Optimal. Leaf size=449 \[ \frac{\sqrt [4]{b} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{4 \sqrt{2} \sqrt [4]{a} (b c-a d)}-\frac{\sqrt [4]{b} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{4 \sqrt{2} \sqrt [4]{a} (b c-a d)}-\frac{\sqrt [4]{d} \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt{c}+\sqrt{d} x^2\right )}{4 \sqrt{2} \sqrt [4]{c} (b c-a d)}+\frac{\sqrt [4]{d} \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt{c}+\sqrt{d} x^2\right )}{4 \sqrt{2} \sqrt [4]{c} (b c-a d)}-\frac{\sqrt [4]{b} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} \sqrt [4]{a} (b c-a d)}+\frac{\sqrt [4]{b} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{2 \sqrt{2} \sqrt [4]{a} (b c-a d)}+\frac{\sqrt [4]{d} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{2 \sqrt{2} \sqrt [4]{c} (b c-a d)}-\frac{\sqrt [4]{d} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}}+1\right )}{2 \sqrt{2} \sqrt [4]{c} (b c-a d)} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.635035, antiderivative size = 449, normalized size of antiderivative = 1., number of steps used = 19, number of rules used = 7, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.318 \[ \frac{\sqrt [4]{b} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{4 \sqrt{2} \sqrt [4]{a} (b c-a d)}-\frac{\sqrt [4]{b} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{4 \sqrt{2} \sqrt [4]{a} (b c-a d)}-\frac{\sqrt [4]{d} \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt{c}+\sqrt{d} x^2\right )}{4 \sqrt{2} \sqrt [4]{c} (b c-a d)}+\frac{\sqrt [4]{d} \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt{c}+\sqrt{d} x^2\right )}{4 \sqrt{2} \sqrt [4]{c} (b c-a d)}-\frac{\sqrt [4]{b} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} \sqrt [4]{a} (b c-a d)}+\frac{\sqrt [4]{b} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{2 \sqrt{2} \sqrt [4]{a} (b c-a d)}+\frac{\sqrt [4]{d} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{2 \sqrt{2} \sqrt [4]{c} (b c-a d)}-\frac{\sqrt [4]{d} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}}+1\right )}{2 \sqrt{2} \sqrt [4]{c} (b c-a d)} \]
Antiderivative was successfully verified.
[In] Int[x^2/((a + b*x^4)*(c + d*x^4)),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 117.405, size = 400, normalized size = 0.89 \[ \frac{\sqrt{2} \sqrt [4]{d} \log{\left (- \sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x + \sqrt{c} + \sqrt{d} x^{2} \right )}}{8 \sqrt [4]{c} \left (a d - b c\right )} - \frac{\sqrt{2} \sqrt [4]{d} \log{\left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x + \sqrt{c} + \sqrt{d} x^{2} \right )}}{8 \sqrt [4]{c} \left (a d - b c\right )} - \frac{\sqrt{2} \sqrt [4]{d} \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}} \right )}}{4 \sqrt [4]{c} \left (a d - b c\right )} + \frac{\sqrt{2} \sqrt [4]{d} \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}} \right )}}{4 \sqrt [4]{c} \left (a d - b c\right )} - \frac{\sqrt{2} \sqrt [4]{b} \log{\left (- \sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x + \sqrt{a} + \sqrt{b} x^{2} \right )}}{8 \sqrt [4]{a} \left (a d - b c\right )} + \frac{\sqrt{2} \sqrt [4]{b} \log{\left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x + \sqrt{a} + \sqrt{b} x^{2} \right )}}{8 \sqrt [4]{a} \left (a d - b c\right )} + \frac{\sqrt{2} \sqrt [4]{b} \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}} \right )}}{4 \sqrt [4]{a} \left (a d - b c\right )} - \frac{\sqrt{2} \sqrt [4]{b} \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}} \right )}}{4 \sqrt [4]{a} \left (a d - b c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**2/(b*x**4+a)/(d*x**4+c),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.259295, size = 340, normalized size = 0.76 \[ \frac{\sqrt [4]{b} \sqrt [4]{c} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )-\sqrt [4]{b} \sqrt [4]{c} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )-2 \sqrt [4]{b} \sqrt [4]{c} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )+2 \sqrt [4]{b} \sqrt [4]{c} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )-\sqrt [4]{a} \sqrt [4]{d} \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt{c}+\sqrt{d} x^2\right )+\sqrt [4]{a} \sqrt [4]{d} \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt{c}+\sqrt{d} x^2\right )+2 \sqrt [4]{a} \sqrt [4]{d} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )-2 \sqrt [4]{a} \sqrt [4]{d} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}}+1\right )}{4 \sqrt{2} \sqrt [4]{a} \sqrt [4]{c} (b c-a d)} \]
Antiderivative was successfully verified.
[In] Integrate[x^2/((a + b*x^4)*(c + d*x^4)),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.003, size = 296, normalized size = 0.7 \[{\frac{\sqrt{2}}{8\,ad-8\,bc}\ln \left ({1 \left ({x}^{2}-\sqrt [4]{{\frac{c}{d}}}x\sqrt{2}+\sqrt{{\frac{c}{d}}} \right ) \left ({x}^{2}+\sqrt [4]{{\frac{c}{d}}}x\sqrt{2}+\sqrt{{\frac{c}{d}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{c}{d}}}}}}+{\frac{\sqrt{2}}{4\,ad-4\,bc}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{c}{d}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{c}{d}}}}}}+{\frac{\sqrt{2}}{4\,ad-4\,bc}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{c}{d}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{c}{d}}}}}}-{\frac{\sqrt{2}}{8\,ad-8\,bc}\ln \left ({1 \left ({x}^{2}-\sqrt [4]{{\frac{a}{b}}}x\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) \left ({x}^{2}+\sqrt [4]{{\frac{a}{b}}}x\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-{\frac{\sqrt{2}}{4\,ad-4\,bc}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-{\frac{\sqrt{2}}{4\,ad-4\,bc}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^2/(b*x^4+a)/(d*x^4+c),x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/((b*x^4 + a)*(d*x^4 + c)),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.253101, size = 1615, normalized size = 3.6 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/((b*x^4 + a)*(d*x^4 + c)),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**2/(b*x**4+a)/(d*x**4+c),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{{\left (b x^{4} + a\right )}{\left (d x^{4} + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/((b*x^4 + a)*(d*x^4 + c)),x, algorithm="giac")
[Out]