Optimal. Leaf size=80 \[ \frac{3 a^2 \left (a+b x^4\right )^{13/4}}{13 b^4}-\frac{a^3 \left (a+b x^4\right )^{9/4}}{9 b^4}+\frac{\left (a+b x^4\right )^{21/4}}{21 b^4}-\frac{3 a \left (a+b x^4\right )^{17/4}}{17 b^4} \]
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Rubi [A] time = 0.0435236, antiderivative size = 80, 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} \[ \frac{3 a^2 \left (a+b x^4\right )^{13/4}}{13 b^4}-\frac{a^3 \left (a+b x^4\right )^{9/4}}{9 b^4}+\frac{\left (a+b x^4\right )^{21/4}}{21 b^4}-\frac{3 a \left (a+b x^4\right )^{17/4}}{17 b^4} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int x^{15} \left (a+b x^4\right )^{5/4} \, dx &=\frac{1}{4} \operatorname{Subst}\left (\int x^3 (a+b x)^{5/4} \, dx,x,x^4\right )\\ &=\frac{1}{4} \operatorname{Subst}\left (\int \left (-\frac{a^3 (a+b x)^{5/4}}{b^3}+\frac{3 a^2 (a+b x)^{9/4}}{b^3}-\frac{3 a (a+b x)^{13/4}}{b^3}+\frac{(a+b x)^{17/4}}{b^3}\right ) \, dx,x,x^4\right )\\ &=-\frac{a^3 \left (a+b x^4\right )^{9/4}}{9 b^4}+\frac{3 a^2 \left (a+b x^4\right )^{13/4}}{13 b^4}-\frac{3 a \left (a+b x^4\right )^{17/4}}{17 b^4}+\frac{\left (a+b x^4\right )^{21/4}}{21 b^4}\\ \end{align*}
Mathematica [A] time = 0.0266784, size = 50, normalized size = 0.62 \[ \frac{\left (a+b x^4\right )^{9/4} \left (288 a^2 b x^4-128 a^3-468 a b^2 x^8+663 b^3 x^{12}\right )}{13923 b^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 47, normalized size = 0.6 \begin{align*} -{\frac{-663\,{b}^{3}{x}^{12}+468\,a{b}^{2}{x}^{8}-288\,{a}^{2}b{x}^{4}+128\,{a}^{3}}{13923\,{b}^{4}} \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 = 0.962003, size = 86, normalized size = 1.08 \begin{align*} \frac{{\left (b x^{4} + a\right )}^{\frac{21}{4}}}{21 \, b^{4}} - \frac{3 \,{\left (b x^{4} + a\right )}^{\frac{17}{4}} a}{17 \, b^{4}} + \frac{3 \,{\left (b x^{4} + a\right )}^{\frac{13}{4}} a^{2}}{13 \, b^{4}} - \frac{{\left (b x^{4} + a\right )}^{\frac{9}{4}} a^{3}}{9 \, b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.71103, size = 166, normalized size = 2.08 \begin{align*} \frac{{\left (663 \, b^{5} x^{20} + 858 \, a b^{4} x^{16} + 15 \, a^{2} b^{3} x^{12} - 20 \, a^{3} b^{2} x^{8} + 32 \, a^{4} b x^{4} - 128 \, a^{5}\right )}{\left (b x^{4} + a\right )}^{\frac{1}{4}}}{13923 \, b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 45.6387, size = 134, normalized size = 1.68 \begin{align*} \begin{cases} - \frac{128 a^{5} \sqrt [4]{a + b x^{4}}}{13923 b^{4}} + \frac{32 a^{4} x^{4} \sqrt [4]{a + b x^{4}}}{13923 b^{3}} - \frac{20 a^{3} x^{8} \sqrt [4]{a + b x^{4}}}{13923 b^{2}} + \frac{5 a^{2} x^{12} \sqrt [4]{a + b x^{4}}}{4641 b} + \frac{22 a x^{16} \sqrt [4]{a + b x^{4}}}{357} + \frac{b x^{20} \sqrt [4]{a + b x^{4}}}{21} & \text{for}\: b \neq 0 \\\frac{a^{\frac{5}{4}} x^{16}}{16} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.10225, size = 181, normalized size = 2.26 \begin{align*} \frac{\frac{21 \,{\left (195 \,{\left (b x^{4} + a\right )}^{\frac{17}{4}} - 765 \,{\left (b x^{4} + a\right )}^{\frac{13}{4}} a + 1105 \,{\left (b x^{4} + a\right )}^{\frac{9}{4}} a^{2} - 663 \,{\left (b x^{4} + a\right )}^{\frac{5}{4}} a^{3}\right )} a}{b^{3}} + \frac{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}}{b^{3}}}{69615 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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