Optimal. Leaf size=101 \[ \frac{6 a^2 \left (a+b x^4\right )^{17/4}}{17 b^5}-\frac{4 a^3 \left (a+b x^4\right )^{13/4}}{13 b^5}+\frac{a^4 \left (a+b x^4\right )^{9/4}}{9 b^5}+\frac{\left (a+b x^4\right )^{25/4}}{25 b^5}-\frac{4 a \left (a+b x^4\right )^{21/4}}{21 b^5} \]
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Rubi [A] time = 0.0539924, 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} \[ \frac{6 a^2 \left (a+b x^4\right )^{17/4}}{17 b^5}-\frac{4 a^3 \left (a+b x^4\right )^{13/4}}{13 b^5}+\frac{a^4 \left (a+b x^4\right )^{9/4}}{9 b^5}+\frac{\left (a+b x^4\right )^{25/4}}{25 b^5}-\frac{4 a \left (a+b x^4\right )^{21/4}}{21 b^5} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int x^{19} \left (a+b x^4\right )^{5/4} \, dx &=\frac{1}{4} \operatorname{Subst}\left (\int x^4 (a+b x)^{5/4} \, dx,x,x^4\right )\\ &=\frac{1}{4} \operatorname{Subst}\left (\int \left (\frac{a^4 (a+b x)^{5/4}}{b^4}-\frac{4 a^3 (a+b x)^{9/4}}{b^4}+\frac{6 a^2 (a+b x)^{13/4}}{b^4}-\frac{4 a (a+b x)^{17/4}}{b^4}+\frac{(a+b x)^{21/4}}{b^4}\right ) \, dx,x,x^4\right )\\ &=\frac{a^4 \left (a+b x^4\right )^{9/4}}{9 b^5}-\frac{4 a^3 \left (a+b x^4\right )^{13/4}}{13 b^5}+\frac{6 a^2 \left (a+b x^4\right )^{17/4}}{17 b^5}-\frac{4 a \left (a+b x^4\right )^{21/4}}{21 b^5}+\frac{\left (a+b x^4\right )^{25/4}}{25 b^5}\\ \end{align*}
Mathematica [A] time = 0.0319342, size = 61, normalized size = 0.6 \[ \frac{\left (a+b x^4\right )^{9/4} \left (7488 a^2 b^2 x^8-4608 a^3 b x^4+2048 a^4-10608 a b^3 x^{12}+13923 b^4 x^{16}\right )}{348075 b^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 58, normalized size = 0.6 \begin{align*}{\frac{13923\,{x}^{16}{b}^{4}-10608\,a{x}^{12}{b}^{3}+7488\,{a}^{2}{x}^{8}{b}^{2}-4608\,{a}^{3}{x}^{4}b+2048\,{a}^{4}}{348075\,{b}^{5}} \left ( b{x}^{4}+a \right ) ^{{\frac{9}{4}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.963809, size = 109, normalized size = 1.08 \begin{align*} \frac{{\left (b x^{4} + a\right )}^{\frac{25}{4}}}{25 \, b^{5}} - \frac{4 \,{\left (b x^{4} + a\right )}^{\frac{21}{4}} a}{21 \, b^{5}} + \frac{6 \,{\left (b x^{4} + a\right )}^{\frac{17}{4}} a^{2}}{17 \, b^{5}} - \frac{4 \,{\left (b x^{4} + a\right )}^{\frac{13}{4}} a^{3}}{13 \, b^{5}} + \frac{{\left (b x^{4} + a\right )}^{\frac{9}{4}} a^{4}}{9 \, b^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.71407, size = 204, normalized size = 2.02 \begin{align*} \frac{{\left (13923 \, b^{6} x^{24} + 17238 \, a b^{5} x^{20} + 195 \, a^{2} b^{4} x^{16} - 240 \, a^{3} b^{3} x^{12} + 320 \, a^{4} b^{2} x^{8} - 512 \, a^{5} b x^{4} + 2048 \, a^{6}\right )}{\left (b x^{4} + a\right )}^{\frac{1}{4}}}{348075 \, b^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 82.9827, size = 156, normalized size = 1.54 \begin{align*} \begin{cases} \frac{2048 a^{6} \sqrt [4]{a + b x^{4}}}{348075 b^{5}} - \frac{512 a^{5} x^{4} \sqrt [4]{a + b x^{4}}}{348075 b^{4}} + \frac{64 a^{4} x^{8} \sqrt [4]{a + b x^{4}}}{69615 b^{3}} - \frac{16 a^{3} x^{12} \sqrt [4]{a + b x^{4}}}{23205 b^{2}} + \frac{a^{2} x^{16} \sqrt [4]{a + b x^{4}}}{1785 b} + \frac{26 a x^{20} \sqrt [4]{a + b x^{4}}}{525} + \frac{b x^{24} \sqrt [4]{a + b x^{4}}}{25} & \text{for}\: b \neq 0 \\\frac{a^{\frac{5}{4}} x^{20}}{20} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.13522, size = 219, normalized size = 2.17 \begin{align*} \frac{\frac{5 \,{\left (3315 \,{\left (b x^{4} + a\right )}^{\frac{21}{4}} - 16380 \,{\left (b x^{4} + a\right )}^{\frac{17}{4}} a + 32130 \,{\left (b x^{4} + a\right )}^{\frac{13}{4}} a^{2} - 30940 \,{\left (b x^{4} + a\right )}^{\frac{9}{4}} a^{3} + 13923 \,{\left (b x^{4} + a\right )}^{\frac{5}{4}} a^{4}\right )} a}{b^{4}} + \frac{13923 \,{\left (b x^{4} + a\right )}^{\frac{25}{4}} - 82875 \,{\left (b x^{4} + a\right )}^{\frac{21}{4}} a + 204750 \,{\left (b x^{4} + a\right )}^{\frac{17}{4}} a^{2} - 267750 \,{\left (b x^{4} + a\right )}^{\frac{13}{4}} a^{3} + 193375 \,{\left (b x^{4} + a\right )}^{\frac{9}{4}} a^{4} - 69615 \,{\left (b x^{4} + a\right )}^{\frac{5}{4}} a^{5}}{b^{4}}}{348075 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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