Optimal. Leaf size=62 \[ -\frac{a^2 \sqrt{a-b x^4}}{2 b^3}-\frac{\left (a-b x^4\right )^{5/2}}{10 b^3}+\frac{a \left (a-b x^4\right )^{3/2}}{3 b^3} \]
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Rubi [A] time = 0.0351539, antiderivative size = 62, 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 = {266, 43} \[ -\frac{a^2 \sqrt{a-b x^4}}{2 b^3}-\frac{\left (a-b x^4\right )^{5/2}}{10 b^3}+\frac{a \left (a-b x^4\right )^{3/2}}{3 b^3} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{x^{11}}{\sqrt{a-b x^4}} \, dx &=\frac{1}{4} \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{a-b x}} \, dx,x,x^4\right )\\ &=\frac{1}{4} \operatorname{Subst}\left (\int \left (\frac{a^2}{b^2 \sqrt{a-b x}}-\frac{2 a \sqrt{a-b x}}{b^2}+\frac{(a-b x)^{3/2}}{b^2}\right ) \, dx,x,x^4\right )\\ &=-\frac{a^2 \sqrt{a-b x^4}}{2 b^3}+\frac{a \left (a-b x^4\right )^{3/2}}{3 b^3}-\frac{\left (a-b x^4\right )^{5/2}}{10 b^3}\\ \end{align*}
Mathematica [A] time = 0.0198333, size = 40, normalized size = 0.65 \[ -\frac{\sqrt{a-b x^4} \left (8 a^2+4 a b x^4+3 b^2 x^8\right )}{30 b^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 37, normalized size = 0.6 \begin{align*} -{\frac{3\,{b}^{2}{x}^{8}+4\,ab{x}^{4}+8\,{a}^{2}}{30\,{b}^{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 = 0.950472, size = 68, normalized size = 1.1 \begin{align*} -\frac{{\left (-b x^{4} + a\right )}^{\frac{5}{2}}}{10 \, b^{3}} + \frac{{\left (-b x^{4} + a\right )}^{\frac{3}{2}} a}{3 \, b^{3}} - \frac{\sqrt{-b x^{4} + a} a^{2}}{2 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.4096, size = 81, normalized size = 1.31 \begin{align*} -\frac{{\left (3 \, b^{2} x^{8} + 4 \, a b x^{4} + 8 \, a^{2}\right )} \sqrt{-b x^{4} + a}}{30 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.69342, size = 70, normalized size = 1.13 \begin{align*} \begin{cases} - \frac{4 a^{2} \sqrt{a - b x^{4}}}{15 b^{3}} - \frac{2 a x^{4} \sqrt{a - b x^{4}}}{15 b^{2}} - \frac{x^{8} \sqrt{a - b x^{4}}}{10 b} & \text{for}\: b \neq 0 \\\frac{x^{12}}{12 \sqrt{a}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.21427, size = 77, normalized size = 1.24 \begin{align*} -\frac{3 \,{\left (b x^{4} - a\right )}^{2} \sqrt{-b x^{4} + a} - 10 \,{\left (-b x^{4} + a\right )}^{\frac{3}{2}} a + 15 \, \sqrt{-b x^{4} + a} a^{2}}{30 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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