Optimal. Leaf size=70 \[ A b^3 x+\frac {1}{3} b^2 x^3 (3 A c+b B)+\frac {1}{7} c^2 x^7 (A c+3 b B)+\frac {3}{5} b c x^5 (A c+b B)+\frac {1}{9} B c^3 x^9 \]
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Rubi [A] time = 0.04, antiderivative size = 70, 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, 373} \[ \frac {1}{3} b^2 x^3 (3 A c+b B)+A b^3 x+\frac {1}{7} c^2 x^7 (A c+3 b B)+\frac {3}{5} b c x^5 (A c+b B)+\frac {1}{9} B c^3 x^9 \]
Antiderivative was successfully verified.
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Rule 373
Rule 1584
Rubi steps
\begin {align*} \int \frac {\left (A+B x^2\right ) \left (b x^2+c x^4\right )^3}{x^6} \, dx &=\int \left (A+B x^2\right ) \left (b+c x^2\right )^3 \, dx\\ &=\int \left (A b^3+b^2 (b B+3 A c) x^2+3 b c (b B+A c) x^4+c^2 (3 b B+A c) x^6+B c^3 x^8\right ) \, dx\\ &=A b^3 x+\frac {1}{3} b^2 (b B+3 A c) x^3+\frac {3}{5} b c (b B+A c) x^5+\frac {1}{7} c^2 (3 b B+A c) x^7+\frac {1}{9} B c^3 x^9\\ \end {align*}
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Mathematica [A] time = 0.01, size = 70, normalized size = 1.00 \[ A b^3 x+\frac {1}{3} b^2 x^3 (3 A c+b B)+\frac {1}{7} c^2 x^7 (A c+3 b B)+\frac {3}{5} b c x^5 (A c+b B)+\frac {1}{9} B c^3 x^9 \]
Antiderivative was successfully verified.
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fricas [A] time = 0.83, size = 70, normalized size = 1.00 \[ \frac {1}{9} \, B c^{3} x^{9} + \frac {1}{7} \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{7} + \frac {3}{5} \, {\left (B b^{2} c + A b c^{2}\right )} x^{5} + A b^{3} x + \frac {1}{3} \, {\left (B b^{3} + 3 \, A b^{2} c\right )} x^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 73, normalized size = 1.04 \[ \frac {1}{9} \, B c^{3} x^{9} + \frac {3}{7} \, B b c^{2} x^{7} + \frac {1}{7} \, A c^{3} x^{7} + \frac {3}{5} \, B b^{2} c x^{5} + \frac {3}{5} \, A b c^{2} x^{5} + \frac {1}{3} \, B b^{3} x^{3} + A b^{2} c x^{3} + A b^{3} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 73, normalized size = 1.04 \[ \frac {B \,c^{3} x^{9}}{9}+\frac {\left (A \,c^{3}+3 B b \,c^{2}\right ) x^{7}}{7}+A \,b^{3} x +\frac {\left (3 A b \,c^{2}+3 B c \,b^{2}\right ) x^{5}}{5}+\frac {\left (3 A c \,b^{2}+B \,b^{3}\right ) x^{3}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.28, size = 70, normalized size = 1.00 \[ \frac {1}{9} \, B c^{3} x^{9} + \frac {1}{7} \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{7} + \frac {3}{5} \, {\left (B b^{2} c + A b c^{2}\right )} x^{5} + A b^{3} x + \frac {1}{3} \, {\left (B b^{3} + 3 \, A b^{2} c\right )} x^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 65, normalized size = 0.93 \[ x^3\,\left (\frac {B\,b^3}{3}+A\,c\,b^2\right )+x^7\,\left (\frac {A\,c^3}{7}+\frac {3\,B\,b\,c^2}{7}\right )+\frac {B\,c^3\,x^9}{9}+A\,b^3\,x+\frac {3\,b\,c\,x^5\,\left (A\,c+B\,b\right )}{5} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.09, size = 76, normalized size = 1.09 \[ A b^{3} x + \frac {B c^{3} x^{9}}{9} + x^{7} \left (\frac {A c^{3}}{7} + \frac {3 B b c^{2}}{7}\right ) + x^{5} \left (\frac {3 A b c^{2}}{5} + \frac {3 B b^{2} c}{5}\right ) + x^{3} \left (A b^{2} c + \frac {B b^{3}}{3}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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