Optimal. Leaf size=87 \[ \frac{1}{3} a^2 x^3 (a C+3 A c)+a^3 A x+\frac{1}{7} c^2 x^7 (3 a C+A c)+\frac{3}{5} a c x^5 (a C+A c)+\frac{B \left (a+c x^2\right )^4}{8 c}+\frac{1}{9} c^3 C x^9 \]
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Rubi [A] time = 0.0578149, antiderivative size = 87, 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} \[ \frac{1}{3} a^2 x^3 (a C+3 A c)+a^3 A x+\frac{1}{7} c^2 x^7 (3 a C+A c)+\frac{3}{5} a c x^5 (a C+A c)+\frac{B \left (a+c x^2\right )^4}{8 c}+\frac{1}{9} c^3 C x^9 \]
Antiderivative was successfully verified.
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Rule 1582
Rule 373
Rubi steps
\begin{align*} \int \left (a+c x^2\right )^3 \left (A+B x+C x^2\right ) \, dx &=\frac{B \left (a+c x^2\right )^4}{8 c}+\int \left (a+c x^2\right )^3 \left (A+C x^2\right ) \, dx\\ &=\frac{B \left (a+c x^2\right )^4}{8 c}+\int \left (a^3 A+a^2 (3 A c+a C) x^2+3 a c (A c+a C) x^4+c^2 (A c+3 a C) x^6+c^3 C x^8\right ) \, dx\\ &=a^3 A x+\frac{1}{3} a^2 (3 A c+a C) x^3+\frac{3}{5} a c (A c+a C) x^5+\frac{1}{7} c^2 (A c+3 a C) x^7+\frac{1}{9} c^3 C x^9+\frac{B \left (a+c x^2\right )^4}{8 c}\\ \end{align*}
Mathematica [A] time = 0.0318585, size = 100, normalized size = 1.15 \[ \frac{1}{20} a^2 c x^3 (20 A+3 x (5 B+4 C x))+\frac{1}{6} a^3 x (6 A+x (3 B+2 C x))+\frac{1}{70} a c^2 x^5 (42 A+5 x (7 B+6 C x))+\frac{1}{504} c^3 x^7 (72 A+7 x (9 B+8 C x)) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.044, size = 111, normalized size = 1.3 \begin{align*}{\frac{{c}^{3}C{x}^{9}}{9}}+{\frac{{c}^{3}B{x}^{8}}{8}}+{\frac{ \left ( A{c}^{3}+3\,a{c}^{2}C \right ){x}^{7}}{7}}+{\frac{a{c}^{2}B{x}^{6}}{2}}+{\frac{ \left ( 3\,aA{c}^{2}+3\,{a}^{2}cC \right ){x}^{5}}{5}}+{\frac{3\,B{a}^{2}c{x}^{4}}{4}}+{\frac{ \left ( 3\,{a}^{2}Ac+{a}^{3}C \right ){x}^{3}}{3}}+{\frac{{a}^{3}B{x}^{2}}{2}}+{a}^{3}Ax \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.22697, size = 146, normalized size = 1.68 \begin{align*} \frac{1}{9} \, C c^{3} x^{9} + \frac{1}{8} \, B c^{3} x^{8} + \frac{1}{2} \, B a c^{2} x^{6} + \frac{3}{4} \, B a^{2} c x^{4} + \frac{1}{7} \,{\left (3 \, C a c^{2} + A c^{3}\right )} x^{7} + \frac{1}{2} \, B a^{3} x^{2} + \frac{3}{5} \,{\left (C a^{2} c + A a c^{2}\right )} x^{5} + A a^{3} x + \frac{1}{3} \,{\left (C a^{3} + 3 \, A a^{2} c\right )} x^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.50704, size = 261, normalized size = 3. \begin{align*} \frac{1}{9} x^{9} c^{3} C + \frac{1}{8} x^{8} c^{3} B + \frac{3}{7} x^{7} c^{2} a C + \frac{1}{7} x^{7} c^{3} A + \frac{1}{2} x^{6} c^{2} a B + \frac{3}{5} x^{5} c a^{2} C + \frac{3}{5} x^{5} c^{2} a A + \frac{3}{4} x^{4} c a^{2} B + \frac{1}{3} x^{3} a^{3} C + x^{3} c a^{2} A + \frac{1}{2} x^{2} a^{3} B + x a^{3} A \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.079793, size = 122, normalized size = 1.4 \begin{align*} A a^{3} x + \frac{B a^{3} x^{2}}{2} + \frac{3 B a^{2} c x^{4}}{4} + \frac{B a c^{2} x^{6}}{2} + \frac{B c^{3} x^{8}}{8} + \frac{C c^{3} x^{9}}{9} + x^{7} \left (\frac{A c^{3}}{7} + \frac{3 C a c^{2}}{7}\right ) + x^{5} \left (\frac{3 A a c^{2}}{5} + \frac{3 C a^{2} c}{5}\right ) + x^{3} \left (A a^{2} c + \frac{C a^{3}}{3}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14382, size = 150, normalized size = 1.72 \begin{align*} \frac{1}{9} \, C c^{3} x^{9} + \frac{1}{8} \, B c^{3} x^{8} + \frac{3}{7} \, C a c^{2} x^{7} + \frac{1}{7} \, A c^{3} x^{7} + \frac{1}{2} \, B a c^{2} x^{6} + \frac{3}{5} \, C a^{2} c x^{5} + \frac{3}{5} \, A a c^{2} x^{5} + \frac{3}{4} \, B a^{2} c x^{4} + \frac{1}{3} \, C a^{3} x^{3} + A a^{2} c x^{3} + \frac{1}{2} \, B a^{3} x^{2} + A a^{3} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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