Optimal. Leaf size=74 \[ -\frac{2 b \left (c+d x^2\right )^{3/2} (b c-a d)}{3 d^3}+\frac{\sqrt{c+d x^2} (b c-a d)^2}{d^3}+\frac{b^2 \left (c+d x^2\right )^{5/2}}{5 d^3} \]
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Rubi [A] time = 0.0535002, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {444, 43} \[ -\frac{2 b \left (c+d x^2\right )^{3/2} (b c-a d)}{3 d^3}+\frac{\sqrt{c+d x^2} (b c-a d)^2}{d^3}+\frac{b^2 \left (c+d x^2\right )^{5/2}}{5 d^3} \]
Antiderivative was successfully verified.
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Rule 444
Rule 43
Rubi steps
\begin{align*} \int \frac{x \left (a+b x^2\right )^2}{\sqrt{c+d x^2}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(a+b x)^2}{\sqrt{c+d x}} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{(-b c+a d)^2}{d^2 \sqrt{c+d x}}-\frac{2 b (b c-a d) \sqrt{c+d x}}{d^2}+\frac{b^2 (c+d x)^{3/2}}{d^2}\right ) \, dx,x,x^2\right )\\ &=\frac{(b c-a d)^2 \sqrt{c+d x^2}}{d^3}-\frac{2 b (b c-a d) \left (c+d x^2\right )^{3/2}}{3 d^3}+\frac{b^2 \left (c+d x^2\right )^{5/2}}{5 d^3}\\ \end{align*}
Mathematica [A] time = 0.0406491, size = 66, normalized size = 0.89 \[ \frac{\sqrt{c+d x^2} \left (15 a^2 d^2+10 a b d \left (d x^2-2 c\right )+b^2 \left (8 c^2-4 c d x^2+3 d^2 x^4\right )\right )}{15 d^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 69, normalized size = 0.9 \begin{align*}{\frac{3\,{b}^{2}{d}^{2}{x}^{4}+10\,ab{d}^{2}{x}^{2}-4\,{b}^{2}cd{x}^{2}+15\,{a}^{2}{d}^{2}-20\,cabd+8\,{b}^{2}{c}^{2}}{15\,{d}^{3}}\sqrt{d{x}^{2}+c}} \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 = 1.35014, size = 151, normalized size = 2.04 \begin{align*} \frac{{\left (3 \, b^{2} d^{2} x^{4} + 8 \, b^{2} c^{2} - 20 \, a b c d + 15 \, a^{2} d^{2} - 2 \,{\left (2 \, b^{2} c d - 5 \, a b d^{2}\right )} x^{2}\right )} \sqrt{d x^{2} + c}}{15 \, d^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.895434, size = 158, normalized size = 2.14 \begin{align*} \begin{cases} \frac{a^{2} \sqrt{c + d x^{2}}}{d} - \frac{4 a b c \sqrt{c + d x^{2}}}{3 d^{2}} + \frac{2 a b x^{2} \sqrt{c + d x^{2}}}{3 d} + \frac{8 b^{2} c^{2} \sqrt{c + d x^{2}}}{15 d^{3}} - \frac{4 b^{2} c x^{2} \sqrt{c + d x^{2}}}{15 d^{2}} + \frac{b^{2} x^{4} \sqrt{c + d x^{2}}}{5 d} & \text{for}\: d \neq 0 \\\frac{\frac{a^{2} x^{2}}{2} + \frac{a b x^{4}}{2} + \frac{b^{2} x^{6}}{6}}{\sqrt{c}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12304, size = 132, normalized size = 1.78 \begin{align*} \frac{3 \,{\left (d x^{2} + c\right )}^{\frac{5}{2}} b^{2} - 10 \,{\left (d x^{2} + c\right )}^{\frac{3}{2}} b^{2} c + 15 \, \sqrt{d x^{2} + c} b^{2} c^{2} + 10 \,{\left (d x^{2} + c\right )}^{\frac{3}{2}} a b d - 30 \, \sqrt{d x^{2} + c} a b c d + 15 \, \sqrt{d x^{2} + c} a^{2} d^{2}}{15 \, d^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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