3.2.79 \(\int \frac {(a+b x^4)^{3/2}}{c+d x^4} \, dx\) [179]

3.2.79.1 Optimal result
3.2.79.2 Mathematica [C] (warning: unable to verify)
3.2.79.3 Rubi [A] (verified)
3.2.79.4 Maple [C] (warning: unable to verify)
3.2.79.5 Fricas [F(-1)]
3.2.79.6 Sympy [F]
3.2.79.7 Maxima [F]
3.2.79.8 Giac [F]
3.2.79.9 Mupad [F(-1)]

3.2.79.1 Optimal result

Integrand size = 21, antiderivative size = 926 \[ \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} \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^{3/4} (3 b c-5 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 )}{6 \sqrt [4]{a} d^2 \sqrt {a+b x^4}}+\frac {\sqrt [4]{b} \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 {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{a} \sqrt {-c} d^2 (b c+a d) \sqrt {a+b x^4}}+\frac {\sqrt [4]{b} \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 {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{a} \sqrt {-c} d^2 (b c+a d) \sqrt {a+b x^4}}+\frac {\left (\sqrt {b} \sqrt {-c}+\sqrt {a} \sqrt {d}\right )^2 (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 d^2 (b c+a d) \sqrt {a+b x^4}}+\frac {\left (\sqrt {b} \sqrt {-c}-\sqrt {a} \sqrt {d}\right )^2 (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 d^2 (b c+a d) \sqrt {a+b x^4}} \]

output
-1/4*(-a*d+b*c)^(3/2)*arctan(x*(-a*d+b*c)^(1/2)/(-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)*arctan(x*(a*d-b*c)^(1/2)/ 
(-c)^(1/4)/d^(1/4)/(b*x^4+a)^(1/2))/(-c)^(3/4)/d^(7/4)+1/3*b*x*(b*x^4+a)^( 
1/2)/d-1/6*b^(3/4)*(-5*a*d+3*b*c)*(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)^(1/ 
2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(b^(1/4)*x/a^(1/ 
4))),1/2*2^(1/2))*(a^(1/2)+x^2*b^(1/2))*((b*x^4+a)/(a^(1/2)+x^2*b^(1/2))^2 
)^(1/2)/a^(1/4)/d^2/(b*x^4+a)^(1/2)+1/4*b^(1/4)*(-a*d+b*c)^2*(cos(2*arctan 
(b^(1/4)*x/a^(1/4)))^2)^(1/2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticF(s 
in(2*arctan(b^(1/4)*x/a^(1/4))),1/2*2^(1/2))*(a^(1/2)+x^2*b^(1/2))*(b^(1/2 
)*(-c)^(1/2)-a^(1/2)*d^(1/2))*((b*x^4+a)/(a^(1/2)+x^2*b^(1/2))^2)^(1/2)/a^ 
(1/4)/d^2/(a*d+b*c)/(-c)^(1/2)/(b*x^4+a)^(1/2)+1/8*(-a*d+b*c)^2*(cos(2*arc 
tan(b^(1/4)*x/a^(1/4)))^2)^(1/2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*Elliptic 
Pi(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^(1/2)+x^2*b^(1/2)) 
*(b^(1/2)*(-c)^(1/2)-a^(1/2)*d^(1/2))^2*((b*x^4+a)/(a^(1/2)+x^2*b^(1/2))^2 
)^(1/2)/a^(1/4)/b^(1/4)/c/d^2/(a*d+b*c)/(b*x^4+a)^(1/2)+1/4*b^(1/4)*(-a*d+ 
b*c)^2*(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)^(1/2)/cos(2*arctan(b^(1/4)*x/a 
^(1/4)))*EllipticF(sin(2*arctan(b^(1/4)*x/a^(1/4))),1/2*2^(1/2))*(a^(1/2)+ 
x^2*b^(1/2))*(b^(1/2)*(-c)^(1/2)+a^(1/2)*d^(1/2))*((b*x^4+a)/(a^(1/2)+x^2* 
b^(1/2))^2)^(1/2)/a^(1/4)/d^2/(a*d+b*c)/(-c)^(1/2)/(b*x^4+a)^(1/2)+1/8*...
 
