\(\int \frac {1}{(d+e x^2)^3 \sqrt {a+c x^4}} \, dx\) [427]

Optimal result

Integrand size = 21, antiderivative size = 729 \[ \int \frac {1}{\left (d+e x^2\right )^3 \sqrt {a+c x^4}} \, dx=-\frac {3 \sqrt {c} e \left (3 c d^2+a e^2\right ) x \sqrt {a+c x^4}}{8 d^2 \left (c d^2+a e^2\right )^2 \left (\sqrt {a}+\sqrt {c} x^2\right )}+\frac {e^2 x \sqrt {a+c x^4}}{4 d \left (c d^2+a e^2\right ) \left (d+e x^2\right )^2}+\frac {3 e^2 \left (3 c d^2+a e^2\right ) x \sqrt {a+c x^4}}{8 d^2 \left (c d^2+a e^2\right )^2 \left (d+e x^2\right )}+\frac {3 \sqrt {e} \left (5 c^2 d^4+2 a c d^2 e^2+a^2 e^4\right ) \arctan \left (\frac {\sqrt {c d^2+a e^2} x}{\sqrt {d} \sqrt {e} \sqrt {a+c x^4}}\right )}{16 d^{5/2} \left (c d^2+a e^2\right )^{5/2}}+\frac {3 \sqrt [4]{a} \sqrt [4]{c} e \left (3 c d^2+a e^2\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{8 d^2 \left (c d^2+a e^2\right )^2 \sqrt {a+c x^4}}+\frac {\sqrt [4]{c} \left (4 c d^2-\sqrt {a} \sqrt {c} d e+3 a e^2\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{8 \sqrt [4]{a} d^2 \left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2+a e^2\right ) \sqrt {a+c x^4}}-\frac {3 \left (\sqrt {c} d+\sqrt {a} e\right ) \left (5 c^2 d^4+2 a c d^2 e^2+a^2 e^4\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticPi}\left (-\frac {\left (\sqrt {c} d-\sqrt {a} e\right )^2}{4 \sqrt {a} \sqrt {c} d e},2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{32 \sqrt [4]{a} \sqrt [4]{c} d^3 \left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2+a e^2\right )^2 \sqrt {a+c x^4}} \] Output:

-3/8*c^(1/2)*e*(a*e^2+3*c*d^2)*x*(c*x^4+a)^(1/2)/d^2/(a*e^2+c*d^2)^2/(a^(1 
/2)+c^(1/2)*x^2)+1/4*e^2*x*(c*x^4+a)^(1/2)/d/(a*e^2+c*d^2)/(e*x^2+d)^2+3/8 
*e^2*(a*e^2+3*c*d^2)*x*(c*x^4+a)^(1/2)/d^2/(a*e^2+c*d^2)^2/(e*x^2+d)+3/16* 
e^(1/2)*(a^2*e^4+2*a*c*d^2*e^2+5*c^2*d^4)*arctan((a*e^2+c*d^2)^(1/2)*x/d^( 
1/2)/e^(1/2)/(c*x^4+a)^(1/2))/d^(5/2)/(a*e^2+c*d^2)^(5/2)+3/8*a^(1/4)*c^(1 
/4)*e*(a*e^2+3*c*d^2)*(a^(1/2)+c^(1/2)*x^2)*((c*x^4+a)/(a^(1/2)+c^(1/2)*x^ 
2)^2)^(1/2)*EllipticE(sin(2*arctan(c^(1/4)*x/a^(1/4))),1/2*2^(1/2))/d^2/(a 
*e^2+c*d^2)^2/(c*x^4+a)^(1/2)+1/8*c^(1/4)*(4*c*d^2-a^(1/2)*c^(1/2)*d*e+3*a 
*e^2)*(a^(1/2)+c^(1/2)*x^2)*((c*x^4+a)/(a^(1/2)+c^(1/2)*x^2)^2)^(1/2)*Inve 
rseJacobiAM(2*arctan(c^(1/4)*x/a^(1/4)),1/2*2^(1/2))/a^(1/4)/d^2/(c^(1/2)* 
d-a^(1/2)*e)/(a*e^2+c*d^2)/(c*x^4+a)^(1/2)-3/32*(c^(1/2)*d+a^(1/2)*e)*(a^2 
*e^4+2*a*c*d^2*e^2+5*c^2*d^4)*(a^(1/2)+c^(1/2)*x^2)*((c*x^4+a)/(a^(1/2)+c^ 
(1/2)*x^2)^2)^(1/2)*EllipticPi(sin(2*arctan(c^(1/4)*x/a^(1/4))),-1/4*(c^(1 
/2)*d-a^(1/2)*e)^2/a^(1/2)/c^(1/2)/d/e,1/2*2^(1/2))/a^(1/4)/c^(1/4)/d^3/(c 
^(1/2)*d-a^(1/2)*e)/(a*e^2+c*d^2)^2/(c*x^4+a)^(1/2)
 

