\(\int \frac {x^4}{(a+b x^4) \sqrt {c+d x^4}} \, dx\) [246]

Optimal result

Integrand size = 24, antiderivative size = 625 \[ \int \frac {x^4}{\left (a+b x^4\right ) \sqrt {c+d x^4}} \, dx=-\frac {\sqrt [4]{-a} \arctan \left (\frac {\sqrt {-b c+a d} x}{\sqrt [4]{-a} \sqrt [4]{b} \sqrt {c+d x^4}}\right )}{4 b^{3/4} \sqrt {-b c+a d}}-\frac {\sqrt [4]{-a} \text {arctanh}\left (\frac {\sqrt {-b c+a d} x}{\sqrt [4]{-a} \sqrt [4]{b} \sqrt {c+d x^4}}\right )}{4 b^{3/4} \sqrt {-b c+a d}}+\frac {c^{3/4} \left (\sqrt {c}+\sqrt {d} x^2\right ) \sqrt {\frac {c+d x^4}{\left (\sqrt {c}+\sqrt {d} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{d} (b c+a d) \sqrt {c+d x^4}}-\frac {\left (\sqrt {b} \sqrt {c}+\sqrt {-a} \sqrt {d}\right ) \left (\sqrt {c}+\sqrt {d} x^2\right ) \sqrt {\frac {c+d x^4}{\left (\sqrt {c}+\sqrt {d} x^2\right )^2}} \operatorname {EllipticPi}\left (-\frac {\left (\sqrt {b} \sqrt {c}-\sqrt {-a} \sqrt {d}\right )^2}{4 \sqrt {-a} \sqrt {b} \sqrt {c} \sqrt {d}},2 \arctan \left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{8 b \sqrt [4]{c} \left (\sqrt {b} \sqrt {c}-\sqrt {-a} \sqrt {d}\right ) \sqrt [4]{d} \sqrt {c+d x^4}}-\frac {\left (\sqrt {b} \sqrt {c}-\sqrt {-a} \sqrt {d}\right ) \left (\sqrt {c}+\sqrt {d} x^2\right ) \sqrt {\frac {c+d x^4}{\left (\sqrt {c}+\sqrt {d} x^2\right )^2}} \operatorname {EllipticPi}\left (\frac {\left (\sqrt {b} \sqrt {c}+\sqrt {-a} \sqrt {d}\right )^2}{4 \sqrt {-a} \sqrt {b} \sqrt {c} \sqrt {d}},2 \arctan \left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{8 b \sqrt [4]{c} \left (\sqrt {b} \sqrt {c}+\sqrt {-a} \sqrt {d}\right ) \sqrt [4]{d} \sqrt {c+d x^4}} \] Output:

-1/4*(-a)^(1/4)*arctan((a*d-b*c)^(1/2)*x/(-a)^(1/4)/b^(1/4)/(d*x^4+c)^(1/2 
))/b^(3/4)/(a*d-b*c)^(1/2)-1/4*(-a)^(1/4)*arctanh((a*d-b*c)^(1/2)*x/(-a)^( 
1/4)/b^(1/4)/(d*x^4+c)^(1/2))/b^(3/4)/(a*d-b*c)^(1/2)+1/2*c^(3/4)*(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))/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))*(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/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))*(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/c^(1/4)/(b^(1/2)*c^(1/2)+(-a)^(1/2)*d^( 
1/2))/d^(1/4)/(d*x^4+c)^(1/2)
 

Mathematica [C] (verified)

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

Time = 10.05 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.10 \[ \int \frac {x^4}{\left (a+b x^4\right ) \sqrt {c+d x^4}} \, dx=\frac {x^5 \sqrt {\frac {c+d x^4}{c}} \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},1,\frac {9}{4},-\frac {d x^4}{c},-\frac {b x^4}{a}\right )}{5 a \sqrt {c+d x^4}} \] Input:

Integrate[x^4/((a + b*x^4)*Sqrt[c + d*x^4]),x]
 

Output:

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

Rubi [A] (warning: unable to verify)

Time = 2.18 (sec) , antiderivative size = 958, normalized size of antiderivative = 1.53, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {983, 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.

