\right )\)

\(\Big \downarrow \) 754

\(\displaystyle 8 a^2 \left (\frac {7}{24} \left (\frac {\left (a x+\sqrt {a^2 x^2-b}\right )^{7/6}}{2 b \left (\left (a x+\sqrt {a^2 x^2-b}\right )^2+b\right )}+\frac {5 \left (\frac {1}{2} \left (-\frac {\int \frac {1}{\sqrt [6]{-b}-\sqrt [3]{a x+\sqrt {a^2 x^2-b}}}d\sqrt [6]{a x+\sqrt {a^2 x^2-b}}}{3 \sqrt [3]{-b}}-\frac {\int \frac {-a x-\sqrt {a^2 x^2-b}+2 \sqrt [12]{-b}}{2 \left (\sqrt [3]{a x+\sqrt {a^2 x^2-b}}-\sqrt [12]{-b} \sqrt [6]{a x+\sqrt {a^2 x^2-b}}+\sqrt [6]{-b}\right )}d\sqrt [6]{a x+\sqrt {a^2 x^2-b}}}{3 (-b)^{5/12}}-\frac {\int \frac {\sqrt [6]{a x+\sqrt {a^2 x^2-b}}+2 \sqrt [12]{-b}}{2 \left (\sqrt [3]{a x+\sqrt {a^2 x^2-b}}+\sqrt [12]{-b} \sqrt [6]{a x+\sqrt {a^2 x^2-b}}+\sqrt [6]{-b}\right )}d\sqrt [6]{a x+\sqrt {a^2 x^2-b}}}{3 (-b)^{5/12}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt [6]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [12]{-b}}\right )}{3 (-b)^{5/12}}+\frac {\int \frac {2 \sqrt [12]{-b}-\sqrt {3} \sqrt [6]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [3]{a x+\sqrt {a^2 x^2-b}}-\sqrt {3} \sqrt [12]{-b} \sqrt [6]{a x+\sqrt {a^2 x^2-b}}+\sqrt [6]{-b}}d\sqrt [6]{a x+\sqrt {a^2 x^2-b}}}{6 (-b)^{5/12}}+\frac {\int \frac {\sqrt {3} \sqrt [6]{a x+\sqrt {a^2 x^2-b}}+2 \sqrt [12]{-b}}{\sqrt [3]{a x+\sqrt {a^2 x^2-b}}+\sqrt {3} \sqrt [12]{-b} \sqrt [6]{a x+\sqrt {a^2 x^2-b}}+\sqrt [6]{-b}}d\sqrt [6]{a x+\sqrt {a^2 x^2-b}}}{6 (-b)^{5/12}}\right )\right )}{2 b}\right )-\frac {\left (a x+\sqrt {a^2 x^2-b}\right )^{7/6}}{4 \left (\left (a x+\sqrt {a^2 x^2-b}\right )^2+b\right )^2}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle 8 a^2 \left (\frac {7}{24} \left (\frac {\left (a x+\sqrt {a^2 x^2-b}\right )^{7/6}}{2 b \left (\left (a x+\sqrt {a^2 x^2-b}\right )^2+b\right )}+\frac {5 \left (\frac {1}{2} \left (-\frac {\int \frac {1}{\sqrt [6]{-b}-\sqrt [3]{a x+\sqrt {a^2 x^2-b}}}d\sqrt [6]{a x+\sqrt {a^2 x^2-b}}}{3 \sqrt [3]{-b}}-\frac {\int \frac {-a x-\sqrt {a^2 x^2-b}+2 \sqrt [12]{-b}}{\sqrt [3]{a x+\sqrt {a^2 x^2-b}}-\sqrt [12]{-b} \sqrt [6]{a x+\sqrt {a^2 x^2-b}}+\sqrt [6]{-b}}d\sqrt [6]{a x+\sqrt {a^2 x^2-b}}}{6 (-b)^{5/12}}-\frac {\int \frac {\sqrt [6]{a x+\sqrt {a^2 x^2-b}}+2 \sqrt [12]{-b}}{\sqrt [3]{a x+\sqrt {a^2 x^2-b}}+\sqrt [12]{-b} \sqrt [6]{a x+\sqrt {a^2 x^2-b}}+\sqrt [6]{-b}}d\sqrt [6]{a x+\sqrt {a^2 x^2-b}}}{6 (-b)^{5/12}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt [6]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [12]{-b}}\right )}{3 (-b)^{5/12}}+\frac {\int \frac {2 \sqrt [12]{-b}-\sqrt {3} \sqrt [6]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [3]{a x+\sqrt {a^2 x^2-b}}-\sqrt {3} \sqrt [12]{-b} \sqrt [6]{a x+\sqrt {a^2 x^2-b}}+\sqrt [6]{-b}}d\sqrt [6]{a x+\sqrt {a^2 x^2-b}}}{6 (-b)^{5/12}}+\frac {\int \frac {\sqrt {3} \sqrt [6]{a x+\sqrt {a^2 x^2-b}}+2 \sqrt [12]{-b}}{\sqrt [3]{a x+\sqrt {a^2 x^2-b}}+\sqrt {3} \sqrt [12]{-b} \sqrt [6]{a x+\sqrt {a^2 x^2-b}}+\sqrt [6]{-b}}d\sqrt [6]{a x+\sqrt {a^2 x^2-b}}}{6 (-b)^{5/12}}\right )\right )}{2 b}\right )-\frac {\left (a x+\sqrt {a^2 x^2-b}\right )^{7/6}}{4 \left (\left (a x+\sqrt {a^2 x^2-b}\right )^2+b\right )^2}\right )\)

