Optimal. Leaf size=71 \[ -\frac {1}{2} \text {RootSum}\left [-\text {$\#$1}^8+\text {$\#$1}^4 a+b\& ,\frac {\text {$\#$1}^3 \log \left (\sqrt [4]{a x^4-b x^2}-\text {$\#$1} x\right )-\text {$\#$1}^3 \log (x)}{2 \text {$\#$1}^4-a}\& \right ] \]
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Rubi [B] time = 1.59, antiderivative size = 607, normalized size of antiderivative = 8.55, number of steps used = 21, number of rules used = 11, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.275, Rules used = {2056, 1269, 1428, 408, 240, 212, 206, 203, 377, 208, 205} \begin {gather*} -\frac {\sqrt {x} \left (-a \sqrt {a^2+4 b}+a^2+2 b\right )^{3/4} \sqrt [4]{a x^2-b} \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 )}{\sqrt {a^2+4 b} \left (a-\sqrt {a^2+4 b}\right )^{3/4} \sqrt [4]{a x^4-b x^2}}+\frac {\sqrt {x} \left (a \sqrt {a^2+4 b}+a^2+2 b\right )^{3/4} \sqrt [4]{a x^2-b} \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 x^4-b x^2}}-\frac {\sqrt {x} \left (-a \sqrt {a^2+4 b}+a^2+2 b\right )^{3/4} \sqrt [4]{a x^2-b} \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 )}{\sqrt {a^2+4 b} \left (a-\sqrt {a^2+4 b}\right )^{3/4} \sqrt [4]{a x^4-b x^2}}+\frac {\sqrt {x} \left (a \sqrt {a^2+4 b}+a^2+2 b\right )^{3/4} \sqrt [4]{a x^2-b} \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 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 408
Rule 1269
Rule 1428
Rule 2056
Rubi steps
\begin {align*} \int \frac {-b+a x^2}{\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 {\left (-b+a x^2\right )^{3/4}}{\sqrt {x} \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 {\left (-b+a x^4\right )^{3/4}}{-b+a x^4+x^8} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{-b x^2+a x^4}}\\ &=\frac {\left (4 \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {\left (-b+a x^4\right )^{3/4}}{a-\sqrt {a^2+4 b}+2 x^4} \, dx,x,\sqrt {x}\right )}{\sqrt {a^2+4 b} \sqrt [4]{-b x^2+a x^4}}-\frac {\left (4 \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {\left (-b+a x^4\right )^{3/4}}{a+\sqrt {a^2+4 b}+2 x^4} \, dx,x,\sqrt {x}\right )}{\sqrt {a^2+4 b} \sqrt [4]{-b x^2+a x^4}}\\ &=-\frac {\left (2 \left (a^2+2 b-a \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 {a^2+4 b} \sqrt [4]{-b x^2+a x^4}}+\frac {\left (2 \left (2 b+a \left (a+\sqrt {a^2+4 b}\right )\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 {a^2+4 b} \sqrt [4]{-b x^2+a x^4}}\\ &=-\frac {\left (2 \left (a^2+2 b-a \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 {a^2+4 b} \sqrt [4]{-b x^2+a x^4}}+\frac {\left (2 \left (2 b+a \left (a+\sqrt {a^2+4 b}\right )\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 {a^2+4 b} \sqrt [4]{-b x^2+a x^4}}\\ &=-\frac {\left (\left (a^2+2 b-a \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^2+4 b} \sqrt {a-\sqrt {a^2+4 b}} \sqrt [4]{-b x^2+a x^4}}-\frac {\left (\left (a^2+2 b-a \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^2+4 b} \sqrt {a-\sqrt {a^2+4 b}} \sqrt [4]{-b x^2+a x^4}}+\frac {\left (\left (2 b+a \left (a+\sqrt {a^2+4 b}\right )\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^2+4 b} \sqrt {a+\sqrt {a^2+4 b}} \sqrt [4]{-b x^2+a x^4}}+\frac {\left (\left (2 b+a \left (a+\sqrt {a^2+4 b}\right )\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^2+4 b} \sqrt {a+\sqrt {a^2+4 b}} \sqrt [4]{-b x^2+a x^4}}\\ &=-\frac {\left (a^2+2 b-a \sqrt {a^2+4 b}\right )^{3/4} \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 )}{\sqrt {a^2+4 b} \left (a-\sqrt {a^2+4 b}\right )^{3/4} \sqrt [4]{-b x^2+a x^4}}+\frac {\left (a^2+2 b+a \sqrt {a^2+4 b}\right )^{3/4} \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 )}{\sqrt {a^2+4 b} \left (a+\sqrt {a^2+4 b}\right )^{3/4} \sqrt [4]{-b x^2+a x^4}}-\frac {\left (a^2+2 b-a \sqrt {a^2+4 b}\right )^{3/4} \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 )}{\sqrt {a^2+4 b} \left (a-\sqrt {a^2+4 b}\right )^{3/4} \sqrt [4]{-b x^2+a x^4}}+\frac {\left (a^2+2 b+a \sqrt {a^2+4 b}\right )^{3/4} \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 )}{\sqrt {a^2+4 b} \left (a+\sqrt {a^2+4 b}\right )^{3/4} \sqrt [4]{-b x^2+a x^4}}\\ \end {align*}
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Mathematica [B] time = 0.81, size = 423, normalized size = 5.96 \begin {gather*} \frac {\left (a x^4-b x^2\right )^{3/4} \left (-\left (-\sqrt {a^2+4 b}-a\right )^{3/4} \log \left (2^{3/4} \sqrt [4]{-\sqrt {a^2+4 b}-a}-2 \sqrt [4]{\frac {b}{x^2}-a}\right )+\left (\sqrt {a^2+4 b}-a\right )^{3/4} \log \left (2^{3/4} \sqrt [4]{\sqrt {a^2+4 b}-a}-2 \sqrt [4]{\frac {b}{x^2}-a}\right )+\left (-\sqrt {a^2+4 b}-a\right )^{3/4} \log \left (2^{3/4} \sqrt [4]{-\sqrt {a^2+4 b}-a}+2 \sqrt [4]{\frac {b}{x^2}-a}\right )-\left (\sqrt {a^2+4 b}-a\right )^{3/4} \log \left (2^{3/4} \sqrt [4]{\sqrt {a^2+4 b}-a}+2 \sqrt [4]{\frac {b}{x^2}-a}\right )-2 \left (-\sqrt {a^2+4 b}-a\right )^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{\frac {2 b}{x^2}-2 a}}{\sqrt [4]{-\sqrt {a^2+4 b}-a}}\right )+2 \left (\sqrt {a^2+4 b}-a\right )^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{\frac {2 b}{x^2}-2 a}}{\sqrt [4]{\sqrt {a^2+4 b}-a}}\right )\right )}{2 x^3 \sqrt {a^2+4 b} \left (\frac {2 b}{x^2}-2 a\right )^{3/4}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.64, size = 71, normalized size = 1.00 \begin {gather*} -\frac {1}{2} \text {RootSum}\left [b+a \text {$\#$1}^4-\text {$\#$1}^8\&,\frac {-\log (x) \text {$\#$1}^3+\log \left (\sqrt [4]{-b x^2+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^3}{-a+2 \text {$\#$1}^4}\&\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 (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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maple [F] time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {a \,x^{2}-b}{\left (x^{4}+a \,x^{2}-b \right ) \left (a \,x^{4}-b \,x^{2}\right )^{\frac {1}{4}}}\, dx\]
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^{2} - 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 = -0.01 \begin {gather*} \int -\frac {b-a\,x^2}{{\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^{2} - 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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