Optimal. Leaf size=101 \[ \frac {\sqrt [4]{a x^4-b x^2} \left (a x^2+4 b\right )}{2 x}+\frac {5}{4} \sqrt [4]{a} b \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b x^2}}\right )-\frac {5}{4} \sqrt [4]{a} b \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b x^2}}\right ) \]
________________________________________________________________________________________
Rubi [A] time = 0.47, antiderivative size = 183, normalized size of antiderivative = 1.81, number of steps used = 16, number of rules used = 9, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.310, Rules used = {2052, 2004, 2032, 329, 331, 298, 203, 206, 2020} \begin {gather*} \frac {1}{2} a x \sqrt [4]{a x^4-b x^2}+\frac {2 b \sqrt [4]{a x^4-b x^2}}{x}+\frac {5 \sqrt [4]{a} b x^{3/2} \left (a x^2-b\right )^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2-b}}\right )}{4 \left (a x^4-b x^2\right )^{3/4}}-\frac {5 \sqrt [4]{a} b x^{3/2} \left (a x^2-b\right )^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2-b}}\right )}{4 \left (a x^4-b x^2\right )^{3/4}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 203
Rule 206
Rule 298
Rule 329
Rule 331
Rule 2004
Rule 2020
Rule 2032
Rule 2052
Rubi steps
\begin {align*} \int \frac {\left (-b+a x^2\right ) \sqrt [4]{-b x^2+a x^4}}{x^2} \, dx &=\int \left (a \sqrt [4]{-b x^2+a x^4}-\frac {b \sqrt [4]{-b x^2+a x^4}}{x^2}\right ) \, dx\\ &=a \int \sqrt [4]{-b x^2+a x^4} \, dx-b \int \frac {\sqrt [4]{-b x^2+a x^4}}{x^2} \, dx\\ &=\frac {2 b \sqrt [4]{-b x^2+a x^4}}{x}+\frac {1}{2} a x \sqrt [4]{-b x^2+a x^4}-\frac {1}{4} (a b) \int \frac {x^2}{\left (-b x^2+a x^4\right )^{3/4}} \, dx-(a b) \int \frac {x^2}{\left (-b x^2+a x^4\right )^{3/4}} \, dx\\ &=\frac {2 b \sqrt [4]{-b x^2+a x^4}}{x}+\frac {1}{2} a x \sqrt [4]{-b x^2+a x^4}-\frac {\left (a b x^{3/2} \left (-b+a x^2\right )^{3/4}\right ) \int \frac {\sqrt {x}}{\left (-b+a x^2\right )^{3/4}} \, dx}{4 \left (-b x^2+a x^4\right )^{3/4}}-\frac {\left (a b x^{3/2} \left (-b+a x^2\right )^{3/4}\right ) \int \frac {\sqrt {x}}{\left (-b+a x^2\right )^{3/4}} \, dx}{\left (-b x^2+a x^4\right )^{3/4}}\\ &=\frac {2 b \sqrt [4]{-b x^2+a x^4}}{x}+\frac {1}{2} a x \sqrt [4]{-b x^2+a x^4}-\frac {\left (a b x^{3/2} \left (-b+a x^2\right )^{3/4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (-b+a x^4\right )^{3/4}} \, dx,x,\sqrt {x}\right )}{2 \left (-b x^2+a x^4\right )^{3/4}}-\frac {\left (2 a b x^{3/2} \left (-b+a x^2\right )^{3/4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (-b+a x^4\right )^{3/4}} \, dx,x,\sqrt {x}\right )}{\left (-b x^2+a x^4\right )^{3/4}}\\ &=\frac {2 b \sqrt [4]{-b x^2+a x^4}}{x}+\frac {1}{2} a x \sqrt [4]{-b x^2+a x^4}-\frac {\left (a b x^{3/2} \left (-b+a x^2\right )^{3/4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{1-a x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{2 \left (-b x^2+a x^4\right )^{3/4}}-\frac {\left (2 a b x^{3/2} \left (-b+a x^2\right )^{3/4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{1-a x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{\left (-b x^2+a x^4\right )^{3/4}}\\ &=\frac {2 b \sqrt [4]{-b x^2+a x^4}}{x}+\frac {1}{2} a x \sqrt [4]{-b x^2+a x^4}-\frac {\left (\sqrt {a} b x^{3/2} \left (-b+a x^2\right )^{3/4}\right ) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {a} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{4 \left (-b x^2+a x^4\right )^{3/4}}+\frac {\left (\sqrt {a} b x^{3/2} \left (-b+a x^2\right )^{3/4}\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {a} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{4 \left (-b x^2+a