Integrand size = 45, antiderivative size = 1186 \[ \int \frac {\sqrt {-b+a^2 x^2}}{\left (c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )^{2/3}} \, dx=\frac {\left (-748701954 b^2 c^7+8135830269 b c^{15}-1204701498 a b^2 c^3 x+3515482215 a b c^{11} x-16271660538 a^2 c^{15} x^2-4687309620 a^3 c^{11} x^3\right ) \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}+\left (820006902 b^2 c^6-2711943423 b c^{14}+1472412942 a b^2 c^2 x-3046751253 a b c^{10} x+5423886846 a^2 c^{14} x^2+4062335004 a^3 c^{10} 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 (-911118780 b^2 c^5+1807962282 b c^{13}-1963217256 a b^2 c x+2708223336 a b c^9 x-3615924564 a^2 c^{13} x^2-3610964448 a^3 c^9 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 (1032601284 b^2 c^4-1406192886 b c^{12}+3272028760 a b^2 x-2450297304 a b c^8 x+2812385772 a^2 c^{12} x^2+3267063072 a^3 c^8 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 (-1204701498 b^2 c^3+1171827405 b c^{11}-16271660538 a c^{15} x-4687309620 a^2 c^{11} x^2\right ) \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}+\left (1472412942 b^2 c^2-1015583751 b c^{10}+5423886846 a c^{14} x+4062335004 a^2 c^{10} 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 (-1963217256 b^2 c+902741112 b c^9-3615924564 a c^{13} x-3610964448 a^2 c^9 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 (3272028760 b^2-816765768 b c^8+2812385772 a c^{12} x+3267063072 a^2 c^8 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 )}{11979231264 a^2 c^8 x \sqrt {-b+a^2 x^2}+5989615632 a c^8 \left (-b+2 a^2 x^2\right )}-\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 )}{19683 \sqrt {3} a c^{26/3}}+\frac {2 \sqrt {3} 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 )}{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 )}{59049 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 )}{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 )}{118098 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 )}{a c^{2/3}} \]
((-16271660538*a^2*c^15*x^2-4687309620*a^3*c^11*x^3+8135830269*b*c^15+3515 482215*a*b*c^11*x-748701954*b^2*c^7-1204701498*a*b^2*c^3*x)*(c+(a*x+(a^2*x ^2-b)^(1/2))^(1/4))^(1/3)+(5423886846*a^2*c^14*x^2+4062335004*a^3*c^10*x^3 -2711943423*b*c^14-3046751253*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))^(1/4)*(c+(a*x+(a^2*x^2-b)^(1/2))^(1/4))^( 1/3)+(-3615924564*a^2*c^13*x^2-3610964448*a^3*c^9*x^3+1807962282*b*c^13+27 08223336*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))^(1/2)*(c+(a*x+(a^2*x^2-b)^(1/2))^(1/4))^(1/3)+(2812385772*a^2*c^1 2*x^2+3267063072*a^3*c^8*x^3-1406192886*b*c^12-2450297304*a*b*c^8*x+103260 1284*b^2*c^4+3272028760*a*b^2*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)*((-16271660538*a*c^15*x-468 7309620*a^2*c^11*x^2+1171827405*b*c^11-1204701498*b^2*c^3)*(c+(a*x+(a^2*x^ 2-b)^(1/2))^(1/4))^(1/3)+(5423886846*a*c^14*x+4062335004*a^2*c^10*x^2-1015 583751*b*c^10+1472412942*b^2*c^2)*(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)+(-3615924564*a*c^13*x-3610964448*a^2*c^9*x^2 +902741112*b*c^9-1963217256*b^2*c)*(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)+(2812385772*a*c^12*x+3267063072*a^2*c^8*x^2 -816765768*b*c^8+3272028760*b^2)*(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)))/(11979231264*a^2*c^8*x*(a^2*x^2-b)^(1/2)+59 89615632*a*c^8*(2*a^2*x^2-b))-21505/59049*b^2*arctan(1/3*3^(1/2)+2/3*(c...