3.2.79.2 Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.

Time = 10.47 (sec) , antiderivative size = 346, normalized size of antiderivative = 0.37 \[ \int \frac {\left (a+b x^4\right )^{3/2}}{c+d x^4} \, dx=\frac {x \left (\frac {b (-3 b c+5 a d) x^4 \sqrt {1+\frac {b x^4}{a}} \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},1,\frac {9}{4},-\frac {b x^4}{a},-\frac {d x^4}{c}\right )}{c}+\frac {5 \left (-5 a c \left (3 a^2 d+a b d x^4+b^2 x^4 \left (c+d x^4\right )\right ) \operatorname {AppellF1}\left (\frac {1}{4},\frac {1}{2},1,\frac {5}{4},-\frac {b x^4}{a},-\frac {d x^4}{c}\right )+2 b x^4 \left (a+b x^4\right ) \left (c+d x^4\right ) \left (2 a d \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},2,\frac {9}{4},-\frac {b x^4}{a},-\frac {d x^4}{c}\right )+b c \operatorname {AppellF1}\left (\frac {5}{4},\frac {3}{2},1,\frac {9}{4},-\frac {b x^4}{a},-\frac {d x^4}{c}\right )\right )\right )}{\left (c+d x^4\right ) \left (-5 a c \operatorname {AppellF1}\left (\frac {1}{4},\frac {1}{2},1,\frac {5}{4},-\frac {b x^4}{a},-\frac {d x^4}{c}\right )+2 x^4 \left (2 a d \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},2,\frac {9}{4},-\frac {b x^4}{a},-\frac {d x^4}{c}\right )+b c \operatorname {AppellF1}\left (\frac {5}{4},\frac {3}{2},1,\frac {9}{4},-\frac {b x^4}{a},-\frac {d x^4}{c}\right )\right )\right )}\right )}{15 d \sqrt {a+b x^4}} \]

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 + (5*(-5*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)] + 2* 
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*AppellF1[5/4, 3/2, 1, 9/4, -((b*x^4)/a), -((d*x^4)/c)])))))/(1 
5*d*Sqrt[a + b*x^4])
 
3.2.79.3 Rubi [A] (verified)

Time = 1.56 (sec) , antiderivative size = 1004, normalized size of antiderivative = 1.08, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.476, Rules used = {933, 25, 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.

\(\displaystyle \int \frac {\left (a+b x^4\right )^{3/2}}{c+d x^4} \, dx\)

\(\Big \downarrow \) 933

\(\displaystyle \frac {\int -\frac {b (3 b c-5 a d) x^4+a (b c-3 a d)}{\sqrt {b x^4+a} \left (d x^4+c\right )}dx}{3 d}+\frac {b x \sqrt {a+b x^4}}{3 d}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {b x \sqrt {a+b x^4}}{3 d}-\frac {\int \frac {b (3 b c-5 a d) x^4+a (b c-3 a d)}{\sqrt {b x^4+a} \left (d x^4+c\right )}dx}{3 d}\)

\(\Big \downarrow \) 1021

\(\displaystyle \frac {b x \sqrt {a+b x^4}}{3 d}-\frac {\frac {b (3 b c-5 a d) \int \frac {1}{\sqrt {b x^4+a}}dx}{d}-\frac {3 (b c-a d)^2 \int \frac {1}{\sqrt {b x^4+a} \left (d x^4+c\right )}dx}{d}}{3 d}\)

\(\Big \downarrow \) 761

\(\displaystyle \frac {b x \sqrt {a+b x^4}}{3 d}-\frac {\frac {b^{3/4} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} (3 b c-5 a d) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{a} d \sqrt {a+b x^4}}-\frac {3 (b c-a d)^2 \int \frac {1}{\sqrt {b x^4+a} \left (d x^4+c\right )}dx}{d}}{3 d}\)

