Optimal. Leaf size=334 \[ -\frac {\text {RootSum}\left [\text {$\#$1}^8-2 \text {$\#$1}^4 a+a^2+a b\& ,\frac {-\text {$\#$1}^4 a^2 \log \left (\sqrt [4]{a x^4+b x^2}-\text {$\#$1} x\right )+\text {$\#$1}^4 a^2 \log (x)+\text {$\#$1}^4 a b \log \left (\sqrt [4]{a x^4+b x^2}-\text {$\#$1} x\right )+\text {$\#$1}^4 b \log \left (\sqrt [4]{a x^4+b x^2}-\text {$\#$1} x\right )-\text {$\#$1}^4 a b \log (x)-\text {$\#$1}^4 b \log (x)+a^3 \log \left (\sqrt [4]{a x^4+b x^2}-\text {$\#$1} x\right )+a^2 b \log \left (\sqrt [4]{a x^4+b x^2}-\text {$\#$1} x\right )-a^3 \log (x)-a^2 b \log (x)}{\text {$\#$1}^3 a-\text {$\#$1}^7}\& \right ]}{4 a}+\frac {\left (4 a^2-b\right ) \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b x^2}}\right )}{4 a^{7/4}}+\frac {\left (b-4 a^2\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b x^2}}\right )}{4 a^{7/4}}+\frac {x \sqrt [4]{a x^4+b x^2}}{2 a} \]
________________________________________________________________________________________
Rubi [A] time = 1.46, antiderivative size = 332, normalized size of antiderivative = 0.99, number of steps used = 17, number of rules used = 11, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.290, Rules used = {2056, 6725, 279, 329, 331, 298, 203, 206, 466, 511, 510} \begin {gather*} -\frac {b \sqrt [4]{a x^4+b x^2} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2+b}}\right )}{4 a^{7/4} \sqrt {x} \sqrt [4]{a x^2+b}}+\frac {b \sqrt [4]{a x^4+b x^2} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2+b}}\right )}{4 a^{7/4} \sqrt {x} \sqrt [4]{a x^2+b}}-\frac {x \left (\frac {\sqrt {-a} a}{\sqrt {b}}+a+1\right ) \sqrt [4]{a x^4+b x^2} F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};-\frac {\sqrt {-a} x^2}{\sqrt {b}},-\frac {a x^2}{b}\right )}{3 a \sqrt [4]{\frac {a x^2}{b}+1}}-\frac {x \left (\frac {(-a)^{3/2}}{\sqrt {b}}+a+1\right ) \sqrt [4]{a x^4+b x^2} F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};\frac {\sqrt {-a} x^2}{\sqrt {b}},-\frac {a x^2}{b}\right )}{3 a \sqrt [4]{\frac {a x^2}{b}+1}}+\frac {x \sqrt [4]{a x^4+b x^2}}{2 a} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
[Out]
Rule 203
Rule 206
Rule 279
Rule 298
Rule 329
Rule 331
Rule 466
Rule 510
Rule 511
Rule 2056
Rule 6725
Rubi steps
\begin {align*} \int \frac {\left (-b-a x^2+x^4\right ) \sqrt [4]{b x^2+a x^4}}{b+a x^4} \, dx &=\frac {\sqrt [4]{b x^2+a x^4} \int \frac {\sqrt {x} \sqrt [4]{b+a x^2} \left (-b-a x^2+x^4\right )}{b+a x^4} \, dx}{\sqrt {x} \sqrt [4]{b+a x^2}}\\ &=\frac {\sqrt [4]{b x^2+a x^4} \int \left (\frac {\sqrt {x} \sqrt [4]{b+a x^2}}{a}-\frac {\sqrt {x} \sqrt [4]{b+a x^2} \left ((1+a) b+a^2 x^2\right )}{a \left (b+a x^4\right )}\right ) \, dx}{\sqrt {x} \sqrt [4]{b+a x^2}}\\ &=\frac {\sqrt [4]{b x^2+a x^4} \int \sqrt {x} \sqrt [4]{b+a x^2} \, dx}{a \sqrt {x} \sqrt [4]{b+a x^2}}-\frac {\sqrt [4]{b x^2+a x^4} \int \frac {\sqrt {x} \sqrt [4]{b+a x^2} \left ((1+a) b+a^2 x^2\right )}{b+a x^4} \, dx}{a \sqrt {x} \sqrt [4]{b+a x^2}}\\ &=\frac {x \sqrt [4]{b x^2+a x^4}}{2 a}-\frac {\sqrt [4]{b x^2+a x^4} \int \left (-\frac {\sqrt {-a} \left (a^2 \sqrt {b}+\sqrt {-a} (1+a) b\right ) \sqrt {x} \sqrt [4]{b+a x^2}}{2 a \sqrt {b} \left (\sqrt {b}-\sqrt {-a} x^2\right )}+\frac {\sqrt {-a} \left (a^2 \sqrt {b}-\sqrt {-a} (1+a) b\right ) \sqrt {x} \sqrt [4]{b+a x^2}}{2 a \sqrt {b} \left (\sqrt {b}+\sqrt {-a} x^2\right )}\right ) \, dx}{a \sqrt {x} \sqrt [4]{b+a x^2}}+\frac {\left (b \sqrt [4]{b x^2+a x^4}\right ) \int \frac {\sqrt {x}}{\left (b+a x^2\right )^{3/4}} \, dx}{4 a \sqrt {x} \sqrt [4]{b+a x^2}}\\ &=\frac {x \sqrt [4]{b x^2+a x^4}}{2 a}-\frac {\left (\left (a \left (\sqrt {-a}+\sqrt {b}\right )+\sqrt {b}\right ) \sqrt [4]{b x^2+a x^4}\right ) \int \frac {\sqrt {x} \sqrt [4]{b+a x^2}}{\sqrt {b}+\sqrt {-a} x^2} \, dx}{2 a \sqrt {x} \sqrt [4]{b+a x^2}}+\frac {\left (\left (\sqrt {-a} a-(1+a) \sqrt {b}\right ) \sqrt [4]{b x^2+a x^4}\right ) \int \frac {\sqrt {x} \sqrt [4]{b+a x^2}}{\sqrt {b}-\sqrt {-a} x^2} \, dx}{2 a \sqrt {x} \sqrt [4]{b+a x^2}}+\frac {\left (b \sqrt [4]{b x^2+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (b+a x^4\right )^{3/4}} \, dx,x,\sqrt {x}\right )}{2 a \sqrt {x} \sqrt [4]{b+a x^2}}\\ &=\frac {x \sqrt [4]{b x^2+a x^4}}{2 a}-\frac {\left (\left (a \left (\sqrt {-a}+\sqrt {b}\right )+\sqrt {b}\right ) \sqrt [4]{b x^2+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2 \sqrt [4]{b+a x^4}}{\sqrt {b}+\sqrt {-a} x^4} \, dx,x,\sqrt {x}\right )}{a \sqrt {x} \sqrt [4]{b+a x^2}}+\frac {\left (\left (\sqrt {-a} a-(1+a) \sqrt {b}\right ) \sqrt [4]{b x^2+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2 \sqrt [4]{b+a x^4}}{\sqrt {b}-\sqrt {-a} x^4} \, dx,x,\sqrt {x}\right )}{a \sqrt {x} \sqrt [4]{b+a x^2}}+\frac {\left (b \sqrt [4]{b x^2+a x^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 a \sqrt {x} \sqrt [4]{b+a x^2}}\\ &=\frac {x \sqrt [4]{b x^2+a x^4}}{2 a}+\frac {\left (b \sqrt [4]{b x^2+a x^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 a^{3/2} \sqrt {x} \sqrt [4]{b+a x^2}}-\frac {\left (b \sqrt [4]{b x^2+a x^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 a^{3/2} \sqrt {x} \sqrt [4]{b+a x^2}}-\frac {\left (\left (a \left (\sqrt {-a}+\sqrt {b}\right )+\sqrt {b}\right ) \sqrt [4]{b x^2+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2 \sqrt [4]{1+\frac {a x^4}{b}}}{\sqrt {b}+\sqrt {-a} x^4} \, dx,x,\sqrt {x}\right )}{a \sqrt {x} \sqrt [4]{1+\frac {a x^2}{b}}}+\frac {\left (\left (\sqrt {-a} a-(1+a) \sqrt {b}\right ) \sqrt [4]{b x^2+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2 \sqrt [4]{1+\frac {a x^4}{b}}}{\sqrt {b}-\sqrt {-a} x^4} \, dx,x,\sqrt {x}\right )}{a \sqrt {x} \sqrt [4]{1+\frac {a x^2}{b}}}\\ &=\frac {x \sqrt [4]{b x^2+a x^4}}{2 a}-\frac {\left (1+\frac {1}{a}+\frac {\sqrt {-a}}{\sqrt {b}}\right ) x \sqrt [4]{b x^2+a x^4} F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};-\frac {\sqrt {-a} x^2}{\sqrt {b}},-\frac {a x^2}{b}\right )}{3 \sqrt [4]{1+\frac {a x^2}{b}}}-\frac {\left (1+\frac {1}{a}-\frac {\sqrt {-a}}{\sqrt {b}}\right ) x \sqrt [4]{b x^2+a x^4} F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};\frac {\sqrt {-a} x^2}{\sqrt {b}},-\frac {a x^2}{b}\right )}{3 \sqrt [4]{1+\frac {a x^2}{b}}}-\frac {b \sqrt [4]{b x^2+a x^4} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{4 a^{7/4} \sqrt {x} \sqrt [4]{b+a x^2}}+\frac {b \sqrt [4]{b x^2+a x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{4 a^{7/4} \sqrt {x} \sqrt [4]{b+a x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 3.33, size = 606, normalized size = 1.81 \begin {gather*} \frac {x \sqrt [4]{x^2 \left (a x^2+b\right )} \left (2-\frac {\left (a+\frac {b}{x^2}\right )^{3/4} \left (\frac {2 \left (a^2-a b-b\right ) \left (\frac {\left (\sqrt {-a}-\sqrt {b}\right ) \left (\tan ^{-1}\left (\frac {\sqrt [4]{a+\frac {b}{x^2}}}{\sqrt [4]{\sqrt {-a} \sqrt {b}+a}}\right )+\tanh ^{-1}\left (\frac {\sqrt [4]{a+\frac {b}{x^2}}}{\sqrt [4]{\sqrt {-a} \sqrt {b}+a}}\right )\right )}{\left (\sqrt {-a} \sqrt {b}+a\right )^{3/4}}-\frac {\left (\sqrt {-a}+\sqrt {b}\right ) \left (\tan ^{-1}\left (\frac {\sqrt [4]{a+\frac {b}{x^2}}}{\sqrt [4]{a-\sqrt {-a} \sqrt {b}}}\right )+\tanh ^{-1}\left (\frac {\sqrt [4]{a+\frac {b}{x^2}}}{\sqrt [4]{a-\sqrt {-a} \sqrt {b}}}\right )\right )}{\left (a-\sqrt {-a} \sqrt {b}\right )^{3/4}}\right )}{\sqrt {b}}+\frac {\left (4 a^2-b\right ) \tan ^{-1}\left (\frac {\sqrt [4]{a+\frac {b}{x^2}}}{\sqrt [4]{a}}\right )}{a^{3/4}}+\frac {\left (4 a^2-b\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{a+\frac {b}{x^2}}}{\sqrt [4]{a}}\right )}{a^{3/4}}+\frac {2 (-a)^{3/2} (a+b) \left (-\left (\sqrt {-a} \sqrt {b}+a\right )^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{a+\frac {b}{x^2}}}{\sqrt [4]{a-\sqrt {-a} \sqrt {b}}}\right )+\left (a-\sqrt {-a} \sqrt {b}\right )^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{a+\frac {b}{x^2}}}{\sqrt [4]{\sqrt {-a} \sqrt {b}+a}}\right )-\left (\sqrt {-a} \sqrt {b}+a\right )^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{a+\frac {b}{x^2}}}{\sqrt [4]{a-\sqrt {-a} \sqrt {b}}}\right )+\left (a-\sqrt {-a} \sqrt {b}\right )^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{a+\frac {b}{x^2}}}{\sqrt [4]{\sqrt {-a} \sqrt {b}+a}}\right )\right )}{\sqrt {b} \left (a-\sqrt {-a} \sqrt {b}\right )^{3/4} \left (\sqrt {-a} \sqrt {b}+a\right )^{3/4}}\right )}{a x^2+b}\right )}{4 a} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 2.51, size = 333, normalized size = 1.00 \begin {gather*} \frac {x \sqrt [4]{b x^2+a x^4}}{2 a}+\frac {\left (4 a^2-b\right ) \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b x^2+a x^4}}\right )}{4 a^{7/4}}+\frac {\left (-4 a^2+b\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b x^2+a x^4}}\right )}{4 a^{7/4}}-\frac {\text {RootSum}\left [a^2+a b-2 a \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {a^3 \log (x)+a^2 b \log (x)-a^3 \log \left (\sqrt [4]{b x^2+a x^4}-x \text {$\#$1}\right )-a^2 b \log \left (\sqrt [4]{b x^2+a x^4}-x \text {$\#$1}\right )-a^2 \log (x) \text {$\#$1}^4+b \log (x) \text {$\#$1}^4+a b \log (x) \text {$\#$1}^4+a^2 \log \left (\sqrt [4]{b x^2+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4-b \log \left (\sqrt [4]{b x^2+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4-a b \log \left (\sqrt [4]{b x^2+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-a \text {$\#$1}^3+\text {$\#$1}^7}\&\right ]}{4 a} \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 [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 (x^{4} - a x^{2} - b\right )}}{a x^{4} + b}\,{d x} \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 \[\int \frac {\left (x^{4}-a \,x^{2}-b \right ) \left (a \,x^{4}+b \,x^{2}\right )^{\frac {1}{4}}}{a \,x^{4}+b}\, dx\]
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 (x^{4} - a x^{2} - b\right )}}{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 {{\left (a\,x^4+b\,x^2\right )}^{1/4}\,\left (-x^4+a\,x^2+b\right )}{a\,x^4+b} \,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 [4]{x^{2} \left (a x^{2} + b\right )} \left (- a x^{2} - b + x^{4}\right )}{a x^{4} + b}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________