Optimal. Leaf size=50 \[ A b^2 x+\frac {1}{5} c x^5 (A c+2 b B)+\frac {1}{3} b x^3 (2 A c+b B)+\frac {1}{7} B c^2 x^7 \]
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Rubi [A] time = 0.03, antiderivative size = 50, 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} \[ A b^2 x+\frac {1}{5} c x^5 (A c+2 b B)+\frac {1}{3} b x^3 (2 A c+b B)+\frac {1}{7} B c^2 x^7 \]
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 )^2}{x^4} \, dx &=\int \left (A+B x^2\right ) \left (b+c x^2\right )^2 \, dx\\ &=\int \left (A b^2+b (b B+2 A c) x^2+c (2 b B+A c) x^4+B c^2 x^6\right ) \, dx\\ &=A b^2 x+\frac {1}{3} b (b B+2 A c) x^3+\frac {1}{5} c (2 b B+A c) x^5+\frac {1}{7} B c^2 x^7\\ \end {align*}
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Mathematica [A] time = 0.01, size = 50, normalized size = 1.00 \[ A b^2 x+\frac {1}{5} c x^5 (A c+2 b B)+\frac {1}{3} b x^3 (2 A c+b B)+\frac {1}{7} B c^2 x^7 \]
Antiderivative was successfully verified.
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fricas [A] time = 0.92, size = 48, normalized size = 0.96 \[ \frac {1}{7} \, B c^{2} x^{7} + \frac {1}{5} \, {\left (2 \, B b c + A c^{2}\right )} x^{5} + A b^{2} x + \frac {1}{3} \, {\left (B b^{2} + 2 \, A b c\right )} x^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 50, normalized size = 1.00 \[ \frac {1}{7} \, B c^{2} x^{7} + \frac {2}{5} \, B b c x^{5} + \frac {1}{5} \, A c^{2} x^{5} + \frac {1}{3} \, B b^{2} x^{3} + \frac {2}{3} \, A b c x^{3} + A b^{2} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 49, normalized size = 0.98 \[ \frac {B \,c^{2} x^{7}}{7}+\frac {\left (A \,c^{2}+2 b B c \right ) x^{5}}{5}+A \,b^{2} x +\frac {\left (2 A b c +B \,b^{2}\right ) x^{3}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.24, size = 48, normalized size = 0.96 \[ \frac {1}{7} \, B c^{2} x^{7} + \frac {1}{5} \, {\left (2 \, B b c + A c^{2}\right )} x^{5} + A b^{2} x + \frac {1}{3} \, {\left (B b^{2} + 2 \, A b c\right )} x^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 48, normalized size = 0.96 \[ x^3\,\left (\frac {B\,b^2}{3}+\frac {2\,A\,c\,b}{3}\right )+x^5\,\left (\frac {A\,c^2}{5}+\frac {2\,B\,b\,c}{5}\right )+\frac {B\,c^2\,x^7}{7}+A\,b^2\,x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.08, size = 53, normalized size = 1.06 \[ A b^{2} x + \frac {B c^{2} x^{7}}{7} + x^{5} \left (\frac {A c^{2}}{5} + \frac {2 B b c}{5}\right ) + x^{3} \left (\frac {2 A b c}{3} + \frac {B b^{2}}{3}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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