Time = 3.16 (sec) , antiderivative size = 1096, normalized size of antiderivative = 0.92 \[ \int \frac {\sqrt {-b+a^2 x^2}}{\left (c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )^{2/3}} \, dx=\frac {\frac {3 c^{2/3} \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}} \left (826686 a c^8 x \left (a x+\sqrt {-b+a^2 x^2}\right ) \left (-19683 c^7-5670 a c^3 x+6561 c^6 \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}+4914 a c^2 x \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}-4374 c^5 \sqrt {a x+\sqrt {-b+a^2 x^2}}-4368 a c x \sqrt {a x+\sqrt {-b+a^2 x^2}}+3402 c^4 \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4}+3952 a x \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4}\right )+5434 b^2 \left (-137781 c^7+150903 c^6 \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}-167670 c^5 \sqrt {a x+\sqrt {-b+a^2 x^2}}+190026 c^4 \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4}-221697 c^3 \left (a x+\sqrt {-b+a^2 x^2}\right )+270963 c^2 \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/4}-361284 c \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/2}+602140 \left (a x+\sqrt {-b+a^2 x^2}\right )^{7/4}\right )-413343 b c^8 \left (-19683 c^7+6561 c^6 \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}-4374 c^5 \sqrt {a x+\sqrt {-b+a^2 x^2}}+3402 c^4 \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4}-2835 c^3 \left (3 a x+\sqrt {-b+a^2 x^2}\right )+2457 c^2 \sqrt [4]{a x+\sqrt {-b+a^2 x^2}} \left (3 a x+\sqrt {-b+a^2 x^2}\right )-2184 c \sqrt {a x+\sqrt {-b+a^2 x^2}} \left (3 a x+\sqrt {-b+a^2 x^2}\right )+1976 \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4} \left (3 a x+\sqrt {-b+a^2 x^2}\right )\right )\right )}{-b+2 a x \left (a x+\sqrt {-b+a^2 x^2}\right )}-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 )+35937693792 \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 )-35937693792 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 )+17968846896 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 )}{17968846896 a c^{26/3}} \]
((3*c^(2/3)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3)*(826686*a*c^8*x*( a*x + Sqrt[-b + a^2*x^2])*(-19683*c^7 - 5670*a*c^3*x + 6561*c^6*(a*x + Sqr t[-b + a^2*x^2])^(1/4) + 4914*a*c^2*x*(a*x + Sqrt[-b + a^2*x^2])^(1/4) - 4 374*c^5*Sqrt[a*x + Sqrt[-b + a^2*x^2]] - 4368*a*c*x*Sqrt[a*x + Sqrt[-b + a ^2*x^2]] + 3402*c^4*(a*x + Sqrt[-b + a^2*x^2])^(3/4) + 3952*a*x*(a*x + Sqr t[-b + a^2*x^2])^(3/4)) + 5434*b^2*(-137781*c^7 + 150903*c^6*(a*x + Sqrt[- b + a^2*x^2])^(1/4) - 167670*c^5*Sqrt[a*x + Sqrt[-b + a^2*x^2]] + 190026*c ^4*(a*x + Sqrt[-b + a^2*x^2])^(3/4) - 221697*c^3*(a*x + Sqrt[-b + a^2*x^2] ) + 270963*c^2*(a*x + Sqrt[-b + a^2*x^2])^(5/4) - 361284*c*(a*x + Sqrt[-b + a^2*x^2])^(3/2) + 602140*(a*x + Sqrt[-b + a^2*x^2])^(7/4)) - 413343*b*c^ 8*(-19683*c^7 + 6561*c^6*(a*x + Sqrt[-b + a^2*x^2])^(1/4) - 4374*c^5*Sqrt[ a*x + Sqrt[-b + a^2*x^2]] + 3402*c^4*(a*x + Sqrt[-b + a^2*x^2])^(3/4) - 28 35*c^3*(3*a*x + Sqrt[-b + a^2*x^2]) + 2457*c^2*(a*x + Sqrt[-b + a^2*x^2])^ (1/4)*(3*a*x + Sqrt[-b + a^2*x^2]) - 2184*c*Sqrt[a*x + Sqrt[-b + a^2*x^2]] *(3*a*x + Sqrt[-b + a^2*x^2]) + 1976*(a*x + Sqrt[-b + a^2*x^2])^(3/4)*(3*a *x + Sqrt[-b + a^2*x^2]))))/(-b + 2*a*x*(a*x + Sqrt[-b + a^2*x^2])) - 6544 057520*Sqrt[3]*b^2*ArcTan[(1 + (2*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^( 1/3))/c^(1/3))/Sqrt[3]] + 35937693792*Sqrt[3]*b*c^8*ArcTan[(1 + (2*(c + (a *x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3))/c^(1/3))/Sqrt[3]] + 6544057520*b^2* Log[-c^(1/3) + (c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3)] - 35937693...
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.
