Optimal. Leaf size=47 \[ -\frac {a^4}{7 x^7}-\frac {4 a^3 b}{5 x^5}-\frac {2 a^2 b^2}{x^3}-\frac {4 a b^3}{x}+b^4 x \]
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Rubi [A] time = 0.03, antiderivative size = 47, 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} \[ -\frac {2 a^2 b^2}{x^3}-\frac {4 a^3 b}{5 x^5}-\frac {a^4}{7 x^7}-\frac {4 a b^3}{x}+b^4 x \]
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^8} \, dx &=\frac {\int \frac {\left (a b+b^2 x^2\right )^4}{x^8} \, dx}{b^4}\\ &=\frac {\int \left (b^8+\frac {a^4 b^4}{x^8}+\frac {4 a^3 b^5}{x^6}+\frac {6 a^2 b^6}{x^4}+\frac {4 a b^7}{x^2}\right ) \, dx}{b^4}\\ &=-\frac {a^4}{7 x^7}-\frac {4 a^3 b}{5 x^5}-\frac {2 a^2 b^2}{x^3}-\frac {4 a b^3}{x}+b^4 x\\ \end {align*}
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Mathematica [A] time = 0.01, size = 47, normalized size = 1.00 \[ -\frac {a^4}{7 x^7}-\frac {4 a^3 b}{5 x^5}-\frac {2 a^2 b^2}{x^3}-\frac {4 a b^3}{x}+b^4 x \]
Antiderivative was successfully verified.
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fricas [A] time = 0.85, size = 48, normalized size = 1.02 \[ \frac {35 \, b^{4} x^{8} - 140 \, a b^{3} x^{6} - 70 \, a^{2} b^{2} x^{4} - 28 \, a^{3} b x^{2} - 5 \, a^{4}}{35 \, x^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 46, normalized size = 0.98 \[ b^{4} x - \frac {140 \, a b^{3} x^{6} + 70 \, a^{2} b^{2} x^{4} + 28 \, a^{3} b x^{2} + 5 \, a^{4}}{35 \, x^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 44, normalized size = 0.94 \[ b^{4} x -\frac {4 a \,b^{3}}{x}-\frac {2 a^{2} b^{2}}{x^{3}}-\frac {4 a^{3} b}{5 x^{5}}-\frac {a^{4}}{7 x^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.45, size = 46, normalized size = 0.98 \[ b^{4} x - \frac {140 \, a b^{3} x^{6} + 70 \, a^{2} b^{2} x^{4} + 28 \, a^{3} b x^{2} + 5 \, a^{4}}{35 \, x^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.19, size = 46, normalized size = 0.98 \[ b^4\,x-\frac {\frac {a^4}{7}+\frac {4\,a^3\,b\,x^2}{5}+2\,a^2\,b^2\,x^4+4\,a\,b^3\,x^6}{x^7} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.30, size = 48, normalized size = 1.02 \[ b^{4} x + \frac {- 5 a^{4} - 28 a^{3} b x^{2} - 70 a^{2} b^{2} x^{4} - 140 a b^{3} x^{6}}{35 x^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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