Integrand size = 68, antiderivative size = 1225 \[ \int \frac {\sqrt {-b+a^2 x^2} \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}} \, dx=\frac {\left (-3272028760 b^3-319355415288 b^2 c^8+1032601284 a b^2 c^4 x+5423886846 a b c^{12} x+6544057520 a^2 b^2 x^2+470457082368 a^2 b c^8 x^2-7231849128 a^3 c^{12} x^3+176421405888 a^4 c^8 x^4\right ) \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}+\left (-748701954 b^2 c^7-10460353203 b c^{15}-1204701498 a b^2 c^3 x-4519905705 a b c^{11} x+20920706406 a^2 c^{15} x^2+6026540940 a^3 c^{11} x^3\right ) \sqrt [4]{a x+\sqrt {-b+a^2 x^2}} \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}+\left (820006902 b^2 c^6+3486784401 b c^{14}+1472412942 a b^2 c^2 x+3917251611 a b c^{10} x-6973568802 a^2 c^{14} x^2-5223002148 a^3 c^{10} x^3\right ) \sqrt {a x+\sqrt {-b+a^2 x^2}} \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}+\left (-911118780 b^2 c^5-2324522934 b c^{13}-1963217256 a b^2 c x-3482001432 a b c^9 x+4649045868 a^2 c^{13} x^2+4642668576 a^3 c^9 x^3\right ) \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4} \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}+\sqrt {-b+a^2 x^2} \left (\left (1032601284 b^2 c^4+1807962282 b c^{12}+6544057520 a b^2 x+558667785312 a b c^8 x-7231849128 a^2 c^{12} x^2+176421405888 a^3 c^8 x^3\right ) \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}+\left (-1204701498 b^2 c^3-1506635235 b c^{11}+20920706406 a c^{15} x+6026540940 a^2 c^{11} x^2\right ) \sqrt [4]{a x+\sqrt {-b+a^2 x^2}} \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}+\left (1472412942 b^2 c^2+1305750537 b c^{10}-6973568802 a c^{14} x-5223002148 a^2 c^{10} x^2\right ) \sqrt {a x+\sqrt {-b+a^2 x^2}} \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}+\left (-1963217256 b^2 c-1160667144 b c^9+4649045868 a c^{13} x+4642668576 a^2 c^9 x^2\right ) \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4} \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}\right )}{161719622064 a c^8 \left (a x+\sqrt {-b+a^2 x^2}\right )^{9/4}}-\frac {21505 b^2 \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt {3} \sqrt [3]{c}}\right )}{531441 \sqrt {3} a c^{26/3}}+\frac {2 b \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt {3} \sqrt [3]{c}}\right )}{\sqrt {3} a c^{2/3}}+\frac {21505 b^2 \log \left (-\sqrt [3]{c}+\sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}\right )}{1594323 a c^{26/3}}-\frac {2 b \log \left (-\sqrt [3]{c}+\sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}\right )}{3 a c^{2/3}}-\frac {21505 b^2 \log \left (c^{2/3}+\sqrt [3]{c} \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}+\left (c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )^{2/3}\right )}{3188646 a c^{26/3}}+\frac {b \log \left (c^{2/3}+\sqrt [3]{c} \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}+\left (c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )^{2/3}\right )}{3 a c^{2/3}} \]
