Optimal. Leaf size=99 \[ -\frac {3 a \text {RootSum}\left [\text {$\#$1}^8-2 \text {$\#$1}^4 a+a^2+a b\& ,\frac {\log \left (\sqrt [4]{a x^4-b x^2}-\text {$\#$1} x\right )-\log (x)}{\text {$\#$1}}\& \right ]}{4 b}-\frac {4 \left (a x^4-b x^2\right )^{3/4} \left (4 a x^2+3 b\right )}{21 b^2 x^5} \]
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Rubi [B] time = 1.58, antiderivative size = 474, normalized size of antiderivative = 4.79, number of steps used = 19, number of rules used = 11, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.290, Rules used = {2056, 6725, 271, 264, 1270, 1529, 1429, 377, 212, 206, 203} \begin {gather*} \frac {16 a \left (b-a x^2\right )}{21 b^2 x \sqrt [4]{a x^4-b x^2}}+\frac {3 a \sqrt {x} \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 b \sqrt [4]{a-\sqrt {-a} \sqrt {b}} \sqrt [4]{a x^4-b x^2}}+\frac {3 a \sqrt {x} \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 b \sqrt [4]{\sqrt {-a} \sqrt {b}+a} \sqrt [4]{a x^4-b x^2}}+\frac {3 a \sqrt {x} \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 b \sqrt [4]{a-\sqrt {-a} \sqrt {b}} \sqrt [4]{a x^4-b x^2}}+\frac {3 a \sqrt {x} \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 b \sqrt [4]{\sqrt {-a} \sqrt {b}+a} \sqrt [4]{a x^4-b x^2}}+\frac {4 \left (b-a x^2\right )}{7 b x^3 \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 264
Rule 271
Rule 377
Rule 1270
Rule 1429
Rule 1529
Rule 2056
Rule 6725
Rubi steps
\begin {align*} \int \frac {-2 b+a x^4}{x^4 \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 {-2 b+a x^4}{x^{9/2} \sqrt [4]{-b+a x^2} \left (b+a x^4\right )} \, dx}{\sqrt [4]{-b x^2+a x^4}}\\ &=\frac {\left (\sqrt {x} \sqrt [4]{-b+a x^2}\right ) \int \left (\frac {1}{x^{9/2} \sqrt [4]{-b+a x^2}}-\frac {3 b}{x^{9/2} \sqrt [4]{-b+a x^2} \left (b+a x^4\right )}\right ) \, dx}{\sqrt [4]{-b x^2+a x^4}}\\ &=\frac {\left (\sqrt {x} \sqrt [4]{-b+a x^2}\right ) \int \frac {1}{x^{9/2} \sqrt [4]{-b+a x^2}} \, dx}{\sqrt [4]{-b x^2+a x^4}}-\frac {\left (3 b \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \int \frac {1}{x^{9/2} \sqrt [4]{-b+a x^2} \left (b+a x^4\right )} \, dx}{\sqrt [4]{-b x^2+a x^4}}\\ &=-\frac {2 \left (b-a x^2\right )}{7 b x^3 \sqrt [4]{-b x^2+a x^4}}+\frac {\left (4 a \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \int \frac {1}{x^{5/2} \sqrt [4]{-b+a x^2}} \, dx}{7 b \sqrt [4]{-b x^2+a x^4}}-\frac {\left (6 b \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{x^8 \sqrt [4]{-b+a x^4} \left (b+a x^8\right )} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{-b x^2+a x^4}}\\ &=-\frac {2 \left (b-a x^2\right )}{7 b x^3 \sqrt [4]{-b x^2+a x^4}}-\frac {8 a \left (b-a x^2\right )}{21 b^2 x \sqrt [4]{-b x^2+a x^4}}-\frac {\left (6 b \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \operatorname {Subst}\left (\int \left (\frac {1}{b x^8 \sqrt [4]{-b+a x^4}}-\frac {a}{b \sqrt [4]{-b+a x^4} \left (b+a x^8\right )}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt [4]{-b x^2+a x^4}}\\ &=-\frac {2 \left (b-a x^2\right )}{7 b x^3 \sqrt [4]{-b x^2+a x^4}}-\frac {8 a \left (b-a x^2\right )}{21 b^2 x \sqrt [4]{-b x^2+a x^4}}-\frac {\left (6 \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{x^8 \sqrt [4]{-b+a x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{-b x^2+a x^4}}+\frac {\left (6 a \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{-b+a x^4} \left (b+a x^8\right )} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{-b x^2+a x^4}}\\ &=\frac {4 \left (b-a x^2\right )}{7 b x^3 \sqrt [4]{-b x^2+a x^4}}-\frac {8 a \left (b-a x^2\right )}{21 b^2 x \sqrt [4]{-b x^2+a x^4}}-\frac {\left (24 a \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{x^4 \sqrt [4]{-b+a x^4}} \, dx,x,\sqrt {x}\right )}{7 b \sqrt [4]{-b x^2+a