Integrand size = 24, antiderivative size = 671 \[ \int \frac {x^4 \sqrt {c+d x^4}}{a+b x^4} \, dx=\frac {x \sqrt {c+d x^4}}{3 b}+\frac {\sqrt [4]{-a} \sqrt {-b c+a d} \arctan \left (\frac {\sqrt {-b c+a d} x}{\sqrt [4]{-a} \sqrt [4]{b} \sqrt {c+d x^4}}\right )}{4 b^{7/4}}+\frac {\sqrt [4]{-a} \sqrt {-b c+a d} \text {arctanh}\left (\frac {\sqrt {-b c+a d} x}{\sqrt [4]{-a} \sqrt [4]{b} \sqrt {c+d x^4}}\right )}{4 b^{7/4}}+\frac {c^{3/4} (b c-2 a d) \left (\sqrt {c}+\sqrt {d} x^2\right ) \sqrt {\frac {c+d x^4}{\left (\sqrt {c}+\sqrt {d} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{3 b \sqrt [4]{d} (b c+a d) \sqrt {c+d x^4}}-\frac {\left (\sqrt {b} \sqrt {c}+\sqrt {-a} \sqrt {d}\right ) (b c-a d) \left (\sqrt {c}+\sqrt {d} x^2\right ) \sqrt {\frac {c+d x^4}{\left (\sqrt {c}+\sqrt {d} x^2\right )^2}} \operatorname {EllipticPi}\left (-\frac {\left (\sqrt {b} \sqrt {c}-\sqrt {-a} \sqrt {d}\right )^2}{4 \sqrt {-a} \sqrt {b} \sqrt {c} \sqrt {d}},2 \arctan \left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{8 b^2 \sqrt [4]{c} \left (\sqrt {b} \sqrt {c}-\sqrt {-a} \sqrt {d}\right ) \sqrt [4]{d} \sqrt {c+d x^4}}-\frac {\left (\sqrt {b} \sqrt {c}-\sqrt {-a} \sqrt {d}\right ) (b c-a d) \left (\sqrt {c}+\sqrt {d} x^2\right ) \sqrt {\frac {c+d x^4}{\left (\sqrt {c}+\sqrt {d} x^2\right )^2}} \operatorname {EllipticPi}\left (\frac {\left (\sqrt {b} \sqrt {c}+\sqrt {-a} \sqrt {d}\right )^2}{4 \sqrt {-a} \sqrt {b} \sqrt {c} \sqrt {d}},2 \arctan \left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{8 b^2 \sqrt [4]{c} \left (\sqrt {b} \sqrt {c}+\sqrt {-a} \sqrt {d}\right ) \sqrt [4]{d} \sqrt {c+d x^4}} \] Output:
1/3*x*(d*x^4+c)^(1/2)/b+1/4*(-a)^(1/4)*(a*d-b*c)^(1/2)*arctan((a*d-b*c)^(1 /2)*x/(-a)^(1/4)/b^(1/4)/(d*x^4+c)^(1/2))/b^(7/4)+1/4*(-a)^(1/4)*(a*d-b*c) ^(1/2)*arctanh((a*d-b*c)^(1/2)*x/(-a)^(1/4)/b^(1/4)/(d*x^4+c)^(1/2))/b^(7/ 4)+1/3*c^(3/4)*(-2*a*d+b*c)*(c^(1/2)+d^(1/2)*x^2)*((d*x^4+c)/(c^(1/2)+d^(1 /2)*x^2)^2)^(1/2)*InverseJacobiAM(2*arctan(d^(1/4)*x/c^(1/4)),1/2*2^(1/2)) /b/d^(1/4)/(a*d+b*c)/(d*x^4+c)^(1/2)-1/8*(b^(1/2)*c^(1/2)+(-a)^(1/2)*d^(1/ 