Optimal. Leaf size=36 \[ \frac{\left (a+b x^2\right )^{3/2}}{3 b^2}-\frac{a \sqrt{a+b x^2}}{b^2} \]
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Rubi [A] time = 0.0225047, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103, Rules used = {5, 266, 43} \[ \frac{\left (a+b x^2\right )^{3/2}}{3 b^2}-\frac{a \sqrt{a+b x^2}}{b^2} \]
Antiderivative was successfully verified.
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Rule 5
Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{x^3}{\sqrt{a+b x^2+(2+2 c-2 (1+c)) x^4}} \, dx &=\int \frac{x^3}{\sqrt{a+b x^2}} \, dx\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x}{\sqrt{a+b x}} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (-\frac{a}{b \sqrt{a+b x}}+\frac{\sqrt{a+b x}}{b}\right ) \, dx,x,x^2\right )\\ &=-\frac{a \sqrt{a+b x^2}}{b^2}+\frac{\left (a+b x^2\right )^{3/2}}{3 b^2}\\ \end{align*}
Mathematica [A] time = 0.013652, size = 27, normalized size = 0.75 \[ \frac{\left (b x^2-2 a\right ) \sqrt{a+b x^2}}{3 b^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 25, normalized size = 0.7 \begin{align*} -{\frac{-b{x}^{2}+2\,a}{3\,{b}^{2}}\sqrt{b{x}^{2}+a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.45639, size = 53, normalized size = 1.47 \begin{align*} \frac{\sqrt{b x^{2} + a}{\left (b x^{2} - 2 \, a\right )}}{3 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.458261, size = 44, normalized size = 1.22 \begin{align*} \begin{cases} - \frac{2 a \sqrt{a + b x^{2}}}{3 b^{2}} + \frac{x^{2} \sqrt{a + b x^{2}}}{3 b} & \text{for}\: b \neq 0 \\\frac{x^{4}}{4 \sqrt{a}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11574, size = 36, normalized size = 1. \begin{align*} \frac{{\left (b x^{2} + a\right )}^{\frac{3}{2}} - 3 \, \sqrt{b x^{2} + a} a}{3 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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