1/161719622064*((-7231849128*a^3*c^12*x^3+176421405888*a^4*c^8*x^4+5423886 846*a*b*c^12*x+470457082368*a^2*b*c^8*x^2-319355415288*b^2*c^8+1032601284* a*b^2*c^4*x+6544057520*a^2*b^2*x^2-3272028760*b^3)*(c+(a*x+(a^2*x^2-b)^(1/ 2))^(1/4))^(1/3)+(20920706406*a^2*c^15*x^2+6026540940*a^3*c^11*x^3-1046035 3203*b*c^15-4519905705*a*b*c^11*x-748701954*b^2*c^7-1204701498*a*b^2*c^3*x )*(a*x+(a^2*x^2-b)^(1/2))^(1/4)*(c+(a*x+(a^2*x^2-b)^(1/2))^(1/4))^(1/3)+(- 6973568802*a^2*c^14*x^2-5223002148*a^3*c^10*x^3+3486784401*b*c^14+39172516 11*a*b*c^10*x+820006902*b^2*c^6+1472412942*a*b^2*c^2*x)*(a*x+(a^2*x^2-b)^( 1/2))^(1/2)*(c+(a*x+(a^2*x^2-b)^(1/2))^(1/4))^(1/3)+(4649045868*a^2*c^13*x ^2+4642668576*a^3*c^9*x^3-2324522934*b*c^13-3482001432*a*b*c^9*x-911118780 *b^2*c^5-1963217256*a*b^2*c*x)*(a*x+(a^2*x^2-b)^(1/2))^(3/4)*(c+(a*x+(a^2* x^2-b)^(1/2))^(1/4))^(1/3)+(a^2*x^2-b)^(1/2)*((-7231849128*a^2*c^12*x^2+17 6421405888*a^3*c^8*x^3+1807962282*b*c^12+558667785312*a*b*c^8*x+1032601284 *b^2*c^4+6544057520*a*b^2*x)*(c+(a*x+(a^2*x^2-b)^(1/2))^(1/4))^(1/3)+(2092 0706406*a*c^15*x+6026540940*a^2*c^11*x^2-1506635235*b*c^11-1204701498*b^2* c^3)*(a*x+(a^2*x^2-b)^(1/2))^(1/4)*(c+(a*x+(a^2*x^2-b)^(1/2))^(1/4))^(1/3) +(-6973568802*a*c^14*x-5223002148*a^2*c^10*x^2+1305750537*b*c^10+147241294 2*b^2*c^2)*(a*x+(a^2*x^2-b)^(1/2))^(1/2)*(c+(a*x+(a^2*x^2-b)^(1/2))^(1/4)) ^(1/3)+(4649045868*a*c^13*x+4642668576*a^2*c^9*x^2-1160667144*b*c^9-196321 7256*b^2*c)*(a*x+(a^2*x^2-b)^(1/2))^(3/4)*(c+(a*x+(a^2*x^2-b)^(1/2))^(1...
Time = 4.35 (sec) , antiderivative size = 1219, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {-b+a^2 x^2} \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}} \, dx=\frac {\frac {3 c^{2/3} \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}} \left (-3272028760 b^3+494 b^2 \left (-646468452 c^8-1515591 c^7 \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}+1659933 c^6 \sqrt {a x+\sqrt {-b+a^2 x^2}}-1844370 c^5 \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4}+2090286 c^4 \left (a x+\sqrt {-b+a^2 x^2}\right )+13247080 a x \left (a x+\sqrt {-b+a^2 x^2}\right )-2438667 c^3 \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/4}+2980593 c^2 \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/2}-3974124 c \left (a x+\sqrt {-b+a^2 x^2}\right )^{7/4}\right )+3188646 a c^8 x \left (a x+\sqrt {-b+a^2 x^2}\right ) \left (55328 a^2 x^2+729 c^5 \sqrt [4]{a x+\sqrt {-b+a^2 x^2}} \left (9 c^2-3 c \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}+2 \sqrt {a x+\sqrt {-b+a^2 x^2}}\right )+14 a c x \left (-162 c^3+135 c^2 \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}-117 c \sqrt {a x+\sqrt {-b+a^2 x^2}}+104 \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4}\right )\right )+531441 b c^8 \left (885248 a^2 x^2+7 a x \left (1458 c^4+150176 \sqrt {-b+a^2 x^2}-1215 c^3 \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}+1053 c^2 \sqrt {a x+\sqrt {-b+a^2 x^2}}-936 c \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4}\right )-3 c \left (-1134 c^3 \sqrt {-b+a^2 x^2}+6561 c^6 \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}+945 c^2 \sqrt {-b+a^2 x^2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}-2187 c^5 \sqrt {a x+\sqrt {-b+a^2 x^2}}-819 c \sqrt {-b+a^2 x^2} \sqrt {a x+\sqrt {-b+a^2 x^2}}+1458 c^4 \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4}+728 \sqrt {-b+a^2 x^2} \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4}\right )\right )\right )}{\left (a x+\sqrt {-b+a^2 x^2}\right )^{9/4}}-6544057520 \sqrt {3} b^2 \arctan \left (\frac {1+\frac {2 \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt [3]{c}}}{\sqrt {3}}\right )+323439244128 \sqrt {3} b c^8 \arctan \left (\frac {1+\frac {2 \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt [3]{c}}}{\sqrt {3}}\right )+6544057520 b^2 \log \left (-\sqrt [3]{c}+\sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}\right )-323439244128 b c^8 \log \left (-\sqrt [3]{c}+\sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}\right )-3272028760 b^2 \log \left (c^{2/3}+\sqrt [3]{c} \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}+\left (c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )^{2/3}\right )+161719622064 b c^8 \log \left (c^{2/3}+\sqrt [3]{c} \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}+\left (c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )^{2/3}\right )}{485158866192 a c^{26/3}} \]
Integrate[(Sqrt[-b + a^2*x^2]*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3) )/(a*x + Sqrt[-b + a^2*x^2])^(1/4),x]
((3*c^(2/3)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3)*(-3272028760*b^3 + 494*b^2*(-646468452*c^8 - 1515591*c^7*(a*x + Sqrt[-b + a^2*x^2])^(1/4) + 1659933*c^6*Sqrt[a*x + Sqrt[-b + a^2*x^2]] - 1844370*c^5*(a*x + Sqrt[-b + a^2*x^2])^(3/4) + 2090286*c^4*(a*x + Sqrt[-b + a^2*x^2]) + 13247080*a*x*( a*x + Sqrt[-b + a^2*x^2]) - 2438667*c^3*(a*x + Sqrt[-b + a^2*x^2])^(5/4) + 2980593*c^2*(a*x + Sqrt[-b + a^2*x^2])^(3/2) - 3974124*c*(a*x + Sqrt[-b + a^2*x^2])^(7/4)) + 3188646*a*c^8*x*(a*x + Sqrt[-b + a^2*x^2])*(55328*a^2* x^2 + 729*c^5*(a*x + Sqrt[-b + a^2*x^2])^(1/4)*(9*c^2 - 3*c*(a*x + Sqrt[-b + a^2*x^2])^(1/4) + 2*Sqrt[a*x + Sqrt[-b + a^2*x^2]]) + 14*a*c*x*(-162*c^ 3 + 135*c^2*(a*x + Sqrt[-b + a^2*x^2])^(1/4) - 117*c*Sqrt[a*x + Sqrt[-b + a^2*x^2]] + 104*(a*x + Sqrt[-b + a^2*x^2])^(3/4))) + 531441*b*c^8*(885248* a^2*x^2 + 7*a*x*(1458*c^4 + 150176*Sqrt[-b + a^2*x^2] - 1215*c^3*(a*x + Sq rt[-b + a^2*x^2])^(1/4) + 1053*c^2*Sqrt[a*x + Sqrt[-b + a^2*x^2]] - 936*c* (a*x + Sqrt[-b + a^2*x^2])^(3/4)) - 3*c*(-1134*c^3*Sqrt[-b + a^2*x^2] + 65 61*c^6*(a*x + Sqrt[-b + a^2*x^2])^(1/4) + 945*c^2*Sqrt[-b + a^2*x^2]*(a*x + Sqrt[-b + a^2*x^2])^(1/4) - 2187*c^5*Sqrt[a*x + Sqrt[-b + a^2*x^2]] - 81 9*c*Sqrt[-b + a^2*x^2]*Sqrt[a*x + Sqrt[-b + a^2*x^2]] + 1458*c^4*(a*x + Sq rt[-b + a^2*x^2])^(3/4) + 728*Sqrt[-b + a^2*x^2]*(a*x + Sqrt[-b + a^2*x^2] )^(3/4)))))/(a*x + Sqrt[-b + a^2*x^2])^(9/4) - 6544057520*Sqrt[3]*b^2*ArcT an[(1 + (2*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3))/c^(1/3))/Sqrt[...
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.
| \(\displaystyle \int \frac {\sqrt {a^2 x^2-b} \sqrt [3]{\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c}}{\sqrt [4]{\sqrt {a^2 x^2-b}+a x}} \, dx\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle \int \frac {\sqrt {a^2 x^2-b} \sqrt [3]{\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c}}{\sqrt [4]{\sqrt {a^2 x^2-b}+a x}}dx\) |
Int[(Sqrt[-b + a^2*x^2]*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3))/(a*x + Sqrt[-b + a^2*x^2])^(1/4),x]
3.32.43.3.1 Defintions of rubi rules used
