Optimal. Leaf size=57 \[ -\frac{a^2}{b^3 \sqrt [4]{a+b x^4}}-\frac{2 a \left (a+b x^4\right )^{3/4}}{3 b^3}+\frac{\left (a+b x^4\right )^{7/4}}{7 b^3} \]
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Rubi [A] time = 0.0334128, antiderivative size = 57, 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} \[ -\frac{a^2}{b^3 \sqrt [4]{a+b x^4}}-\frac{2 a \left (a+b x^4\right )^{3/4}}{3 b^3}+\frac{\left (a+b x^4\right )^{7/4}}{7 b^3} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{x^{11}}{\left (a+b x^4\right )^{5/4}} \, dx &=\frac{1}{4} \operatorname{Subst}\left (\int \frac{x^2}{(a+b x)^{5/4}} \, dx,x,x^4\right )\\ &=\frac{1}{4} \operatorname{Subst}\left (\int \left (\frac{a^2}{b^2 (a+b x)^{5/4}}-\frac{2 a}{b^2 \sqrt [4]{a+b x}}+\frac{(a+b x)^{3/4}}{b^2}\right ) \, dx,x,x^4\right )\\ &=-\frac{a^2}{b^3 \sqrt [4]{a+b x^4}}-\frac{2 a \left (a+b x^4\right )^{3/4}}{3 b^3}+\frac{\left (a+b x^4\right )^{7/4}}{7 b^3}\\ \end{align*}
Mathematica [A] time = 0.0173317, size = 39, normalized size = 0.68 \[ \frac{-32 a^2-8 a b x^4+3 b^2 x^8}{21 b^3 \sqrt [4]{a+b x^4}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 36, normalized size = 0.6 \begin{align*} -{\frac{-3\,{b}^{2}{x}^{8}+8\,ab{x}^{4}+32\,{a}^{2}}{21\,{b}^{3}}{\frac{1}{\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 = 0.980901, size = 63, normalized size = 1.11 \begin{align*} \frac{{\left (b x^{4} + a\right )}^{\frac{7}{4}}}{7 \, b^{3}} - \frac{2 \,{\left (b x^{4} + a\right )}^{\frac{3}{4}} a}{3 \, b^{3}} - \frac{a^{2}}{{\left (b x^{4} + a\right )}^{\frac{1}{4}} b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.48, size = 101, normalized size = 1.77 \begin{align*} \frac{{\left (3 \, b^{2} x^{8} - 8 \, a b x^{4} - 32 \, a^{2}\right )}{\left (b x^{4} + a\right )}^{\frac{3}{4}}}{21 \,{\left (b^{4} x^{4} + a b^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.00822, size = 68, normalized size = 1.19 \begin{align*} \begin{cases} - \frac{32 a^{2}}{21 b^{3} \sqrt [4]{a + b x^{4}}} - \frac{8 a x^{4}}{21 b^{2} \sqrt [4]{a + b x^{4}}} + \frac{x^{8}}{7 b \sqrt [4]{a + b x^{4}}} & \text{for}\: b \neq 0 \\\frac{x^{12}}{12 a^{\frac{5}{4}}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.0869, size = 58, normalized size = 1.02 \begin{align*} \frac{3 \,{\left (b x^{4} + a\right )}^{\frac{7}{4}} - 14 \,{\left (b x^{4} + a\right )}^{\frac{3}{4}} a - \frac{21 \, a^{2}}{{\left (b x^{4} + a\right )}^{\frac{1}{4}}}}{21 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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