Optimal. Leaf size=155 \[ \frac {\left (8 a^{9/4}-a^{5/4} b\right ) \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{2^{3/4} b^2}-\frac {\left (8 a^{9/4}-a^{5/4} b\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{2^{3/4} b^2}+\frac {\sqrt [4]{a x^4-b} \left (80 a^2 x^8-9 a b x^8-16 a b x^4+8 b^2\right )}{9 b^2 x^9} \]
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Rubi [C] time = 0.70, antiderivative size = 232, normalized size of antiderivative = 1.50, number of steps used = 12, number of rules used = 10, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {6725, 271, 264, 277, 331, 298, 203, 206, 511, 510} \begin {gather*} \frac {a^{5/4} (8 a-b) \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{2 b^2}-\frac {a^{5/4} (8 a-b) \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{2 b^2}+\frac {a^2 x^3 (8 a-b) \sqrt [4]{a x^4-b} F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};-\frac {a x^4}{b},\frac {a x^4}{b}\right )}{3 b^3 \sqrt [4]{1-\frac {a x^4}{b}}}+\frac {a (8 a-b) \sqrt [4]{a x^4-b}}{b^2 x}+\frac {8 a \left (a x^4-b\right )^{5/4}}{9 b^2 x^5}-\frac {8 \left (a x^4-b\right )^{5/4}}{9 b x^9} \end {gather*}
Warning: Unable to verify antiderivative.
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Rule 203
Rule 206
Rule 264
Rule 271
Rule 277
Rule 298
Rule 331
Rule 510
Rule 511
Rule 6725
Rubi steps
\begin {align*} \int \frac {\sqrt [4]{-b+a x^4} \left (-8 b+a x^8\right )}{x^{10} \left (b+a x^4\right )} \, dx &=\int \left (-\frac {8 \sqrt [4]{-b+a x^4}}{x^{10}}+\frac {8 a \sqrt [4]{-b+a x^4}}{b x^6}-\frac {a (8 a-b) \sqrt [4]{-b+a x^4}}{b^2 x^2}+\frac {a^2 (8 a-b) x^2 \sqrt [4]{-b+a x^4}}{b^2 \left (b+a x^4\right )}\right ) \, dx\\ &=-\left (8 \int \frac {\sqrt [4]{-b+a x^4}}{x^{10}} \, dx\right )-\frac {(a (8 a-b)) \int \frac {\sqrt [4]{-b+a x^4}}{x^2} \, dx}{b^2}+\frac {\left (a^2 (8 a-b)\right ) \int \frac {x^2 \sqrt [4]{-b+a x^4}}{b+a x^4} \, dx}{b^2}+\frac {(8 a) \int \frac {\sqrt [4]{-b+a x^4}}{x^6} \, dx}{b}\\ &=\frac {a (8 a-b) \sqrt [4]{-b+a x^4}}{b^2 x}-\frac {8 \left (-b+a x^4\right )^{5/4}}{9 b x^9}+\frac {8 a \left (-b+a x^4\right )^{5/4}}{5 b^2 x^5}-\frac {\left (a^2 (8 a-b)\right ) \int \frac {x^2}{\left (-b+a x^4\right )^{3/4}} \, dx}{b^2}-\frac {(32 a) \int \frac {\sqrt [4]{-b+a x^4}}{x^6} \, dx}{9 b}+\frac {\left (a^2 (8 a-b) \sqrt [4]{-b+a x^4}\right ) \int \frac {x^2 \sqrt [4]{1-\frac {a x^4}{b}}}{b+a x^4} \, dx}{b^2 \sqrt [4]{1-\frac {a x^4}{b}}}\\ &=\frac {a (8 a-b) \sqrt [4]{-b+a x^4}}{b^2 x}-\frac {8 \left (-b+a x^4\right )^{5/4}}{9 b x^9}+\frac {8 a \left (-b+a x^4\right )^{5/4}}{9 b^2 x^5}+\frac {a^2 (8 a-b) x^3 \sqrt [4]{-b+a x^4} F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};-\frac {a x^4}{b},\frac {a x^4}{b}\right )}{3 b^3 \sqrt [4]{1-\frac {a x^4}{b}}}-\frac {\left (a^2 (8 a-b)\right ) \operatorname {Subst}\left (\int \frac {x^2}{1-a x^4} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{b^2}\\ &=\frac {a (8 a-b) \sqrt [4]{-b+a x^4}}{b^2 x}-\frac {8 \left (-b+a x^4\right )^{5/4}}{9 b x^9}+\frac {8 a \left (-b+a x^4\right )^{5/4}}{9 b^2 x^5}+\frac {a^2 (8 a-b) x^3 \sqrt [4]{-b+a x^4} F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};-\frac {a x^4}{b},\frac {a x^4}{b}\right )}{3 b^3 \sqrt [4]{1-\frac {a x^4}{b}}}-\frac {\left (a^{3/2} (8 a-b)\right ) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {a} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{2 b^2}+\frac {\left (a^{3/2} (8 a-b)\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {a} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{2 b^2}\\ &=\frac {a (8 a-b) \sqrt [4]{-b+a x^4}}{b^2 x}-\frac {8 \left (-b+a x^4\right )^{5/4}}{9 b x^9}+\frac {8 a \left (-b+a x^4\right )^{5/4}}{9 b^2 x^5}+\frac {a^2 (8 a-b) x^3 \sqrt [4]{-b+a x^4} F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};-\frac {a x^4}{b},\frac {a x^4}{b}\right )}{3 b^3 \sqrt [4]{1-\frac {a x^4}{b}}}+\frac {a^{5/4} (8 a-b) \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{2 b^2}-\frac {a^{5/4} (8 a-b) \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{2 b^2}\\ \end {align*}
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Mathematica [C] time = 0.33, size = 129, normalized size = 0.83 \begin {gather*} \frac {\left (a x^4-b\right ) \left (80 a^2 x^8-a b \left (9 x^4+16\right ) x^4+8 b^2\right )+\frac {6 a^2 x^{12} (b-8 a) \left (1-\frac {a x^4}{b}\right )^{3/4} \, _2F_1\left (\frac {3}{4},\frac {3}{4};\frac {7}{4};\frac {2 a x^4}{a x^4+b}\right )}{\left (\frac {a x^4}{b}+1\right )^{3/4}}}{9 b^2 x^9 \left (a x^4-b\right )^{3/4}} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.49, size = 155, normalized size = 1.00 \begin {gather*} \frac {\left (8 a^{9/4}-a^{5/4} b\right ) \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{2^{3/4} b^2}-\frac {\left (8 a^{9/4}-a^{5/4} b\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{2^{3/4} b^2}+\frac {\sqrt [4]{a x^4-b} \left (80 a^2 x^8-9 a b x^8-16 a b x^4+8 b^2\right )}{9 b^2 x^9} \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^{8} - 8 \, b\right )} {\left (a x^{4} - b\right )}^{\frac {1}{4}}}{{\left (a x^{4} + b\right )} x^{10}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.43, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a \,x^{4}-b \right )^{\frac {1}{4}} \left (a \,x^{8}-8 b \right )}{x^{10} \left (a \,x^{4}+b \right )}\, dx \end {gather*}
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^{8} - 8 \, b\right )} {\left (a x^{4} - b\right )}^{\frac {1}{4}}}{{\left (a x^{4} + b\right )} x^{10}}\,{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\right )}^{1/4}\,\left (8\,b-a\,x^8\right )}{x^{10}\,\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] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt [4]{a x^{4} - b} \left (a x^{8} - 8 b\right )}{x^{10} \left (a x^{4} + b\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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