Optimal. Leaf size=112 \[ -\frac{b \left (c+d x^2\right )^{5/2} (3 b c-2 a d)}{5 d^4}+\frac{\left (c+d x^2\right )^{3/2} (b c-a d) (3 b c-a d)}{3 d^4}-\frac{c \sqrt{c+d x^2} (b c-a d)^2}{d^4}+\frac{b^2 \left (c+d x^2\right )^{7/2}}{7 d^4} \]
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Rubi [A] time = 0.0889361, antiderivative size = 112, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {446, 77} \[ -\frac{b \left (c+d x^2\right )^{5/2} (3 b c-2 a d)}{5 d^4}+\frac{\left (c+d x^2\right )^{3/2} (b c-a d) (3 b c-a d)}{3 d^4}-\frac{c \sqrt{c+d x^2} (b c-a d)^2}{d^4}+\frac{b^2 \left (c+d x^2\right )^{7/2}}{7 d^4} \]
Antiderivative was successfully verified.
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Rule 446
Rule 77
Rubi steps
\begin{align*} \int \frac{x^3 \left (a+b x^2\right )^2}{\sqrt{c+d x^2}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x (a+b x)^2}{\sqrt{c+d x}} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (-\frac{c (b c-a d)^2}{d^3 \sqrt{c+d x}}+\frac{(b c-a d) (3 b c-a d) \sqrt{c+d x}}{d^3}-\frac{b (3 b c-2 a d) (c+d x)^{3/2}}{d^3}+\frac{b^2 (c+d x)^{5/2}}{d^3}\right ) \, dx,x,x^2\right )\\ &=-\frac{c (b c-a d)^2 \sqrt{c+d x^2}}{d^4}+\frac{(b c-a d) (3 b c-a d) \left (c+d x^2\right )^{3/2}}{3 d^4}-\frac{b (3 b c-2 a d) \left (c+d x^2\right )^{5/2}}{5 d^4}+\frac{b^2 \left (c+d x^2\right )^{7/2}}{7 d^4}\\ \end{align*}
Mathematica [A] time = 0.0708376, size = 99, normalized size = 0.88 \[ \frac{\sqrt{c+d x^2} \left (35 a^2 d^2 \left (d x^2-2 c\right )+14 a b d \left (8 c^2-4 c d x^2+3 d^2 x^4\right )-3 b^2 \left (-8 c^2 d x^2+16 c^3+6 c d^2 x^4-5 d^3 x^6\right )\right )}{105 d^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 108, normalized size = 1. \begin{align*} -{\frac{-15\,{b}^{2}{x}^{6}{d}^{3}-42\,ab{d}^{3}{x}^{4}+18\,{b}^{2}c{d}^{2}{x}^{4}-35\,{a}^{2}{d}^{3}{x}^{2}+56\,abc{d}^{2}{x}^{2}-24\,{b}^{2}{c}^{2}d{x}^{2}+70\,{a}^{2}c{d}^{2}-112\,ab{c}^{2}d+48\,{b}^{2}{c}^{3}}{105\,{d}^{4}}\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 = 1.33319, size = 231, normalized size = 2.06 \begin{align*} \frac{{\left (15 \, b^{2} d^{3} x^{6} - 48 \, b^{2} c^{3} + 112 \, a b c^{2} d - 70 \, a^{2} c d^{2} - 6 \,{\left (3 \, b^{2} c d^{2} - 7 \, a b d^{3}\right )} x^{4} +{\left (24 \, b^{2} c^{2} d - 56 \, a b c d^{2} + 35 \, a^{2} d^{3}\right )} x^{2}\right )} \sqrt{d x^{2} + c}}{105 \, d^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.48448, size = 240, normalized size = 2.14 \begin{align*} \begin{cases} - \frac{2 a^{2} c \sqrt{c + d x^{2}}}{3 d^{2}} + \frac{a^{2} x^{2} \sqrt{c + d x^{2}}}{3 d} + \frac{16 a b c^{2} \sqrt{c + d x^{2}}}{15 d^{3}} - \frac{8 a b c x^{2} \sqrt{c + d x^{2}}}{15 d^{2}} + \frac{2 a b x^{4} \sqrt{c + d x^{2}}}{5 d} - \frac{16 b^{2} c^{3} \sqrt{c + d x^{2}}}{35 d^{4}} + \frac{8 b^{2} c^{2} x^{2} \sqrt{c + d x^{2}}}{35 d^{3}} - \frac{6 b^{2} c x^{4} \sqrt{c + d x^{2}}}{35 d^{2}} + \frac{b^{2} x^{6} \sqrt{c + d x^{2}}}{7 d} & \text{for}\: d \neq 0 \\\frac{\frac{a^{2} x^{4}}{4} + \frac{a b x^{6}}{3} + \frac{b^{2} x^{8}}{8}}{\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 = 1.13078, size = 203, normalized size = 1.81 \begin{align*} \frac{15 \,{\left (d x^{2} + c\right )}^{\frac{7}{2}} b^{2} - 63 \,{\left (d x^{2} + c\right )}^{\frac{5}{2}} b^{2} c + 105 \,{\left (d x^{2} + c\right )}^{\frac{3}{2}} b^{2} c^{2} - 105 \, \sqrt{d x^{2} + c} b^{2} c^{3} + 42 \,{\left (d x^{2} + c\right )}^{\frac{5}{2}} a b d - 140 \,{\left (d x^{2} + c\right )}^{\frac{3}{2}} a b c d + 210 \, \sqrt{d x^{2} + c} a b c^{2} d + 35 \,{\left (d x^{2} + c\right )}^{\frac{3}{2}} a^{2} d^{2} - 105 \, \sqrt{d x^{2} + c} a^{2} c d^{2}}{105 \, d^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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