Integrand size = 68, antiderivative size = 803 \[ \int \frac {\sqrt {-b+a^2 x^2}}{\sqrt [3]{a x+\sqrt {-b+a^2 x^2}} \sqrt [4]{c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}} \, dx=\frac {\left (363738375 b^3-3081830400 b^2 c^6+238761600 a b^2 c^3 x-1056964608 a b c^9 x-727476750 a^2 b^2 x^2+5752750080 a^2 b c^6 x^2+1409286144 a^3 c^9 x^3\right ) \left (c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}\right )^{3/4}+\left (214016000 b^2 c^5-536870912 b c^{11}-258658400 a b^2 c^2 x+968884224 a b c^8 x+1073741824 a^2 c^{11} x^2-1291845632 a^3 c^8 x^3\right ) \sqrt [3]{a x+\sqrt {-b+a^2 x^2}} \left (c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}\right )^{3/4}+\left (-224716800 b^2 c^4+402653184 b c^{10}+290990700 a b^2 c x-908328960 a b c^7 x-805306368 a^2 c^{10} x^2+1211105280 a^3 c^7 x^3\right ) \left (a x+\sqrt {-b+a^2 x^2}\right )^{2/3} \left (c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}\right )^{3/4}+\sqrt {-b+a^2 x^2} \left (\left (238761600 b^2 c^3-352321536 b c^9-727476750 a b^2 x+5752750080 a b c^6 x+1409286144 a^2 c^9 x^2\right ) \left (c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}\right )^{3/4}+\left (-258658400 b^2 c^2+322961408 b c^8+1073741824 a c^{11} x-1291845632 a^2 c^8 x^2\right ) \sqrt [3]{a x+\sqrt {-b+a^2 x^2}} \left (c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}\right )^{3/4}+\left (290990700 b^2 c-302776320 b c^7-805306368 a c^{10} x+1211105280 a^2 c^7 x^2\right ) \left (a x+\sqrt {-b+a^2 x^2}\right )^{2/3} \left (c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}\right )^{3/4}\right )}{1917583360 a c^7 \left (a x+\sqrt {-b+a^2 x^2}\right )^{7/3}}-\frac {49725 b^2 \arctan \left (\frac {\sqrt [4]{c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt [4]{c}}\right )}{524288 a c^{29/4}}+\frac {3 b \arctan \left (\frac {\sqrt [4]{c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt [4]{c}}\right )}{4 a c^{5/4}}+\frac {49725 b^2 \text {arctanh}\left (\frac {\sqrt [4]{c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt [4]{c}}\right )}{524288 a c^{29/4}}-\frac {3 b \text {arctanh}\left (\frac {\sqrt [4]{c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt [4]{c}}\right )}{4 a c^{5/4}} \]
1/1917583360*((1409286144*a^3*c^9*x^3-1056964608*a*b*c^9*x+5752750080*a^2* b*c^6*x^2-3081830400*b^2*c^6+238761600*a*b^2*c^3*x-727476750*a^2*b^2*x^2+3 63738375*b^3)*(c+(a*x+(a^2*x^2-b)^(1/2))^(1/3))^(3/4)+(1073741824*a^2*c^11 *x^2-1291845632*a^3*c^8*x^3-536870912*b*c^11+968884224*a*b*c^8*x+214016000 *b^2*c^5-258658400*a*b^2*c^2*x)*(a*x+(a^2*x^2-b)^(1/2))^(1/3)*(c+(a*x+(a^2 *x^2-b)^(1/2))^(1/3))^(3/4)+(-805306368*a^2*c^10*x^2+1211105280*a^3*c^7*x^ 3+402653184*b*c^10-908328960*a*b*c^7*x-224716800*b^2*c^4+290990700*a*b^2*c *x)*(a*x+(a^2*x^2-b)^(1/2))^(2/3)*(c+(a*x+(a^2*x^2-b)^(1/2))^(1/3))^(3/4)+ (a^2*x^2-b)^(1/2)*((1409286144*a^2*c^9*x^2-352321536*b*c^9+5752750080*a*b* c^6*x+238761600*b^2*c^3-727476750*a*b^2*x)*(c+(a*x+(a^2*x^2-b)^(1/2))^(1/3 ))^(3/4)+(1073741824*a*c^11*x-1291845632*a^2*c^8*x^2+322961408*b*c^8-25865 8400*b^2*c^2)*(a*x+(a^2*x^2-b)^(1/2))^(1/3)*(c+(a*x+(a^2*x^2-b)^(1/2))^(1/ 3))^(3/4)+(-805306368*a*c^10*x+1211105280*a^2*c^7*x^2-302776320*b*c^7+2909 90700*b^2*c)*(a*x+(a^2*x^2-b)^(1/2))^(2/3)*(c+(a*x+(a^2*x^2-b)^(1/2))^(1/3 ))^(3/4)))/a/c^7/(a*x+(a^2*x^2-b)^(1/2))^(7/3)-49725/524288*b^2*arctan((c+ (a*x+(a^2*x^2-b)^(1/2))^(1/3))^(1/4)/c^(1/4))/a/c^(29/4)+3/4*b*arctan((c+( a*x+(a^2*x^2-b)^(1/2))^(1/3))^(1/4)/c^(1/4))/a/c^(5/4)+49725/524288*b^2*ar ctanh((c+(a*x+(a^2*x^2-b)^(1/2))^(1/3))^(1/4)/c^(1/4))/a/c^(29/4)-3/4*b*ar ctanh((c+(a*x+(a^2*x^2-b)^(1/2))^(1/3))^(1/4)/c^(1/4))/a/c^(5/4)
Time = 2.62 (sec) , antiderivative size = 803, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {-b+a^2 x^2}}{\sqrt [3]{a x+\sqrt {-b+a^2 x^2}} \sqrt [4]{c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}} \, dx=\frac {\left (363738375 b^3-3081830400 b^2 c^6+238761600 a b^2 c^3 x-1056964608 a b c^9 x-727476750 a^2 b^2 x^2+5752750080 a^2 b c^6 x^2+1409286144 a^3 c^9 x^3\right ) \left (c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}\right )^{3/4}+\left (214016000 b^2 c^5-536870912 b c^{11}-258658400 a b^2 c^2 x+968884224 a b c^8 x+1073741824 a^2 c^{11} x^2-1291845632 a^3 c^8 x^3\right ) \sqrt [3]{a x+\sqrt {-b+a^2 x^2}} \left (c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}\right )^{3/4}+\left (-224716800 b^2 c^4+402653184 b c^{10}+290990700 a b^2 c x-908328960 a b c^7 x-805306368 a^2 c^{10} x^2+1211105280 a^3 c^7 x^3\right ) \left (a x+\sqrt {-b+a^2 x^2}\right )^{2/3} \left (c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}\right )^{3/4}+\sqrt {-b+a^2 x^2} \left (\left (238761600 b^2 c^3-352321536 b c^9-727476750 a b^2 x+5752750080 a b c^6 x+1409286144 a^2 c^9 x^2\right ) \left (c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}\right )^{3/4}+\left (-258658400 b^2 c^2+322961408 b c^8+1073741824 a c^{11} x-1291845632 a^2 c^8 x^2\right ) \sqrt [3]{a x+\sqrt {-b+a^2 x^2}} \left (c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}\right )^{3/4}+\left (290990700 b^2 c-302776320 b c^7-805306368 a c^{10} x+1211105280 a^2 c^7 x^2\right ) \left (a x+\sqrt {-b+a^2 x^2}\right )^{2/3} \left (c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}\right )^{3/4}\right )}{1917583360 a c^7 \left (a x+\sqrt {-b+a^2 x^2}\right )^{7/3}}-\frac {49725 b^2 \arctan \left (\frac {\sqrt [4]{c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt [4]{c}}\right )}{524288 a c^{29/4}}+\frac {3 b \arctan \left (\frac {\sqrt [4]{c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt [4]{c}}\right )}{4 a c^{5/4}}+\frac {49725 b^2 \text {arctanh}\left (\frac {\sqrt [4]{c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt [4]{c}}\right )}{524288 a c^{29/4}}-\frac {3 b \text {arctanh}\left (\frac {\sqrt [4]{c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt [4]{c}}\right )}{4 a c^{5/4}} \]