\(\Big \downarrow \) 925

\(\displaystyle \frac {b x \sqrt {a+b x^4}}{3 d}-\frac {\frac {b^{3/4} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} (3 b c-5 a d) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{a} d \sqrt {a+b x^4}}-\frac {3 (b c-a d)^2 \left (\frac {\int \frac {1}{\left (1-\frac {\sqrt {d} x^2}{\sqrt {-c}}\right ) \sqrt {b x^4+a}}dx}{2 c}+\frac {\int \frac {1}{\left (\frac {\sqrt {d} x^2}{\sqrt {-c}}+1\right ) \sqrt {b x^4+a}}dx}{2 c}\right )}{d}}{3 d}\)

\(\Big \downarrow \) 1541

\(\displaystyle \frac {b x \sqrt {a+b x^4}}{3 d}-\frac {\frac {b^{3/4} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} (3 b c-5 a d) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{a} d \sqrt {a+b x^4}}-\frac {3 (b c-a d)^2 \left (\frac {\frac {\sqrt {b} \left (\sqrt {a} \sqrt {-c} \sqrt {d}+\sqrt {b} c\right ) \int \frac {1}{\sqrt {b x^4+a}}dx}{a d+b c}-\frac {\sqrt {a} \sqrt {d} \left (\sqrt {b} \sqrt {-c}-\sqrt {a} \sqrt {d}\right ) \int \frac {\sqrt {b} x^2+\sqrt {a}}{\sqrt {a} \left (1-\frac {\sqrt {d} x^2}{\sqrt {-c}}\right ) \sqrt {b x^4+a}}dx}{a d+b c}}{2 c}+\frac {\frac {\sqrt {b} c \left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {-c}}+\sqrt {b}\right ) \int \frac {1}{\sqrt {b x^4+a}}dx}{a d+b c}+\frac {\sqrt {a} \sqrt {d} \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {-c}\right ) \int \frac {\sqrt {b} x^2+\sqrt {a}}{\sqrt {a} \left (\frac {\sqrt {d} x^2}{\sqrt {-c}}+1\right ) \sqrt {b x^4+a}}dx}{a d+b c}}{2 c}\right )}{d}}{3 d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {b x \sqrt {a+b x^4}}{3 d}-\frac {\frac {b^{3/4} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} (3 b c-5 a d) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{a} d \sqrt {a+b x^4}}-\frac {3 (b c-a d)^2 \left (\frac {\frac {\sqrt {b} \left (\sqrt {a} \sqrt {-c} \sqrt {d}+\sqrt {b} c\right ) \int \frac {1}{\sqrt {b x^4+a}}dx}{a d+b c}-\frac {\sqrt {d} \left (\sqrt {b} \sqrt {-c}-\sqrt {a} \sqrt {d}\right ) \int \frac {\sqrt {b} x^2+\sqrt {a}}{\left (1-\frac {\sqrt {d} x^2}{\sqrt {-c}}\right ) \sqrt {b x^4+a}}dx}{a d+b c}}{2 c}+\frac {\frac {\sqrt {b} c \left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {-c}}+\sqrt {b}\right ) \int \frac {1}{\sqrt {b x^4+a}}dx}{a d+b c}+\frac {\sqrt {d} \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {-c}\right ) \int \frac {\sqrt {b} x^2+\sqrt {a}}{\left (\frac {\sqrt {d} x^2}{\sqrt {-c}}+1\right ) \sqrt {b x^4+a}}dx}{a d+b c}}{2 c}\right )}{d}}{3 d}\)

