Optimal. Leaf size=34 \[ -\frac {a^3}{5 x^5}-\frac {a^2 b}{x^3}-\frac {3 a b^2}{x}+b^3 x \]
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Rubi [A]
time = 0.01, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {276}
\begin {gather*} -\frac {a^3}{5 x^5}-\frac {a^2 b}{x^3}-\frac {3 a b^2}{x}+b^3 x \end {gather*}
Antiderivative was successfully verified.
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Rule 276
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^3}{x^6} \, dx &=\int \left (b^3+\frac {a^3}{x^6}+\frac {3 a^2 b}{x^4}+\frac {3 a b^2}{x^2}\right ) \, dx\\ &=-\frac {a^3}{5 x^5}-\frac {a^2 b}{x^3}-\frac {3 a b^2}{x}+b^3 x\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 34, normalized size = 1.00 \begin {gather*} -\frac {a^3}{5 x^5}-\frac {a^2 b}{x^3}-\frac {3 a b^2}{x}+b^3 x \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.01, size = 33, normalized size = 0.97
method | result | size |
default | \(-\frac {a^{3}}{5 x^{5}}-\frac {a^{2} b}{x^{3}}-\frac {3 a \,b^{2}}{x}+b^{3} x\) | \(33\) |
risch | \(b^{3} x +\frac {-3 a \,b^{2} x^{4}-a^{2} b \,x^{2}-\frac {1}{5} a^{3}}{x^{5}}\) | \(35\) |
gosper | \(-\frac {-5 b^{3} x^{6}+15 a \,b^{2} x^{4}+5 a^{2} b \,x^{2}+a^{3}}{5 x^{5}}\) | \(36\) |
norman | \(\frac {b^{3} x^{6}-3 a \,b^{2} x^{4}-a^{2} b \,x^{2}-\frac {1}{5} a^{3}}{x^{5}}\) | \(36\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 33, normalized size = 0.97 \begin {gather*} b^{3} x - \frac {15 \, a b^{2} x^{4} + 5 \, a^{2} b x^{2} + a^{3}}{5 \, x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.79, size = 37, normalized size = 1.09 \begin {gather*} \frac {5 \, b^{3} x^{6} - 15 \, a b^{2} x^{4} - 5 \, a^{2} b x^{2} - a^{3}}{5 \, x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.07, size = 34, normalized size = 1.00 \begin {gather*} b^{3} x + \frac {- a^{3} - 5 a^{2} b x^{2} - 15 a b^{2} x^{4}}{5 x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.63, size = 33, normalized size = 0.97 \begin {gather*} b^{3} x - \frac {15 \, a b^{2} x^{4} + 5 \, a^{2} b x^{2} + a^{3}}{5 \, x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.03, size = 34, normalized size = 1.00 \begin {gather*} b^3\,x-\frac {\frac {a^3}{5}+a^2\,b\,x^2+3\,a\,b^2\,x^4}{x^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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