Integrand size = 24, antiderivative size = 1005 \[ \int \frac {x^{13}}{\left (a+b x^8\right ) \sqrt {c+d x^8}} \, dx=\frac {x^2 \sqrt {c+d x^8}}{2 b \sqrt {d} \left (\sqrt {c}+\sqrt {d} x^4\right )}+\frac {(-a)^{3/4} \arctan \left (\frac {\sqrt {b c-a d} x^2}{\sqrt [4]{-a} \sqrt [4]{b} \sqrt {c+d x^8}}\right )}{8 b^{5/4} \sqrt {b c-a d}}-\frac {(-a)^{3/4} \arctan \left (\frac {\sqrt {-b c+a d} x^2}{\sqrt [4]{-a} \sqrt [4]{b} \sqrt {c+d x^8}}\right )}{8 b^{5/4} \sqrt {-b c+a d}}-\frac {\sqrt [4]{c} \left (\sqrt {c}+\sqrt {d} x^4\right ) \sqrt {\frac {c+d x^8}{\left (\sqrt {c}+\sqrt {d} x^4\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{d} x^2}{\sqrt [4]{c}}\right )|\frac {1}{2}\right )}{2 b d^{3/4} \sqrt {c+d x^8}}+\frac {\sqrt [4]{c} \left (\sqrt {c}+\sqrt {d} x^4\right ) \sqrt {\frac {c+d x^8}{\left (\sqrt {c}+\sqrt {d} x^4\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} x^2}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{4 b d^{3/4} \sqrt {c+d x^8}}+\frac {a \left (\sqrt {c}-\frac {\sqrt {-a} \sqrt {d}}{\sqrt {b}}\right ) \sqrt [4]{d} \left (\sqrt {c}+\sqrt {d} x^4\right ) \sqrt {\frac {c+d x^8}{\left (\sqrt {c}+\sqrt {d} x^4\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} x^2}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{8 b \sqrt [4]{c} (b c+a d) \sqrt {c+d x^8}}+\frac {a \left (\sqrt {c}+\frac {\sqrt {-a} \sqrt {d}}{\sqrt {b}}\right ) \sqrt [4]{d} \left (\sqrt {c}+\sqrt {d} x^4\right ) \sqrt {\frac {c+d x^8}{\left (\sqrt {c}+\sqrt {d} x^4\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} x^2}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{8 b \sqrt [4]{c} (b c+a d) \sqrt {c+d x^8}}+\frac {\sqrt {-a} \left (\sqrt {b} \sqrt {c}+\sqrt {-a} \sqrt {d}\right )^2 \left (\sqrt {c}+\sqrt {d} x^4\right ) \sqrt {\frac {c+d x^8}{\left (\sqrt {c}+\sqrt {d} x^4\right )^2}} \operatorname {EllipticPi}\left (-\frac {\left (\sqrt {b} \sqrt {c}-\sqrt {-a} \sqrt {d}\right )^2}{4 \sqrt {-a} \sqrt {b} \sqrt {c} \sqrt {d}},2 \arctan \left (\frac {\sqrt [4]{d} x^2}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{16 b^{3/2} \sqrt [4]{c} \sqrt [4]{d} (b c+a d) \sqrt {c+d x^8}}-\frac {\sqrt {-a} \left (\sqrt {b} \sqrt {c}-\sqrt {-a} \sqrt {d}\right )^2 \left (\sqrt {c}+\sqrt {d} x^4\right ) \sqrt {\frac {c+d x^8}{\left (\sqrt {c}+\sqrt {d} x^4\right )^2}} \operatorname {EllipticPi}\left (\frac {\left (\sqrt {b} \sqrt {c}+\sqrt {-a} \sqrt {d}\right )^2}{4 \sqrt {-a} \sqrt {b} \sqrt {c} \sqrt {d}},2 \arctan \left (\frac {\sqrt [4]{d} x^2}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{16 b^{3/2} \sqrt [4]{c} \sqrt [4]{d} (b c+a d) \sqrt {c+d x^8}} \]
