Optimal. Leaf size=35 \[ \frac{a}{b^2 \sqrt [4]{a+b x^4}}+\frac{\left (a+b x^4\right )^{3/4}}{3 b^2} \]
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Rubi [A] time = 0.0213331, antiderivative size = 35, 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} \[ \frac{a}{b^2 \sqrt [4]{a+b x^4}}+\frac{\left (a+b x^4\right )^{3/4}}{3 b^2} \]
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 )^{5/4}} \, dx &=\frac{1}{4} \operatorname{Subst}\left (\int \frac{x}{(a+b x)^{5/4}} \, dx,x,x^4\right )\\ &=\frac{1}{4} \operatorname{Subst}\left (\int \left (-\frac{a}{b (a+b x)^{5/4}}+\frac{1}{b \sqrt [4]{a+b x}}\right ) \, dx,x,x^4\right )\\ &=\frac{a}{b^2 \sqrt [4]{a+b x^4}}+\frac{\left (a+b x^4\right )^{3/4}}{3 b^2}\\ \end{align*}
Mathematica [A] time = 0.010949, size = 27, normalized size = 0.77 \[ \frac{4 a+b x^4}{3 b^2 \sqrt [4]{a+b x^4}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 24, normalized size = 0.7 \begin{align*}{\frac{b{x}^{4}+4\,a}{3\,{b}^{2}}{\frac{1}{\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 = 0.979252, size = 39, normalized size = 1.11 \begin{align*} \frac{{\left (b x^{4} + a\right )}^{\frac{3}{4}}}{3 \, b^{2}} + \frac{a}{{\left (b x^{4} + a\right )}^{\frac{1}{4}} b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.47996, size = 74, normalized size = 2.11 \begin{align*} \frac{{\left (b x^{4} + 4 \, a\right )}{\left (b x^{4} + a\right )}^{\frac{3}{4}}}{3 \,{\left (b^{3} x^{4} + a b^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.13166, size = 44, normalized size = 1.26 \begin{align*} \begin{cases} \frac{4 a}{3 b^{2} \sqrt [4]{a + b x^{4}}} + \frac{x^{4}}{3 b \sqrt [4]{a + b x^{4}}} & \text{for}\: b \neq 0 \\\frac{x^{8}}{8 a^{\frac{5}{4}}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.07925, size = 36, normalized size = 1.03 \begin{align*} \frac{{\left (b x^{4} + a\right )}^{\frac{3}{4}} + \frac{3 \, a}{{\left (b x^{4} + a\right )}^{\frac{1}{4}}}}{3 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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