Optimal. Leaf size=47 \[ -\frac {a^2}{7 x^7}-\frac {2 a b}{5 x^5}-\frac {b^2+2 a c}{3 x^3}-\frac {2 b c}{x}+c^2 x \]
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Rubi [A]
time = 0.02, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {1122}
\begin {gather*} -\frac {a^2}{7 x^7}-\frac {2 a c+b^2}{3 x^3}-\frac {2 a b}{5 x^5}-\frac {2 b c}{x}+c^2 x \end {gather*}
Antiderivative was successfully verified.
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Rule 1122
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2+c x^4\right )^2}{x^8} \, dx &=\int \left (c^2+\frac {a^2}{x^8}+\frac {2 a b}{x^6}+\frac {b^2+2 a c}{x^4}+\frac {2 b c}{x^2}\right ) \, dx\\ &=-\frac {a^2}{7 x^7}-\frac {2 a b}{5 x^5}-\frac {b^2+2 a c}{3 x^3}-\frac {2 b c}{x}+c^2 x\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 49, normalized size = 1.04 \begin {gather*} -\frac {a^2}{7 x^7}-\frac {2 a b}{5 x^5}+\frac {-b^2-2 a c}{3 x^3}-\frac {2 b c}{x}+c^2 x \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 42, normalized size = 0.89
method | result | size |
default | \(c^{2} x -\frac {2 a b}{5 x^{5}}-\frac {2 b c}{x}-\frac {2 a c +b^{2}}{3 x^{3}}-\frac {a^{2}}{7 x^{7}}\) | \(42\) |
risch | \(c^{2} x +\frac {-2 b c \,x^{6}+\left (-\frac {2 a c}{3}-\frac {b^{2}}{3}\right ) x^{4}-\frac {2 a b \,x^{2}}{5}-\frac {a^{2}}{7}}{x^{7}}\) | \(45\) |
norman | \(\frac {c^{2} x^{8}-2 b c \,x^{6}+\left (-\frac {2 a c}{3}-\frac {b^{2}}{3}\right ) x^{4}-\frac {2 a b \,x^{2}}{5}-\frac {a^{2}}{7}}{x^{7}}\) | \(46\) |
gosper | \(-\frac {-105 c^{2} x^{8}+210 b c \,x^{6}+70 a c \,x^{4}+35 b^{2} x^{4}+42 a b \,x^{2}+15 a^{2}}{105 x^{7}}\) | \(49\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 44, normalized size = 0.94 \begin {gather*} c^{2} x - \frac {210 \, b c x^{6} + 35 \, {\left (b^{2} + 2 \, a c\right )} x^{4} + 42 \, a b x^{2} + 15 \, a^{2}}{105 \, x^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 46, normalized size = 0.98 \begin {gather*} \frac {105 \, c^{2} x^{8} - 210 \, b c x^{6} - 35 \, {\left (b^{2} + 2 \, a c\right )} x^{4} - 42 \, a b x^{2} - 15 \, a^{2}}{105 \, x^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.36, size = 46, normalized size = 0.98 \begin {gather*} c^{2} x + \frac {- 15 a^{2} - 42 a b x^{2} - 210 b c x^{6} + x^{4} \left (- 70 a c - 35 b^{2}\right )}{105 x^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.39, size = 46, normalized size = 0.98 \begin {gather*} c^{2} x - \frac {210 \, b c x^{6} + 35 \, b^{2} x^{4} + 70 \, a c x^{4} + 42 \, a b x^{2} + 15 \, a^{2}}{105 \, x^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.17, size = 45, normalized size = 0.96 \begin {gather*} c^2\,x-\frac {\frac {a^2}{7}+x^4\,\left (\frac {b^2}{3}+\frac {2\,a\,c}{3}\right )+\frac {2\,a\,b\,x^2}{5}+2\,b\,c\,x^6}{x^7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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