Optimal. Leaf size=36 \[ \frac{\left (a+b x^4\right )^{5/4}}{5 b^2}-\frac{a \sqrt [4]{a+b x^4}}{b^2} \]
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Rubi [A] time = 0.0211734, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {266, 43} \[ \frac{\left (a+b x^4\right )^{5/4}}{5 b^2}-\frac{a \sqrt [4]{a+b x^4}}{b^2} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{x^7}{\left (a+b x^4\right )^{3/4}} \, dx &=\frac{1}{4} \operatorname{Subst}\left (\int \frac{x}{(a+b x)^{3/4}} \, dx,x,x^4\right )\\ &=\frac{1}{4} \operatorname{Subst}\left (\int \left (-\frac{a}{b (a+b x)^{3/4}}+\frac{\sqrt [4]{a+b x}}{b}\right ) \, dx,x,x^4\right )\\ &=-\frac{a \sqrt [4]{a+b x^4}}{b^2}+\frac{\left (a+b x^4\right )^{5/4}}{5 b^2}\\ \end{align*}
Mathematica [A] time = 0.0129924, size = 27, normalized size = 0.75 \[ \frac{\left (b x^4-4 a\right ) \sqrt [4]{a+b x^4}}{5 b^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 25, normalized size = 0.7 \begin{align*} -{\frac{-b{x}^{4}+4\,a}{5\,{b}^{2}}\sqrt [4]{b{x}^{4}+a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.947952, size = 41, normalized size = 1.14 \begin{align*} \frac{{\left (b x^{4} + a\right )}^{\frac{5}{4}}}{5 \, b^{2}} - \frac{{\left (b x^{4} + a\right )}^{\frac{1}{4}} a}{b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.4701, size = 55, normalized size = 1.53 \begin{align*} \frac{{\left (b x^{4} + a\right )}^{\frac{1}{4}}{\left (b x^{4} - 4 \, a\right )}}{5 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.15406, size = 44, normalized size = 1.22 \begin{align*} \begin{cases} - \frac{4 a \sqrt [4]{a + b x^{4}}}{5 b^{2}} + \frac{x^{4} \sqrt [4]{a + b x^{4}}}{5 b} & \text{for}\: b \neq 0 \\\frac{x^{8}}{8 a^{\frac{3}{4}}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.08946, size = 36, normalized size = 1. \begin{align*} \frac{{\left (b x^{4} + a\right )}^{\frac{5}{4}} - 5 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}} a}{5 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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