\(\int \frac {\sqrt {a-c x^4}}{\sqrt {d+e x^2}} \, dx\) [438]

Optimal result

Integrand size = 24, antiderivative size = 458 \[ \int \frac {\sqrt {a-c x^4}}{\sqrt {d+e x^2}} \, dx=\frac {\sqrt {d+e x^2} \sqrt {a-c x^4}}{2 e x}+\frac {c \left (d+\frac {\sqrt {a} e}{\sqrt {c}}\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right )|\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{2 e \sqrt {d+e x^2} \sqrt {a-c x^4}}+\frac {\sqrt {a} \sqrt {c} \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right ),\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{2 \sqrt {d+e x^2} \sqrt {a-c x^4}}+\frac {c d \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right ),\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{2 e \sqrt {d+e x^2} \sqrt {a-c x^4}} \] Output:

1/2*(e*x^2+d)^(1/2)*(-c*x^4+a)^(1/2)/e/x+1/2*c*(d+a^(1/2)*e/c^(1/2))*(1-a/ 
c/x^4)^(1/2)*x^3*(a^(1/2)*(e*x^2+d)/(c^(1/2)*d+a^(1/2)*e)/x^2)^(1/2)*Ellip 
ticE(1/2*(1-a^(1/2)/c^(1/2)/x^2)^(1/2)*2^(1/2),2^(1/2)*(d/(d+a^(1/2)*e/c^( 
1/2)))^(1/2))/e/(e*x^2+d)^(1/2)/(-c*x^4+a)^(1/2)+1/2*a^(1/2)*c^(1/2)*(1-a/ 
c/x^4)^(1/2)*x^3*(a^(1/2)*(e*x^2+d)/(c^(1/2)*d+a^(1/2)*e)/x^2)^(1/2)*Ellip 
ticF(1/2*(1-a^(1/2)/c^(1/2)/x^2)^(1/2)*2^(1/2),2^(1/2)*(d/(d+a^(1/2)*e/c^( 
1/2)))^(1/2))/(e*x^2+d)^(1/2)/(-c*x^4+a)^(1/2)+1/2*c*d*(1-a/c/x^4)^(1/2)*x 
^3*(a^(1/2)*(e*x^2+d)/(c^(1/2)*d+a^(1/2)*e)/x^2)^(1/2)*EllipticPi(1/2*(1-a 
^(1/2)/c^(1/2)/x^2)^(1/2)*2^(1/2),2,2^(1/2)*(d/(d+a^(1/2)*e/c^(1/2)))^(1/2 
))/e/(e*x^2+d)^(1/2)/(-c*x^4+a)^(1/2)
 

Mathematica [F]

\[ \int \frac {\sqrt {a-c x^4}}{\sqrt {d+e x^2}} \, dx=\int \frac {\sqrt {a-c x^4}}{\sqrt {d+e x^2}} \, dx \] Input:

Integrate[Sqrt[a - c*x^4]/Sqrt[d + e*x^2],x]
 

Output:

Integrate[Sqrt[a - c*x^4]/Sqrt[d + e*x^2], x]
 

Rubi [A] (warning: unable to verify)

Time = 2.00 (sec) , antiderivative size = 519, normalized size of antiderivative = 1.13, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {1566, 1803, 628, 2351, 600, 509, 508, 327, 512, 511, 321, 633, 632, 186, 413, 412}

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.

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

\(\Big \downarrow \) 1566

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

\(\Big \downarrow \) 1803

\(\displaystyle -\frac {\sqrt {a-c x^4} \sqrt {\frac {d}{x^2}+e} \int \frac {\sqrt {\frac {a}{x^4}-c} x^4}{\sqrt {\frac {d}{x^2}+e}}d\frac {1}{x^2}}{2 x \sqrt {\frac {a}{x^4}-c} \sqrt {d+e x^2}}\)