1/8*(-a)^(3/4)*arctan(x^2*(-a*d+b*c)^(1/2)/(-a)^(1/4)/b^(1/4)/(d*x^8+c)^(1 /2))/b^(5/4)/(-a*d+b*c)^(1/2)-1/8*(-a)^(3/4)*arctan(x^2*(a*d-b*c)^(1/2)/(- a)^(1/4)/b^(1/4)/(d*x^8+c)^(1/2))/b^(5/4)/(a*d-b*c)^(1/2)+1/2*x^2*(d*x^8+c )^(1/2)/b/d^(1/2)/(c^(1/2)+x^4*d^(1/2))-1/2*c^(1/4)*(cos(2*arctan(d^(1/4)* x^2/c^(1/4)))^2)^(1/2)/cos(2*arctan(d^(1/4)*x^2/c^(1/4)))*EllipticE(sin(2* arctan(d^(1/4)*x^2/c^(1/4))),1/2*2^(1/2))*(c^(1/2)+x^4*d^(1/2))*((d*x^8+c) /(c^(1/2)+x^4*d^(1/2))^2)^(1/2)/b/d^(3/4)/(d*x^8+c)^(1/2)+1/4*c^(1/4)*(cos (2*arctan(d^(1/4)*x^2/c^(1/4)))^2)^(1/2)/cos(2*arctan(d^(1/4)*x^2/c^(1/4)) )*EllipticF(sin(2*arctan(d^(1/4)*x^2/c^(1/4))),1/2*2^(1/2))*(c^(1/2)+x^4*d ^(1/2))*((d*x^8+c)/(c^(1/2)+x^4*d^(1/2))^2)^(1/2)/b/d^(3/4)/(d*x^8+c)^(1/2 )-1/16*(cos(2*arctan(d^(1/4)*x^2/c^(1/4)))^2)^(1/2)/cos(2*arctan(d^(1/4)*x ^2/c^(1/4)))*EllipticPi(sin(2*arctan(d^(1/4)*x^2/c^(1/4))),1/4*(b^(1/2)*c^ (1/2)+(-a)^(1/2)*d^(1/2))^2/(-a)^(1/2)/b^(1/2)/c^(1/2)/d^(1/2),1/2*2^(1/2) )*(-a)^(1/2)*(c^(1/2)+x^4*d^(1/2))*(b^(1/2)*c^(1/2)-(-a)^(1/2)*d^(1/2))^2* ((d*x^8+c)/(c^(1/2)+x^4*d^(1/2))^2)^(1/2)/b^(3/2)/c^(1/4)/d^(1/4)/(a*d+b*c )/(d*x^8+c)^(1/2)+1/16*(cos(2*arctan(d^(1/4)*x^2/c^(1/4)))^2)^(1/2)/cos(2* arctan(d^(1/4)*x^2/c^(1/4)))*EllipticPi(sin(2*arctan(d^(1/4)*x^2/c^(1/4))) ,-1/4*(b^(1/2)*c^(1/2)-(-a)^(1/2)*d^(1/2))^2/(-a)^(1/2)/b^(1/2)/c^(1/2)/d^ (1/2),1/2*2^(1/2))*(-a)^(1/2)*(c^(1/2)+x^4*d^(1/2))*(b^(1/2)*c^(1/2)+(-a)^ (1/2)*d^(1/2))^2*((d*x^8+c)/(c^(1/2)+x^4*d^(1/2))^2)^(1/2)/b^(3/2)/c^(1...
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 10.07 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.06 \[ \int \frac {x^{13}}{\left (a+b x^8\right ) \sqrt {c+d x^8}} \, dx=\frac {x^{14} \sqrt {\frac {c+d x^8}{c}} \operatorname {AppellF1}\left (\frac {7}{4},\frac {1}{2},1,\frac {11}{4},-\frac {d x^8}{c},-\frac {b x^8}{a}\right )}{14 a \sqrt {c+d x^8}} \]
(x^14*Sqrt[(c + d*x^8)/c]*AppellF1[7/4, 1/2, 1, 11/4, -((d*x^8)/c), -((b*x ^8)/a)])/(14*a*Sqrt[c + d*x^8])
Time = 1.57 (sec) , antiderivative size = 1111, normalized size of antiderivative = 1.11, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {965, 983, 834, 27, 761, 993, 1510, 1541, 27, 761, 2221, 2223}
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.
