Optimal. Leaf size=243 \[ \frac {1}{2} \text {RootSum}\left [\text {$\#$1}^8-3 \text {$\#$1}^4 a+2 a^2-b\& ,\frac {-\text {$\#$1}^4 a \log \left (\sqrt [4]{a x^4-b x^2}-\text {$\#$1} x\right )-\text {$\#$1}^4 \log \left (\sqrt [4]{a x^4-b x^2}-\text {$\#$1} x\right )+\text {$\#$1}^4 a \log (x)+\text {$\#$1}^4 \log (x)+2 a^2 \log \left (\sqrt [4]{a x^4-b x^2}-\text {$\#$1} x\right )+a \log \left (\sqrt [4]{a x^4-b x^2}-\text {$\#$1} x\right )-2 a^2 \log (x)-a \log (x)}{3 \text {$\#$1} a-2 \text {$\#$1}^5}\& \right ]+a^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b x^2}}\right )+a^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b x^2}}\right ) \]
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Rubi [B] time = 2.77, antiderivative size = 805, normalized size of antiderivative = 3.31, number of steps used = 18, number of rules used = 10, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.256, Rules used = {2056, 6715, 6728, 240, 212, 206, 203, 377, 208, 205} \begin {gather*} \frac {a^{3/4} \sqrt {x} \sqrt [4]{a x^2-b} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2-b}}\right )}{\sqrt [4]{a x^4-b x^2}}-\frac {\left (a^2-\frac {a^3+2 b a+2 b}{\sqrt {a^2+4 b}}\right ) \sqrt {x} \sqrt [4]{a x^2-b} \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 )}{\left (a-\sqrt {a^2+4 b}\right )^{3/4} \sqrt [4]{a^2-\sqrt {a^2+4 b} a-2 b} \sqrt [4]{a x^4-b x^2}}-\frac {\left (a^2+\frac {a^3+2 b a+2 b}{\sqrt {a^2+4 b}}\right ) \sqrt {x} \sqrt [4]{a x^2-b} \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 )}{\left (a+\sqrt {a^2+4 b}\right )^{3/4} \sqrt [4]{a^2+\sqrt {a^2+4 b} a-2 b} \sqrt [4]{a x^4-b x^2}}+\frac {a^{3/4} \sqrt {x} \sqrt [4]{a x^2-b} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2-b}}\right )}{\sqrt [4]{a x^4-b x^2}}-\frac {\left (a^2-\frac {a^3+2 b a+2 b}{\sqrt {a^2+4 b}}\right ) \sqrt {x} \sqrt [4]{a x^2-b} \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 )}{\left (a-\sqrt {a^2+4 b}\right )^{3/4} \sqrt [4]{a^2-\sqrt {a^2+4 b} a-2 b} \sqrt [4]{a x^4-b x^2}}-\frac {\left (a^2+\frac {a^3+2 b a+2 b}{\sqrt {a^2+4 b}}\right ) \sqrt {x} \sqrt [4]{a x^2-b} \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 )}{\left (a+\sqrt {a^2+4 b}\right )^{3/4} \sqrt [4]{a^2+\sqrt {a^2+4 b} a-2 b} \sqrt [4]{a x^4-b x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 203
Rule 205
Rule 206
Rule 208
Rule 212
Rule 240
Rule 377
Rule 2056
Rule 6715
Rule 6728
Rubi steps
\begin {align*} \int \frac {b+a x^4}{\left (-b-a x^2+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^4}{\sqrt {x} \sqrt [4]{-b+a x^2} \left (-b-a x^2+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^8}{\sqrt [4]{-b+a x^4} \left (-b-a x^4+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 {a}{\sqrt [4]{-b+a x^4}}+\frac {(1+a) b+a^2 x^4}{\sqrt [4]{-b+a x^4} \left (-b-a x^4+x^8\right )}\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 \frac {(1+a) b+a^2 x^4}{\sqrt [4]{-b+a x^4} \left (-b-a x^4+x^8\right )} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{-b x^2+a x^4}}+\frac {\left (2 a \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{-b+a x^4}} \, 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 {a^2+\frac {a^3+2 b+2 a b}{\sqrt {a^2+4 b}}}{\left (-a-\sqrt {a^2+4 b}+2 x^4\right ) \sqrt [4]{-b+a x^4}}+\frac {a^2-\frac {a^3+2 b+2 a b}{\sqrt {a^2+4 b}}}{\left (-a+\sqrt {a^2+4 b}+2 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 (2 a \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{1-a x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{\sqrt [4]{-b x^2+a x^4}}\\ &=\frac {\left (a \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {a} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{\sqrt [4]{-b x^2+a