Integrand size = 22, antiderivative size = 653 \[ \int \frac {\left (a+b x^4\right )^{3/2}}{c-d x^4} \, dx=-\frac {b x \sqrt {a+b x^4}}{3 d}+\frac {(b c+a d)^{3/2} \arctan \left (\frac {\sqrt {b c+a d} x}{\sqrt [4]{c} \sqrt [4]{d} \sqrt {a+b x^4}}\right )}{4 c^{3/4} d^{7/4}}+\frac {(b c+a d)^{3/2} \text {arctanh}\left (\frac {\sqrt {b c+a d} x}{\sqrt [4]{c} \sqrt [4]{d} \sqrt {a+b x^4}}\right )}{4 c^{3/4} d^{7/4}}+\frac {2 a^{3/4} b^{3/4} (b c+2 a d) \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{3 d (b c-a d) \sqrt {a+b x^4}}-\frac {\left (\sqrt {b} \sqrt {c}+\sqrt {a} \sqrt {d}\right ) (b c+a d)^2 \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \operatorname {EllipticPi}\left (-\frac {\left (\sqrt {b} \sqrt {c}-\sqrt {a} \sqrt {d}\right )^2}{4 \sqrt {a} \sqrt {b} \sqrt {c} \sqrt {d}},2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{8 \sqrt [4]{a} \sqrt [4]{b} c \left (\sqrt {b} \sqrt {c}-\sqrt {a} \sqrt {d}\right ) d^2 \sqrt {a+b x^4}}-\frac {\left (\sqrt {b} \sqrt {c}-\sqrt {a} \sqrt {d}\right ) (b c+a d)^2 \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \operatorname {EllipticPi}\left (\frac {\left (\sqrt {b} \sqrt {c}+\sqrt {a} \sqrt {d}\right )^2}{4 \sqrt {a} \sqrt {b} \sqrt {c} \sqrt {d}},2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{8 \sqrt [4]{a} \sqrt [4]{b} c \left (\sqrt {b} \sqrt {c}+\sqrt {a} \sqrt {d}\right ) d^2 \sqrt {a+b x^4}} \] Output:
-1/3*b*x*(b*x^4+a)^(1/2)/d+1/4*(a*d+b*c)^(3/2)*arctan((a*d+b*c)^(1/2)*x/c^ (1/4)/d^(1/4)/(b*x^4+a)^(1/2))/c^(3/4)/d^(7/4)+1/4*(a*d+b*c)^(3/2)*arctanh ((a*d+b*c)^(1/2)*x/c^(1/4)/d^(1/4)/(b*x^4+a)^(1/2))/c^(3/4)/d^(7/4)+2/3*a^ (3/4)*b^(3/4)*(2*a*d+b*c)*(a^(1/2)+b^(1/2)*x^2)*((b*x^4+a)/(a^(1/2)+b^(1/2 )*x^2)^2)^(1/2)*InverseJacobiAM(2*arctan(b^(1/4)*x/a^(1/4)),1/2*2^(1/2))/d /(-a*d+b*c)/(b*x^4+a)^(1/2)-1/8*(b^(1/2)*c^(1/2)+a^(1/2)*d^(1/2))*(a*d+b*c )^2*(a^(1/2)+b^(1/2)*x^2)*((b*x^4+a)/(a^(1/2)+b^(1/2)*x^2)^2)^(1/2)*Ellipt icPi(sin(2*arctan(b^(1/4)*x/a^(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^(1/4)/b^(1/4)/c/(b^(1/ 2)*c^(1/2)-a^(1/2)*d^(1/2))/d^2/(b*x^4+a)^(1/2)-1/8*(b^(1/2)*c^(1/2)-a^(1/ 2)*d^(1/2))*(a*d+b*c)^2*(a^(1/2)+b^(1/2)*x^2)*((b*x^4+a)/(a^(1/2)+b^(1/2)* x^2)^2)^(1/2)*EllipticPi(sin(2*arctan(b^(1/4)*x/a^(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/ 4)/b^(1/4)/c/(b^(1/2)*c^(1/2)+a^(1/2)*d^(1/2))/d^2/(b*x^4+a)^(1/2)
