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 = 0.01, 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, 200}
\begin {gather*} 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 {gather*}
Antiderivative was successfully verified.
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Rule 28
Rule 200
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 \begin {gather*} 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 {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.01, size = 44, normalized size = 0.86
method | result | size |
default | \(a^{4} x +\frac {4}{3} a^{3} b \,x^{3}+\frac {6}{5} a^{2} b^{2} x^{5}+\frac {4}{7} a \,b^{3} x^{7}+\frac {1}{9} b^{4} x^{9}\) | \(44\) |
norman | \(a^{4} x +\frac {4}{3} a^{3} b \,x^{3}+\frac {6}{5} a^{2} b^{2} x^{5}+\frac {4}{7} a \,b^{3} x^{7}+\frac {1}{9} b^{4} x^{9}\) | \(44\) |
risch | \(a^{4} x +\frac {4}{3} a^{3} b \,x^{3}+\frac {6}{5} a^{2} b^{2} x^{5}+\frac {4}{7} a \,b^{3} x^{7}+\frac {1}{9} b^{4} x^{9}\) | \(44\) |
gosper | \(\frac {x \left (35 b^{4} x^{8}+180 a \,b^{3} x^{6}+378 a^{2} b^{2} x^{4}+420 a^{3} b \,x^{2}+315 a^{4}\right )}{315}\) | \(47\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 55, normalized size = 1.08 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.33, size = 43, normalized size = 0.84 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.01, size = 49, normalized size = 0.96 \begin {gather*} 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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 5.41, size = 43, normalized size = 0.84 \begin {gather*} \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 {gather*}
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 \begin {gather*} 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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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