Integrand size = 42, antiderivative size = 725 \[ \int \frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{x \left (-b+a^2 x^2\right )^{3/2}} \, dx=\frac {\left (a x+\sqrt {-b+a^2 x^2}\right )^{5/4}}{b \left (b-a^2 x^2-a x \sqrt {-b+a^2 x^2}\right )}-\frac {\arctan \left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{b}}\right )}{2 b^{11/8}}-\frac {\sqrt {2+\sqrt {2}} \arctan \left (\frac {\left (\sqrt {\frac {2}{2-\sqrt {2}}} \sqrt [8]{b}-\frac {2 \sqrt [8]{b}}{\sqrt {2-\sqrt {2}}}\right ) \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{-\sqrt [4]{b}+\sqrt {a x+\sqrt {-b+a^2 x^2}}}\right )}{b^{11/8}}-\frac {\arctan \left (\frac {\sqrt {2} \sqrt [8]{b} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{-\sqrt [4]{b}+\sqrt {a x+\sqrt {-b+a^2 x^2}}}\right )}{2 \sqrt {2} b^{11/8}}-\frac {\sqrt {2-\sqrt {2}} \arctan \left (\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{b} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{-\sqrt [4]{b}+\sqrt {a x+\sqrt {-b+a^2 x^2}}}\right )}{b^{11/8}}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{b}}\right )}{2 b^{11/8}}+\frac {\text {arctanh}\left (\frac {\frac {\sqrt [8]{b}}{\sqrt {2}}+\frac {\sqrt {a x+\sqrt {-b+a^2 x^2}}}{\sqrt {2} \sqrt [8]{b}}}{\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}\right )}{2 \sqrt {2} b^{11/8}}-\frac {\sqrt {2+\sqrt {2}} \text {arctanh}\left (\frac {\frac {\sqrt [8]{b}}{\sqrt {2-\sqrt {2}}}+\frac {\sqrt {a x+\sqrt {-b+a^2 x^2}}}{\sqrt {2-\sqrt {2}} \sqrt [8]{b}}}{\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}\right )}{b^{11/8}}+\frac {\sqrt {2-\sqrt {2}} \text {arctanh}\left (\frac {\frac {\sqrt [8]{b}}{\sqrt {2+\sqrt {2}}}+\frac {\sqrt {a x+\sqrt {-b+a^2 x^2}}}{\sqrt {2+\sqrt {2}} \sqrt [8]{b}}}{\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}\right )}{b^{11/8}} \]
(a*x+(a^2*x^2-b)^(1/2))^(5/4)/b/(b-a^2*x^2-(a^2*x^2-b)^(1/2)*a*x)-1/2*arct an((a*x+(a^2*x^2-b)^(1/2))^(1/4)/b^(1/8))/b^(11/8)-(2+2^(1/2))^(1/2)*arcta n((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^(11/8)-1 /4*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^(11/8)-(2-2^(1/2))^(1/2)*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^(11/8)-1/2*arctanh((a*x+(a^2*x^2-b)^(1/2))^(1/4)/b^(1/8) )/b^(11/8)+1/4*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^(11/8)-(2+2^ (1/2))^(1/2)*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^(11/8)+(2- 2^(1/2))^(1/2)*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^(11/8)
Time = 2.35 (sec) , antiderivative size = 624, normalized size of antiderivative = 0.86 \[ \int \frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{x \left (-b+a^2 x^2\right )^{3/2}} \, dx=\frac {-\frac {4 b^{3/8} \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/4}}{-b+a x \left (a x+\sqrt {-b+a^2 x^2}\right )}-2 \arctan \left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{b}}\right )+\sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt [8]{b} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [4]{b}-\sqrt {a x+\sqrt {-b+a^2 x^2}}}\right )+4 \sqrt {2-\sqrt {2}} \arctan \left (\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{b} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [4]{b}-\sqrt {a x+\sqrt {-b+a^2 x^2}}}\right )+4 \sqrt {2+\sqrt {2}} \arctan \left (\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{b} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{-\sqrt [4]{b}+\sqrt {a x+\sqrt {-b+a^2 x^2}}}\right )-2 \text {arctanh}\left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{b}}\right )+\sqrt {2} \text {arctanh}\left (\frac {\sqrt [4]{b}+\sqrt {a x+\sqrt {-b+a^2 