2))*(-a*d+b*c)*(c^(1/2)+d^(1/2)*x^2)*((d*x^4+c)/(c^(1/2)+d^(1/2)*x^2)^2)^( 1/2)*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))/b^2/c^( 1/4)/(b^(1/2)*c^(1/2)-(-a)^(1/2)*d^(1/2))/d^(1/4)/(d*x^4+c)^(1/2)-1/8*(b^( 1/2)*c^(1/2)-(-a)^(1/2)*d^(1/2))*(-a*d+b*c)*(c^(1/2)+d^(1/2)*x^2)*((d*x^4+ c)/(c^(1/2)+d^(1/2)*x^2)^2)^(1/2)*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))/b^2/c^(1/4)/(b^(1/2)*c^(1/2)+(-a)^(1/2)*d^(1/2))/d^(1 /4)/(d*x^4+c)^(1/2)
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 10.33 (sec) , antiderivative size = 241, normalized size of antiderivative = 0.36 \[ \int \frac {x^4 \sqrt {c+d x^4}}{a+b x^4} \, dx=\frac {x \left (\frac {(2 b c-3 a d) x^4 \sqrt {1+\frac {d x^4}{c}} \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},1,\frac {9}{4},-\frac {d x^4}{c},-\frac {b x^4}{a}\right )}{a}+5 \left (c+d x^4+\frac {5 a^2 c^2 \operatorname {AppellF1}\left (\frac {1}{4},\frac {1}{2},1,\frac {5}{4},-\frac {d x^4}{c},-\frac {b x^4}{a}\right )}{\left (a+b x^4\right ) \left (-5 a c \operatorname {AppellF1}\left (\frac {1}{4},\frac {1}{2},1,\frac {5}{4},-\frac {d x^4}{c},-\frac {b x^4}{a}\right )+2 x^4 \left (2 b c \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},2,\frac {9}{4},-\frac {d x^4}{c},-\frac {b x^4}{a}\right )+a d \operatorname {AppellF1}\left (\frac {5}{4},\frac {3}{2},1,\frac {9}{4},-\frac {d x^4}{c},-\frac {b x^4}{a}\right )\right )\right )}\right )\right )}{15 b \sqrt {c+d x^4}} \] Input:
Integrate[(x^4*Sqrt[c + d*x^4])/(a + b*x^4),x]
Output:
(x*(((2*b*c - 3*a*d)*x^4*Sqrt[1 + (d*x^4)/c]*AppellF1[5/4, 1/2, 1, 9/4, -( (d*x^4)/c), -((b*x^4)/a)])/a + 5*(c + d*x^4 + (5*a^2*c^2*AppellF1[1/4, 1/2 , 1, 5/4, -((d*x^4)/c), -((b*x^4)/a)])/((a + b*x^4)*(-5*a*c*AppellF1[1/4, 1/2, 1, 5/4, -((d*x^4)/c), -((b*x^4)/a)] + 2*x^4*(2*b*c*AppellF1[5/4, 1/2, 2, 9/4, -((d*x^4)/c), -((b*x^4)/a)] + a*d*AppellF1[5/4, 3/2, 1, 9/4, -((d *x^4)/c), -((b*x^4)/a)]))))))/(15*b*Sqrt[c + d*x^4])
Time = 2.64 (sec) , antiderivative size = 1002, normalized size of antiderivative = 1.49, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {978, 1021, 761, 925, 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.