x^4\right )^{3/4}}-\frac {\left (\sqrt {a} b x^{3/2} \left (-b+a x^2\right )^{3/4}\right ) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {a} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{\left (-b x^2+a x^4\right )^{3/4}}+\frac {\left (\sqrt {a} b x^{3/2} \left (-b+a x^2\right )^{3/4}\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {a} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{\left (-b x^2+a x^4\right )^{3/4}}\\ &=\frac {2 b \sqrt [4]{-b x^2+a x^4}}{x}+\frac {1}{2} a x \sqrt [4]{-b x^2+a x^4}+\frac {5 \sqrt [4]{a} b x^{3/2} \left (-b+a x^2\right )^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{4 \left (-b x^2+a x^4\right )^{3/4}}-\frac {5 \sqrt [4]{a} b x^{3/2} \left (-b+a x^2\right )^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{4 \left (-b x^2+a x^4\right )^{3/4}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.02, size = 55, normalized size = 0.54 \begin {gather*} \frac {2 b \sqrt [4]{a x^4-b x^2} \, _2F_1\left (-\frac {5}{4},-\frac {1}{4};\frac {3}{4};\frac {a x^2}{b}\right )}{x \sqrt [4]{1-\frac {a x^2}{b}}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.33, size = 101, normalized size = 1.00 \begin {gather*} \frac {\left (4 b+a x^2\right ) \sqrt [4]{-b x^2+a x^4}}{2 x}+\frac {5}{4} \sqrt [4]{a} b \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b x^2+a x^4}}\right )-\frac {5}{4} \sqrt [4]{a} b \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b x^2+a x^4}}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.20, size = 229, normalized size = 2.27 \begin {gather*} \frac {8 \, {\left (a - \frac {b}{x^{2}}\right )}^{\frac {1}{4}} a b x^{2} - 10 \, \sqrt {2} \left (-a\right )^{\frac {1}{4}} b^{2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} + 2 \, {\left (a - \frac {b}{x^{2}}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right ) - 10 \, \sqrt {2} \left (-a\right )^{\frac {1}{4}} b^{2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} - 2 \, {\left (a - \frac {b}{x^{2}}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right ) - 5 \, \sqrt {2} \left (-a\right )^{\frac {1}{4}} b^{2} \log \left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} {\left (a - \frac {b}{x^{2}}\right )}^{\frac {1}{4}} + \sqrt {-a} + \sqrt {a - \frac {b}{x^{2}}}\right ) + 5 \, \sqrt {2} \left (-a\right )^{\frac {1}{4}} b^{2} \log \left (-\sqrt {2} \left (-a\right )^{\frac {1}{4}} {\left (a - \frac {b}{x^{2}}\right )}^{\frac {1}{4}} + \sqrt {-a} + \sqrt {a - \frac {b}{x^{2}}}\right ) + 32 \, {\left (a - \frac {b}{x^{2}}\right )}^{\frac {1}{4}} b^{2}}{16 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\left (a \,x^{2}-b \right ) \left (a \,x^{4}-b \,x^{2}\right )^{\frac {1}{4}}}{x^{2}}\, dx\]
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 {{\left (a x^{4} - b x^{2}\right )}^{\frac {1}{4}} {\left (a x^{2} - b\right )}}{x^{2}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.36, size = 91, normalized size = 0.90 \begin {gather*} \frac {2\,a\,x\,{\left (a\,x^4-b\,x^2\right )}^{1/4}\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{4},\frac {3}{4};\ \frac {7}{4};\ \frac {a\,x^2}{b}\right )}{3\,{\left (1-\frac {a\,x^2}{b}\right )}^{1/4}}+\frac {2\,b\,{\left (a\,x^4-b\,x^2\right )}^{1/4}\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{4},-\frac {1}{4};\ \frac {3}{4};\ \frac {a\,x^2}{b}\right )}{x\,{\left (1-\frac {a\,x^2}{b}\right )}^{1/4}} \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 [4]{x^{2} \left (a x^{2} - b\right )} \left (a x^{2} - b\right )}{x^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________