Optimal. Leaf size=105 \[ -\frac {1}{4} \text {RootSum}\left [\text {$\#$1}^8-2 \text {$\#$1}^4 a+a^2+a b\& ,\frac {\text {$\#$1}^4 \log \left (\sqrt [4]{a x^4+b x^2}-\text {$\#$1} x\right )+\text {$\#$1}^4 (-\log (x))-2 a \log \left (\sqrt [4]{a x^4+b x^2}-\text {$\#$1} x\right )+2 a \log (x)}{\text {$\#$1} a-\text {$\#$1}^5}\& \right ] \]
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Rubi [B] time = 1.33, antiderivative size = 445, normalized size of antiderivative = 4.24, number of steps used = 12, number of rules used = 7, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.206, Rules used = {2056, 6715, 6725, 377, 212, 206, 203} \begin {gather*} \frac {\sqrt {x} \left (\sqrt {-a}-\sqrt {b}\right ) \sqrt [4]{a x^2+b} \tan ^{-1}\left (\frac {\sqrt {x} \sqrt [4]{a-\sqrt {-a} \sqrt {b}}}{\sqrt [4]{a x^2+b}}\right )}{2 \sqrt {b} \sqrt [4]{a-\sqrt {-a} \sqrt {b}} \sqrt [4]{a x^4+b x^2}}-\frac {\sqrt {x} \left (\sqrt {-a}+\sqrt {b}\right ) \sqrt [4]{a x^2+b} \tan ^{-1}\left (\frac {\sqrt {x} \sqrt [4]{\sqrt {-a} \sqrt {b}+a}}{\sqrt [4]{a x^2+b}}\right )}{2 \sqrt {b} \sqrt [4]{\sqrt {-a} \sqrt {b}+a} \sqrt [4]{a x^4+b x^2}}+\frac {\sqrt {x} \left (\sqrt {-a}-\sqrt {b}\right ) \sqrt [4]{a x^2+b} \tanh ^{-1}\left (\frac {\sqrt {x} \sqrt [4]{a-\sqrt {-a} \sqrt {b}}}{\sqrt [4]{a x^2+b}}\right )}{2 \sqrt {b} \sqrt [4]{a-\sqrt {-a} \sqrt {b}} \sqrt [4]{a x^4+b x^2}}-\frac {\sqrt {x} \left (\sqrt {-a}+\sqrt {b}\right ) \sqrt [4]{a x^2+b} \tanh ^{-1}\left (\frac {\sqrt {x} \sqrt [4]{\sqrt {-a} \sqrt {b}+a}}{\sqrt [4]{a x^2+b}}\right )}{2 \sqrt {b} \sqrt [4]{\sqrt {-a} \sqrt {b}+a} \sqrt [4]{a x^4+b x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 203
Rule 206
Rule 212
Rule 377
Rule 2056
Rule 6715
Rule 6725
Rubi steps
\begin {align*} \int \frac {-b+a x^2}{\left (b+a x^4\right ) \sqrt [4]{b x^2+a x^4}} \, dx &=\frac {\left (\sqrt {x} \sqrt [4]{b+a x^2}\right ) \int \frac {-b+a x^2}{\sqrt {x} \sqrt [4]{b+a x^2} \left (b+a x^4\right )} \, dx}{\sqrt [4]{b x^2+a x^4}}\\ &=\frac {\left (2 \sqrt {x} \sqrt [4]{b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {-b+a x^4}{\sqrt [4]{b+a x^4} \left (b+a x^8\right )} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{b x^2+a x^4}}\\ &=\frac {\left (2 \sqrt {x} \sqrt [4]{b+a x^2}\right ) \operatorname {Subst}\left (\int \left (-\frac {\sqrt {-a} \left (a \sqrt {b}-\sqrt {-a} b\right )}{2 a \sqrt {b} \left (\sqrt {b}-\sqrt {-a} x^4\right ) \sqrt [4]{b+a x^4}}+\frac {\sqrt {-a} \left (a \sqrt {b}+\sqrt {-a} b\right )}{2 a \sqrt {b} \left (\sqrt {b}+\sqrt {-a} x^4\right ) \sqrt [4]{b+a x^4}}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt [4]{b x^2+a x^4}}\\ &=\frac {\left (\left (\sqrt {-a}-\sqrt {b}\right ) \sqrt {x} \sqrt [4]{b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\sqrt {b}+\sqrt {-a} x^4\right ) \sqrt [4]{b+a x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{b x^2+a x^4}}-\frac {\left (\left (\sqrt {-a}+\sqrt {b}\right ) \sqrt {x} \sqrt [4]{b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\sqrt {b}-\sqrt {-a} x^4\right ) \sqrt [4]{b+a x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{b x^2+a x^4}}\\ &=\frac {\left (\left (\sqrt {-a}-\sqrt {b}\right ) \sqrt {x} \sqrt [4]{b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b}-\left (a \sqrt {b}-\sqrt {-a} b\right ) x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{\sqrt [4]{b x^2+a x^4}}-\frac {\left (\left (\sqrt {-a}+\sqrt {b}\right ) \sqrt {x} \sqrt [4]{b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b}-\left (a \sqrt {b}+\sqrt {-a} b\right ) x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{\sqrt [4]{b x^2+a x^4}}\\ &=\frac {\left (\left (\sqrt {-a}-\sqrt {b}\right ) \sqrt {x} \sqrt [4]{b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {a-\sqrt {-a} \sqrt {b}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{2 \sqrt {b} \sqrt [4]{b x^2+a x^4}}+\frac {\left (\left (\sqrt {-a}-\sqrt {b}\right ) \sqrt {x} \sqrt [4]{b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {a-\sqrt {-a} \sqrt {b}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{2 \sqrt {b} \sqrt [4]{b x^2+a x^4}}-\frac {\left (\left (\sqrt {-a}+\sqrt {b}\right ) \sqrt {x} \sqrt [4]{b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {a+\sqrt {-a} \sqrt {b}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{2 \sqrt {b} \sqrt [4]{b x^2+a x^4}}-\frac {\left (\left (\sqrt {-a}+\sqrt {b}\right ) \sqrt {x} \sqrt [4]{b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {a+\sqrt {-a} \sqrt {b}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{2 \sqrt {b} \sqrt [4]{b x^2+a x^4}}\\ &=\frac {\left (\sqrt {-a}-\sqrt {b}\right ) \sqrt {x} \sqrt [4]{b+a x^2} \tan ^{-1}\left (\frac {\sqrt [4]{a-\sqrt {-a} \sqrt {b}} \sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{2 \sqrt [4]{a-\sqrt {-a} \sqrt {b}} \sqrt {b} \sqrt [4]{b x^2+a x^4}}-\frac {\left (\sqrt {-a}+\sqrt {b}\right ) \sqrt {x} \sqrt [4]{b+a x^2} \tan ^{-1}\left (\frac {\sqrt [4]{a+\sqrt {-a} \sqrt {b}} \sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{2 \sqrt [4]{a+\sqrt {-a} \sqrt {b}} \sqrt {b} \sqrt [4]{b x^2+a x^4}}+\frac {\left (\sqrt {-a}-\sqrt {b}\right ) \sqrt {x} \sqrt [4]{b+a x^2} \tanh ^{-1}\left (\frac {\sqrt [4]{a-\sqrt {-a} \sqrt {b}} \sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{2 \sqrt [4]{a-\sqrt {-a} \sqrt {b}} \sqrt {b} \sqrt [4]{b x^2+a x^4}}-\frac {\left (\sqrt {-a}+\sqrt {b}\right ) \sqrt {x} \sqrt [4]{b+a x^2} \tanh ^{-1}\left (\frac {\sqrt [4]{a+\sqrt {-a} \sqrt {b}} \sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{2 \sqrt [4]{a+\sqrt {-a} \sqrt {b}} \sqrt {b} \sqrt [4]{b x^2+a x^4}}\\ \end {align*}
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Mathematica [B] time = 1.25, size = 448, normalized size = 4.27 \begin {gather*} \frac {x \sqrt [4]{a+\frac {b}{x^2}} \left (\sqrt {b} \sqrt [4]{\sqrt {-a} \sqrt {b}+a} \tan ^{-1}\left (\frac {\sqrt [4]{a+\frac {b}{x^2}}}{\sqrt [4]{a-\sqrt {-a} \sqrt {b}}}\right )-\sqrt {-a} \sqrt [4]{\sqrt {-a} \sqrt {b}+a} \tan ^{-1}\left (\frac {\sqrt [4]{a+\frac {b}{x^2}}}{\sqrt [4]{a-\sqrt {-a} \sqrt {b}}}\right )+\sqrt {b} \sqrt [4]{a-\sqrt {-a} \sqrt {b}} \tan ^{-1}\left (\frac {\sqrt [4]{a+\frac {b}{x^2}}}{\sqrt [4]{\sqrt {-a} \sqrt {b}+a}}\right )+\sqrt {-a} \sqrt [4]{a-\sqrt {-a} \sqrt {b}} \tan ^{-1}\left (\frac {\sqrt [4]{a+\frac {b}{x^2}}}{\sqrt [4]{\sqrt {-a} \sqrt {b}+a}}\right )+\left (\sqrt {-a}-\sqrt {b}\right ) \sqrt [4]{\sqrt {-a} \sqrt {b}+a} \tanh ^{-1}\left (\frac {\sqrt [4]{a+\frac {b}{x^2}}}{\sqrt [4]{a-\sqrt {-a} \sqrt {b}}}\right )-\left (\sqrt {-a}+\sqrt {b}\right ) \sqrt [4]{a-\sqrt {-a} \sqrt {b}} \tanh ^{-1}\left (\frac {\sqrt [4]{a+\frac {b}{x^2}}}{\sqrt [4]{\sqrt {-a} \sqrt {b}+a}}\right )\right )}{2 \sqrt {b} \sqrt [4]{a-\sqrt {-a} \sqrt {b}} \sqrt [4]{\sqrt {-a} \sqrt {b}+a} \sqrt [4]{x^2 \left (a x^2+b\right )}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.55, size = 104, normalized size = 0.99 \begin {gather*} -\frac {1}{4} \text {RootSum}\left [a^2+a b-2 a \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-2 a \log (x)+2 a \log \left (\sqrt [4]{b x^2+a x^4}-x \text {$\#$1}\right )+\log (x) \text {$\#$1}^4-\log \left (\sqrt [4]{b x^2+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-a \text {$\#$1}+\text {$\#$1}^5}\&\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] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a x^{2} - b}{{\left (a x^{4} + b x^{2}\right )}^{\frac {1}{4}} {\left (a x^{4} + b\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {a \,x^{2}-b}{\left (a \,x^{4}+b \right ) \left (a \,x^{4}+b \,x^{2}\right )^{\frac {1}{4}}}\, dx\]
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 {a x^{2} - b}{{\left (a x^{4} + b x^{2}\right )}^{\frac {1}{4}} {\left (a x^{4} + b\right )}}\,{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 = -0.01 \begin {gather*} \int -\frac {b-a\,x^2}{\left (a\,x^4+b\right )\,{\left (a\,x^4+b\,x^2\right )}^{1/4}} \,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 {a x^{2} - b}{\sqrt [4]{x^{2} \left (a x^{2} + b\right )} \left (a x^{4} + b\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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