Optimal. Leaf size=38 \[ \frac{\left (a+b x^4\right )^{9/4}}{9 b^2}-\frac{a \left (a+b x^4\right )^{5/4}}{5 b^2} \]
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Rubi [A] time = 0.0225679, antiderivative size = 38, 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} \[ \frac{\left (a+b x^4\right )^{9/4}}{9 b^2}-\frac{a \left (a+b x^4\right )^{5/4}}{5 b^2} \]
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 = 0.0136701, size = 28, normalized size = 0.74 \[ \frac{\left (a+b x^4\right )^{5/4} \left (5 b x^4-4 a\right )}{45 b^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 25, 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 = 0.949801, size = 41, 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 = 1.42787, size = 78, normalized size = 2.05 \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 = 2.2081, size = 63, normalized size = 1.66 \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 = 1.13719, size = 39, normalized size = 1.03 \begin{align*} \frac{5 \,{\left (b x^{4} + a\right )}^{\frac{9}{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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