Optimal. Leaf size=104 \[ \frac{x \sqrt{b x^2+c x^4} (4 b B-3 A c)}{3 b c^2}-\frac{2 \sqrt{b x^2+c x^4} (4 b B-3 A c)}{3 c^3 x}-\frac{x^5 (b B-A c)}{b c \sqrt{b x^2+c x^4}} \]
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Rubi [A] time = 0.199874, antiderivative size = 104, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {2037, 2016, 1588} \[ \frac{x \sqrt{b x^2+c x^4} (4 b B-3 A c)}{3 b c^2}-\frac{2 \sqrt{b x^2+c x^4} (4 b B-3 A c)}{3 c^3 x}-\frac{x^5 (b B-A c)}{b c \sqrt{b x^2+c x^4}} \]
Antiderivative was successfully verified.
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Rule 2037
Rule 2016
Rule 1588
Rubi steps
\begin{align*} \int \frac{x^6 \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^{3/2}} \, dx &=-\frac{(b B-A c) x^5}{b c \sqrt{b x^2+c x^4}}+\frac{(4 b B-3 A c) \int \frac{x^4}{\sqrt{b x^2+c x^4}} \, dx}{b c}\\ &=-\frac{(b B-A c) x^5}{b c \sqrt{b x^2+c x^4}}+\frac{(4 b B-3 A c) x \sqrt{b x^2+c x^4}}{3 b c^2}-\frac{(2 (4 b B-3 A c)) \int \frac{x^2}{\sqrt{b x^2+c x^4}} \, dx}{3 c^2}\\ &=-\frac{(b B-A c) x^5}{b c \sqrt{b x^2+c x^4}}-\frac{2 (4 b B-3 A c) \sqrt{b x^2+c x^4}}{3 c^3 x}+\frac{(4 b B-3 A c) x \sqrt{b x^2+c x^4}}{3 b c^2}\\ \end{align*}
Mathematica [A] time = 0.0381198, size = 60, normalized size = 0.58 \[ \frac{x \left (b \left (6 A c-4 B c x^2\right )+c^2 x^2 \left (3 A+B x^2\right )-8 b^2 B\right )}{3 c^3 \sqrt{x^2 \left (b+c x^2\right )}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 66, normalized size = 0.6 \begin{align*}{\frac{ \left ( c{x}^{2}+b \right ) \left ( B{c}^{2}{x}^{4}+3\,A{x}^{2}{c}^{2}-4\,B{x}^{2}bc+6\,Abc-8\,B{b}^{2} \right ){x}^{3}}{3\,{c}^{3}} \left ( c{x}^{4}+b{x}^{2} \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.1939, size = 80, normalized size = 0.77 \begin{align*} \frac{{\left (c x^{2} + 2 \, b\right )} A}{\sqrt{c x^{2} + b} c^{2}} + \frac{{\left (c^{2} x^{4} - 4 \, b c x^{2} - 8 \, b^{2}\right )} B}{3 \, \sqrt{c x^{2} + b} c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.48939, size = 139, normalized size = 1.34 \begin{align*} \frac{{\left (B c^{2} x^{4} - 8 \, B b^{2} + 6 \, A b c -{\left (4 \, B b c - 3 \, A c^{2}\right )} x^{2}\right )} \sqrt{c x^{4} + b x^{2}}}{3 \,{\left (c^{4} x^{3} + b c^{3} x\right )}} \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 \frac{x^{6} \left (A + B x^{2}\right )}{\left (x^{2} \left (b + c x^{2}\right )\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (B x^{2} + A\right )} x^{6}}{{\left (c x^{4} + b x^{2}\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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