\(\Big \downarrow \) 219

\(\displaystyle 8 a^2 \left (\frac {7}{24} \left (\frac {5 \left (\frac {1}{2} \left (\frac {\int \frac {2 \sqrt [12]{-b}-\sqrt {3} \sqrt [6]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [3]{a x+\sqrt {a^2 x^2-b}}-\sqrt {3} \sqrt [12]{-b} \sqrt [6]{a x+\sqrt {a^2 x^2-b}}+\sqrt [6]{-b}}d\sqrt [6]{a x+\sqrt {a^2 x^2-b}}}{6 (-b)^{5/12}}+\frac {\int \frac {\sqrt {3} \sqrt [6]{a x+\sqrt {a^2 x^2-b}}+2 \sqrt [12]{-b}}{\sqrt [3]{a x+\sqrt {a^2 x^2-b}}+\sqrt {3} \sqrt [12]{-b} \sqrt [6]{a x+\sqrt {a^2 x^2-b}}+\sqrt [6]{-b}}d\sqrt [6]{a x+\sqrt {a^2 x^2-b}}}{6 (-b)^{5/12}}+\frac {\arctan \left (\frac {\sqrt [6]{\sqrt {a^2 x^2-b}+a x}}{\sqrt [12]{-b}}\right )}{3 (-b)^{5/12}}\right )+\frac {1}{2} \left (-\frac {\int \frac {-a x-\sqrt {a^2 x^2-b}+2 \sqrt [12]{-b}}{\sqrt [3]{a x+\sqrt {a^2 x^2-b}}-\sqrt [12]{-b} \sqrt [6]{a x+\sqrt {a^2 x^2-b}}+\sqrt [6]{-b}}d\sqrt [6]{a x+\sqrt {a^2 x^2-b}}}{6 (-b)^{5/12}}-\frac {\int \frac {\sqrt [6]{a x+\sqrt {a^2 x^2-b}}+2 \sqrt [12]{-b}}{\sqrt [3]{a x+\sqrt {a^2 x^2-b}}+\sqrt [12]{-b} \sqrt [6]{a x+\sqrt {a^2 x^2-b}}+\sqrt [6]{-b}}d\sqrt [6]{a x+\sqrt {a^2 x^2-b}}}{6 (-b)^{5/12}}-\frac {\text {arctanh}\left (\frac {\sqrt [6]{\sqrt {a^2 x^2-b}+a x}}{\sqrt [12]{-b}}\right )}{3 (-b)^{5/12}}\right )\right )}{2 b}+\frac {\left (\sqrt {a^2 x^2-b}+a x\right )^{7/6}}{2 b \left (\left (\sqrt {a^2 x^2-b}+a x\right )^2+b\right )}\right )-\frac {\left (\sqrt {a^2 x^2-b}+a x\right )^{7/6}}{4 \left (\left (\sqrt {a^2 x^2-b}+a x\right )^2+b\right )^2}\right )\)

\(\Big \downarrow \) 1142

\(\displaystyle 8 a^2 \left (\frac {7}{24} \left (\frac {\left (a x+\sqrt {a^2 x^2-b}\right )^{7/6}}{2 b \left (\left (a x+\sqrt {a^2 x^2-b}\right )^2+b\right )}+\frac {5 \left (\frac {1}{2} \left (-\frac {\text {arctanh}\left (\frac {\sqrt [6]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [12]{-b}}\right )}{3 (-b)^{5/12}}-\frac {\frac {3}{2} \sqrt [12]{-b} \int \frac {1}{\sqrt [3]{a x+\sqrt {a^2 x^2-b}}-\sqrt [12]{-b} \sqrt [6]{a x+\sqrt {a^2 x^2-b}}+\sqrt [6]{-b}}d\sqrt [6]{a x+\sqrt {a^2 x^2-b}}-\frac {1}{2} \int -\frac {\sqrt [12]{-b}-2 \sqrt [6]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [3]{a x+\sqrt {a^2 x^2-b}}-\sqrt [12]{-b} \sqrt [6]{a x+\sqrt {a^2 x^2-b}}+\sqrt [6]{-b}}d\sqrt [6]{a x+\sqrt {a^2 x^2-b}}}{6 (-b)^{5/12}}-\frac {\frac {3}{2} \sqrt [12]{-b} \int \frac {1}{\sqrt [3]{a x+\sqrt {a^2 x^2-b}}+\sqrt [12]{-b} \sqrt [6]{a x+\sqrt {a^2 x^2-b}}+\sqrt [6]{-b}}d\sqrt [6]{a x+\sqrt {a^2 x^2-b}}+\frac {1}{2} \int \frac {2 \sqrt [6]{a x+\sqrt {a^2 x^2-b}}+\sqrt [12]{-b}}{\sqrt [3]{a x+\sqrt {a^2 x^2-b}}+\sqrt [12]{-b} \sqrt [6]{a x+\sqrt {a^2 x^2-b}}+\sqrt [6]{-b}}d\sqrt [6]{a x+\sqrt {a^2 x^2-b}}}{6 (-b)^{5/12}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt [6]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [12]{-b}}\right )}{3 (-b)^{5/12}}+\frac {\frac {1}{2} \sqrt [12]{-b} \int \frac {1}{\sqrt [3]{a x+\sqrt {a^2 x^2-b}}-\sqrt {3} \sqrt [12]{-b} \sqrt [6]{a x+\sqrt {a^2 x^2-b}}+\sqrt [6]{-b}}d\sqrt [6]{a x+\sqrt {a^2 x^2-b}}-\frac {1}{2} \sqrt {3} \int -\frac {\sqrt {3} \sqrt [12]{-b}-2 \sqrt [6]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [3]{a x+\sqrt {a^2 x^2-b}}-\sqrt {3} \sqrt [12]{-b} \sqrt [6]{a x+\sqrt {a^2 x^2-b}}+\sqrt [6]{-b}}d\sqrt [6]{a x+\sqrt {a^2 x^2-b}}}{6 (-b)^{5/12}}+\frac {\frac {1}{2} \sqrt [12]{-b} \int \frac {1}{\sqrt [3]{a x+\sqrt {a^2 x^2-b}}+\sqrt {3} \sqrt [12]{-b} \sqrt [6]{a x+\sqrt {a^2 x^2-b}}+\sqrt [6]{-b}}d\sqrt [6]{a x+\sqrt {a^2 x^2-b}}+\frac {1}{2} \sqrt {3} \int \frac {2 \sqrt [6]{a x+\sqrt {a^2 x^2-b}}+\sqrt {3} \sqrt [12]{-b}}{\sqrt [3]{a x+\sqrt {a^2 x^2-b}}+\sqrt {3} \sqrt [12]{-b} \sqrt [6]{a x+\sqrt {a^2 x^2-b}}+\sqrt [6]{-b}}d\sqrt [6]{a x+\sqrt {a^2 x^2-b}}}{6 (-b)^{5/12}}\right )\right )}{2 b}\right )-\frac {\left (a x+\sqrt {a^2 x^2-b}\right )^{7/6}}{4 \left (\left (a x+\sqrt {a^2 x^2-b}\right )^2+b\right )^2}\right )\)

