Optimal. Leaf size=44 \[ \frac{4 b \left (a+b x^4\right )^{3/4}}{21 a^2 x^3}-\frac{\left (a+b x^4\right )^{3/4}}{7 a x^7} \]
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Rubi [A] time = 0.0101368, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {271, 264} \[ \frac{4 b \left (a+b x^4\right )^{3/4}}{21 a^2 x^3}-\frac{\left (a+b x^4\right )^{3/4}}{7 a x^7} \]
Antiderivative was successfully verified.
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Rule 271
Rule 264
Rubi steps
\begin{align*} \int \frac{1}{x^8 \sqrt [4]{a+b x^4}} \, dx &=-\frac{\left (a+b x^4\right )^{3/4}}{7 a x^7}-\frac{(4 b) \int \frac{1}{x^4 \sqrt [4]{a+b x^4}} \, dx}{7 a}\\ &=-\frac{\left (a+b x^4\right )^{3/4}}{7 a x^7}+\frac{4 b \left (a+b x^4\right )^{3/4}}{21 a^2 x^3}\\ \end{align*}
Mathematica [A] time = 0.01283, size = 31, normalized size = 0.7 \[ \frac{\left (a+b x^4\right )^{3/4} \left (4 b x^4-3 a\right )}{21 a^2 x^7} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 28, normalized size = 0.6 \begin{align*} -{\frac{-4\,b{x}^{4}+3\,a}{21\,{a}^{2}{x}^{7}} \left ( b{x}^{4}+a \right ) ^{{\frac{3}{4}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01794, size = 47, normalized size = 1.07 \begin{align*} \frac{\frac{7 \,{\left (b x^{4} + a\right )}^{\frac{3}{4}} b}{x^{3}} - \frac{3 \,{\left (b x^{4} + a\right )}^{\frac{7}{4}}}{x^{7}}}{21 \, a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.51174, size = 68, normalized size = 1.55 \begin{align*} \frac{{\left (4 \, b x^{4} - 3 \, a\right )}{\left (b x^{4} + a\right )}^{\frac{3}{4}}}{21 \, a^{2} x^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.85342, size = 70, normalized size = 1.59 \begin{align*} - \frac{3 b^{\frac{3}{4}} \left (\frac{a}{b x^{4}} + 1\right )^{\frac{3}{4}} \Gamma \left (- \frac{7}{4}\right )}{16 a x^{4} \Gamma \left (\frac{1}{4}\right )} + \frac{b^{\frac{7}{4}} \left (\frac{a}{b x^{4}} + 1\right )^{\frac{3}{4}} \Gamma \left (- \frac{7}{4}\right )}{4 a^{2} \Gamma \left (\frac{1}{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{1}{{\left (b x^{4} + a\right )}^{\frac{1}{4}} x^{8}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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