| \(\displaystyle \int \frac {x^{13}}{\left (a+b x^8\right ) \sqrt {c+d x^8}} \, dx\) |
| \(\Big \downarrow \) 965 |
| \(\displaystyle \frac {1}{2} \int \frac {x^{12}}{\left (b x^8+a\right ) \sqrt {d x^8+c}}dx^2\) |
| \(\Big \downarrow \) 983 |
| \(\displaystyle \frac {1}{2} \left (\frac {\int \frac {x^4}{\sqrt {d x^8+c}}dx^2}{b}-\frac {a \int \frac {x^4}{\left (b x^8+a\right ) \sqrt {d x^8+c}}dx^2}{b}\right )\) |
| \(\Big \downarrow \) 834 |
| \(\displaystyle \frac {1}{2} \left (\frac {\frac {\sqrt {c} \int \frac {1}{\sqrt {d x^8+c}}dx^2}{\sqrt {d}}-\frac {\sqrt {c} \int \frac {\sqrt {c}-\sqrt {d} x^4}{\sqrt {c} \sqrt {d x^8+c}}dx^2}{\sqrt {d}}}{b}-\frac {a \int \frac {x^4}{\left (b x^8+a\right ) \sqrt {d x^8+c}}dx^2}{b}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \left (\frac {\frac {\sqrt {c} \int \frac {1}{\sqrt {d x^8+c}}dx^2}{\sqrt {d}}-\frac {\int \frac {\sqrt {c}-\sqrt {d} x^4}{\sqrt {d x^8+c}}dx^2}{\sqrt {d}}}{b}-\frac {a \int \frac {x^4}{\left (b x^8+a\right ) \sqrt {d x^8+c}}dx^2}{b}\right )\) |
| \(\Big \downarrow \) 761 |
| \(\displaystyle \frac {1}{2} \left (\frac {\frac {\sqrt [4]{c} \left (\sqrt {c}+\sqrt {d} x^4\right ) \sqrt {\frac {c+d x^8}{\left (\sqrt {c}+\sqrt {d} x^4\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} x^2}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{2 d^{3/4} \sqrt {c+d x^8}}-\frac {\int \frac {\sqrt {c}-\sqrt {d} x^4}{\sqrt {d x^8+c}}dx^2}{\sqrt {d}}}{b}-\frac {a \int \frac {x^4}{\left (b x^8+a\right ) \sqrt {d x^8+c}}dx^2}{b}\right )\) |
| \(\Big \downarrow \) 993 |
| \(\displaystyle \frac {1}{2} \left (\frac {\frac {\sqrt [4]{c} \left (\sqrt {c}+\sqrt {d} x^4\right ) \sqrt {\frac {c+d x^8}{\left (\sqrt {c}+\sqrt {d} x^4\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} x^2}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{2 d^{3/4} \sqrt {c+d x^8}}-\frac {\int \frac {\sqrt {c}-\sqrt {d} x^4}{\sqrt {d x^8+c}}dx^2}{\sqrt {d}}}{b}-\frac {a \left (\frac {\int \frac {1}{\left (\sqrt {b} x^4+\sqrt {-a}\right ) \sqrt {d x^8+c}}dx^2}{2 \sqrt {b}}-\frac {\int \frac {1}{\left (\sqrt {-a}-\sqrt {b} x^4\right ) \sqrt {d x^8+c}}dx^2}{2 \sqrt {b}}\right )}{b}\right )\) |
| \(\Big \downarrow \) 1510 |
| \(\displaystyle \frac {1}{2} \left (\frac {\frac {\sqrt [4]{c} \left (\sqrt {c}+\sqrt {d} x^4\right ) \sqrt {\frac {c+d x^8}{\left (\sqrt {c}+\sqrt {d} x^4\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} x^2}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{2 d^{3/4} \sqrt {c+d x^8}}-\frac {\frac {\sqrt [4]{c} \left (\sqrt {c}+\sqrt {d} x^4\right ) \sqrt {\frac {c+d x^8}{\left (\sqrt {c}+\sqrt {d} x^4\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{d} x^2}{\sqrt [4]{c}}\right )|\frac {1}{2}\right )}{\sqrt [4]{d} \sqrt {c+d x^8}}-\frac {x^2 \sqrt {c+d x^8}}{\sqrt {c}+\sqrt {d} x^4}}{\sqrt {d}}}{b}-\frac {a \left (\frac {\int \frac {1}{\left (\sqrt {b} x^4+\sqrt {-a}\right ) \sqrt {d x^8+c}}dx^2}{2 \sqrt {b}}-\frac {\int \frac {1}{\left (\sqrt {-a}-\sqrt {b} x^4\right ) \sqrt {d x^8+c}}dx^2}{2 \sqrt {b}}\right )}{b}\right )\) |