x^2}}}{\sqrt {2} \sqrt [8]{b} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}\right )+4 \sqrt {2-\sqrt {2}} \text {arctanh}\left (\frac {\sqrt {1-\frac {1}{\sqrt {2}}} \left (\sqrt [4]{b}+\sqrt {a x+\sqrt {-b+a^2 x^2}}\right )}{\sqrt [8]{b} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}\right )-4 \sqrt {2+\sqrt {2}} \text {arctanh}\left (\frac {\sqrt {1+\frac {1}{\sqrt {2}}} \left (\sqrt [4]{b}+\sqrt {a x+\sqrt {-b+a^2 x^2}}\right )}{\sqrt [8]{b} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}\right )}{4 b^{11/8}} \]
((-4*b^(3/8)*(a*x + Sqrt[-b + a^2*x^2])^(5/4))/(-b + a*x*(a*x + Sqrt[-b + a^2*x^2])) - 2*ArcTan[(a*x + Sqrt[-b + a^2*x^2])^(1/4)/b^(1/8)] + Sqrt[2]* 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]])] + 4*Sqrt[2 - Sqrt[2]]*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]])] + 4*Sqrt[2 + Sqrt[2]]*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]])] - 2*ArcTanh[(a*x + Sqrt[-b + a^2*x^2])^(1/4)/b^(1/8)] + Sqrt[2]*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))] + 4*Sqrt[2 - Sqrt[2]]*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))] - 4*Sqrt[2 + Sqrt[2]]*ArcTanh[(Sqrt[1 + 1/Sqrt[2]]*(b^(1/4) + S qrt[a*x + Sqrt[-b + a^2*x^2]]))/(b^(1/8)*(a*x + Sqrt[-b + a^2*x^2])^(1/4)) ])/(4*b^(11/8))
Time = 0.83 (sec) , antiderivative size = 763, normalized size of antiderivative = 1.05, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.119, Rules used = {2545, 368, 971, 1054, 2009}
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.
| \(\displaystyle \int \frac {\sqrt [4]{\sqrt {a^2 x^2-b}+a x}}{x \left (a^2 x^2-b\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 2545 |
| \(\displaystyle 8 \int \frac {\left (a x+\sqrt {a^2 x^2-b}\right )^{9/4}}{\left (b-\left (a x+\sqrt {a^2 x^2-b}\right )^2\right )^2 \left (\left (a x+\sqrt {a^2 x^2-b}\right )^2+b\right )}d\left (a x+\sqrt {a^2 x^2-b}\right )\) |
| \(\Big \downarrow \) 368 |
| \(\displaystyle 32 \int \frac {\left (a x+\sqrt {a^2 x^2-b}\right )^3}{\left (b-\left (a x+\sqrt {a^2 x^2-b}\right )^2\right )^2 \left (\left (a x+\sqrt {a^2 x^2-b}\right )^2+b\right )}d\sqrt [4]{a x+\sqrt {a^2 x^2-b}}\) |
| \(\Big \downarrow \) 971 |
| \(\displaystyle 32 \left (\frac {\left (\sqrt {a^2 x^2-b}+a x\right )^{5/4}}{16 b \left (b-\left (\sqrt {a^2 x^2-b}+a x\right )^2\right )}-\frac {\int \frac {\left (a x+\sqrt {a^2 x^2-b}\right ) \left (5 b-3 \left (a x+\sqrt {a^2 x^2-b}\right )^2\right )}{\left (b-\left (a x+\sqrt {a^2 x^2-b}\right )^2\right ) \left (\left (a x+\sqrt {a^2 x^2-b}\right )^2+b\right )}d\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{16 b}\right )\) |
| \(\Big \downarrow \) 1054 |
| \(\displaystyle 32 \left (\frac {\left (\sqrt {a^2 x^2-b}+a x\right )^{5/4}}{16 b \left (b-\left (\sqrt {a^2 x^2-b}+a x\right )^2\right )}-\frac {\int \left (\frac {4 \left (a x+\sqrt {a^2 x^2-b}\right )}{\left (a x+\sqrt {a^2 x^2-b}\right )^2+b}-\frac {a x+\sqrt {a^2 x^2-b}}{\left (a x+\sqrt {a^2 x^2-b}\right )^2-b}\right )d\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{16 b}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 32 \left (\frac {\left (\sqrt {a^2 x^2-b}+a x\right )^{5/4}}{16 b \left (b-\left (\sqrt {a^2 x^2-b}+a x\right )^2\right )}-\frac {\frac {\arctan \left (\frac {\sqrt [4]{\sqrt {a^2 x^2-b}+a x}}{\sqrt [8]{b}}\right )}{4 b^{3/8}}+\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{\sqrt {a^2 x^2-b}+a x}}{\sqrt [8]{b}}\right )}{4 \sqrt {2} b^{3/8}}-\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{\sqrt {a^2 x^2-b}+a x}}{\sqrt [8]{b}}+1\right )}{4 \sqrt {2} b^{3/8}}-\frac {\arctan \left (\frac {\sqrt [4]{\sqrt {a^2 x^2-b}+a x}}{\sqrt [8]{-b}}\right )}{(-b)^{3/8}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{\sqrt {a^2 x^2-b}+a x}}{\sqrt [8]{-b}}\right )}{\sqrt {2} (-b)^{3/8}}+\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{\sqrt {a^2 x^2-b}+a x}}{\sqrt [8]{-b}}+1\right )}{\sqrt {2} (-b)^{3/8}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{\sqrt {a^2 x^2-b}+a x}}{\sqrt [8]{b}}\right )}{4 b^{3/8}}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{\sqrt {a^2 x^2-b}+a x}}{\sqrt [8]{-b}}\right )}{(-b)^{3/8}}+\frac {\log \left (\sqrt {\sqrt {a^2 x^2-b}+a x}-\sqrt {2} \sqrt [8]{b} \sqrt [4]{\sqrt {a^2 x^2-b}+a x}+\sqrt [4]{b}\right )}{8 \sqrt {2} b^{3/8}}-\frac {\log \left (\sqrt {\sqrt {a^2 x^2-b}+a x}+\sqrt {2} \sqrt [8]{b} \sqrt [4]{\sqrt {a^2 x^2-b}+a x}+\sqrt [4]{b}\right )}{8 \sqrt {2} b^{3/8}}-\frac {\log \left (\sqrt {\sqrt {a^2 x^2-b}+a x}-\sqrt {2} \sqrt [8]{-b} \sqrt [4]{\sqrt {a^2 x^2-b}+a x}+\sqrt [4]{-b}\right )}{2 \sqrt {2} (-b)^{3/8}}+\frac {\log \left (\sqrt {\sqrt {a^2 x^2-b}+a x}+\sqrt {2} \sqrt [8]{-b} \sqrt [4]{\sqrt {a^2 x^2-b}+a x}+\sqrt [4]{-b}\right )}{2 \sqrt {2} (-b)^{3/8}}}{16 b}\right )\) |
32*((a*x + Sqrt[-b + a^2*x^2])^(5/4)/(16*b*(b - (a*x + Sqrt[-b + a^2*x^2]) ^2)) - (-(ArcTan[(a*x + Sqrt[-b + a^2*x^2])^(1/4)/(-b)^(1/8)]/(-b)^(3/8)) + ArcTan[(a*x + Sqrt[-b + a^2*x^2])^(1/4)/b^(1/8)]/(4*b^(3/8)) - ArcTan[1 - (Sqrt[2]*(a*x + Sqrt[-b + a^2*x^2])^(1/4))/(-b)^(1/8)]/(Sqrt[2]*(-b)^(3/ 8)) + ArcTan[1 + (Sqrt[2]*(a*x + Sqrt[-b + a^2*x^2])^(1/4))/(-b)^(1/8)]/(S qrt[2]*(-b)^(3/8)) + ArcTan[1 - (Sqrt[2]*(a*x + Sqrt[-b + a^2*x^2])^(1/4)) /b^(1/8)]/(4*Sqrt[2]*b^(3/8)) - ArcTan[1 + (Sqrt[2]*(a*x + Sqrt[-b + a^2*x ^2])^(1/4))/b^(1/8)]/(4*Sqrt[2]*b^(3/8)) - ArcTanh[(a*x + Sqrt[-b + a^2*x^ 2])^(1/4)/(-b)^(1/8)]/(-b)^(3/8) + ArcTanh[(a*x + Sqrt[-b + a^2*x^2])^(1/4 )/b^(1/8)]/(4*b^(3/8)) - 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)^(3/8 )) + 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)^(3/8)) + Log[b^(1/4) - S qrt[2]*b^(1/8)*(a*x + Sqrt[-b + a^2*x^2])^(1/4) + Sqrt[a*x + Sqrt[-b + a^2 *x^2]]]/(8*Sqrt[2]*b^(3/8)) - 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]]]/(8*Sqrt[2]*b^(3/8)))/ (16*b))
3.32.24.3.1 Defintions of rubi rules used
Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_) , x_Symbol] :> With[{k = Denominator[m]}, Simp[k/e Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(k*2)/e^2))^p*(c + d*(x^(k*2)/e^2))^q, x], x, (e*x)^(1/k)], x]] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b*c - a*d, 0] && FractionQ[m ] && IntegerQ[p]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_ ))^(q_), x_Symbol] :> Simp[e^(n - 1)*(e*x)^(m - n + 1)*(a + b*x^n)^(p + 1)* ((c + d*x^n)^(q + 1)/(n*(b*c - a*d)*(p + 1))), x] - Simp[e^n/(n*(b*c - a*d) *(p + 1)) Int[(e*x)^(m - n)*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(m - n + 1) + d*(m + n*(p + q + 1) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e , q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[p, -1] && GeQ[n, m - n + 1] && GtQ[m - n + 1, 0] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]
Int[(((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((e_) + (f_.)*(x_)^(n _)))/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Int[ExpandIntegrand[(g*x)^m*(a + b*x^n)^p*((e + f*x^n)/(c + d*x^n)), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && IGtQ[n, 0]
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])
\[\int \frac {\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{4}}}{x \left (a^{2} x^{2}-b \right )^{\frac {3}{2}}}d x\]
Result contains complex when optimal does not.