\(\Big \downarrow \) 628

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

\(\Big \downarrow \) 2351

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

\(\Big \downarrow \) 600

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

\(\Big \downarrow \) 509

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

\(\Big \downarrow \) 508

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

\(\Big \downarrow \) 327

\(\displaystyle -\frac {\sqrt {a-c x^4} \sqrt {\frac {d}{x^2}+e} \left (\frac {1}{2} \left (a \int \frac {1}{\sqrt {\frac {a}{x^4}-c} \sqrt {\frac {d}{x^2}+e}}d\frac {1}{x^2}+\frac {c d \int \frac {x^2}{\sqrt {\frac {a}{x^4}-c} \sqrt {\frac {d}{x^2}+e}}d\frac {1}{x^2}}{e}-\frac {2 \sqrt {a} \sqrt {c} \sqrt {1-\frac {a}{c x^4}} \sqrt {\frac {d}{x^2}+e} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right )|\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{e \sqrt {\frac {a}{x^4}-c} \sqrt {\frac {\sqrt {a} \left (\frac {d}{x^2}+e\right )}{\sqrt {a} e+\sqrt {c} d}}}\right )-\frac {x^2 \sqrt {\frac {a}{x^4}-c} \sqrt {\frac {d}{x^2}+e}}{e}\right )}{2 x \sqrt {\frac {a}{x^4}-c} \sqrt {d+e x^2}}\)

\(\Big \downarrow \) 512

\(\displaystyle -\frac {\sqrt {a-c x^4} \sqrt {\frac {d}{x^2}+e} \left (\frac {1}{2} \left (\frac {a \sqrt {1-\frac {a}{c x^4}} \int \frac {1}{\sqrt {1-\frac {a}{c x^4}} \sqrt {\frac {d}{x^2}+e}}d\frac {1}{x^2}}{\sqrt {\frac {a}{x^4}-c}}+\frac {c d \int \frac {x^2}{\sqrt {\frac {a}{x^4}-c} \sqrt {\frac {d}{x^2}+e}}d\frac {1}{x^2}}{e}-\frac {2 \sqrt {a} \sqrt {c} \sqrt {1-\frac {a}{c x^4}} \sqrt {\frac {d}{x^2}+e} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right )|\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{e \sqrt {\frac {a}{x^4}-c} \sqrt {\frac {\sqrt {a} \left (\frac {d}{x^2}+e\right )}{\sqrt {a} e+\sqrt {c} d}}}\right )-\frac {x^2 \sqrt {\frac {a}{x^4}-c} \sqrt {\frac {d}{x^2}+e}}{e}\right )}{2 x \sqrt {\frac {a}{x^4}-c} \sqrt {d+e x^2}}\)

\(\Big \downarrow \) 511

\(\displaystyle -\frac {\sqrt {a-c x^4} \sqrt {\frac {d}{x^2}+e} \left (\frac {1}{2} \left (-\frac {2 \sqrt {a} \sqrt {c} \sqrt {1-\frac {a}{c x^4}} \sqrt {\frac {\sqrt {a} \left (\frac {d}{x^2}+e\right )}{\sqrt {a} e+\sqrt {c} d}} \int \frac {1}{\sqrt {1-\frac {1}{x^4}} \sqrt {1-\frac {2 d}{\left (d+\frac {\sqrt {a} e}{\sqrt {c}}\right ) x^4}}}d\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}}{\sqrt {\frac {a}{x^4}-c} \sqrt {\frac {d}{x^2}+e}}+\frac {c d \int \frac {x^2}{\sqrt {\frac {a}{x^4}-c} \sqrt {\frac {d}{x^2}+e}}d\frac {1}{x^2}}{e}-\frac {2 \sqrt {a} \sqrt {c} \sqrt {1-\frac {a}{c x^4}} \sqrt {\frac {d}{x^2}+e} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right )|\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{e \sqrt {\frac {a}{x^4}-c} \sqrt {\frac {\sqrt {a} \left (\frac {d}{x^2}+e\right )}{\sqrt {a} e+\sqrt {c} d}}}\right )-\frac {x^2 \sqrt {\frac {a}{x^4}-c} \sqrt {\frac {d}{x^2}+e}}{e}\right )}{2 x \sqrt {\frac {a}{x^4}-c} \sqrt {d+e x^2}}\)

\(\Big \downarrow \) 321

\(\displaystyle -\frac {\sqrt {a-c x^4} \sqrt {\frac {d}{x^2}+e} \left (\frac {1}{2} \left (\frac {c d \int \frac {x^2}{\sqrt {\frac {a}{x^4}-c} \sqrt {\frac {d}{x^2}+e}}d\frac {1}{x^2}}{e}-\frac {2 \sqrt {a} \sqrt {c} \sqrt {1-\frac {a}{c x^4}} \sqrt {\frac {\sqrt {a} \left (\frac {d}{x^2}+e\right )}{\sqrt {a} e+\sqrt {c} d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right ),\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{\sqrt {\frac {a}{x^4}-c} \sqrt {\frac {d}{x^2}+e}}-\frac {2 \sqrt {a} \sqrt {c} \sqrt {1-\frac {a}{c x^4}} \sqrt {\frac {d}{x^2}+e} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right )|\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{e \sqrt {\frac {a}{x^4}-c} \sqrt {\frac {\sqrt {a} \left (\frac {d}{x^2}+e\right )}{\sqrt {a} e+\sqrt {c} d}}}\right )-\frac {x^2 \sqrt {\frac {a}{x^4}-c} \sqrt {\frac {d}{x^2}+e}}{e}\right )}{2 x \sqrt {\frac {a}{x^4}-c} \sqrt {d+e x^2}}\)

