Integrand size = 24, antiderivative size = 786 \[ \int \frac {x^2 \sqrt {c+d x^4}}{a+b x^4} \, dx=\frac {\sqrt {d} x \sqrt {c+d x^4}}{b \left (\sqrt {c}+\sqrt {d} x^2\right )}+\frac {\sqrt {-\frac {b c-a d}{\sqrt {-a} \sqrt {b}}} \arctan \left (\frac {\sqrt {-\frac {b c-a d}{\sqrt {-a} \sqrt {b}}} x}{\sqrt {c+d x^4}}\right )}{4 b}+\frac {\sqrt {\frac {b c-a d}{\sqrt {-a} \sqrt {b}}} \arctan \left (\frac {\sqrt {\frac {b c-a d}{\sqrt {-a} \sqrt {b}}} x}{\sqrt {c+d x^4}}\right )}{4 b}-\frac {\sqrt [4]{c} \sqrt [4]{d} \left (\sqrt {c}+\sqrt {d} x^2\right ) \sqrt {\frac {c+d x^4}{\left (\sqrt {c}+\sqrt {d} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac {1}{2}\right )}{b \sqrt {c+d x^4}}+\frac {a \sqrt [4]{c} d^{5/4} \left (\sqrt {c}+\sqrt {d} x^2\right ) \sqrt {\frac {c+d x^4}{\left (\sqrt {c}+\sqrt {d} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{b (b c+a d) \sqrt {c+d x^4}}-\frac {\left (\sqrt {b} \sqrt {c}-\sqrt {-a} \sqrt {d}\right ) (b c-a d) \left (\sqrt {c}+\sqrt {d} x^2\right ) \sqrt {\frac {c+d x^4}{\left (\sqrt {c}+\sqrt {d} x^2\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}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{8 b^{3/2} \sqrt [4]{c} \left (\sqrt {-a} \sqrt {b} \sqrt {c}-a \sqrt {d}\right ) \sqrt [4]{d} \sqrt {c+d x^4}}+\frac {\left (\sqrt {b} \sqrt {c}+\sqrt {-a} \sqrt {d}\right ) (b c-a d) \left (\sqrt {c}+\sqrt {d} x^2\right ) \sqrt {\frac {c+d x^4}{\left (\sqrt {c}+\sqrt {d} x^2\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}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{8 b^{3/2} \sqrt [4]{c} \left (\sqrt {-a} \sqrt {b} \sqrt {c}+a \sqrt {d}\right ) \sqrt [4]{d} \sqrt {c+d x^4}} \]
x*d^(1/2)*(d*x^4+c)^(1/2)/b/(c^(1/2)+x^2*d^(1/2))+1/4*arctan(x*((a*d-b*c)/ (-a)^(1/2)/b^(1/2))^(1/2)/(d*x^4+c)^(1/2))*((a*d-b*c)/(-a)^(1/2)/b^(1/2))^ (1/2)/b+1/4*arctan(x*((-a*d+b*c)/(-a)^(1/2)/b^(1/2))^(1/2)/(d*x^4+c)^(1/2) )*((-a*d+b*c)/(-a)^(1/2)/b^(1/2))^(1/2)/b-c^(1/4)*d^(1/4)*(cos(2*arctan(d^ (1/4)*x/c^(1/4)))^2)^(1/2)/cos(2*arctan(d^(1/4)*x/c^(1/4)))*EllipticE(sin( 2*arctan(d^(1/4)*x/c^(1/4))),1/2*2^(1/2))*(c^(1/2)+x^2*d^(1/2))*((d*x^4+c) /(c^(1/2)+x^2*d^(1/2))^2)^(1/2)/b/(d*x^4+c)^(1/2)+a*c^(1/4)*d^(5/4)*(cos(2 *arctan(d^(1/4)*x/c^(1/4)))^2)^(1/2)/cos(2*arctan(d^(1/4)*x/c^(1/4)))*Elli pticF(sin(2*arctan(d^(1/4)*x/c^(1/4))),1/2*2^(1/2))*(c^(1/2)+x^2*d^(1/2))* ((d*x^4+c)/(c^(1/2)+x^2*d^(1/2))^2)^(1/2)/b/(a*d+b*c)/(d*x^4+c)^(1/2)-1/8* (-a*d+b*c)*(cos(2*arctan(d^(1/4)*x/c^(1/4)))^2)^(1/2)/cos(2*arctan(d^(1/4) *x/c^(1/4)))*EllipticPi(sin(2*arctan(d^(1/4)*x/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))* (c^(1/2)+x^2*d^(1/2))*(b^(1/2)*c^(1/2)-(-a)^(1/2)*d^(1/2))*((d*x^4+c)/(c^( 