Optimal. Leaf size=62 \[ -\frac{a^2 \left (a-b x^4\right )^{5/4}}{5 b^3}-\frac{\left (a-b x^4\right )^{13/4}}{13 b^3}+\frac{2 a \left (a-b x^4\right )^{9/4}}{9 b^3} \]
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Rubi [A] time = 0.0342731, 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} \[ -\frac{a^2 \left (a-b x^4\right )^{5/4}}{5 b^3}-\frac{\left (a-b x^4\right )^{13/4}}{13 b^3}+\frac{2 a \left (a-b x^4\right )^{9/4}}{9 b^3} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int x^{11} \sqrt [4]{a-b x^4} \, dx &=\frac{1}{4} \operatorname{Subst}\left (\int x^2 \sqrt [4]{a-b x} \, dx,x,x^4\right )\\ &=\frac{1}{4} \operatorname{Subst}\left (\int \left (\frac{a^2 \sqrt [4]{a-b x}}{b^2}-\frac{2 a (a-b x)^{5/4}}{b^2}+\frac{(a-b x)^{9/4}}{b^2}\right ) \, dx,x,x^4\right )\\ &=-\frac{a^2 \left (a-b x^4\right )^{5/4}}{5 b^3}+\frac{2 a \left (a-b x^4\right )^{9/4}}{9 b^3}-\frac{\left (a-b x^4\right )^{13/4}}{13 b^3}\\ \end{align*}
Mathematica [A] time = 0.0187155, size = 40, normalized size = 0.65 \[ -\frac{\left (a-b x^4\right )^{5/4} \left (32 a^2+40 a b x^4+45 b^2 x^8\right )}{585 b^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 37, normalized size = 0.6 \begin{align*} -{\frac{45\,{b}^{2}{x}^{8}+40\,ab{x}^{4}+32\,{a}^{2}}{585\,{b}^{3}} \left ( -b{x}^{4}+a \right ) ^{{\frac{5}{4}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.963633, size = 68, normalized size = 1.1 \begin{align*} -\frac{{\left (-b x^{4} + a\right )}^{\frac{13}{4}}}{13 \, b^{3}} + \frac{2 \,{\left (-b x^{4} + a\right )}^{\frac{9}{4}} a}{9 \, b^{3}} - \frac{{\left (-b x^{4} + a\right )}^{\frac{5}{4}} a^{2}}{5 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.40807, size = 109, normalized size = 1.76 \begin{align*} \frac{{\left (45 \, b^{3} x^{12} - 5 \, a b^{2} x^{8} - 8 \, a^{2} b x^{4} - 32 \, a^{3}\right )}{\left (-b x^{4} + a\right )}^{\frac{1}{4}}}{585 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.82563, size = 87, normalized size = 1.4 \begin{align*} \begin{cases} - \frac{32 a^{3} \sqrt [4]{a - b x^{4}}}{585 b^{3}} - \frac{8 a^{2} x^{4} \sqrt [4]{a - b x^{4}}}{585 b^{2}} - \frac{a x^{8} \sqrt [4]{a - b x^{4}}}{117 b} + \frac{x^{12} \sqrt [4]{a - b x^{4}}}{13} & \text{for}\: b \neq 0 \\\frac{\sqrt [4]{a} x^{12}}{12} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17633, size = 92, normalized size = 1.48 \begin{align*} \frac{45 \,{\left (b x^{4} - a\right )}^{3}{\left (-b x^{4} + a\right )}^{\frac{1}{4}} + 130 \,{\left (b x^{4} - a\right )}^{2}{\left (-b x^{4} + a\right )}^{\frac{1}{4}} a - 117 \,{\left (-b x^{4} + a\right )}^{\frac{5}{4}} a^{2}}{585 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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