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 = 0.03, antiderivative size = 62, normalized size of antiderivative = 1.00, 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 43
Rule 266
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*}
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Mathematica [A] time = 0.02, 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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fricas [A] time = 0.85, size = 47, normalized size = 0.76 \[ \frac {{\left (45 \, b^{3} x^{12} - 5 \, a b^{2} x^{8} - 8 \, a^{2} b x^{4} - 32 \, a^{3}\right )} {\left (-b x^{4} + a\right )}^{\frac {1}{4}}}{585 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 68, normalized size = 1.10 \[ \frac {45 \, {\left (b x^{4} - a\right )}^{3} {\left (-b x^{4} + a\right )}^{\frac {1}{4}} + 130 \, {\left (b x^{4} - a\right )}^{2} {\left (-b x^{4} + a\right )}^{\frac {1}{4}} a - 117 \, {\left (-b x^{4} + a\right )}^{\frac {5}{4}} a^{2}}{585 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 37, normalized size = 0.60 \[ -\frac {\left (-b \,x^{4}+a \right )^{\frac {5}{4}} \left (45 b^{2} x^{8}+40 a b \,x^{4}+32 a^{2}\right )}{585 b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.35, size = 50, normalized size = 0.81 \[ -\frac {{\left (-b x^{4} + a\right )}^{\frac {13}{4}}}{13 \, b^{3}} + \frac {2 \, {\left (-b x^{4} + a\right )}^{\frac {9}{4}} a}{9 \, b^{3}} - \frac {{\left (-b x^{4} + a\right )}^{\frac {5}{4}} a^{2}}{5 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.13, size = 46, normalized size = 0.74 \[ -{\left (a-b\,x^4\right )}^{1/4}\,\left (\frac {32\,a^3}{585\,b^3}-\frac {x^{12}}{13}+\frac {a\,x^8}{117\,b}+\frac {8\,a^2\,x^4}{585\,b^2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 6.25, size = 87, normalized size = 1.40 \[ \begin {cases} - \frac {32 a^{3} \sqrt [4]{a - b x^{4}}}{585 b^{3}} - \frac {8 a^{2} x^{4} \sqrt [4]{a - b x^{4}}}{585 b^{2}} - \frac {a x^{8} \sqrt [4]{a - b x^{4}}}{117 b} + \frac {x^{12} \sqrt [4]{a - b x^{4}}}{13} & \text {for}\: b \neq 0 \\\frac {\sqrt [4]{a} x^{12}}{12} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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