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 10.51 (sec) , antiderivative size = 343, normalized size of antiderivative = 0.53 \[ \int \frac {\left (a+b x^4\right )^{3/2}}{c-d x^4} \, dx=\frac {x \left (\frac {b (3 b c+5 a d) x^4 \sqrt {1+\frac {b x^4}{a}} \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},1,\frac {9}{4},-\frac {b x^4}{a},\frac {d x^4}{c}\right )}{c}+\frac {-25 a c \left (3 a^2 d+a b d x^4+b^2 x^4 \left (-c+d x^4\right )\right ) \operatorname {AppellF1}\left (\frac {1}{4},\frac {1}{2},1,\frac {5}{4},-\frac {b x^4}{a},\frac {d x^4}{c}\right )+10 b x^4 \left (a+b x^4\right ) \left (-c+d x^4\right ) \left (-2 a d \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},2,\frac {9}{4},-\frac {b x^4}{a},\frac {d x^4}{c}\right )+b c \operatorname {AppellF1}\left (\frac {5}{4},\frac {3}{2},1,\frac {9}{4},-\frac {b x^4}{a},\frac {d x^4}{c}\right )\right )}{\left (c-d x^4\right ) \left (-5 a c \operatorname {AppellF1}\left (\frac {1}{4},\frac {1}{2},1,\frac {5}{4},-\frac {b x^4}{a},\frac {d x^4}{c}\right )+2 x^4 \left (-2 a d \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},2,\frac {9}{4},-\frac {b x^4}{a},\frac {d x^4}{c}\right )+b c \operatorname {AppellF1}\left (\frac {5}{4},\frac {3}{2},1,\frac {9}{4},-\frac {b x^4}{a},\frac {d x^4}{c}\right )\right )\right )}\right )}{15 d \sqrt {a+b x^4}} \] Input:
Integrate[(a + b*x^4)^(3/2)/(c - d*x^4),x]
Output:
(x*((b*(3*b*c + 5*a*d)*x^4*Sqrt[1 + (b*x^4)/a]*AppellF1[5/4, 1/2, 1, 9/4, -((b*x^4)/a), (d*x^4)/c])/c + (-25*a*c*(3*a^2*d + a*b*d*x^4 + b^2*x^4*(-c + d*x^4))*AppellF1[1/4, 1/2, 1, 5/4, -((b*x^4)/a), (d*x^4)/c] + 10*b*x^4*( a + b*x^4)*(-c + d*x^4)*(-2*a*d*AppellF1[5/4, 1/2, 2, 9/4, -((b*x^4)/a), ( d*x^4)/c] + b*c*AppellF1[5/4, 3/2, 1, 9/4, -((b*x^4)/a), (d*x^4)/c]))/((c - d*x^4)*(-5*a*c*AppellF1[1/4, 1/2, 1, 5/4, -((b*x^4)/a), (d*x^4)/c] + 2*x ^4*(-2*a*d*AppellF1[5/4, 1/2, 2, 9/4, -((b*x^4)/a), (d*x^4)/c] + b*c*Appel lF1[5/4, 3/2, 1, 9/4, -((b*x^4)/a), (d*x^4)/c])))))/(15*d*Sqrt[a + b*x^4])
Time = 2.22 (sec) , antiderivative size = 946, normalized size of antiderivative = 1.45, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {933, 25, 1021, 761, 925, 27, 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.
