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 = 0.040093, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, 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 1582
Rule 373
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*}
Mathematica [A] time = 0.0289469, 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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Maple [A] time = 0.046, size = 75, normalized size = 1.1 \begin{align*}{\frac{{c}^{2}C{x}^{7}}{7}}+{\frac{B{c}^{2}{x}^{6}}{6}}+{\frac{ \left ( A{c}^{2}+2\,acC \right ){x}^{5}}{5}}+{\frac{aBc{x}^{4}}{2}}+{\frac{ \left ( 2\,aAc+{a}^{2}C \right ){x}^{3}}{3}}+{\frac{{a}^{2}B{x}^{2}}{2}}+{a}^{2}Ax \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01937, size = 100, normalized size = 1.49 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.50887, size = 185, normalized size = 2.76 \begin{align*} \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 \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.077304, size = 83, normalized size = 1.24 \begin{align*} 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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15864, size = 103, normalized size = 1.54 \begin{align*} \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 \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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