\(\Big \downarrow \) 633

\(\displaystyle -\frac {\sqrt {a-c x^4} \sqrt {\frac {d}{x^2}+e} \left (\frac {1}{2} \left (\frac {c d \sqrt {1-\frac {a}{c x^4}} \int \frac {x^2}{\sqrt {1-\frac {a}{c x^4}} \sqrt {\frac {d}{x^2}+e}}d\frac {1}{x^2}}{e \sqrt {\frac {a}{x^4}-c}}-\frac {2 \sqrt {a} \sqrt {c} \sqrt {1-\frac {a}{c x^4}} \sqrt {\frac {\sqrt {a} \left (\frac {d}{x^2}+e\right )}{\sqrt {a} e+\sqrt {c} d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right ),\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{\sqrt {\frac {a}{x^4}-c} \sqrt {\frac {d}{x^2}+e}}-\frac {2 \sqrt {a} \sqrt {c} \sqrt {1-\frac {a}{c x^4}} \sqrt {\frac {d}{x^2}+e} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right )|\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{e \sqrt {\frac {a}{x^4}-c} \sqrt {\frac {\sqrt {a} \left (\frac {d}{x^2}+e\right )}{\sqrt {a} e+\sqrt {c} d}}}\right )-\frac {x^2 \sqrt {\frac {a}{x^4}-c} \sqrt {\frac {d}{x^2}+e}}{e}\right )}{2 x \sqrt {\frac {a}{x^4}-c} \sqrt {d+e x^2}}\)

\(\Big \downarrow \) 632

\(\displaystyle -\frac {\sqrt {a-c x^4} \sqrt {\frac {d}{x^2}+e} \left (\frac {1}{2} \left (\frac {c d \sqrt {1-\frac {a}{c x^4}} \int \frac {x^2}{\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}} \sqrt {\frac {\sqrt {a}}{\sqrt {c} x^2}+1} \sqrt {\frac {d}{x^2}+e}}d\frac {1}{x^2}}{e \sqrt {\frac {a}{x^4}-c}}-\frac {2 \sqrt {a} \sqrt {c} \sqrt {1-\frac {a}{c x^4}} \sqrt {\frac {\sqrt {a} \left (\frac {d}{x^2}+e\right )}{\sqrt {a} e+\sqrt {c} d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right ),\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{\sqrt {\frac {a}{x^4}-c} \sqrt {\frac {d}{x^2}+e}}-\frac {2 \sqrt {a} \sqrt {c} \sqrt {1-\frac {a}{c x^4}} \sqrt {\frac {d}{x^2}+e} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right )|\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{e \sqrt {\frac {a}{x^4}-c} \sqrt {\frac {\sqrt {a} \left (\frac {d}{x^2}+e\right )}{\sqrt {a} e+\sqrt {c} d}}}\right )-\frac {x^2 \sqrt {\frac {a}{x^4}-c} \sqrt {\frac {d}{x^2}+e}}{e}\right )}{2 x \sqrt {\frac {a}{x^4}-c} \sqrt {d+e x^2}}\)

