Optimal. Leaf size=73 \[ -\frac {A b^3}{9 x^9}-\frac {b^2 (3 A c+b B)}{7 x^7}-\frac {c^2 (A c+3 b B)}{3 x^3}-\frac {3 b c (A c+b B)}{5 x^5}-\frac {B c^3}{x} \]
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Rubi [A] time = 0.05, antiderivative size = 73, normalized size of antiderivative = 1.00, 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} \[ -\frac {b^2 (3 A c+b B)}{7 x^7}-\frac {A b^3}{9 x^9}-\frac {c^2 (A c+3 b B)}{3 x^3}-\frac {3 b c (A c+b B)}{5 x^5}-\frac {B c^3}{x} \]
Antiderivative was successfully verified.
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Rule 448
Rule 1584
Rubi steps
\begin {align*} \int \frac {\left (A+B x^2\right ) \left (b x^2+c x^4\right )^3}{x^{16}} \, dx &=\int \frac {\left (A+B x^2\right ) \left (b+c x^2\right )^3}{x^{10}} \, dx\\ &=\int \left (\frac {A b^3}{x^{10}}+\frac {b^2 (b B+3 A c)}{x^8}+\frac {3 b c (b B+A c)}{x^6}+\frac {c^2 (3 b B+A c)}{x^4}+\frac {B c^3}{x^2}\right ) \, dx\\ &=-\frac {A b^3}{9 x^9}-\frac {b^2 (b B+3 A c)}{7 x^7}-\frac {3 b c (b B+A c)}{5 x^5}-\frac {c^2 (3 b B+A c)}{3 x^3}-\frac {B c^3}{x}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 73, normalized size = 1.00 \[ -\frac {A b^3}{9 x^9}-\frac {b^2 (3 A c+b B)}{7 x^7}-\frac {c^2 (A c+3 b B)}{3 x^3}-\frac {3 b c (A c+b B)}{5 x^5}-\frac {B c^3}{x} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.82, size = 75, normalized size = 1.03 \[ -\frac {315 \, B c^{3} x^{8} + 105 \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{6} + 189 \, {\left (B b^{2} c + A b c^{2}\right )} x^{4} + 35 \, A b^{3} + 45 \, {\left (B b^{3} + 3 \, A b^{2} c\right )} x^{2}}{315 \, x^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 79, normalized size = 1.08 \[ -\frac {315 \, B c^{3} x^{8} + 315 \, B b c^{2} x^{6} + 105 \, A c^{3} x^{6} + 189 \, B b^{2} c x^{4} + 189 \, A b c^{2} x^{4} + 45 \, B b^{3} x^{2} + 135 \, A b^{2} c x^{2} + 35 \, A b^{3}}{315 \, x^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 66, normalized size = 0.90 \[ -\frac {B \,c^{3}}{x}-\frac {\left (A c +3 b B \right ) c^{2}}{3 x^{3}}-\frac {3 \left (A c +b B \right ) b c}{5 x^{5}}-\frac {A \,b^{3}}{9 x^{9}}-\frac {\left (3 A c +b B \right ) b^{2}}{7 x^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.38, size = 75, normalized size = 1.03 \[ -\frac {315 \, B c^{3} x^{8} + 105 \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{6} + 189 \, {\left (B b^{2} c + A b c^{2}\right )} x^{4} + 35 \, A b^{3} + 45 \, {\left (B b^{3} + 3 \, A b^{2} c\right )} x^{2}}{315 \, x^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 74, normalized size = 1.01 \[ -\frac {x^4\,\left (\frac {3\,B\,b^2\,c}{5}+\frac {3\,A\,b\,c^2}{5}\right )+\frac {A\,b^3}{9}+x^2\,\left (\frac {B\,b^3}{7}+\frac {3\,A\,c\,b^2}{7}\right )+x^6\,\left (\frac {A\,c^3}{3}+B\,b\,c^2\right )+B\,c^3\,x^8}{x^9} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.45, size = 83, normalized size = 1.14 \[ \frac {- 35 A b^{3} - 315 B c^{3} x^{8} + x^{6} \left (- 105 A c^{3} - 315 B b c^{2}\right ) + x^{4} \left (- 189 A b c^{2} - 189 B b^{2} c\right ) + x^{2} \left (- 135 A b^{2} c - 45 B b^{3}\right )}{315 x^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
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