Integrand size = 68, antiderivative size = 674 \[ \int \frac {\sqrt {-b+a^2 x^2} \sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [4]{c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}} \, dx=\frac {\left (-1860655104 b^2 c^4+2013265920 b c^{10}+2409402996 a b^2 c x-4541644800 a b c^7 x-4026531840 a^2 c^{10} x^2+6055526400 a^3 c^7 x^3\right ) \left (c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}\right )^{3/4}+\left (1976946048 b^2 c^3-1761607680 b c^9-3011753745 a b^2 x+4314562560 a b c^6 x+3523215360 a^2 c^9 x^2-5752750080 a^3 c^6 x^3\right ) \sqrt [3]{a x+\sqrt {-b+a^2 x^2}} \left (c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}\right )^{3/4}+\left (-2141691552 b^2 c^2+1614807040 b c^8-28943523840 a b c^5 x+2684354560 a c^{11} x-3229614080 a^2 c^8 x^2+5513052160 a^3 c^5 x^3\right ) \left (a x+\sqrt {-b+a^2 x^2}\right )^{2/3} \left (c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}\right )^{3/4}+\sqrt {-b+a^2 x^2} \left (\left (2409402996 b^2 c-1513881600 b c^7-4026531840 a c^{10} x+6055526400 a^2 c^7 x^2\right ) \left (c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}\right )^{3/4}+\left (-3011753745 b^2+1438187520 b c^6+3523215360 a c^9 x-5752750080 a^2 c^6 x^2\right ) \sqrt [3]{a x+\sqrt {-b+a^2 x^2}} \left (c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}\right )^{3/4}+\left (-26186997760 b c^5+2684354560 c^{11}-3229614080 a c^8 x+5513052160 a^2 c^5 x^2\right ) \left (a x+\sqrt {-b+a^2 x^2}\right )^{2/3} \left (c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}\right )^{3/4}\right )}{12404367360 a c^5 \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/3}}-\frac {1989 b^2 \arctan \left (\frac {\sqrt [4]{c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt [4]{c}}\right )}{16384 a c^{21/4}}+\frac {1989 b^2 \text {arctanh}\left (\frac {\sqrt [4]{c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt [4]{c}}\right )}{16384 a c^{21/4}} \]
1/12404367360*((-4026531840*a^2*c^10*x^2+6055526400*a^3*c^7*x^3+2013265920 *b*c^10-4541644800*a*b*c^7*x-1860655104*b^2*c^4+2409402996*a*b^2*c*x)*(c+( a*x+(a^2*x^2-b)^(1/2))^(1/3))^(3/4)+(3523215360*a^2*c^9*x^2-5752750080*a^3 *c^6*x^3-1761607680*b*c^9+4314562560*a*b*c^6*x+1976946048*b^2*c^3-30117537 45*a*b^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)+(2684354560*a*c^11*x-3229614080*a^2*c^8*x^2+5513052160*a^3*c^5*x^3 +1614807040*b*c^8-28943523840*a*b*c^5*x-2141691552*b^2*c^2)*(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)* ((-4026531840*a*c^10*x+6055526400*a^2*c^7*x^2-1513881600*b*c^7+2409402996* b^2*c)*(c+(a*x+(a^2*x^2-b)^(1/2))^(1/3))^(3/4)+(3523215360*a*c^9*x-5752750 080*a^2*c^6*x^2+1438187520*b*c^6-3011753745*b^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)+(2684354560*c^11-3229614080*a *c^8*x+5513052160*a^2*c^5*x^2-26186997760*b*c^5)*(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^5/(a*x+(a^2*x^2-b)^(1/2 ))^(5/3)-1989/16384*b^2*arctan((c+(a*x+(a^2*x^2-b)^(1/2))^(1/3))^(1/4)/c^( 1/4))/a/c^(21/4)+1989/16384*b^2*arctanh((c+(a*x+(a^2*x^2-b)^(1/2))^(1/3))^ (1/4)/c^(1/4))/a/c^(21/4)