\(\Big \downarrow \) 186

\(\displaystyle -\frac {\sqrt {a-c x^4} \sqrt {\frac {d}{x^2}+e} \left (\frac {1}{2} \left (-\frac {2 c d \sqrt {1-\frac {a}{c x^4}} \int \frac {1}{\left (1-\frac {1}{x^4}\right ) \sqrt {2-\frac {1}{x^4}} \sqrt {\frac {\sqrt {c} d}{\sqrt {a}}-\frac {\sqrt {c} d}{\sqrt {a} x^4}+e}}d\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{e \sqrt {\frac {a}{x^4}-c}}-\frac {2 \sqrt {a} \sqrt {c} \sqrt {1-\frac {a}{c x^4}} \sqrt {\frac {\sqrt {a} \left (\frac {d}{x^2}+e\right )}{\sqrt {a} e+\sqrt {c} d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right ),\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{\sqrt {\frac {a}{x^4}-c} \sqrt {\frac {d}{x^2}+e}}-\frac {2 \sqrt {a} \sqrt {c} \sqrt {1-\frac {a}{c x^4}} \sqrt {\frac {d}{x^2}+e} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right )|\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{e \sqrt {\frac {a}{x^4}-c} \sqrt {\frac {\sqrt {a} \left (\frac {d}{x^2}+e\right )}{\sqrt {a} e+\sqrt {c} d}}}\right )-\frac {x^2 \sqrt {\frac {a}{x^4}-c} \sqrt {\frac {d}{x^2}+e}}{e}\right )}{2 x \sqrt {\frac {a}{x^4}-c} \sqrt {d+e x^2}}\)

\(\Big \downarrow \) 413

\(\displaystyle -\frac {\sqrt {a-c x^4} \sqrt {\frac {d}{x^2}+e} \left (\frac {1}{2} \left (-\frac {2 c d \sqrt {1-\frac {a}{c x^4}} \sqrt {1-\frac {\sqrt {c} d}{x^4 \left (\sqrt {a} e+\sqrt {c} d\right )}} \int \frac {1}{\left (1-\frac {1}{x^4}\right ) \sqrt {2-\frac {1}{x^4}} \sqrt {1-\frac {\sqrt {c} d}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^4}}}d\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{e \sqrt {\frac {a}{x^4}-c} \sqrt {-\frac {\sqrt {c} d}{\sqrt {a} x^4}+\frac {\sqrt {c} d}{\sqrt {a}}+e}}-\frac {2 \sqrt {a} \sqrt {c} \sqrt {1-\frac {a}{c x^4}} \sqrt {\frac {\sqrt {a} \left (\frac {d}{x^2}+e\right )}{\sqrt {a} e+\sqrt {c} d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right ),\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{\sqrt {\frac {a}{x^4}-c} \sqrt {\frac {d}{x^2}+e}}-\frac {2 \sqrt {a} \sqrt {c} \sqrt {1-\frac {a}{c x^4}} \sqrt {\frac {d}{x^2}+e} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right )|\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{e \sqrt {\frac {a}{x^4}-c} \sqrt {\frac {\sqrt {a} \left (\frac {d}{x^2}+e\right )}{\sqrt {a} e+\sqrt {c} d}}}\right )-\frac {x^2 \sqrt {\frac {a}{x^4}-c} \sqrt {\frac {d}{x^2}+e}}{e}\right )}{2 x \sqrt {\frac {a}{x^4}-c} \sqrt {d+e x^2}}\)

\(\Big \downarrow \) 412

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

Input:

Int[Sqrt[a - c*x^4]/Sqrt[d + e*x^2],x]
 

Output:

-1/2*(Sqrt[e + d/x^2]*Sqrt[a - c*x^4]*(-((Sqrt[-c + a/x^4]*Sqrt[e + d/x^2] 
*x^2)/e) + ((-2*Sqrt[a]*Sqrt[c]*Sqrt[1 - a/(c*x^4)]*Sqrt[e + d/x^2]*Ellipt 
icE[ArcSin[Sqrt[1 - Sqrt[a]/(Sqrt[c]*x^2)]/Sqrt[2]], (2*d)/(d + (Sqrt[a]*e 
)/Sqrt[c])])/(e*Sqrt[-c + a/x^4]*Sqrt[(Sqrt[a]*(e + d/x^2))/(Sqrt[c]*d + S 
qrt[a]*e)]) - (2*Sqrt[a]*Sqrt[c]*Sqrt[1 - a/(c*x^4)]*Sqrt[(Sqrt[a]*(e + d/ 
x^2))/(Sqrt[c]*d + Sqrt[a]*e)]*EllipticF[ArcSin[Sqrt[1 - Sqrt[a]/(Sqrt[c]* 
x^2)]/Sqrt[2]], (2*d)/(d + (Sqrt[a]*e)/Sqrt[c])])/(Sqrt[-c + a/x^4]*Sqrt[e 
 + d/x^2]) - (2*c*d*Sqrt[1 - a/(c*x^4)]*Sqrt[1 - (Sqrt[c]*d)/((Sqrt[c]*d + 
 Sqrt[a]*e)*x^4)]*EllipticPi[2, ArcSin[Sqrt[1 - Sqrt[a]/(Sqrt[c]*x^2)]/Sqr 
t[2]], (2*Sqrt[c]*d)/(Sqrt[c]*d + Sqrt[a]*e)])/(e*Sqrt[-c + a/x^4]*Sqrt[(S 
qrt[c]*d)/Sqrt[a] + e - (Sqrt[c]*d)/(Sqrt[a]*x^4)]))/2))/(Sqrt[-c + a/x^4] 
*x*Sqrt[d + e*x^2])
 

