Optimal. Leaf size=59 \[ \frac{a^2 \left (a+b x^4\right )^{3/4}}{3 b^3}+\frac{\left (a+b x^4\right )^{11/4}}{11 b^3}-\frac{2 a \left (a+b x^4\right )^{7/4}}{7 b^3} \]
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Rubi [A] time = 0.0327176, 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} \[ \frac{a^2 \left (a+b x^4\right )^{3/4}}{3 b^3}+\frac{\left (a+b x^4\right )^{11/4}}{11 b^3}-\frac{2 a \left (a+b x^4\right )^{7/4}}{7 b^3} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{x^{11}}{\sqrt [4]{a+b x^4}} \, dx &=\frac{1}{4} \operatorname{Subst}\left (\int \frac{x^2}{\sqrt [4]{a+b x}} \, dx,x,x^4\right )\\ &=\frac{1}{4} \operatorname{Subst}\left (\int \left (\frac{a^2}{b^2 \sqrt [4]{a+b x}}-\frac{2 a (a+b x)^{3/4}}{b^2}+\frac{(a+b x)^{7/4}}{b^2}\right ) \, dx,x,x^4\right )\\ &=\frac{a^2 \left (a+b x^4\right )^{3/4}}{3 b^3}-\frac{2 a \left (a+b x^4\right )^{7/4}}{7 b^3}+\frac{\left (a+b x^4\right )^{11/4}}{11 b^3}\\ \end{align*}
Mathematica [A] time = 0.0186086, size = 39, normalized size = 0.66 \[ \frac{\left (a+b x^4\right )^{3/4} \left (32 a^2-24 a b x^4+21 b^2 x^8\right )}{231 b^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 36, normalized size = 0.6 \begin{align*}{\frac{21\,{b}^{2}{x}^{8}-24\,ab{x}^{4}+32\,{a}^{2}}{231\,{b}^{3}} \left ( b{x}^{4}+a \right ) ^{{\frac{3}{4}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01119, size = 63, normalized size = 1.07 \begin{align*} \frac{{\left (b x^{4} + a\right )}^{\frac{11}{4}}}{11 \, b^{3}} - \frac{2 \,{\left (b x^{4} + a\right )}^{\frac{7}{4}} a}{7 \, b^{3}} + \frac{{\left (b x^{4} + a\right )}^{\frac{3}{4}} a^{2}}{3 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.49564, size = 86, normalized size = 1.46 \begin{align*} \frac{{\left (21 \, b^{2} x^{8} - 24 \, a b x^{4} + 32 \, a^{2}\right )}{\left (b x^{4} + a\right )}^{\frac{3}{4}}}{231 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.48652, size = 68, normalized size = 1.15 \begin{align*} \begin{cases} \frac{32 a^{2} \left (a + b x^{4}\right )^{\frac{3}{4}}}{231 b^{3}} - \frac{8 a x^{4} \left (a + b x^{4}\right )^{\frac{3}{4}}}{77 b^{2}} + \frac{x^{8} \left (a + b x^{4}\right )^{\frac{3}{4}}}{11 b} & \text{for}\: b \neq 0 \\\frac{x^{12}}{12 \sqrt [4]{a}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16014, size = 58, normalized size = 0.98 \begin{align*} \frac{21 \,{\left (b x^{4} + a\right )}^{\frac{11}{4}} - 66 \,{\left (b x^{4} + a\right )}^{\frac{7}{4}} a + 77 \,{\left (b x^{4} + a\right )}^{\frac{3}{4}} a^{2}}{231 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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