Integrate[Sqrt[-b + a^2*x^2]/((a*x + Sqrt[-b + a^2*x^2])^(1/3)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(1/4)),x]
((363738375*b^3 - 3081830400*b^2*c^6 + 238761600*a*b^2*c^3*x - 1056964608* a*b*c^9*x - 727476750*a^2*b^2*x^2 + 5752750080*a^2*b*c^6*x^2 + 1409286144* a^3*c^9*x^3)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(3/4) + (214016000*b^2 *c^5 - 536870912*b*c^11 - 258658400*a*b^2*c^2*x + 968884224*a*b*c^8*x + 10 73741824*a^2*c^11*x^2 - 1291845632*a^3*c^8*x^3)*(a*x + Sqrt[-b + a^2*x^2]) ^(1/3)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(3/4) + (-224716800*b^2*c^4 + 402653184*b*c^10 + 290990700*a*b^2*c*x - 908328960*a*b*c^7*x - 805306368 *a^2*c^10*x^2 + 1211105280*a^3*c^7*x^3)*(a*x + Sqrt[-b + a^2*x^2])^(2/3)*( c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(3/4) + Sqrt[-b + a^2*x^2]*((2387616 00*b^2*c^3 - 352321536*b*c^9 - 727476750*a*b^2*x + 5752750080*a*b*c^6*x + 1409286144*a^2*c^9*x^2)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(3/4) + (-2 58658400*b^2*c^2 + 322961408*b*c^8 + 1073741824*a*c^11*x - 1291845632*a^2* c^8*x^2)*(a*x + Sqrt[-b + a^2*x^2])^(1/3)*(c + (a*x + Sqrt[-b + a^2*x^2])^ (1/3))^(3/4) + (290990700*b^2*c - 302776320*b*c^7 - 805306368*a*c^10*x + 1 211105280*a^2*c^7*x^2)*(a*x + Sqrt[-b + a^2*x^2])^(2/3)*(c + (a*x + Sqrt[- b + a^2*x^2])^(1/3))^(3/4)))/(1917583360*a*c^7*(a*x + Sqrt[-b + a^2*x^2])^ (7/3)) - (49725*b^2*ArcTan[(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(1/4)/c^ (1/4)])/(524288*a*c^(29/4)) + (3*b*ArcTan[(c + (a*x + Sqrt[-b + a^2*x^2])^ (1/3))^(1/4)/c^(1/4)])/(4*a*c^(5/4)) + (49725*b^2*ArcTanh[(c + (a*x + Sqrt [-b + a^2*x^2])^(1/3))^(1/4)/c^(1/4)])/(524288*a*c^(29/4)) - (3*b*ArcTa...
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.
| \(\displaystyle \int \frac {\sqrt {a^2 x^2-b}}{\sqrt [3]{\sqrt {a^2 x^2-b}+a x} \sqrt [4]{\sqrt [3]{\sqrt {a^2 x^2-b}+a x}+c}} \, dx\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle \int \frac {\sqrt {a^2 x^2-b}}{\sqrt [3]{\sqrt {a^2 x^2-b}+a x} \sqrt [4]{\sqrt [3]{\sqrt {a^2 x^2-b}+a x}+c}}dx\) |
Int[Sqrt[-b + a^2*x^2]/((a*x + Sqrt[-b + a^2*x^2])^(1/3)*(c + (a*x + Sqrt[ -b + a^2*x^2])^(1/3))^(1/4)),x]
3.32.32.3.1 Defintions of rubi rules used
\[\int \frac {\sqrt {a^{2} x^{2}-b}}{\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{3}} {\left (c +\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{3}}\right )}^{\frac {1}{4}}}d x\]
int((a^2*x^2-b)^(1/2)/(a*x+(a^2*x^2-b)^(1/2))^(1/3)/(c+(a*x+(a^2*x^2-b)^(1 /2))^(1/3))^(1/4),x)
int((a^2*x^2-b)^(1/2)/(a*x+(a^2*x^2-b)^(1/2))^(1/3)/(c+(a*x+(a^2*x^2-b)^(1 /2))^(1/3))^(1/4),x)
Result contains complex when optimal does not.