1/2)+x^2*d^(1/2))^2)^(1/2)/b^(3/2)/c^(1/4)/d^(1/4)/((-a)^(1/2)*b^(1/2)*c^( 1/2)-a*d^(1/2))/(d*x^4+c)^(1/2)+1/8*(-a*d+b*c)*(cos(2*arctan(d^(1/4)*x/c^( 1/4)))^2)^(1/2)/cos(2*arctan(d^(1/4)*x/c^(1/4)))*EllipticPi(sin(2*arctan(d ^(1/4)*x/c^(1/4))),-1/4*c^(1/2)*(b^(1/2)-(-a)^(1/2)*d^(1/2)/c^(1/2))^2/(-a )^(1/2)/b^(1/2)/d^(1/2),1/2*2^(1/2))*(c^(1/2)+x^2*d^(1/2))*(b^(1/2)*c^(1/2 )+(-a)^(1/2)*d^(1/2))*((d*x^4+c)/(c^(1/2)+x^2*d^(1/2))^2)^(1/2)/b^(3/2)...
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 10.04 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.08 \[ \int \frac {x^2 \sqrt {c+d x^4}}{a+b x^4} \, dx=\frac {x^3 \sqrt {c+d x^4} \operatorname {AppellF1}\left (\frac {3}{4},-\frac {1}{2},1,\frac {7}{4},-\frac {d x^4}{c},-\frac {b x^4}{a}\right )}{3 a \sqrt {\frac {c+d x^4}{c}}} \]
(x^3*Sqrt[c + d*x^4]*AppellF1[3/4, -1/2, 1, 7/4, -((d*x^4)/c), -((b*x^4)/a )])/(3*a*Sqrt[(c + d*x^4)/c])
Time = 1.59 (sec) , antiderivative size = 1096, normalized size of antiderivative = 1.39, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.458, Rules used = {994, 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.
| \(\displaystyle \int \frac {x^2 \sqrt {c+d x^4}}{a+b x^4} \, dx\) |
| \(\Big \downarrow \) 994 |
| \(\displaystyle \frac {(b c-a d) \int \frac {x^2}{\left (b x^4+a\right ) \sqrt {d x^4+c}}dx}{b}+\frac {d \int \frac {x^2}{\sqrt {d x^4+c}}dx}{b}\) |
| \(\Big \downarrow \) 834 |
| \(\displaystyle \frac {(b c-a d) \int \frac {x^2}{\left (b x^4+a\right ) \sqrt {d x^4+c}}dx}{b}+\frac {d \left (\frac {\sqrt {c} \int \frac {1}{\sqrt {d x^4+c}}dx}{\sqrt {d}}-\frac {\sqrt {c} \int \frac {\sqrt {c}-\sqrt {d} x^2}{\sqrt {c} \sqrt {d x^4+c}}dx}{\sqrt {d}}\right )}{b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(b c-a d) \int \frac {x^2}{\left (b x^4+a\right ) \sqrt {d x^4+c}}dx}{b}+\frac {d \left (\frac {\sqrt {c} \int \frac {1}{\sqrt {d x^4+c}}dx}{\sqrt {d}}-\frac {\int \frac {\sqrt {c}-\sqrt {d} x^2}{\sqrt {d x^4+c}}dx}{\sqrt {d}}\right )}{b}\) |
| \(\Big \downarrow \) 761 |
| \(\displaystyle \frac {(b c-a d) \int \frac {x^2}{\left (b x^4+a\right ) \sqrt {d x^4+c}}dx}{b}+\frac {d \left (\frac {\sqrt [4]{c} \left (\sqrt {c}+\sqrt {d} x^2\right ) \sqrt {\frac {c+d x^4}{\left (\sqrt {c}+\sqrt {d} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{2 d^{3/4} \sqrt {c+d x^4}}-\frac {\int \frac {\sqrt {c}-\sqrt {d} x^2}{\sqrt {d x^4+c}}dx}{\sqrt {d}}\right )}{b}\) |
