Optimal. Leaf size=39 \[ \frac {x}{5 a \left (a+b x^4\right )^{5/4}}+\frac {4 x}{5 a^2 \sqrt [4]{a+b x^4}} \]
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Rubi [A]
time = 0.00, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {198, 197}
\begin {gather*} \frac {4 x}{5 a^2 \sqrt [4]{a+b x^4}}+\frac {x}{5 a \left (a+b x^4\right )^{5/4}} \end {gather*}
Antiderivative was successfully verified.
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Rule 197
Rule 198
Rubi steps
\begin {align*} \int \frac {1}{\left (a+b x^4\right )^{9/4}} \, dx &=\frac {x}{5 a \left (a+b x^4\right )^{5/4}}+\frac {4 \int \frac {1}{\left (a+b x^4\right )^{5/4}} \, dx}{5 a}\\ &=\frac {x}{5 a \left (a+b x^4\right )^{5/4}}+\frac {4 x}{5 a^2 \sqrt [4]{a+b x^4}}\\ \end {align*}
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Mathematica [A]
time = 0.25, size = 29, normalized size = 0.74 \begin {gather*} \frac {5 a x+4 b x^5}{5 a^2 \left (a+b x^4\right )^{5/4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.17, size = 26, normalized size = 0.67
method | result | size |
gosper | \(\frac {x \left (4 b \,x^{4}+5 a \right )}{5 \left (b \,x^{4}+a \right )^{\frac {5}{4}} a^{2}}\) | \(26\) |
trager | \(\frac {x \left (4 b \,x^{4}+5 a \right )}{5 \left (b \,x^{4}+a \right )^{\frac {5}{4}} a^{2}}\) | \(26\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 31, normalized size = 0.79 \begin {gather*} -\frac {{\left (b - \frac {5 \, {\left (b x^{4} + a\right )}}{x^{4}}\right )} x^{5}}{5 \, {\left (b x^{4} + a\right )}^{\frac {5}{4}} a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 47, normalized size = 1.21 \begin {gather*} \frac {{\left (4 \, b x^{5} + 5 \, a x\right )} {\left (b x^{4} + a\right )}^{\frac {3}{4}}}{5 \, {\left (a^{2} b^{2} x^{8} + 2 \, a^{3} b x^{4} + a^{4}\right )}} \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. 126 vs.
\(2 (32) = 64\).
time = 0.61, size = 126, normalized size = 3.23 \begin {gather*} \frac {5 a x \Gamma \left (\frac {1}{4}\right )}{16 a^{\frac {13}{4}} \sqrt [4]{1 + \frac {b x^{4}}{a}} \Gamma \left (\frac {9}{4}\right ) + 16 a^{\frac {9}{4}} b x^{4} \sqrt [4]{1 + \frac {b x^{4}}{a}} \Gamma \left (\frac {9}{4}\right )} + \frac {4 b x^{5} \Gamma \left (\frac {1}{4}\right )}{16 a^{\frac {13}{4}} \sqrt [4]{1 + \frac {b x^{4}}{a}} \Gamma \left (\frac {9}{4}\right ) + 16 a^{\frac {9}{4}} b x^{4} \sqrt [4]{1 + \frac {b x^{4}}{a}} \Gamma \left (\frac {9}{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 = 1.06, size = 28, normalized size = 0.72 \begin {gather*} \frac {4\,x\,\left (b\,x^4+a\right )+a\,x}{5\,a^2\,{\left (b\,x^4+a\right )}^{5/4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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