Optimal. Leaf size=158 \[ x \sqrt [4]{b x^2+a x^4}-\frac {b \text {ArcTan}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b x^2+a x^4}}\right )}{2 a^{3/4}}+\frac {b \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b x^2+a x^4}}\right )}{2 a^{3/4}}+\frac {3}{8} b \text {RootSum}\left [2 a^2-a b-4 a \text {$\#$1}^4+2 \text {$\#$1}^8\& ,\frac {-\log (x) \text {$\#$1}+\log \left (\sqrt [4]{b x^2+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}}{-a+\text {$\#$1}^4}\& \right ] \]
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Rubi [A]
time = 0.78, antiderivative size = 293, normalized size of antiderivative = 1.85, number of steps
used = 16, number of rules used = 12, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.324, Rules used = {2081, 6857,
285, 335, 338, 304, 209, 212, 1284, 1543, 525, 524} \begin {gather*} -\frac {b \sqrt [4]{a x^4+b x^2} \text {ArcTan}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2+b}}\right )}{2 a^{3/4} \sqrt {x} \sqrt [4]{a x^2+b}}+\frac {b \sqrt [4]{a x^4+b x^2} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2+b}}\right )}{2 a^{3/4} \sqrt {x} \sqrt [4]{a x^2+b}}-\frac {x \sqrt [4]{a x^4+b x^2} F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};-\frac {\sqrt {a} x^2}{\sqrt {2} \sqrt {b}},-\frac {a x^2}{b}\right )}{2 \sqrt [4]{\frac {a x^2}{b}+1}}-\frac {x \sqrt [4]{a x^4+b x^2} F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};\frac {\sqrt {a} x^2}{\sqrt {2} \sqrt {b}},-\frac {a x^2}{b}\right )}{2 \sqrt [4]{\frac {a x^2}{b}+1}}+x \sqrt [4]{a x^4+b x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 212
Rule 285
Rule 304
Rule 335
Rule 338
Rule 524
Rule 525
Rule 1284
Rule 1543
Rule 2081
Rule 6857
Rubi steps
\begin {align*} \int \frac {\sqrt [4]{b x^2+a x^4} \left (-b+2 a x^4\right )}{-2 b+a x^4} \, dx &=\frac {\sqrt [4]{b x^2+a x^4} \int \frac {\sqrt {x} \sqrt [4]{b+a x^2} \left (-b+2 a x^4\right )}{-2 b+a x^4} \, dx}{\sqrt {x} \sqrt [4]{b+a x^2}}\\ &=\frac {\sqrt [4]{b x^2+a x^4} \int \left (2 \sqrt {x} \sqrt [4]{b+a x^2}+\frac {3 b \sqrt {x} \sqrt [4]{b+a x^2}}{-2 b+a x^4}\right ) \, dx}{\sqrt {x} \sqrt [4]{b+a x^2}}\\ &=\frac {\left (2 \sqrt [4]{b x^2+a x^4}\right ) \int \sqrt {x} \sqrt [4]{b+a x^2} \, dx}{\sqrt {x} \sqrt [4]{b+a x^2}}+\frac {\left (3 b \sqrt [4]{b x^2+a x^4}\right ) \int \frac {\sqrt {x} \sqrt [4]{b+a x^2}}{-2 b+a x^4} \, dx}{\sqrt {x} \sqrt [4]{b+a x^2}}\\ &=x \sqrt [4]{b x^2+a x^4}+\frac {\left (b \sqrt [4]{b x^2+a x^4}\right ) \int \frac {\sqrt {x}}{\left (b+a x^2\right )^{3/4}} \, dx}{2 \sqrt {x} \sqrt [4]{b+a x^2}}+\frac {\left (6 b \sqrt [4]{b x^2+a x^4}\right ) \text {Subst}\left (\int \frac {x^2 \sqrt [4]{b+a x^4}}{-2 b+a x^8} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt [4]{b+a x^2}}\\ &=x \sqrt [4]{b x^2+a x^4}+\frac {\left (b \sqrt [4]{b x^2+a x^4}\right ) \text {Subst}\left (\int \frac {x^2}{\left (b+a x^4\right )^{3/4}} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt [4]{b+a x^2}}+\frac {\left (6 b \sqrt [4]{b x^2+a x^4}\right ) \text {Subst}\left (\int \left (-\frac {\sqrt {a} x^2 \sqrt [4]{b+a x^4}}{2 \sqrt {2} \sqrt {b} \left (\sqrt {2} \sqrt {a} \sqrt {b}-a x^4\right )}-\frac {\sqrt {a} x^2 \sqrt [4]{b+a x^4}}{2 \sqrt {2} \sqrt {b} \left (\sqrt {2} \sqrt {a} \sqrt {b}+a x^4\right )}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt [4]{b+a x^2}}\\ &=x \sqrt [4]{b x^2+a x^4}-\frac {\left (3 \sqrt {a} \sqrt {b} \sqrt [4]{b x^2+a x^4}\right ) \text {Subst}\left (\int \frac {x^2 \sqrt [4]{b+a