\[\int \frac {\sqrt {a^{2} x^{2}-b}\, {\left (c +\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{4}}\right )}^{\frac {1}{3}}}{\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{4}}}d x\]
int((a^2*x^2-b)^(1/2)*(c+(a*x+(a^2*x^2-b)^(1/2))^(1/4))^(1/3)/(a*x+(a^2*x^ 2-b)^(1/2))^(1/4),x)
int((a^2*x^2-b)^(1/2)*(c+(a*x+(a^2*x^2-b)^(1/2))^(1/4))^(1/3)/(a*x+(a^2*x^ 2-b)^(1/2))^(1/4),x)
Time = 0.36 (sec) , antiderivative size = 719, normalized size of antiderivative = 0.59 \[ \int \frac {\sqrt {-b+a^2 x^2} \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}} \, dx=\frac {304304 \, \sqrt {3} {\left (1062882 \, b^{2} c^{9} - 21505 \, b^{3} c\right )} \sqrt {-\left (-c^{2}\right )^{\frac {1}{3}}} \arctan \left (-\frac {\sqrt {3} \left (-c^{2}\right )^{\frac {1}{3}} c \sqrt {-\left (-c^{2}\right )^{\frac {1}{3}}} - 2 \, \sqrt {3} \left (-c^{2}\right )^{\frac {2}{3}} {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {1}{3}} \sqrt {-\left (-c^{2}\right )^{\frac {1}{3}}}}{3 \, c^{2}}\right ) + 152152 \, {\left (1062882 \, b^{2} c^{8} - 21505 \, b^{3}\right )} \left (-c^{2}\right )^{\frac {2}{3}} \log \left ({\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {2}{3}} c - \left (-c^{2}\right )^{\frac {1}{3}} c + \left (-c^{2}\right )^{\frac {2}{3}} {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {1}{3}}\right ) - 304304 \, {\left (1062882 \, b^{2} c^{8} - 21505 \, b^{3}\right )} \left (-c^{2}\right )^{\frac {2}{3}} \log \left ({\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {1}{3}} c - \left (-c^{2}\right )^{\frac {2}{3}}\right ) + 3 \, {\left (10460353203 \, b c^{17} - 1497403908 \, a^{2} b c^{9} x^{2} + 748701954 \, b^{2} c^{9} + 567 \, {\left (2657205 \, a b c^{13} - 2124694 \, a b^{2} c^{5}\right )} x - 2 \, {\left (35937693792 \, a^{3} c^{10} x^{3} + 903981141 \, b c^{14} - 1032601284 \, a^{2} b c^{6} x^{2} + 516300642 \, b^{2} c^{6} - 6916 \, {\left (28874961 \, a b c^{10} + 236555 \, a b^{2} c^{2}\right )} x - 988 \, {\left (36374184 \, a^{2} c^{10} x^{2} - 161617113 \, b c^{10} - 1045143 \, a b c^{6} x - 1655885 \, b^{2} c^{2}\right )} \sqrt {a^{2} x^{2} - b}\right )} {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {3}{4}} + 567 \, {\left (2657205 \, b c^{13} + 2640924 \, a b c^{9} x + 2124694 \, b^{2} c^{5}\right )} \sqrt {a^{2} x^{2} - b} + 6 \, {\left (387420489 \, b c^{15} - 303706260 \, a^{2} b c^{7} x^{2} + 151853130 \, b^{2} c^{7} + 364 \, {\left (531441 \, a b c^{11} - 898909 \, a b^{2} c^{3}\right )} x + 52 \, {\left (3720087 \, b c^{11} + 5840505 \, a b c^{7} x + 6292363 \, b^{2} c^{3}\right )} \sqrt {a^{2} x^{2} - b}\right )} \sqrt {a x + \sqrt {a^{2} x^{2} - b}} - 9 \, {\left (387420489 \, b c^{16} - 182223756 \, a^{2} b c^{8} x^{2} + 91111878 \, b^{2} c^{8} + 91 \, {\left (1594323 \, a b c^{12} - 1797818 \, a b^{2} c^{4}\right )} x + 13 \, {\left (11160261 \, b c^{12} + 14017212 \, a b c^{8} x + 12584726 \, b^{2} c^{4}\right )} \sqrt {a^{2} x^{2} - b}\right )} {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )} {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {1}{3}}}{485158866192 \, a b c^{10}} \]