| \(\Big \downarrow \) 1541 |
| \(\displaystyle \frac {1}{2} \left (\frac {\frac {\sqrt [4]{c} \left (\sqrt {c}+\sqrt {d} x^4\right ) \sqrt {\frac {c+d x^8}{\left (\sqrt {c}+\sqrt {d} x^4\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} x^2}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{2 d^{3/4} \sqrt {c+d x^8}}-\frac {\frac {\sqrt [4]{c} \left (\sqrt {c}+\sqrt {d} x^4\right ) \sqrt {\frac {c+d x^8}{\left (\sqrt {c}+\sqrt {d} x^4\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{d} x^2}{\sqrt [4]{c}}\right )|\frac {1}{2}\right )}{\sqrt [4]{d} \sqrt {c+d x^8}}-\frac {x^2 \sqrt {c+d x^8}}{\sqrt {c}+\sqrt {d} x^4}}{\sqrt {d}}}{b}-\frac {a \left (\frac {\frac {\sqrt {b} \sqrt {c} \left (\sqrt {-a} \sqrt {d}+\sqrt {b} \sqrt {c}\right ) \int \frac {\sqrt {d} x^4+\sqrt {c}}{\sqrt {c} \left (\sqrt {b} x^4+\sqrt {-a}\right ) \sqrt {d x^8+c}}dx^2}{a d+b c}-\frac {\sqrt {d} \left (\sqrt {-a} \sqrt {d}+\sqrt {b} \sqrt {c}\right ) \int \frac {1}{\sqrt {d x^8+c}}dx^2}{a d+b c}}{2 \sqrt {b}}-\frac {\frac {\sqrt {d} \left (\sqrt {b} \sqrt {c}-\sqrt {-a} \sqrt {d}\right ) \int \frac {1}{\sqrt {d x^8+c}}dx^2}{a d+b c}+\frac {\sqrt {b} \sqrt {c} \left (\sqrt {b} \sqrt {c}-\sqrt {-a} \sqrt {d}\right ) \int \frac {\sqrt {d} x^4+\sqrt {c}}{\sqrt {c} \left (\sqrt {-a}-\sqrt {b} x^4\right ) \sqrt {d x^8+c}}dx^2}{a d+b c}}{2 \sqrt {b}}\right )}{b}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \left (\frac {\frac {\sqrt [4]{c} \left (\sqrt {c}+\sqrt {d} x^4\right ) \sqrt {\frac {c+d x^8}{\left (\sqrt {c}+\sqrt {d} x^4\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} x^2}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{2 d^{3/4} \sqrt {c+d x^8}}-\frac {\frac {\sqrt [4]{c} \left (\sqrt {c}+\sqrt {d} x^4\right ) \sqrt {\frac {c+d x^8}{\left (\sqrt {c}+\sqrt {d} x^4\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{d} x^2}{\sqrt [4]{c}}\right )|\frac {1}{2}\right )}{\sqrt [4]{d} \sqrt {c+d x^8}}-\frac {x^2 \sqrt {c+d x^8}}{\sqrt {c}+\sqrt {d} x^4}}{\sqrt {d}}}{b}-\frac {a \left (\frac {\frac {\sqrt {b} \left (\sqrt {-a} \sqrt {d}+\sqrt {b} \sqrt {c}\right ) \int \frac {\sqrt {d} x^4+\sqrt {c}}{\left (\sqrt {b} x^4+\sqrt {-a}\right ) \sqrt {d x^8+c}}dx^2}{a d+b c}-\frac {\sqrt {d} \left (\sqrt {-a} \sqrt {d}+\sqrt {b} \sqrt {c}\right ) \int \frac {1}{\sqrt {d x^8+c}}dx^2}{a d+b c}}{2 \sqrt {b}}-\frac {\frac {\sqrt {d} \left (\sqrt {b} \sqrt {c}-\sqrt {-a} \sqrt {d}\right ) \int \frac {1}{\sqrt {d x^8+c}}dx^2}{a d+b c}+\frac {\sqrt {b} \left (\sqrt {b} \sqrt {c}-\sqrt {-a} \sqrt {d}\right ) \int \frac {\sqrt {d} x^4+\sqrt {c}}{\left (\sqrt {-a}-\sqrt {b} x^4\right ) \sqrt {d x^8+c}}dx^2}{a d+b c}}{2 \sqrt {b}}\right )}{b}\right )\) |
