Optimal. Leaf size=92 \[ -\frac {\left (a+b x^4\right )^{9/4}}{21 a x^{21}}+\frac {4 b \left (a+b x^4\right )^{9/4}}{119 a^2 x^{17}}-\frac {32 b^2 \left (a+b x^4\right )^{9/4}}{1547 a^3 x^{13}}+\frac {128 b^3 \left (a+b x^4\right )^{9/4}}{13923 a^4 x^9} \]
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Rubi [A]
time = 0.02, 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 = {277, 270}
\begin {gather*} \frac {128 b^3 \left (a+b x^4\right )^{9/4}}{13923 a^4 x^9}-\frac {32 b^2 \left (a+b x^4\right )^{9/4}}{1547 a^3 x^{13}}+\frac {4 b \left (a+b x^4\right )^{9/4}}{119 a^2 x^{17}}-\frac {\left (a+b x^4\right )^{9/4}}{21 a x^{21}} \end {gather*}
Antiderivative was successfully verified.
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Rule 270
Rule 277
Rubi steps
\begin {align*} \int \frac {\left (a+b x^4\right )^{5/4}}{x^{22}} \, dx &=-\frac {\left (a+b x^4\right )^{9/4}}{21 a x^{21}}-\frac {(4 b) \int \frac {\left (a+b x^4\right )^{5/4}}{x^{18}} \, dx}{7 a}\\ &=-\frac {\left (a+b x^4\right )^{9/4}}{21 a x^{21}}+\frac {4 b \left (a+b x^4\right )^{9/4}}{119 a^2 x^{17}}+\frac {\left (32 b^2\right ) \int \frac {\left (a+b x^4\right )^{5/4}}{x^{14}} \, dx}{119 a^2}\\ &=-\frac {\left (a+b x^4\right )^{9/4}}{21 a x^{21}}+\frac {4 b \left (a+b x^4\right )^{9/4}}{119 a^2 x^{17}}-\frac {32 b^2 \left (a+b x^4\right )^{9/4}}{1547 a^3 x^{13}}-\frac {\left (128 b^3\right ) \int \frac {\left (a+b x^4\right )^{5/4}}{x^{10}} \, dx}{1547 a^3}\\ &=-\frac {\left (a+b x^4\right )^{9/4}}{21 a x^{21}}+\frac {4 b \left (a+b x^4\right )^{9/4}}{119 a^2 x^{17}}-\frac {32 b^2 \left (a+b x^4\right )^{9/4}}{1547 a^3 x^{13}}+\frac {128 b^3 \left (a+b x^4\right )^{9/4}}{13923 a^4 x^9}\\ \end {align*}
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Mathematica [A]
time = 0.16, size = 53, normalized size = 0.58 \begin {gather*} \frac {\left (a+b x^4\right )^{9/4} \left (-663 a^3+468 a^2 b x^4-288 a b^2 x^8+128 b^3 x^{12}\right )}{13923 a^4 x^{21}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.18, size = 50, normalized size = 0.54
method | result | size |
gosper | \(-\frac {\left (b \,x^{4}+a \right )^{\frac {9}{4}} \left (-128 b^{3} x^{12}+288 a \,b^{2} x^{8}-468 a^{2} b \,x^{4}+663 a^{3}\right )}{13923 x^{21} a^{4}}\) | \(50\) |
trager | \(-\frac {\left (-128 b^{5} x^{20}+32 a \,b^{4} x^{16}-20 a^{2} b^{3} x^{12}+15 a^{3} b^{2} x^{8}+858 a^{4} b \,x^{4}+663 a^{5}\right ) \left (b \,x^{4}+a \right )^{\frac {1}{4}}}{13923 x^{21} a^{4}}\) | \(72\) |
risch | \(-\frac {\left (-128 b^{5} x^{20}+32 a \,b^{4} x^{16}-20 a^{2} b^{3} x^{12}+15 a^{3} b^{2} x^{8}+858 a^{4} b \,x^{4}+663 a^{5}\right ) \left (b \,x^{4}+a \right )^{\frac {1}{4}}}{13923 x^{21} a^{4}}\) | \(72\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 69, normalized size = 0.75 \begin {gather*} \frac {\frac {1547 \, {\left (b x^{4} + a\right )}^{\frac {9}{4}} b^{3}}{x^{9}} - \frac {3213 \, {\left (b x^{4} + a\right )}^{\frac {13}{4}} b^{2}}{x^{13}} + \frac {2457 \, {\left (b x^{4} + a\right )}^{\frac {17}{4}} b}{x^{17}} - \frac {663 \, {\left (b x^{4} + a\right )}^{\frac {21}{4}}}{x^{21}}}{13923 \, a^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 71, normalized size = 0.77 \begin {gather*} \frac {{\left (128 \, b^{5} x^{20} - 32 \, a b^{4} x^{16} + 20 \, a^{2} b^{3} x^{12} - 15 \, a^{3} b^{2} x^{8} - 858 \, a^{4} b x^{4} - 663 \, a^{5}\right )} {\left (b x^{4} + a\right )}^{\frac {1}{4}}}{13923 \, a^{4} x^{21}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 954 vs.