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 10.88 (sec) , antiderivative size = 332, normalized size of antiderivative = 0.46 \[ \int \frac {1}{\left (d+e x^2\right )^3 \sqrt {a+c x^4}} \, dx=\frac {\frac {d e^2 x \left (a+c x^4\right ) \left (a e^2 \left (5 d+3 e x^2\right )+c d^2 \left (11 d+9 e x^2\right )\right )}{\left (d+e x^2\right )^2}+\frac {\sqrt {1+\frac {c x^4}{a}} \left (-3 \sqrt {a} \sqrt {c} d e \left (3 c d^2+a e^2\right ) E\left (\left .i \text {arcsinh}\left (\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} x\right )\right |-1\right )+i \left (\sqrt {c} d \left (7 c^{3/2} d^3-9 i \sqrt {a} c d^2 e+a \sqrt {c} d e^2-3 i a^{3/2} e^3\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} x\right ),-1\right )-3 \left (5 c^2 d^4+2 a c d^2 e^2+a^2 e^4\right ) \operatorname {EllipticPi}\left (-\frac {i \sqrt {a} e}{\sqrt {c} d},i \text {arcsinh}\left (\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} x\right ),-1\right )\right )\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}}}{8 d^3 \left (c d^2+a e^2\right )^2 \sqrt {a+c x^4}} \] Input:

Integrate[1/((d + e*x^2)^3*Sqrt[a + c*x^4]),x]
 

Output:

((d*e^2*x*(a + c*x^4)*(a*e^2*(5*d + 3*e*x^2) + c*d^2*(11*d + 9*e*x^2)))/(d 
 + e*x^2)^2 + (Sqrt[1 + (c*x^4)/a]*(-3*Sqrt[a]*Sqrt[c]*d*e*(3*c*d^2 + a*e^ 
2)*EllipticE[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[a]]*x], -1] + I*(Sqrt[c]*d*(7 
*c^(3/2)*d^3 - (9*I)*Sqrt[a]*c*d^2*e + a*Sqrt[c]*d*e^2 - (3*I)*a^(3/2)*e^3 
)*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[a]]*x], -1] - 3*(5*c^2*d^4 + 2 
*a*c*d^2*e^2 + a^2*e^4)*EllipticPi[((-I)*Sqrt[a]*e)/(Sqrt[c]*d), I*ArcSinh 
[Sqrt[(I*Sqrt[c])/Sqrt[a]]*x], -1])))/Sqrt[(I*Sqrt[c])/Sqrt[a]])/(8*d^3*(c 
*d^2 + a*e^2)^2*Sqrt[a + c*x^4])
 

Rubi [A] (verified)

Time = 2.80 (sec) , antiderivative size = 700, normalized size of antiderivative = 0.96, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.524, Rules used = {1552, 25, 2211, 25, 2233, 27, 1510, 2227, 27, 761, 2221}

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.

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

\(\Big \downarrow \) 1552

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2211

\(\displaystyle \frac {\frac {3 e^2 x \sqrt {a+c x^4} \left (a e^2+3 c d^2\right )}{2 d \left (d+e x^2\right ) \left (a e^2+c d^2\right )}-\frac {\int -\frac {8 c^2 d^4+5 a c e^2 d^2-4 c e \left (4 c d^2+a e^2\right ) x^2 d+3 a^2 e^4-3 c e^2 \left (3 c d^2+a e^2\right ) x^4}{\left (e x^2+d\right ) \sqrt {c x^4+a}}dx}{2 d \left (a e^2+c d^2\right )}}{4 d \left (a e^2+c d^2\right )}+\frac {e^2 x \sqrt {a+c x^4}}{4 d \left (d+e x^2\right )^2 \left (a e^2+c d^2\right )}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\int \frac {8 c^2 d^4+5 a c e^2 d^2-4 c e \left (4 c d^2+a e^2\right ) x^2 d+3 a^2 e^4-3 c e^2 \left (3 c d^2+a e^2\right ) x^4}{\left (e x^2+d\right ) \sqrt {c x^4+a}}dx}{2 d \left (a e^2+c d^2\right )}+\frac {3 e^2 x \sqrt {a+c x^4} \left (a e^2+3 c d^2\right )}{2 d \left (d+e x^2\right ) \left (a e^2+c d^2\right )}}{4 d \left (a e^2+c d^2\right )}+\frac {e^2 x \sqrt {a+c x^4}}{4 d \left (d+e x^2\right )^2 \left (a e^2+c d^2\right )}\)

