Optimal. Leaf size=40 \[ \frac{\left (a-b x^4\right )^{9/4}}{9 b^2}-\frac{a \left (a-b x^4\right )^{5/4}}{5 b^2} \]
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Rubi [A] time = 0.0230262, antiderivative size = 40, normalized size of antiderivative = 1., 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 = {266, 43} \[ \frac{\left (a-b x^4\right )^{9/4}}{9 b^2}-\frac{a \left (a-b x^4\right )^{5/4}}{5 b^2} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int x^7 \sqrt [4]{a-b x^4} \, dx &=\frac{1}{4} \operatorname{Subst}\left (\int x \sqrt [4]{a-b x} \, dx,x,x^4\right )\\ &=\frac{1}{4} \operatorname{Subst}\left (\int \left (\frac{a \sqrt [4]{a-b x}}{b}-\frac{(a-b x)^{5/4}}{b}\right ) \, dx,x,x^4\right )\\ &=-\frac{a \left (a-b x^4\right )^{5/4}}{5 b^2}+\frac{\left (a-b x^4\right )^{9/4}}{9 b^2}\\ \end{align*}
Mathematica [A] time = 0.0136176, size = 29, normalized size = 0.72 \[ -\frac{\left (a-b x^4\right )^{5/4} \left (4 a+5 b x^4\right )}{45 b^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 26, normalized size = 0.7 \begin{align*} -{\frac{5\,b{x}^{4}+4\,a}{45\,{b}^{2}} \left ( -b{x}^{4}+a \right ) ^{{\frac{5}{4}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.956828, size = 43, normalized size = 1.08 \begin{align*} \frac{{\left (-b x^{4} + a\right )}^{\frac{9}{4}}}{9 \, b^{2}} - \frac{{\left (-b x^{4} + a\right )}^{\frac{5}{4}} a}{5 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.4545, size = 80, normalized size = 2. \begin{align*} \frac{{\left (5 \, b^{2} x^{8} - a b x^{4} - 4 \, a^{2}\right )}{\left (-b x^{4} + a\right )}^{\frac{1}{4}}}{45 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.5992, size = 63, normalized size = 1.58 \begin{align*} \begin{cases} - \frac{4 a^{2} \sqrt [4]{a - b x^{4}}}{45 b^{2}} - \frac{a x^{4} \sqrt [4]{a - b x^{4}}}{45 b} + \frac{x^{8} \sqrt [4]{a - b x^{4}}}{9} & \text{for}\: b \neq 0 \\\frac{\sqrt [4]{a} x^{8}}{8} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14779, size = 57, normalized size = 1.42 \begin{align*} \frac{5 \,{\left (b x^{4} - a\right )}^{2}{\left (-b x^{4} + a\right )}^{\frac{1}{4}} - 9 \,{\left (-b x^{4} + a\right )}^{\frac{5}{4}} a}{45 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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