\(\Big \downarrow \) 761

\(\displaystyle \frac {b x \sqrt {a+b x^4}}{3 d}-\frac {\frac {b^{3/4} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} (3 b c-5 a d) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{a} d \sqrt {a+b x^4}}-\frac {3 (b c-a d)^2 \left (\frac {\frac {\sqrt [4]{b} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \left (\sqrt {a} \sqrt {-c} \sqrt {d}+\sqrt {b} c\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{a} \sqrt {a+b x^4} (a d+b c)}-\frac {\sqrt {d} \left (\sqrt {b} \sqrt {-c}-\sqrt {a} \sqrt {d}\right ) \int \frac {\sqrt {b} x^2+\sqrt {a}}{\left (1-\frac {\sqrt {d} x^2}{\sqrt {-c}}\right ) \sqrt {b x^4+a}}dx}{a d+b c}}{2 c}+\frac {\frac {\sqrt {d} \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {-c}\right ) \int \frac {\sqrt {b} x^2+\sqrt {a}}{\left (\frac {\sqrt {d} x^2}{\sqrt {-c}}+1\right ) \sqrt {b x^4+a}}dx}{a d+b c}+\frac {\sqrt [4]{b} c \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {-c}}+\sqrt {b}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{a} \sqrt {a+b x^4} (a d+b c)}}{2 c}\right )}{d}}{3 d}\)

\(\Big \downarrow \) 2221

\(\displaystyle \frac {b x \sqrt {b x^4+a}}{3 d}-\frac {\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}}-\frac {3 (b c-a d)^2 \left (\frac {\frac {\sqrt [4]{b} \left (\sqrt {b} c+\sqrt {a} \sqrt {-c} \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 {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{a} (b c+a d) \sqrt {b x^4+a}}-\frac {\left (\sqrt {b} \sqrt {-c}-\sqrt {a} \sqrt {d}\right ) \sqrt {d} \left (\frac {\sqrt [4]{-c} \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]{d} \sqrt {b c-a d}}+\frac {\left (\sqrt {a}-\frac {\sqrt {b} \sqrt {-c}}{\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 )}{b c+a d}}{2 c}+\frac {\frac {\sqrt [4]{b} c \left (\sqrt {b}+\frac {\sqrt {a} \sqrt {d}}{\sqrt {-c}}\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 {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{a} (b c+a d) \sqrt {b x^4+a}}+\frac {\left (\sqrt {b} \sqrt {-c}+\sqrt {a} \sqrt {d}\right ) \sqrt {d} \int \frac {\sqrt {b} x^2+\sqrt {a}}{\left (\frac {\sqrt {d} x^2}{\sqrt {-c}}+1\right ) \sqrt {b x^4+a}}dx}{b c+a d}}{2 c}\right )}{d}}{3 d}\)

\(\Big \downarrow \) 2223

\(\displaystyle \frac {b x \sqrt {b x^4+a}}{3 d}-\frac {\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}}-\frac {3 (b c-a d)^2 \left (\frac {\frac {\sqrt [4]{b} c \left (\sqrt {b}+\frac {\sqrt {a} \sqrt {d}}{\sqrt {-c}}\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 {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{a} (b c+a d) \sqrt {b x^4+a}}+\frac {\left (\sqrt {b} \sqrt {-c}+\sqrt {a} \sqrt {d}\right ) \sqrt {d} \left (\frac {\left (\sqrt {a}+\frac {\sqrt {b} \sqrt {-c}}{\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}}-\frac {(-c)^{3/4} \left (\sqrt {b}-\frac {\sqrt {a} \sqrt {d}}{\sqrt {-c}}\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]{d} \sqrt {b c-a d}}\right )}{b c+a d}}{2 c}+\frac {\frac {\sqrt [4]{b} \left (\sqrt {b} c+\sqrt {a} \sqrt {-c} \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 {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{a} (b c+a d) \sqrt {b x^4+a}}-\frac {\left (\sqrt {b} \sqrt {-c}-\sqrt {a} \sqrt {d}\right ) \sqrt {d} \left (\frac {\sqrt [4]{-c} \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]{d} \sqrt {b c-a d}}+\frac {\left (\sqrt {a}-\frac {\sqrt {b} \sqrt {-c}}{\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 )}{b c+a d}}{2 c}\right )}{d}}{3 d}\)

input
Int[(a + b*x^4)^(3/2)/(c + d*x^4),x]
 