x^4}}{\sqrt {2} \sqrt {a} \sqrt {b}-a x^4} \, dx,x,\sqrt {x}\right )}{\sqrt {2} \sqrt {x} \sqrt [4]{b+a x^2}}-\frac {\left (3 \sqrt {a} \sqrt {b} \sqrt [4]{b x^2+a x^4}\right ) \text {Subst}\left (\int \frac {x^2 \sqrt [4]{b+a x^4}}{\sqrt {2} \sqrt {a} \sqrt {b}+a x^4} \, dx,x,\sqrt {x}\right )}{\sqrt {2} \sqrt {x} \sqrt [4]{b+a x^2}}+\frac {\left (b \sqrt [4]{b x^2+a x^4}\right ) \text {Subst}\left (\int \frac {x^2}{1-a x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{\sqrt {x} \sqrt [4]{b+a x^2}}\\ &=x \sqrt [4]{b x^2+a x^4}+\frac {\left (b \sqrt [4]{b x^2+a x^4}\right ) \text {Subst}\left (\int \frac {1}{1-\sqrt {a} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{2 \sqrt {a} \sqrt {x} \sqrt [4]{b+a x^2}}-\frac {\left (b \sqrt [4]{b x^2+a x^4}\right ) \text {Subst}\left (\int \frac {1}{1+\sqrt {a} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{2 \sqrt {a} \sqrt {x} \sqrt [4]{b+a x^2}}-\frac {\left (3 \sqrt {a} \sqrt {b} \sqrt [4]{b x^2+a x^4}\right ) \text {Subst}\left (\int \frac {x^2 \sqrt [4]{1+\frac {a x^4}{b}}}{\sqrt {2} \sqrt {a} \sqrt {b}-a x^4} \, dx,x,\sqrt {x}\right )}{\sqrt {2} \sqrt {x} \sqrt [4]{1+\frac {a x^2}{b}}}-\frac {\left (3 \sqrt {a} \sqrt {b} \sqrt [4]{b x^2+a x^4}\right ) \text {Subst}\left (\int \frac {x^2 \sqrt [4]{1+\frac {a x^4}{b}}}{\sqrt {2} \sqrt {a} \sqrt {b}+a x^4} \, dx,x,\sqrt {x}\right )}{\sqrt {2} \sqrt {x} \sqrt [4]{1+\frac {a x^2}{b}}}\\ &=x \sqrt [4]{b x^2+a x^4}-\frac {x \sqrt [4]{b x^2+a x^4} F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};-\frac {\sqrt {a} x^2}{\sqrt {2} \sqrt {b}},-\frac {a x^2}{b}\right )}{2 \sqrt [4]{1+\frac {a x^2}{b}}}-\frac {x \sqrt [4]{b x^2+a x^4} F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};\frac {\sqrt {a} x^2}{\sqrt {2} \sqrt {b}},-\frac {a x^2}{b}\right )}{2 \sqrt [4]{1+\frac {a x^2}{b}}}-\frac {b \sqrt [4]{b x^2+a x^4} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{2 a^{3/4} \sqrt {x} \sqrt [4]{b+a x^2}}+\frac {b \sqrt [4]{b x^2+a x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{2 a^{3/4} \sqrt {x} \sqrt [4]{b+a x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.44, size = 197, normalized size = 1.25 \begin {gather*} \frac {x^{3/2} \left (b+a x^2\right )^{3/4} \left (8 a^{3/4} x^{3/2} \sqrt [4]{b+a x^2}-4 b \text {ArcTan}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b+a x^2}}\right )+4 b \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b+a x^2}}\right )+3 a^{3/4} b \text {RootSum}\left [2 a^2-a b-4 a \text {$\#$1}^4+2 \text {$\#$1}^8\&,\frac {-\log \left (\sqrt {x}\right ) \text {$\#$1}+\log \left (\sqrt [4]{b+a x^2}-\sqrt {x} \text {$\#$1}\right ) \text {$\#$1}}{-a+\text {$\#$1}^4}\&\right ]\right )}{8 a^{3/4} \left (x^2 \left (b+a x^2\right )\right )^{3/4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\left (a \,x^{4}+b \,x^{2}\right )^{\frac {1}{4}} \left (2 a \,x^{4}-b \right )}{a \,x^{4}-2 b}\, dx\]
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] Timed out
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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt [4]{x^{2} \left (a x^{2} + b\right )} \left (2 a x^{4} - b\right )}{a x^{4} - 2 b}\, dx \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*} \text {could not integrate} \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 {\left (b-2\,a\,x^4\right )\,{\left (a\,x^4+b\,x^2\right )}^{1/4}}{2\,b-a\,x^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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