Time = 0.36 (sec) , antiderivative size = 1010, normalized size of antiderivative = 1.26 \[ \int \frac {\sqrt {-b+a^2 x^2}}{\sqrt [3]{a x+\sqrt {-b+a^2 x^2}} \sqrt [4]{c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}} \, dx=\text {Too large to display} \]
integrate((a^2*x^2-b)^(1/2)/(a*x+(a^2*x^2-b)^(1/2))^(1/3)/(c+(a*x+(a^2*x^2 -b)^(1/2))^(1/3))^(1/4),x, algorithm="fricas")
-1/7670333440*(21945*a*b*c^7*((295147905179352825856*b^4*c^24 - 1492943276 47331942400*b^5*c^18 + 28319017190031360000*b^6*c^12 - 2387429351424000000 *b^7*c^6 + 75476916312890625*b^8)/(a^4*c^29))^(1/4)*log(27*a^3*c^22*((2951 47905179352825856*b^4*c^24 - 149294327647331942400*b^5*c^18 + 283190171900 31360000*b^6*c^12 - 2387429351424000000*b^7*c^6 + 75476916312890625*b^8)/( a^4*c^29))^(3/4) + 27*(2251799813685248*b^3*c^18 - 854268995174400*b^4*c^1 2 + 108028477440000*b^5*c^6 - 4553660109375*b^6)*(c + (a*x + sqrt(a^2*x^2 - b))^(1/3))^(1/4)) - 21945*I*a*b*c^7*((295147905179352825856*b^4*c^24 - 1 49294327647331942400*b^5*c^18 + 28319017190031360000*b^6*c^12 - 2387429351 424000000*b^7*c^6 + 75476916312890625*b^8)/(a^4*c^29))^(1/4)*log(27*I*a^3* c^22*((295147905179352825856*b^4*c^24 - 149294327647331942400*b^5*c^18 + 2 8319017190031360000*b^6*c^12 - 2387429351424000000*b^7*c^6 + 7547691631289 0625*b^8)/(a^4*c^29))^(3/4) + 27*(2251799813685248*b^3*c^18 - 854268995174 400*b^4*c^12 + 108028477440000*b^5*c^6 - 4553660109375*b^6)*(c + (a*x + sq rt(a^2*x^2 - b))^(1/3))^(1/4)) + 21945*I*a*b*c^7*((295147905179352825856*b ^4*c^24 - 149294327647331942400*b^5*c^18 + 28319017190031360000*b^6*c^12 - 2387429351424000000*b^7*c^6 + 75476916312890625*b^8)/(a^4*c^29))^(1/4)*lo g(-27*I*a^3*c^22*((295147905179352825856*b^4*c^24 - 149294327647331942400* b^5*c^18 + 28319017190031360000*b^6*c^12 - 2387429351424000000*b^7*c^6 + 7 5476916312890625*b^8)/(a^4*c^29))^(3/4) + 27*(2251799813685248*b^3*c^18...
\[ \int \frac {\sqrt {-b+a^2 x^2}}{\sqrt [3]{a x+\sqrt {-b+a^2 x^2}} \sqrt [4]{c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}} \, dx=\int \frac {\sqrt {a^{2} x^{2} - b}}{\sqrt [4]{c + \sqrt [3]{a x + \sqrt {a^{2} x^{2} - b}}} \sqrt [3]{a x + \sqrt {a^{2} x^{2} - b}}}\, dx \]
integrate((a**2*x**2-b)**(1/2)/(a*x+(a**2*x**2-b)**(1/2))**(1/3)/(c+(a*x+( a**2*x**2-b)**(1/2))**(1/3))**(1/4),x)
Integral(sqrt(a**2*x**2 - b)/((c + (a*x + sqrt(a**2*x**2 - b))**(1/3))**(1 /4)*(a*x + sqrt(a**2*x**2 - b))**(1/3)), x)
\[ \int \frac {\sqrt {-b+a^2 x^2}}{\sqrt [3]{a x+\sqrt {-b+a^2 x^2}} \sqrt [4]{c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}} \, dx=\int { \frac {\sqrt {a^{2} x^{2} - b}}{{\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{3}} {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{3}}\right )}^{\frac {1}{4}}} \,d x } \]
integrate((a^2*x^2-b)^(1/2)/(a*x+(a^2*x^2-b)^(1/2))^(1/3)/(c+(a*x+(a^2*x^2 -b)^(1/2))^(1/3))^(1/4),x, algorithm="maxima")
integrate(sqrt(a^2*x^2 - b)/((a*x + sqrt(a^2*x^2 - b))^(1/3)*(c + (a*x + s qrt(a^2*x^2 - b))^(1/3))^(1/4)), x)
Timed out. \[ \int \frac {\sqrt {-b+a^2 x^2}}{\sqrt [3]{a x+\sqrt {-b+a^2 x^2}} \sqrt [4]{c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}} \, dx=\text {Timed out} \]
integrate((a^2*x^2-b)^(1/2)/(a*x+(a^2*x^2-b)^(1/2))^(1/3)/(c+(a*x+(a^2*x^2 -b)^(1/2))^(1/3))^(1/4),x, algorithm="giac")
Timed out. \[ \int \frac {\sqrt {-b+a^2 x^2}}{\sqrt [3]{a x+\sqrt {-b+a^2 x^2}} \sqrt [4]{c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}} \, dx=\int \frac {\sqrt {a^2\,x^2-b}}{{\left (a\,x+\sqrt {a^2\,x^2-b}\right )}^{1/3}\,{\left (c+{\left (a\,x+\sqrt {a^2\,x^2-b}\right )}^{1/3}\right )}^{1/4}} \,d x \]
int((a^2*x^2 - b)^(1/2)/((a*x + (a^2*x^2 - b)^(1/2))^(1/3)*(c + (a*x + (a^ 2*x^2 - b)^(1/2))^(1/3))^(1/4)),x)