Optimal. Leaf size=68 \[ -\frac{32 b^2 \left (a+b x^4\right )^{7/4}}{1155 a^3 x^7}+\frac{8 b \left (a+b x^4\right )^{7/4}}{165 a^2 x^{11}}-\frac{\left (a+b x^4\right )^{7/4}}{15 a x^{15}} \]
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Rubi [A] time = 0.0195247, antiderivative size = 68, 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 = {271, 264} \[ -\frac{32 b^2 \left (a+b x^4\right )^{7/4}}{1155 a^3 x^7}+\frac{8 b \left (a+b x^4\right )^{7/4}}{165 a^2 x^{11}}-\frac{\left (a+b x^4\right )^{7/4}}{15 a x^{15}} \]
Antiderivative was successfully verified.
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Rule 271
Rule 264
Rubi steps
\begin{align*} \int \frac{\left (a+b x^4\right )^{3/4}}{x^{16}} \, dx &=-\frac{\left (a+b x^4\right )^{7/4}}{15 a x^{15}}-\frac{(8 b) \int \frac{\left (a+b x^4\right )^{3/4}}{x^{12}} \, dx}{15 a}\\ &=-\frac{\left (a+b x^4\right )^{7/4}}{15 a x^{15}}+\frac{8 b \left (a+b x^4\right )^{7/4}}{165 a^2 x^{11}}+\frac{\left (32 b^2\right ) \int \frac{\left (a+b x^4\right )^{3/4}}{x^8} \, dx}{165 a^2}\\ &=-\frac{\left (a+b x^4\right )^{7/4}}{15 a x^{15}}+\frac{8 b \left (a+b x^4\right )^{7/4}}{165 a^2 x^{11}}-\frac{32 b^2 \left (a+b x^4\right )^{7/4}}{1155 a^3 x^7}\\ \end{align*}
Mathematica [A] time = 0.0103439, size = 42, normalized size = 0.62 \[ -\frac{\left (a+b x^4\right )^{7/4} \left (77 a^2-56 a b x^4+32 b^2 x^8\right )}{1155 a^3 x^{15}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 39, normalized size = 0.6 \begin{align*} -{\frac{32\,{b}^{2}{x}^{8}-56\,ab{x}^{4}+77\,{a}^{2}}{1155\,{a}^{3}{x}^{15}} \left ( b{x}^{4}+a \right ) ^{{\frac{7}{4}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.996411, size = 70, normalized size = 1.03 \begin{align*} -\frac{\frac{165 \,{\left (b x^{4} + a\right )}^{\frac{7}{4}} b^{2}}{x^{7}} - \frac{210 \,{\left (b x^{4} + a\right )}^{\frac{11}{4}} b}{x^{11}} + \frac{77 \,{\left (b x^{4} + a\right )}^{\frac{15}{4}}}{x^{15}}}{1155 \, a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.76497, size = 123, normalized size = 1.81 \begin{align*} -\frac{{\left (32 \, b^{3} x^{12} - 24 \, a b^{2} x^{8} + 21 \, a^{2} b x^{4} + 77 \, a^{3}\right )}{\left (b x^{4} + a\right )}^{\frac{3}{4}}}{1155 \, a^{3} x^{15}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 7.12324, size = 520, normalized size = 7.65 \begin{align*} \frac{77 a^{5} b^{\frac{19}{4}} \left (\frac{a}{b x^{4}} + 1\right )^{\frac{3}{4}} \Gamma \left (- \frac{15}{4}\right )}{64 a^{5} b^{4} x^{12} \Gamma \left (- \frac{3}{4}\right ) + 128 a^{4} b^{5} x^{16} \Gamma \left (- \frac{3}{4}\right ) + 64 a^{3} b^{6} x^{20} \Gamma \left (- \frac{3}{4}\right )} + \frac{175 a^{4} b^{\frac{23}{4}} x^{4} \left (\frac{a}{b x^{4}} + 1\right )^{\frac{3}{4}} \Gamma \left (- \frac{15}{4}\right )}{64 a^{5} b^{4} x^{12} \Gamma \left (- \frac{3}{4}\right ) + 128 a^{4} b^{5} x^{16} \Gamma \left (- \frac{3}{4}\right ) + 64 a^{3} b^{6} x^{20} \Gamma \left (- \frac{3}{4}\right )} + \frac{95 a^{3} b^{\frac{27}{4}} x^{8} \left (\frac{a}{b x^{4}} + 1\right )^{\frac{3}{4}} \Gamma \left (- \frac{15}{4}\right )}{64 a^{5} b^{4} x^{12} \Gamma \left (- \frac{3}{4}\right ) + 128 a^{4} b^{5} x^{16} \Gamma \left (- \frac{3}{4}\right ) + 64 a^{3} b^{6} x^{20} \Gamma \left (- \frac{3}{4}\right )} + \frac{5 a^{2} b^{\frac{31}{4}} x^{12} \left (\frac{a}{b x^{4}} + 1\right )^{\frac{3}{4}} \Gamma \left (- \frac{15}{4}\right )}{64 a^{5} b^{4} x^{12} \Gamma \left (- \frac{3}{4}\right ) + 128 a^{4} b^{5} x^{16} \Gamma \left (- \frac{3}{4}\right ) + 64 a^{3} b^{6} x^{20} \Gamma \left (- \frac{3}{4}\right )} + \frac{40 a b^{\frac{35}{4}} x^{16} \left (\frac{a}{b x^{4}} + 1\right )^{\frac{3}{4}} \Gamma \left (- \frac{15}{4}\right )}{64 a^{5} b^{4} x^{12} \Gamma \left (- \frac{3}{4}\right ) + 128 a^{4} b^{5} x^{16} \Gamma \left (- \frac{3}{4}\right ) + 64 a^{3} b^{6} x^{20} \Gamma \left (- \frac{3}{4}\right )} + \frac{32 b^{\frac{39}{4}} x^{20} \left (\frac{a}{b x^{4}} + 1\right )^{\frac{3}{4}} \Gamma \left (- \frac{15}{4}\right )}{64 a^{5} b^{4} x^{12} \Gamma \left (- \frac{3}{4}\right ) + 128 a^{4} b^{5} x^{16} \Gamma \left (- \frac{3}{4}\right ) + 64 a^{3} b^{6} x^{20} \Gamma \left (- \frac{3}{4}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x^{4} + a\right )}^{\frac{3}{4}}}{x^{16}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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