Optimal. Leaf size=68 \[ -\frac{b^2 (3 A c+b B)}{3 x^3}-\frac{A b^3}{5 x^5}+c^2 x (A c+3 b B)-\frac{3 b c (A c+b B)}{x}+\frac{1}{3} B c^3 x^3 \]
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Rubi [A] time = 0.0495744, antiderivative size = 68, 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 = {1584, 448} \[ -\frac{b^2 (3 A c+b B)}{3 x^3}-\frac{A b^3}{5 x^5}+c^2 x (A c+3 b B)-\frac{3 b c (A c+b B)}{x}+\frac{1}{3} B c^3 x^3 \]
Antiderivative was successfully verified.
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Rule 1584
Rule 448
Rubi steps
\begin{align*} \int \frac{\left (A+B x^2\right ) \left (b x^2+c x^4\right )^3}{x^{12}} \, dx &=\int \frac{\left (A+B x^2\right ) \left (b+c x^2\right )^3}{x^6} \, dx\\ &=\int \left (c^2 (3 b B+A c)+\frac{A b^3}{x^6}+\frac{b^2 (b B+3 A c)}{x^4}+\frac{3 b c (b B+A c)}{x^2}+B c^3 x^2\right ) \, dx\\ &=-\frac{A b^3}{5 x^5}-\frac{b^2 (b B+3 A c)}{3 x^3}-\frac{3 b c (b B+A c)}{x}+c^2 (3 b B+A c) x+\frac{1}{3} B c^3 x^3\\ \end{align*}
Mathematica [A] time = 0.0244062, size = 68, normalized size = 1. \[ -\frac{b^2 (3 A c+b B)}{3 x^3}-\frac{A b^3}{5 x^5}+c^2 x (A c+3 b B)-\frac{3 b c (A c+b B)}{x}+\frac{1}{3} B c^3 x^3 \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 64, normalized size = 0.9 \begin{align*}{\frac{B{c}^{3}{x}^{3}}{3}}+A{c}^{3}x+3\,Bb{c}^{2}x-{\frac{{b}^{2} \left ( 3\,Ac+Bb \right ) }{3\,{x}^{3}}}-{\frac{A{b}^{3}}{5\,{x}^{5}}}-3\,{\frac{bc \left ( Ac+Bb \right ) }{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.10651, size = 99, normalized size = 1.46 \begin{align*} \frac{1}{3} \, B c^{3} x^{3} +{\left (3 \, B b c^{2} + A c^{3}\right )} x - \frac{45 \,{\left (B b^{2} c + A b c^{2}\right )} x^{4} + 3 \, A b^{3} + 5 \,{\left (B b^{3} + 3 \, A b^{2} c\right )} x^{2}}{15 \, x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.472586, size = 162, normalized size = 2.38 \begin{align*} \frac{5 \, B c^{3} x^{8} + 15 \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{6} - 45 \,{\left (B b^{2} c + A b c^{2}\right )} x^{4} - 3 \, A b^{3} - 5 \,{\left (B b^{3} + 3 \, A b^{2} c\right )} x^{2}}{15 \, x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.789304, size = 75, normalized size = 1.1 \begin{align*} \frac{B c^{3} x^{3}}{3} + x \left (A c^{3} + 3 B b c^{2}\right ) - \frac{3 A b^{3} + x^{4} \left (45 A b c^{2} + 45 B b^{2} c\right ) + x^{2} \left (15 A b^{2} c + 5 B b^{3}\right )}{15 x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.23963, size = 101, normalized size = 1.49 \begin{align*} \frac{1}{3} \, B c^{3} x^{3} + 3 \, B b c^{2} x + A c^{3} x - \frac{45 \, B b^{2} c x^{4} + 45 \, A b c^{2} x^{4} + 5 \, B b^{3} x^{2} + 15 \, A b^{2} c x^{2} + 3 \, A b^{3}}{15 \, x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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