Optimal. Leaf size=59 \[ \frac{a^2 \left (a+b x^4\right )^{5/4}}{5 b^3}+\frac{\left (a+b x^4\right )^{13/4}}{13 b^3}-\frac{2 a \left (a+b x^4\right )^{9/4}}{9 b^3} \]
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Rubi [A] time = 0.0339833, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, 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 = 0.0169775, size = 39, normalized size = 0.66 \[ \frac{\left (a+b x^4\right )^{5/4} \left (32 a^2-40 a b x^4+45 b^2 x^8\right )}{585 b^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 36, 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 = 0.963285, size = 63, normalized size = 1.07 \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 = 1.44149, size = 108, normalized size = 1.83 \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 = 6.41015, size = 87, normalized size = 1.47 \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 = 1.15427, size = 58, normalized size = 0.98 \begin{align*} \frac{45 \,{\left (b x^{4} + a\right )}^{\frac{13}{4}} - 130 \,{\left (b x^{4} + a\right )}^{\frac{9}{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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