Optimal. Leaf size=38 \[ \frac{a}{2 b^2 \sqrt{a+b x^4}}+\frac{\sqrt{a+b x^4}}{2 b^2} \]
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Rubi [A] time = 0.0221735, 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} \[ \frac{a}{2 b^2 \sqrt{a+b x^4}}+\frac{\sqrt{a+b x^4}}{2 b^2} \]
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 )^{3/2}} \, dx &=\frac{1}{4} \operatorname{Subst}\left (\int \frac{x}{(a+b x)^{3/2}} \, dx,x,x^4\right )\\ &=\frac{1}{4} \operatorname{Subst}\left (\int \left (-\frac{a}{b (a+b x)^{3/2}}+\frac{1}{b \sqrt{a+b x}}\right ) \, dx,x,x^4\right )\\ &=\frac{a}{2 b^2 \sqrt{a+b x^4}}+\frac{\sqrt{a+b x^4}}{2 b^2}\\ \end{align*}
Mathematica [A] time = 0.0111201, size = 27, normalized size = 0.71 \[ \frac{2 a+b x^4}{2 b^2 \sqrt{a+b x^4}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 24, normalized size = 0.6 \begin{align*}{\frac{b{x}^{4}+2\,a}{2\,{b}^{2}}{\frac{1}{\sqrt{b{x}^{4}+a}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.954405, size = 41, normalized size = 1.08 \begin{align*} \frac{\sqrt{b x^{4} + a}}{2 \, b^{2}} + \frac{a}{2 \, \sqrt{b x^{4} + a} b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.47133, size = 72, normalized size = 1.89 \begin{align*} \frac{{\left (b x^{4} + 2 \, a\right )} \sqrt{b x^{4} + a}}{2 \,{\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 = 1.51455, size = 41, normalized size = 1.08 \begin{align*} \begin{cases} \frac{a}{b^{2} \sqrt{a + b x^{4}}} + \frac{x^{4}}{2 b \sqrt{a + b x^{4}}} & \text{for}\: b \neq 0 \\\frac{x^{8}}{8 a^{\frac{3}{2}}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10081, size = 35, normalized size = 0.92 \begin{align*} \frac{\sqrt{b x^{4} + a} + \frac{a}{\sqrt{b x^{4} + a}}}{2 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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