Optimal. Leaf size=56 \[ a^2 A c x^3+a^3 A x+\frac{3}{5} a A c^2 x^5+\frac{B \left (a+c x^2\right )^4}{8 c}+\frac{1}{7} A c^3 x^7 \]
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Rubi [A] time = 0.0221575, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {641, 194} \[ a^2 A c x^3+a^3 A x+\frac{3}{5} a A c^2 x^5+\frac{B \left (a+c x^2\right )^4}{8 c}+\frac{1}{7} A c^3 x^7 \]
Antiderivative was successfully verified.
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Rule 641
Rule 194
Rubi steps
\begin{align*} \int (A+B x) \left (a+c x^2\right )^3 \, dx &=\frac{B \left (a+c x^2\right )^4}{8 c}+A \int \left (a+c x^2\right )^3 \, dx\\ &=\frac{B \left (a+c x^2\right )^4}{8 c}+A \int \left (a^3+3 a^2 c x^2+3 a c^2 x^4+c^3 x^6\right ) \, dx\\ &=a^3 A x+a^2 A c x^3+\frac{3}{5} a A c^2 x^5+\frac{1}{7} A c^3 x^7+\frac{B \left (a+c x^2\right )^4}{8 c}\\ \end{align*}
Mathematica [A] time = 0.0024509, size = 85, normalized size = 1.52 \[ a^2 A c x^3+a^3 A x+\frac{3}{4} a^2 B c x^4+\frac{1}{2} a^3 B x^2+\frac{3}{5} a A c^2 x^5+\frac{1}{2} a B c^2 x^6+\frac{1}{7} A c^3 x^7+\frac{1}{8} B c^3 x^8 \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 74, normalized size = 1.3 \begin{align*}{\frac{B{c}^{3}{x}^{8}}{8}}+{\frac{A{c}^{3}{x}^{7}}{7}}+{\frac{aB{c}^{2}{x}^{6}}{2}}+{\frac{3\,aA{c}^{2}{x}^{5}}{5}}+{\frac{3\,{a}^{2}Bc{x}^{4}}{4}}+{a}^{2}Ac{x}^{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 = 0.998353, size = 99, normalized size = 1.77 \begin{align*} \frac{1}{8} \, B c^{3} x^{8} + \frac{1}{7} \, A c^{3} x^{7} + \frac{1}{2} \, B a c^{2} x^{6} + \frac{3}{5} \, A a c^{2} x^{5} + \frac{3}{4} \, B a^{2} c x^{4} + 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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Fricas [A] time = 1.58575, size = 169, normalized size = 3.02 \begin{align*} \frac{1}{8} x^{8} c^{3} B + \frac{1}{7} x^{7} c^{3} A + \frac{1}{2} x^{6} c^{2} a B + \frac{3}{5} x^{5} c^{2} a A + \frac{3}{4} x^{4} c a^{2} B + 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 = 0.097281, size = 85, normalized size = 1.52 \begin{align*} A a^{3} x + A a^{2} c x^{3} + \frac{3 A a c^{2} x^{5}}{5} + \frac{A c^{3} x^{7}}{7} + \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16705, size = 99, normalized size = 1.77 \begin{align*} \frac{1}{8} \, B c^{3} x^{8} + \frac{1}{7} \, A c^{3} x^{7} + \frac{1}{2} \, B a c^{2} x^{6} + \frac{3}{5} \, A a c^{2} x^{5} + \frac{3}{4} \, B a^{2} c x^{4} + 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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