Optimal. Leaf size=50 \[ -\frac {a^4}{5 x^5}-\frac {4 a^3 b}{3 x^3}-\frac {6 a^2 b^2}{x}+4 a b^3 x+\frac {b^4 x^3}{3} \]
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Rubi [A] time = 0.03, antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {28, 270} \[ -\frac {6 a^2 b^2}{x}-\frac {4 a^3 b}{3 x^3}-\frac {a^4}{5 x^5}+4 a b^3 x+\frac {b^4 x^3}{3} \]
Antiderivative was successfully verified.
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Rule 28
Rule 270
Rubi steps
\begin {align*} \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^2}{x^6} \, dx &=\frac {\int \frac {\left (a b+b^2 x^2\right )^4}{x^6} \, dx}{b^4}\\ &=\frac {\int \left (4 a b^7+\frac {a^4 b^4}{x^6}+\frac {4 a^3 b^5}{x^4}+\frac {6 a^2 b^6}{x^2}+b^8 x^2\right ) \, dx}{b^4}\\ &=-\frac {a^4}{5 x^5}-\frac {4 a^3 b}{3 x^3}-\frac {6 a^2 b^2}{x}+4 a b^3 x+\frac {b^4 x^3}{3}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 50, normalized size = 1.00 \[ -\frac {a^4}{5 x^5}-\frac {4 a^3 b}{3 x^3}-\frac {6 a^2 b^2}{x}+4 a b^3 x+\frac {b^4 x^3}{3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.75, size = 48, normalized size = 0.96 \[ \frac {5 \, b^{4} x^{8} + 60 \, a b^{3} x^{6} - 90 \, a^{2} b^{2} x^{4} - 20 \, a^{3} b x^{2} - 3 \, a^{4}}{15 \, x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 47, normalized size = 0.94 \[ \frac {1}{3} \, b^{4} x^{3} + 4 \, a b^{3} x - \frac {90 \, a^{2} b^{2} x^{4} + 20 \, a^{3} b x^{2} + 3 \, a^{4}}{15 \, x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 45, normalized size = 0.90 \[ \frac {b^{4} x^{3}}{3}+4 a \,b^{3} x -\frac {6 a^{2} b^{2}}{x}-\frac {4 a^{3} b}{3 x^{3}}-\frac {a^{4}}{5 x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.37, size = 47, normalized size = 0.94 \[ \frac {1}{3} \, b^{4} x^{3} + 4 \, a b^{3} x - \frac {90 \, a^{2} b^{2} x^{4} + 20 \, a^{3} b x^{2} + 3 \, a^{4}}{15 \, x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 47, normalized size = 0.94 \[ \frac {b^4\,x^3}{3}-\frac {\frac {a^4}{5}+\frac {4\,a^3\,b\,x^2}{3}+6\,a^2\,b^2\,x^4}{x^5}+4\,a\,b^3\,x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.23, size = 49, normalized size = 0.98 \[ 4 a b^{3} x + \frac {b^{4} x^{3}}{3} + \frac {- 3 a^{4} - 20 a^{3} b x^{2} - 90 a^{2} b^{2} x^{4}}{15 x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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