| \(\displaystyle \int \frac {\sqrt {a^2 x^2-b}}{\left (\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c\right )^{2/3}} \, dx\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle \int \frac {\sqrt {a^2 x^2-b}}{\left (\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c\right )^{2/3}}dx\) |
3.32.41.3.1 Defintions of rubi rules used
\[\int \frac {\sqrt {a^{2} x^{2}-b}}{{\left (c +\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{4}}\right )}^{\frac {2}{3}}}d x\]
Time = 0.39 (sec) , antiderivative size = 644, normalized size of antiderivative = 0.54 \[ \int \frac {\sqrt {-b+a^2 x^2}}{\left (c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )^{2/3}} \, dx=\frac {304304 \, \sqrt {3} {\left (118098 \, b c^{9} - 21505 \, b^{2} 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 (118098 \, b c^{8} - 21505 \, b^{2}\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 (118098 \, b c^{8} - 21505 \, b^{2}\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 (8135830269 \, c^{17} + 1497403908 \, a^{2} c^{9} x^{2} - 748701954 \, b c^{9} + 567 \, {\left (2066715 \, a c^{13} + 2124694 \, a b c^{5}\right )} x - 2 \, {\left (703096443 \, c^{14} + 1032601284 \, a^{2} c^{6} x^{2} - 516300642 \, b c^{6} + 6916 \, {\left (59049 \, a c^{10} + 236555 \, a b c^{2}\right )} x + 988 \, {\left (413343 \, c^{10} - 1045143 \, a c^{6} x - 1655885 \, b 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 (2066715 \, c^{13} - 2640924 \, a c^{9} x - 2124694 \, b c^{5}\right )} \sqrt {a^{2} x^{2} - b} + 6 \, {\left (301327047 \, c^{15} + 303706260 \, a^{2} c^{7} x^{2} - 151853130 \, b c^{7} + 364 \, {\left (413343 \, a c^{11} + 898909 \, a b c^{3}\right )} x + 52 \, {\left (2893401 \, c^{11} - 5840505 \, a c^{7} x - 6292363 \, b c^{3}\right )} \sqrt {a^{2} x^{2} - b}\right )} \sqrt {a x + \sqrt {a^{2} x^{2} - b}} - 9 \, {\left (301327047 \, c^{16} + 182223756 \, a^{2} c^{8} x^{2} - 91111878 \, b c^{8} + 91 \, {\left (1240029 \, a c^{12} + 1797818 \, a b c^{4}\right )} x + 13 \, {\left (8680203 \, c^{12} - 14017212 \, a c^{8} x - 12584726 \, b 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}}}{17968846896 \, a c^{10}} \]
1/17968846896*(304304*sqrt(3)*(118098*b*c^9 - 21505*b^2*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*(118098*b*c^8 - 21505*b^2)*(-c^2)^(2/3)*log((c + (a*x + s qrt(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*(118098*b*c^8 - 21505*b^2)*(- c^2)^(2/3)*log((c + (a*x + sqrt(a^2*x^2 - b))^(1/4))^(1/3)*c - (-c^2)^(2/3 )) - 3*(8135830269*c^17 + 1497403908*a^2*c^9*x^2 - 748701954*b*c^9 + 567*( 2066715*a*c^13 + 2124694*a*b*c^5)*x - 2*(703096443*c^14 + 1032601284*a^2*c ^6*x^2 - 516300642*b*c^6 + 6916*(59049*a*c^10 + 236555*a*b*c^2)*x + 988*(4 13343*c^10 - 1045143*a*c^6*x - 1655885*b*c^2)*sqrt(a^2*x^2 - b))*(a*x + sq rt(a^2*x^2 - b))^(3/4) + 567*(2066715*c^13 - 2640924*a*c^9*x - 2124694*b*c ^5)*sqrt(a^2*x^2 - b) + 6*(301327047*c^15 + 303706260*a^2*c^7*x^2 - 151853 130*b*c^7 + 364*(413343*a*c^11 + 898909*a*b*c^3)*x + 52*(2893401*c^11 - 58 40505*a*c^7*x - 6292363*b*c^3)*sqrt(a^2*x^2 - b))*sqrt(a*x + sqrt(a^2*x^2 - b)) - 9*(301327047*c^16 + 182223756*a^2*c^8*x^2 - 91111878*b*c^8 + 91*(1 240029*a*c^12 + 1797818*a*b*c^4)*x + 13*(8680203*c^12 - 14017212*a*c^8*x - 12584726*b*c^4)*sqrt(a^2*x^2 - b))*(a*x + sqrt(a^2*x^2 - b))^(1/4))*(c + (a*x + sqrt(a^2*x^2 - b))^(1/4))^(1/3))/(a*c^10)
\[ \int \frac {\sqrt {-b+a^2 x^2}}{\left (c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )^{2/3}} \, dx=\int \frac {\sqrt {a^{2} x^{2} - b}}{\left (c + \sqrt [4]{a x + \sqrt {a^{2} x^{2} - b}}\right )^{\frac {2}{3}}}\, dx \]
\[ \int \frac {\sqrt {-b+a^2 x^2}}{\left (c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )^{2/3}} \, dx=\int { \frac {\sqrt {a^{2} x^{2} - b}}{{\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {2}{3}}} \,d x } \]
Timed out. \[ \int \frac {\sqrt {-b+a^2 x^2}}{\left (c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )^{2/3}} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {\sqrt {-b+a^2 x^2}}{\left (c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )^{2/3}} \, dx=\int \frac {\sqrt {a^2\,x^2-b}}{{\left (c+{\left (a\,x+\sqrt {a^2\,x^2-b}\right )}^{1/4}\right )}^{2/3}} \,d x \]