| \(\Big \downarrow \) 761 |
| \(\displaystyle \frac {1}{2} \left (\frac {\frac {\sqrt [4]{c} \left (\sqrt {c}+\sqrt {d} x^4\right ) \sqrt {\frac {c+d x^8}{\left (\sqrt {c}+\sqrt {d} x^4\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} x^2}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{2 d^{3/4} \sqrt {c+d x^8}}-\frac {\frac {\sqrt [4]{c} \left (\sqrt {c}+\sqrt {d} x^4\right ) \sqrt {\frac {c+d x^8}{\left (\sqrt {c}+\sqrt {d} x^4\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{d} x^2}{\sqrt [4]{c}}\right )|\frac {1}{2}\right )}{\sqrt [4]{d} \sqrt {c+d x^8}}-\frac {x^2 \sqrt {c+d x^8}}{\sqrt {c}+\sqrt {d} x^4}}{\sqrt {d}}}{b}-\frac {a \left (\frac {\frac {\sqrt {b} \left (\sqrt {-a} \sqrt {d}+\sqrt {b} \sqrt {c}\right ) \int \frac {\sqrt {d} x^4+\sqrt {c}}{\left (\sqrt {b} x^4+\sqrt {-a}\right ) \sqrt {d x^8+c}}dx^2}{a d+b c}-\frac {\sqrt [4]{d} \left (\sqrt {c}+\sqrt {d} x^4\right ) \sqrt {\frac {c+d x^8}{\left (\sqrt {c}+\sqrt {d} x^4\right )^2}} \left (\sqrt {-a} \sqrt {d}+\sqrt {b} \sqrt {c}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} x^2}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{c} \sqrt {c+d x^8} (a d+b c)}}{2 \sqrt {b}}-\frac {\frac {\sqrt {b} \left (\sqrt {b} \sqrt {c}-\sqrt {-a} \sqrt {d}\right ) \int \frac {\sqrt {d} x^4+\sqrt {c}}{\left (\sqrt {-a}-\sqrt {b} x^4\right ) \sqrt {d x^8+c}}dx^2}{a d+b c}+\frac {\sqrt [4]{d} \left (\sqrt {c}+\sqrt {d} x^4\right ) \sqrt {\frac {c+d x^8}{\left (\sqrt {c}+\sqrt {d} x^4\right )^2}} \left (\sqrt {b} \sqrt {c}-\sqrt {-a} \sqrt {d}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} x^2}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{c} \sqrt {c+d x^8} (a d+b c)}}{2 \sqrt {b}}\right )}{b}\right )\) |
| \(\Big \downarrow \) 2221 |
| \(\displaystyle \frac {1}{2} \left (\frac {\frac {\sqrt [4]{c} \left (\sqrt {d} x^4+\sqrt {c}\right ) \sqrt {\frac {d x^8+c}{\left (\sqrt {d} x^4+\sqrt {c}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} x^2}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{2 d^{3/4} \sqrt {d x^8+c}}-\frac {\frac {\sqrt [4]{c} \left (\sqrt {d} x^4+\sqrt {c}\right ) \sqrt {\frac {d x^8+c}{\left (\sqrt {d} x^4+\sqrt {c}\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{d} x^2}{\sqrt [4]{c}}\right )|\frac {1}{2}\right )}{\sqrt [4]{d} \sqrt {d x^8+c}}-\frac {x^2 \sqrt {d x^8+c}}{\sqrt {d} x^4+\sqrt {c}}}{\sqrt {d}}}{b}-\frac {a \left (\frac {\frac {\sqrt {b} \left (\sqrt {b} \sqrt {c}+\sqrt {-a} \sqrt {d}\right ) \left (\frac {\left (\sqrt {b} \sqrt {c}-\sqrt {-a} \sqrt {d}\right ) \arctan \left (\frac {\sqrt {b c-a d} x^2}{\sqrt [4]{-a} \sqrt [4]{b} \sqrt {d x^8+c}}\right )}{2 \sqrt [4]{-a} \sqrt [4]{b} \sqrt {b c-a d}}+\frac {\left (\frac {\sqrt {c}}{\sqrt {-a}}+\frac {\sqrt {d}}{\sqrt {b}}\right ) \left (\sqrt {d} x^4+\sqrt {c}\right ) \sqrt {\frac {d x^8+c}{\left (\sqrt {d} x^4+\sqrt {c}\right )^2}} \operatorname {EllipticPi}\left (-\frac {\sqrt {c} \left (\sqrt {b}-\frac {\sqrt {-a} \sqrt {d}}{\sqrt {c}}\right )^2}{4 \sqrt {-a} \sqrt {b} \sqrt {d}},2 \arctan \left (\frac {\sqrt [4]{d} x^2}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{c} \sqrt [4]{d} \sqrt {d x^8+c}}\right )}{b c+a d}-\frac {\left (\sqrt {b} \sqrt {c}+\sqrt {-a} \sqrt {d}\right ) \sqrt [4]{d} \left (\sqrt {d} x^4+\sqrt {c}\right ) \sqrt {\frac {d x^8+c}{\left (\sqrt {d} x^4+\sqrt {c}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} x^2}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{c} (b c+a d) \sqrt {d x^8+c}}}{2 \sqrt {b}}-\frac {\frac {\left (\sqrt {b} \sqrt {c}-\sqrt {-a} \sqrt {d}\right ) \sqrt [4]{d} \left (\sqrt {d} x^4+\sqrt {c}\right ) \sqrt {\frac {d x^8+c}{\left (\sqrt {d} x^4+\sqrt {c}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} x^2}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{c} (b c+a d) \sqrt {d x^8+c}}+\frac {\sqrt {b} \left (\sqrt {b} \sqrt {c}-\sqrt {-a} \sqrt {d}\right ) \int \frac {\sqrt {d} x^4+\sqrt {c}}{\left (\sqrt {-a}-\sqrt {b} x^4\right ) \sqrt {d x^8+c}}dx^2}{b c+a d}}{2 \sqrt {b}}\right )}{b}\right )\) |
| \(\Big \downarrow \) 2223 |
| \(\displaystyle \frac {1}{2} \left (\frac {\frac {\sqrt [4]{c} \left (\sqrt {d} x^4+\sqrt {c}\right ) \sqrt {\frac {d x^8+c}{\left (\sqrt {d} x^4+\sqrt {c}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} x^2}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{2 d^{3/4} \sqrt {d x^8+c}}-\frac {\frac {\sqrt [4]{c} \left (\sqrt {d} x^4+\sqrt {c}\right ) \sqrt {\frac {d x^8+c}{\left (\sqrt {d} x^4+\sqrt {c}\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{d} x^2}{\sqrt [4]{c}}\right )|\frac {1}{2}\right )}{\sqrt [4]{d} \sqrt {d x^8+c}}-\frac {x^2 \sqrt {d x^8+c}}{\sqrt {d} x^4+\sqrt {c}}}{\sqrt {d}}}{b}-\frac {a \left (\frac {\frac {\sqrt {b} \left (\sqrt {b} \sqrt {c}+\sqrt {-a} \sqrt {d}\right ) \left (\frac {\left (\sqrt {b} \sqrt {c}-\sqrt {-a} \sqrt {d}\right ) \arctan \left (\frac {\sqrt {b c-a d} x^2}{\sqrt [4]{-a} \sqrt [4]{b} \sqrt {d x^8+c}}\right )}{2 \sqrt [4]{-a} \sqrt [4]{b} \sqrt {b c-a d}}+\frac {\left (\frac {\sqrt {c}}{\sqrt {-a}}+\frac {\sqrt {d}}{\sqrt {b}}\right ) \left (\sqrt {d} x^4+\sqrt {c}\right ) \sqrt {\frac {d x^8+c}{\left (\sqrt {d} x^4+\sqrt {c}\right )^2}} \operatorname {EllipticPi}\left (-\frac {\sqrt {c} \left (\sqrt {b}-\frac {\sqrt {-a} \sqrt {d}}{\sqrt {c}}\right )^2}{4 \sqrt {-a} \sqrt {b} \sqrt {d}},2 \arctan \left (\frac {\sqrt [4]{d} x^2}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{c} \sqrt [4]{d} \sqrt {d x^8+c}}\right )}{b c+a d}-\frac {\left (\sqrt {b} \sqrt {c}+\sqrt {-a} \sqrt {d}\right ) \sqrt [4]{d} \left (\sqrt {d} x^4+\sqrt {c}\right ) \sqrt {\frac {d x^8+c}{\left (\sqrt {d} x^4+\sqrt {c}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} x^2}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{c} (b c+a d) \sqrt {d x^8+c}}}{2 \sqrt {b}}-\frac {\frac {\left (\sqrt {b} \sqrt {c}-\sqrt {-a} \sqrt {d}\right ) \sqrt [4]{d} \left (\sqrt {d} x^4+\sqrt {c}\right ) \sqrt {\frac {d x^8+c}{\left (\sqrt {d} x^4+\sqrt {c}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} x^2}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{c} (b c+a d) \sqrt {d x^8+c}}+\frac {\sqrt {b} \left (\sqrt {b} \sqrt {c}-\sqrt {-a} \sqrt {d}\right ) \left (\frac {\left (\sqrt {b} \sqrt {c}+\sqrt {-a} \sqrt {d}\right ) \text {arctanh}\left (\frac {\sqrt {b c-a d} x^2}{\sqrt [4]{-a} \sqrt [4]{b} \sqrt {d x^8+c}}\right )}{2 \sqrt [4]{-a} \sqrt [4]{b} \sqrt {b c-a d}}-\frac {\left (\frac {\sqrt {c} a}{(-a)^{3/2}}+\frac {\sqrt {d}}{\sqrt {b}}\right ) \left (\sqrt {d} x^4+\sqrt {c}\right ) \sqrt {\frac {d x^8+c}{\left (\sqrt {d} x^4+\sqrt {c}\right )^2}} \operatorname {EllipticPi}\left (\frac {\left (\sqrt {b} \sqrt {c}+\sqrt {-a} \sqrt {d}\right )^2}{4 \sqrt {-a} \sqrt {b} \sqrt {c} \sqrt {d}},2 \arctan \left (\frac {\sqrt [4]{d} x^2}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{c} \sqrt [4]{d} \sqrt {d x^8+c}}\right )}{b c+a d}}{2 \sqrt {b}}\right )}{b}\right )\) |
((-((-((x^2*Sqrt[c + d*x^8])/(Sqrt[c] + Sqrt[d]*x^4)) + (c^(1/4)*(Sqrt[c] + Sqrt[d]*x^4)*Sqrt[(c + d*x^8)/(Sqrt[c] + Sqrt[d]*x^4)^2]*EllipticE[2*Arc Tan[(d^(1/4)*x^2)/c^(1/4)], 1/2])/(d^(1/4)*Sqrt[c + d*x^8]))/Sqrt[d]) + (c ^(1/4)*(Sqrt[c] + Sqrt[d]*x^4)*Sqrt[(c + d*x^8)/(Sqrt[c] + Sqrt[d]*x^4)^2] *EllipticF[2*ArcTan[(d^(1/4)*x^2)/c^(1/4)], 1/2])/(2*d^(3/4)*Sqrt[c + d*x^ 8]))/b - (a*(-1/2*(((Sqrt[b]*Sqrt[c] - Sqrt[-a]*Sqrt[d])*d^(1/4)*(Sqrt[c] + Sqrt[d]*x^4)*Sqrt[(c + d*x^8)/(Sqrt[c] + Sqrt[d]*x^4)^2]*EllipticF[2*Arc Tan[(d^(1/4)*x^2)/c^(1/4)], 1/2])/(2*c^(1/4)*(b*c + a*d)*Sqrt[c + d*x^8]) + (Sqrt[b]*(Sqrt[b]*Sqrt[c] - Sqrt[-a]*Sqrt[d])*(((Sqrt[b]*Sqrt[c] + Sqrt[ -a]*Sqrt[d])*ArcTanh[(Sqrt[b*c - a*d]*x^2)/((-a)^(1/4)*b^(1/4)*Sqrt[c + d* x^8])])/(2*(-a)^(1/4)*b^(1/4)*Sqrt[b*c - a*d]) - (((a*Sqrt[c])/(-a)^(3/2) + Sqrt[d]/Sqrt[b])*(Sqrt[c] + Sqrt[d]*x^4)*Sqrt[(c + d*x^8)/(Sqrt[c] + Sqr t[d]*x^4)^2]*EllipticPi[(Sqrt[b]*Sqrt[c] + Sqrt[-a]*Sqrt[d])^2/(4*Sqrt[-a] *Sqrt[b]*Sqrt[c]*Sqrt[d]), 2*ArcTan[(d^(1/4)*x^2)/c^(1/4)], 1/2])/(4*c^(1/ 4)*d^(1/4)*Sqrt[c + d*x^8])))/(b*c + a*d))/Sqrt[b] + (-1/2*((Sqrt[b]*Sqrt[ c] + Sqrt[-a]*Sqrt[d])*d^(1/4)*(Sqrt[c] + Sqrt[d]*x^4)*Sqrt[(c + d*x^8)/(S qrt[c] + Sqrt[d]*x^4)^2]*EllipticF[2*ArcTan[(d^(1/4)*x^2)/c^(1/4)], 1/2])/ (c^(1/4)*(b*c + a*d)*Sqrt[c + d*x^8]) + (Sqrt[b]*(Sqrt[b]*Sqrt[c] + Sqrt[- a]*Sqrt[d])*(((Sqrt[b]*Sqrt[c] - Sqrt[-a]*Sqrt[d])*ArcTan[(Sqrt[b*c - a*d] *x^2)/((-a)^(1/4)*b^(1/4)*Sqrt[c + d*x^8])])/(2*(-a)^(1/4)*b^(1/4)*Sqrt...
