Optimal. Leaf size=61 \[ \frac{B \left (b x^2+c x^4\right )^{3/2}}{5 c x}-\frac{\left (b x^2+c x^4\right )^{3/2} (2 b B-5 A c)}{15 c^2 x^3} \]
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Rubi [A] time = 0.018547, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {1145, 2000} \[ \frac{B \left (b x^2+c x^4\right )^{3/2}}{5 c x}-\frac{\left (b x^2+c x^4\right )^{3/2} (2 b B-5 A c)}{15 c^2 x^3} \]
Antiderivative was successfully verified.
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Rule 1145
Rule 2000
Rubi steps
\begin{align*} \int \left (A+B x^2\right ) \sqrt{b x^2+c x^4} \, dx &=\frac{B \left (b x^2+c x^4\right )^{3/2}}{5 c x}-\frac{(2 b B-5 A c) \int \sqrt{b x^2+c x^4} \, dx}{5 c}\\ &=-\frac{(2 b B-5 A c) \left (b x^2+c x^4\right )^{3/2}}{15 c^2 x^3}+\frac{B \left (b x^2+c x^4\right )^{3/2}}{5 c x}\\ \end{align*}
Mathematica [A] time = 0.0253083, size = 41, normalized size = 0.67 \[ \frac{\left (x^2 \left (b+c x^2\right )\right )^{3/2} \left (5 A c-2 b B+3 B c x^2\right )}{15 c^2 x^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 45, normalized size = 0.7 \begin{align*}{\frac{ \left ( c{x}^{2}+b \right ) \left ( 3\,Bc{x}^{2}+5\,Ac-2\,Bb \right ) }{15\,{c}^{2}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 = 1.20372, size = 69, normalized size = 1.13 \begin{align*} \frac{{\left (c x^{2} + b\right )}^{\frac{3}{2}} A}{3 \, c} + \frac{{\left (3 \, c^{2} x^{4} + b c x^{2} - 2 \, b^{2}\right )} \sqrt{c x^{2} + b} B}{15 \, c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.961428, size = 124, normalized size = 2.03 \begin{align*} \frac{{\left (3 \, B c^{2} x^{4} - 2 \, B b^{2} + 5 \, A b c +{\left (B b c + 5 \, A c^{2}\right )} x^{2}\right )} \sqrt{c x^{4} + b x^{2}}}{15 \, c^{2} 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 \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 = 1.21547, size = 99, normalized size = 1.62 \begin{align*} \frac{5 \,{\left (c x^{2} + b\right )}^{\frac{3}{2}} A \mathrm{sgn}\left (x\right ) + \frac{{\left (3 \,{\left (c x^{2} + b\right )}^{\frac{5}{2}} - 5 \,{\left (c x^{2} + b\right )}^{\frac{3}{2}} b\right )} B \mathrm{sgn}\left (x\right )}{c}}{15 \, c} + \frac{{\left (2 \, B b^{\frac{5}{2}} - 5 \, A b^{\frac{3}{2}} c\right )} \mathrm{sgn}\left (x\right )}{15 \, c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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