Optimal. Leaf size=74 \[ \frac{a^3}{b^4 \sqrt [4]{a+b x^4}}+\frac{a^2 \left (a+b x^4\right )^{3/4}}{b^4}-\frac{3 a \left (a+b x^4\right )^{7/4}}{7 b^4}+\frac{\left (a+b x^4\right )^{11/4}}{11 b^4} \]
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Rubi [A] time = 0.0439708, antiderivative size = 74, 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} \[ \frac{a^3}{b^4 \sqrt [4]{a+b x^4}}+\frac{a^2 \left (a+b x^4\right )^{3/4}}{b^4}-\frac{3 a \left (a+b x^4\right )^{7/4}}{7 b^4}+\frac{\left (a+b x^4\right )^{11/4}}{11 b^4} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{x^{15}}{\left (a+b x^4\right )^{5/4}} \, dx &=\frac{1}{4} \operatorname{Subst}\left (\int \frac{x^3}{(a+b x)^{5/4}} \, dx,x,x^4\right )\\ &=\frac{1}{4} \operatorname{Subst}\left (\int \left (-\frac{a^3}{b^3 (a+b x)^{5/4}}+\frac{3 a^2}{b^3 \sqrt [4]{a+b x}}-\frac{3 a (a+b x)^{3/4}}{b^3}+\frac{(a+b x)^{7/4}}{b^3}\right ) \, dx,x,x^4\right )\\ &=\frac{a^3}{b^4 \sqrt [4]{a+b x^4}}+\frac{a^2 \left (a+b x^4\right )^{3/4}}{b^4}-\frac{3 a \left (a+b x^4\right )^{7/4}}{7 b^4}+\frac{\left (a+b x^4\right )^{11/4}}{11 b^4}\\ \end{align*}
Mathematica [A] time = 0.0213592, size = 50, normalized size = 0.68 \[ \frac{32 a^2 b x^4+128 a^3-12 a b^2 x^8+7 b^3 x^{12}}{77 b^4 \sqrt [4]{a+b x^4}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 47, normalized size = 0.6 \begin{align*}{\frac{7\,{b}^{3}{x}^{12}-12\,a{b}^{2}{x}^{8}+32\,{a}^{2}b{x}^{4}+128\,{a}^{3}}{77\,{b}^{4}}{\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 = 1.00307, size = 84, normalized size = 1.14 \begin{align*} \frac{{\left (b x^{4} + a\right )}^{\frac{11}{4}}}{11 \, b^{4}} - \frac{3 \,{\left (b x^{4} + a\right )}^{\frac{7}{4}} a}{7 \, b^{4}} + \frac{{\left (b x^{4} + a\right )}^{\frac{3}{4}} a^{2}}{b^{4}} + \frac{a^{3}}{{\left (b x^{4} + a\right )}^{\frac{1}{4}} b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.42926, size = 128, normalized size = 1.73 \begin{align*} \frac{{\left (7 \, b^{3} x^{12} - 12 \, a b^{2} x^{8} + 32 \, a^{2} b x^{4} + 128 \, a^{3}\right )}{\left (b x^{4} + a\right )}^{\frac{3}{4}}}{77 \,{\left (b^{5} x^{4} + a b^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 8.62131, size = 92, normalized size = 1.24 \begin{align*} \begin{cases} \frac{128 a^{3}}{77 b^{4} \sqrt [4]{a + b x^{4}}} + \frac{32 a^{2} x^{4}}{77 b^{3} \sqrt [4]{a + b x^{4}}} - \frac{12 a x^{8}}{77 b^{2} \sqrt [4]{a + b x^{4}}} + \frac{x^{12}}{11 b \sqrt [4]{a + b x^{4}}} & \text{for}\: b \neq 0 \\\frac{x^{16}}{16 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 = 1.09573, size = 77, normalized size = 1.04 \begin{align*} \frac{7 \,{\left (b x^{4} + a\right )}^{\frac{11}{4}} - 33 \,{\left (b x^{4} + a\right )}^{\frac{7}{4}} a + 77 \,{\left (b x^{4} + a\right )}^{\frac{3}{4}} a^{2} + \frac{77 \, a^{3}}{{\left (b x^{4} + a\right )}^{\frac{1}{4}}}}{77 \, b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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