Optimal. Leaf size=55 \[ \frac{1}{5} a^2 A x^5+\frac{1}{7} b x^7 (2 a B+A b)+\frac{1}{6} a x^6 (a B+2 A b)+\frac{1}{8} b^2 B x^8 \]
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Rubi [A] time = 0.0483102, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, Rules used = {27, 76} \[ \frac{1}{5} a^2 A x^5+\frac{1}{7} b x^7 (2 a B+A b)+\frac{1}{6} a x^6 (a B+2 A b)+\frac{1}{8} b^2 B x^8 \]
Antiderivative was successfully verified.
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Rule 27
Rule 76
Rubi steps
\begin{align*} \int x^4 (A+B x) \left (a^2+2 a b x+b^2 x^2\right ) \, dx &=\int x^4 (a+b x)^2 (A+B x) \, dx\\ &=\int \left (a^2 A x^4+a (2 A b+a B) x^5+b (A b+2 a B) x^6+b^2 B x^7\right ) \, dx\\ &=\frac{1}{5} a^2 A x^5+\frac{1}{6} a (2 A b+a B) x^6+\frac{1}{7} b (A b+2 a B) x^7+\frac{1}{8} b^2 B x^8\\ \end{align*}
Mathematica [A] time = 0.0079606, size = 55, normalized size = 1. \[ \frac{1}{5} a^2 A x^5+\frac{1}{7} b x^7 (2 a B+A b)+\frac{1}{6} a x^6 (a B+2 A b)+\frac{1}{8} b^2 B x^8 \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 52, normalized size = 1. \begin{align*}{\frac{{b}^{2}B{x}^{8}}{8}}+{\frac{ \left ( A{b}^{2}+2\,abB \right ){x}^{7}}{7}}+{\frac{ \left ( 2\,Aab+B{a}^{2} \right ){x}^{6}}{6}}+{\frac{{a}^{2}A{x}^{5}}{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01765, size = 69, normalized size = 1.25 \begin{align*} \frac{1}{8} \, B b^{2} x^{8} + \frac{1}{5} \, A a^{2} x^{5} + \frac{1}{7} \,{\left (2 \, B a b + A b^{2}\right )} x^{7} + \frac{1}{6} \,{\left (B a^{2} + 2 \, A a b\right )} x^{6} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.07104, size = 128, normalized size = 2.33 \begin{align*} \frac{1}{8} x^{8} b^{2} B + \frac{2}{7} x^{7} b a B + \frac{1}{7} x^{7} b^{2} A + \frac{1}{6} x^{6} a^{2} B + \frac{1}{3} x^{6} b a A + \frac{1}{5} x^{5} a^{2} A \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.085482, size = 54, normalized size = 0.98 \begin{align*} \frac{A a^{2} x^{5}}{5} + \frac{B b^{2} x^{8}}{8} + x^{7} \left (\frac{A b^{2}}{7} + \frac{2 B a b}{7}\right ) + x^{6} \left (\frac{A a b}{3} + \frac{B a^{2}}{6}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1157, size = 72, normalized size = 1.31 \begin{align*} \frac{1}{8} \, B b^{2} x^{8} + \frac{2}{7} \, B a b x^{7} + \frac{1}{7} \, A b^{2} x^{7} + \frac{1}{6} \, B a^{2} x^{6} + \frac{1}{3} \, A a b x^{6} + \frac{1}{5} \, A a^{2} x^{5} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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