Optimal. Leaf size=51 \[ \frac{6}{5} a^2 b^2 x^5+\frac{4}{3} a^3 b x^3+a^4 x+\frac{4}{7} a b^3 x^7+\frac{b^4 x^9}{9} \]
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Rubi [A] time = 0.0238017, antiderivative size = 51, 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 = {28, 194} \[ \frac{6}{5} a^2 b^2 x^5+\frac{4}{3} a^3 b x^3+a^4 x+\frac{4}{7} a b^3 x^7+\frac{b^4 x^9}{9} \]
Antiderivative was successfully verified.
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Rule 28
Rule 194
Rubi steps
\begin{align*} \int \left (a^2+2 a b x^2+b^2 x^4\right )^2 \, dx &=\frac{\int \left (a b+b^2 x^2\right )^4 \, dx}{b^4}\\ &=\frac{\int \left (a^4 b^4+4 a^3 b^5 x^2+6 a^2 b^6 x^4+4 a b^7 x^6+b^8 x^8\right ) \, dx}{b^4}\\ &=a^4 x+\frac{4}{3} a^3 b x^3+\frac{6}{5} a^2 b^2 x^5+\frac{4}{7} a b^3 x^7+\frac{b^4 x^9}{9}\\ \end{align*}
Mathematica [A] time = 0.0012044, size = 51, normalized size = 1. \[ \frac{6}{5} a^2 b^2 x^5+\frac{4}{3} a^3 b x^3+a^4 x+\frac{4}{7} a b^3 x^7+\frac{b^4 x^9}{9} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.04, size = 44, normalized size = 0.9 \begin{align*}{a}^{4}x+{\frac{4\,{a}^{3}b{x}^{3}}{3}}+{\frac{6\,{a}^{2}{b}^{2}{x}^{5}}{5}}+{\frac{4\,a{b}^{3}{x}^{7}}{7}}+{\frac{{b}^{4}{x}^{9}}{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.976963, size = 74, normalized size = 1.45 \begin{align*} \frac{1}{9} \, b^{4} x^{9} + \frac{4}{7} \, a b^{3} x^{7} + \frac{4}{5} \, a^{2} b^{2} x^{5} + a^{4} x + \frac{2}{15} \,{\left (3 \, b^{2} x^{5} + 10 \, a b x^{3}\right )} a^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.4885, size = 96, normalized size = 1.88 \begin{align*} \frac{1}{9} x^{9} b^{4} + \frac{4}{7} x^{7} b^{3} a + \frac{6}{5} x^{5} b^{2} a^{2} + \frac{4}{3} x^{3} b a^{3} + x a^{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.068264, size = 49, normalized size = 0.96 \begin{align*} a^{4} x + \frac{4 a^{3} b x^{3}}{3} + \frac{6 a^{2} b^{2} x^{5}}{5} + \frac{4 a b^{3} x^{7}}{7} + \frac{b^{4} x^{9}}{9} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16168, size = 58, normalized size = 1.14 \begin{align*} \frac{1}{9} \, b^{4} x^{9} + \frac{4}{7} \, a b^{3} x^{7} + \frac{6}{5} \, a^{2} b^{2} x^{5} + \frac{4}{3} \, a^{3} b x^{3} + a^{4} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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