Optimal. Leaf size=247 \[ \frac {1}{2} \text {RootSum}\left [\text {$\#$1}^8-\text {$\#$1}^4 a+b\& ,\frac {\text {$\#$1}^4 a^2 \log \left (\sqrt [4]{a x^4+b x^2}-\text {$\#$1} x\right )+\text {$\#$1}^4 \left (-a^2\right ) \log (x)-\text {$\#$1}^4 b \log \left (\sqrt [4]{a x^4+b x^2}-\text {$\#$1} x\right )+\text {$\#$1}^4 b \log (x)-a b \log \left (\sqrt [4]{a x^4+b x^2}-\text {$\#$1} x\right )+a b \log (x)}{\text {$\#$1}^3 a-2 \text {$\#$1}^7}\& \right ]+\frac {\left (4 a^2-b\right ) \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b x^2}}\right )}{4 a^{3/4}}+\frac {\left (b-4 a^2\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b x^2}}\right )}{4 a^{3/4}}+\frac {1}{2} x \sqrt [4]{a x^4+b x^2} \]
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Rubi [B] time = 3.89, antiderivative size = 1403, normalized size of antiderivative = 5.68, number of steps used = 30, number of rules used = 13, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.419, Rules used = {2056, 1269, 1516, 459, 331, 298, 203, 206, 6728, 1528, 494, 205, 208} \begin {gather*} \frac {1}{2} \sqrt [4]{a x^4+b x^2} x+\frac {\left (4 a^2-b\right ) \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^{3/4} \sqrt [4]{a x^2+b} \sqrt {x}}-\frac {2 \left (a^2-b\right ) b \sqrt [4]{a x^4+b x^2} \tan ^{-1}\left (\frac {\sqrt [4]{a^2-\sqrt {a^2-4 b} a-2 b} \sqrt {x}}{\sqrt [4]{a-\sqrt {a^2-4 b}} \sqrt [4]{a x^2+b}}\right )}{\sqrt [4]{a-\sqrt {a^2-4 b}} \sqrt {a^2-4 b} \left (a^2-\sqrt {a^2-4 b} a-2 b\right )^{3/4} \sqrt [4]{a x^2+b} \sqrt {x}}+\frac {a \left (a-\sqrt {a^2-4 b}\right )^{3/4} \left (a^2-2 b\right ) \sqrt [4]{a x^4+b x^2} \tan ^{-1}\left (\frac {\sqrt [4]{a^2-\sqrt {a^2-4 b} a-2 b} \sqrt {x}}{\sqrt [4]{a-\sqrt {a^2-4 b}} \sqrt [4]{a x^2+b}}\right )}{\sqrt {a^2-4 b} \left (a^2-\sqrt {a^2-4 b} a-2 b\right )^{3/4} \sqrt [4]{a x^2+b} \sqrt {x}}+\frac {2 \left (a^2-b\right ) b \sqrt [4]{a x^4+b x^2} \tan ^{-1}\left (\frac {\sqrt [4]{a^2+\sqrt {a^2-4 b} a-2 b} \sqrt {x}}{\sqrt [4]{a+\sqrt {a^2-4 b}} \sqrt [4]{a x^2+b}}\right )}{\sqrt [4]{a+\sqrt {a^2-4 b}} \sqrt {a^2-4 b} \left (a^2+\sqrt {a^2-4 b} a-2 b\right )^{3/4} \sqrt [4]{a x^2+b} \sqrt {x}}-\frac {a \left (a+\sqrt {a^2-4 b}\right )^{3/4} \left (a^2-2 b\right ) \sqrt [4]{a x^4+b x^2} \tan ^{-1}\left (\frac {\sqrt [4]{a^2+\sqrt {a^2-4 b} a-2 b} \sqrt {x}}{\sqrt [4]{a+\sqrt {a^2-4 b}} \sqrt [4]{a x^2+b}}\right )}{\sqrt {a^2-4 b} \left (a^2+\sqrt {a^2-4 b} a-2 b\right )^{3/4} \sqrt [4]{a x^2+b} \sqrt {x}}-\frac {\left (4 a^2-b\right ) \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^{3/4} \sqrt [4]{a x^2+b} \sqrt {x}}+\frac {2 \left (a^2-b\right ) b \sqrt [4]{a x^4+b x^2} \tanh ^{-1}\left (\frac {\sqrt [4]{a^2-\sqrt {a^2-4 b} a-2 b} \sqrt {x}}{\sqrt [4]{a-\sqrt {a^2-4 b}} \sqrt [4]{a x^2+b}}\right )}{\sqrt [4]{a-\sqrt {a^2-4 b}} \sqrt {a^2-4 b} \left (a^2-\sqrt {a^2-4 b} a-2 b\right )^{3/4} \sqrt [4]{a x^2+b} \sqrt {x}}-\frac {a \left (a-\sqrt {a^2-4 b}\right )^{3/4} \left (a^2-2 b\right ) \sqrt [4]{a x^4+b x^2} \tanh ^{-1}\left (\frac {\sqrt [4]{a^2-\sqrt {a^2-4 b} a-2 b} \sqrt {x}}{\sqrt [4]{a-\sqrt {a^2-4 b}} \sqrt [4]{a x^2+b}}\right )}{\sqrt {a^2-4 b} \left (a^2-\sqrt {a^2-4 b} a-2 b\right )^{3/4} \sqrt [4]{a x^2+b} \sqrt {x}}-\frac {2 \left (a^2-b\right ) b \sqrt [4]{a x^4+b x^2} \tanh ^{-1}\left (\frac {\sqrt [4]{a^2+\sqrt {a^2-4 b} a-2 b} \sqrt {x}}{\sqrt [4]{a+\sqrt {a^2-4 b}} \sqrt [4]{a x^2+b}}\right )}{\sqrt [4]{a+\sqrt {a^2-4 b}} \sqrt {a^2-4 b} \left (a^2+\sqrt {a^2-4 b} a-2 b\right )^{3/4} \sqrt [4]{a x^2+b} \sqrt {x}}+\frac {a \left (a+\sqrt {a^2-4 b}\right )^{3/4} \left (a^2-2 b\right ) \sqrt [4]{a x^4+b x^2} \tanh ^{-1}\left (\frac {\sqrt [4]{a^2+\sqrt {a^2-4 b} a-2 b} \sqrt {x}}{\sqrt [4]{a+\sqrt {a^2-4 b}} \sqrt [4]{a x^2+b}}\right )}{\sqrt {a^2-4 b} \left (a^2+\sqrt {a^2-4 b} a-2 b\right )^{3/4} \sqrt [4]{a x^2+b} \sqrt {x}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 203
Rule 205
Rule 206
Rule 208
Rule 298
Rule 331
Rule 459
Rule 494
Rule 1269
Rule 1516
Rule 1528
Rule 2056
Rule 6728
Rubi steps
\begin {align*} \int \frac {x^4 \sqrt [4]{b x^2+a x^4}}{b+a x^2+x^4} \, dx &=\frac {\sqrt [4]{b x^2+a x^4} \int \frac {x^{9/2} \sqrt [4]{b+a x^2}}{b+a x^2+x^4} \, dx}{\sqrt {x} \sqrt [4]{b+a x^2}}\\ &=\frac {\left (2 \sqrt [4]{b x^2+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^{10} \sqrt [4]{b+a x^4}}{b+a x^4+x^8} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt [4]{b+a x^2}}\\ &=\frac {\left (2 \sqrt [4]{b x^2+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2 \left (-a^2+b+a x^4\right )}{\left (b+a x^4\right )^{3/4}} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt [4]{b+a x^2}}-\frac {\left (2 \sqrt [4]{b x^2+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2 \left (-\left (\left (a^2-b\right ) b\right )-a \left (a^2-2 b\right ) x^4\right )}{\left (b+a x^4\right )^{3/4} \left (b+a x^4+x^8\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt [4]{b+a x^2}}\\ &=\frac {1}{2} x \sqrt [4]{b x^2+a x^4}-\frac {\left (2 \sqrt [4]{b x^2+a x^4}\right ) \operatorname {Subst}\left (\int \left (-\frac {\left (a^2-b\right ) b x^2}{\left (b+a x^4\right )^{3/4} \left (b+a x^4+x^8\right )}-\frac {a \left (a^2-2 b\right ) x^6}{\left (b+a x^4\right )^{3/4} \left (b+a x^4+x^8\right )}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt [4]{b+a x^2}}-\frac {\left (\left (4 a^2-b\right ) \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 \sqrt {x} \sqrt [4]{b+a x^2}}\\ &=\frac {1}{2} x \sqrt [4]{b x^2+a x^4}+\frac {\left (2 a \left (a^2-2 b\right ) \sqrt [4]{b x^2+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^6}{\left (b+a x^4\right )^{3/4} \left (b+a x^4+x^8\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt [4]{b+a x^2}}-\frac {\left (\left (4 a^2-b\right ) \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 \sqrt {x} \sqrt [4]{b+a x^2}}+\frac {\left (2 \left (a^2-b\right ) 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} \left (b+a x^4+x^8\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt [4]{b+a x^2}}\\ &=\frac {1}{2} x \sqrt [4]{b x^2+a x^4}+\frac {\left (2 a \left (a^2-2 b\right ) \sqrt [4]{b x^2+a x^4}\right ) \operatorname {Subst}\left (\int \left (-\frac {\left (-a+\sqrt {a^2-4 b}\right ) x^2}{\sqrt {a^2-4 b} \left (-a+\sqrt {a^2-4 b}-2 x^4\right ) \left (b+a x^4\right )^{3/4}}+\frac {\left (a+\sqrt {a^2-4 b}\right ) x^2}{\sqrt {a^2-4 b} \left (a+\sqrt {a^2-4 b}+2 x^4\right ) \left (b+a x^4\right )^{3/4}}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt [4]{b+a x^2}}-\frac {\left (\left (4 a^2-b\right ) \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 \sqrt {a} \sqrt {x} \sqrt [4]{b+a x^2}}+\frac {\left (\left (4 a^2-b\right ) \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 \sqrt {a} \sqrt {x} \sqrt [4]{b+a x^2}}+\frac {\left (2 \left (a^2-b\right ) b \sqrt [4]{b x^2+a x^4}\right ) \operatorname {Subst}\left (\int \left (\frac {2 x^2}{\sqrt {a^2-4 b} \left (a-\sqrt {a^2-4 b}+2 x^4\right ) \left (b+a x^4\right )^{3/4}}-\frac {2 x^2}{\sqrt {a^2-4 b} \left (a+\sqrt {a^2-4 b}+2 x^4\right ) \left (b+a x^4\right )^{3/4}}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt [4]{b+a x^2}}\\ &=\frac {1}{2} x \sqrt [4]{b x^2+a x^4}+\frac {\left (4 a^2-b\right ) \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^{3/4} \sqrt {x} \sqrt [4]{b+a x^2}}-\frac {\left (4 a^2-b\right ) \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^{3/4} \sqrt {x} \sqrt [4]{b+a x^2}}-\frac {\left (2 a \left (1-\frac {a}{\sqrt {a^2-4 b}}\right ) \left (a^2-2 b\right ) \sqrt [4]{b x^2+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (-a+\sqrt {a^2-4 b}-2 x^4\right ) \left (b+a x^4\right )^{3/4}} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt [4]{b+a x^2}}+\frac {\left (2 a \left (1+\frac {a}{\sqrt {a^2-4 b}}\right ) \left (a^2-2 b\right ) \sqrt [4]{b x^2+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (a+\sqrt {a^2-4 b}+2 x^4\right ) \left (b+a x^4\right )^{3/4}} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt [4]{b+a x^2}}+\frac {\left (4 \left (a^2-b\right ) b \sqrt [4]{b x^2+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (a-\sqrt {a^2-4 b}+2 x^4\right ) \left (b+a x^4\right )^{3/4}} \, dx,x,\sqrt {x}\right )}{\sqrt {a^2-4 b} \sqrt {x} \sqrt [4]{b+a x^2}}-\frac {\left (4 \left (a^2-b\right ) b \sqrt [4]{b x^2+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (a+\sqrt {a^2-4 b}+2 x^4\right ) \left (b+a x^4\right )^{3/4}} \, dx,x,\sqrt {x}\right )}{\sqrt {a^2-4 b} \sqrt {x} \sqrt [4]{b+a x^2}}\\ &=\frac {1}{2} x \sqrt [4]{b x^2+a x^4}+\frac {\left (4 a^2-b\right ) \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^{3/4} \sqrt {x} \sqrt [4]{b+a x^2}}-\frac {\left (4 a^2-b\right ) \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^{3/4} \sqrt {x} \sqrt [4]{b+a x^2}}-\frac {\left (2 a \left (1-\frac {a}{\sqrt {a^2-4 b}}\right ) \left (a^2-2 b\right ) \sqrt [4]{b x^2+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{-a+\sqrt {a^2-4 b}-\left (a \left (-a+\sqrt {a^2-4 b}\right )+2 b\right ) x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{\sqrt {x} \sqrt [4]{b+a x^2}}+\frac {\left (2 a \left (1+\frac {a}{\sqrt {a^2-4 b}}\right ) \left (a^2-2 b\right ) \sqrt [4]{b x^2+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{a+\sqrt {a^2-4 b}-\left (a \left (a+\sqrt {a^2-4 b}\right )-2 b\right ) x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{\sqrt {x} \sqrt [4]{b+a x^2}}+\frac {\left (4 \left (a^2-b\right ) b \sqrt [4]{b x^2+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{a-\sqrt {a^2-4 b}-\left (a \left (a-\sqrt {a^2-4 b}\right )-2 b\right ) x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{\sqrt {a^2-4 b} \sqrt {x} \sqrt [4]{b+a x^2}}-\frac {\left (4 \left (a^2-b\right ) b \sqrt [4]{b x^2+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{a+\sqrt {a^2-4 b}-\left (a \left (a+\sqrt {a^2-4 b}\right )-2 b\right ) x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{\sqrt {a^2-4 b} \sqrt {x} \sqrt [4]{b+a x^2}}\\ &=\frac {1}{2} x \sqrt [4]{b x^2+a x^4}+\frac {\left (4 a^2-b\right ) \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^{3/4} \sqrt {x} \sqrt [4]{b+a x^2}}-\frac {\left (4 a^2-b\right ) \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^{3/4} \sqrt {x} \sqrt [4]{b+a x^2}}+\frac {\left (a \left (1-\frac {a}{\sqrt {a^2-4 b}}\right ) \left (a^2-2 b\right ) \sqrt [4]{b x^2+a x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a-\sqrt {a^2-4 b}}-\sqrt {a^2-a \sqrt {a^2-4 b}-2 b} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{\sqrt {a^2-a \sqrt {a^2-4 b}-2 b} \sqrt {x} \sqrt [4]{b+a x^2}}-\frac {\left (a \left (1-\frac {a}{\sqrt {a^2-4 b}}\right ) \left (a^2-2 b\right ) \sqrt [4]{b x^2+a x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a-\sqrt {a^2-4 b}}+\sqrt {a^2-a \sqrt {a^2-4 b}-2 b} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{\sqrt {a^2-a \sqrt {a^2-4 b}-2 b} \sqrt {x} \sqrt [4]{b+a x^2}}+\frac {\left (a \left (1+\frac {a}{\sqrt {a^2-4 b}}\right ) \left (a^2-2 b\right ) \sqrt [4]{b x^2+a x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+\sqrt {a^2-4 b}}-\sqrt {a^2+a \sqrt {a^2-4 b}-2 b} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{\sqrt {a^2+a \sqrt {a^2-4 b}-2 b} \sqrt {x} \sqrt [4]{b+a x^2}}-\frac {\left (a \left (1+\frac {a}{\sqrt {a^2-4 b}}\right ) \left (a^2-2 b\right ) \sqrt [4]{b x^2+a x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+\sqrt {a^2-4 b}}+\sqrt {a^2+a \sqrt {a^2-4 b}-2 b} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{\sqrt {a^2+a \sqrt {a^2-4 b}-2 b} \sqrt {x} \sqrt [4]{b+a x^2}}+\frac {\left (2 \left (a^2-b\right ) b \sqrt [4]{b x^2+a x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a-\sqrt {a^2-4 b}}-\sqrt {a^2-a \sqrt {a^2-4 b}-2 b} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{\sqrt {a^2-4 b} \sqrt {a^2-a \sqrt {a^2-4 b}-2 b} \sqrt {x} \sqrt [4]{b+a x^2}}-\frac {\left (2 \left (a^2-b\right ) b \sqrt [4]{b x^2+a x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a-\sqrt {a^2-4 b}}+\sqrt {a^2-a \sqrt {a^2-4 b}-2 b} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{\sqrt {a^2-4 b} \sqrt {a^2-a \sqrt {a^2-4 b}-2 b} \sqrt {x} \sqrt [4]{b+a x^2}}-\frac {\left (2 \left (a^2-b\right ) b \sqrt [4]{b x^2+a x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+\sqrt {a^2-4 b}}-\sqrt {a^2+a \sqrt {a^2-4 b}-2 b} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{\sqrt {a^2-4 b} \sqrt {a^2+a \sqrt {a^2-4 b}-2 b} \sqrt {x} \sqrt [4]{b+a x^2}}+\frac {\left (2 \left (a^2-b\right ) b \sqrt [4]{b x^2+a x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+\sqrt {a^2-4 b}}+\sqrt {a^2+a \sqrt {a^2-4 b}-2 b} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{\sqrt {a^2-4 b} \sqrt {a^2+a \sqrt {a^2-4 b}-2 b} \sqrt {x} \sqrt [4]{b+a x^2}}\\ &=\frac {1}{2} x \sqrt [4]{b x^2+a x^4}+\frac {\left (4 a^2-b\right ) \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^{3/4} \sqrt {x} \sqrt [4]{b+a x^2}}-\frac {a \left (1-\frac {a}{\sqrt {a^2-4 b}}\right ) \left (a^2-2 b\right ) \sqrt [4]{b x^2+a x^4} \tan ^{-1}\left (\frac {\sqrt [4]{a^2-a \sqrt {a^2-4 b}-2 b} \sqrt {x}}{\sqrt [4]{a-\sqrt {a^2-4 b}} \sqrt [4]{b+a x^2}}\right )}{\sqrt [4]{a-\sqrt {a^2-4 b}} \left (a^2-a \sqrt {a^2-4 b}-2 b\right )^{3/4} \sqrt {x} \sqrt [4]{b+a x^2}}-\frac {2 \left (a^2-b\right ) b \sqrt [4]{b x^2+a x^4} \tan ^{-1}\left (\frac {\sqrt [4]{a^2-a \sqrt {a^2-4 b}-2 b} \sqrt {x}}{\sqrt [4]{a-\sqrt {a^2-4 b}} \sqrt [4]{b+a x^2}}\right )}{\sqrt [4]{a-\sqrt {a^2-4 b}} \sqrt {a^2-4 b} \left (a^2-a \sqrt {a^2-4 b}-2 b\right )^{3/4} \sqrt {x} \sqrt [4]{b+a x^2}}-\frac {a \left (a+\sqrt {a^2-4 b}\right )^{3/4} \left (a^2-2 b\right ) \sqrt [4]{b x^2+a x^4} \tan ^{-1}\left (\frac {\sqrt [4]{a^2+a \sqrt {a^2-4 b}-2 b} \sqrt {x}}{\sqrt [4]{a+\sqrt {a^2-4 b}} \sqrt [4]{b+a x^2}}\right )}{\sqrt {a^2-4 b} \left (a^2+a \sqrt {a^2-4 b}-2 b\right )^{3/4} \sqrt {x} \sqrt [4]{b+a x^2}}+\frac {2 \left (a^2-b\right ) b \sqrt [4]{b x^2+a x^4} \tan ^{-1}\left (\frac {\sqrt [4]{a^2+a \sqrt {a^2-4 b}-2 b} \sqrt {x}}{\sqrt [4]{a+\sqrt {a^2-4 b}} \sqrt [4]{b+a x^2}}\right )}{\sqrt [4]{a+\sqrt {a^2-4 b}} \sqrt {a^2-4 b} \left (a^2+a \sqrt {a^2-4 b}-2 b\right )^{3/4} \sqrt {x} \sqrt [4]{b+a x^2}}-\frac {\left (4 a^2-b\right ) \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^{3/4} \sqrt {x} \sqrt [4]{b+a x^2}}+\frac {a \left (1-\frac {a}{\sqrt {a^2-4 b}}\right ) \left (a^2-2 b\right ) \sqrt [4]{b x^2+a x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{a^2-a \sqrt {a^2-4 b}-2 b} \sqrt {x}}{\sqrt [4]{a-\sqrt {a^2-4 b}} \sqrt [4]{b+a x^2}}\right )}{\sqrt [4]{a-\sqrt {a^2-4 b}} \left (a^2-a \sqrt {a^2-4 b}-2 b\right )^{3/4} \sqrt {x} \sqrt [4]{b+a x^2}}+\frac {2 \left (a^2-b\right ) b \sqrt [4]{b x^2+a x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{a^2-a \sqrt {a^2-4 b}-2 b} \sqrt {x}}{\sqrt [4]{a-\sqrt {a^2-4 b}} \sqrt [4]{b+a x^2}}\right )}{\sqrt [4]{a-\sqrt {a^2-4 b}} \sqrt {a^2-4 b} \left (a^2-a \sqrt {a^2-4 b}-2 b\right )^{3/4} \sqrt {x} \sqrt [4]{b+a x^2}}+\frac {a \left (a+\sqrt {a^2-4 b}\right )^{3/4} \left (a^2-2 b\right ) \sqrt [4]{b x^2+a x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{a^2+a \sqrt {a^2-4 b}-2 b} \sqrt {x}}{\sqrt [4]{a+\sqrt {a^2-4 b}} \sqrt [4]{b+a x^2}}\right )}{\sqrt {a^2-4 b} \left (a^2+a \sqrt {a^2-4 b}-2 b\right )^{3/4} \sqrt {x} \sqrt [4]{b+a x^2}}-\frac {2 \left (a^2-b\right ) b \sqrt [4]{b x^2+a x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{a^2+a \sqrt {a^2-4 b}-2 b} \sqrt {x}}{\sqrt [4]{a+\sqrt {a^2-4 b}} \sqrt [4]{b+a x^2}}\right )}{\sqrt [4]{a+\sqrt {a^2-4 b}} \sqrt {a^2-4 b} \left (a^2+a \sqrt {a^2-4 b}-2 b\right )^{3/4} \sqrt {x} \sqrt [4]{b+a x^2}}\\ \end {align*}