Defintions of rubi rules used

rule 186

Int[1/(((a_.) + (b_.)*(x_))*Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*(x_ 
)]*Sqrt[(g_.) + (h_.)*(x_)]), x_] :> Simp[-2   Subst[Int[1/(Simp[b*c - a*d 
- b*x^2, x]*Sqrt[Simp[(d*e - c*f)/d + f*(x^2/d), x]]*Sqrt[Simp[(d*g - c*h)/ 
d + h*(x^2/d), x]]), x], x, Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f, 
g, h}, x] && GtQ[(d*e - c*f)/d, 0]
 

rule 321

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S 
imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c 
/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 
0] &&  !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
 

rule 327

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ 
(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) 
)], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
 

rule 412

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x 
_)^2]), x_Symbol] :> Simp[(1/(a*Sqrt[c]*Sqrt[e]*Rt[-d/c, 2]))*EllipticPi[b* 
(c/(a*d)), ArcSin[Rt[-d/c, 2]*x], c*(f/(d*e))], x] /; FreeQ[{a, b, c, d, e, 
 f}, x] &&  !GtQ[d/c, 0] && GtQ[c, 0] && GtQ[e, 0] &&  !( !GtQ[f/e, 0] && S 
implerSqrtQ[-f/e, -d/c])
 

rule 413

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x 
_)^2]), x_Symbol] :> Simp[Sqrt[1 + (d/c)*x^2]/Sqrt[c + d*x^2]   Int[1/((a + 
 b*x^2)*Sqrt[1 + (d/c)*x^2]*Sqrt[e + f*x^2]), x], x] /; FreeQ[{a, b, c, d, 
e, f}, x] &&  !GtQ[c, 0]
 

rule 508

Int[Sqrt[(c_) + (d_.)*(x_)]/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> With[{q 
 = Rt[-b/a, 2]}, Simp[-2*(Sqrt[c + d*x]/(Sqrt[a]*q*Sqrt[q*((c + d*x)/(d + c 
*q))]))   Subst[Int[Sqrt[1 - 2*d*(x^2/(d + c*q))]/Sqrt[1 - x^2], x], x, Sqr 
t[(1 - q*x)/2]], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && GtQ[a, 0]
 

rule 509

Int[Sqrt[(c_) + (d_.)*(x_)]/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[Sq 
rt[1 + b*(x^2/a)]/Sqrt[a + b*x^2]   Int[Sqrt[c + d*x]/Sqrt[1 + b*(x^2/a)], 
x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] &&  !GtQ[a, 0]
 

rule 511

Int[1/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Wit 
h[{q = Rt[-b/a, 2]}, Simp[-2*(Sqrt[q*((c + d*x)/(d + c*q))]/(Sqrt[a]*q*Sqrt 
[c + d*x]))   Subst[Int[1/(Sqrt[1 - 2*d*(x^2/(d + c*q))]*Sqrt[1 - x^2]), x] 
, x, Sqrt[(1 - q*x)/2]], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && GtQ[ 
a, 0]
 

rule 512

Int[1/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Sim 
p[Sqrt[1 + b*(x^2/a)]/Sqrt[a + b*x^2]   Int[1/(Sqrt[c + d*x]*Sqrt[1 + b*(x^ 
2/a)]), x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] &&  !GtQ[a, 0]
 

rule 600

Int[((A_.) + (B_.)*(x_))/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2] 
), x_Symbol] :> Simp[B/d   Int[Sqrt[c + d*x]/Sqrt[a + b*x^2], x], x] - Simp 
[(B*c - A*d)/d   Int[1/(Sqrt[c + d*x]*Sqrt[a + b*x^2]), x], x] /; FreeQ[{a, 
 b, c, d, A, B}, x] && NegQ[b/a]
 