\(\Big \downarrow \) 2233

\(\displaystyle \frac {\frac {\frac {\int \frac {c e \left (8 c^2 d^4+5 a c e^2 d^2-3 \sqrt {a} \sqrt {c} e \left (3 c d^2+a e^2\right ) d+3 a^2 e^4-\sqrt {c} e \left (7 c^{3/2} d^3+9 \sqrt {a} c e d^2+a \sqrt {c} e^2 d+3 a^{3/2} e^3\right ) x^2\right )}{\left (e x^2+d\right ) \sqrt {c x^4+a}}dx}{c e}+3 \sqrt {a} \sqrt {c} e \left (a e^2+3 c d^2\right ) \int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {a} \sqrt {c x^4+a}}dx}{2 d \left (a e^2+c d^2\right )}+\frac {3 e^2 x \sqrt {a+c x^4} \left (a e^2+3 c d^2\right )}{2 d \left (d+e x^2\right ) \left (a e^2+c d^2\right )}}{4 d \left (a e^2+c d^2\right )}+\frac {e^2 x \sqrt {a+c x^4}}{4 d \left (d+e x^2\right )^2 \left (a e^2+c d^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int \frac {8 c^2 d^4+5 a c e^2 d^2-3 \sqrt {a} \sqrt {c} e \left (3 c d^2+a e^2\right ) d+3 a^2 e^4-\sqrt {c} e \left (7 c^{3/2} d^3+9 \sqrt {a} c e d^2+a \sqrt {c} e^2 d+3 a^{3/2} e^3\right ) x^2}{\left (e x^2+d\right ) \sqrt {c x^4+a}}dx+3 \sqrt {c} e \left (a e^2+3 c d^2\right ) \int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {c x^4+a}}dx}{2 d \left (a e^2+c d^2\right )}+\frac {3 e^2 x \sqrt {a+c x^4} \left (a e^2+3 c d^2\right )}{2 d \left (d+e x^2\right ) \left (a e^2+c d^2\right )}}{4 d \left (a e^2+c d^2\right )}+\frac {e^2 x \sqrt {a+c x^4}}{4 d \left (d+e x^2\right )^2 \left (a e^2+c d^2\right )}\)

\(\Big \downarrow \) 1510

\(\displaystyle \frac {\frac {\int \frac {8 c^2 d^4+5 a c e^2 d^2-3 \sqrt {a} \sqrt {c} e \left (3 c d^2+a e^2\right ) d+3 a^2 e^4-\sqrt {c} e \left (7 c^{3/2} d^3+9 \sqrt {a} c e d^2+a \sqrt {c} e^2 d+3 a^{3/2} e^3\right ) x^2}{\left (e x^2+d\right ) \sqrt {c x^4+a}}dx+3 \sqrt {c} e \left (a e^2+3 c d^2\right ) \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{\sqrt [4]{c} \sqrt {a+c x^4}}-\frac {x \sqrt {a+c x^4}}{\sqrt {a}+\sqrt {c} x^2}\right )}{2 d \left (a e^2+c d^2\right )}+\frac {3 e^2 x \sqrt {a+c x^4} \left (a e^2+3 c d^2\right )}{2 d \left (d+e x^2\right ) \left (a e^2+c d^2\right )}}{4 d \left (a e^2+c d^2\right )}+\frac {e^2 x \sqrt {a+c x^4}}{4 d \left (d+e x^2\right )^2 \left (a e^2+c d^2\right )}\)

