Integrand size = 42, antiderivative size = 526 \[ \int \frac {\left (-b+a^2 x^2\right )^{3/2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{x} \, dx=\frac {4 \sqrt {-b+a^2 x^2} \left (418 a b^2 x-561 a^3 b x^3+132 a^5 x^5\right )+4 \left (-152 b^3+682 a^2 b^2 x^2-627 a^4 b x^4+132 a^6 x^6\right )}{429 \left (a x+\sqrt {-b+a^2 x^2}\right )^{11/4}}+\sqrt {2+\sqrt {2}} b^{13/8} \arctan \left (\frac {\left (\sqrt {\frac {2}{2-\sqrt {2}}} \sqrt [8]{b}-\frac {2 \sqrt [8]{b}}{\sqrt {2-\sqrt {2}}}\right ) \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{-\sqrt [4]{b}+\sqrt {a x+\sqrt {-b+a^2 x^2}}}\right )+\sqrt {2-\sqrt {2}} b^{13/8} \arctan \left (\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{b} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{-\sqrt [4]{b}+\sqrt {a x+\sqrt {-b+a^2 x^2}}}\right )+\sqrt {2+\sqrt {2}} b^{13/8} \text {arctanh}\left (\frac {\frac {\sqrt [8]{b}}{\sqrt {2-\sqrt {2}}}+\frac {\sqrt {a x+\sqrt {-b+a^2 x^2}}}{\sqrt {2-\sqrt {2}} \sqrt [8]{b}}}{\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}\right )-\sqrt {2-\sqrt {2}} b^{13/8} \text {arctanh}\left (\frac {\frac {\sqrt [8]{b}}{\sqrt {2+\sqrt {2}}}+\frac {\sqrt {a x+\sqrt {-b+a^2 x^2}}}{\sqrt {2+\sqrt {2}} \sqrt [8]{b}}}{\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}\right ) \]
1/429*(4*(a^2*x^2-b)^(1/2)*(132*a^5*x^5-561*a^3*b*x^3+418*a*b^2*x)+528*a^6 *x^6-2508*a^4*b*x^4+2728*a^2*b^2*x^2-608*b^3)/(a*x+(a^2*x^2-b)^(1/2))^(11/ 4)+(2+2^(1/2))^(1/2)*b^(13/8)*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)))+(2-2^(1/2))^(1/2)*b^(13/8)*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)))+(2+2^(1/2))^(1/2)*b^(13/8)*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))-(2-2^(1/2))^(1/2)*b^(13/8)*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))
Time = 1.43 (sec) , antiderivative size = 469, normalized size of antiderivative = 0.89 \[ \int \frac {\left (-b+a^2 x^2\right )^{3/2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{x} \, dx=\frac {4 \left (-152 b^3+682 a^2 b^2 x^2-627 a^4 b x^4+132 a^6 x^6+11 \sqrt {-b+a^2 x^2} \left (38 a b^2 x-51 a^3 b x^3+12 a^5 x^5\right )\right )}{429 \left (a x+\sqrt {-b+a^2 x^2}\right )^{11/4}}+\sqrt {2+\sqrt {2}} b^{13/8} \arctan \left (\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{b} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [4]{b}-\sqrt {a x+\sqrt {-b+a^2 x^2}}}\right )+\sqrt {2-\sqrt {2}} b^{13/8} \arctan \left (\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{b} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{-\sqrt [4]{b}+\sqrt {a x+\sqrt {-b+a^2 x^2}}}\right )-\sqrt {2-\sqrt {2}} b^{13/8} \text {arctanh}\left (\frac {\sqrt {1-\frac {1}{\sqrt {2}}} \left (\sqrt [4]{b}+\sqrt {a x+\sqrt {-b+a^2 x^2}}\right )}{\sqrt [8]{b} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}\right )+\sqrt {2+\sqrt {2}} b^{13/8} \text {arctanh}\left (\frac {\sqrt {1+\frac {1}{\sqrt {2}}} \left (\sqrt [4]{b}+\sqrt {a x+\sqrt {-b+a^2 x^2}}\right )}{\sqrt [8]{b} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}\right ) \]
(4*(-152*b^3 + 682*a^2*b^2*x^2 - 627*a^4*b*x^4 + 132*a^6*x^6 + 11*Sqrt[-b + a^2*x^2]*(38*a*b^2*x - 51*a^3*b*x^3 + 12*a^5*x^5)))/(429*(a*x + Sqrt[-b + a^2*x^2])^(11/4)) + Sqrt[2 + Sqrt[2]]*b^(13/8)*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]])] + Sqrt[2 - Sqrt[2]]*b^(13/8)*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]])] - Sqrt[2 - Sqrt[2]]*b^(13/8)*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))] + Sqrt[2 + Sqrt[2]]*b^(13/8)*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) )]
Time = 0.63 (sec) , antiderivative size = 471, normalized size of antiderivative = 0.90, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2545, 368, 961, 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.
