Optimal. Leaf size=88 \[ \frac {a^2 \text {RootSum}\left [\text {$\#$1}^8-2 \text {$\#$1}^4 a+2 a^2\& ,\frac {\log \left (\sqrt [4]{a x^4+b}-\text {$\#$1} x\right )-\log (x)}{\text {$\#$1}^3 a-\text {$\#$1}^7}\& \right ]}{4 b}+\frac {\sqrt [4]{a x^4+b}}{b x} \]
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Rubi [B] time = 0.84, antiderivative size = 226, normalized size of antiderivative = 2.57, number of steps used = 14, number of rules used = 10, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.244, Rules used = {1586, 6725, 277, 331, 298, 203, 206, 1529, 511, 510} \begin {gather*} \frac {a x^3 \sqrt [4]{a x^4+b} F_1\left (\frac {3}{4};1,-\frac {5}{4};\frac {7}{4};-\frac {\sqrt {-a^2} x^4}{b},-\frac {a x^4}{b}\right )}{6 b^2 \sqrt [4]{\frac {a x^4}{b}+1}}+\frac {a x^3 \sqrt [4]{a x^4+b} F_1\left (\frac {3}{4};1,-\frac {5}{4};\frac {7}{4};\frac {\sqrt {-a^2} x^4}{b},-\frac {a x^4}{b}\right )}{6 b^2 \sqrt [4]{\frac {a x^4}{b}+1}}+\frac {\sqrt [4]{a x^4+b}}{b x}+\frac {\sqrt [4]{a} \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{2 b}-\frac {\sqrt [4]{a} \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{2 b} \end {gather*}
Warning: Unable to verify antiderivative.
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Rule 203
Rule 206
Rule 277
Rule 298
Rule 331
Rule 510
Rule 511
Rule 1529
Rule 1586
Rule 6725
Rubi steps
\begin {align*} \int \frac {-b^2+a^2 x^8}{x^2 \left (b+a x^4\right )^{3/4} \left (b^2+a^2 x^8\right )} \, dx &=\int \frac {\left (-b+a x^4\right ) \sqrt [4]{b+a x^4}}{x^2 \left (b^2+a^2 x^8\right )} \, dx\\ &=\int \left (-\frac {\sqrt [4]{b+a x^4}}{b x^2}+\frac {a x^2 \left (b+a x^4\right )^{5/4}}{b \left (b^2+a^2 x^8\right )}\right ) \, dx\\ &=-\frac {\int \frac {\sqrt [4]{b+a x^4}}{x^2} \, dx}{b}+\frac {a \int \frac {x^2 \left (b+a x^4\right )^{5/4}}{b^2+a^2 x^8} \, dx}{b}\\ &=\frac {\sqrt [4]{b+a x^4}}{b x}-\frac {a \int \frac {x^2}{\left (b+a x^4\right )^{3/4}} \, dx}{b}+\frac {a \int \left (-\frac {a^2 x^2 \left (b+a x^4\right )^{5/4}}{2 \sqrt {-a^2} b \left (\sqrt {-a^2} b-a^2 x^4\right )}-\frac {a^2 x^2 \left (b+a x^4\right )^{5/4}}{2 \sqrt {-a^2} b \left (\sqrt {-a^2} b+a^2 x^4\right )}\right ) \, dx}{b}\\ &=\frac {\sqrt [4]{b+a x^4}}{b x}+\frac {\left (a \sqrt {-a^2}\right ) \int \frac {x^2 \left (b+a x^4\right )^{5/4}}{\sqrt {-a^2} b-a^2 x^4} \, dx}{2 b^2}+\frac {\left (a \sqrt {-a^2}\right ) \int \frac {x^2 \left (b+a x^4\right )^{5/4}}{\sqrt {-a^2} b+a^2 x^4} \, dx}{2 b^2}-\frac {a \operatorname {Subst}\left (\int \frac {x^2}{1-a x^4} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )}{b}\\ &=\frac {\sqrt [4]{b+a x^4}}{b x}-\frac {\sqrt {a} \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {a} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )}{2 b}+\frac {\sqrt {a} \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {a} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )}{2 b}+\frac {\left (a \sqrt {-a^2} \sqrt [4]{b+a x^4}\right ) \int \frac {x^2 \left (1+\frac {a x^4}{b}\right )^{5/4}}{\sqrt {-a^2} b-a^2 x^4} \, dx}{2 b \sqrt [4]{1+\frac {a x^4}{b}}}+\frac {\left (a \sqrt {-a^2} \sqrt [4]{b+a x^4}\right ) \int \frac {x^2 \left (1+\frac {a x^4}{b}\right )^{5/4}}{\sqrt {-a^2} b+a^2 x^4} \, dx}{2 b \sqrt [4]{1+\frac {a x^4}{b}}}\\ &=\frac {\sqrt [4]{b+a x^4}}{b x}+\frac {a x^3 \sqrt [4]{b+a x^4} F_1\left (\frac {3}{4};1,-\frac {5}{4};\frac {7}{4};-\frac {\sqrt {-a^2} x^4}{b},-\frac {a x^4}{b}\right )}{6 b^2 \sqrt [4]{1+\frac {a x^4}{b}}}+\frac {a x^3 \sqrt [4]{b+a x^4} F_1\left (\frac {3}{4};1,-\frac {5}{4};\frac {7}{4};\frac {\sqrt {-a^2} x^4}{b},-\frac {a x^4}{b}\right )}{6 b^2 \sqrt [4]{1+\frac {a x^4}{b}}}+\frac {\sqrt [4]{a} \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{2 b}-\frac {\sqrt [4]{a} \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{2 b}\\ \end {align*}
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Mathematica [F] time = 0.22, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-b^2+a^2 x^8}{x^2 \left (b+a x^4\right )^{3/4} \left (b^2+a^2 x^8\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 0.63, size = 87, normalized size = 0.99 \begin {gather*} \frac {\sqrt [4]{b+a x^4}}{b x}+\frac {a^2 \text {RootSum}\left [2 a^2-2 a \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {\log (x)-\log \left (\sqrt [4]{b+a x^4}-x \text {$\#$1}\right )}{-a \text {$\#$1}^3+\text {$\#$1}^7}\&\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^{2} x^{8} - b^{2}}{{\left (a^{2} x^{8} + b^{2}\right )} {\left (a x^{4} + b\right )}^{\frac {3}{4}} x^{2}}\,{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 \[\int \frac {a^{2} x^{8}-b^{2}}{x^{2} \left (a \,x^{4}+b \right )^{\frac {3}{4}} \left (a^{2} x^{8}+b^{2}\right )}\, dx\]
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 {a^{2} x^{8} - b^{2}}{{\left (a^{2} x^{8} + b^{2}\right )} {\left (a x^{4} + b\right )}^{\frac {3}{4}} x^{2}}\,{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^2-a^2\,x^8}{x^2\,\left (a^2\,x^8+b^2\right )\,{\left (a\,x^4+b\right )}^{3/4}} \,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 {\left (a x^{4} - b\right ) \sqrt [4]{a x^{4} + b}}{x^{2} \left (a^{2} x^{8} + b^{2}\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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