Optimal. Leaf size=105 \[ \frac {a \text {RootSum}\left [\text {$\#$1}^8-2 \text {$\#$1}^4 a+a^2+a b\& ,\frac {\text {$\#$1} \log \left (\sqrt [4]{a x^4+b x^2}-\text {$\#$1} x\right )-\text {$\#$1} \log (x)}{\text {$\#$1}^4-a}\& \right ]}{4 b}-\frac {2 \left (a x^2+b\right ) \sqrt [4]{a x^4+b x^2}}{5 b^2 x^3} \]
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Rubi [A] time = 1.18, antiderivative size = 201, normalized size of antiderivative = 1.91, number of steps used = 14, number of rules used = 9, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.321, Rules used = {2056, 1270, 1521, 271, 264, 6725, 1529, 511, 510} \begin {gather*} -\frac {a 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 {b}},-\frac {a x^2}{b}\right )}{3 b^2 \sqrt [4]{\frac {a x^2}{b}+1}}-\frac {a 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 {b}},-\frac {a x^2}{b}\right )}{3 b^2 \sqrt [4]{\frac {a x^2}{b}+1}}-\frac {2 a \sqrt [4]{a x^4+b x^2}}{5 b^2 x}-\frac {2 \sqrt [4]{a x^4+b x^2}}{5 b x^3} \end {gather*}
Warning: Unable to verify antiderivative.
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Rule 264
Rule 271
Rule 510
Rule 511
Rule 1270
Rule 1521
Rule 1529
Rule 2056
Rule 6725
Rubi steps
\begin {align*} \int \frac {\sqrt [4]{b x^2+a x^4}}{x^4 \left (b+a x^4\right )} \, dx &=\frac {\sqrt [4]{b x^2+a x^4} \int \frac {\sqrt [4]{b+a x^2}}{x^{7/2} \left (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 ) \operatorname {Subst}\left (\int \frac {\sqrt [4]{b+a x^4}}{x^6 \left (b+a x^8\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt [4]{b+a x^2}}\\ &=\frac {\left (2 \sqrt [4]{b x^2+a x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{x^6 \left (b+a x^4\right )^{3/4}} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt [4]{b+a x^2}}+\frac {\left (2 \sqrt [4]{b x^2+a x^4}\right ) \operatorname {Subst}\left (\int \frac {a b-a b x^4}{x^2 \left (b+a x^4\right )^{3/4} \left (b+a x^8\right )} \, dx,x,\sqrt {x}\right )}{b \sqrt {x} \sqrt [4]{b+a x^2}}\\ &=-\frac {2 \sqrt [4]{b x^2+a x^4}}{5 b x^3}+\frac {\left (2 \sqrt [4]{b x^2+a x^4}\right ) \operatorname {Subst}\left (\int \left (\frac {a}{x^2 \left (b+a x^4\right )^{3/4}}-\frac {a x^2 \sqrt [4]{b+a x^4}}{b+a x^8}\right ) \, dx,x,\sqrt {x}\right )}{b \sqrt {x} \sqrt [4]{b+a x^2}}-\frac {\left (8 a \sqrt [4]{b x^2+a x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{x^2 \left (b+a x^4\right )^{3/4}} \, dx,x,\sqrt {x}\right )}{5 b \sqrt {x} \sqrt [4]{b+a x^2}}\\ &=-\frac {2 \sqrt [4]{b x^2+a x^4}}{5 b x^3}+\frac {8 a \sqrt [4]{b x^2+a x^4}}{5 b^2 x}+\frac {\left (2 a \sqrt [4]{b x^2+a x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{x^2 \left (b+a x^4\right )^{3/4}} \, dx,x,\sqrt {x}\right )}{b \sqrt {x} \sqrt [4]{b+a x^2}}-\frac {\left (2 a \sqrt [4]{b x^2+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2 \sqrt [4]{b+a x^4}}{b+a x^8} \, dx,x,\sqrt {x}\right )}{b \sqrt {x} \sqrt [4]{b+a x^2}}\\ &=-\frac {2 \sqrt [4]{b x^2+a x^4}}{5 b x^3}-\frac {2 a \sqrt [4]{b x^2+a x^4}}{5 b^2 x}-\frac {\left (2 a \sqrt [4]{b x^2+a x^4}\right ) \operatorname {Subst}\left (\int \left (-\frac {a x^2 \sqrt [4]{b+a x^4}}{2 \sqrt {-a} \sqrt {b} \left (\sqrt {-a} \sqrt {b}-a x^4\right )}-\frac {a x^2 \sqrt [4]{b+a x^4}}{2 \sqrt {-a} \sqrt {b} \left (\sqrt {-a} \sqrt {b}+a x^4\right )}\right ) \, dx,x,\sqrt {x}\right )}{b \sqrt {x} \sqrt [4]{b+a x^2}}\\ &=-\frac {2 \sqrt [4]{b x^2+a x^4}}{5 b x^3}-\frac {2 a \sqrt [4]{b x^2+a x^4}}{5 b^2 x}-\frac {\left (\sqrt {-a} a \sqrt [4]{b x^2+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2 \sqrt [4]{b+a x^4}}{\sqrt {-a} \sqrt {b}-a x^4} \, dx,x,\sqrt {x}\right )}{b^{3/2} \sqrt {x} \sqrt [4]{b+a x^2}}-\frac {\left (\sqrt {-a} a \sqrt [4]{b x^2+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2 \sqrt [4]{b+a x^4}}{\sqrt {-a} \sqrt {b}+a x^4} \, dx,x,\sqrt {x}\right )}{b^{3/2} \sqrt {x} \sqrt [4]{b+a x^2}}\\ &=-\frac {2 \sqrt [4]{b x^2+a x^4}}{5 b x^3}-\frac {2 a \sqrt [4]{b x^2+a x^4}}{5 b^2 x}-\frac {\left (\sqrt {-a} a \sqrt [4]{b x^2+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2 \sqrt [4]{1+\frac {a x^4}{b}}}{\sqrt {-a} \sqrt {b}-a x^4} \, dx,x,\sqrt {x}\right )}{b^{3/2} \sqrt {x} \sqrt [4]{1+\frac {a x^2}{b}}}-\frac {\left (\sqrt {-a} a \sqrt [4]{b x^2+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2 \sqrt [4]{1+\frac {a x^4}{b}}}{\sqrt {-a} \sqrt {b}+a x^4} \, dx,x,\sqrt {x}\right )}{b^{3/2} \sqrt {x} \sqrt [4]{1+\frac {a x^2}{b}}}\\ &=-\frac {2 \sqrt [4]{b x^2+a x^4}}{5 b x^3}-\frac {2 a \sqrt [4]{b x^2+a x^4}}{5 b^2 x}-\frac {a 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 {b}},-\frac {a x^2}{b}\right )}{3 b^2 \sqrt [4]{1+\frac {a x^2}{b}}}-\frac {a 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 {b}},-\frac {a x^2}{b}\right )}{3 b^2 \sqrt [4]{1+\frac {a x^2}{b}}}\\ \end {align*}
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Mathematica [F] time = 1.48, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt [4]{b x^2+a x^4}}{x^4 \left (b+a x^4\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 0.00, size = 105, normalized size = 1.00 \begin {gather*} -\frac {2 \left (b+a x^2\right ) \sqrt [4]{b x^2+a x^4}}{5 b^2 x^3}+\frac {a \text {RootSum}\left [a^2+a b-2 a \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log (x) \text {$\#$1}+\log \left (\sqrt [4]{b x^2+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}}{-a+\text {$\#$1}^4}\&\right ]}{4 b} \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 {{\left (a x^{4} + b x^{2}\right )}^{\frac {1}{4}}}{{\left (a x^{4} + b\right )} x^{4}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.00, size = 0, normalized size = 0.00 \[\int \frac {\left (a \,x^{4}+b \,x^{2}\right )^{\frac {1}{4}}}{x^{4} \left (a \,x^{4}+b \right )}\, dx\]
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 {{\left (a x^{4} + b x^{2}\right )}^{\frac {1}{4}}}{{\left (a x^{4} + b\right )} x^{4}}\,{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 {{\left (a\,x^4+b\,x^2\right )}^{1/4}}{x^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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sympy [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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