Optimal. Leaf size=67 \[ a^2 A x+\frac {1}{5} c x^5 (2 a C+A c)+\frac {1}{3} a x^3 (a C+2 A c)+\frac {B \left (a+c x^2\right )^3}{6 c}+\frac {1}{7} c^2 C x^7 \]
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Rubi [A] time = 0.04, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {1582, 373} \[ a^2 A x+\frac {1}{5} c x^5 (2 a C+A c)+\frac {1}{3} a x^3 (a C+2 A c)+\frac {B \left (a+c x^2\right )^3}{6 c}+\frac {1}{7} c^2 C x^7 \]
Antiderivative was successfully verified.
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Rule 373
Rule 1582
Rubi steps
\begin {align*} \int \left (a+c x^2\right )^2 \left (A+B x+C x^2\right ) \, dx &=\frac {B \left (a+c x^2\right )^3}{6 c}+\int \left (a+c x^2\right )^2 \left (A+C x^2\right ) \, dx\\ &=\frac {B \left (a+c x^2\right )^3}{6 c}+\int \left (a^2 A+a (2 A c+a C) x^2+c (A c+2 a C) x^4+c^2 C x^6\right ) \, dx\\ &=a^2 A x+\frac {1}{3} a (2 A c+a C) x^3+\frac {1}{5} c (A c+2 a C) x^5+\frac {1}{7} c^2 C x^7+\frac {B \left (a+c x^2\right )^3}{6 c}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 69, normalized size = 1.03 \[ \frac {1}{210} x \left (35 a^2 (6 A+x (3 B+2 C x))+7 a c x^2 (20 A+3 x (5 B+4 C x))+c^2 x^4 (42 A+5 x (7 B+6 C x))\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.90, size = 76, normalized size = 1.13 \[ \frac {1}{7} x^{7} c^{2} C + \frac {1}{6} x^{6} c^{2} B + \frac {2}{5} x^{5} c a C + \frac {1}{5} x^{5} c^{2} A + \frac {1}{2} x^{4} c a B + \frac {1}{3} x^{3} a^{2} C + \frac {2}{3} x^{3} c a A + \frac {1}{2} x^{2} a^{2} B + x a^{2} A \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 76, normalized size = 1.13 \[ \frac {1}{7} \, C c^{2} x^{7} + \frac {1}{6} \, B c^{2} x^{6} + \frac {2}{5} \, C a c x^{5} + \frac {1}{5} \, A c^{2} x^{5} + \frac {1}{2} \, B a c x^{4} + \frac {1}{3} \, C a^{2} x^{3} + \frac {2}{3} \, A a c x^{3} + \frac {1}{2} \, B a^{2} x^{2} + A a^{2} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 75, normalized size = 1.12 \[ \frac {C \,c^{2} x^{7}}{7}+\frac {B \,c^{2} x^{6}}{6}+\frac {B a c \,x^{4}}{2}+\frac {B \,a^{2} x^{2}}{2}+\frac {\left (A \,c^{2}+2 a c C \right ) x^{5}}{5}+A \,a^{2} x +\frac {\left (2 a c A +a^{2} C \right ) x^{3}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 74, normalized size = 1.10 \[ \frac {1}{7} \, C c^{2} x^{7} + \frac {1}{6} \, B c^{2} x^{6} + \frac {1}{2} \, B a c x^{4} + \frac {1}{5} \, {\left (2 \, C a c + A c^{2}\right )} x^{5} + \frac {1}{2} \, B a^{2} x^{2} + A a^{2} x + \frac {1}{3} \, {\left (C a^{2} + 2 \, A a c\right )} x^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 74, normalized size = 1.10 \[ x^3\,\left (\frac {C\,a^2}{3}+\frac {2\,A\,c\,a}{3}\right )+x^5\,\left (\frac {A\,c^2}{5}+\frac {2\,C\,a\,c}{5}\right )+\frac {B\,a^2\,x^2}{2}+\frac {B\,c^2\,x^6}{6}+\frac {C\,c^2\,x^7}{7}+A\,a^2\,x+\frac {B\,a\,c\,x^4}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.08, size = 83, normalized size = 1.24 \[ A a^{2} x + \frac {B a^{2} x^{2}}{2} + \frac {B a c x^{4}}{2} + \frac {B c^{2} x^{6}}{6} + \frac {C c^{2} x^{7}}{7} + x^{5} \left (\frac {A c^{2}}{5} + \frac {2 C a c}{5}\right ) + x^{3} \left (\frac {2 A a c}{3} + \frac {C a^{2}}{3}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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