\(\Big \downarrow \) 2227

\(\displaystyle \frac {\frac {-\frac {3 \sqrt {a} e \left (a^2 e^4+2 a c d^2 e^2+5 c^2 d^4\right ) \int \frac {\sqrt {c} x^2+\sqrt {a}}{\sqrt {a} \left (e x^2+d\right ) \sqrt {c x^4+a}}dx}{\sqrt {c} d-\sqrt {a} e}+\frac {2 \sqrt {c} \left (a e^2+c d^2\right ) \left (-\sqrt {a} \sqrt {c} d e+3 a e^2+4 c d^2\right ) \int \frac {1}{\sqrt {c x^4+a}}dx}{\sqrt {c} d-\sqrt {a} e}+3 \sqrt {c} e \left (a e^2+3 c d^2\right ) \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{\sqrt [4]{c} \sqrt {a+c x^4}}-\frac {x \sqrt {a+c x^4}}{\sqrt {a}+\sqrt {c} x^2}\right )}{2 d \left (a e^2+c d^2\right )}+\frac {3 e^2 x \sqrt {a+c x^4} \left (a e^2+3 c d^2\right )}{2 d \left (d+e x^2\right ) \left (a e^2+c d^2\right )}}{4 d \left (a e^2+c d^2\right )}+\frac {e^2 x \sqrt {a+c x^4}}{4 d \left (d+e x^2\right )^2 \left (a e^2+c d^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {-\frac {3 e \left (a^2 e^4+2 a c d^2 e^2+5 c^2 d^4\right ) \int \frac {\sqrt {c} x^2+\sqrt {a}}{\left (e x^2+d\right ) \sqrt {c x^4+a}}dx}{\sqrt {c} d-\sqrt {a} e}+\frac {2 \sqrt {c} \left (a e^2+c d^2\right ) \left (-\sqrt {a} \sqrt {c} d e+3 a e^2+4 c d^2\right ) \int \frac {1}{\sqrt {c x^4+a}}dx}{\sqrt {c} d-\sqrt {a} e}+3 \sqrt {c} e \left (a e^2+3 c d^2\right ) \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{\sqrt [4]{c} \sqrt {a+c x^4}}-\frac {x \sqrt {a+c x^4}}{\sqrt {a}+\sqrt {c} x^2}\right )}{2 d \left (a e^2+c d^2\right )}+\frac {3 e^2 x \sqrt {a+c x^4} \left (a e^2+3 c d^2\right )}{2 d \left (d+e x^2\right ) \left (a e^2+c d^2\right )}}{4 d \left (a e^2+c d^2\right )}+\frac {e^2 x \sqrt {a+c x^4}}{4 d \left (d+e x^2\right )^2 \left (a e^2+c d^2\right )}\)

\(\Big \downarrow \) 761

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

\(\Big \downarrow \) 2221

\(\displaystyle \frac {\frac {-\frac {3 e \left (a^2 e^4+2 a c d^2 e^2+5 c^2 d^4\right ) \left (\frac {\left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (\sqrt {a} e+\sqrt {c} d\right ) \operatorname {EllipticPi}\left (-\frac {\sqrt {a} \left (\frac {\sqrt {c} d}{\sqrt {a}}-e\right )^2}{4 \sqrt {c} d e},2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{a} \sqrt [4]{c} d e \sqrt {a+c x^4}}-\frac {\left (\sqrt {c} d-\sqrt {a} e\right ) \arctan \left (\frac {x \sqrt {a e^2+c d^2}}{\sqrt {d} \sqrt {e} \sqrt {a+c x^4}}\right )}{2 \sqrt {d} \sqrt {e} \sqrt {a e^2+c d^2}}\right )}{\sqrt {c} d-\sqrt {a} e}+\frac {\sqrt [4]{c} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (a e^2+c d^2\right ) \left (-\sqrt {a} \sqrt {c} d e+3 a e^2+4 c d^2\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{\sqrt [4]{a} \sqrt {a+c x^4} \left (\sqrt {c} d-\sqrt {a} e\right )}+3 \sqrt {c} e \left (a e^2+3 c d^2\right ) \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{\sqrt [4]{c} \sqrt {a+c x^4}}-\frac {x \sqrt {a+c x^4}}{\sqrt {a}+\sqrt {c} x^2}\right )}{2 d \left (a e^2+c d^2\right )}+\frac {3 e^2 x \sqrt {a+c x^4} \left (a e^2+3 c d^2\right )}{2 d \left (d+e x^2\right ) \left (a e^2+c d^2\right )}}{4 d \left (a e^2+c d^2\right )}+\frac {e^2 x \sqrt {a+c x^4}}{4 d \left (d+e x^2\right )^2 \left (a e^2+c d^2\right )}\)