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

\(\Big \downarrow \) 983

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

\(\Big \downarrow \) 761

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

\(\Big \downarrow \) 925

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

\(\Big \downarrow \) 1541

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 761

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

\(\Big \downarrow \) 2221

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

\(\Big \downarrow \) 2223

\(\displaystyle \frac {\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}}-\frac {a \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}\)

Input:

Int[x^4/((a + b*x^4)*Sqrt[c + d*x^4]),x]
 

Output:

((Sqrt[c] + Sqrt[d]*x^2)*Sqrt[(c + d*x^4)/(Sqrt[c] + Sqrt[d]*x^2)^2]*Ellip 
ticF[2*ArcTan[(d^(1/4)*x)/c^(1/4)], 1/2])/(2*b*c^(1/4)*d^(1/4)*Sqrt[c + d* 
x^4]) - (a*(((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*ArcTa 
n[(d^(1/4)*x)/c^(1/4)], 1/2])/(2*c^(1/4)*(b*c + a*d)*Sqrt[c + d*x^4]) + (S 
qrt[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])*(Sqrt[c] + Sqrt[d]*x^2)*Sqrt[(c + d*x^4)/(Sqrt[c] + Sqrt[d]*x^2) 
^2]*EllipticPi[-1/4*(Sqrt[b]*Sqrt[c] - Sqrt[-a]*Sqrt[d])^2/(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))/(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] - (Sqrt[-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]*Sqr 
t[d])^2/(4*Sqrt[-a]*Sqrt[b]*Sqrt[c]*Sqrt[d]), 2*ArcTan[(d^(1/4)*x)/c^(1...
 

Defintions of rubi rules used

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 983

Int[(((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_)^(n_))^(q_.))/((a_) + (b_.)*(x_)^( 
n_)), x_Symbol] :> Simp[e^n/b   Int[(e*x)^(m - n)*(c + d*x^n)^q, x], x] - S 
imp[a*(e^n/b)   Int[(e*x)^(m - n)*((c + d*x^n)^q/(a + b*x^n)), x], x] /; Fr 
eeQ[{a, b, c, d, e, m, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LeQ[n, 
m, 2*n - 1] && IntBinomialQ[a, b, c, d, e, m, n, -1, q, x]
 

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)]
 

Maple [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 1.12 (sec) , antiderivative size = 265, normalized size of antiderivative = 0.42

method	result	size
default	\(\frac {\sqrt {1-\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \sqrt {1+\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}, i\right )}{b \sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}\, \sqrt {d \,x^{4}+c}}-\frac {a \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (\textit {\_Z}^{4} b +a \right )}{\sum }\frac {-\frac {\operatorname {arctanh}\left (\frac {2 d \,x^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}+2 c}{2 \sqrt {\frac {-a d +c b}{b}}\, \sqrt {d \,x^{4}+c}}\right )}{\sqrt {\frac {-a d +c b}{b}}}+\frac {2 \underline {\hspace {1.25 ex}}\alpha ^{3} b \sqrt {1-\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \sqrt {1+\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}, \frac {i \sqrt {c}\, \underline {\hspace {1.25 ex}}\alpha ^{2} b}{\sqrt {d}\, a}, \frac {\sqrt {-\frac {i \sqrt {d}}{\sqrt {c}}}}{\sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}}\right )}{\sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}\, a \sqrt {d \,x^{4}+c}}}{\underline {\hspace {1.25 ex}}\alpha ^{3}}\right )}{8 b^{2}}\)	\(265\)
elliptic	\(\frac {\sqrt {1-\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \sqrt {1+\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}, i\right )}{b \sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}\, \sqrt {d \,x^{4}+c}}-\frac {a \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (\textit {\_Z}^{4} b +a \right )}{\sum }\frac {-\frac {\operatorname {arctanh}\left (\frac {2 d \,x^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}+2 c}{2 \sqrt {\frac {-a d +c b}{b}}\, \sqrt {d \,x^{4}+c}}\right )}{\sqrt {\frac {-a d +c b}{b}}}+\frac {2 \underline {\hspace {1.25 ex}}\alpha ^{3} b \sqrt {1-\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \sqrt {1+\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}, \frac {i \sqrt {c}\, \underline {\hspace {1.25 ex}}\alpha ^{2} b}{\sqrt {d}\, a}, \frac {\sqrt {-\frac {i \sqrt {d}}{\sqrt {c}}}}{\sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}}\right )}{\sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}\, a \sqrt {d \,x^{4}+c}}}{\underline {\hspace {1.25 ex}}\alpha ^{3}}\right )}{8 b^{2}}\)	\(265\)

