Optimal. Leaf size=98 \[ \frac{2 a^2 \left (a+b x^4\right )^{9/4}}{3 b^5}-\frac{4 a^3 \left (a+b x^4\right )^{5/4}}{5 b^5}+\frac{a^4 \sqrt [4]{a+b x^4}}{b^5}+\frac{\left (a+b x^4\right )^{17/4}}{17 b^5}-\frac{4 a \left (a+b x^4\right )^{13/4}}{13 b^5} \]
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Rubi [A] time = 0.0543862, antiderivative size = 98, 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} \[ \frac{2 a^2 \left (a+b x^4\right )^{9/4}}{3 b^5}-\frac{4 a^3 \left (a+b x^4\right )^{5/4}}{5 b^5}+\frac{a^4 \sqrt [4]{a+b x^4}}{b^5}+\frac{\left (a+b x^4\right )^{17/4}}{17 b^5}-\frac{4 a \left (a+b x^4\right )^{13/4}}{13 b^5} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{x^{19}}{\left (a+b x^4\right )^{3/4}} \, dx &=\frac{1}{4} \operatorname{Subst}\left (\int \frac{x^4}{(a+b x)^{3/4}} \, dx,x,x^4\right )\\ &=\frac{1}{4} \operatorname{Subst}\left (\int \left (\frac{a^4}{b^4 (a+b x)^{3/4}}-\frac{4 a^3 \sqrt [4]{a+b x}}{b^4}+\frac{6 a^2 (a+b x)^{5/4}}{b^4}-\frac{4 a (a+b x)^{9/4}}{b^4}+\frac{(a+b x)^{13/4}}{b^4}\right ) \, dx,x,x^4\right )\\ &=\frac{a^4 \sqrt [4]{a+b x^4}}{b^5}-\frac{4 a^3 \left (a+b x^4\right )^{5/4}}{5 b^5}+\frac{2 a^2 \left (a+b x^4\right )^{9/4}}{3 b^5}-\frac{4 a \left (a+b x^4\right )^{13/4}}{13 b^5}+\frac{\left (a+b x^4\right )^{17/4}}{17 b^5}\\ \end{align*}
Mathematica [A] time = 0.0275568, size = 61, normalized size = 0.62 \[ \frac{\sqrt [4]{a+b x^4} \left (320 a^2 b^2 x^8-512 a^3 b x^4+2048 a^4-240 a b^3 x^{12}+195 b^4 x^{16}\right )}{3315 b^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 58, normalized size = 0.6 \begin{align*}{\frac{195\,{x}^{16}{b}^{4}-240\,a{x}^{12}{b}^{3}+320\,{a}^{2}{x}^{8}{b}^{2}-512\,{a}^{3}{x}^{4}b+2048\,{a}^{4}}{3315\,{b}^{5}}\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 = 0.984462, size = 108, normalized size = 1.1 \begin{align*} \frac{{\left (b x^{4} + a\right )}^{\frac{17}{4}}}{17 \, b^{5}} - \frac{4 \,{\left (b x^{4} + a\right )}^{\frac{13}{4}} a}{13 \, b^{5}} + \frac{2 \,{\left (b x^{4} + a\right )}^{\frac{9}{4}} a^{2}}{3 \, b^{5}} - \frac{4 \,{\left (b x^{4} + a\right )}^{\frac{5}{4}} a^{3}}{5 \, b^{5}} + \frac{{\left (b x^{4} + a\right )}^{\frac{1}{4}} a^{4}}{b^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.45558, size = 144, normalized size = 1.47 \begin{align*} \frac{{\left (195 \, b^{4} x^{16} - 240 \, a b^{3} x^{12} + 320 \, a^{2} b^{2} x^{8} - 512 \, a^{3} b x^{4} + 2048 \, a^{4}\right )}{\left (b x^{4} + a\right )}^{\frac{1}{4}}}{3315 \, b^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 19.4907, size = 116, normalized size = 1.18 \begin{align*} \begin{cases} \frac{2048 a^{4} \sqrt [4]{a + b x^{4}}}{3315 b^{5}} - \frac{512 a^{3} x^{4} \sqrt [4]{a + b x^{4}}}{3315 b^{4}} + \frac{64 a^{2} x^{8} \sqrt [4]{a + b x^{4}}}{663 b^{3}} - \frac{16 a x^{12} \sqrt [4]{a + b x^{4}}}{221 b^{2}} + \frac{x^{16} \sqrt [4]{a + b x^{4}}}{17 b} & \text{for}\: b \neq 0 \\\frac{x^{20}}{20 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 = 1.09238, size = 96, normalized size = 0.98 \begin{align*} \frac{195 \,{\left (b x^{4} + a\right )}^{\frac{17}{4}} - 1020 \,{\left (b x^{4} + a\right )}^{\frac{13}{4}} a + 2210 \,{\left (b x^{4} + a\right )}^{\frac{9}{4}} a^{2} - 2652 \,{\left (b x^{4} + a\right )}^{\frac{5}{4}} a^{3} + 3315 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}} a^{4}}{3315 \, b^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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