Time = 0.31 (sec) , antiderivative size = 979, normalized size of antiderivative = 1.35 \[ \int \frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{x \left (-b+a^2 x^2\right )^{3/2}} \, dx=\text {Too large to display} \]
1/8*(4*sqrt(2)*(-(I + 1)*a^2*b*x^2 + (I + 1)*b^2)*(-1/b^11)^(1/8)*log((I + 1)*sqrt(2)*b^7*(-1/b^11)^(5/8) + 2*(a*x + sqrt(a^2*x^2 - b))^(1/4)) + 4*s qrt(2)*((I - 1)*a^2*b*x^2 - (I - 1)*b^2)*(-1/b^11)^(1/8)*log(-(I - 1)*sqrt (2)*b^7*(-1/b^11)^(5/8) + 2*(a*x + sqrt(a^2*x^2 - b))^(1/4)) + 4*sqrt(2)*( -(I - 1)*a^2*b*x^2 + (I - 1)*b^2)*(-1/b^11)^(1/8)*log((I - 1)*sqrt(2)*b^7* (-1/b^11)^(5/8) + 2*(a*x + sqrt(a^2*x^2 - b))^(1/4)) + 4*sqrt(2)*((I + 1)* a^2*b*x^2 - (I + 1)*b^2)*(-1/b^11)^(1/8)*log(-(I + 1)*sqrt(2)*b^7*(-1/b^11 )^(5/8) + 2*(a*x + sqrt(a^2*x^2 - b))^(1/4)) + sqrt(2)*((I + 1)*a^2*b*x^2 - (I + 1)*b^2)*(b^(-11))^(1/8)*log((1/2*I + 1/2)*sqrt(2)*b^7*(b^(-11))^(5/ 8) + (a*x + sqrt(a^2*x^2 - b))^(1/4)) + sqrt(2)*(-(I - 1)*a^2*b*x^2 + (I - 1)*b^2)*(b^(-11))^(1/8)*log(-(1/2*I - 1/2)*sqrt(2)*b^7*(b^(-11))^(5/8) + (a*x + sqrt(a^2*x^2 - b))^(1/4)) + sqrt(2)*((I - 1)*a^2*b*x^2 - (I - 1)*b^ 2)*(b^(-11))^(1/8)*log((1/2*I - 1/2)*sqrt(2)*b^7*(b^(-11))^(5/8) + (a*x + sqrt(a^2*x^2 - b))^(1/4)) + sqrt(2)*(-(I + 1)*a^2*b*x^2 + (I + 1)*b^2)*(b^ (-11))^(1/8)*log(-(1/2*I + 1/2)*sqrt(2)*b^7*(b^(-11))^(5/8) + (a*x + sqrt( a^2*x^2 - b))^(1/4)) + 8*(a^2*b*x^2 - b^2)*(-1/b^11)^(1/8)*log(b^7*(-1/b^1 1)^(5/8) + (a*x + sqrt(a^2*x^2 - b))^(1/4)) - 8*(-I*a^2*b*x^2 + I*b^2)*(-1 /b^11)^(1/8)*log(I*b^7*(-1/b^11)^(5/8) + (a*x + sqrt(a^2*x^2 - b))^(1/4)) - 8*(I*a^2*b*x^2 - I*b^2)*(-1/b^11)^(1/8)*log(-I*b^7*(-1/b^11)^(5/8) + (a* x + sqrt(a^2*x^2 - b))^(1/4)) - 8*(a^2*b*x^2 - b^2)*(-1/b^11)^(1/8)*log...
\[ \int \frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{x \left (-b+a^2 x^2\right )^{3/2}} \, dx=\int \frac {\sqrt [4]{a x + \sqrt {a^{2} x^{2} - b}}}{x \left (a^{2} x^{2} - b\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{x \left (-b+a^2 x^2\right )^{3/2}} \, dx=\int { \frac {{\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}}{{\left (a^{2} x^{2} - b\right )}^{\frac {3}{2}} x} \,d x } \]
Timed out. \[ \int \frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{x \left (-b+a^2 x^2\right )^{3/2}} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{x \left (-b+a^2 x^2\right )^{3/2}} \, dx=\int \frac {{\left (a\,x+\sqrt {a^2\,x^2-b}\right )}^{1/4}}{x\,{\left (a^2\,x^2-b\right )}^{3/2}} \,d x \]