Input:

int(x^4/(b*x^4+a)/(d*x^4+c)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

1/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)-1/8*a/b^2*sum(1/_alpha^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))
 

Fricas [F]

\[ \int \frac {x^4}{\left (a+b x^4\right ) \sqrt {c+d x^4}} \, dx=\int { \frac {x^{4}}{{\left (b x^{4} + a\right )} \sqrt {d x^{4} + c}} \,d x } \] Input:

integrate(x^4/(b*x^4+a)/(d*x^4+c)^(1/2),x, algorithm="fricas")
 

Output:

integral(sqrt(d*x^4 + c)*x^4/(b*d*x^8 + (b*c + a*d)*x^4 + a*c), x)
 

Sympy [F]

\[ \int \frac {x^4}{\left (a+b x^4\right ) \sqrt {c+d x^4}} \, dx=\int \frac {x^{4}}{\left (a + b x^{4}\right ) \sqrt {c + d x^{4}}}\, dx \] Input:

integrate(x**4/(b*x**4+a)/(d*x**4+c)**(1/2),x)
 

Output:

Integral(x**4/((a + b*x**4)*sqrt(c + d*x**4)), x)
 

Maxima [F]

\[ \int \frac {x^4}{\left (a+b x^4\right ) \sqrt {c+d x^4}} \, dx=\int { \frac {x^{4}}{{\left (b x^{4} + a\right )} \sqrt {d x^{4} + c}} \,d x } \] Input:

integrate(x^4/(b*x^4+a)/(d*x^4+c)^(1/2),x, algorithm="maxima")
 

Output:

integrate(x^4/((b*x^4 + a)*sqrt(d*x^4 + c)), x)
 

Giac [F]

\[ \int \frac {x^4}{\left (a+b x^4\right ) \sqrt {c+d x^4}} \, dx=\int { \frac {x^{4}}{{\left (b x^{4} + a\right )} \sqrt {d x^{4} + c}} \,d x } \] Input:

integrate(x^4/(b*x^4+a)/(d*x^4+c)^(1/2),x, algorithm="giac")
 

Output:

integrate(x^4/((b*x^4 + a)*sqrt(d*x^4 + c)), x)
 

Mupad [F(-1)]

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

int(x^4/((a + b*x^4)*(c + d*x^4)^(1/2)),x)
 

Output:

int(x^4/((a + b*x^4)*(c + d*x^4)^(1/2)), x)
 

Reduce [F]

\[ \int \frac {x^4}{\left (a+b x^4\right ) \sqrt {c+d x^4}} \, dx=\int \frac {\sqrt {d \,x^{4}+c}\, x^{4}}{b d \,x^{8}+a d \,x^{4}+b c \,x^{4}+a c}d x \] Input:

int(x^4/(b*x^4+a)/(d*x^4+c)^(1/2),x)
 

Output:

int((sqrt(c + d*x**4)*x**4)/(a*c + a*d*x**4 + b*c*x**4 + b*d*x**8),x)