3.9.100.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 4]}, Simp[( 1 + q^2*x^2)*(Sqrt[(a + b*x^4)/(a*(1 + q^2*x^2)^2)]/(2*q*Sqrt[a + b*x^4]))* EllipticF[2*ArcTan[q*x], 1/2], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]
Int[(x_)^2/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 2]}, S imp[1/q Int[1/Sqrt[a + b*x^4], x], x] - Simp[1/q Int[(1 - q*x^2)/Sqrt[a + b*x^4], x], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Simp[1/k Subst[Int[x^((m + 1)/k - 1)*(a + b*x^(n/k))^p*(c + d*x^(n/k))^q, x], x, x^k], x] /; k != 1] /; Free Q[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && IntegerQ[m]
Int[(((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_)^(n_))^(q_.))/((a_) + (b_.)*(x_)^( n_)), x_Symbol] :> Simp[e^n/b Int[(e*x)^(m - n)*(c + d*x^n)^q, x], x] - S imp[a*(e^n/b) Int[(e*x)^(m - n)*((c + d*x^n)^q/(a + b*x^n)), x], x] /; Fr eeQ[{a, b, c, d, e, m, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LeQ[n, m, 2*n - 1] && IntBinomialQ[a, b, c, d, e, m, n, -1, q, x]
Int[(x_)^2/(((a_) + (b_.)*(x_)^4)*Sqrt[(c_) + (d_.)*(x_)^4]), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}, Simp[s/(2* b) Int[1/((r + s*x^2)*Sqrt[c + d*x^4]), x], x] - Simp[s/(2*b) Int[1/((r - s*x^2)*Sqrt[c + d*x^4]), x], x]] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c/a, 4]}, Simp[(-d)*x*(Sqrt[a + c*x^4]/(a*(1 + q^2*x^2))), x] + Simp[d* (1 + q^2*x^2)*(Sqrt[(a + c*x^4)/(a*(1 + q^2*x^2)^2)]/(q*Sqrt[a + c*x^4]))*E llipticE[2*ArcTan[q*x], 1/2], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, c, d, e }, x] && PosQ[c/a]
Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> With[ {q = Rt[c/a, 2]}, Simp[(c*d + a*e*q)/(c*d^2 - a*e^2) Int[1/Sqrt[a + c*x^4 ], x], x] - Simp[(a*e*(e + d*q))/(c*d^2 - a*e^2) Int[(1 + q*x^2)/((d + e* x^2)*Sqrt[a + c*x^4]), x], x]] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e ^2, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[c/a]
Int[((A_) + (B_.)*(x_)^2)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]) , x_Symbol] :> With[{q = Rt[B/A, 2]}, Simp[(-(B*d - A*e))*(ArcTan[Rt[c*(d/e ) + a*(e/d), 2]*(x/Sqrt[a + c*x^4])]/(2*d*e*Rt[c*(d/e) + a*(e/d), 2])), x] + Simp[(B*d + A*e)*(1 + q^2*x^2)*(Sqrt[(a + c*x^4)/(a*(1 + q^2*x^2)^2)]/(4* d*e*q*Sqrt[a + c*x^4]))*EllipticPi[-(e - d*q^2)^2/(4*d*e*q^2), 2*ArcTan[q*x ], 1/2], x]] /; FreeQ[{a, c, d, e, A, B}, x] && NeQ[c*d^2 - a*e^2, 0] && Po sQ[c/a] && EqQ[c*A^2 - a*B^2, 0] && PosQ[B/A] && PosQ[c*(d/e) + a*(e/d)]
Int[((A_) + (B_.)*(x_)^2)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]) , x_Symbol] :> With[{q = Rt[B/A, 2]}, Simp[(-(B*d - A*e))*(ArcTanh[Rt[(-c)* (d/e) - a*(e/d), 2]*(x/Sqrt[a + c*x^4])]/(2*d*e*Rt[(-c)*(d/e) - a*(e/d), 2] )), x] + Simp[(B*d + A*e)*(1 + q^2*x^2)*(Sqrt[(a + c*x^4)/(a*(1 + q^2*x^2)^ 2)]/(4*d*e*q*Sqrt[a + c*x^4]))*EllipticPi[-(e - d*q^2)^2/(4*d*e*q^2), 2*Arc Tan[q*x], 1/2], x]] /; FreeQ[{a, c, d, e, A, B}, x] && NeQ[c*d^2 - a*e^2, 0 ] && PosQ[c/a] && EqQ[c*A^2 - a*B^2, 0] && PosQ[B/A] && NegQ[c*(d/e) + a*(e /d)]
\[\int \frac {x^{13}}{\left (b \,x^{8}+a \right ) \sqrt {d \,x^{8}+c}}d x\]
\[ \int \frac {x^{13}}{\left (a+b x^8\right ) \sqrt {c+d x^8}} \, dx=\int { \frac {x^{13}}{{\left (b x^{8} + a\right )} \sqrt {d x^{8} + c}} \,d x } \]
\[ \int \frac {x^{13}}{\left (a+b x^8\right ) \sqrt {c+d x^8}} \, dx=\int \frac {x^{13}}{\left (a + b x^{8}\right ) \sqrt {c + d x^{8}}}\, dx \]
\[ \int \frac {x^{13}}{\left (a+b x^8\right ) \sqrt {c+d x^8}} \, dx=\int { \frac {x^{13}}{{\left (b x^{8} + a\right )} \sqrt {d x^{8} + c}} \,d x } \]
\[ \int \frac {x^{13}}{\left (a+b x^8\right ) \sqrt {c+d x^8}} \, dx=\int { \frac {x^{13}}{{\left (b x^{8} + a\right )} \sqrt {d x^{8} + c}} \,d x } \]
Timed out. \[ \int \frac {x^{13}}{\left (a+b x^8\right ) \sqrt {c+d x^8}} \, dx=\int \frac {x^{13}}{\left (b\,x^8+a\right )\,\sqrt {d\,x^8+c}} \,d x \]