Optimal. Leaf size=73 \[ -\frac{2 b \sqrt{c+d x^2} (b c-a d)}{d^3}-\frac{(b c-a d)^2}{d^3 \sqrt{c+d x^2}}+\frac{b^2 \left (c+d x^2\right )^{3/2}}{3 d^3} \]
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Rubi [A] time = 0.0556265, antiderivative size = 73, 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} \[ -\frac{2 b \sqrt{c+d x^2} (b c-a d)}{d^3}-\frac{(b c-a d)^2}{d^3 \sqrt{c+d x^2}}+\frac{b^2 \left (c+d x^2\right )^{3/2}}{3 d^3} \]
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}{\left (c+d x^2\right )^{3/2}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(a+b x)^2}{(c+d x)^{3/2}} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{(-b c+a d)^2}{d^2 (c+d x)^{3/2}}-\frac{2 b (b c-a d)}{d^2 \sqrt{c+d x}}+\frac{b^2 \sqrt{c+d x}}{d^2}\right ) \, dx,x,x^2\right )\\ &=-\frac{(b c-a d)^2}{d^3 \sqrt{c+d x^2}}-\frac{2 b (b c-a d) \sqrt{c+d x^2}}{d^3}+\frac{b^2 \left (c+d x^2\right )^{3/2}}{3 d^3}\\ \end{align*}
Mathematica [A] time = 0.0382335, size = 65, normalized size = 0.89 \[ \frac{-3 a^2 d^2+6 a b d \left (2 c+d x^2\right )+b^2 \left (-8 c^2-4 c d x^2+d^2 x^4\right )}{3 d^3 \sqrt{c+d x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 69, normalized size = 1. \begin{align*} -{\frac{-{b}^{2}{d}^{2}{x}^{4}-6\,ab{d}^{2}{x}^{2}+4\,{b}^{2}cd{x}^{2}+3\,{a}^{2}{d}^{2}-12\,cabd+8\,{b}^{2}{c}^{2}}{3\,{d}^{3}}{\frac{1}{\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 = 1.35808, size = 165, normalized size = 2.26 \begin{align*} \frac{{\left (b^{2} d^{2} x^{4} - 8 \, b^{2} c^{2} + 12 \, a b c d - 3 \, a^{2} d^{2} - 2 \,{\left (2 \, b^{2} c d - 3 \, a b d^{2}\right )} x^{2}\right )} \sqrt{d x^{2} + c}}{3 \,{\left (d^{4} x^{2} + c d^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.01098, size = 155, normalized size = 2.12 \begin{align*} \begin{cases} - \frac{a^{2}}{d \sqrt{c + d x^{2}}} + \frac{4 a b c}{d^{2} \sqrt{c + d x^{2}}} + \frac{2 a b x^{2}}{d \sqrt{c + d x^{2}}} - \frac{8 b^{2} c^{2}}{3 d^{3} \sqrt{c + d x^{2}}} - \frac{4 b^{2} c x^{2}}{3 d^{2} \sqrt{c + d x^{2}}} + \frac{b^{2} x^{4}}{3 d \sqrt{c + d x^{2}}} & \text{for}\: d \neq 0 \\\frac{\frac{a^{2} x^{2}}{2} + \frac{a b x^{4}}{2} + \frac{b^{2} x^{6}}{6}}{c^{\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 = 1.13702, size = 108, normalized size = 1.48 \begin{align*} \frac{{\left (d x^{2} + c\right )}^{\frac{3}{2}} b^{2} - 6 \, \sqrt{d x^{2} + c} b^{2} c + 6 \, \sqrt{d x^{2} + c} a b d - \frac{3 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )}}{\sqrt{d x^{2} + c}}}{3 \, d^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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