| \(\displaystyle \int \frac {x^4 \sqrt {c+d x^4}}{a+b x^4} \, dx\) |
| \(\Big \downarrow \) 978 |
| \(\displaystyle \frac {x \sqrt {c+d x^4}}{3 b}-\frac {\int \frac {a c-(2 b c-3 a d) x^4}{\left (b x^4+a\right ) \sqrt {d x^4+c}}dx}{3 b}\) |
| \(\Big \downarrow \) 1021 |
| \(\displaystyle \frac {x \sqrt {c+d x^4}}{3 b}-\frac {\frac {3 a (b c-a d) \int \frac {1}{\left (b x^4+a\right ) \sqrt {d x^4+c}}dx}{b}-\frac {(2 b c-3 a d) \int \frac {1}{\sqrt {d x^4+c}}dx}{b}}{3 b}\) |
| \(\Big \downarrow \) 761 |
| \(\displaystyle \frac {x \sqrt {c+d x^4}}{3 b}-\frac {\frac {3 a (b c-a d) \int \frac {1}{\left (b x^4+a\right ) \sqrt {d x^4+c}}dx}{b}-\frac {\left (\sqrt {c}+\sqrt {d} x^2\right ) \sqrt {\frac {c+d x^4}{\left (\sqrt {c}+\sqrt {d} x^2\right )^2}} (2 b c-3 a d) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{2 b \sqrt [4]{c} \sqrt [4]{d} \sqrt {c+d x^4}}}{3 b}\) |
| \(\Big \downarrow \) 925 |
| \(\displaystyle \frac {x \sqrt {c+d x^4}}{3 b}-\frac {\frac {3 a (b c-a d) \left (\frac {\int \frac {1}{\left (1-\frac {\sqrt {b} x^2}{\sqrt {-a}}\right ) \sqrt {d x^4+c}}dx}{2 a}+\frac {\int \frac {1}{\left (\frac {\sqrt {b} x^2}{\sqrt {-a}}+1\right ) \sqrt {d x^4+c}}dx}{2 a}\right )}{b}-\frac {\left (\sqrt {c}+\sqrt {d} x^2\right ) \sqrt {\frac {c+d x^4}{\left (\sqrt {c}+\sqrt {d} x^2\right )^2}} (2 b c-3 a d) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{2 b \sqrt [4]{c} \sqrt [4]{d} \sqrt {c+d x^4}}}{3 b}\) |
| \(\Big \downarrow \) 1541 |
| \(\displaystyle \frac {x \sqrt {c+d x^4}}{3 b}-\frac {\frac {3 a (b c-a d) \left (\frac {\frac {\sqrt {d} \left (\sqrt {-a} \sqrt {b} \sqrt {c}+a \sqrt {d}\right ) \int \frac {1}{\sqrt {d x^4+c}}dx}{a d+b c}+\frac {\sqrt {b} \sqrt {c} \left (\sqrt {b} \sqrt {c}-\sqrt {-a} \sqrt {d}\right ) \int \frac {\sqrt {d} x^2+\sqrt {c}}{\sqrt {c} \left (1-\frac {\sqrt {b} x^2}{\sqrt {-a}}\right ) \sqrt {d x^4+c}}dx}{a d+b c}}{2 a}+\frac {\frac {a \sqrt {d} \left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {-a}}+\sqrt {d}\right ) \int \frac {1}{\sqrt {d x^4+c}}dx}{a d+b c}+\frac {\sqrt {b} \sqrt {c} \left (\sqrt {-a} \sqrt {d}+\sqrt {b} \sqrt {c}\right ) \int \frac {\sqrt {d} x^2+\sqrt {c}}{\sqrt {c} \left (\frac {\sqrt {b} x^2}{\sqrt {-a}}+1\right ) \sqrt {d x^4+c}}dx}{a d+b c}}{2 a}\right )}{b}-\frac {\left (\sqrt {c}+\sqrt {d} x^2\right ) \sqrt {\frac {c+d x^4}{\left (\sqrt {c}+\sqrt {d} x^2\right )^2}} (2 b c-3 a d) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{2 b \sqrt [4]{c} \sqrt [4]{d} \sqrt {c+d x^4}}}{3 b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x \sqrt {c+d x^4}}{3 b}-\frac {\frac {3 a (b c-a d) \left (\frac {\frac {\sqrt {d} \left (\sqrt {-a} \sqrt {b} \sqrt {c}+a \sqrt {d}\right ) \int \frac {1}{\sqrt {d x^4+c}}dx}{a d+b c}+\frac {\sqrt {b} \left (\sqrt {b} \sqrt {c}-\sqrt {-a} \sqrt {d}\right ) \int \frac {\sqrt {d} x^2+\sqrt {c}}{\left (1-\frac {\sqrt {b} x^2}{\sqrt {-a}}\right ) \sqrt {d x^4+c}}dx}{a d+b c}}{2 a}+\frac {\frac {a \sqrt {d} \left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {-a}}+\sqrt {d}\right ) \int \frac {1}{\sqrt {d x^4+c}}dx}{a d+b c}+\frac {\sqrt {b} \left (\sqrt {-a} \sqrt {d}+\sqrt {b} \sqrt {c}\right ) \int \frac {\sqrt {d} x^2+\sqrt {c}}{\left (\frac {\sqrt {b} x^2}{\sqrt {-a}}+1\right ) \sqrt {d x^4+c}}dx}{a d+b c}}{2 a}\right )}{b}-\frac {\left (\sqrt {c}+\sqrt {d} x^2\right ) \sqrt {\frac {c+d x^4}{\left (\sqrt {c}+\sqrt {d} x^2\right )^2}} (2 b c-3 a d) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{2 b \sqrt [4]{c} \sqrt [4]{d} \sqrt {c+d x^4}}}{3 b}\) |
| \(\Big \downarrow \) 761 |
| \(\displaystyle \frac {x \sqrt {c+d x^4}}{3 b}-\frac {\frac {3 a (b c-a d) \left (\frac {\frac {\sqrt {b} \left (\sqrt {b} \sqrt {c}-\sqrt {-a} \sqrt {d}\right ) \int \frac {\sqrt {d} x^2+\sqrt {c}}{\left (1-\frac {\sqrt {b} x^2}{\sqrt {-a}}\right ) \sqrt {d x^4+c}}dx}{a d+b c}+\frac {\sqrt [4]{d} \left (\sqrt {c}+\sqrt {d} x^2\right ) \sqrt {\frac {c+d x^4}{\left (\sqrt {c}+\sqrt {d} x^2\right )^2}} \left (\sqrt {-a} \sqrt {b} \sqrt {c}+a \sqrt {d}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{c} \sqrt {c+d x^4} (a d+b c)}}{2 a}+\frac {\frac {\sqrt {b} \left (\sqrt {-a} \sqrt {d}+\sqrt {b} \sqrt {c}\right ) \int \frac {\sqrt {d} x^2+\sqrt {c}}{\left (\frac {\sqrt {b} x^2}{\sqrt {-a}}+1\right ) \sqrt {d x^4+c}}dx}{a d+b c}+\frac {a \sqrt [4]{d} \left (\sqrt {c}+\sqrt {d} x^2\right ) \sqrt {\frac {c+d x^4}{\left (\sqrt {c}+\sqrt {d} x^2\right )^2}} \left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {-a}}+\sqrt {d}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{c} \sqrt {c+d x^4} (a d+b c)}}{2 a}\right )}{b}-\frac {\left (\sqrt {c}+\sqrt {d} x^2\right ) \sqrt {\frac {c+d x^4}{\left (\sqrt {c}+\sqrt {d} x^2\right )^2}} (2 b c-3 a d) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{2 b \sqrt [4]{c} \sqrt [4]{d} \sqrt {c+d x^4}}}{3 b}\) |
| \(\Big \downarrow \) 2221 |
| \(\displaystyle \frac {x \sqrt {d x^4+c}}{3 b}-\frac {\frac {3 a (b c-a d) \left (\frac {\frac {a \left (\frac {\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 {(-a)^{3/4} \left (\frac {\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]{b} \sqrt {b c-a d}}+\frac {\left (\sqrt {c}+\frac {\sqrt {-a} \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 a}+\frac {\frac {\left (\sqrt {d} a+\sqrt {-a} \sqrt {b} \sqrt {c}\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 (1-\frac {\sqrt {b} x^2}{\sqrt {-a}}\right ) \sqrt {d x^4+c}}dx}{b c+a d}}{2 a}\right )}{b}-\frac {(2 b c-3 a 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 b \sqrt [4]{c} \sqrt [4]{d} \sqrt {d x^4+c}}}{3 b}\) |
| \(\Big \downarrow \) 2223 |