\(2 (85) = 170\).
time = 2.97, size = 954, normalized size = 10.37 \begin {gather*} - \frac {1989 a^{8} b^{\frac {37}{4}} \sqrt [4]{\frac {a}{b x^{4}} + 1} \Gamma \left (- \frac {21}{4}\right )}{256 a^{7} b^{9} x^{20} \Gamma \left (- \frac {5}{4}\right ) + 768 a^{6} b^{10} x^{24} \Gamma \left (- \frac {5}{4}\right ) + 768 a^{5} b^{11} x^{28} \Gamma \left (- \frac {5}{4}\right ) + 256 a^{4} b^{12} x^{32} \Gamma \left (- \frac {5}{4}\right )} - \frac {8541 a^{7} b^{\frac {41}{4}} x^{4} \sqrt [4]{\frac {a}{b x^{4}} + 1} \Gamma \left (- \frac {21}{4}\right )}{256 a^{7} b^{9} x^{20} \Gamma \left (- \frac {5}{4}\right ) + 768 a^{6} b^{10} x^{24} \Gamma \left (- \frac {5}{4}\right ) + 768 a^{5} b^{11} x^{28} \Gamma \left (- \frac {5}{4}\right ) + 256 a^{4} b^{12} x^{32} \Gamma \left (- \frac {5}{4}\right )} - \frac {13734 a^{6} b^{\frac {45}{4}} x^{8} \sqrt [4]{\frac {a}{b x^{4}} + 1} \Gamma \left (- \frac {21}{4}\right )}{256 a^{7} b^{9} x^{20} \Gamma \left (- \frac {5}{4}\right ) + 768 a^{6} b^{10} x^{24} \Gamma \left (- \frac {5}{4}\right ) + 768 a^{5} b^{11} x^{28} \Gamma \left (- \frac {5}{4}\right ) + 256 a^{4} b^{12} x^{32} \Gamma \left (- \frac {5}{4}\right )} - \frac {9786 a^{5} b^{\frac {49}{4}} x^{12} \sqrt [4]{\frac {a}{b x^{4}} + 1} \Gamma \left (- \frac {21}{4}\right )}{256 a^{7} b^{9} x^{20} \Gamma \left (- \frac {5}{4}\right ) + 768 a^{6} b^{10} x^{24} \Gamma \left (- \frac {5}{4}\right ) + 768 a^{5} b^{11} x^{28} \Gamma \left (- \frac {5}{4}\right ) + 256 a^{4} b^{12} x^{32} \Gamma \left (- \frac {5}{4}\right )} - \frac {2625 a^{4} b^{\frac {53}{4}} x^{16} \sqrt [4]{\frac {a}{b x^{4}} + 1} \Gamma \left (- \frac {21}{4}\right )}{256 a^{7} b^{9} x^{20} \Gamma \left (- \frac {5}{4}\right ) + 768 a^{6} b^{10} x^{24} \Gamma \left (- \frac {5}{4}\right ) + 768 a^{5} b^{11} x^{28} \Gamma \left (- \frac {5}{4}\right ) + 256 a^{4} b^{12} x^{32} \Gamma \left (- \frac {5}{4}\right )} + \frac {231 a^{3} b^{\frac {57}{4}} x^{20} \sqrt [4]{\frac {a}{b x^{4}} + 1} \Gamma \left (- \frac {21}{4}\right )}{256 a^{7} b^{9} x^{20} \Gamma \left (- \frac {5}{4}\right ) + 768 a^{6} b^{10} x^{24} \Gamma \left (- \frac {5}{4}\right ) + 768 a^{5} b^{11} x^{28} \Gamma \left (- \frac {5}{4}\right ) + 256 a^{4} b^{12} x^{32} \Gamma \left (- \frac {5}{4}\right )} + \frac {924 a^{2} b^{\frac {61}{4}} x^{24} \sqrt [4]{\frac {a}{b x^{4}} + 1} \Gamma \left (- \frac {21}{4}\right )}{256 a^{7} b^{9} x^{20} \Gamma \left (- \frac {5}{4}\right ) + 768 a^{6} b^{10} x^{24} \Gamma \left (- \frac {5}{4}\right ) + 768 a^{5} b^{11} x^{28} \Gamma \left (- \frac {5}{4}\right ) + 256 a^{4} b^{12} x^{32} \Gamma \left (- \frac {5}{4}\right )} + \frac {1056 a b^{\frac {65}{4}} x^{28} \sqrt [4]{\frac {a}{b x^{4}} + 1} \Gamma \left (- \frac {21}{4}\right )}{256 a^{7} b^{9} x^{20} \Gamma \left (- \frac {5}{4}\right ) + 768 a^{6} b^{10} x^{24} \Gamma \left (- \frac {5}{4}\right ) + 768 a^{5} b^{11} x^{28} \Gamma \left (- \frac {5}{4}\right ) + 256 a^{4} b^{12} x^{32} \Gamma \left (- \frac {5}{4}\right )} + \frac {384 b^{\frac {69}{4}} x^{32} \sqrt [4]{\frac {a}{b x^{4}} + 1} \Gamma \left (- \frac {21}{4}\right )}{256 a^{7} b^{9} x^{20} \Gamma \left (- \frac {5}{4}\right ) + 768 a^{6} b^{10} x^{24} \Gamma \left (- \frac {5}{4}\right ) + 768 a^{5} b^{11} x^{28} \Gamma \left (- \frac {5}{4}\right ) + 256 a^{4} b^{12} x^{32} \Gamma \left (- \frac {5}{4}\right )} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.52, size = 111, normalized size = 1.21 \begin {gather*} \frac {128\,b^5\,{\left (b\,x^4+a\right )}^{1/4}}{13923\,a^4\,x}-\frac {22\,b\,{\left (b\,x^4+a\right )}^{1/4}}{357\,x^{17}}-\frac {a\,{\left (b\,x^4+a\right )}^{1/4}}{21\,x^{21}}-\frac {32\,b^4\,{\left (b\,x^4+a\right )}^{1/4}}{13923\,a^3\,x^5}+\frac {20\,b^3\,{\left (b\,x^4+a\right )}^{1/4}}{13923\,a^2\,x^9}-\frac {5\,b^2\,{\left (b\,x^4+a\right )}^{1/4}}{4641\,a\,x^{13}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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