output
(b*x*Sqrt[a + b*x^4])/(3*d) - ((b^(3/4)*(3*b*c - 5*a*d)*(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)*d*Sqrt[a + b*x^4]) - (3*(b*c - a*d)^2*(( 
(b^(1/4)*c*(Sqrt[b] + (Sqrt[a]*Sqrt[d])/Sqrt[-c])*(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)*(b*c + a*d)*Sqrt[a + b*x^4]) + ((Sqrt[b]*Sqrt[ 
-c] + Sqrt[a]*Sqrt[d])*Sqrt[d]*(-1/2*((-c)^(3/4)*(Sqrt[b] - (Sqrt[a]*Sqrt[ 
d])/Sqrt[-c])*ArcTanh[(Sqrt[b*c - a*d]*x)/((-c)^(1/4)*d^(1/4)*Sqrt[a + b*x 
^4])])/(d^(1/4)*Sqrt[b*c - a*d]) + ((Sqrt[a] + (Sqrt[b]*Sqrt[-c])/Sqrt[d]) 
*(Sqrt[a] + Sqrt[b]*x^2)*Sqrt[(a + b*x^4)/(Sqrt[a] + Sqrt[b]*x^2)^2]*Ellip 
ticPi[-1/4*(Sqrt[b]*Sqrt[-c] - Sqrt[a]*Sqrt[d])^2/(Sqrt[a]*Sqrt[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])))/(b*c + a*d))/(2*c) + ((b^(1/4)*(Sqrt[b]*c + Sqrt[a]*Sqrt[-c]* 
Sqrt[d])*(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)*(b*c + a*d)*S 
qrt[a + b*x^4]) - ((Sqrt[b]*Sqrt[-c] - Sqrt[a]*Sqrt[d])*Sqrt[d]*(((-c)^(1/ 
4)*(Sqrt[b]*Sqrt[-c] + Sqrt[a]*Sqrt[d])*ArcTan[(Sqrt[b*c - a*d]*x)/((-c)^( 
1/4)*d^(1/4)*Sqrt[a + b*x^4])])/(2*d^(1/4)*Sqrt[b*c - a*d]) + ((Sqrt[a] - 
(Sqrt[b]*Sqrt[-c])/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...
 

3.2.79.3.1 Defintions of rubi rules used

rule 25
Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27
Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 761
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]
 

rule 925
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]
 

rule 933
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]
 

rule 1021
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]
 

rule 1541
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]
 

rule 2221
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)]
 

rule 2223
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)]
 
3.2.79.4 Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 6.44 (sec) , antiderivative size = 319, normalized size of antiderivative = 0.34