Input:

Int[1/((d + e*x^2)^3*Sqrt[a + c*x^4]),x]
 

Output:

(e^2*x*Sqrt[a + c*x^4])/(4*d*(c*d^2 + a*e^2)*(d + e*x^2)^2) + ((3*e^2*(3*c 
*d^2 + a*e^2)*x*Sqrt[a + c*x^4])/(2*d*(c*d^2 + a*e^2)*(d + e*x^2)) + (3*Sq 
rt[c]*e*(3*c*d^2 + a*e^2)*(-((x*Sqrt[a + c*x^4])/(Sqrt[a] + Sqrt[c]*x^2)) 
+ (a^(1/4)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2 
)^2]*EllipticE[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(c^(1/4)*Sqrt[a + c*x^ 
4])) + (c^(1/4)*(c*d^2 + a*e^2)*(4*c*d^2 - Sqrt[a]*Sqrt[c]*d*e + 3*a*e^2)* 
(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*Ellipt 
icF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(a^(1/4)*(Sqrt[c]*d - Sqrt[a]*e)* 
Sqrt[a + c*x^4]) - (3*e*(5*c^2*d^4 + 2*a*c*d^2*e^2 + a^2*e^4)*(-1/2*((Sqrt 
[c]*d - Sqrt[a]*e)*ArcTan[(Sqrt[c*d^2 + a*e^2]*x)/(Sqrt[d]*Sqrt[e]*Sqrt[a 
+ c*x^4])])/(Sqrt[d]*Sqrt[e]*Sqrt[c*d^2 + a*e^2]) + ((Sqrt[c]*d + Sqrt[a]* 
e)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*Ell 
ipticPi[-1/4*(Sqrt[a]*((Sqrt[c]*d)/Sqrt[a] - e)^2)/(Sqrt[c]*d*e), 2*ArcTan 
[(c^(1/4)*x)/a^(1/4)], 1/2])/(4*a^(1/4)*c^(1/4)*d*e*Sqrt[a + c*x^4])))/(Sq 
rt[c]*d - Sqrt[a]*e))/(2*d*(c*d^2 + a*e^2)))/(4*d*(c*d^2 + a*e^2))
 

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 1510

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> With[{q = 
 Rt[c/a, 4]}, Simp[(-d)*x*(Sqrt[a + c*x^4]/(a*(1 + q^2*x^2))), x] + Simp[d* 
(1 + q^2*x^2)*(Sqrt[(a + c*x^4)/(a*(1 + q^2*x^2)^2)]/(q*Sqrt[a + c*x^4]))*E 
llipticE[2*ArcTan[q*x], 1/2], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, c, d, e 
}, x] && PosQ[c/a]
 

rule 1552

Int[((d_) + (e_.)*(x_)^2)^(q_)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> Simp 
[(-e^2)*x*(d + e*x^2)^(q + 1)*(Sqrt[a + c*x^4]/(2*d*(q + 1)*(c*d^2 + a*e^2) 
)), x] + Simp[1/(2*d*(q + 1)*(c*d^2 + a*e^2))   Int[((d + e*x^2)^(q + 1)/Sq 
rt[a + c*x^4])*Simp[a*e^2*(2*q + 3) + 2*c*d^2*(q + 1) - 2*e*c*d*(q + 1)*x^2 
 + c*e^2*(2*q + 5)*x^4, x], x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + 
 a*e^2, 0] && ILtQ[q, -1]
 

rule 2211

Int[((P4x_)*((d_) + (e_.)*(x_)^2)^(q_))/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol 
] :> With[{A = Coeff[P4x, x, 0], B = Coeff[P4x, x, 2], C = Coeff[P4x, x, 4] 
}, Simp[(-(C*d^2 - B*d*e + A*e^2))*x*(d + e*x^2)^(q + 1)*(Sqrt[a + c*x^4]/( 
2*d*(q + 1)*(c*d^2 + a*e^2))), x] + Simp[1/(2*d*(q + 1)*(c*d^2 + a*e^2)) 
Int[((d + e*x^2)^(q + 1)/Sqrt[a + c*x^4])*Simp[a*d*(C*d - B*e) + A*(a*e^2*( 
2*q + 3) + 2*c*d^2*(q + 1)) + 2*d*(B*c*d - A*c*e + a*C*e)*(q + 1)*x^2 + c*( 
C*d^2 - B*d*e + A*e^2)*(2*q + 5)*x^4, x], x], x]] /; FreeQ[{a, c, d, e}, x] 
 && PolyQ[P4x, x^2] && LeQ[Expon[P4x, x], 4] && ILtQ[q, -1]
 

