Optimal. Leaf size=210 \[ -\frac {\tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x}{\sqrt {a x^4+b}+\sqrt {a} x^2+\sqrt {b}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {6-4 \sqrt {2}} \sqrt [4]{a} \sqrt [4]{b} x}{\sqrt {a x^4+b}+\sqrt {a} x^2+\sqrt {b}}\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {6+4 \sqrt {2}} \sqrt [4]{a} \sqrt [4]{b} x}{\sqrt {a x^4+b}+\sqrt {a} x^2+\sqrt {b}}\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}} \]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 97, normalized size of antiderivative = 0.46, number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {404, 212, 206, 203} \begin {gather*} -\frac {\tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x}{\sqrt {a x^4+b}}\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x}{\sqrt {a x^4+b}}\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 203
Rule 206
Rule 212
Rule 404
Rubi steps
\begin {align*} \int \frac {\sqrt {b+a x^4}}{-b+a x^4} \, dx &=-\operatorname {Subst}\left (\int \frac {1}{1-4 a b x^4} \, dx,x,\frac {x}{\sqrt {b+a x^4}}\right )\\ &=-\left (\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{1-2 \sqrt {a} \sqrt {b} x^2} \, dx,x,\frac {x}{\sqrt {b+a x^4}}\right )\right )-\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{1+2 \sqrt {a} \sqrt {b} x^2} \, dx,x,\frac {x}{\sqrt {b+a x^4}}\right )\\ &=-\frac {\tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x}{\sqrt {b+a x^4}}\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x}{\sqrt {b+a x^4}}\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.21, size = 152, normalized size = 0.72 \begin {gather*} -\frac {5 b x \sqrt {a x^4+b} F_1\left (\frac {1}{4};-\frac {1}{2},1;\frac {5}{4};-\frac {a x^4}{b},\frac {a x^4}{b}\right )}{\left (b-a x^4\right ) \left (2 a x^4 \left (2 F_1\left (\frac {5}{4};-\frac {1}{2},2;\frac {9}{4};-\frac {a x^4}{b},\frac {a x^4}{b}\right )+F_1\left (\frac {5}{4};\frac {1}{2},1;\frac {9}{4};-\frac {a x^4}{b},\frac {a x^4}{b}\right )\right )+5 b F_1\left (\frac {1}{4};-\frac {1}{2},1;\frac {5}{4};-\frac {a x^4}{b},\frac {a x^4}{b}\right )\right )} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.37, size = 97, normalized size = 0.46 \begin {gather*} -\frac {\tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x}{\sqrt {a x^4+b}}\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x}{\sqrt {a x^4+b}}\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.77, size = 259, normalized size = 1.23 \begin {gather*} \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (\frac {1}{a b}\right )^{\frac {1}{4}} \arctan \left (\frac {\left (\frac {1}{4}\right )^{\frac {1}{4}} \sqrt {a x^{4} + b} \left (\frac {1}{a b}\right )^{\frac {1}{4}} - \frac {\left (\frac {1}{4}\right )^{\frac {1}{4}} a x^{2} \left (\frac {1}{a b}\right )^{\frac {1}{4}} + 2 \, \left (\frac {1}{4}\right )^{\frac {3}{4}} a b \left (\frac {1}{a b}\right )^{\frac {3}{4}}}{\sqrt {a}}}{x}\right ) - \frac {1}{4} \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (\frac {1}{a b}\right )^{\frac {1}{4}} \log \left (\frac {4 \, \left (\frac {1}{4}\right )^{\frac {3}{4}} a b x^{3} \left (\frac {1}{a b}\right )^{\frac {3}{4}} + 2 \, \left (\frac {1}{4}\right )^{\frac {1}{4}} b x \left (\frac {1}{a b}\right )^{\frac {1}{4}} + \sqrt {a x^{4} + b} {\left (x^{2} + b \sqrt {\frac {1}{a b}}\right )}}{a x^{4} - b}\right ) + \frac {1}{4} \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (\frac {1}{a b}\right )^{\frac {1}{4}} \log \left (-\frac {4 \, \left (\frac {1}{4}\right )^{\frac {3}{4}} a b x^{3} \left (\frac {1}{a b}\right )^{\frac {3}{4}} + 2 \, \left (\frac {1}{4}\right )^{\frac {1}{4}} b x \left (\frac {1}{a b}\right )^{\frac {1}{4}} - \sqrt {a x^{4} + b} {\left (x^{2} + b \sqrt {\frac {1}{a b}}\right )}}{a x^{4} - b}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {a x^{4} + b}}{a x^{4} - b}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.04, size = 97, normalized size = 0.46 \begin {gather*} \frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {a \,x^{4}+b}\, \sqrt {2}}{2 x \left (a b \right )^{\frac {1}{4}}}\right )}{4 \left (a b \right )^{\frac {1}{4}}}-\frac {\sqrt {2}\, \ln \left (\frac {\frac {\sqrt {a \,x^{4}+b}\, \sqrt {2}}{2 x}+\left (a b \right )^{\frac {1}{4}}}{\frac {\sqrt {a \,x^{4}+b}\, \sqrt {2}}{2 x}-\left (a b \right )^{\frac {1}{4}}}\right )}{8 \left (a b \right )^{\frac {1}{4}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {a x^{4} + b}}{a x^{4} - b}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} -\int \frac {\sqrt {a\,x^4+b}}{b-a\,x^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {a x^{4} + b}}{a x^{4} - b}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________