Optimal. Leaf size=56 \[ \frac{a^2 \sqrt [4]{a+b x^4}}{b^3}+\frac{\left (a+b x^4\right )^{9/4}}{9 b^3}-\frac{2 a \left (a+b x^4\right )^{5/4}}{5 b^3} \]
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Rubi [A] time = 0.0322967, antiderivative size = 56, 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} \[ \frac{a^2 \sqrt [4]{a+b x^4}}{b^3}+\frac{\left (a+b x^4\right )^{9/4}}{9 b^3}-\frac{2 a \left (a+b x^4\right )^{5/4}}{5 b^3} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{x^{11}}{\left (a+b x^4\right )^{3/4}} \, dx &=\frac{1}{4} \operatorname{Subst}\left (\int \frac{x^2}{(a+b x)^{3/4}} \, dx,x,x^4\right )\\ &=\frac{1}{4} \operatorname{Subst}\left (\int \left (\frac{a^2}{b^2 (a+b x)^{3/4}}-\frac{2 a \sqrt [4]{a+b x}}{b^2}+\frac{(a+b x)^{5/4}}{b^2}\right ) \, dx,x,x^4\right )\\ &=\frac{a^2 \sqrt [4]{a+b x^4}}{b^3}-\frac{2 a \left (a+b x^4\right )^{5/4}}{5 b^3}+\frac{\left (a+b x^4\right )^{9/4}}{9 b^3}\\ \end{align*}
Mathematica [A] time = 0.0168346, size = 39, normalized size = 0.7 \[ \frac{\sqrt [4]{a+b x^4} \left (32 a^2-8 a b x^4+5 b^2 x^8\right )}{45 b^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 36, normalized size = 0.6 \begin{align*}{\frac{5\,{b}^{2}{x}^{8}-8\,ab{x}^{4}+32\,{a}^{2}}{45\,{b}^{3}}\sqrt [4]{b{x}^{4}+a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.990486, size = 62, normalized size = 1.11 \begin{align*} \frac{{\left (b x^{4} + a\right )}^{\frac{9}{4}}}{9 \, b^{3}} - \frac{2 \,{\left (b x^{4} + a\right )}^{\frac{5}{4}} a}{5 \, b^{3}} + \frac{{\left (b x^{4} + a\right )}^{\frac{1}{4}} a^{2}}{b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.45409, size = 82, normalized size = 1.46 \begin{align*} \frac{{\left (5 \, b^{2} x^{8} - 8 \, a b x^{4} + 32 \, a^{2}\right )}{\left (b x^{4} + a\right )}^{\frac{1}{4}}}{45 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.84998, size = 68, normalized size = 1.21 \begin{align*} \begin{cases} \frac{32 a^{2} \sqrt [4]{a + b x^{4}}}{45 b^{3}} - \frac{8 a x^{4} \sqrt [4]{a + b x^{4}}}{45 b^{2}} + \frac{x^{8} \sqrt [4]{a + b x^{4}}}{9 b} & \text{for}\: b \neq 0 \\\frac{x^{12}}{12 a^{\frac{3}{4}}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09452, size = 58, normalized size = 1.04 \begin{align*} \frac{5 \,{\left (b x^{4} + a\right )}^{\frac{9}{4}} - 18 \,{\left (b x^{4} + a\right )}^{\frac{5}{4}} a + 45 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}} a^{2}}{45 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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