Optimal. Leaf size=72 \[ -\frac {1}{4} \text {RootSum}\left [\text {$\#$1}^8-2 \text {$\#$1}^4 a+a^2+a b\& ,\frac {\text {$\#$1}^3 \log \left (\sqrt [4]{a x^4+b x^2}-\text {$\#$1} x\right )-\text {$\#$1}^3 \log (x)}{\text {$\#$1}^4-a}\& \right ] \]
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Rubi [B] time = 0.72, antiderivative size = 417, normalized size of antiderivative = 5.79, number of steps used = 21, number of rules used = 9, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.281, Rules used = {2056, 1270, 1429, 408, 240, 212, 206, 203, 377} \begin {gather*} -\frac {\sqrt {x} \left (a-\sqrt {-a} \sqrt {b}\right )^{3/4} \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 {-a} \sqrt {b} \sqrt [4]{a x^4+b x^2}}+\frac {\sqrt {x} \left (\sqrt {-a} \sqrt {b}+a\right )^{3/4} \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 {-a} \sqrt {b} \sqrt [4]{a x^4+b x^2}}-\frac {\sqrt {x} \left (a-\sqrt {-a} \sqrt {b}\right )^{3/4} \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 {-a} \sqrt {b} \sqrt [4]{a x^4+b x^2}}+\frac {\sqrt {x} \left (\sqrt {-a} \sqrt {b}+a\right )^{3/4} \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 {-a} \sqrt {b} \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 240
Rule 377
Rule 408
Rule 1270
Rule 1429
Rule 2056
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 {\left (b+a x^2\right )^{3/4}}{\sqrt {x} \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 {\left (b+a x^4\right )^{3/4}}{b+a x^8} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{b x^2+a x^4}}\\ &=\frac {\left (\sqrt {-a} \sqrt {x} \sqrt [4]{b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {\left (b+a x^4\right )^{3/4}}{\sqrt {-a} \sqrt {b}-a x^4} \, dx,x,\sqrt {x}\right )}{\sqrt {b} \sqrt [4]{b x^2+a x^4}}+\frac {\left (\sqrt {-a} \sqrt {x} \sqrt [4]{b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {\left (b+a x^4\right )^{3/4}}{\sqrt {-a} \sqrt {b}+a x^4} \, dx,x,\sqrt {x}\right )}{\sqrt {b} \sqrt [4]{b x^2+a x^4}}\\ &=-\frac {\left (\sqrt {-a} \left (\sqrt {-a}-\sqrt {b}\right ) \sqrt {x} \sqrt [4]{b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\sqrt {-a} \sqrt {b}+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 (\sqrt {-a} \left (\sqrt {-a}+\sqrt {b}\right ) \sqrt {x} \sqrt [4]{b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\sqrt {-a} \sqrt {b}-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 (\sqrt {-a} \left (\sqrt {-a}-\sqrt {b}\right ) \sqrt {x} \sqrt [4]{b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-a} \sqrt {b}-\left (\sqrt {-a} a \sqrt {b}-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 (\sqrt {-a} \left (\sqrt {-a}+\sqrt {b}\right ) \sqrt {x} \sqrt [4]{b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-a} \sqrt {b}-\left (\sqrt {-a} a \sqrt {b}+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 (a-\sqrt {-a} \sqrt {b}\right )^{3/4} \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 {-a} \sqrt {b} \sqrt [4]{b x^2+a x^4}}+\frac {\left (a+\sqrt {-a} \sqrt {b}\right )^{3/4} \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 {-a} \sqrt {b} \sqrt [4]{b x^2+a x^4}}-\frac {\left (a-\sqrt {-a} \sqrt {b}\right )^{3/4} \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 {-a} \sqrt {b} \sqrt [4]{b x^2+a x^4}}+\frac {\left (a+\sqrt {-a} \sqrt {b}\right )^{3/4} \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 {-a} \sqrt {b} \sqrt [4]{b x^2+a x^4}}\\ \end {align*}
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Mathematica [B] time = 0.85, size = 245, normalized size = 3.40 \begin {gather*} -\frac {\left (x^2 \left (a x^2+b\right )\right )^{3/4} \left (\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 )}{\sqrt [4]{a-\sqrt {-a} \sqrt {b}}}-\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 )}{\sqrt [4]{\sqrt {-a} \sqrt {b}+a}}\right )}{2 \sqrt {b} x^3 \left (a+\frac {b}{x^2}\right )^{3/4}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.59, size = 72, normalized size = 1.00 \begin {gather*} -\frac {1}{4} \text {RootSum}\left [a^2+a b-2 a \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log (x) \text {$\#$1}^3+\log \left (\sqrt [4]{b x^2+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^3}{-a+\text {$\#$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 (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 = 0.03, 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 {a\,x^2+b}{\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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