| \(\Big \downarrow \) 993 |
| \(\displaystyle \frac {(b c-a d) \left (\frac {\int \frac {1}{\left (\sqrt {b} x^2+\sqrt {-a}\right ) \sqrt {d x^4+c}}dx}{2 \sqrt {b}}-\frac {\int \frac {1}{\left (\sqrt {-a}-\sqrt {b} x^2\right ) \sqrt {d x^4+c}}dx}{2 \sqrt {b}}\right )}{b}+\frac {d \left (\frac {\sqrt [4]{c} \left (\sqrt {c}+\sqrt {d} x^2\right ) \sqrt {\frac {c+d x^4}{\left (\sqrt {c}+\sqrt {d} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{2 d^{3/4} \sqrt {c+d x^4}}-\frac {\int \frac {\sqrt {c}-\sqrt {d} x^2}{\sqrt {d x^4+c}}dx}{\sqrt {d}}\right )}{b}\) |
| \(\Big \downarrow \) 1510 |
| \(\displaystyle \frac {(b c-a d) \left (\frac {\int \frac {1}{\left (\sqrt {b} x^2+\sqrt {-a}\right ) \sqrt {d x^4+c}}dx}{2 \sqrt {b}}-\frac {\int \frac {1}{\left (\sqrt {-a}-\sqrt {b} x^2\right ) \sqrt {d x^4+c}}dx}{2 \sqrt {b}}\right )}{b}+\frac {d \left (\frac {\sqrt [4]{c} \left (\sqrt {c}+\sqrt {d} x^2\right ) \sqrt {\frac {c+d x^4}{\left (\sqrt {c}+\sqrt {d} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{2 d^{3/4} \sqrt {c+d x^4}}-\frac {\frac {\sqrt [4]{c} \left (\sqrt {c}+\sqrt {d} x^2\right ) \sqrt {\frac {c+d x^4}{\left (\sqrt {c}+\sqrt {d} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac {1}{2}\right )}{\sqrt [4]{d} \sqrt {c+d x^4}}-\frac {x \sqrt {c+d x^4}}{\sqrt {c}+\sqrt {d} x^2}}{\sqrt {d}}\right )}{b}\) |
| \(\Big \downarrow \) 1541 |
| \(\displaystyle \frac {(b c-a d) \left (\frac {\frac {\sqrt {b} \sqrt {c} \left (\sqrt {-a} \sqrt {d}+\sqrt {b} \sqrt {c}\right ) \int \frac {\sqrt {d} x^2+\sqrt {c}}{\sqrt {c} \left (\sqrt {b} x^2+\sqrt {-a}\right ) \sqrt {d x^4+c}}dx}{a d+b c}-\frac {\sqrt {d} \left (\sqrt {-a} \sqrt {d}+\sqrt {b} \sqrt {c}\right ) \int \frac {1}{\sqrt {d x^4+c}}dx}{a d+b c}}{2 \sqrt {b}}-\frac {\frac {\sqrt {d} \left (\sqrt {b} \sqrt {c}-\sqrt {-a} \sqrt {d}\right ) \int \frac {1}{\sqrt {d x^4+c}}dx}{a d+b c}+\frac {\sqrt {b} \sqrt {c} \left (\sqrt {b} \sqrt {c}-\sqrt {-a} \sqrt {d}\right ) \int \frac {\sqrt {d} x^2+\sqrt {c}}{\sqrt {c} \left (\sqrt {-a}-\sqrt {b} x^2\right ) \sqrt {d x^4+c}}dx}{a d+b c}}{2 \sqrt {b}}\right )}{b}+\frac {d \left (\frac {\sqrt [4]{c} \left (\sqrt {c}+\sqrt {d} x^2\right ) \sqrt {\frac {c+d x^4}{\left (\sqrt {c}+\sqrt {d} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{2 d^{3/4} \sqrt {c+d x^4}}-\frac {\frac {\sqrt [4]{c} \left (\sqrt {c}+\sqrt {d} x^2\right ) \sqrt {\frac {c+d x^4}{\left (\sqrt {c}+\sqrt {d} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac {1}{2}\right )}{\sqrt [4]{d} \sqrt {c+d