Optimal. Leaf size=16 \[ \frac {\left (a+b x^2\right )^3}{6 b} \]
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Rubi [A]
time = 0.00, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {1598, 267}
\begin {gather*} \frac {\left (a+b x^2\right )^3}{6 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 267
Rule 1598
Rubi steps
\begin {align*} \int \frac {\left (a x+b x^3\right )^2}{x} \, dx &=\int x \left (a+b x^2\right )^2 \, dx\\ &=\frac {\left (a+b x^2\right )^3}{6 b}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 16, normalized size = 1.00 \begin {gather*} \frac {\left (a+b x^2\right )^3}{6 b} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.35, size = 15, normalized size = 0.94
method | result | size |
default | \(\frac {\left (b \,x^{2}+a \right )^{3}}{6 b}\) | \(15\) |
norman | \(\frac {1}{6} b^{2} x^{6}+\frac {1}{2} a b \,x^{4}+\frac {1}{2} a^{2} x^{2}\) | \(25\) |
gosper | \(\frac {x^{2} \left (b^{2} x^{4}+3 a b \,x^{2}+3 a^{2}\right )}{6}\) | \(26\) |
risch | \(\frac {b^{2} x^{6}}{6}+\frac {a b \,x^{4}}{2}+\frac {a^{2} x^{2}}{2}+\frac {a^{3}}{6 b}\) | \(33\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 24, normalized size = 1.50 \begin {gather*} \frac {1}{6} \, b^{2} x^{6} + \frac {1}{2} \, a b x^{4} + \frac {1}{2} \, a^{2} x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.82, size = 24, normalized size = 1.50 \begin {gather*} \frac {1}{6} \, b^{2} x^{6} + \frac {1}{2} \, a b x^{4} + \frac {1}{2} \, a^{2} x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 24 vs.
\(2 (10) = 20\).
time = 0.01, size = 24, normalized size = 1.50 \begin {gather*} \frac {a^{2} x^{2}}{2} + \frac {a b x^{4}}{2} + \frac {b^{2} x^{6}}{6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.00, size = 24, normalized size = 1.50 \begin {gather*} \frac {1}{6} \, b^{2} x^{6} + \frac {1}{2} \, a b x^{4} + \frac {1}{2} \, a^{2} x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.03, size = 24, normalized size = 1.50 \begin {gather*} \frac {a^2\,x^2}{2}+\frac {a\,b\,x^4}{2}+\frac {b^2\,x^6}{6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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