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 2227

Int[((A_) + (B_.)*(x_)^2)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]) 
, x_Symbol] :> With[{q = Rt[c/a, 2]}, Simp[(A*(c*d + a*e*q) - a*B*(e + d*q) 
)/(c*d^2 - a*e^2)   Int[1/Sqrt[a + c*x^4], x], x] + Simp[a*(B*d - 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, A, B}, x] && NeQ[c*d^2 - a*e^2, 0] && PosQ[c/a] 
 && NeQ[c*A^2 - a*B^2, 0]
 

rule 2233

Int[(P4x_)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> 
With[{q = Rt[c/a, 2], A = Coeff[P4x, x, 0], B = Coeff[P4x, x, 2], C = Coeff 
[P4x, x, 4]}, Simp[-C/(e*q)   Int[(1 - q*x^2)/Sqrt[a + c*x^4], x], x] + Sim 
p[1/(c*e)   Int[(A*c*e + a*C*d*q + (B*c*e - C*(c*d - a*e*q))*x^2)/((d + e*x 
^2)*Sqrt[a + c*x^4]), x], x]] /; FreeQ[{a, c, d, e}, x] && PolyQ[P4x, x^2, 
2] && NeQ[c*d^2 - a*e^2, 0] && PosQ[c/a]
 

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 1.10 (sec) , antiderivative size = 1018, normalized size of antiderivative = 1.40

method	result	size
default	\(\text {Expression too large to display}\)	\(1018\)
elliptic	\(\text {Expression too large to display}\)	\(1018\)

Input:

int(1/(e*x^2+d)^3/(c*x^4+a)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

1/4*e^2*x*(c*x^4+a)^(1/2)/d/(a*e^2+c*d^2)/(e*x^2+d)^2+3/8*e^2*(a*e^2+3*c*d 
^2)*x*(c*x^4+a)^(1/2)/d^2/(a*e^2+c*d^2)^2/(e*x^2+d)-1/8*c/d/(a*e^2+c*d^2)^ 
2/(I/a^(1/2)*c^(1/2))^(1/2)*(1-I/a^(1/2)*c^(1/2)*x^2)^(1/2)*(1+I/a^(1/2)*c 
^(1/2)*x^2)^(1/2)/(c*x^4+a)^(1/2)*EllipticF(x*(I/a^(1/2)*c^(1/2))^(1/2),I) 
*a*e^2-7/8*c^2*d/(a*e^2+c*d^2)^2/(I/a^(1/2)*c^(1/2))^(1/2)*(1-I/a^(1/2)*c^ 
(1/2)*x^2)^(1/2)*(1+I/a^(1/2)*c^(1/2)*x^2)^(1/2)/(c*x^4+a)^(1/2)*EllipticF 
(x*(I/a^(1/2)*c^(1/2))^(1/2),I)-9/8*I*c^(3/2)*e/(a*e^2+c*d^2)^2*a^(1/2)/(I 
/a^(1/2)*c^(1/2))^(1/2)*(1-I/a^(1/2)*c^(1/2)*x^2)^(1/2)*(1+I/a^(1/2)*c^(1/ 
2)*x^2)^(1/2)/(c*x^4+a)^(1/2)*EllipticF(x*(I/a^(1/2)*c^(1/2))^(1/2),I)+3/8 
*I*c^(1/2)*e^3/d^2/(a*e^2+c*d^2)^2*a^(3/2)/(I/a^(1/2)*c^(1/2))^(1/2)*(1-I/ 
a^(1/2)*c^(1/2)*x^2)^(1/2)*(1+I/a^(1/2)*c^(1/2)*x^2)^(1/2)/(c*x^4+a)^(1/2) 
*EllipticE(x*(I/a^(1/2)*c^(1/2))^(1/2),I)-3/8*I*c^(1/2)*e^3/d^2/(a*e^2+c*d 
^2)^2*a^(3/2)/(I/a^(1/2)*c^(1/2))^(1/2)*(1-I/a^(1/2)*c^(1/2)*x^2)^(1/2)*(1 
+I/a^(1/2)*c^(1/2)*x^2)^(1/2)/(c*x^4+a)^(1/2)*EllipticF(x*(I/a^(1/2)*c^(1/ 
2))^(1/2),I)+9/8*I*c^(3/2)*e/(a*e^2+c*d^2)^2*a^(1/2)/(I/a^(1/2)*c^(1/2))^( 
1/2)*(1-I/a^(1/2)*c^(1/2)*x^2)^(1/2)*(1+I/a^(1/2)*c^(1/2)*x^2)^(1/2)/(c*x^ 
4+a)^(1/2)*EllipticE(x*(I/a^(1/2)*c^(1/2))^(1/2),I)+3/8/d^3/(a*e^2+c*d^2)^ 
2*e^4/(I/a^(1/2)*c^(1/2))^(1/2)*(1-I/a^(1/2)*c^(1/2)*x^2)^(1/2)*(1+I/a^(1/ 
2)*c^(1/2)*x^2)^(1/2)/(c*x^4+a)^(1/2)*EllipticPi(x*(I/a^(1/2)*c^(1/2))^(1/ 
2),I*a^(1/2)/c^(1/2)/d*e,(-I/a^(1/2)*c^(1/2))^(1/2)/(I/a^(1/2)*c^(1/2))...
 