x^4}}-\frac {x \sqrt {c+d x^4}}{\sqrt {c}+\sqrt {d} x^2}}{\sqrt {d}}\right )}{b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(b c-a d) \left (\frac {\frac {\sqrt {b} \left (\sqrt {-a} \sqrt {d}+\sqrt {b} \sqrt {c}\right ) \int \frac {\sqrt {d} x^2+\sqrt {c}}{\left (\sqrt {b} x^2+\sqrt {-a}\right ) \sqrt {d x^4+c}}dx}{a d+b c}-\frac {\sqrt {d} \left (\sqrt {-a} \sqrt {d}+\sqrt {b} \sqrt {c}\right ) \int \frac {1}{\sqrt {d x^4+c}}dx}{a d+b c}}{2 \sqrt {b}}-\frac {\frac {\sqrt {d} \left (\sqrt {b} \sqrt {c}-\sqrt {-a} \sqrt {d}\right ) \int \frac {1}{\sqrt {d x^4+c}}dx}{a d+b c}+\frac {\sqrt {b} \left (\sqrt {b} \sqrt {c}-\sqrt {-a} \sqrt {d}\right ) \int \frac {\sqrt {d} x^2+\sqrt {c}}{\left (\sqrt {-a}-\sqrt {b} x^2\right ) \sqrt {d x^4+c}}dx}{a d+b c}}{2 \sqrt {b}}\right )}{b}+\frac {d \left (\frac {\sqrt [4]{c} \left (\sqrt {c}+\sqrt {d} x^2\right ) \sqrt {\frac {c+d x^4}{\left (\sqrt {c}+\sqrt {d} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{2 d^{3/4} \sqrt {c+d x^4}}-\frac {\frac {\sqrt [4]{c} \left (\sqrt {c}+\sqrt {d} x^2\right ) \sqrt {\frac {c+d x^4}{\left (\sqrt {c}+\sqrt {d} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac {1}{2}\right )}{\sqrt [4]{d} \sqrt {c+d x^4}}-\frac {x \sqrt {c+d x^4}}{\sqrt {c}+\sqrt {d} x^2}}{\sqrt {d}}\right )}{b}\) |
| \(\Big \downarrow \) 761 |
| \(\displaystyle \frac {(b c-a d) \left (\frac {\frac {\sqrt {b} \left (\sqrt {-a} \sqrt {d}+\sqrt {b} \sqrt {c}\right ) \int \frac {\sqrt {d} x^2+\sqrt {c}}{\left (\sqrt {b} x^2+\sqrt {-a}\right ) \sqrt {d x^4+c}}dx}{a d+b c}-\frac {\sqrt [4]{d} \left (\sqrt {c}+\sqrt {d} x^2\right ) \sqrt {\frac {c+d x^4}{\left (\sqrt {c}+\sqrt {d} x^2\right )^2}} \left (\sqrt {-a} \sqrt {d}+\sqrt {b} \sqrt {c}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{c} \sqrt {c+d x^4} (a d+b c)}}{2 \sqrt {b}}-\frac {\frac {\sqrt {b} \left (\sqrt {b} \sqrt {c}-\sqrt {-a} \sqrt {d}\right ) \int \frac {\sqrt {d} x^2+\sqrt {c}}{\left (\sqrt {-a}-\sqrt {b} x^2\right ) \sqrt {d x^4+c}}dx}{a d+b c}+\frac {\sqrt [4]{d} \left (\sqrt {c}+\sqrt {d} x^2\right ) \sqrt {\frac {c+d x^4}{\left (\sqrt {c}+\sqrt {d} x^2\right )^2}} \left (\sqrt {b} \sqrt {c}-\sqrt {-a} \sqrt {d}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{c} \sqrt {c+d x^4} (a d+b c)}}{2 \sqrt {b}}\right )}{b}+\frac {d \left (\frac {\sqrt [4]{c} \left (\sqrt {c}+\sqrt {d} x^2\right ) \sqrt {\frac {c+d x^4}{\left (\sqrt {c}+\sqrt {d} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{2 d^{3/4} \sqrt {c+d x^4}}-\frac {\frac {\sqrt [4]{c} \left (\sqrt {c}+\sqrt {d} x^2\right ) \sqrt {\frac {c+d x^4}{\left (\sqrt {c}+\sqrt {d} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac {1}{2}\right )}{\sqrt [4]{d} \sqrt {c+d x^4}}-\frac {x \sqrt {c+d x^4}}{\sqrt {c}+\sqrt {d} x^2}}{\sqrt {d}}\right )}{b}\) |
