Optimal. Leaf size=131 \[ -\frac{8 b^2 \left (b x^2+c x^4\right )^{3/2} (2 b B-3 A c)}{315 c^4 x^3}-\frac{x \left (b x^2+c x^4\right )^{3/2} (2 b B-3 A c)}{21 c^2}+\frac{4 b \left (b x^2+c x^4\right )^{3/2} (2 b B-3 A c)}{105 c^3 x}+\frac{B x^3 \left (b x^2+c x^4\right )^{3/2}}{9 c} \]
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Rubi [A] time = 0.219836, antiderivative size = 131, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {2039, 2016, 2000} \[ -\frac{8 b^2 \left (b x^2+c x^4\right )^{3/2} (2 b B-3 A c)}{315 c^4 x^3}-\frac{x \left (b x^2+c x^4\right )^{3/2} (2 b B-3 A c)}{21 c^2}+\frac{4 b \left (b x^2+c x^4\right )^{3/2} (2 b B-3 A c)}{105 c^3 x}+\frac{B x^3 \left (b x^2+c x^4\right )^{3/2}}{9 c} \]
Antiderivative was successfully verified.
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Rule 2039
Rule 2016
Rule 2000
Rubi steps
\begin{align*} \int x^4 \left (A+B x^2\right ) \sqrt{b x^2+c x^4} \, dx &=\frac{B x^3 \left (b x^2+c x^4\right )^{3/2}}{9 c}-\frac{(6 b B-9 A c) \int x^4 \sqrt{b x^2+c x^4} \, dx}{9 c}\\ &=-\frac{(2 b B-3 A c) x \left (b x^2+c x^4\right )^{3/2}}{21 c^2}+\frac{B x^3 \left (b x^2+c x^4\right )^{3/2}}{9 c}+\frac{(4 b (2 b B-3 A c)) \int x^2 \sqrt{b x^2+c x^4} \, dx}{21 c^2}\\ &=\frac{4 b (2 b B-3 A c) \left (b x^2+c x^4\right )^{3/2}}{105 c^3 x}-\frac{(2 b B-3 A c) x \left (b x^2+c x^4\right )^{3/2}}{21 c^2}+\frac{B x^3 \left (b x^2+c x^4\right )^{3/2}}{9 c}-\frac{\left (8 b^2 (2 b B-3 A c)\right ) \int \sqrt{b x^2+c x^4} \, dx}{105 c^3}\\ &=-\frac{8 b^2 (2 b B-3 A c) \left (b x^2+c x^4\right )^{3/2}}{315 c^4 x^3}+\frac{4 b (2 b B-3 A c) \left (b x^2+c x^4\right )^{3/2}}{105 c^3 x}-\frac{(2 b B-3 A c) x \left (b x^2+c x^4\right )^{3/2}}{21 c^2}+\frac{B x^3 \left (b x^2+c x^4\right )^{3/2}}{9 c}\\ \end{align*}
Mathematica [A] time = 0.0596196, size = 82, normalized size = 0.63 \[ \frac{\left (x^2 \left (b+c x^2\right )\right )^{3/2} \left (24 b^2 c \left (A+B x^2\right )-6 b c^2 x^2 \left (6 A+5 B x^2\right )+5 c^3 x^4 \left (9 A+7 B x^2\right )-16 b^3 B\right )}{315 c^4 x^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 91, normalized size = 0.7 \begin{align*}{\frac{ \left ( c{x}^{2}+b \right ) \left ( 35\,B{c}^{3}{x}^{6}+45\,A{x}^{4}{c}^{3}-30\,B{x}^{4}b{c}^{2}-36\,A{x}^{2}b{c}^{2}+24\,B{x}^{2}{b}^{2}c+24\,A{b}^{2}c-16\,B{b}^{3} \right ) }{315\,{c}^{4}x}\sqrt{c{x}^{4}+b{x}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.20731, size = 143, normalized size = 1.09 \begin{align*} \frac{{\left (15 \, c^{3} x^{6} + 3 \, b c^{2} x^{4} - 4 \, b^{2} c x^{2} + 8 \, b^{3}\right )} \sqrt{c x^{2} + b} A}{105 \, c^{3}} + \frac{{\left (35 \, c^{4} x^{8} + 5 \, b c^{3} x^{6} - 6 \, b^{2} c^{2} x^{4} + 8 \, b^{3} c x^{2} - 16 \, b^{4}\right )} \sqrt{c x^{2} + b} B}{315 \, c^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.11853, size = 230, normalized size = 1.76 \begin{align*} \frac{{\left (35 \, B c^{4} x^{8} + 5 \,{\left (B b c^{3} + 9 \, A c^{4}\right )} x^{6} - 16 \, B b^{4} + 24 \, A b^{3} c - 3 \,{\left (2 \, B b^{2} c^{2} - 3 \, A b c^{3}\right )} x^{4} + 4 \,{\left (2 \, B b^{3} c - 3 \, A b^{2} c^{2}\right )} x^{2}\right )} \sqrt{c x^{4} + b x^{2}}}{315 \, c^{4} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{4} \sqrt{x^{2} \left (b + c x^{2}\right )} \left (A + B x^{2}\right )\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16876, size = 180, normalized size = 1.37 \begin{align*} \frac{\frac{3 \,{\left (15 \,{\left (c x^{2} + b\right )}^{\frac{7}{2}} - 42 \,{\left (c x^{2} + b\right )}^{\frac{5}{2}} b + 35 \,{\left (c x^{2} + b\right )}^{\frac{3}{2}} b^{2}\right )} A \mathrm{sgn}\left (x\right )}{c^{2}} + \frac{{\left (35 \,{\left (c x^{2} + b\right )}^{\frac{9}{2}} - 135 \,{\left (c x^{2} + b\right )}^{\frac{7}{2}} b + 189 \,{\left (c x^{2} + b\right )}^{\frac{5}{2}} b^{2} - 105 \,{\left (c x^{2} + b\right )}^{\frac{3}{2}} b^{3}\right )} B \mathrm{sgn}\left (x\right )}{c^{3}}}{315 \, c} + \frac{8 \,{\left (2 \, B b^{\frac{9}{2}} - 3 \, A b^{\frac{7}{2}} c\right )} \mathrm{sgn}\left (x\right )}{315 \, c^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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