Optimal. Leaf size=73 \[ \frac{6}{5} a^2 A c^2 x^5+\frac{4}{3} a^3 A c x^3+a^4 A x+\frac{4}{7} a A c^3 x^7+\frac{B \left (a+c x^2\right )^5}{10 c}+\frac{1}{9} A c^4 x^9 \]
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Rubi [A] time = 0.0250958, antiderivative size = 73, 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} \[ \frac{6}{5} a^2 A c^2 x^5+\frac{4}{3} a^3 A c x^3+a^4 A x+\frac{4}{7} a A c^3 x^7+\frac{B \left (a+c x^2\right )^5}{10 c}+\frac{1}{9} A c^4 x^9 \]
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 )^4 \, dx &=\frac{B \left (a+c x^2\right )^5}{10 c}+A \int \left (a+c x^2\right )^4 \, dx\\ &=\frac{B \left (a+c x^2\right )^5}{10 c}+A \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 A x+\frac{4}{3} a^3 A c x^3+\frac{6}{5} a^2 A c^2 x^5+\frac{4}{7} a A c^3 x^7+\frac{1}{9} A c^4 x^9+\frac{B \left (a+c x^2\right )^5}{10 c}\\ \end{align*}
Mathematica [A] time = 0.0029984, size = 110, normalized size = 1.51 \[ \frac{6}{5} a^2 A c^2 x^5+\frac{4}{3} a^3 A c x^3+a^4 A x+a^2 B c^2 x^6+a^3 B c x^4+\frac{1}{2} a^4 B x^2+\frac{4}{7} a A c^3 x^7+\frac{1}{2} a B c^3 x^8+\frac{1}{9} A c^4 x^9+\frac{1}{10} B c^4 x^{10} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 97, normalized size = 1.3 \begin{align*}{\frac{B{c}^{4}{x}^{10}}{10}}+{\frac{A{c}^{4}{x}^{9}}{9}}+{\frac{aB{c}^{3}{x}^{8}}{2}}+{\frac{4\,aA{c}^{3}{x}^{7}}{7}}+{a}^{2}B{c}^{2}{x}^{6}+{\frac{6\,{a}^{2}A{c}^{2}{x}^{5}}{5}}+{a}^{3}Bc{x}^{4}+{\frac{4\,{a}^{3}Ac{x}^{3}}{3}}+{\frac{{a}^{4}B{x}^{2}}{2}}+{a}^{4}Ax \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.03723, size = 130, normalized size = 1.78 \begin{align*} \frac{1}{10} \, B c^{4} x^{10} + \frac{1}{9} \, A c^{4} x^{9} + \frac{1}{2} \, B a c^{3} x^{8} + \frac{4}{7} \, A a c^{3} x^{7} + B a^{2} c^{2} x^{6} + \frac{6}{5} \, A a^{2} c^{2} x^{5} + B a^{3} c x^{4} + \frac{4}{3} \, A a^{3} c x^{3} + \frac{1}{2} \, B a^{4} x^{2} + A a^{4} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.2478, size = 220, normalized size = 3.01 \begin{align*} \frac{1}{10} x^{10} c^{4} B + \frac{1}{9} x^{9} c^{4} A + \frac{1}{2} x^{8} c^{3} a B + \frac{4}{7} x^{7} c^{3} a A + x^{6} c^{2} a^{2} B + \frac{6}{5} x^{5} c^{2} a^{2} A + x^{4} c a^{3} B + \frac{4}{3} x^{3} c a^{3} A + \frac{1}{2} x^{2} a^{4} B + x a^{4} A \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.118268, size = 112, normalized size = 1.53 \begin{align*} A a^{4} x + \frac{4 A a^{3} c x^{3}}{3} + \frac{6 A a^{2} c^{2} x^{5}}{5} + \frac{4 A a c^{3} x^{7}}{7} + \frac{A c^{4} x^{9}}{9} + \frac{B a^{4} x^{2}}{2} + B a^{3} c x^{4} + B a^{2} c^{2} x^{6} + \frac{B a c^{3} x^{8}}{2} + \frac{B c^{4} x^{10}}{10} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12445, size = 130, normalized size = 1.78 \begin{align*} \frac{1}{10} \, B c^{4} x^{10} + \frac{1}{9} \, A c^{4} x^{9} + \frac{1}{2} \, B a c^{3} x^{8} + \frac{4}{7} \, A a c^{3} x^{7} + B a^{2} c^{2} x^{6} + \frac{6}{5} \, A a^{2} c^{2} x^{5} + B a^{3} c x^{4} + \frac{4}{3} \, A a^{3} c x^{3} + \frac{1}{2} \, B a^{4} x^{2} + A a^{4} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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