Optimal. Leaf size=77 \[ -\frac{2 b \left (c+d x^2\right )^{5/2} (b c-a d)}{5 d^3}+\frac{\left (c+d x^2\right )^{3/2} (b c-a d)^2}{3 d^3}+\frac{b^2 \left (c+d x^2\right )^{7/2}}{7 d^3} \]
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Rubi [A] time = 0.0595577, antiderivative size = 77, 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} \[ -\frac{2 b \left (c+d x^2\right )^{5/2} (b c-a d)}{5 d^3}+\frac{\left (c+d x^2\right )^{3/2} (b c-a d)^2}{3 d^3}+\frac{b^2 \left (c+d x^2\right )^{7/2}}{7 d^3} \]
Antiderivative was successfully verified.
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Rule 444
Rule 43
Rubi steps
\begin{align*} \int x \left (a+b x^2\right )^2 \sqrt{c+d x^2} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int (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 \sqrt{c+d x}}{d^2}-\frac{2 b (b c-a d) (c+d x)^{3/2}}{d^2}+\frac{b^2 (c+d x)^{5/2}}{d^2}\right ) \, dx,x,x^2\right )\\ &=\frac{(b c-a d)^2 \left (c+d x^2\right )^{3/2}}{3 d^3}-\frac{2 b (b c-a d) \left (c+d x^2\right )^{5/2}}{5 d^3}+\frac{b^2 \left (c+d x^2\right )^{7/2}}{7 d^3}\\ \end{align*}
Mathematica [A] time = 0.0419449, size = 67, normalized size = 0.87 \[ \frac{\left (c+d x^2\right )^{3/2} \left (35 a^2 d^2+14 a b d \left (3 d x^2-2 c\right )+b^2 \left (8 c^2-12 c d x^2+15 d^2 x^4\right )\right )}{105 d^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 69, normalized size = 0.9 \begin{align*}{\frac{15\,{b}^{2}{d}^{2}{x}^{4}+42\,ab{d}^{2}{x}^{2}-12\,{b}^{2}cd{x}^{2}+35\,{a}^{2}{d}^{2}-28\,cabd+8\,{b}^{2}{c}^{2}}{105\,{d}^{3}} \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}} \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 = 1.59331, size = 225, normalized size = 2.92 \begin{align*} \frac{{\left (15 \, b^{2} d^{3} x^{6} + 8 \, b^{2} c^{3} - 28 \, a b c^{2} d + 35 \, a^{2} c d^{2} + 3 \,{\left (b^{2} c d^{2} + 14 \, a b d^{3}\right )} x^{4} -{\left (4 \, b^{2} c^{2} d - 14 \, a b c d^{2} - 35 \, a^{2} d^{3}\right )} x^{2}\right )} \sqrt{d x^{2} + c}}{105 \, d^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.790295, size = 226, normalized size = 2.94 \begin{align*} \begin{cases} \frac{a^{2} c \sqrt{c + d x^{2}}}{3 d} + \frac{a^{2} x^{2} \sqrt{c + d x^{2}}}{3} - \frac{4 a b c^{2} \sqrt{c + d x^{2}}}{15 d^{2}} + \frac{2 a b c x^{2} \sqrt{c + d x^{2}}}{15 d} + \frac{2 a b x^{4} \sqrt{c + d x^{2}}}{5} + \frac{8 b^{2} c^{3} \sqrt{c + d x^{2}}}{105 d^{3}} - \frac{4 b^{2} c^{2} x^{2} \sqrt{c + d x^{2}}}{105 d^{2}} + \frac{b^{2} c x^{4} \sqrt{c + d x^{2}}}{35 d} + \frac{b^{2} x^{6} \sqrt{c + d x^{2}}}{7} & \text{for}\: d \neq 0 \\\sqrt{c} \left (\frac{a^{2} x^{2}}{2} + \frac{a b x^{4}}{2} + \frac{b^{2} x^{6}}{6}\right ) & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.54361, size = 130, normalized size = 1.69 \begin{align*} \frac{35 \,{\left (d x^{2} + c\right )}^{\frac{3}{2}} a^{2} + \frac{14 \,{\left (3 \,{\left (d x^{2} + c\right )}^{\frac{5}{2}} - 5 \,{\left (d x^{2} + c\right )}^{\frac{3}{2}} c\right )} a b}{d} + \frac{{\left (15 \,{\left (d x^{2} + c\right )}^{\frac{7}{2}} - 42 \,{\left (d x^{2} + c\right )}^{\frac{5}{2}} c + 35 \,{\left (d x^{2} + c\right )}^{\frac{3}{2}} c^{2}\right )} b^{2}}{d^{2}}}{105 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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