x^4}}+\frac {\left (a \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {a} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{\sqrt [4]{-b x^2+a x^4}}+\frac {\left (2 \left (a^2-\frac {a^3+2 b+2 a b}{\sqrt {a^2+4 b}}\right ) \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-a+\sqrt {a^2+4 b}+2 x^4\right ) \sqrt [4]{-b+a x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{-b x^2+a x^4}}+\frac {\left (2 \left (a^2+\frac {a^3+2 b+2 a b}{\sqrt {a^2+4 b}}\right ) \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-a-\sqrt {a^2+4 b}+2 x^4\right ) \sqrt [4]{-b+a x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{-b x^2+a x^4}}\\ &=\frac {a^{3/4} \sqrt {x} \sqrt [4]{-b+a x^2} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{\sqrt [4]{-b x^2+a x^4}}+\frac {a^{3/4} \sqrt {x} \sqrt [4]{-b+a x^2} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{\sqrt [4]{-b x^2+a x^4}}+\frac {\left (2 \left (a^2-\frac {a^3+2 b+2 a b}{\sqrt {a^2+4 b}}\right ) \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{-a+\sqrt {a^2+4 b}-\left (2 b+a \left (-a+\sqrt {a^2+4 b}\right )\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 (2 \left (a^2+\frac {a^3+2 b+2 a b}{\sqrt {a^2+4 b}}\right ) \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{-a-\sqrt {a^2+4 b}-\left (2 b+a \left (-a-\sqrt {a^2+4 b}\right )\right ) x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{\sqrt [4]{-b x^2+a x^4}}\\ &=\frac {a^{3/4} \sqrt {x} \sqrt [4]{-b+a x^2} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{\sqrt [4]{-b x^2+a x^4}}+\frac {a^{3/4} \sqrt {x} \sqrt [4]{-b+a x^2} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{\sqrt [4]{-b x^2+a x^4}}-\frac {\left (\left (a^2-\frac {a^3+2 b+2 a b}{\sqrt {a^2+4 b}}\right ) \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a-\sqrt {a^2+4 b}}-\sqrt {a^2-2 b-a \sqrt {a^2+4 b}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{\sqrt {a-\sqrt {a^2+4 b}} \sqrt [4]{-b x^2+a x^4}}-\frac {\left (\left (a^2-\frac {a^3+2 b+2 a b}{\sqrt {a^2+4 b}}\right ) \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a-\sqrt {a^2+4 b}}+\sqrt {a^2-2 b-a \sqrt {a^2+4 b}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{\sqrt {a-\sqrt {a^2+4 b}} \sqrt [4]{-b x^2+a x^4}}-\frac {\left (\left (a^2+\frac {a^3+2 b+2 a b}{\sqrt {a^2+4 b}}\right ) \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+\sqrt {a^2+4 b}}-\sqrt {a^2-2 b+a \sqrt {a^2+4 b}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{\sqrt {a+\sqrt {a^2+4 b}} \sqrt [4]{-b x^2+a x^4}}-\frac {\left (\left (a^2+\frac {a^3+2 b+2 a b}{\sqrt {a^2+4 b}}\right ) \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+\sqrt {a^2+4 b}}+\sqrt {a^2-2 b+a \sqrt {a^2+4 b}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{\sqrt {a+\sqrt {a^2+4 b}} \sqrt [4]{-b x^2+a x^4}}\\ &=\frac {a^{3/4} \sqrt {x} \sqrt [4]{-b+a x^2} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{\sqrt [4]{-b x^2+a x^4}}-\frac {\left (a^2-\frac {a^3+2 b+2 a b}{\sqrt {a^2+4 b}}\right ) \sqrt {x} \sqrt [4]{-b+a x^2} \tan ^{-1}\left (\frac {\sqrt [4]{a^2-2 b-a \sqrt {a^2+4 b}} \sqrt {x}}{\sqrt [4]{a-\sqrt {a^2+4 b}} \sqrt [4]{-b+a x^2}}\right )}{\left (a-\sqrt {a^2+4 b}\right )^{3/4} \sqrt [4]{a^2-2 b-a \sqrt {a^2+4 b}} \sqrt [4]{-b x^2+a x^4}}-\frac {\left (a^2+\frac {a^3+2 b+2 a b}{\sqrt {a^2+4 b}}\right ) \sqrt {x} \sqrt [4]{-b+a x^2} \tan ^{-1}\left (\frac {\sqrt [4]{a^2-2 b+a \sqrt {a^2+4 b}} \sqrt {x}}{\sqrt [4]{a+\sqrt {a^2+4 b}} \sqrt [4]{-b+a x^2}}\right )}{\left (a+\sqrt {a^2+4 b}\right )^{3/4} \sqrt [4]{a^2-2 b+a \sqrt {a^2+4 b}} \sqrt [4]{-b x^2+a x^4}}+\frac {a^{3/4} \sqrt {x} \sqrt [4]{-b+a x^2} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{\sqrt [4]{-b x^2+a x^4}}-\frac {\left (a^2-\frac {a^3+2 b+2 a b}{\sqrt {a^2+4 b}}\right ) \sqrt {x} \sqrt [4]{-b+a x^2} \tanh ^{-1}\left (\frac {\sqrt [4]{a^2-2 b-a \sqrt {a^2+4 b}} \sqrt {x}}{\sqrt [4]{a-\sqrt {a^2+4 b}} \sqrt [4]{-b+a x^2}}\right )}{\left (a-\sqrt {a^2+4 b}\right )^{3/4} \sqrt [4]{a^2-2 b-a \sqrt {a^2+4 b}} \sqrt [4]{-b x^2+a x^4}}-\frac {\left (a^2+\frac {a^3+2 b+2 a b}{\sqrt {a^2+4 b}}\right ) \sqrt {x} \sqrt [4]{-b+a x^2} \tanh ^{-1}\left (\frac {\sqrt [4]{a^2-2 b+a \sqrt {a^2+4 b}} \sqrt {x}}{\sqrt [4]{a+\sqrt {a^2+4 b}} \sqrt [4]{-b+a x^2}}\right )}{\left (a+\sqrt {a^2+4 b}\right )^{3/4} \sqrt [4]{a^2-2 b+a \sqrt {a^2+4 b}} \sqrt [4]{-b x^2+a x^4}}\\ \end {align*}
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Mathematica [B] time = 2.00, size = 656, normalized size = 2.70 \begin {gather*} \frac {\sqrt {x} \sqrt [4]{a x^2-b} \left (a^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2-b}}\right )+a^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2-b}}\right )-\frac {\left (a^2-\frac {a^3+2 a b+2 b}{\sqrt {a^2+4 b}}\right ) \tan ^{-1}\left (\frac {\sqrt {x} \sqrt [4]{-a \sqrt {a^2+4 b}+a^2-2 b}}{\sqrt [4]{a-\sqrt {a^2+4 b}} \sqrt [4]{a x^2-b}}\right )}{\left (a-\sqrt {a^2+4 b}\right )^{3/4} \sqrt [4]{-a \sqrt {a^2+4 b}+a^2-2 b}}-\frac {\left (a^3+a^2 \sqrt {a^2+4 b}+2 a b+2 b\right ) \tan ^{-1}\left (\frac {\sqrt {x} \sqrt [4]{a \sqrt {a^2+4 b}+a^2-2 b}}{\sqrt [4]{\sqrt {a^2+4 b}+a} \sqrt [4]{a x^2-b}}\right )}{\sqrt {a^2+4 b} \left (\sqrt {a^2+4 b}+a\right )^{3/4} \sqrt [4]{a \sqrt {a^2+4 b}+a^2-2 b}}-\frac {\left (a^2-\frac {a^3+2 a b+2 b}{\sqrt {a^2+4 b}}\right ) \tanh ^{-1}\left (\frac {\sqrt {x} \sqrt [4]{-a \sqrt {a^2+4 b}+a^2-2 b}}{\sqrt [4]{a-\sqrt {a^2+4 b}} \sqrt [4]{a x^2-b}}\right )}{\left (a-\sqrt {a^2+4 b}\right )^{3/4} \sqrt [4]{-a \sqrt {a^2+4 b}+a^2-2 b}}-\frac {\left (a^3+a^2 \sqrt {a^2+4 b}+2 a b+2 b\right ) \tanh ^{-1}\left (\frac {\sqrt {x} \sqrt [4]{a \sqrt {a^2+4 b}+a^2-2 b}}{\sqrt [4]{\sqrt {a^2+4 b}+a} \sqrt [4]{a x^2-b}}\right )}{\sqrt {a^2+4 b} \left (\sqrt {a^2+4 b}+a\right )^{3/4} \sqrt [4]{a \sqrt {a^2+4 b}+a^2-2 b}}\right )}{\sqrt [4]{a x^4-b x^2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 3.15, size = 243, normalized size = 1.00 \begin {gather*} a^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b x^2+a x^4}}\right )+a^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b x^2+a x^4}}\right )+\frac {1}{2} \text {RootSum}\left [2 a^2-b-3 a \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {a \log (x)+2 a^2 \log (x)-a \log \left (\sqrt [4]{-b x^2+a x^4}-x \text {$\#$1}\right )-2 a^2 \log \left (\sqrt [4]{-b x^2+a x^4}-x \text {$\#$1}\right )-\log (x) \text {$\#$1}^4-a \log (x) \text {$\#$1}^4+\log \left (\sqrt [4]{-b x^2+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4+a \log \left (\sqrt [4]{-b x^2+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-3 a \text {$\#$1}+2 \text {$\#$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(-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 = 0.04, size = 0, normalized size = 0.00 \[\int \frac {a \,x^{4}+b}{\left (x^{4}-a \,x^{2}-b \right ) \left (a \,x^{4}-b \,x^{2}\right )^{\frac {1}{4}}}\, dx\]
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^{4} + b}{{\left (a x^{4} - b x^{2}\right )}^{\frac {1}{4}} {\left (x^{4} - a x^{2} - 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 = -0.00 \begin {gather*} \int -\frac {a\,x^4+b}{{\left (a\,x^4-b\,x^2\right )}^{1/4}\,\left (-x^4+a\,x^2+b\right )} \,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^{4} + b}{\sqrt [4]{x^{2} \left (a x^{2} - b\right )} \left (- a x^{2} - b + x^{4}\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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