Optimal. Leaf size=75 \[ \frac {1}{5} A b^3 x^5+\frac {1}{7} b^2 x^7 (3 A c+b B)+\frac {1}{11} c^2 x^{11} (A c+3 b B)+\frac {1}{3} b c x^9 (A c+b B)+\frac {1}{13} B c^3 x^{13} \]
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Rubi [A] time = 0.06, antiderivative size = 75, 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} \[ \frac {1}{7} b^2 x^7 (3 A c+b B)+\frac {1}{5} A b^3 x^5+\frac {1}{11} c^2 x^{11} (A c+3 b B)+\frac {1}{3} b c x^9 (A c+b B)+\frac {1}{13} B c^3 x^{13} \]
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^2} \, dx &=\int x^4 \left (A+B x^2\right ) \left (b+c x^2\right )^3 \, dx\\ &=\int \left (A b^3 x^4+b^2 (b B+3 A c) x^6+3 b c (b B+A c) x^8+c^2 (3 b B+A c) x^{10}+B c^3 x^{12}\right ) \, dx\\ &=\frac {1}{5} A b^3 x^5+\frac {1}{7} b^2 (b B+3 A c) x^7+\frac {1}{3} b c (b B+A c) x^9+\frac {1}{11} c^2 (3 b B+A c) x^{11}+\frac {1}{13} B c^3 x^{13}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 75, normalized size = 1.00 \[ \frac {1}{5} A b^3 x^5+\frac {1}{7} b^2 x^7 (3 A c+b B)+\frac {1}{11} c^2 x^{11} (A c+3 b B)+\frac {1}{3} b c x^9 (A c+b B)+\frac {1}{13} B c^3 x^{13} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.77, size = 73, normalized size = 0.97 \[ \frac {1}{13} \, B c^{3} x^{13} + \frac {1}{11} \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{11} + \frac {1}{3} \, {\left (B b^{2} c + A b c^{2}\right )} x^{9} + \frac {1}{5} \, A b^{3} x^{5} + \frac {1}{7} \, {\left (B b^{3} + 3 \, A b^{2} c\right )} x^{7} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 77, normalized size = 1.03 \[ \frac {1}{13} \, B c^{3} x^{13} + \frac {3}{11} \, B b c^{2} x^{11} + \frac {1}{11} \, A c^{3} x^{11} + \frac {1}{3} \, B b^{2} c x^{9} + \frac {1}{3} \, A b c^{2} x^{9} + \frac {1}{7} \, B b^{3} x^{7} + \frac {3}{7} \, A b^{2} c x^{7} + \frac {1}{5} \, A b^{3} x^{5} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 76, normalized size = 1.01 \[ \frac {B \,c^{3} x^{13}}{13}+\frac {\left (A \,c^{3}+3 B b \,c^{2}\right ) x^{11}}{11}+\frac {A \,b^{3} x^{5}}{5}+\frac {\left (3 A b \,c^{2}+3 B c \,b^{2}\right ) x^{9}}{9}+\frac {\left (3 A c \,b^{2}+B \,b^{3}\right ) x^{7}}{7} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.34, size = 73, normalized size = 0.97 \[ \frac {1}{13} \, B c^{3} x^{13} + \frac {1}{11} \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{11} + \frac {1}{3} \, {\left (B b^{2} c + A b c^{2}\right )} x^{9} + \frac {1}{5} \, A b^{3} x^{5} + \frac {1}{7} \, {\left (B b^{3} + 3 \, A b^{2} c\right )} x^{7} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 69, normalized size = 0.92 \[ x^7\,\left (\frac {B\,b^3}{7}+\frac {3\,A\,c\,b^2}{7}\right )+x^{11}\,\left (\frac {A\,c^3}{11}+\frac {3\,B\,b\,c^2}{11}\right )+\frac {A\,b^3\,x^5}{5}+\frac {B\,c^3\,x^{13}}{13}+\frac {b\,c\,x^9\,\left (A\,c+B\,b\right )}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.08, size = 80, normalized size = 1.07 \[ \frac {A b^{3} x^{5}}{5} + \frac {B c^{3} x^{13}}{13} + x^{11} \left (\frac {A c^{3}}{11} + \frac {3 B b c^{2}}{11}\right ) + x^{9} \left (\frac {A b c^{2}}{3} + \frac {B b^{2} c}{3}\right ) + x^{7} \left (\frac {3 A b^{2} c}{7} + \frac {B b^{3}}{7}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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