Optimal. Leaf size=51 \[ a^4 x+\frac {4}{3} a^3 b x^3+\frac {6}{5} a^2 b^2 x^5+\frac {4}{7} a b^3 x^7+\frac {b^4 x^9}{9} \]
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Rubi [A] time = 0.02, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, 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*}
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Mathematica [A] time = 0.00, size = 51, normalized size = 1.00 \[ a^4 x+\frac {4}{3} a^3 b x^3+\frac {6}{5} a^2 b^2 x^5+\frac {4}{7} a b^3 x^7+\frac {b^4 x^9}{9} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.71, size = 43, normalized size = 0.84 \[ \frac {1}{9} x^{9} b^{4} + \frac {4}{7} x^{7} b^{3} a + \frac {6}{5} x^{5} b^{2} a^{2} + \frac {4}{3} x^{3} b a^{3} + x a^{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 43, normalized size = 0.84 \[ \frac {1}{9} \, b^{4} x^{9} + \frac {4}{7} \, a b^{3} x^{7} + \frac {6}{5} \, a^{2} b^{2} x^{5} + \frac {4}{3} \, a^{3} b x^{3} + a^{4} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 44, normalized size = 0.86 \[ \frac {1}{9} b^{4} x^{9}+\frac {4}{7} a \,b^{3} x^{7}+\frac {6}{5} a^{2} b^{2} x^{5}+\frac {4}{3} a^{3} b \,x^{3}+a^{4} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.28, size = 55, normalized size = 1.08 \[ \frac {1}{9} \, b^{4} x^{9} + \frac {4}{7} \, a b^{3} x^{7} + \frac {4}{5} \, a^{2} b^{2} x^{5} + a^{4} x + \frac {2}{15} \, {\left (3 \, b^{2} x^{5} + 10 \, a b x^{3}\right )} a^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.02, size = 43, normalized size = 0.84 \[ a^4\,x+\frac {4\,a^3\,b\,x^3}{3}+\frac {6\,a^2\,b^2\,x^5}{5}+\frac {4\,a\,b^3\,x^7}{7}+\frac {b^4\,x^9}{9} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.08, size = 49, normalized size = 0.96 \[ a^{4} x + \frac {4 a^{3} b x^{3}}{3} + \frac {6 a^{2} b^{2} x^{5}}{5} + \frac {4 a b^{3} x^{7}}{7} + \frac {b^{4} x^{9}}{9} \]
Verification of antiderivative is not currently implemented for this CAS.
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