Fricas [F(-1)]

Timed out. \[ \int \frac {1}{\left (d+e x^2\right )^3 \sqrt {a+c x^4}} \, dx=\text {Timed out} \] Input:

integrate(1/(e*x^2+d)^3/(c*x^4+a)^(1/2),x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F]

\[ \int \frac {1}{\left (d+e x^2\right )^3 \sqrt {a+c x^4}} \, dx=\int \frac {1}{\sqrt {a + c x^{4}} \left (d + e x^{2}\right )^{3}}\, dx \] Input:

integrate(1/(e*x**2+d)**3/(c*x**4+a)**(1/2),x)
 

Output:

Integral(1/(sqrt(a + c*x**4)*(d + e*x**2)**3), x)
 

Maxima [F]

\[ \int \frac {1}{\left (d+e x^2\right )^3 \sqrt {a+c x^4}} \, dx=\int { \frac {1}{\sqrt {c x^{4} + a} {\left (e x^{2} + d\right )}^{3}} \,d x } \] Input:

integrate(1/(e*x^2+d)^3/(c*x^4+a)^(1/2),x, algorithm="maxima")
 

Output:

integrate(1/(sqrt(c*x^4 + a)*(e*x^2 + d)^3), x)
 

Giac [F]

\[ \int \frac {1}{\left (d+e x^2\right )^3 \sqrt {a+c x^4}} \, dx=\int { \frac {1}{\sqrt {c x^{4} + a} {\left (e x^{2} + d\right )}^{3}} \,d x } \] Input:

integrate(1/(e*x^2+d)^3/(c*x^4+a)^(1/2),x, algorithm="giac")
 

Output:

integrate(1/(sqrt(c*x^4 + a)*(e*x^2 + d)^3), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\left (d+e x^2\right )^3 \sqrt {a+c x^4}} \, dx=\int \frac {1}{\sqrt {c\,x^4+a}\,{\left (e\,x^2+d\right )}^3} \,d x \] Input:

int(1/((a + c*x^4)^(1/2)*(d + e*x^2)^3),x)
 

Output:

int(1/((a + c*x^4)^(1/2)*(d + e*x^2)^3), x)
 

Reduce [F]

\[ \int \frac {1}{\left (d+e x^2\right )^3 \sqrt {a+c x^4}} \, dx=\int \frac {\sqrt {c \,x^{4}+a}}{c \,e^{3} x^{10}+3 c d \,e^{2} x^{8}+a \,e^{3} x^{6}+3 c \,d^{2} e \,x^{6}+3 a d \,e^{2} x^{4}+c \,d^{3} x^{4}+3 a \,d^{2} e \,x^{2}+a \,d^{3}}d x \] Input:

int(1/(e*x^2+d)^3/(c*x^4+a)^(1/2),x)
 

Output:

int(sqrt(a + c*x**4)/(a*d**3 + 3*a*d**2*e*x**2 + 3*a*d*e**2*x**4 + a*e**3* 
x**6 + c*d**3*x**4 + 3*c*d**2*e*x**6 + 3*c*d*e**2*x**8 + c*e**3*x**10),x)