\(\Big \downarrow \) 25

\(\displaystyle 8 a^2 \left (\frac {7}{24} \left (\frac {\left (a x+\sqrt {a^2 x^2-b}\right )^{7/6}}{2 b \left (\left (a x+\sqrt {a^2 x^2-b}\right )^2+b\right )}+\frac {5 \left (\frac {1}{2} \left (-\frac {\text {arctanh}\left (\frac {\sqrt [6]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [12]{-b}}\right )}{3 (-b)^{5/12}}-\frac {\frac {3}{2} \sqrt [12]{-b} \int \frac {1}{\sqrt [3]{a x+\sqrt {a^2 x^2-b}}-\sqrt [12]{-b} \sqrt [6]{a x+\sqrt {a^2 x^2-b}}+\sqrt [6]{-b}}d\sqrt [6]{a x+\sqrt {a^2 x^2-b}}+\frac {1}{2} \int \frac {\sqrt [12]{-b}-2 \sqrt [6]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [3]{a x+\sqrt {a^2 x^2-b}}-\sqrt [12]{-b} \sqrt [6]{a x+\sqrt {a^2 x^2-b}}+\sqrt [6]{-b}}d\sqrt [6]{a x+\sqrt {a^2 x^2-b}}}{6 (-b)^{5/12}}-\frac {\frac {3}{2} \sqrt [12]{-b} \int \frac {1}{\sqrt [3]{a x+\sqrt {a^2 x^2-b}}+\sqrt [12]{-b} \sqrt [6]{a x+\sqrt {a^2 x^2-b}}+\sqrt [6]{-b}}d\sqrt [6]{a x+\sqrt {a^2 x^2-b}}+\frac {1}{2} \int \frac {2 \sqrt [6]{a x+\sqrt {a^2 x^2-b}}+\sqrt [12]{-b}}{\sqrt [3]{a x+\sqrt {a^2 x^2-b}}+\sqrt [12]{-b} \sqrt [6]{a x+\sqrt {a^2 x^2-b}}+\sqrt [6]{-b}}d\sqrt [6]{a x+\sqrt {a^2 x^2-b}}}{6 (-b)^{5/12}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt [6]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [12]{-b}}\right )}{3 (-b)^{5/12}}+\frac {\frac {1}{2} \sqrt [12]{-b} \int \frac {1}{\sqrt [3]{a x+\sqrt {a^2 x^2-b}}-\sqrt {3} \sqrt [12]{-b} \sqrt [6]{a x+\sqrt {a^2 x^2-b}}+\sqrt [6]{-b}}d\sqrt [6]{a x+\sqrt {a^2 x^2-b}}+\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3} \sqrt [12]{-b}-2 \sqrt [6]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [3]{a x+\sqrt {a^2 x^2-b}}-\sqrt {3} \sqrt [12]{-b} \sqrt [6]{a x+\sqrt {a^2 x^2-b}}+\sqrt [6]{-b}}d\sqrt [6]{a x+\sqrt {a^2 x^2-b}}}{6 (-b)^{5/12}}+\frac {\frac {1}{2} \sqrt [12]{-b} \int \frac {1}{\sqrt [3]{a x+\sqrt {a^2 x^2-b}}+\sqrt {3} \sqrt [12]{-b} \sqrt [6]{a x+\sqrt {a^2 x^2-b}}+\sqrt [6]{-b}}d\sqrt [6]{a x+\sqrt {a^2 x^2-b}}+\frac {1}{2} \sqrt {3} \int \frac {2 \sqrt [6]{a x+\sqrt {a^2 x^2-b}}+\sqrt {3} \sqrt [12]{-b}}{\sqrt [3]{a x+\sqrt {a^2 x^2-b}}+\sqrt {3} \sqrt [12]{-b} \sqrt [6]{a x+\sqrt {a^2 x^2-b}}+\sqrt [6]{-b}}d\sqrt [6]{a x+\sqrt {a^2 x^2-b}}}{6 (-b)^{5/12}}\right )\right )}{2 b}\right )-\frac {\left (a x+\sqrt {a^2 x^2-b}\right )^{7/6}}{4 \left (\left (a x+\sqrt {a^2 x^2-b}\right )^2+b\right )^2}\right )\)