Time = 1.98 (sec) , antiderivative size = 637, normalized size of antiderivative = 0.95 \[ \int \frac {\sqrt {-b+a^2 x^2} \sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [4]{c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}} \, dx=\frac {\frac {2 \sqrt [4]{c} \left (c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}\right )^{3/4} \left (-302841 b^2 \left (6144 c^4-6528 c^3 \sqrt [3]{a x+\sqrt {-b+a^2 x^2}}+7072 c^2 \left (a x+\sqrt {-b+a^2 x^2}\right )^{2/3}-7956 c \left (a x+\sqrt {-b+a^2 x^2}\right )+9945 \left (a x+\sqrt {-b+a^2 x^2}\right )^{4/3}\right )-40960 b c^5 \left (-49152 c^5+43008 c^4 \sqrt [3]{a x+\sqrt {-b+a^2 x^2}}-39424 c^3 \left (a x+\sqrt {-b+a^2 x^2}\right )^{2/3}+36960 c^2 \left (3 a x+\sqrt {-b+a^2 x^2}\right )-35112 c \sqrt [3]{a x+\sqrt {-b+a^2 x^2}} \left (3 a x+\sqrt {-b+a^2 x^2}\right )+33649 \left (a x+\sqrt {-b+a^2 x^2}\right )^{2/3} \left (21 a x+19 \sqrt {-b+a^2 x^2}\right )\right )+163840 c^5 \left (a x+\sqrt {-b+a^2 x^2}\right ) \left (16384 c^6 \left (a x+\sqrt {-b+a^2 x^2}\right )^{2/3}-256 a c^3 x \left (96 c^2-84 c \sqrt [3]{a x+\sqrt {-b+a^2 x^2}}+77 \left (a x+\sqrt {-b+a^2 x^2}\right )^{2/3}\right )+77 a^2 x^2 \left (480 c^2-456 c \sqrt [3]{a x+\sqrt {-b+a^2 x^2}}+437 \left (a x+\sqrt {-b+a^2 x^2}\right )^{2/3}\right )\right )\right )}{\left (a x+\sqrt {-b+a^2 x^2}\right )^{5/3}}-3011753745 b^2 \arctan \left (\frac {\sqrt [4]{c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt [4]{c}}\right )+3011753745 b^2 \text {arctanh}\left (\frac {\sqrt [4]{c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt [4]{c}}\right )}{24808734720 a c^{21/4}} \]
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]
((2*c^(1/4)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(3/4)*(-302841*b^2*(614 4*c^4 - 6528*c^3*(a*x + Sqrt[-b + a^2*x^2])^(1/3) + 7072*c^2*(a*x + Sqrt[- b + a^2*x^2])^(2/3) - 7956*c*(a*x + Sqrt[-b + a^2*x^2]) + 9945*(a*x + Sqrt [-b + a^2*x^2])^(4/3)) - 40960*b*c^5*(-49152*c^5 + 43008*c^4*(a*x + Sqrt[- b + a^2*x^2])^(1/3) - 39424*c^3*(a*x + Sqrt[-b + a^2*x^2])^(2/3) + 36960*c ^2*(3*a*x + Sqrt[-b + a^2*x^2]) - 35112*c*(a*x + Sqrt[-b + a^2*x^2])^(1/3) *(3*a*x + Sqrt[-b + a^2*x^2]) + 33649*(a*x + Sqrt[-b + a^2*x^2])^(2/3)*(21 *a*x + 19*Sqrt[-b + a^2*x^2])) + 163840*c^5*(a*x + Sqrt[-b + a^2*x^2])*(16 384*c^6*(a*x + Sqrt[-b + a^2*x^2])^(2/3) - 256*a*c^3*x*(96*c^2 - 84*c*(a*x + Sqrt[-b + a^2*x^2])^(1/3) + 77*(a*x + Sqrt[-b + a^2*x^2])^(2/3)) + 77*a ^2*x^2*(480*c^2 - 456*c*(a*x + Sqrt[-b + a^2*x^2])^(1/3) + 437*(a*x + Sqrt [-b + a^2*x^2])^(2/3)))))/(a*x + Sqrt[-b + a^2*x^2])^(5/3) - 3011753745*b^ 2*ArcTan[(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(1/4)/c^(1/4)] + 301175374 5*b^2*ArcTanh[(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(1/4)/c^(1/4)])/(2480 8734720*a*c^(21/4))
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]
3.32.17.3.1 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 = 0.42 (sec) , antiderivative size = 560, normalized size of antiderivative = 0.83 \[ \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 {3011753745 \, a c^{5} \left (\frac {b^{8}}{a^{4} c^{21}}\right )^{\frac {1}{4}} \log \left (7868724669 \, a^{3} c^{16} \left (\frac {b^{8}}{a^{4} c^{21}}\right )^{\frac {3}{4}} + 7868724669 \, b^{6} {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{3}}\right )}^{\frac {1}{4}}\right ) - 3011753745 i \, a c^{5} \left (\frac {b^{8}}{a^{4} c^{21}}\right )^{\frac {1}{4}} \log \left (7868724669 i \, a^{3} c^{16} \left (\frac {b^{8}}{a^{4} c^{21}}\right )^{\frac {3}{4}} + 7868724669 \, b^{6} {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{3}}\right )}^{\frac {1}{4}}\right ) + 3011753745 i \, a c^{5} \left (\frac {b^{8}}{a^{4} c^{21}}\right )^{\frac {1}{4}} \log \left (-7868724669 i \, a^{3} c^{16} \left (\frac {b^{8}}{a^{4} c^{21}}\right )^{\frac {3}{4}} + 7868724669 \, b^{6} {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{3}}\right )}^{\frac {1}{4}}\right ) - 3011753745 \, a c^{5} \left (\frac {b^{8}}{a^{4} c^{21}}\right )^{\frac {1}{4}} \log \left (-7868724669 \, a^{3} c^{16} \left (\frac {b^{8}}{a^{4} c^{21}}\right )^{\frac {3}{4}} + 7868724669 \, b^{6} {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{3}}\right )}^{\frac {1}{4}}\right ) + 4 \, {\left (2684354560 \, c^{11} + 2756526080 \, a^{2} c^{5} x^{2} - 26186997760 \, b c^{5} - 2464 \, {\left (655360 \, a c^{8} + 869193 \, a b c^{2}\right )} x + 21 \, {\left (83886080 \, c^{9} + 188280576 \, a^{2} c^{3} x^{2} - 94140288 \, b c^{3} - 1045 \, {\left (65536 \, a c^{6} + 137241 \, a b\right )} x - 209 \, {\left (327680 \, c^{6} + 900864 \, a c^{3} x - 686205 \, b\right )} \sqrt {a^{2} x^{2} - b}\right )} {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {2}{3}} - 2464 \, {\left (655360 \, c^{8} - 1118720 \, a c^{5} x - 869193 \, b c^{2}\right )} \sqrt {a^{2} x^{2} - b} - 12 \, {\left (167772160 \, c^{10} + 310109184 \, a^{2} c^{4} x^{2} - 155054592 \, b c^{4} - 77 \, {\left (1638400 \, a c^{7} + 2607579 \, a b c\right )} x - 77 \, {\left (1638400 \, c^{7} + 4027392 \, a c^{4} x - 2607579 \, b c\right )} \sqrt {a^{2} x^{2} - b}\right )} {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{3}}\right )} {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{3}}\right )}^{\frac {3}{4}}}{49617469440 \, a c^{5}} \]
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")
1/49617469440*(3011753745*a*c^5*(b^8/(a^4*c^21))^(1/4)*log(7868724669*a^3* c^16*(b^8/(a^4*c^21))^(3/4) + 7868724669*b^6*(c + (a*x + sqrt(a^2*x^2 - b) )^(1/3))^(1/4)) - 3011753745*I*a*c^5*(b^8/(a^4*c^21))^(1/4)*log(7868724669 *I*a^3*c^16*(b^8/(a^4*c^21))^(3/4) + 7868724669*b^6*(c + (a*x + sqrt(a^2*x ^2 - b))^(1/3))^(1/4)) + 3011753745*I*a*c^5*(b^8/(a^4*c^21))^(1/4)*log(-78 68724669*I*a^3*c^16*(b^8/(a^4*c^21))^(3/4) + 7868724669*b^6*(c + (a*x + sq rt(a^2*x^2 - b))^(1/3))^(1/4)) - 3011753745*a*c^5*(b^8/(a^4*c^21))^(1/4)*l og(-7868724669*a^3*c^16*(b^8/(a^4*c^21))^(3/4) + 7868724669*b^6*(c + (a*x + sqrt(a^2*x^2 - b))^(1/3))^(1/4)) + 4*(2684354560*c^11 + 2756526080*a^2*c ^5*x^2 - 26186997760*b*c^5 - 2464*(655360*a*c^8 + 869193*a*b*c^2)*x + 21*( 83886080*c^9 + 188280576*a^2*c^3*x^2 - 94140288*b*c^3 - 1045*(65536*a*c^6 + 137241*a*b)*x - 209*(327680*c^6 + 900864*a*c^3*x - 686205*b)*sqrt(a^2*x^ 2 - b))*(a*x + sqrt(a^2*x^2 - b))^(2/3) - 2464*(655360*c^8 - 1118720*a*c^5 *x - 869193*b*c^2)*sqrt(a^2*x^2 - b) - 12*(167772160*c^10 + 310109184*a^2* c^4*x^2 - 155054592*b*c^4 - 77*(1638400*a*c^7 + 2607579*a*b*c)*x - 77*(163 8400*c^7 + 4027392*a*c^4*x - 2607579*b*c)*sqrt(a^2*x^2 - b))*(a*x + sqrt(a ^2*x^2 - b))^(1/3))*(c + (a*x + sqrt(a^2*x^2 - b))^(1/3))^(3/4))/(a*c^5)
\[ \int \frac {\sqrt {-b+a^2 x^2} \sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [4]{c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}} \, dx=\int \frac {\sqrt [3]{a x + \sqrt {a^{2} x^{2} - b}} \sqrt {a^{2} x^{2} - b}}{\sqrt [4]{c + \sqrt [3]{a x + \sqrt {a^{2} x^{2} - b}}}}\, dx \]
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((a*x + sqrt(a**2*x**2 - b))**(1/3)*sqrt(a**2*x**2 - b)/(c + (a*x + sqrt(a**2*x**2 - b))**(1/3))**(1/4), 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 + sq rt(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. \[ \int \frac {\sqrt {-b+a^2 x^2} \sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [4]{c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}} \, dx=\int \frac {{\left (a\,x+\sqrt {a^2\,x^2-b}\right )}^{1/3}\,\sqrt {a^2\,x^2-b}}{{\left (c+{\left (a\,x+\sqrt {a^2\,x^2-b}\right )}^{1/3}\right )}^{1/4}} \,d x \]
int(((a*x + (a^2*x^2 - b)^(1/2))^(1/3)*(a^2*x^2 - b)^(1/2))/(c + (a*x + (a ^2*x^2 - b)^(1/2))^(1/3))^(1/4),x)