Optimal. Leaf size=101 \[ \frac{6 a^2 \left (a+b x^4\right )^{11/4}}{11 b^5}-\frac{4 a^3 \left (a+b x^4\right )^{7/4}}{7 b^5}+\frac{a^4 \left (a+b x^4\right )^{3/4}}{3 b^5}+\frac{\left (a+b x^4\right )^{19/4}}{19 b^5}-\frac{4 a \left (a+b x^4\right )^{15/4}}{15 b^5} \]
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Rubi [A] time = 0.0534013, antiderivative size = 101, 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} \[ \frac{6 a^2 \left (a+b x^4\right )^{11/4}}{11 b^5}-\frac{4 a^3 \left (a+b x^4\right )^{7/4}}{7 b^5}+\frac{a^4 \left (a+b x^4\right )^{3/4}}{3 b^5}+\frac{\left (a+b x^4\right )^{19/4}}{19 b^5}-\frac{4 a \left (a+b x^4\right )^{15/4}}{15 b^5} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{x^{19}}{\sqrt [4]{a+b x^4}} \, dx &=\frac{1}{4} \operatorname{Subst}\left (\int \frac{x^4}{\sqrt [4]{a+b x}} \, dx,x,x^4\right )\\ &=\frac{1}{4} \operatorname{Subst}\left (\int \left (\frac{a^4}{b^4 \sqrt [4]{a+b x}}-\frac{4 a^3 (a+b x)^{3/4}}{b^4}+\frac{6 a^2 (a+b x)^{7/4}}{b^4}-\frac{4 a (a+b x)^{11/4}}{b^4}+\frac{(a+b x)^{15/4}}{b^4}\right ) \, dx,x,x^4\right )\\ &=\frac{a^4 \left (a+b x^4\right )^{3/4}}{3 b^5}-\frac{4 a^3 \left (a+b x^4\right )^{7/4}}{7 b^5}+\frac{6 a^2 \left (a+b x^4\right )^{11/4}}{11 b^5}-\frac{4 a \left (a+b x^4\right )^{15/4}}{15 b^5}+\frac{\left (a+b x^4\right )^{19/4}}{19 b^5}\\ \end{align*}
Mathematica [A] time = 0.0291817, size = 61, normalized size = 0.6 \[ \frac{\left (a+b x^4\right )^{3/4} \left (1344 a^2 b^2 x^8-1536 a^3 b x^4+2048 a^4-1232 a b^3 x^{12}+1155 b^4 x^{16}\right )}{21945 b^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 58, normalized size = 0.6 \begin{align*}{\frac{1155\,{x}^{16}{b}^{4}-1232\,a{x}^{12}{b}^{3}+1344\,{a}^{2}{x}^{8}{b}^{2}-1536\,{a}^{3}{x}^{4}b+2048\,{a}^{4}}{21945\,{b}^{5}} \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 = 1.02766, size = 109, normalized size = 1.08 \begin{align*} \frac{{\left (b x^{4} + a\right )}^{\frac{19}{4}}}{19 \, b^{5}} - \frac{4 \,{\left (b x^{4} + a\right )}^{\frac{15}{4}} a}{15 \, b^{5}} + \frac{6 \,{\left (b x^{4} + a\right )}^{\frac{11}{4}} a^{2}}{11 \, b^{5}} - \frac{4 \,{\left (b x^{4} + a\right )}^{\frac{7}{4}} a^{3}}{7 \, b^{5}} + \frac{{\left (b x^{4} + a\right )}^{\frac{3}{4}} a^{4}}{3 \, b^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.42504, size = 151, normalized size = 1.5 \begin{align*} \frac{{\left (1155 \, b^{4} x^{16} - 1232 \, a b^{3} x^{12} + 1344 \, a^{2} b^{2} x^{8} - 1536 \, a^{3} b x^{4} + 2048 \, a^{4}\right )}{\left (b x^{4} + a\right )}^{\frac{3}{4}}}{21945 \, b^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 21.0584, size = 116, normalized size = 1.15 \begin{align*} \begin{cases} \frac{2048 a^{4} \left (a + b x^{4}\right )^{\frac{3}{4}}}{21945 b^{5}} - \frac{512 a^{3} x^{4} \left (a + b x^{4}\right )^{\frac{3}{4}}}{7315 b^{4}} + \frac{64 a^{2} x^{8} \left (a + b x^{4}\right )^{\frac{3}{4}}}{1045 b^{3}} - \frac{16 a x^{12} \left (a + b x^{4}\right )^{\frac{3}{4}}}{285 b^{2}} + \frac{x^{16} \left (a + b x^{4}\right )^{\frac{3}{4}}}{19 b} & \text{for}\: b \neq 0 \\\frac{x^{20}}{20 \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 = 1.59247, size = 96, normalized size = 0.95 \begin{align*} \frac{1155 \,{\left (b x^{4} + a\right )}^{\frac{19}{4}} - 5852 \,{\left (b x^{4} + a\right )}^{\frac{15}{4}} a + 11970 \,{\left (b x^{4} + a\right )}^{\frac{11}{4}} a^{2} - 12540 \,{\left (b x^{4} + a\right )}^{\frac{7}{4}} a^{3} + 7315 \,{\left (b x^{4} + a\right )}^{\frac{3}{4}} a^{4}}{21945 \, b^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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