Optimal. Leaf size=66 \[ -\frac{b^2 (3 A c+b B)}{5 x^5}-\frac{A b^3}{7 x^7}-\frac{c^2 (A c+3 b B)}{x}-\frac{b c (A c+b B)}{x^3}+B c^3 x \]
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Rubi [A] time = 0.0495761, antiderivative size = 66, 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} \[ -\frac{b^2 (3 A c+b B)}{5 x^5}-\frac{A b^3}{7 x^7}-\frac{c^2 (A c+3 b B)}{x}-\frac{b c (A c+b B)}{x^3}+B c^3 x \]
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^{14}} \, dx &=\int \frac{\left (A+B x^2\right ) \left (b+c x^2\right )^3}{x^8} \, dx\\ &=\int \left (B c^3+\frac{A b^3}{x^8}+\frac{b^2 (b B+3 A c)}{x^6}+\frac{3 b c (b B+A c)}{x^4}+\frac{c^2 (3 b B+A c)}{x^2}\right ) \, dx\\ &=-\frac{A b^3}{7 x^7}-\frac{b^2 (b B+3 A c)}{5 x^5}-\frac{b c (b B+A c)}{x^3}-\frac{c^2 (3 b B+A c)}{x}+B c^3 x\\ \end{align*}
Mathematica [A] time = 0.0274184, size = 66, normalized size = 1. \[ -\frac{b^2 (3 A c+b B)}{5 x^5}-\frac{A b^3}{7 x^7}-\frac{c^2 (A c+3 b B)}{x}-\frac{b c (A c+b B)}{x^3}+B c^3 x \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 63, normalized size = 1. \begin{align*} -{\frac{A{b}^{3}}{7\,{x}^{7}}}-{\frac{{b}^{2} \left ( 3\,Ac+Bb \right ) }{5\,{x}^{5}}}-{\frac{bc \left ( Ac+Bb \right ) }{{x}^{3}}}-{\frac{{c}^{2} \left ( Ac+3\,Bb \right ) }{x}}+B{c}^{3}x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.19158, size = 99, normalized size = 1.5 \begin{align*} B c^{3} x - \frac{35 \,{\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} + 5 \, A b^{3} + 7 \,{\left (B b^{3} + 3 \, A b^{2} c\right )} x^{2}}{35 \, x^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.483685, size = 163, normalized size = 2.47 \begin{align*} \frac{35 \, B c^{3} x^{8} - 35 \,{\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} - 5 \, A b^{3} - 7 \,{\left (B b^{3} + 3 \, A b^{2} c\right )} x^{2}}{35 \, x^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.60242, size = 75, normalized size = 1.14 \begin{align*} B c^{3} x - \frac{5 A b^{3} + x^{6} \left (35 A c^{3} + 105 B b c^{2}\right ) + x^{4} \left (35 A b c^{2} + 35 B b^{2} c\right ) + x^{2} \left (21 A b^{2} c + 7 B b^{3}\right )}{35 x^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.27154, size = 104, normalized size = 1.58 \begin{align*} B c^{3} x - \frac{105 \, B b c^{2} x^{6} + 35 \, A c^{3} x^{6} + 35 \, B b^{2} c x^{4} + 35 \, A b c^{2} x^{4} + 7 \, B b^{3} x^{2} + 21 \, A b^{2} c x^{2} + 5 \, A b^{3}}{35 \, x^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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