| \(\Big \downarrow \) 2221 |
| \(\displaystyle \frac {d \left (\frac {\sqrt [4]{c} \left (\sqrt {d} x^2+\sqrt {c}\right ) \sqrt {\frac {d x^4+c}{\left (\sqrt {d} x^2+\sqrt {c}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{2 d^{3/4} \sqrt {d x^4+c}}-\frac {\frac {\sqrt [4]{c} \left (\sqrt {d} x^2+\sqrt {c}\right ) \sqrt {\frac {d x^4+c}{\left (\sqrt {d} x^2+\sqrt {c}\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac {1}{2}\right )}{\sqrt [4]{d} \sqrt {d x^4+c}}-\frac {x \sqrt {d x^4+c}}{\sqrt {d} x^2+\sqrt {c}}}{\sqrt {d}}\right )}{b}+\frac {(b c-a d) \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}{\sqrt [4]{-a} \sqrt [4]{b} \sqrt {d x^4+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^2+\sqrt {c}\right ) \sqrt {\frac {d x^4+c}{\left (\sqrt {d} x^2+\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}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{c} \sqrt [4]{d} \sqrt {d x^4+c}}\right )}{b c+a d}-\frac {\left (\sqrt {b} \sqrt {c}+\sqrt {-a} \sqrt {d}\right ) \sqrt [4]{d} \left (\sqrt {d} x^2+\sqrt {c}\right ) \sqrt {\frac {d x^4+c}{\left (\sqrt {d} x^2+\sqrt {c}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{c} (b c+a d) \sqrt {d x^4+c}}}{2 \sqrt {b}}-\frac {\frac {\left (\sqrt {b} \sqrt {c}-\sqrt {-a} \sqrt {d}\right ) \sqrt [4]{d} \left (\sqrt {d} x^2+\sqrt {c}\right ) \sqrt {\frac {d x^4+c}{\left (\sqrt {d} x^2+\sqrt {c}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{c} (b c+a d) \sqrt {d x^4+c}}+\frac {\sqrt {b} \left (\sqrt {b} \sqrt {c}-\sqrt {-a} \sqrt {d}\right ) \int \frac {\sqrt {d} x^2+\sqrt {c}}{\left (\sqrt {-a}-\sqrt {b} x^2\right ) \sqrt {d x^4+c}}dx}{b c+a d}}{2 \sqrt {b}}\right )}{b}\) |
| \(\Big \downarrow \) 2223 |
| \(\displaystyle \frac {d \left (\frac {\sqrt [4]{c} \left (\sqrt {d} x^2+\sqrt {c}\right ) \sqrt {\frac {d x^4+c}{\left (\sqrt {d} x^2+\sqrt {c}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{2 d^{3/4} \sqrt {d x^4+c}}-\frac {\frac {\sqrt [4]{c} \left (\sqrt {d} x^2+\sqrt {c}\right ) \sqrt {\frac {d x^4+c}{\left (\sqrt {d} x^2+\sqrt {c}\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac {1}{2}\right )}{\sqrt [4]{d} \sqrt {d x^4+c}}-\frac {x \sqrt {d x^4+c}}{\sqrt {d} x^2+\sqrt {c}}}{\sqrt {d}}\right )}{b}+\frac {(b c-a d) \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}{\sqrt [4]{-a} \sqrt [4]{b} \sqrt {d x^4+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^2+\sqrt {c}\right ) \sqrt {\frac {d x^4+c}{\left (\sqrt {d} x^2+\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}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{c} \sqrt [4]{d} \sqrt {d x^4+c}}\right )}{b c+a d}-\frac {\left (\sqrt {b} \sqrt {c}+\sqrt {-a} \sqrt {d}\right ) \sqrt [4]{d} \left (\sqrt {d} x^2+\sqrt {c}\right ) \sqrt {\frac {d x^4+c}{\left (\sqrt {d} x^2+\sqrt {c}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{c} (b c+a d) \sqrt {d x^4+c}}}{2 \sqrt {b}}-\frac {\frac {\left (\sqrt {b} \sqrt {c}-\sqrt {-a} \sqrt {d}\right ) \sqrt [4]{d} \left (\sqrt {d} x^2+\sqrt {c}\right ) \sqrt {\frac {d x^4+c}{\left (\sqrt {d} x^2+\sqrt {c}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{c} (b c+a d) \sqrt {d x^4+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}{\sqrt [4]{-a} \sqrt [4]{b} \sqrt {d x^4+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^2+\sqrt {c}\right ) \sqrt {\frac {d x^4+c}{\left (\sqrt {d} x^2+\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}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{c} \sqrt [4]{d} \sqrt {d x^4+c}}\right )}{b c+a d}}{2 \sqrt {b}}\right )}{b}\) |
(d*(-((-((x*Sqrt[c + d*x^4])/(Sqrt[c] + Sqrt[d]*x^2)) + (c^(1/4)*(Sqrt[c] + Sqrt[d]*x^2)*Sqrt[(c + d*x^4)/(Sqrt[c] + Sqrt[d]*x^2)^2]*EllipticE[2*Arc Tan[(d^(1/4)*x)/c^(1/4)], 1/2])/(d^(1/4)*Sqrt[c + d*x^4]))/Sqrt[d]) + (c^( 1/4)*(Sqrt[c] + Sqrt[d]*x^2)*Sqrt[(c + d*x^4)/(Sqrt[c] + Sqrt[d]*x^2)^2]*E llipticF[2*ArcTan[(d^(1/4)*x)/c^(1/4)], 1/2])/(2*d^(3/4)*Sqrt[c + d*x^4])) )/b + ((b*c - a*d)*(-1/2*(((Sqrt[b]*Sqrt[c] - Sqrt[-a]*Sqrt[d])*d^(1/4)*(S qrt[c] + Sqrt[d]*x^2)*Sqrt[(c + d*x^4)/(Sqrt[c] + Sqrt[d]*x^2)^2]*Elliptic F[2*ArcTan[(d^(1/4)*x)/c^(1/4)], 1/2])/(2*c^(1/4)*(b*c + a*d)*Sqrt[c + d*x ^4]) + (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)/((-a)^(1/4)*b^(1/4)*Sqrt[c + d*x^4])])/(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^2)*Sqrt[(c + d*x^4)/(Sqrt[c] + Sqrt[d]*x^2)^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)/c^(1/4)], 1/2])/(4*c^(1 /4)*d^(1/4)*Sqrt[c + d*x^4])))/(b*c + a*d))/Sqrt[b] + (-1/2*((Sqrt[b]*Sqrt [c] + Sqrt[-a]*Sqrt[d])*d^(1/4)*(Sqrt[c] + Sqrt[d]*x^2)*Sqrt[(c + d*x^4)/( Sqrt[c] + Sqrt[d]*x^2)^2]*EllipticF[2*ArcTan[(d^(1/4)*x)/c^(1/4)], 1/2])/( c^(1/4)*(b*c + a*d)*Sqrt[c + d*x^4]) + (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)/((-a)^(1/4)*b^(1/4)*Sqrt[c + d*x^4])])/(2*(-a)^(1/4)*b^(1/4)*Sqrt[b*...