\(\Big \downarrow \) 1082

\(\displaystyle 8 a^2 \left (\frac {7}{24} \left (\frac {\left (a x+\sqrt {a^2 x^2-b}\right )^{7/6}}{2 b \left (\left (a x+\sqrt {a^2 x^2-b}\right )^2+b\right )}+\frac {5 \left (\frac {1}{2} \left (-\frac {\text {arctanh}\left (\frac {\sqrt [6]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [12]{-b}}\right )}{3 (-b)^{5/12}}-\frac {3 \int \frac {1}{-\sqrt [3]{a x+\sqrt {a^2 x^2-b}}-3}d\left (1-\frac {2 \sqrt [6]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [12]{-b}}\right )+\frac {1}{2} \int \frac {\sqrt [12]{-b}-2 \sqrt [6]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [3]{a x+\sqrt {a^2 x^2-b}}-\sqrt [12]{-b} \sqrt [6]{a x+\sqrt {a^2 x^2-b}}+\sqrt [6]{-b}}d\sqrt [6]{a x+\sqrt {a^2 x^2-b}}}{6 (-b)^{5/12}}-\frac {\frac {1}{2} \int \frac {2 \sqrt [6]{a x+\sqrt {a^2 x^2-b}}+\sqrt [12]{-b}}{\sqrt [3]{a x+\sqrt {a^2 x^2-b}}+\sqrt [12]{-b} \sqrt [6]{a x+\sqrt {a^2 x^2-b}}+\sqrt [6]{-b}}d\sqrt [6]{a x+\sqrt {a^2 x^2-b}}-3 \int \frac {1}{-\sqrt [3]{a x+\sqrt {a^2 x^2-b}}-3}d\left (\frac {2 \sqrt [6]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [12]{-b}}+1\right )}{6 (-b)^{5/12}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt [6]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [12]{-b}}\right )}{3 (-b)^{5/12}}+\frac {\frac {\int \frac {1}{-\sqrt [3]{a x+\sqrt {a^2 x^2-b}}-\frac {1}{3}}d\left (1-\frac {2 \sqrt [6]{a x+\sqrt {a^2 x^2-b}}}{\sqrt {3} \sqrt [12]{-b}}\right )}{\sqrt {3}}+\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3} \sqrt [12]{-b}-2 \sqrt [6]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [3]{a x+\sqrt {a^2 x^2-b}}-\sqrt {3} \sqrt [12]{-b} \sqrt [6]{a x+\sqrt {a^2 x^2-b}}+\sqrt [6]{-b}}d\sqrt [6]{a x+\sqrt {a^2 x^2-b}}}{6 (-b)^{5/12}}+\frac {\frac {1}{2} \sqrt {3} \int \frac {2 \sqrt [6]{a x+\sqrt {a^2 x^2-b}}+\sqrt {3} \sqrt [12]{-b}}{\sqrt [3]{a x+\sqrt {a^2 x^2-b}}+\sqrt {3} \sqrt [12]{-b} \sqrt [6]{a x+\sqrt {a^2 x^2-b}}+\sqrt [6]{-b}}d\sqrt [6]{a x+\sqrt {a^2 x^2-b}}-\frac {\int \frac {1}{-\sqrt [3]{a x+\sqrt {a^2 x^2-b}}-\frac {1}{3}}d\left (\frac {2 \sqrt [6]{a x+\sqrt {a^2 x^2-b}}}{\sqrt {3} \sqrt [12]{-b}}+1\right )}{\sqrt {3}}}{6 (-b)^{5/12}}\right )\right )}{2 b}\right )-\frac {\left (a x+\sqrt {a^2 x^2-b}\right )^{7/6}}{4 \left (\left (a x+\sqrt {a^2 x^2-b}\right )^2+b\right )^2}\right )\)

