Optimal. Leaf size=30 \[ \frac {a^2 x^4}{4}+\frac {2}{5} a b x^5+\frac {b^2 x^6}{6} \]
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Rubi [A]
time = 0.01, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {1598, 45}
\begin {gather*} \frac {a^2 x^4}{4}+\frac {2}{5} a b x^5+\frac {b^2 x^6}{6} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 1598
Rubi steps
\begin {align*} \int \frac {\left (a x^2+b x^3\right )^2}{x} \, dx &=\int x^3 (a+b x)^2 \, dx\\ &=\int \left (a^2 x^3+2 a b x^4+b^2 x^5\right ) \, dx\\ &=\frac {a^2 x^4}{4}+\frac {2}{5} a b x^5+\frac {b^2 x^6}{6}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 30, normalized size = 1.00 \begin {gather*} \frac {a^2 x^4}{4}+\frac {2}{5} a b x^5+\frac {b^2 x^6}{6} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.34, size = 25, normalized size = 0.83
method | result | size |
gosper | \(\frac {x^{4} \left (10 b^{2} x^{2}+24 a b x +15 a^{2}\right )}{60}\) | \(25\) |
default | \(\frac {1}{4} a^{2} x^{4}+\frac {2}{5} a b \,x^{5}+\frac {1}{6} b^{2} x^{6}\) | \(25\) |
norman | \(\frac {1}{4} a^{2} x^{4}+\frac {2}{5} a b \,x^{5}+\frac {1}{6} b^{2} x^{6}\) | \(25\) |
risch | \(\frac {1}{4} a^{2} x^{4}+\frac {2}{5} a b \,x^{5}+\frac {1}{6} b^{2} x^{6}\) | \(25\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 24, normalized size = 0.80 \begin {gather*} \frac {1}{6} \, b^{2} x^{6} + \frac {2}{5} \, a b x^{5} + \frac {1}{4} \, a^{2} x^{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.21, size = 24, normalized size = 0.80 \begin {gather*} \frac {1}{6} \, b^{2} x^{6} + \frac {2}{5} \, a b x^{5} + \frac {1}{4} \, a^{2} x^{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.01, size = 26, normalized size = 0.87 \begin {gather*} \frac {a^{2} x^{4}}{4} + \frac {2 a b x^{5}}{5} + \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 = 1.38, size = 24, normalized size = 0.80 \begin {gather*} \frac {1}{6} \, b^{2} x^{6} + \frac {2}{5} \, a b x^{5} + \frac {1}{4} \, a^{2} x^{4} \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 = 0.80 \begin {gather*} \frac {a^2\,x^4}{4}+\frac {2\,a\,b\,x^5}{5}+\frac {b^2\,x^6}{6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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