x^4}}+\frac {\left (3 \sqrt {-a} a \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 {b} \sqrt [4]{-b x^2+a x^4}}+\frac {\left (3 \sqrt {-a} a \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 {b} \sqrt [4]{-b x^2+a x^4}}\\ &=\frac {4 \left (b-a x^2\right )}{7 b x^3 \sqrt [4]{-b x^2+a x^4}}+\frac {16 a \left (b-a x^2\right )}{21 b^2 x \sqrt [4]{-b x^2+a x^4}}+\frac {\left (3 \sqrt {-a} a \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 {b} \sqrt [4]{-b x^2+a x^4}}+\frac {\left (3 \sqrt {-a} a \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 {b} \sqrt [4]{-b x^2+a x^4}}\\ &=\frac {4 \left (b-a x^2\right )}{7 b x^3 \sqrt [4]{-b x^2+a x^4}}+\frac {16 a \left (b-a x^2\right )}{21 b^2 x \sqrt [4]{-b x^2+a x^4}}+\frac {\left (3 a \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 b \sqrt [4]{-b x^2+a x^4}}+\frac {\left (3 a \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 b \sqrt [4]{-b x^2+a x^4}}+\frac {\left (3 a \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 b \sqrt [4]{-b x^2+a x^4}}+\frac {\left (3 a \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 b \sqrt [4]{-b x^2+a x^4}}\\ &=\frac {4 \left (b-a x^2\right )}{7 b x^3 \sqrt [4]{-b x^2+a x^4}}+\frac {16 a \left (b-a x^2\right )}{21 b^2 x \sqrt [4]{-b x^2+a x^4}}+\frac {3 a \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 [4]{a-\sqrt {-a} \sqrt {b}} b \sqrt [4]{-b x^2+a x^4}}+\frac {3 a \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 [4]{a+\sqrt {-a} \sqrt {b}} b \sqrt [4]{-b x^2+a x^4}}+\frac {3 a \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 [4]{a-\sqrt {-a} \sqrt {b}} b \sqrt [4]{-b x^2+a x^4}}+\frac {3 a \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 [4]{a+\sqrt {-a} \sqrt {b}} b \sqrt [4]{-b x^2+a x^4}}\\ \end {align*}
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Mathematica [B] time = 2.96, size = 361, normalized size = 3.65 \begin {gather*} \frac {\sqrt {x} \left (\frac {8 \left (b-a x^2\right ) \left (4 a x^2+3 b\right )}{21 b^2 x^{7/2}}+\frac {3 (-a)^{7/8} \sqrt {x} \sqrt [4]{\frac {b}{x^2}-a} \left (\sqrt [4]{\sqrt {-a}+\sqrt {b}} \tan ^{-1}\left (\frac {\sqrt [4]{\frac {b}{x^2}-a}}{\sqrt [8]{-a} \sqrt [4]{\sqrt {-a}-\sqrt {b}}}\right )+\sqrt [4]{\sqrt {-a}-\sqrt {b}} \tan ^{-1}\left (\frac {\sqrt [4]{\frac {b}{x^2}-a}}{\sqrt [8]{-a} \sqrt [4]{\sqrt {-a}+\sqrt {b}}}\right )-\sqrt [4]{\sqrt {-a}+\sqrt {b}} \tanh ^{-1}\left (\frac {\sqrt [4]{\frac {b}{x^2}-a}}{\sqrt [8]{-a} \sqrt [4]{\sqrt {-a}-\sqrt {b}}}\right )-\sqrt [4]{\sqrt {-a}-\sqrt {b}} \tanh ^{-1}\left (\frac {\sqrt [4]{\frac {b}{x^2}-a}}{\sqrt [8]{-a} \sqrt [4]{\sqrt {-a}+\sqrt {b}}}\right )\right )}{b \sqrt [4]{\sqrt {-a}-\sqrt {b}} \sqrt [4]{\sqrt {-a}+\sqrt {b}}}\right )}{2 \sqrt [4]{a x^4-b x^2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 1.36, size = 99, normalized size = 1.00 \begin {gather*} -\frac {4 \left (3 b+4 a x^2\right ) \left (-b x^2+a x^4\right )^{3/4}}{21 b^2 x^5}-\frac {3 a \text {RootSum}\left [a^2+a b-2 a \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log (x)+\log \left (\sqrt [4]{-b x^2+a x^4}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\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 {a x^{4} - 2 \, b}{{\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 = 0.01, size = 0, normalized size = 0.00 \[\int \frac {a \,x^{4}-2 b}{x^{4} \left (a \,x^{4}+b \right ) \left (a \,x^{4}-b \,x^{2}\right )^{\frac {1}{4}}}\, dx\]
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^{4} - 2 \, b}{{\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 {2\,b-a\,x^4}{x^4\,\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^{4} - 2 b}{x^{4} \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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