\(\Big \downarrow \) 217

\(\displaystyle 8 a^2 \left (\frac {7}{24} \left (\frac {\left (a x+\sqrt {a^2 x^2-b}\right )^{7/6}}{2 b \left (\left (a x+\sqrt {a^2 x^2-b}\right )^2+b\right )}+\frac {5 \left (\frac {1}{2} \left (-\frac {\text {arctanh}\left (\frac {\sqrt [6]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [12]{-b}}\right )}{3 (-b)^{5/12}}-\frac {\frac {1}{2} \int \frac {\sqrt [12]{-b}-2 \sqrt [6]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [3]{a x+\sqrt {a^2 x^2-b}}-\sqrt [12]{-b} \sqrt [6]{a x+\sqrt {a^2 x^2-b}}+\sqrt [6]{-b}}d\sqrt [6]{a x+\sqrt {a^2 x^2-b}}-\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [6]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [12]{-b}}}{\sqrt {3}}\right )}{6 (-b)^{5/12}}-\frac {\sqrt {3} \arctan \left (\frac {\frac {2 \sqrt [6]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [12]{-b}}+1}{\sqrt {3}}\right )+\frac {1}{2} \int \frac {2 \sqrt [6]{a x+\sqrt {a^2 x^2-b}}+\sqrt [12]{-b}}{\sqrt [3]{a x+\sqrt {a^2 x^2-b}}+\sqrt [12]{-b} \sqrt [6]{a x+\sqrt {a^2 x^2-b}}+\sqrt [6]{-b}}d\sqrt [6]{a x+\sqrt {a^2 x^2-b}}}{6 (-b)^{5/12}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt [6]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [12]{-b}}\right )}{3 (-b)^{5/12}}+\frac {\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3} \sqrt [12]{-b}-2 \sqrt [6]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [3]{a x+\sqrt {a^2 x^2-b}}-\sqrt {3} \sqrt [12]{-b} \sqrt [6]{a x+\sqrt {a^2 x^2-b}}+\sqrt [6]{-b}}d\sqrt [6]{a x+\sqrt {a^2 x^2-b}}-\arctan \left (\sqrt {3} \left (1-\frac {2 \sqrt [6]{a x+\sqrt {a^2 x^2-b}}}{\sqrt {3} \sqrt [12]{-b}}\right )\right )}{6 (-b)^{5/12}}+\frac {\arctan \left (\sqrt {3} \left (\frac {2 \sqrt [6]{a x+\sqrt {a^2 x^2-b}}}{\sqrt {3} \sqrt [12]{-b}}+1\right )\right )+\frac {1}{2} \sqrt {3} \int \frac {2 \sqrt [6]{a x+\sqrt {a^2 x^2-b}}+\sqrt {3} \sqrt [12]{-b}}{\sqrt [3]{a x+\sqrt {a^2 x^2-b}}+\sqrt {3} \sqrt [12]{-b} \sqrt [6]{a x+\sqrt {a^2 x^2-b}}+\sqrt [6]{-b}}d\sqrt [6]{a x+\sqrt {a^2 x^2-b}}}{6 (-b)^{5/12}}\right )\right )}{2 b}\right )-\frac {\left (a x+\sqrt {a^2 x^2-b}\right )^{7/6}}{4 \left (\left (a x+\sqrt {a^2 x^2-b}\right )^2+b\right )^2}\right )\)

\(\Big \downarrow \) 1103

\(\displaystyle 8 a^2 \left (\frac {7}{24} \left (\frac {5 \left (\frac {1}{2} \left (-\frac {-\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [6]{\sqrt {a^2 x^2-b}+a x}}{\sqrt [12]{-b}}}{\sqrt {3}}\right )-\frac {1}{2} \log \left (\sqrt [3]{\sqrt {a^2 x^2-b}+a x}-\sqrt [12]{-b} \sqrt [6]{\sqrt {a^2 x^2-b}+a x}+\sqrt [6]{-b}\right )}{6 (-b)^{5/12}}-\frac {\sqrt {3} \arctan \left (\frac {\frac {2 \sqrt [6]{\sqrt {a^2 x^2-b}+a x}}{\sqrt [12]{-b}}+1}{\sqrt {3}}\right )+\frac {1}{2} \log \left (\sqrt [3]{\sqrt {a^2 x^2-b}+a x}+\sqrt [12]{-b} \sqrt [6]{\sqrt {a^2 x^2-b}+a x}+\sqrt [6]{-b}\right )}{6 (-b)^{5/12}}-\frac {\text {arctanh}\left (\frac {\sqrt [6]{\sqrt {a^2 x^2-b}+a x}}{\sqrt [12]{-b}}\right )}{3 (-b)^{5/12}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt [6]{\sqrt {a^2 x^2-b}+a x}}{\sqrt [12]{-b}}\right )}{3 (-b)^{5/12}}+\frac {-\arctan \left (\sqrt {3} \left (1-\frac {2 \sqrt [6]{\sqrt {a^2 x^2-b}+a x}}{\sqrt {3} \sqrt [12]{-b}}\right )\right )-\frac {1}{2} \sqrt {3} \log \left (\sqrt [3]{\sqrt {a^2 x^2-b}+a x}-\sqrt {3} \sqrt [12]{-b} \sqrt [6]{\sqrt {a^2 x^2-b}+a x}+\sqrt [6]{-b}\right )}{6 (-b)^{5/12}}+\frac {\arctan \left (\sqrt {3} \left (\frac {2 \sqrt [6]{\sqrt {a^2 x^2-b}+a x}}{\sqrt {3} \sqrt [12]{-b}}+1\right )\right )+\frac {1}{2} \sqrt {3} \log \left (\sqrt [3]{\sqrt {a^2 x^2-b}+a x}+\sqrt {3} \sqrt [12]{-b} \sqrt [6]{\sqrt {a^2 x^2-b}+a x}+\sqrt [6]{-b}\right )}{6 (-b)^{5/12}}\right )\right )}{2 b}+\frac {\left (\sqrt {a^2 x^2-b}+a x\right )^{7/6}}{2 b \left (\left (\sqrt {a^2 x^2-b}+a x\right )^2+b\right )}\right )-\frac {\left (\sqrt {a^2 x^2-b}+a x\right )^{7/6}}{4 \left (\left (\sqrt {a^2 x^2-b}+a x\right )^2+b\right )^2}\right )\)

Input:

Int[(a*x + Sqrt[-b + a^2*x^2])^(1/6)/(x^3*Sqrt[-b + a^2*x^2]),x]
 

Output:

8*a^2*(-1/4*(a*x + Sqrt[-b + a^2*x^2])^(7/6)/(b + (a*x + Sqrt[-b + a^2*x^2 
])^2)^2 + (7*((a*x + Sqrt[-b + a^2*x^2])^(7/6)/(2*b*(b + (a*x + Sqrt[-b + 
a^2*x^2])^2)) + (5*((-1/3*ArcTanh[(a*x + Sqrt[-b + a^2*x^2])^(1/6)/(-b)^(1 
/12)]/(-b)^(5/12) - (-(Sqrt[3]*ArcTan[(1 - (2*(a*x + Sqrt[-b + a^2*x^2])^( 
1/6))/(-b)^(1/12))/Sqrt[3]]) - Log[(-b)^(1/6) - (-b)^(1/12)*(a*x + Sqrt[-b 
 + a^2*x^2])^(1/6) + (a*x + Sqrt[-b + a^2*x^2])^(1/3)]/2)/(6*(-b)^(5/12)) 
- (Sqrt[3]*ArcTan[(1 + (2*(a*x + Sqrt[-b + a^2*x^2])^(1/6))/(-b)^(1/12))/S 
qrt[3]] + Log[(-b)^(1/6) + (-b)^(1/12)*(a*x + Sqrt[-b + a^2*x^2])^(1/6) + 
(a*x + Sqrt[-b + a^2*x^2])^(1/3)]/2)/(6*(-b)^(5/12)))/2 + (ArcTan[(a*x + S 
qrt[-b + a^2*x^2])^(1/6)/(-b)^(1/12)]/(3*(-b)^(5/12)) + (-ArcTan[Sqrt[3]*( 
1 - (2*(a*x + Sqrt[-b + a^2*x^2])^(1/6))/(Sqrt[3]*(-b)^(1/12)))] - (Sqrt[3 
]*Log[(-b)^(1/6) - Sqrt[3]*(-b)^(1/12)*(a*x + Sqrt[-b + a^2*x^2])^(1/6) + 
(a*x + Sqrt[-b + a^2*x^2])^(1/3)])/2)/(6*(-b)^(5/12)) + (ArcTan[Sqrt[3]*(1 
 + (2*(a*x + Sqrt[-b + a^2*x^2])^(1/6))/(Sqrt[3]*(-b)^(1/12)))] + (Sqrt[3] 
*Log[(-b)^(1/6) + Sqrt[3]*(-b)^(1/12)*(a*x + Sqrt[-b + a^2*x^2])^(1/6) + ( 
a*x + Sqrt[-b + a^2*x^2])^(1/3)])/2)/(6*(-b)^(5/12)))/2))/(2*b)))/24)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 216

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])
 

rule 217

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])
 

rule 219

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])
 

rule 252

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]
 

rule 253

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]
 

rule 266

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]
 

rule 753

Int[((a_) + (b_.)*(x_)^(n_))^(-1), x_Symbol] :> Module[{r = Numerator[Rt[a/ 
b, n]], s = Denominator[Rt[a/b, n]], k, u, v}, Simp[u = Int[(r - s*Cos[(2*k 
 - 1)*(Pi/n)]*x)/(r^2 - 2*r*s*Cos[(2*k - 1)*(Pi/n)]*x + s^2*x^2), x] + Int[ 
(r + s*Cos[(2*k - 1)*(Pi/n)]*x)/(r^2 + 2*r*s*Cos[(2*k - 1)*(Pi/n)]*x + s^2* 
x^2), x]; 2*(r^2/(a*n))   Int[1/(r^2 + s^2*x^2), x] + 2*(r/(a*n))   Sum[u, 
{k, 1, (n - 2)/4}], x]] /; FreeQ[{a, b}, x] && IGtQ[(n - 2)/4, 0] && PosQ[a 
/b]
 

rule 754

Int[((a_) + (b_.)*(x_)^(n_))^(-1), x_Symbol] :> Module[{r = Numerator[Rt[-a 
/b, n]], s = Denominator[Rt[-a/b, n]], k, u}, Simp[u = Int[(r - s*Cos[(2*k* 
Pi)/n]*x)/(r^2 - 2*r*s*Cos[(2*k*Pi)/n]*x + s^2*x^2), x] + Int[(r + s*Cos[(2 
*k*Pi)/n]*x)/(r^2 + 2*r*s*Cos[(2*k*Pi)/n]*x + s^2*x^2), x]; 2*(r^2/(a*n)) 
 Int[1/(r^2 - s^2*x^2), x] + 2*(r/(a*n))   Sum[u, {k, 1, (n - 2)/4}], x]] / 
; FreeQ[{a, b}, x] && IGtQ[(n - 2)/4, 0] && NegQ[a/b]
 

rule 830

Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{r = Numerator[Rt 
[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}, Simp[s/(2*b)   Int[x^(m - n/2)/( 
r + s*x^(n/2)), x], x] - Simp[s/(2*b)   Int[x^(m - n/2)/(r - s*x^(n/2)), x] 
, x]] /; FreeQ[{a, b}, x] && IGtQ[n/4, 0] && IGtQ[m, 0] && LeQ[n/2, m] && L 
tQ[m, n] &&  !GtQ[a/b, 0]
 

rule 1082

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]
 

rule 1103

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]
 

rule 1142

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[(2*c*d - b*e)/(2*c)   Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) 
Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
 

rule 2545

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])
 

Maple [F]

\[\int \frac {\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{6}}}{x^{3} \sqrt {a^{2} x^{2}-b}}d x\]

Input:

int((a*x+(a^2*x^2-b)^(1/2))^(1/6)/x^3/(a^2*x^2-b)^(1/2),x)
 

Output:

int((a*x+(a^2*x^2-b)^(1/2))^(1/6)/x^3/(a^2*x^2-b)^(1/2),x)
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.76 (sec) , antiderivative size = 1065, normalized size of antiderivative = 1.57 \[ \int \frac {\sqrt [6]{a x+\sqrt {-b+a^2 x^2}}}{x^3 \sqrt {-b+a^2 x^2}} \, dx=\text {Too large to display} \] Input:

integrate((a*x+(a^2*x^2-b)^(1/2))^(1/6)/x^3/(a^2*x^2-b)^(1/2),x, algorithm 
="fricas")
 

Output:

-1/144*(70*(-a^24/b^17)^(1/12)*b*x^2*(1/2*I*sqrt(3) + 1/2)^(3/2)*log(64339 
296875*(a*x + sqrt(a^2*x^2 - b))^(1/6)*a^14 + 64339296875*(-a^24/b^17)^(7/ 
12)*b^10*(1/2*I*sqrt(3) + 1/2)^(3/2)) - 70*(-a^24/b^17)^(1/12)*b*x^2*(1/2* 
I*sqrt(3) + 1/2)^(3/2)*log(64339296875*(a*x + sqrt(a^2*x^2 - b))^(1/6)*a^1 
4 - 64339296875*(-a^24/b^17)^(7/12)*b^10*(1/2*I*sqrt(3) + 1/2)^(3/2)) - 35 
*(-a^24/b^17)^(1/12)*b*x^2*(-I*sqrt(3) - 1)*log(64339296875*(a*x + sqrt(a^ 
2*x^2 - b))^(1/6)*a^14 + 64339296875/2*(-a^24/b^17)^(7/12)*b^10*(-I*sqrt(3 
) - 1)) + 35*(-a^24/b^17)^(1/12)*b*x^2*(-I*sqrt(3) - 1)*log(64339296875*(a 
*x + sqrt(a^2*x^2 - b))^(1/6)*a^14 - 64339296875/2*(-a^24/b^17)^(7/12)*b^1 
0*(-I*sqrt(3) - 1)) + 70*(-a^24/b^17)^(1/12)*b*x^2*sqrt(1/2*I*sqrt(3) + 1/ 
2)*log(64339296875*(a*x + sqrt(a^2*x^2 - b))^(1/6)*a^14 + 64339296875*(-a^ 
24/b^17)^(7/12)*b^10*sqrt(1/2*I*sqrt(3) + 1/2)) - 70*(-a^24/b^17)^(1/12)*b 
*x^2*sqrt(1/2*I*sqrt(3) + 1/2)*log(64339296875*(a*x + sqrt(a^2*x^2 - b))^( 
1/6)*a^14 - 64339296875*(-a^24/b^17)^(7/12)*b^10*sqrt(1/2*I*sqrt(3) + 1/2) 
) - 70*(-a^24/b^17)^(1/12)*b*x^2*log(64339296875*(a*x + sqrt(a^2*x^2 - b)) 
^(1/6)*a^14 + 64339296875*(-a^24/b^17)^(7/12)*b^10) + 70*(-a^24/b^17)^(1/1 
2)*b*x^2*log(64339296875*(a*x + sqrt(a^2*x^2 - b))^(1/6)*a^14 - 6433929687 
5*(-a^24/b^17)^(7/12)*b^10) + 70*((-a^24/b^17)^(1/12)*b*x^2*(1/2*I*sqrt(3) 
 + 1/2)^(3/2) - (-a^24/b^17)^(1/12)*b*x^2*sqrt(1/2*I*sqrt(3) + 1/2))*log(6 
4339296875*(a*x + sqrt(a^2*x^2 - b))^(1/6)*a^14 + 64339296875*(-a^24/b^...
 

Sympy [F]

\[ \int \frac {\sqrt [6]{a x+\sqrt {-b+a^2 x^2}}}{x^3 \sqrt {-b+a^2 x^2}} \, dx=\int \frac {\sqrt [6]{a x + \sqrt {a^{2} x^{2} - b}}}{x^{3} \sqrt {a^{2} x^{2} - b}}\, dx \] Input:

integrate((a*x+(a**2*x**2-b)**(1/2))**(1/6)/x**3/(a**2*x**2-b)**(1/2),x)
 

Output:

Integral((a*x + sqrt(a**2*x**2 - b))**(1/6)/(x**3*sqrt(a**2*x**2 - b)), x)
 

Maxima [F]

\[ \int \frac {\sqrt [6]{a x+\sqrt {-b+a^2 x^2}}}{x^3 \sqrt {-b+a^2 x^2}} \, dx=\int { \frac {{\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{6}}}{\sqrt {a^{2} x^{2} - b} x^{3}} \,d x } \] Input:

integrate((a*x+(a^2*x^2-b)^(1/2))^(1/6)/x^3/(a^2*x^2-b)^(1/2),x, algorithm 
="maxima")
 

Output:

integrate((a*x + sqrt(a^2*x^2 - b))^(1/6)/(sqrt(a^2*x^2 - b)*x^3), x)
 

Giac [F(-1)]

Timed out. \[ \int \frac {\sqrt [6]{a x+\sqrt {-b+a^2 x^2}}}{x^3 \sqrt {-b+a^2 x^2}} \, dx=\text {Timed out} \] Input:

integrate((a*x+(a^2*x^2-b)^(1/2))^(1/6)/x^3/(a^2*x^2-b)^(1/2),x, algorithm 
="giac")
 

Output:

Timed out
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt [6]{a x+\sqrt {-b+a^2 x^2}}}{x^3 \sqrt {-b+a^2 x^2}} \, dx=\int \frac {{\left (a\,x+\sqrt {a^2\,x^2-b}\right )}^{1/6}}{x^3\,\sqrt {a^2\,x^2-b}} \,d x \] Input:

int((a*x + (a^2*x^2 - b)^(1/2))^(1/6)/(x^3*(a^2*x^2 - b)^(1/2)),x)
 

Output:

int((a*x + (a^2*x^2 - b)^(1/2))^(1/6)/(x^3*(a^2*x^2 - b)^(1/2)), x)
 