| \(\displaystyle \int \frac {\left (a+b x^4\right )^{3/2}}{c-d x^4} \, dx\) |
| \(\Big \downarrow \) 933 |
| \(\displaystyle -\frac {\int -\frac {b (3 b c+5 a d) x^4+a (b c+3 a d)}{\sqrt {b x^4+a} \left (c-d x^4\right )}dx}{3 d}-\frac {b x \sqrt {a+b x^4}}{3 d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {b (3 b c+5 a d) x^4+a (b c+3 a d)}{\sqrt {b x^4+a} \left (c-d x^4\right )}dx}{3 d}-\frac {b x \sqrt {a+b x^4}}{3 d}\) |
| \(\Big \downarrow \) 1021 |
| \(\displaystyle \frac {\frac {3 (a d+b c)^2 \int \frac {1}{\sqrt {b x^4+a} \left (c-d x^4\right )}dx}{d}-\frac {b (5 a d+3 b c) \int \frac {1}{\sqrt {b x^4+a}}dx}{d}}{3 d}-\frac {b x \sqrt {a+b x^4}}{3 d}\) |
| \(\Big \downarrow \) 761 |
| \(\displaystyle \frac {\frac {3 (a d+b c)^2 \int \frac {1}{\sqrt {b x^4+a} \left (c-d x^4\right )}dx}{d}-\frac {b^{3/4} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} (5 a d+3 b c) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{a} d \sqrt {a+b x^4}}}{3 d}-\frac {b x \sqrt {a+b x^4}}{3 d}\) |
| \(\Big \downarrow \) 925 |
| \(\displaystyle \frac {\frac {3 (a d+b c)^2 \left (\frac {\int \frac {\sqrt {c}}{\left (\sqrt {c}-\sqrt {d} x^2\right ) \sqrt {b x^4+a}}dx}{2 c}+\frac {\int \frac {\sqrt {c}}{\left (\sqrt {d} x^2+\sqrt {c}\right ) \sqrt {b x^4+a}}dx}{2 c}\right )}{d}-\frac {b^{3/4} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} (5 a d+3 b c) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{a} d \sqrt {a+b x^4}}}{3 d}-\frac {b x \sqrt {a+b x^4}}{3 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {3 (a d+b c)^2 \left (\frac {\int \frac {1}{\left (\sqrt {c}-\sqrt {d} x^2\right ) \sqrt {b x^4+a}}dx}{2 \sqrt {c}}+\frac {\int \frac {1}{\left (\sqrt {d} x^2+\sqrt {c}\right ) \sqrt {b x^4+a}}dx}{2 \sqrt {c}}\right )}{d}-\frac {b^{3/4} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} (5 a d+3 b c) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{a} d \sqrt {a+b x^4}}}{3 d}-\frac {b x \sqrt {a+b x^4}}{3 d}\) |
| \(\Big \downarrow \) 1541 |
| \(\displaystyle \frac {\frac {3 (a d+b c)^2 \left (\frac {\frac {\sqrt {b} \int \frac {1}{\sqrt {b x^4+a}}dx}{\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {c}}+\frac {\sqrt {a} \sqrt {d} \int \frac {\sqrt {b} x^2+\sqrt {a}}{\sqrt {a} \left (\sqrt {c}-\sqrt {d} x^2\right ) \sqrt {b x^4+a}}dx}{\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {c}}}{2 \sqrt {c}}+\frac {\frac {\sqrt {b} \int \frac {1}{\sqrt {b x^4+a}}dx}{\sqrt {b} \sqrt {c}-\sqrt {a} \sqrt {d}}-\frac {\sqrt {a} \sqrt {d} \int \frac {\sqrt {b} x^2+\sqrt {a}}{\sqrt {a} \left (\sqrt {d} x^2+\sqrt {c}\right ) \sqrt {b x^4+a}}dx}{\sqrt {b} \sqrt {c}-\sqrt {a} \sqrt {d}}}{2 \sqrt {c}}\right )}{d}-\frac {b^{3/4} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} (5 a d+3 