Optimal. Leaf size=51 \[ \frac{6}{5} a^2 c^2 x^5+\frac{4}{3} a^3 c x^3+a^4 x+\frac{4}{7} a c^3 x^7+\frac{c^4 x^9}{9} \]
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Rubi [A] time = 0.0173736, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {194} \[ \frac{6}{5} a^2 c^2 x^5+\frac{4}{3} a^3 c x^3+a^4 x+\frac{4}{7} a c^3 x^7+\frac{c^4 x^9}{9} \]
Antiderivative was successfully verified.
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Rule 194
Rubi steps
\begin{align*} \int \left (a+c x^2\right )^4 \, dx &=\int \left (a^4+4 a^3 c x^2+6 a^2 c^2 x^4+4 a c^3 x^6+c^4 x^8\right ) \, dx\\ &=a^4 x+\frac{4}{3} a^3 c x^3+\frac{6}{5} a^2 c^2 x^5+\frac{4}{7} a c^3 x^7+\frac{c^4 x^9}{9}\\ \end{align*}
Mathematica [A] time = 0.0017068, size = 51, normalized size = 1. \[ \frac{6}{5} a^2 c^2 x^5+\frac{4}{3} a^3 c x^3+a^4 x+\frac{4}{7} a c^3 x^7+\frac{c^4 x^9}{9} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.045, size = 44, normalized size = 0.9 \begin{align*}{a}^{4}x+{\frac{4\,{a}^{3}c{x}^{3}}{3}}+{\frac{6\,{a}^{2}{c}^{2}{x}^{5}}{5}}+{\frac{4\,a{c}^{3}{x}^{7}}{7}}+{\frac{{c}^{4}{x}^{9}}{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.16844, size = 58, normalized size = 1.14 \begin{align*} \frac{1}{9} \, c^{4} x^{9} + \frac{4}{7} \, a c^{3} x^{7} + \frac{6}{5} \, a^{2} c^{2} x^{5} + \frac{4}{3} \, a^{3} c x^{3} + a^{4} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.79691, size = 96, normalized size = 1.88 \begin{align*} \frac{1}{9} x^{9} c^{4} + \frac{4}{7} x^{7} c^{3} a + \frac{6}{5} x^{5} c^{2} a^{2} + \frac{4}{3} x^{3} c a^{3} + x a^{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.074117, size = 49, normalized size = 0.96 \begin{align*} a^{4} x + \frac{4 a^{3} c x^{3}}{3} + \frac{6 a^{2} c^{2} x^{5}}{5} + \frac{4 a c^{3} x^{7}}{7} + \frac{c^{4} x^{9}}{9} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.23238, size = 58, normalized size = 1.14 \begin{align*} \frac{1}{9} \, c^{4} x^{9} + \frac{4}{7} \, a c^{3} x^{7} + \frac{6}{5} \, a^{2} c^{2} x^{5} + \frac{4}{3} \, a^{3} c x^{3} + a^{4} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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