| \(\displaystyle \frac {x \sqrt {d x^4+c}}{3 b}-\frac {\frac {3 a (b c-a d) \left (\frac {\frac {a \left (\frac {\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 {(-a)^{3/4} \left (\frac {\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]{b} \sqrt {b c-a d}}+\frac {\left (\sqrt {c}+\frac {\sqrt {-a} \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 a}+\frac {\frac {\left (\sqrt {d} a+\sqrt {-a} \sqrt {b} \sqrt {c}\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 {\sqrt [4]{-a} \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]{b} \sqrt {b c-a d}}+\frac {\left (\sqrt {c}-\frac {\sqrt {-a} \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 a}\right )}{b}-\frac {(2 b c-3 a 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 b \sqrt [4]{c} \sqrt [4]{d} \sqrt {d x^4+c}}}{3 b}\) |
Input:
Int[(x^4*Sqrt[c + d*x^4])/(a + b*x^4),x]
Output:
(x*Sqrt[c + d*x^4])/(3*b) - (-1/2*((2*b*c - 3*a*d)*(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])/(b*c^(1/4)*d^(1/4)*Sqrt[c + d*x^4]) + (3*a*(b*c - a*d)*( ((a*((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])/(2*c^(1/4)*(b*c + a*d)*Sqrt[c + d*x^4]) + (Sqrt[b]*(Sqrt [b]*Sqrt[c] + Sqrt[-a]*Sqrt[d])*(((-a)^(3/4)*((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*b^(1/4)*Sqrt[b*c - a*d]) + ((Sqrt[c] + (Sqrt[-a]*Sqrt[d])/Sqrt[b])*(S qrt[c] + Sqrt[d]*x^2)*Sqrt[(c + d*x^4)/(Sqrt[c] + Sqrt[d]*x^2)^2]*Elliptic Pi[-1/4*(Sqrt[b]*Sqrt[c] - Sqrt[-a]*Sqrt[d])^2/(Sqrt[-a]*Sqrt[b]*Sqrt[c]*S qrt[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))/(2*a) + (((Sqrt[-a]*Sqrt[b]*Sqrt[c] + 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])/(2*c^(1/4)*(b*c + a*d)*Sqrt [c + d*x^4]) + (Sqrt[b]*(Sqrt[b]*Sqrt[c] - Sqrt[-a]*Sqrt[d])*(((-a)^(1/4)* (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*b^(1/4)*Sqrt[b*c - a*d]) + ((Sqrt[c] - (S qrt[-a]*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/(...
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[1/(Sqrt[(a_) + (b_.)*(x_)^4]*((c_) + (d_.)*(x_)^4)), x_Symbol] :> Simp[ 1/(2*c) Int[1/(Sqrt[a + b*x^4]*(1 - Rt[-d/c, 2]*x^2)), x], x] + Simp[1/(2 *c) Int[1/(Sqrt[a + b*x^4]*(1 + Rt[-d/c, 2]*x^2)), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_ ))^(q_), x_Symbol] :> Simp[e^(n - 1)*(e*x)^(m - n + 1)*(a + b*x^n)^(p + 1)* ((c + d*x^n)^q/(b*(m + n*(p + q) + 1))), x] - Simp[e^n/(b*(m + n*(p + q) + 1)) Int[(e*x)^(m - n)*(a + b*x^n)^p*(c + d*x^n)^(q - 1)*Simp[a*c*(m - n + 1) + (a*d*(m - n + 1) - n*q*(b*c - a*d))*x^n, x], x], x] /; FreeQ[{a, b, c , d, e, p}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && GtQ[q, 0] && GtQ[m - n + 1, 0] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]
Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*Sqrt[(c_) + (d_.)*(x _)^(n_)]), x_Symbol] :> Simp[f/b Int[1/Sqrt[c + d*x^n], x], x] + Simp[(b* e - a*f)/b Int[1/((a + b*x^n)*Sqrt[c + d*x^n]), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x]
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 complex when optimal does not.