| \(\displaystyle \int \frac {\left (a^2 x^2-b\right )^{3/2} \sqrt [4]{\sqrt {a^2 x^2-b}+a x}}{x} \, dx\) |
| \(\Big \downarrow \) 2545 |
| \(\displaystyle \frac {1}{8} \int \frac {\left (b-\left (a x+\sqrt {a^2 x^2-b}\right )^2\right )^4}{\left (a x+\sqrt {a^2 x^2-b}\right )^{15/4} \left (\left (a x+\sqrt {a^2 x^2-b}\right )^2+b\right )}d\left (a x+\sqrt {a^2 x^2-b}\right )\) |
| \(\Big \downarrow \) 368 |
| \(\displaystyle \frac {1}{2} \int \frac {\left (b-\left (a x+\sqrt {a^2 x^2-b}\right )^2\right )^4}{\left (a x+\sqrt {a^2 x^2-b}\right )^3 \left (\left (a x+\sqrt {a^2 x^2-b}\right )^2+b\right )}d\sqrt [4]{a x+\sqrt {a^2 x^2-b}}\) |
| \(\Big \downarrow \) 961 |
| \(\displaystyle \frac {1}{2} \int \left (\frac {b^3}{\left (a x+\sqrt {a^2 x^2-b}\right )^3}-\frac {5 b^2}{a x+\sqrt {a^2 x^2-b}}+\frac {16 \left (a x+\sqrt {a^2 x^2-b}\right ) b^2}{\left (a x+\sqrt {a^2 x^2-b}\right )^2+b}-5 \left (a x+\sqrt {a^2 x^2-b}\right ) b+\left (a x+\sqrt {a^2 x^2-b}\right )^3\right )d\sqrt [4]{a x+\sqrt {a^2 x^2-b}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{2} \left (-4 (-b)^{13/8} \arctan \left (\frac {\sqrt [4]{\sqrt {a^2 x^2-b}+a x}}{\sqrt [8]{-b}}\right )-2 \sqrt {2} (-b)^{13/8} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{\sqrt {a^2 x^2-b}+a x}}{\sqrt [8]{-b}}\right )+2 \sqrt {2} (-b)^{13/8} \arctan \left (\frac {\sqrt {2} \sqrt [4]{\sqrt {a^2 x^2-b}+a x}}{\sqrt [8]{-b}}+1\right )-4 (-b)^{13/8} \text {arctanh}\left (\frac {\sqrt [4]{\sqrt {a^2 x^2-b}+a x}}{\sqrt [8]{-b}}\right )-\frac {b^3}{11 \left (\sqrt {a^2 x^2-b}+a x\right )^{11/4}}+\frac {5 b^2}{3 \left (\sqrt {a^2 x^2-b}+a x\right )^{3/4}}+\frac {1}{13} \left (\sqrt {a^2 x^2-b}+a x\right )^{13/4}-b \left (\sqrt {a^2 x^2-b}+a x\right )^{5/4}-\sqrt {2} (-b)^{13/8} \log \left (\sqrt {\sqrt {a^2 x^2-b}+a x}-\sqrt {2} \sqrt [8]{-b} \sqrt [4]{\sqrt {a^2 x^2-b}+a x}+\sqrt [4]{-b}\right )+\sqrt {2} (-b)^{13/8} \log \left (\sqrt {\sqrt {a^2 x^2-b}+a x}+\sqrt {2} \sqrt [8]{-b} \sqrt [4]{\sqrt {a^2 x^2-b}+a x}+\sqrt [4]{-b}\right )\right )\) |
(-1/11*b^3/(a*x + Sqrt[-b + a^2*x^2])^(11/4) + (5*b^2)/(3*(a*x + Sqrt[-b + a^2*x^2])^(3/4)) - b*(a*x + Sqrt[-b + a^2*x^2])^(5/4) + (a*x + Sqrt[-b + a^2*x^2])^(13/4)/13 - 4*(-b)^(13/8)*ArcTan[(a*x + Sqrt[-b + a^2*x^2])^(1/4 )/(-b)^(1/8)] - 2*Sqrt[2]*(-b)^(13/8)*ArcTan[1 - (Sqrt[2]*(a*x + Sqrt[-b + a^2*x^2])^(1/4))/(-b)^(1/8)] + 2*Sqrt[2]*(-b)^(13/8)*ArcTan[1 + (Sqrt[2]* (a*x + Sqrt[-b + a^2*x^2])^(1/4))/(-b)^(1/8)] - 4*(-b)^(13/8)*ArcTanh[(a*x + Sqrt[-b + a^2*x^2])^(1/4)/(-b)^(1/8)] - Sqrt[2]*(-b)^(13/8)*Log[(-b)^(1 /4) - Sqrt[2]*(-b)^(1/8)*(a*x + Sqrt[-b + a^2*x^2])^(1/4) + Sqrt[a*x + Sqr t[-b + a^2*x^2]]] + Sqrt[2]*(-b)^(13/8)*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
3.31.89.3.1 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_)), x_Symbol] :> Int[ExpandIntegrand[(e*x)^m*((a + b*x^n)^p/(c + d*x^n)), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && IGtQ[p, 0] && (IntegerQ[m] || IGtQ[2*(m + 1), 0] || !RationalQ[m])
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])
\[\int \frac {\left (a^{2} x^{2}-b \right )^{\frac {3}{2}} \left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{4}}}{x}d x\]
Result contains complex when optimal does not.