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Mathematica [A] time = 0.46, size = 367, normalized size = 1.49 \begin {gather*} \frac {\sqrt [4]{x^2 \left (a x^2+b\right )} \left (3 \sqrt {a^2-4 b} \left (4 a^2-b\right ) \left (a x^2+b\right ) \left (\tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2+b}}\right )-\tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2+b}}\right )\right )-4 a^{3/4} x^{3/2} \left (a^3-a^2 \sqrt {a^2-4 b}+b \sqrt {a^2-4 b}-3 a b\right ) \sqrt [4]{a x^2+b} \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};-\frac {\left (\frac {2}{a-\sqrt {a^2-4 b}}-\frac {a}{b}\right ) b x^2}{a x^2+b}\right )+4 a^{3/4} x^{3/2} \left (a^3+a^2 \sqrt {a^2-4 b}-b \sqrt {a^2-4 b}-3 a b\right ) \sqrt [4]{a x^2+b} \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};-\frac {\left (\frac {2}{a+\sqrt {a^2-4 b}}-\frac {a}{b}\right ) b x^2}{a x^2+b}\right )\right )}{12 a^{3/4} \sqrt {x} \sqrt {a^2-4 b} \left (a x^2+b\right )^{5/4}}+\frac {1}{2} x \sqrt [4]{x^2 \left (a x^2+b\right )} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.00, size = 248, normalized size = 1.00 \begin {gather*} \frac {1}{2} x \sqrt [4]{b x^2+a x^4}+\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^{3/4}}+\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^{3/4}}+\frac {1}{2} \text {RootSum}\left [b-a \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-a b \log (x)+a b \log \left (\sqrt [4]{b x^2+a x^4}-x \text {$\#$1}\right )+a^2 \log (x) \text {$\#$1}^4-b \log (x) \text {$\#$1}^4-a^2 \log \left (\sqrt [4]{b x^2+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4+b \log \left (\sqrt [4]{b x^2+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-a \text {$\#$1}^3+2 \text {$\#$1}^7}\&\right ] \end {gather*}
Antiderivative was successfully verified.
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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.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.00, size = 0, normalized size = 0.00 \[\int \frac {x^{4} \left (a \,x^{4}+b \,x^{2}\right )^{\frac {1}{4}}}{x^{4}+a \,x^{2}+b}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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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}} x^{4}}{x^{4} + a x^{2} + b}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^4\,{\left (a\,x^4+b\,x^2\right )}^{1/4}}{x^4+a\,x^2+b} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{4} \sqrt [4]{x^{2} \left (a x^{2} + b\right )}}{a x^{2} + b + x^{4}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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