b c) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{a} d \sqrt {a+b x^4}}}{3 d}-\frac {b x \sqrt {a+b x^4}}{3 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {3 (a d+b c)^2 \left (\frac {\frac {\sqrt {b} \int \frac {1}{\sqrt {b x^4+a}}dx}{\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {c}}+\frac {\sqrt {d} \int \frac {\sqrt {b} x^2+\sqrt {a}}{\left (\sqrt {c}-\sqrt {d} x^2\right ) \sqrt {b x^4+a}}dx}{\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {c}}}{2 \sqrt {c}}+\frac {\frac {\sqrt {b} \int \frac {1}{\sqrt {b x^4+a}}dx}{\sqrt {b} \sqrt {c}-\sqrt {a} \sqrt {d}}-\frac {\sqrt {d} \int \frac {\sqrt {b} x^2+\sqrt {a}}{\left (\sqrt {d} x^2+\sqrt {c}\right ) \sqrt {b x^4+a}}dx}{\sqrt {b} \sqrt {c}-\sqrt {a} \sqrt {d}}}{2 \sqrt {c}}\right )}{d}-\frac {b^{3/4} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} (5 a d+3 b c) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{a} d \sqrt {a+b x^4}}}{3 d}-\frac {b x \sqrt {a+b x^4}}{3 d}\) |
| \(\Big \downarrow \) 761 |
| \(\displaystyle \frac {\frac {3 (a d+b c)^2 \left (\frac {\frac {\sqrt {d} \int \frac {\sqrt {b} x^2+\sqrt {a}}{\left (\sqrt {c}-\sqrt {d} x^2\right ) \sqrt {b x^4+a}}dx}{\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {c}}+\frac {\sqrt [4]{b} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{a} \sqrt {a+b x^4} \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {c}\right )}}{2 \sqrt {c}}+\frac {\frac {\sqrt [4]{b} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{a} \sqrt {a+b x^4} \left (\sqrt {b} \sqrt {c}-\sqrt {a} \sqrt {d}\right )}-\frac {\sqrt {d} \int \frac {\sqrt {b} x^2+\sqrt {a}}{\left (\sqrt {d} x^2+\sqrt {c}\right ) \sqrt {b x^4+a}}dx}{\sqrt {b} \sqrt {c}-\sqrt {a} \sqrt {d}}}{2 \sqrt {c}}\right )}{d}-\frac {b^{3/4} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} (5 a d+3 b c) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{a} d \sqrt {a+b x^4}}}{3 d}-\frac {b x \sqrt {a+b x^4}}{3 d}\) |
| \(\Big \downarrow \) 2221 |
| \(\displaystyle \frac {\frac {3 (a d+b c)^2 \left (\frac {\frac {\sqrt {d} \int \frac {\sqrt {b} x^2+\sqrt {a}}{\left (\sqrt {c}-\sqrt {d} x^2\right ) \sqrt {b x^4+a}}dx}{\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {c}}+\frac {\sqrt [4]{b} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{a} \sqrt {a+b x^4} \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {c}\right )}}{2 \sqrt {c}}+\frac {\frac {\sqrt [4]{b} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{a} \sqrt {a+b x^4} \left (\sqrt {b} \sqrt {c}-\sqrt {a} \sqrt {d}\right )}-\frac {\sqrt {d} \left (\frac {\left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \left (\frac {\sqrt {a}}{\sqrt {c}}+\frac {\sqrt {b}}{\sqrt {d}}\right ) \operatorname {EllipticPi}\left (-\frac {\sqrt {a} \left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a}}-\sqrt {d}\right )^2}{4 \sqrt {b} \sqrt {c} \sqrt {d}},2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{a} \sqrt [4]{b} \sqrt {a+b x^4}}-\frac {\left (\sqrt {b} \sqrt {c}-\sqrt {a} \sqrt {d}\right ) \arctan \left (\frac {x \sqrt {a d+b c}}{\sqrt [4]{c} \sqrt [4]{d} \sqrt {a+b x^4}}\right )}{2 \sqrt [4]{c} \sqrt [4]{d} \sqrt {a d+b c}}\right )}{\sqrt {b} \sqrt {c}-\sqrt {a} \sqrt {d}}}{2 \sqrt {c}}\right )}{d}-\frac {b^{3/4} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} (5 a d+3 b c) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{a} d \sqrt {a+b x^4}}}{3 d}-\frac {b x \sqrt {a+b x^4}}{3 d}\) |
| \(\Big \downarrow \) 2223 |
| \(\displaystyle \frac {\frac {3 (b c+a d)^2 \left (\frac {\frac {\sqrt [4]{b} \left (\sqrt {b} x^2+\sqrt {a}\right ) \sqrt {\frac {b x^4+a}{\left (\sqrt {b} x^2+\sqrt {a}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{a} \left (\sqrt {b} \sqrt {c}-\sqrt {a} \sqrt {d}\right ) \sqrt {b x^4+a}}-\frac {\sqrt {d} \left (\frac {\left (\frac {\sqrt {a}}{\sqrt {c}}+\frac {\sqrt {b}}{\sqrt {d}}\right ) \left (\sqrt {b} x^2+\sqrt {a}\right ) \sqrt {\frac {b x^4+a}{\left (\sqrt {b} x^2+\sqrt {a}\right )^2}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} \left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a}}-\sqrt {d}\right )^2}{4 \sqrt {b} \sqrt {c} \sqrt {d}},2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{a} \sqrt [4]{b} \sqrt {b x^4+a}}-\frac {\left (\sqrt {b} \sqrt {c}-\sqrt {a} \sqrt {d}\right ) \arctan \left (\frac {\sqrt {b c+a d} x}{\sqrt [4]{c} \sqrt [4]{d} \sqrt {b x^4+a}}\right )}{2 \sqrt [4]{c} \sqrt [4]{d} \sqrt {b c+a d}}\right )}{\sqrt {b} \sqrt {c}-\sqrt {a} \sqrt {d}}}{2 \sqrt {c}}+\frac {\frac {\sqrt [4]{b} \left (\sqrt {b} x^2+\sqrt {a}\right ) \sqrt {\frac {b x^4+a}{\left (\sqrt {b} x^2+\sqrt {a}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{a} \left (\sqrt {b} \sqrt {c}+\sqrt {a} \sqrt {d}\right ) \sqrt {b x^4+a}}+\frac {\sqrt {d} \left (\frac {\left (\sqrt {b} \sqrt {c}+\sqrt {a} \sqrt {d}\right ) \text {arctanh}\left (\frac {\sqrt {b c+a d} x}{\sqrt [4]{c} \sqrt [4]{d} \sqrt {b x^4+a}}\right )}{2 \sqrt [4]{c} \sqrt [4]{d} \sqrt {b c+a d}}+\frac {\left (\frac {\sqrt {a}}{\sqrt {c}}-\frac {\sqrt {b}}{\sqrt {d}}\right ) \left (\sqrt {b} x^2+\sqrt {a}\right ) \sqrt {\frac {b x^4+a}{\left (\sqrt {b} x^2+\sqrt {a}\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]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{a} \sqrt [4]{b} \sqrt {b x^4+a}}\right )}{\sqrt {b} \sqrt {c}+\sqrt {a} \sqrt {d}}}{2 \sqrt {c}}\right )}{d}-\frac {b^{3/4} (3 b c+5 a d) \left (\sqrt {b} x^2+\sqrt {a}\right ) \sqrt {\frac {b x^4+a}{\left (\sqrt {b} x^2+\sqrt {a}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{a} d \sqrt {b x^4+a}}}{3 d}-\frac {b x \sqrt {b x^4+a}}{3 d}\) |
