Optimal. Leaf size=32 \[ \frac{a}{b^2 \sqrt{a+b x^2}}+\frac{\sqrt{a+b x^2}}{b^2} \]
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Rubi [A] time = 0.0223081, antiderivative size = 32, 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} \[ \frac{a}{b^2 \sqrt{a+b x^2}}+\frac{\sqrt{a+b x^2}}{b^2} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{x^3}{\left (a+b x^2\right )^{3/2}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x}{(a+b x)^{3/2}} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (-\frac{a}{b (a+b x)^{3/2}}+\frac{1}{b \sqrt{a+b x}}\right ) \, dx,x,x^2\right )\\ &=\frac{a}{b^2 \sqrt{a+b x^2}}+\frac{\sqrt{a+b x^2}}{b^2}\\ \end{align*}
Mathematica [A] time = 0.0108361, size = 24, normalized size = 0.75 \[ \frac{2 a+b x^2}{b^2 \sqrt{a+b x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 23, normalized size = 0.7 \begin{align*}{\frac{b{x}^{2}+2\,a}{{b}^{2}}{\frac{1}{\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 = 1.29934, size = 66, normalized size = 2.06 \begin{align*} \frac{{\left (b x^{2} + 2 \, a\right )} \sqrt{b x^{2} + a}}{b^{3} x^{2} + a b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.558642, size = 41, normalized size = 1.28 \begin{align*} \begin{cases} \frac{2 a}{b^{2} \sqrt{a + b x^{2}}} + \frac{x^{2}}{b \sqrt{a + b x^{2}}} & \text{for}\: b \neq 0 \\\frac{x^{4}}{4 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 = 2.11577, size = 34, normalized size = 1.06 \begin{align*} \frac{\sqrt{b x^{2} + a} + \frac{a}{\sqrt{b x^{2} + a}}}{b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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