integrate((a^2*x^2-b)^(1/2)*(c+(a*x+(a^2*x^2-b)^(1/2))^(1/4))^(1/3)/(a*x+( a^2*x^2-b)^(1/2))^(1/4),x, algorithm="fricas")
1/485158866192*(304304*sqrt(3)*(1062882*b^2*c^9 - 21505*b^3*c)*sqrt(-(-c^2 )^(1/3))*arctan(-1/3*(sqrt(3)*(-c^2)^(1/3)*c*sqrt(-(-c^2)^(1/3)) - 2*sqrt( 3)*(-c^2)^(2/3)*(c + (a*x + sqrt(a^2*x^2 - b))^(1/4))^(1/3)*sqrt(-(-c^2)^( 1/3)))/c^2) + 152152*(1062882*b^2*c^8 - 21505*b^3)*(-c^2)^(2/3)*log((c + ( a*x + sqrt(a^2*x^2 - b))^(1/4))^(2/3)*c - (-c^2)^(1/3)*c + (-c^2)^(2/3)*(c + (a*x + sqrt(a^2*x^2 - b))^(1/4))^(1/3)) - 304304*(1062882*b^2*c^8 - 215 05*b^3)*(-c^2)^(2/3)*log((c + (a*x + sqrt(a^2*x^2 - b))^(1/4))^(1/3)*c - ( -c^2)^(2/3)) + 3*(10460353203*b*c^17 - 1497403908*a^2*b*c^9*x^2 + 74870195 4*b^2*c^9 + 567*(2657205*a*b*c^13 - 2124694*a*b^2*c^5)*x - 2*(35937693792* a^3*c^10*x^3 + 903981141*b*c^14 - 1032601284*a^2*b*c^6*x^2 + 516300642*b^2 *c^6 - 6916*(28874961*a*b*c^10 + 236555*a*b^2*c^2)*x - 988*(36374184*a^2*c ^10*x^2 - 161617113*b*c^10 - 1045143*a*b*c^6*x - 1655885*b^2*c^2)*sqrt(a^2 *x^2 - b))*(a*x + sqrt(a^2*x^2 - b))^(3/4) + 567*(2657205*b*c^13 + 2640924 *a*b*c^9*x + 2124694*b^2*c^5)*sqrt(a^2*x^2 - b) + 6*(387420489*b*c^15 - 30 3706260*a^2*b*c^7*x^2 + 151853130*b^2*c^7 + 364*(531441*a*b*c^11 - 898909* a*b^2*c^3)*x + 52*(3720087*b*c^11 + 5840505*a*b*c^7*x + 6292363*b^2*c^3)*s qrt(a^2*x^2 - b))*sqrt(a*x + sqrt(a^2*x^2 - b)) - 9*(387420489*b*c^16 - 18 2223756*a^2*b*c^8*x^2 + 91111878*b^2*c^8 + 91*(1594323*a*b*c^12 - 1797818* a*b^2*c^4)*x + 13*(11160261*b*c^12 + 14017212*a*b*c^8*x + 12584726*b^2*c^4 )*sqrt(a^2*x^2 - b))*(a*x + sqrt(a^2*x^2 - b))^(1/4))*(c + (a*x + sqrt(...
\[ \int \frac {\sqrt {-b+a^2 x^2} \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}} \, dx=\int \frac {\sqrt [3]{c + \sqrt [4]{a x + \sqrt {a^{2} x^{2} - b}}} \sqrt {a^{2} x^{2} - b}}{\sqrt [4]{a x + \sqrt {a^{2} x^{2} - b}}}\, dx \]
integrate((a**2*x**2-b)**(1/2)*(c+(a*x+(a**2*x**2-b)**(1/2))**(1/4))**(1/3 )/(a*x+(a**2*x**2-b)**(1/2))**(1/4),x)
Integral((c + (a*x + sqrt(a**2*x**2 - b))**(1/4))**(1/3)*sqrt(a**2*x**2 - b)/(a*x + sqrt(a**2*x**2 - b))**(1/4), x)
\[ \int \frac {\sqrt {-b+a^2 x^2} \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}} \, dx=\int { \frac {\sqrt {a^{2} x^{2} - b} {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {1}{3}}}{{\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}} \,d x } \]
integrate((a^2*x^2-b)^(1/2)*(c+(a*x+(a^2*x^2-b)^(1/2))^(1/4))^(1/3)/(a*x+( a^2*x^2-b)^(1/2))^(1/4),x, algorithm="maxima")
integrate(sqrt(a^2*x^2 - b)*(c + (a*x + sqrt(a^2*x^2 - b))^(1/4))^(1/3)/(a *x + sqrt(a^2*x^2 - b))^(1/4), x)
Timed out. \[ \int \frac {\sqrt {-b+a^2 x^2} \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}} \, dx=\text {Timed out} \]
integrate((a^2*x^2-b)^(1/2)*(c+(a*x+(a^2*x^2-b)^(1/2))^(1/4))^(1/3)/(a*x+( a^2*x^2-b)^(1/2))^(1/4),x, algorithm="giac")
Timed out. \[ \int \frac {\sqrt {-b+a^2 x^2} \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}} \, dx=\int \frac {{\left (c+{\left (a\,x+\sqrt {a^2\,x^2-b}\right )}^{1/4}\right )}^{1/3}\,\sqrt {a^2\,x^2-b}}{{\left (a\,x+\sqrt {a^2\,x^2-b}\right )}^{1/4}} \,d x \]
int(((c + (a*x + (a^2*x^2 - b)^(1/2))^(1/4))^(1/3)*(a^2*x^2 - b)^(1/2))/(a *x + (a^2*x^2 - b)^(1/2))^(1/4),x)