Optimal. Leaf size=71 \[ -\frac {\left (a-b x^4\right )^{5/4}}{13 a x^{13}}-\frac {8 b \left (a-b x^4\right )^{5/4}}{117 a^2 x^9}-\frac {32 b^2 \left (a-b x^4\right )^{5/4}}{585 a^3 x^5} \]
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Rubi [A]
time = 0.01, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {277, 270}
\begin {gather*} -\frac {32 b^2 \left (a-b x^4\right )^{5/4}}{585 a^3 x^5}-\frac {8 b \left (a-b x^4\right )^{5/4}}{117 a^2 x^9}-\frac {\left (a-b x^4\right )^{5/4}}{13 a x^{13}} \end {gather*}
Antiderivative was successfully verified.
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Rule 270
Rule 277
Rubi steps
\begin {align*} \int \frac {\sqrt [4]{a-b x^4}}{x^{14}} \, dx &=-\frac {\left (a-b x^4\right )^{5/4}}{13 a x^{13}}+\frac {(8 b) \int \frac {\sqrt [4]{a-b x^4}}{x^{10}} \, dx}{13 a}\\ &=-\frac {\left (a-b x^4\right )^{5/4}}{13 a x^{13}}-\frac {8 b \left (a-b x^4\right )^{5/4}}{117 a^2 x^9}+\frac {\left (32 b^2\right ) \int \frac {\sqrt [4]{a-b x^4}}{x^6} \, dx}{117 a^2}\\ &=-\frac {\left (a-b x^4\right )^{5/4}}{13 a x^{13}}-\frac {8 b \left (a-b x^4\right )^{5/4}}{117 a^2 x^9}-\frac {32 b^2 \left (a-b x^4\right )^{5/4}}{585 a^3 x^5}\\ \end {align*}
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Mathematica [A]
time = 0.14, size = 43, normalized size = 0.61 \begin {gather*} \frac {\left (a-b x^4\right )^{5/4} \left (-45 a^2-40 a b x^4-32 b^2 x^8\right )}{585 a^3 x^{13}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.18, size = 40, normalized size = 0.56
method | result | size |
gosper | \(-\frac {\left (-b \,x^{4}+a \right )^{\frac {5}{4}} \left (32 b^{2} x^{8}+40 a b \,x^{4}+45 a^{2}\right )}{585 x^{13} a^{3}}\) | \(40\) |
trager | \(-\frac {\left (-32 b^{3} x^{12}-8 a \,b^{2} x^{8}-5 a^{2} b \,x^{4}+45 a^{3}\right ) \left (-b \,x^{4}+a \right )^{\frac {1}{4}}}{585 x^{13} a^{3}}\) | \(51\) |
risch | \(-\frac {\left (-b \,x^{4}+a \right )^{\frac {1}{4}} \left (\left (-b \,x^{4}+a \right )^{3}\right )^{\frac {1}{4}} \left (-32 b^{3} x^{12}-8 a \,b^{2} x^{8}-5 a^{2} b \,x^{4}+45 a^{3}\right )}{585 x^{13} \left (-\left (b \,x^{4}-a \right )^{3}\right )^{\frac {1}{4}} a^{3}}\) | \(78\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 55, normalized size = 0.77 \begin {gather*} -\frac {\frac {117 \, {\left (-b x^{4} + a\right )}^{\frac {5}{4}} b^{2}}{x^{5}} + \frac {130 \, {\left (-b x^{4} + a\right )}^{\frac {9}{4}} b}{x^{9}} + \frac {45 \, {\left (-b x^{4} + a\right )}^{\frac {13}{4}}}{x^{13}}}{585 \, a^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 50, normalized size = 0.70 \begin {gather*} \frac {{\left (32 \, b^{3} x^{12} + 8 \, a b^{2} x^{8} + 5 \, a^{2} b x^{4} - 45 \, a^{3}\right )} {\left (-b x^{4} + a\right )}^{\frac {1}{4}}}{585 \, a^{3} x^{13}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 1.09, size = 1090, normalized size = 15.35 \begin {gather*} \begin {cases} \frac {45 a^{5} b^{\frac {17}{4}} \sqrt [4]{\frac {a}{b x^{4}} - 1} \Gamma \left (- \frac {13}{4}\right )}{64 a^{5} b^{4} x^{12} \Gamma \left (- \frac {1}{4}\right ) - 128 a^{4} b^{5} x^{16} \Gamma \left (- \frac {1}{4}\right ) + 64 a^{3} b^{6} x^{20} \Gamma \left (- \frac {1}{4}\right )} - \frac {95 a^{4} b^{\frac {21}{4}} x^{4} \sqrt [4]{\frac {a}{b x^{4}} - 1} \Gamma \left (- \frac {13}{4}\right )}{64 a^{5} b^{4} x^{12} \Gamma \left (- \frac {1}{4}\right ) - 128 a^{4} b^{5} x^{16} \Gamma \left (- \frac {1}{4}\right ) + 64 a^{3} b^{6} x^{20} \Gamma \left (- \frac {1}{4}\right )} + \frac {47 a^{3} b^{\frac {25}{4}} x^{8} \sqrt [4]{\frac {a}{b x^{4}} - 1} \Gamma \left (- \frac {13}{4}\right )}{64 a^{5} b^{4} x^{12} \Gamma \left (- \frac {1}{4}\right ) - 128 a^{4} b^{5} x^{16} \Gamma \left (- \frac {1}{4}\right ) + 64 a^{3} b^{6} x^{20} \Gamma \left (- \frac {1}{4}\right )} - \frac {21 a^{2} b^{\frac {29}{4}} x^{12} \sqrt [4]{\frac {a}{b x^{4}} - 1} \Gamma \left (- \frac {13}{4}\right )}{64 a^{5} b^{4} x^{12} \Gamma \left (- \frac {1}{4}\right ) - 128 a^{4} b^{5} x^{16} \Gamma \left (- \frac {1}{4}\right ) + 64 a^{3} b^{6} x^{20} \Gamma \left (- \frac {1}{4}\right )} + \frac {56 a b^{\frac {33}{4}} x^{16} \sqrt [4]{\frac {a}{b x^{4}} - 1} \Gamma \left (- \frac {13}{4}\right )}{64 a^{5} b^{4} x^{12} \Gamma \left (- \frac {1}{4}\right ) - 128 a^{4} b^{5} x^{16} \Gamma \left (- \frac {1}{4}\right ) + 64 a^{3} b^{6} x^{20} \Gamma \left (- \frac {1}{4}\right )} - \frac {32 b^{\frac {37}{4}} x^{20} \sqrt [4]{\frac {a}{b x^{4}} - 1} \Gamma \left (- \frac {13}{4}\right )}{64 a^{5} b^{4} x^{12} \Gamma \left (- \frac {1}{4}\right ) - 128 a^{4} b^{5} x^{16} \Gamma \left (- \frac {1}{4}\right ) + 64 a^{3} b^{6} x^{20} \Gamma \left (- \frac {1}{4}\right )} & \text {for}\: \left |{\frac {a}{b x^{4}}}\right | > 1 \\\frac {45 a^{5} b^{\frac {17}{4}} \sqrt [4]{- \frac {a}{b x^{4}} + 1} e^{\frac {i \pi }{4}} \Gamma \left (- \frac {13}{4}\right )}{64 a^{5} b^{4} x^{12} \Gamma \left (- \frac {1}{4}\right ) - 128 a^{4} b^{5} x^{16} \Gamma \left (- \frac {1}{4}\right ) + 64 a^{3} b^{6} x^{20} \Gamma \left (- \frac {1}{4}\right )} - \frac {95 a^{4} b^{\frac {21}{4}} x^{4} \sqrt [4]{- \frac {a}{b x^{4}} + 1} e^{\frac {i \pi }{4}} \Gamma \left (- \frac {13}{4}\right )}{64 a^{5} b^{4} x^{12} \Gamma \left (- \frac {1}{4}\right ) - 128 a^{4} b^{5} x^{16} \Gamma \left (- \frac {1}{4}\right ) + 64 a^{3} b^{6} x^{20} \Gamma \left (- \frac {1}{4}\right )} + \frac {47 a^{3} b^{\frac {25}{4}} x^{8} \sqrt [4]{- \frac {a}{b x^{4}} + 1} e^{\frac {i \pi }{4}} \Gamma \left (- \frac {13}{4}\right )}{64 a^{5} b^{4} x^{12} \Gamma \left (- \frac {1}{4}\right ) - 128 a^{4} b^{5} x^{16} \Gamma \left (- \frac {1}{4}\right ) + 64 a^{3} b^{6} x^{20} \Gamma \left (- \frac {1}{4}\right )} - \frac {21 a^{2} b^{\frac {29}{4}} x^{12} \sqrt [4]{- \frac {a}{b x^{4}} + 1} e^{\frac {i \pi }{4}} \Gamma \left (- \frac {13}{4}\right )}{64 a^{5} b^{4} x^{12} \Gamma \left (- \frac {1}{4}\right ) - 128 a^{4} b^{5} x^{16} \Gamma \left (- \frac {1}{4}\right ) + 64 a^{3} b^{6} x^{20} \Gamma \left (- \frac {1}{4}\right )} + \frac {56 a b^{\frac {33}{4}} x^{16} \sqrt [4]{- \frac {a}{b x^{4}} + 1} e^{\frac {i \pi }{4}} \Gamma \left (- \frac {13}{4}\right )}{64 a^{5} b^{4} x^{12} \Gamma \left (- \frac {1}{4}\right ) - 128 a^{4} b^{5} x^{16} \Gamma \left (- \frac {1}{4}\right ) + 64 a^{3} b^{6} x^{20} \Gamma \left (- \frac {1}{4}\right )} - \frac {32 b^{\frac {37}{4}} x^{20} \sqrt [4]{- \frac {a}{b x^{4}} + 1} e^{\frac {i \pi }{4}} \Gamma \left (- \frac {13}{4}\right )}{64 a^{5} b^{4} x^{12} \Gamma \left (- \frac {1}{4}\right ) - 128 a^{4} b^{5} x^{16} \Gamma \left (- \frac {1}{4}\right ) + 64 a^{3} b^{6} x^{20} \Gamma \left (- \frac {1}{4}\right )} & \text {otherwise} \end {cases} \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 = 1.48, size = 77, normalized size = 1.08 \begin {gather*} \frac {b\,{\left (a-b\,x^4\right )}^{1/4}}{117\,a\,x^9}-\frac {{\left (a-b\,x^4\right )}^{1/4}}{13\,x^{13}}+\frac {32\,b^3\,{\left (a-b\,x^4\right )}^{1/4}}{585\,a^3\,x}+\frac {8\,b^2\,{\left (a-b\,x^4\right )}^{1/4}}{585\,a^2\,x^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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