Input:
Int[(a + b*x^4)^(3/2)/(c - d*x^4),x]
Output:
-1/3*(b*x*Sqrt[a + b*x^4])/d + (-1/2*(b^(3/4)*(3*b*c + 5*a*d)*(Sqrt[a] + S qrt[b]*x^2)*Sqrt[(a + b*x^4)/(Sqrt[a] + Sqrt[b]*x^2)^2]*EllipticF[2*ArcTan [(b^(1/4)*x)/a^(1/4)], 1/2])/(a^(1/4)*d*Sqrt[a + b*x^4]) + (3*(b*c + a*d)^ 2*(((b^(1/4)*(Sqrt[a] + Sqrt[b]*x^2)*Sqrt[(a + b*x^4)/(Sqrt[a] + Sqrt[b]*x ^2)^2]*EllipticF[2*ArcTan[(b^(1/4)*x)/a^(1/4)], 1/2])/(2*a^(1/4)*(Sqrt[b]* Sqrt[c] - Sqrt[a]*Sqrt[d])*Sqrt[a + b*x^4]) - (Sqrt[d]*(-1/2*((Sqrt[b]*Sqr t[c] - Sqrt[a]*Sqrt[d])*ArcTan[(Sqrt[b*c + a*d]*x)/(c^(1/4)*d^(1/4)*Sqrt[a + b*x^4])])/(c^(1/4)*d^(1/4)*Sqrt[b*c + a*d]) + ((Sqrt[a]/Sqrt[c] + Sqrt[ b]/Sqrt[d])*(Sqrt[a] + Sqrt[b]*x^2)*Sqrt[(a + b*x^4)/(Sqrt[a] + Sqrt[b]*x^ 2)^2]*EllipticPi[-1/4*(Sqrt[a]*((Sqrt[b]*Sqrt[c])/Sqrt[a] - Sqrt[d])^2)/(S qrt[b]*Sqrt[c]*Sqrt[d]), 2*ArcTan[(b^(1/4)*x)/a^(1/4)], 1/2])/(4*a^(1/4)*b ^(1/4)*Sqrt[a + b*x^4])))/(Sqrt[b]*Sqrt[c] - Sqrt[a]*Sqrt[d]))/(2*Sqrt[c]) + ((b^(1/4)*(Sqrt[a] + Sqrt[b]*x^2)*Sqrt[(a + b*x^4)/(Sqrt[a] + Sqrt[b]*x ^2)^2]*EllipticF[2*ArcTan[(b^(1/4)*x)/a^(1/4)], 1/2])/(2*a^(1/4)*(Sqrt[b]* Sqrt[c] + Sqrt[a]*Sqrt[d])*Sqrt[a + b*x^4]) + (Sqrt[d]*(((Sqrt[b]*Sqrt[c] + Sqrt[a]*Sqrt[d])*ArcTanh[(Sqrt[b*c + a*d]*x)/(c^(1/4)*d^(1/4)*Sqrt[a + b *x^4])])/(2*c^(1/4)*d^(1/4)*Sqrt[b*c + a*d]) + ((Sqrt[a]/Sqrt[c] - Sqrt[b] /Sqrt[d])*(Sqrt[a] + Sqrt[b]*x^2)*Sqrt[(a + b*x^4)/(Sqrt[a] + Sqrt[b]*x^2) ^2]*EllipticPi[(Sqrt[b]*Sqrt[c] + Sqrt[a]*Sqrt[d])^2/(4*Sqrt[a]*Sqrt[b]*Sq rt[c]*Sqrt[d]), 2*ArcTan[(b^(1/4)*x)/a^(1/4)], 1/2])/(4*a^(1/4)*b^(1/4)...
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[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[d*x*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q - 1)/(b*(n*(p + q) + 1))), x] + Simp[1/(b*(n*(p + q) + 1)) Int[(a + b*x^n)^p*(c + d*x^n)^(q - 2)*Sim p[c*(b*c*(n*(p + q) + 1) - a*d) + d*(b*c*(n*(p + 2*q - 1) + 1) - a*d*(n*(q - 1) + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d , 0] && GtQ[q, 1] && NeQ[n*(p + q) + 1, 0] && !IGtQ[p, 1] && IntBinomialQ[ a, b, c, d, 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 = 4.45 (sec) , antiderivative size = 317, normalized size of antiderivative = 0.49
| method | result | size |
| risch | \(-\frac {b x \sqrt {b \,x^{4}+a}}{3 d}-\frac {\frac {b \left (5 a d +3 b c \right ) \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{d \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}+\frac {3 \left (a^{2} d^{2}+2 a b c d +b^{2} c^{2}\right ) \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (\textit {\_Z}^{4} d -c \right )}{\sum }\frac {-\frac {\operatorname {arctanh}\left (\frac {2 b \,x^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}+2 a}{2 \sqrt {\frac {a d +b c}{d}}\, \sqrt {b \,x^{4}+a}}\right )}{\sqrt {\frac {a d +b c}{d}}}-\frac {2 \underline {\hspace {1.25 ex}}\alpha ^{3} d \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, -\frac {i \sqrt {a}\, \underline {\hspace {1.25 ex}}\alpha ^{2} d}{\sqrt {b}\, c}, \frac {\sqrt {-\frac {i \sqrt {b}}{\sqrt {a}}}}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}}\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, c \sqrt {b \,x^{4}+a}}}{\underline {\hspace {1.25 ex}}\alpha ^{3}}\right )}{8 d^{2}}}{3 d}\) | \(317\) |
| default | \(-\frac {b x \sqrt {b \,x^{4}+a}}{3 d}+\frac {\left (-\frac {b \left (2 a d +b c \right )}{d^{2}}+\frac {b a}{3 d}\right ) \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}-\frac {\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (\textit {\_Z}^{4} d -c \right )}{\sum }\frac {\left (a^{2} d^{2}+2 a b c d +b^{2} c^{2}\right ) \left (-\frac {\operatorname {arctanh}\left (\frac {2 b \,x^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}+2 a}{2 \sqrt {\frac {a d +b c}{d}}\, \sqrt {b \,x^{4}+a}}\right )}{\sqrt {\frac {a d +b c}{d}}}-\frac {2 \underline {\hspace {1.25 ex}}\alpha ^{3} d \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, -\frac {i \sqrt {a}\, \underline {\hspace {1.25 ex}}\alpha ^{2} d}{\sqrt {b}\, c}, \frac {\sqrt {-\frac {i \sqrt {b}}{\sqrt {a}}}}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}}\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, c \sqrt {b \,x^{4}+a}}\right )}{\underline {\hspace {1.25 ex}}\alpha ^{3}}}{8 d^{3}}\) | \(320\) |
| elliptic | \(-\frac {b x \sqrt {b \,x^{4}+a}}{3 d}+\frac {\left (-\frac {b \left (2 a d +b c \right )}{d^{2}}+\frac {b a}{3 d}\right ) \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}-\frac {\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (\textit {\_Z}^{4} d -c \right )}{\sum }\frac {\left (a^{2} d^{2}+2 a b c d +b^{2} c^{2}\right ) \left (-\frac {\operatorname {arctanh}\left (\frac {2 b \,x^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}+2 a}{2 \sqrt {\frac {a d +b c}{d}}\, \sqrt {b \,x^{4}+a}}\right )}{\sqrt {\frac {a d +b c}{d}}}-\frac {2 \underline {\hspace {1.25 ex}}\alpha ^{3} d \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, -\frac {i \sqrt {a}\, \underline {\hspace {1.25 ex}}\alpha ^{2} d}{\sqrt {b}\, c}, \frac {\sqrt {-\frac {i \sqrt {b}}{\sqrt {a}}}}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}}\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, c \sqrt {b \,x^{4}+a}}\right )}{\underline {\hspace {1.25 ex}}\alpha ^{3}}}{8 d^{3}}\) | \(320\) |
Input:
int((b*x^4+a)^(3/2)/(-d*x^4+c),x,method=_RETURNVERBOSE)
Output:
