Optimal. Leaf size=73 \[ a^6 x+2 a^5 b x^3+3 a^4 b^2 x^5+\frac {20}{7} a^3 b^3 x^7+\frac {5}{3} a^2 b^4 x^9+\frac {6}{11} a b^5 x^{11}+\frac {b^6 x^{13}}{13} \]
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Rubi [A] time = 0.03, antiderivative size = 73, 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} \[ \frac {5}{3} a^2 b^4 x^9+\frac {20}{7} a^3 b^3 x^7+3 a^4 b^2 x^5+2 a^5 b x^3+a^6 x+\frac {6}{11} a b^5 x^{11}+\frac {b^6 x^{13}}{13} \]
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 )^3 \, dx &=\frac {\int \left (a b+b^2 x^2\right )^6 \, dx}{b^6}\\ &=\frac {\int \left (a^6 b^6+6 a^5 b^7 x^2+15 a^4 b^8 x^4+20 a^3 b^9 x^6+15 a^2 b^{10} x^8+6 a b^{11} x^{10}+b^{12} x^{12}\right ) \, dx}{b^6}\\ &=a^6 x+2 a^5 b x^3+3 a^4 b^2 x^5+\frac {20}{7} a^3 b^3 x^7+\frac {5}{3} a^2 b^4 x^9+\frac {6}{11} a b^5 x^{11}+\frac {b^6 x^{13}}{13}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 73, normalized size = 1.00 \[ a^6 x+2 a^5 b x^3+3 a^4 b^2 x^5+\frac {20}{7} a^3 b^3 x^7+\frac {5}{3} a^2 b^4 x^9+\frac {6}{11} a b^5 x^{11}+\frac {b^6 x^{13}}{13} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.91, size = 65, normalized size = 0.89 \[ \frac {1}{13} x^{13} b^{6} + \frac {6}{11} x^{11} b^{5} a + \frac {5}{3} x^{9} b^{4} a^{2} + \frac {20}{7} x^{7} b^{3} a^{3} + 3 x^{5} b^{2} a^{4} + 2 x^{3} b a^{5} + x a^{6} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 65, normalized size = 0.89 \[ \frac {1}{13} \, b^{6} x^{13} + \frac {6}{11} \, a b^{5} x^{11} + \frac {5}{3} \, a^{2} b^{4} x^{9} + \frac {20}{7} \, a^{3} b^{3} x^{7} + 3 \, a^{4} b^{2} x^{5} + 2 \, a^{5} b x^{3} + a^{6} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 66, normalized size = 0.90 \[ \frac {1}{13} b^{6} x^{13}+\frac {6}{11} a \,b^{5} x^{11}+\frac {5}{3} a^{2} b^{4} x^{9}+\frac {20}{7} a^{3} b^{3} x^{7}+3 a^{4} b^{2} x^{5}+2 a^{5} b \,x^{3}+a^{6} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.34, size = 100, normalized size = 1.37 \[ \frac {1}{13} \, b^{6} x^{13} + \frac {6}{11} \, a b^{5} x^{11} + \frac {4}{3} \, a^{2} b^{4} x^{9} + \frac {8}{7} \, a^{3} b^{3} x^{7} + a^{6} x + \frac {1}{5} \, {\left (3 \, b^{2} x^{5} + 10 \, a b x^{3}\right )} a^{4} + \frac {1}{105} \, {\left (35 \, b^{4} x^{9} + 180 \, a b^{3} x^{7} + 252 \, a^{2} b^{2} x^{5}\right )} a^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 65, normalized size = 0.89 \[ a^6\,x+2\,a^5\,b\,x^3+3\,a^4\,b^2\,x^5+\frac {20\,a^3\,b^3\,x^7}{7}+\frac {5\,a^2\,b^4\,x^9}{3}+\frac {6\,a\,b^5\,x^{11}}{11}+\frac {b^6\,x^{13}}{13} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.08, size = 73, normalized size = 1.00 \[ a^{6} x + 2 a^{5} b x^{3} + 3 a^{4} b^{2} x^{5} + \frac {20 a^{3} b^{3} x^{7}}{7} + \frac {5 a^{2} b^{4} x^{9}}{3} + \frac {6 a b^{5} x^{11}}{11} + \frac {b^{6} x^{13}}{13} \]
Verification of antiderivative is not currently implemented for this CAS.
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