Optimal. Leaf size=92 \[ \frac {128 b^3 \left (a+b x^4\right )^{3/4}}{1155 a^4 x^3}-\frac {32 b^2 \left (a+b x^4\right )^{3/4}}{385 a^3 x^7}+\frac {4 b \left (a+b x^4\right )^{3/4}}{55 a^2 x^{11}}-\frac {\left (a+b x^4\right )^{3/4}}{15 a x^{15}} \]
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Rubi [A] time = 0.03, antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {271, 264} \[ \frac {128 b^3 \left (a+b x^4\right )^{3/4}}{1155 a^4 x^3}-\frac {32 b^2 \left (a+b x^4\right )^{3/4}}{385 a^3 x^7}+\frac {4 b \left (a+b x^4\right )^{3/4}}{55 a^2 x^{11}}-\frac {\left (a+b x^4\right )^{3/4}}{15 a x^{15}} \]
Antiderivative was successfully verified.
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Rule 264
Rule 271
Rubi steps
\begin {align*} \int \frac {1}{x^{16} \sqrt [4]{a+b x^4}} \, dx &=-\frac {\left (a+b x^4\right )^{3/4}}{15 a x^{15}}-\frac {(4 b) \int \frac {1}{x^{12} \sqrt [4]{a+b x^4}} \, dx}{5 a}\\ &=-\frac {\left (a+b x^4\right )^{3/4}}{15 a x^{15}}+\frac {4 b \left (a+b x^4\right )^{3/4}}{55 a^2 x^{11}}+\frac {\left (32 b^2\right ) \int \frac {1}{x^8 \sqrt [4]{a+b x^4}} \, dx}{55 a^2}\\ &=-\frac {\left (a+b x^4\right )^{3/4}}{15 a x^{15}}+\frac {4 b \left (a+b x^4\right )^{3/4}}{55 a^2 x^{11}}-\frac {32 b^2 \left (a+b x^4\right )^{3/4}}{385 a^3 x^7}-\frac {\left (128 b^3\right ) \int \frac {1}{x^4 \sqrt [4]{a+b x^4}} \, dx}{385 a^3}\\ &=-\frac {\left (a+b x^4\right )^{3/4}}{15 a x^{15}}+\frac {4 b \left (a+b x^4\right )^{3/4}}{55 a^2 x^{11}}-\frac {32 b^2 \left (a+b x^4\right )^{3/4}}{385 a^3 x^7}+\frac {128 b^3 \left (a+b x^4\right )^{3/4}}{1155 a^4 x^3}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 53, normalized size = 0.58 \[ \frac {\left (a+b x^4\right )^{3/4} \left (-77 a^3+84 a^2 b x^4-96 a b^2 x^8+128 b^3 x^{12}\right )}{1155 a^4 x^{15}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.79, size = 49, normalized size = 0.53 \[ \frac {{\left (128 \, b^{3} x^{12} - 96 \, a b^{2} x^{8} + 84 \, a^{2} b x^{4} - 77 \, a^{3}\right )} {\left (b x^{4} + a\right )}^{\frac {3}{4}}}{1155 \, a^{4} x^{15}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b x^{4} + a\right )}^{\frac {1}{4}} x^{16}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 50, normalized size = 0.54 \[ -\frac {\left (b \,x^{4}+a \right )^{\frac {3}{4}} \left (-128 b^{3} x^{12}+96 a \,b^{2} x^{8}-84 a^{2} b \,x^{4}+77 a^{3}\right )}{1155 a^{4} x^{15}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.35, size = 69, normalized size = 0.75 \[ \frac {\frac {385 \, {\left (b x^{4} + a\right )}^{\frac {3}{4}} b^{3}}{x^{3}} - \frac {495 \, {\left (b x^{4} + a\right )}^{\frac {7}{4}} b^{2}}{x^{7}} + \frac {315 \, {\left (b x^{4} + a\right )}^{\frac {11}{4}} b}{x^{11}} - \frac {77 \, {\left (b x^{4} + a\right )}^{\frac {15}{4}}}{x^{15}}}{1155 \, a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.35, size = 76, normalized size = 0.83 \[ \frac {4\,b\,{\left (b\,x^4+a\right )}^{3/4}}{55\,a^2\,x^{11}}-\frac {{\left (b\,x^4+a\right )}^{3/4}}{15\,a\,x^{15}}+\frac {128\,b^3\,{\left (b\,x^4+a\right )}^{3/4}}{1155\,a^4\,x^3}-\frac {32\,b^2\,{\left (b\,x^4+a\right )}^{3/4}}{385\,a^3\,x^7} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 5.36, size = 692, normalized size = 7.52 \[ - \frac {231 a^{6} b^{\frac {39}{4}} \left (\frac {a}{b x^{4}} + 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {15}{4}\right )}{256 a^{7} b^{9} x^{12} \Gamma \left (\frac {1}{4}\right ) + 768 a^{6} b^{10} x^{16} \Gamma \left (\frac {1}{4}\right ) + 768 a^{5} b^{11} x^{20} \Gamma \left (\frac {1}{4}\right ) + 256 a^{4} b^{12} x^{24} \Gamma \left (\frac {1}{4}\right )} - \frac {441 a^{5} b^{\frac {43}{4}} x^{4} \left (\frac {a}{b x^{4}} + 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {15}{4}\right )}{256 a^{7} b^{9} x^{12} \Gamma \left (\frac {1}{4}\right ) + 768 a^{6} b^{10} x^{16} \Gamma \left (\frac {1}{4}\right ) + 768 a^{5} b^{11} x^{20} \Gamma \left (\frac {1}{4}\right ) + 256 a^{4} b^{12} x^{24} \Gamma \left (\frac {1}{4}\right )} - \frac {225 a^{4} b^{\frac {47}{4}} x^{8} \left (\frac {a}{b x^{4}} + 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {15}{4}\right )}{256 a^{7} b^{9} x^{12} \Gamma \left (\frac {1}{4}\right ) + 768 a^{6} b^{10} x^{16} \Gamma \left (\frac {1}{4}\right ) + 768 a^{5} b^{11} x^{20} \Gamma \left (\frac {1}{4}\right ) + 256 a^{4} b^{12} x^{24} \Gamma \left (\frac {1}{4}\right )} + \frac {45 a^{3} b^{\frac {51}{4}} x^{12} \left (\frac {a}{b x^{4}} + 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {15}{4}\right )}{256 a^{7} b^{9} x^{12} \Gamma \left (\frac {1}{4}\right ) + 768 a^{6} b^{10} x^{16} \Gamma \left (\frac {1}{4}\right ) + 768 a^{5} b^{11} x^{20} \Gamma \left (\frac {1}{4}\right ) + 256 a^{4} b^{12} x^{24} \Gamma \left (\frac {1}{4}\right )} + \frac {540 a^{2} b^{\frac {55}{4}} x^{16} \left (\frac {a}{b x^{4}} + 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {15}{4}\right )}{256 a^{7} b^{9} x^{12} \Gamma \left (\frac {1}{4}\right ) + 768 a^{6} b^{10} x^{16} \Gamma \left (\frac {1}{4}\right ) + 768 a^{5} b^{11} x^{20} \Gamma \left (\frac {1}{4}\right ) + 256 a^{4} b^{12} x^{24} \Gamma \left (\frac {1}{4}\right )} + \frac {864 a b^{\frac {59}{4}} x^{20} \left (\frac {a}{b x^{4}} + 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {15}{4}\right )}{256 a^{7} b^{9} x^{12} \Gamma \left (\frac {1}{4}\right ) + 768 a^{6} b^{10} x^{16} \Gamma \left (\frac {1}{4}\right ) + 768 a^{5} b^{11} x^{20} \Gamma \left (\frac {1}{4}\right ) + 256 a^{4} b^{12} x^{24} \Gamma \left (\frac {1}{4}\right )} + \frac {384 b^{\frac {63}{4}} x^{24} \left (\frac {a}{b x^{4}} + 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {15}{4}\right )}{256 a^{7} b^{9} x^{12} \Gamma \left (\frac {1}{4}\right ) + 768 a^{6} b^{10} x^{16} \Gamma \left (\frac {1}{4}\right ) + 768 a^{5} b^{11} x^{20} \Gamma \left (\frac {1}{4}\right ) + 256 a^{4} b^{12} x^{24} \Gamma \left (\frac {1}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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