-1/3*b*x*(b*x^4+a)^(1/2)/d-1/3/d*(b*(5*a*d+3*b*c)/d/(I/a^(1/2)*b^(1/2))^(1 /2)*(1-I*b^(1/2)*x^2/a^(1/2))^(1/2)*(1+I*b^(1/2)*x^2/a^(1/2))^(1/2)/(b*x^4 +a)^(1/2)*EllipticF(x*(I/a^(1/2)*b^(1/2))^(1/2),I)+3/8*(a^2*d^2+2*a*b*c*d+ b^2*c^2)/d^2*sum(1/_alpha^3*(-1/((a*d+b*c)/d)^(1/2)*arctanh(1/2*(2*_alpha^ 2*b*x^2+2*a)/((a*d+b*c)/d)^(1/2)/(b*x^4+a)^(1/2))-2/(I/a^(1/2)*b^(1/2))^(1 /2)*_alpha^3*d/c*(1-I*b^(1/2)*x^2/a^(1/2))^(1/2)*(1+I*b^(1/2)*x^2/a^(1/2)) ^(1/2)/(b*x^4+a)^(1/2)*EllipticPi(x*(I/a^(1/2)*b^(1/2))^(1/2),-I*a^(1/2)/b ^(1/2)*_alpha^2/c*d,(-I/a^(1/2)*b^(1/2))^(1/2)/(I/a^(1/2)*b^(1/2))^(1/2))) ,_alpha=RootOf(_Z^4*d-c)))
Timed out. \[ \int \frac {\left (a+b x^4\right )^{3/2}}{c-d x^4} \, dx=\text {Timed out} \] Input:
integrate((b*x^4+a)^(3/2)/(-d*x^4+c),x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {\left (a+b x^4\right )^{3/2}}{c-d x^4} \, dx=- \int \frac {a \sqrt {a + b x^{4}}}{- c + d x^{4}}\, dx - \int \frac {b x^{4} \sqrt {a + b x^{4}}}{- c + d x^{4}}\, dx \] Input:
integrate((b*x**4+a)**(3/2)/(-d*x**4+c),x)
Output:
-Integral(a*sqrt(a + b*x**4)/(-c + d*x**4), x) - Integral(b*x**4*sqrt(a + b*x**4)/(-c + d*x**4), x)
\[ \int \frac {\left (a+b x^4\right )^{3/2}}{c-d x^4} \, dx=\int { -\frac {{\left (b x^{4} + a\right )}^{\frac {3}{2}}}{d x^{4} - c} \,d x } \] Input:
integrate((b*x^4+a)^(3/2)/(-d*x^4+c),x, algorithm="maxima")
Output:
-integrate((b*x^4 + a)^(3/2)/(d*x^4 - c), x)
\[ \int \frac {\left (a+b x^4\right )^{3/2}}{c-d x^4} \, dx=\int { -\frac {{\left (b x^{4} + a\right )}^{\frac {3}{2}}}{d x^{4} - c} \,d x } \] Input:
integrate((b*x^4+a)^(3/2)/(-d*x^4+c),x, algorithm="giac")
Output:
integrate(-(b*x^4 + a)^(3/2)/(d*x^4 - c), x)
Timed out. \[ \int \frac {\left (a+b x^4\right )^{3/2}}{c-d x^4} \, dx=\int \frac {{\left (b\,x^4+a\right )}^{3/2}}{c-d\,x^4} \,d x \] Input:
int((a + b*x^4)^(3/2)/(c - d*x^4),x)
Output:
int((a + b*x^4)^(3/2)/(c - d*x^4), x)
\[ \int \frac {\left (a+b x^4\right )^{3/2}}{c-d x^4} \, dx=\frac {-\sqrt {b \,x^{4}+a}\, b x +3 \left (\int \frac {\sqrt {b \,x^{4}+a}}{-b d \,x^{8}-a d \,x^{4}+b c \,x^{4}+a c}d x \right ) a^{2} d +\left (\int \frac {\sqrt {b \,x^{4}+a}}{-b d \,x^{8}-a d \,x^{4}+b c \,x^{4}+a c}d x \right ) a b c +5 \left (\int \frac {\sqrt {b \,x^{4}+a}\, x^{4}}{-b d \,x^{8}-a d \,x^{4}+b c \,x^{4}+a c}d x \right ) a b d +3 \left (\int \frac {\sqrt {b \,x^{4}+a}\, x^{4}}{-b d \,x^{8}-a d \,x^{4}+b c \,x^{4}+a c}d x \right ) b^{2} c}{3 d} \] Input:
int((b*x^4+a)^(3/2)/(-d*x^4+c),x)
Output:
( - sqrt(a + b*x**4)*b*x + 3*int(sqrt(a + b*x**4)/(a*c - a*d*x**4 + b*c*x* *4 - b*d*x**8),x)*a**2*d + int(sqrt(a + b*x**4)/(a*c - a*d*x**4 + b*c*x**4 - b*d*x**8),x)*a*b*c + 5*int((sqrt(a + b*x**4)*x**4)/(a*c - a*d*x**4 + b* c*x**4 - b*d*x**8),x)*a*b*d + 3*int((sqrt(a + b*x**4)*x**4)/(a*c - a*d*x** 4 + b*c*x**4 - b*d*x**8),x)*b**2*c)/(3*d)