rule 628

Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*Sqrt[(a_) + (b_.)*(x_)^2], x 
_Symbol] :> Simp[c^(n - 1/2)*(e*x)^(m + 1)*Sqrt[c + d*x]*(Sqrt[a + b*x^2]/( 
e*(m + 1))), x] - Simp[1/(2*e*(m + 1))   Int[((e*x)^(m + 1)/(Sqrt[c + d*x]* 
Sqrt[a + b*x^2]))*ExpandToSum[(2*a*c^(n + 1/2)*(m + 1) + a*c^(n - 1/2)*d*(2 
*m + 3)*x + 2*b*c^(n + 1/2)*(m + 2)*x^2 + b*c^(n - 1/2)*d*(2*m + 5)*x^3 - 2 
*a*(m + 1)*(c + d*x)^(n + 1/2) - 2*b*(m + 1)*x^2*(c + d*x)^(n + 1/2))/x, x] 
, x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[n + 3/2, 0] && LtQ[m, -1] && 
IntegerQ[2*m]
 

rule 632

Int[1/((x_)*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] : 
> With[{q = Rt[-b/a, 2]}, Simp[1/Sqrt[a]   Int[1/(x*Sqrt[c + d*x]*Sqrt[1 - 
q*x]*Sqrt[1 + q*x]), x], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && GtQ[ 
a, 0]
 

rule 633

Int[1/((x_)*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] : 
> Simp[Sqrt[1 + b*(x^2/a)]/Sqrt[a + b*x^2]   Int[1/(x*Sqrt[c + d*x]*Sqrt[1 
+ b*(x^2/a)]), x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] &&  !GtQ[a, 0]
 

rule 1566

Int[Sqrt[(a_) + (c_.)*(x_)^4]/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[ 
Sqrt[e + d/x^2]*(Sqrt[a + c*x^4]/(x*Sqrt[d + e*x^2]*Sqrt[c + a/x^4]))   Int 
[(x*Sqrt[c + a/x^4])/Sqrt[e + d/x^2], x], x] /; FreeQ[{a, c, d, e}, x] && N 
eQ[c*d^2 + a*e^2, 0]
 

rule 1803

Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q 
_.), x_Symbol] :> Simp[1/n   Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(d + e*x 
)^q*(a + c*x^2)^p, x], x, x^n], x] /; FreeQ[{a, c, d, e, m, n, p, q}, x] && 
 EqQ[n2, 2*n] && IntegerQ[Simplify[(m + 1)/n]]
 

rule 2351

Int[((Px_)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_.))/(x_), x_S 
ymbol] :> Int[PolynomialQuotient[Px, x, x]*(c + d*x)^n*(a + b*x^2)^p, x] + 
Simp[PolynomialRemainder[Px, x, x]   Int[(c + d*x)^n*((a + b*x^2)^p/x), x], 
 x] /; FreeQ[{a, b, c, d, n, p}, x] && PolynomialQ[Px, x]
 

Maple [F]

\[\int \frac {\sqrt {-c \,x^{4}+a}}{\sqrt {e \,x^{2}+d}}d x\]

Input:

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

Output:

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

Fricas [F]

\[ \int \frac {\sqrt {a-c x^4}}{\sqrt {d+e x^2}} \, dx=\int { \frac {\sqrt {-c x^{4} + a}}{\sqrt {e x^{2} + d}} \,d x } \] Input:

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

Output:

integral(sqrt(-c*x^4 + a)/sqrt(e*x^2 + d), x)
 

Sympy [F]

\[ \int \frac {\sqrt {a-c x^4}}{\sqrt {d+e x^2}} \, dx=\int \frac {\sqrt {a - c x^{4}}}{\sqrt {d + e x^{2}}}\, dx \] Input:

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

Output:

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

Maxima [F]

\[ \int \frac {\sqrt {a-c x^4}}{\sqrt {d+e x^2}} \, dx=\int { \frac {\sqrt {-c x^{4} + a}}{\sqrt {e x^{2} + d}} \,d x } \] Input:

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

Output:

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

Giac [F]

\[ \int \frac {\sqrt {a-c x^4}}{\sqrt {d+e x^2}} \, dx=\int { \frac {\sqrt {-c x^{4} + a}}{\sqrt {e x^{2} + d}} \,d x } \] Input:

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {a-c x^4}}{\sqrt {d+e x^2}} \, dx=\int \frac {\sqrt {a-c\,x^4}}{\sqrt {e\,x^2+d}} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {\sqrt {a-c x^4}}{\sqrt {d+e x^2}} \, dx=\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}}{e \,x^{2}+d}d x \] Input:

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

Output:

int((sqrt(d + e*x**2)*sqrt(a - c*x**4))/(d + e*x**2),x)