Reduce [F]

\[ \int \frac {\sqrt [6]{a x+\sqrt {-b+a^2 x^2}}}{x^3 \sqrt {-b+a^2 x^2}} \, dx=\frac {4 \sqrt {a^{2} x^{2}-b}\, \left (\sqrt {a^{2} x^{2}-b}+a x \right )^{\frac {5}{6}} b -2 \left (\sqrt {a^{2} x^{2}-b}+a x \right )^{\frac {2}{3}} \left (\int \frac {\left (\sqrt {a^{2} x^{2}-b}+a x \right )^{\frac {1}{6}}}{2 \sqrt {a^{2} x^{2}-b}\, a^{3} x^{5}-2 \sqrt {a^{2} x^{2}-b}\, a b \,x^{3}+2 a^{4} x^{6}-3 a^{2} b \,x^{4}+b^{2} x^{2}}d x \right ) a \,b^{3} x^{2}-4 \left (\sqrt {a^{2} x^{2}-b}+a x \right )^{\frac {2}{3}} \left (\int \frac {\left (\sqrt {a^{2} x^{2}-b}+a x \right )^{\frac {1}{6}}}{2 \sqrt {a^{2} x^{2}-b}\, a^{3} x^{3}-2 \sqrt {a^{2} x^{2}-b}\, a b x +2 a^{4} x^{4}-3 a^{2} b \,x^{2}+b^{2}}d x \right ) a^{3} b^{2} x^{2}-2 \left (\sqrt {a^{2} x^{2}-b}+a x \right )^{\frac {2}{3}} \left (\int \frac {\sqrt {a^{2} x^{2}-b}\, \left (\sqrt {a^{2} x^{2}-b}+a x \right )^{\frac {1}{6}}}{2 \sqrt {a^{2} x^{2}-b}\, a^{3} x^{6}-2 \sqrt {a^{2} x^{2}-b}\, a b \,x^{4}+2 a^{4} x^{7}-3 a^{2} b \,x^{5}+b^{2} x^{3}}d x \right ) b^{3} x^{2}-4 \left (\sqrt {a^{2} x^{2}-b}+a x \right )^{\frac {2}{3}} \left (\int \frac {\sqrt {a^{2} x^{2}-b}\, \left (\sqrt {a^{2} x^{2}-b}+a x \right )^{\frac {1}{6}}}{2 \sqrt {a^{2} x^{2}-b}\, a^{3} x^{4}-2 \sqrt {a^{2} x^{2}-b}\, a b \,x^{2}+2 a^{4} x^{5}-3 a^{2} b \,x^{3}+b^{2} x}d x \right ) a^{2} b^{2} x^{2}+3 \sqrt {b}\, \left (\sqrt {a^{2} x^{2}-b}+a x \right )^{\frac {5}{6}} \mathrm {log}\left (-\sqrt {b}+a x \right ) a^{2} x^{2}-3 \sqrt {b}\, \left (\sqrt {a^{2} x^{2}-b}+a x \right )^{\frac {5}{6}} \mathrm {log}\left (\sqrt {b}+a x \right ) a^{2} x^{2}}{10 \left (\sqrt {a^{2} x^{2}-b}+a x \right )^{\frac {2}{3}} b^{2} x^{2}} \] Input:

int((a*x+(a^2*x^2-b)^(1/2))^(1/6)/x^3/(a^2*x^2-b)^(1/2),x)
 

Output:

(4*sqrt(a**2*x**2 - b)*(sqrt(a**2*x**2 - b) + a*x)**(5/6)*b - 2*(sqrt(a**2 
*x**2 - b) + a*x)**(2/3)*int((sqrt(a**2*x**2 - b) + a*x)**(1/6)/(2*sqrt(a* 
*2*x**2 - b)*a**3*x**5 - 2*sqrt(a**2*x**2 - b)*a*b*x**3 + 2*a**4*x**6 - 3* 
a**2*b*x**4 + b**2*x**2),x)*a*b**3*x**2 - 4*(sqrt(a**2*x**2 - b) + a*x)**( 
2/3)*int((sqrt(a**2*x**2 - b) + a*x)**(1/6)/(2*sqrt(a**2*x**2 - b)*a**3*x* 
*3 - 2*sqrt(a**2*x**2 - b)*a*b*x + 2*a**4*x**4 - 3*a**2*b*x**2 + b**2),x)* 
a**3*b**2*x**2 - 2*(sqrt(a**2*x**2 - b) + a*x)**(2/3)*int((sqrt(a**2*x**2 
- b)*(sqrt(a**2*x**2 - b) + a*x)**(1/6))/(2*sqrt(a**2*x**2 - b)*a**3*x**6 
- 2*sqrt(a**2*x**2 - b)*a*b*x**4 + 2*a**4*x**7 - 3*a**2*b*x**5 + b**2*x**3 
),x)*b**3*x**2 - 4*(sqrt(a**2*x**2 - b) + a*x)**(2/3)*int((sqrt(a**2*x**2 
- b)*(sqrt(a**2*x**2 - b) + a*x)**(1/6))/(2*sqrt(a**2*x**2 - b)*a**3*x**4 
- 2*sqrt(a**2*x**2 - b)*a*b*x**2 + 2*a**4*x**5 - 3*a**2*b*x**3 + b**2*x),x 
)*a**2*b**2*x**2 + 3*sqrt(b)*(sqrt(a**2*x**2 - b) + a*x)**(5/6)*log( - sqr 
t(b) + a*x)*a**2*x**2 - 3*sqrt(b)*(sqrt(a**2*x**2 - b) + a*x)**(5/6)*log(s 
qrt(b) + a*x)*a**2*x**2)/(10*(sqrt(a**2*x**2 - b) + a*x)**(2/3)*b**2*x**2)