Time = 0.30 (sec) , antiderivative size = 432, normalized size of antiderivative = 0.82 \[ \int \frac {\left (-b+a^2 x^2\right )^{3/2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{x} \, dx=\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} \left (-b^{13}\right )^{\frac {1}{8}} \log \left (2 \, {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}} b^{8} + \left (i + 1\right ) \, \sqrt {2} \left (-b^{13}\right )^{\frac {5}{8}}\right ) - \left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} \left (-b^{13}\right )^{\frac {1}{8}} \log \left (2 \, {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}} b^{8} - \left (i - 1\right ) \, \sqrt {2} \left (-b^{13}\right )^{\frac {5}{8}}\right ) + \left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} \left (-b^{13}\right )^{\frac {1}{8}} \log \left (2 \, {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}} b^{8} + \left (i - 1\right ) \, \sqrt {2} \left (-b^{13}\right )^{\frac {5}{8}}\right ) - \left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} \left (-b^{13}\right )^{\frac {1}{8}} \log \left (2 \, {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}} b^{8} - \left (i + 1\right ) \, \sqrt {2} \left (-b^{13}\right )^{\frac {5}{8}}\right ) - \frac {4}{429} \, {\left (3 \, a^{3} x^{3} - 38 \, a b x - 4 \, {\left (9 \, a^{2} x^{2} - 38 \, b\right )} \sqrt {a^{2} x^{2} - b}\right )} {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}} - \left (-b^{13}\right )^{\frac {1}{8}} \log \left ({\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}} b^{8} + \left (-b^{13}\right )^{\frac {5}{8}}\right ) - i \, \left (-b^{13}\right )^{\frac {1}{8}} \log \left ({\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}} b^{8} + i \, \left (-b^{13}\right )^{\frac {5}{8}}\right ) + i \, \left (-b^{13}\right )^{\frac {1}{8}} \log \left ({\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}} b^{8} - i \, \left (-b^{13}\right )^{\frac {5}{8}}\right ) + \left (-b^{13}\right )^{\frac {1}{8}} \log \left ({\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}} b^{8} - \left (-b^{13}\right )^{\frac {5}{8}}\right ) \]
(1/2*I + 1/2)*sqrt(2)*(-b^13)^(1/8)*log(2*(a*x + sqrt(a^2*x^2 - b))^(1/4)* b^8 + (I + 1)*sqrt(2)*(-b^13)^(5/8)) - (1/2*I - 1/2)*sqrt(2)*(-b^13)^(1/8) *log(2*(a*x + sqrt(a^2*x^2 - b))^(1/4)*b^8 - (I - 1)*sqrt(2)*(-b^13)^(5/8) ) + (1/2*I - 1/2)*sqrt(2)*(-b^13)^(1/8)*log(2*(a*x + sqrt(a^2*x^2 - b))^(1 /4)*b^8 + (I - 1)*sqrt(2)*(-b^13)^(5/8)) - (1/2*I + 1/2)*sqrt(2)*(-b^13)^( 1/8)*log(2*(a*x + sqrt(a^2*x^2 - b))^(1/4)*b^8 - (I + 1)*sqrt(2)*(-b^13)^( 5/8)) - 4/429*(3*a^3*x^3 - 38*a*b*x - 4*(9*a^2*x^2 - 38*b)*sqrt(a^2*x^2 - b))*(a*x + sqrt(a^2*x^2 - b))^(1/4) - (-b^13)^(1/8)*log((a*x + sqrt(a^2*x^ 2 - b))^(1/4)*b^8 + (-b^13)^(5/8)) - I*(-b^13)^(1/8)*log((a*x + sqrt(a^2*x ^2 - b))^(1/4)*b^8 + I*(-b^13)^(5/8)) + I*(-b^13)^(1/8)*log((a*x + sqrt(a^ 2*x^2 - b))^(1/4)*b^8 - I*(-b^13)^(5/8)) + (-b^13)^(1/8)*log((a*x + sqrt(a ^2*x^2 - b))^(1/4)*b^8 - (-b^13)^(5/8))
\[ \int \frac {\left (-b+a^2 x^2\right )^{3/2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{x} \, dx=\int \frac {\sqrt [4]{a x + \sqrt {a^{2} x^{2} - b}} \left (a^{2} x^{2} - b\right )^{\frac {3}{2}}}{x}\, dx \]
\[ \int \frac {\left (-b+a^2 x^2\right )^{3/2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{x} \, dx=\int { \frac {{\left (a^{2} x^{2} - b\right )}^{\frac {3}{2}} {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}}{x} \,d x } \]
Timed out. \[ \int \frac {\left (-b+a^2 x^2\right )^{3/2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{x} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {\left (-b+a^2 x^2\right )^{3/2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{x} \, dx=\int \frac {{\left (a\,x+\sqrt {a^2\,x^2-b}\right )}^{1/4}\,{\left (a^2\,x^2-b\right )}^{3/2}}{x} \,d x \]