3.8.97.3.1 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_)^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[((x_)^2*Sqrt[(c_) + (d_.)*(x_)^4])/((a_) + (b_.)*(x_)^4), x_Symbol] :> Simp[d/b Int[x^2/Sqrt[c + d*x^4], x], x] + Simp[(b*c - a*d)/b Int[x^2/( (a + b*x^4)*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)]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 4.68 (sec) , antiderivative size = 299, normalized size of antiderivative = 0.38
| method | result | size |
| default | \(\frac {i \sqrt {d}\, \sqrt {c}\, \sqrt {1-\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \sqrt {1+\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \left (F\left (x \sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}, i\right )-E\left (x \sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}, i\right )\right )}{b \sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}\, \sqrt {d \,x^{4}+c}}-\frac {\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (\textit {\_Z}^{4} b +a \right )}{\sum }\frac {\left (a d -b c \right ) \left (-\frac {\operatorname {arctanh}\left (\frac {2 d \,x^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}+2 c}{2 \sqrt {\frac {-a d +b c}{b}}\, \sqrt {d \,x^{4}+c}}\right )}{\sqrt {\frac {-a d +b c}{b}}}+\frac {2 \underline {\hspace {1.25 ex}}\alpha ^{3} b \sqrt {1-\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \sqrt {1+\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \Pi \left (x \sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}, \frac {i \sqrt {c}\, \underline {\hspace {1.25 ex}}\alpha ^{2} b}{\sqrt {d}\, a}, \frac {\sqrt {-\frac {i \sqrt {d}}{\sqrt {c}}}}{\sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}}\right )}{\sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}\, a \sqrt {d \,x^{4}+c}}\right )}{\underline {\hspace {1.25 ex}}\alpha }}{8 b^{2}}\) | \(299\) |
| elliptic | \(\frac {i \sqrt {d}\, \sqrt {c}\, \sqrt {1-\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \sqrt {1+\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \left (F\left (x \sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}, i\right )-E\left (x \sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}, i\right )\right )}{b \sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}\, \sqrt {d \,x^{4}+c}}-\frac {\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (\textit {\_Z}^{4} b +a \right )}{\sum }\frac {\left (a d -b c \right ) \left (-\frac {\operatorname {arctanh}\left (\frac {2 d \,x^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}+2 c}{2 \sqrt {\frac {-a d +b c}{b}}\, \sqrt {d \,x^{4}+c}}\right )}{\sqrt {\frac {-a d +b c}{b}}}+\frac {2 \underline {\hspace {1.25 ex}}\alpha ^{3} b \sqrt {1-\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \sqrt {1+\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \Pi \left (x \sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}, \frac {i \sqrt {c}\, \underline {\hspace {1.25 ex}}\alpha ^{2} b}{\sqrt {d}\, a}, \frac {\sqrt {-\frac {i \sqrt {d}}{\sqrt {c}}}}{\sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}}\right )}{\sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}\, a \sqrt {d \,x^{4}+c}}\right )}{\underline {\hspace {1.25 ex}}\alpha }}{8 b^{2}}\) | \(299\) |
I*d^(1/2)/b*c^(1/2)/(I/c^(1/2)*d^(1/2))^(1/2)*(1-I/c^(1/2)*d^(1/2)*x^2)^(1 /2)*(1+I/c^(1/2)*d^(1/2)*x^2)^(1/2)/(d*x^4+c)^(1/2)*(EllipticF(x*(I/c^(1/2 )*d^(1/2))^(1/2),I)-EllipticE(x*(I/c^(1/2)*d^(1/2))^(1/2),I))-1/8/b^2*sum( (a*d-b*c)/_alpha*(-1/((-a*d+b*c)/b)^(1/2)*arctanh(1/2*(2*_alpha^2*d*x^2+2* c)/((-a*d+b*c)/b)^(1/2)/(d*x^4+c)^(1/2))+2/(I/c^(1/2)*d^(1/2))^(1/2)*_alph a^3*b/a*(1-I/c^(1/2)*d^(1/2)*x^2)^(1/2)*(1+I/c^(1/2)*d^(1/2)*x^2)^(1/2)/(d *x^4+c)^(1/2)*EllipticPi(x*(I/c^(1/2)*d^(1/2))^(1/2),I*c^(1/2)/d^(1/2)*_al pha^2/a*b,(-I/c^(1/2)*d^(1/2))^(1/2)/(I/c^(1/2)*d^(1/2))^(1/2))),_alpha=Ro otOf(_Z^4*b+a))
Timed out. \[ \int \frac {x^2 \sqrt {c+d x^4}}{a+b x^4} \, dx=\text {Timed out} \]
\[ \int \frac {x^2 \sqrt {c+d x^4}}{a+b x^4} \, dx=\int \frac {x^{2} \sqrt {c + d x^{4}}}{a + b x^{4}}\, dx \]
\[ \int \frac {x^2 \sqrt {c+d x^4}}{a+b x^4} \, dx=\int { \frac {\sqrt {d x^{4} + c} x^{2}}{b x^{4} + a} \,d x } \]
\[ \int \frac {x^2 \sqrt {c+d x^4}}{a+b x^4} \, dx=\int { \frac {\sqrt {d x^{4} + c} x^{2}}{b x^{4} + a} \,d x } \]
Timed out. \[ \int \frac {x^2 \sqrt {c+d x^4}}{a+b x^4} \, dx=\int \frac {x^2\,\sqrt {d\,x^4+c}}{b\,x^4+a} \,d x \]