method result size
risch \(\frac {b x \sqrt {b \,x^{4}+a}}{3 d}+\frac {\frac {b \left (5 a d -3 b c \right ) \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{d \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}+\frac {\left (3 a^{2} d^{2}-6 a b c d +3 b^{2} c^{2}\right ) \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (d \,\textit {\_Z}^{4}+c \right )}{\sum }\frac {-\frac {\operatorname {arctanh}\left (\frac {2 b \,x^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}+2 a}{2 \sqrt {\frac {a d -b c}{d}}\, \sqrt {b \,x^{4}+a}}\right )}{\sqrt {\frac {a d -b c}{d}}}+\frac {2 \underline {\hspace {1.25 ex}}\alpha ^{3} d \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \Pi \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, \frac {i \sqrt {a}\, \underline {\hspace {1.25 ex}}\alpha ^{2} d}{\sqrt {b}\, c}, \frac {\sqrt {-\frac {i \sqrt {b}}{\sqrt {a}}}}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}}\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, c \sqrt {b \,x^{4}+a}}}{\underline {\hspace {1.25 ex}}\alpha ^{3}}\right )}{8 d^{2}}}{3 d}\) \(319\)
default \(\frac {b x \sqrt {b \,x^{4}+a}}{3 d}+\frac {\left (\frac {b \left (2 a d -b c \right )}{d^{2}}-\frac {b a}{3 d}\right ) \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}-\frac {\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (d \,\textit {\_Z}^{4}+c \right )}{\sum }\frac {\left (-a^{2} d^{2}+2 a b c d -b^{2} c^{2}\right ) \left (-\frac {\operatorname {arctanh}\left (\frac {2 b \,x^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}+2 a}{2 \sqrt {\frac {a d -b c}{d}}\, \sqrt {b \,x^{4}+a}}\right )}{\sqrt {\frac {a d -b c}{d}}}+\frac {2 \underline {\hspace {1.25 ex}}\alpha ^{3} d \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \Pi \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, \frac {i \sqrt {a}\, \underline {\hspace {1.25 ex}}\alpha ^{2} d}{\sqrt {b}\, c}, \frac {\sqrt {-\frac {i \sqrt {b}}{\sqrt {a}}}}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}}\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, c \sqrt {b \,x^{4}+a}}\right )}{\underline {\hspace {1.25 ex}}\alpha ^{3}}}{8 d^{3}}\) \(322\)
elliptic \(\frac {b x \sqrt {b \,x^{4}+a}}{3 d}+\frac {\left (\frac {b \left (2 a d -b c \right )}{d^{2}}-\frac {b a}{3 d}\right ) \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}-\frac {\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (d \,\textit {\_Z}^{4}+c \right )}{\sum }\frac {\left (-a^{2} d^{2}+2 a b c d -b^{2} c^{2}\right ) \left (-\frac {\operatorname {arctanh}\left (\frac {2 b \,x^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}+2 a}{2 \sqrt {\frac {a d -b c}{d}}\, \sqrt {b \,x^{4}+a}}\right )}{\sqrt {\frac {a d -b c}{d}}}+\frac {2 \underline {\hspace {1.25 ex}}\alpha ^{3} d \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \Pi \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, \frac {i \sqrt {a}\, \underline {\hspace {1.25 ex}}\alpha ^{2} d}{\sqrt {b}\, c}, \frac {\sqrt {-\frac {i \sqrt {b}}{\sqrt {a}}}}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}}\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, c \sqrt {b \,x^{4}+a}}\right )}{\underline {\hspace {1.25 ex}}\alpha ^{3}}}{8 d^{3}}\) \(322\)

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/a^(1/2)*b^(1/2)*x^2)^(1/2)*(1+I/a^(1/2)*b^(1/2)*x^2)^(1/2)/(b*x^4+ 
a)^(1/2)*EllipticF(x*(I/a^(1/2)*b^(1/2))^(1/2),I)+1/8*(3*a^2*d^2-6*a*b*c*d 
+3*b^2*c^2)/d^2*sum(1/_alpha^3*(-1/(1/d*(a*d-b*c))^(1/2)*arctanh(1/2*(2*_a 
lpha^2*b*x^2+2*a)/(1/d*(a*d-b*c))^(1/2)/(b*x^4+a)^(1/2))+2/(I/a^(1/2)*b^(1 
/2))^(1/2)*_alpha^3*d/c*(1-I/a^(1/2)*b^(1/2)*x^2)^(1/2)*(1+I/a^(1/2)*b^(1/ 
2)*x^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)))
 
3.2.79.5 Fricas [F(-1)]

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
 
3.2.79.6 Sympy [F]

\[ \int \frac {\left (a+b x^4\right )^{3/2}}{c+d x^4} \, dx=\int \frac {\left (a + b x^{4}\right )^{\frac {3}{2}}}{c + d x^{4}}\, dx \]

input
integrate((b*x**4+a)**(3/2)/(d*x**4+c),x)
 
output
Integral((a + b*x**4)**(3/2)/(c + d*x**4), x)
 
3.2.79.7 Maxima [F]

\[ \int \frac {\left (a+b x^4\right )^{3/2}}{c+d x^4} \, dx=\int { \frac {{\left (b x^{4} + a\right )}^{\frac {3}{2}}}{d x^{4} + c} \,d x } \]

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)
 
3.2.79.8 Giac [F]

\[ \int \frac {\left (a+b x^4\right )^{3/2}}{c+d x^4} \, dx=\int { \frac {{\left (b x^{4} + a\right )}^{\frac {3}{2}}}{d x^{4} + c} \,d x } \]

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)
 
3.2.79.9 Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b x^4\right )^{3/2}}{c+d x^4} \, dx=\int \frac {{\left (b\,x^4+a\right )}^{3/2}}{d\,x^4+c} \,d x \]

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)