Optimal. Leaf size=65 \[ b^2 x (3 A c+b B)-\frac{A b^3}{x}+\frac{1}{5} c^2 x^5 (A c+3 b B)+b c x^3 (A c+b B)+\frac{1}{7} B c^3 x^7 \]
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Rubi [A] time = 0.0454822, antiderivative size = 65, 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} \[ b^2 x (3 A c+b B)-\frac{A b^3}{x}+\frac{1}{5} c^2 x^5 (A c+3 b B)+b c x^3 (A c+b B)+\frac{1}{7} B c^3 x^7 \]
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^8} \, dx &=\int \frac{\left (A+B x^2\right ) \left (b+c x^2\right )^3}{x^2} \, dx\\ &=\int \left (b^2 (b B+3 A c)+\frac{A b^3}{x^2}+3 b c (b B+A c) x^2+c^2 (3 b B+A c) x^4+B c^3 x^6\right ) \, dx\\ &=-\frac{A b^3}{x}+b^2 (b B+3 A c) x+b c (b B+A c) x^3+\frac{1}{5} c^2 (3 b B+A c) x^5+\frac{1}{7} B c^3 x^7\\ \end{align*}
Mathematica [A] time = 0.0222336, size = 65, normalized size = 1. \[ b^2 x (3 A c+b B)-\frac{A b^3}{x}+\frac{1}{5} c^2 x^5 (A c+3 b B)+b c x^3 (A c+b B)+\frac{1}{7} B c^3 x^7 \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 71, normalized size = 1.1 \begin{align*}{\frac{B{c}^{3}{x}^{7}}{7}}+{\frac{A{x}^{5}{c}^{3}}{5}}+{\frac{3\,B{x}^{5}b{c}^{2}}{5}}+A{x}^{3}b{c}^{2}+B{x}^{3}{b}^{2}c+3\,A{b}^{2}cx+B{b}^{3}x-{\frac{A{b}^{3}}{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.16804, size = 93, normalized size = 1.43 \begin{align*} \frac{1}{7} \, B c^{3} x^{7} + \frac{1}{5} \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{5} +{\left (B b^{2} c + A b c^{2}\right )} x^{3} - \frac{A b^{3}}{x} +{\left (B b^{3} + 3 \, A b^{2} c\right )} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.481409, size = 161, normalized size = 2.48 \begin{align*} \frac{5 \, B c^{3} x^{8} + 7 \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{6} + 35 \,{\left (B b^{2} c + A b c^{2}\right )} x^{4} - 35 \, A b^{3} + 35 \,{\left (B b^{3} + 3 \, A b^{2} c\right )} x^{2}}{35 \, x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.313686, size = 68, normalized size = 1.05 \begin{align*} - \frac{A b^{3}}{x} + \frac{B c^{3} x^{7}}{7} + x^{5} \left (\frac{A c^{3}}{5} + \frac{3 B b c^{2}}{5}\right ) + x^{3} \left (A b c^{2} + B b^{2} c\right ) + x \left (3 A b^{2} c + B b^{3}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1633, size = 95, normalized size = 1.46 \begin{align*} \frac{1}{7} \, B c^{3} x^{7} + \frac{3}{5} \, B b c^{2} x^{5} + \frac{1}{5} \, A c^{3} x^{5} + B b^{2} c x^{3} + A b c^{2} x^{3} + B b^{3} x + 3 \, A b^{2} c x - \frac{A b^{3}}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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