Optimal. Leaf size=71 \[ -\frac{4 b^2 \sqrt{a-b x^4}}{15 a^3 x^2}-\frac{2 b \sqrt{a-b x^4}}{15 a^2 x^6}-\frac{\sqrt{a-b x^4}}{10 a x^{10}} \]
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Rubi [A] time = 0.0187645, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {271, 264} \[ -\frac{4 b^2 \sqrt{a-b x^4}}{15 a^3 x^2}-\frac{2 b \sqrt{a-b x^4}}{15 a^2 x^6}-\frac{\sqrt{a-b x^4}}{10 a x^{10}} \]
Antiderivative was successfully verified.
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Rule 271
Rule 264
Rubi steps
\begin{align*} \int \frac{1}{x^{11} \sqrt{a-b x^4}} \, dx &=-\frac{\sqrt{a-b x^4}}{10 a x^{10}}+\frac{(4 b) \int \frac{1}{x^7 \sqrt{a-b x^4}} \, dx}{5 a}\\ &=-\frac{\sqrt{a-b x^4}}{10 a x^{10}}-\frac{2 b \sqrt{a-b x^4}}{15 a^2 x^6}+\frac{\left (8 b^2\right ) \int \frac{1}{x^3 \sqrt{a-b x^4}} \, dx}{15 a^2}\\ &=-\frac{\sqrt{a-b x^4}}{10 a x^{10}}-\frac{2 b \sqrt{a-b x^4}}{15 a^2 x^6}-\frac{4 b^2 \sqrt{a-b x^4}}{15 a^3 x^2}\\ \end{align*}
Mathematica [A] time = 0.0085303, size = 43, normalized size = 0.61 \[ -\frac{\sqrt{a-b x^4} \left (3 a^2+4 a b x^4+8 b^2 x^8\right )}{30 a^3 x^{10}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 40, normalized size = 0.6 \begin{align*} -{\frac{8\,{b}^{2}{x}^{8}+4\,ab{x}^{4}+3\,{a}^{2}}{30\,{x}^{10}{a}^{3}}\sqrt{-b{x}^{4}+a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.967473, size = 74, normalized size = 1.04 \begin{align*} -\frac{\frac{15 \, \sqrt{-b x^{4} + a} b^{2}}{x^{2}} + \frac{10 \,{\left (-b x^{4} + a\right )}^{\frac{3}{2}} b}{x^{6}} + \frac{3 \,{\left (-b x^{4} + a\right )}^{\frac{5}{2}}}{x^{10}}}{30 \, a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.65597, size = 90, normalized size = 1.27 \begin{align*} -\frac{{\left (8 \, b^{2} x^{8} + 4 \, a b x^{4} + 3 \, a^{2}\right )} \sqrt{-b x^{4} + a}}{30 \, a^{3} x^{10}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 2.63956, size = 612, normalized size = 8.62 \begin{align*} \begin{cases} - \frac{3 a^{4} b^{\frac{9}{2}} \sqrt{\frac{a}{b x^{4}} - 1}}{30 a^{5} b^{4} x^{8} - 60 a^{4} b^{5} x^{12} + 30 a^{3} b^{6} x^{16}} + \frac{2 a^{3} b^{\frac{11}{2}} x^{4} \sqrt{\frac{a}{b x^{4}} - 1}}{30 a^{5} b^{4} x^{8} - 60 a^{4} b^{5} x^{12} + 30 a^{3} b^{6} x^{16}} - \frac{3 a^{2} b^{\frac{13}{2}} x^{8} \sqrt{\frac{a}{b x^{4}} - 1}}{30 a^{5} b^{4} x^{8} - 60 a^{4} b^{5} x^{12} + 30 a^{3} b^{6} x^{16}} + \frac{12 a b^{\frac{15}{2}} x^{12} \sqrt{\frac{a}{b x^{4}} - 1}}{30 a^{5} b^{4} x^{8} - 60 a^{4} b^{5} x^{12} + 30 a^{3} b^{6} x^{16}} - \frac{8 b^{\frac{17}{2}} x^{16} \sqrt{\frac{a}{b x^{4}} - 1}}{30 a^{5} b^{4} x^{8} - 60 a^{4} b^{5} x^{12} + 30 a^{3} b^{6} x^{16}} & \text{for}\: \frac{\left |{a}\right |}{\left |{b}\right | \left |{x^{4}}\right |} > 1 \\- \frac{3 i a^{4} b^{\frac{9}{2}} \sqrt{- \frac{a}{b x^{4}} + 1}}{30 a^{5} b^{4} x^{8} - 60 a^{4} b^{5} x^{12} + 30 a^{3} b^{6} x^{16}} + \frac{2 i a^{3} b^{\frac{11}{2}} x^{4} \sqrt{- \frac{a}{b x^{4}} + 1}}{30 a^{5} b^{4} x^{8} - 60 a^{4} b^{5} x^{12} + 30 a^{3} b^{6} x^{16}} - \frac{3 i a^{2} b^{\frac{13}{2}} x^{8} \sqrt{- \frac{a}{b x^{4}} + 1}}{30 a^{5} b^{4} x^{8} - 60 a^{4} b^{5} x^{12} + 30 a^{3} b^{6} x^{16}} + \frac{12 i a b^{\frac{15}{2}} x^{12} \sqrt{- \frac{a}{b x^{4}} + 1}}{30 a^{5} b^{4} x^{8} - 60 a^{4} b^{5} x^{12} + 30 a^{3} b^{6} x^{16}} - \frac{8 i b^{\frac{17}{2}} x^{16} \sqrt{- \frac{a}{b x^{4}} + 1}}{30 a^{5} b^{4} x^{8} - 60 a^{4} b^{5} x^{12} + 30 a^{3} b^{6} x^{16}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11868, size = 80, normalized size = 1.13 \begin{align*} -\frac{3 \,{\left (b - \frac{a}{x^{4}}\right )}^{2} \sqrt{-b + \frac{a}{x^{4}}} + 15 \, b^{2} \sqrt{-b + \frac{a}{x^{4}}} + 10 \, b{\left (-b + \frac{a}{x^{4}}\right )}^{\frac{3}{2}}}{30 \, a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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