Time = 4.01 (sec) , antiderivative size = 303, normalized size of antiderivative = 0.45
| method | result | size |
| risch | \(\frac {x \sqrt {d \,x^{4}+c}}{3 b}-\frac {\frac {\left (3 a d -2 c b \right ) \sqrt {1-\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \sqrt {1+\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}, i\right )}{b \sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}\, \sqrt {d \,x^{4}+c}}-\frac {3 a \left (a d -c b \right ) \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (\textit {\_Z}^{4} b +a \right )}{\sum }\frac {-\frac {\operatorname {arctanh}\left (\frac {2 d \,x^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}+2 c}{2 \sqrt {\frac {-a d +c b}{b}}\, \sqrt {d \,x^{4}+c}}\right )}{\sqrt {\frac {-a d +c b}{b}}}+\frac {2 \underline {\hspace {1.25 ex}}\alpha ^{3} b \sqrt {1-\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \sqrt {1+\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}, \frac {i \sqrt {c}\, \underline {\hspace {1.25 ex}}\alpha ^{2} b}{\sqrt {d}\, a}, \frac {\sqrt {-\frac {i \sqrt {d}}{\sqrt {c}}}}{\sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}}\right )}{\sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}\, a \sqrt {d \,x^{4}+c}}}{\underline {\hspace {1.25 ex}}\alpha ^{3}}\right )}{8 b^{2}}}{3 b}\) | \(303\) |
| elliptic | \(\frac {x \sqrt {d \,x^{4}+c}}{3 b}+\frac {\left (-\frac {a d -c b}{b^{2}}-\frac {c}{3 b}\right ) \sqrt {1-\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \sqrt {1+\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}, i\right )}{\sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}\, \sqrt {d \,x^{4}+c}}+\frac {a \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (\textit {\_Z}^{4} b +a \right )}{\sum }\frac {\left (a d -c b \right ) \left (-\frac {\operatorname {arctanh}\left (\frac {2 d \,x^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}+2 c}{2 \sqrt {\frac {-a d +c b}{b}}\, \sqrt {d \,x^{4}+c}}\right )}{\sqrt {\frac {-a d +c b}{b}}}+\frac {2 \underline {\hspace {1.25 ex}}\alpha ^{3} b \sqrt {1-\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \sqrt {1+\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}, \frac {i \sqrt {c}\, \underline {\hspace {1.25 ex}}\alpha ^{2} b}{\sqrt {d}\, a}, \frac {\sqrt {-\frac {i \sqrt {d}}{\sqrt {c}}}}{\sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}}\right )}{\sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}\, a \sqrt {d \,x^{4}+c}}\right )}{\underline {\hspace {1.25 ex}}\alpha ^{3}}\right )}{8 b^{3}}\) | \(305\) |
| default | \(\frac {\frac {x \sqrt {d \,x^{4}+c}}{3}+\frac {2 c \sqrt {1-\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \sqrt {1+\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}, i\right )}{3 \sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}\, \sqrt {d \,x^{4}+c}}}{b}-\frac {a \left (\frac {d \sqrt {1-\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \sqrt {1+\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}, i\right )}{b \sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}\, \sqrt {d \,x^{4}+c}}-\frac {\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (\textit {\_Z}^{4} b +a \right )}{\sum }\frac {\left (a d -c b \right ) \left (-\frac {\operatorname {arctanh}\left (\frac {2 d \,x^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}+2 c}{2 \sqrt {\frac {-a d +c b}{b}}\, \sqrt {d \,x^{4}+c}}\right )}{\sqrt {\frac {-a d +c b}{b}}}+\frac {2 \underline {\hspace {1.25 ex}}\alpha ^{3} b \sqrt {1-\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \sqrt {1+\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}, \frac {i \sqrt {c}\, \underline {\hspace {1.25 ex}}\alpha ^{2} b}{\sqrt {d}\, a}, \frac {\sqrt {-\frac {i \sqrt {d}}{\sqrt {c}}}}{\sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}}\right )}{\sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}\, a \sqrt {d \,x^{4}+c}}\right )}{\underline {\hspace {1.25 ex}}\alpha ^{3}}}{8 b^{2}}\right )}{b}\) | \(368\) |
Input:
int(x^4*(d*x^4+c)^(1/2)/(b*x^4+a),x,method=_RETURNVERBOSE)
Output:
1/3*x*(d*x^4+c)^(1/2)/b-1/3/b*((3*a*d-2*b*c)/b/(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)-3/8*a*(a*d-b*c)/b^2*sum(1/_a lpha^3*(-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)*_alpha^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)*_alpha^2/a*b, (-I/c^(1/2)*d^(1/2))^(1/2)/(I/c^(1/2)*d^(1/2))^(1/2))),_alpha=RootOf(_Z^4* b+a)))
\[ \int \frac {x^4 \sqrt {c+d x^4}}{a+b x^4} \, dx=\int { \frac {\sqrt {d x^{4} + c} x^{4}}{b x^{4} + a} \,d x } \] Input:
integrate(x^4*(d*x^4+c)^(1/2)/(b*x^4+a),x, algorithm="fricas")
Output:
integral(sqrt(d*x^4 + c)*x^4/(b*x^4 + a), x)
\[ \int \frac {x^4 \sqrt {c+d x^4}}{a+b x^4} \, dx=\int \frac {x^{4} \sqrt {c + d x^{4}}}{a + b x^{4}}\, dx \] Input:
integrate(x**4*(d*x**4+c)**(1/2)/(b*x**4+a),x)
Output:
Integral(x**4*sqrt(c + d*x**4)/(a + b*x**4), x)
\[ \int \frac {x^4 \sqrt {c+d x^4}}{a+b x^4} \, dx=\int { \frac {\sqrt {d x^{4} + c} x^{4}}{b x^{4} + a} \,d x } \] Input:
integrate(x^4*(d*x^4+c)^(1/2)/(b*x^4+a),x, algorithm="maxima")
Output:
integrate(sqrt(d*x^4 + c)*x^4/(b*x^4 + a), x)
\[ \int \frac {x^4 \sqrt {c+d x^4}}{a+b x^4} \, dx=\int { \frac {\sqrt {d x^{4} + c} x^{4}}{b x^{4} + a} \,d x } \] Input:
integrate(x^4*(d*x^4+c)^(1/2)/(b*x^4+a),x, algorithm="giac")
Output:
integrate(sqrt(d*x^4 + c)*x^4/(b*x^4 + a), x)
Timed out. \[ \int \frac {x^4 \sqrt {c+d x^4}}{a+b x^4} \, dx=\int \frac {x^4\,\sqrt {d\,x^4+c}}{b\,x^4+a} \,d x \] Input:
int((x^4*(c + d*x^4)^(1/2))/(a + b*x^4),x)
Output:
int((x^4*(c + d*x^4)^(1/2))/(a + b*x^4), x)
\[ \int \frac {x^4 \sqrt {c+d x^4}}{a+b x^4} \, dx=\frac {\sqrt {d \,x^{4}+c}\, x -\left (\int \frac {\sqrt {d \,x^{4}+c}}{b d \,x^{8}+a d \,x^{4}+b c \,x^{4}+a c}d x \right ) a c -3 \left (\int \frac {\sqrt {d \,x^{4}+c}\, x^{4}}{b d \,x^{8}+a d \,x^{4}+b c \,x^{4}+a c}d x \right ) a d +2 \left (\int \frac {\sqrt {d \,x^{4}+c}\, x^{4}}{b d \,x^{8}+a d \,x^{4}+b c \,x^{4}+a c}d x \right ) b c}{3 b} \] Input:
int(x^4*(d*x^4+c)^(1/2)/(b*x^4+a),x)
Output:
(sqrt(c + d*x**4)*x - int(sqrt(c + d*x**4)/(a*c + a*d*x**4 + b*c*x**4 + b* d*x**8),x)*a*c - 3*int((sqrt(c + d*x**4)*x**4)/(a*c + a*d*x**4 + b*c*x**4 + b*d*x**8),x)*a*d + 2*int((sqrt(c + d*x**4)*x**4)/(a*c + a*d*x**4 + b*c*x **4 + b*d*x**8),x)*b*c)/(3*b)