Optimal. Leaf size=17 \[ \frac {1}{3} \log \left (a+b x^3+c x^6\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {1482, 642}
\begin {gather*} \frac {1}{3} \log \left (a+b x^3+c x^6\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 642
Rule 1482
Rubi steps
\begin {align*} \int \frac {x^2 \left (b+2 c x^3\right )}{a+b x^3+c x^6} \, dx &=\frac {1}{3} \text {Subst}\left (\int \frac {b+2 c x}{a+b x+c x^2} \, dx,x,x^3\right )\\ &=\frac {1}{3} \log \left (a+b x^3+c x^6\right )\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 17, normalized size = 1.00 \begin {gather*} \frac {1}{3} \log \left (a+b x^3+c x^6\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 16, normalized size = 0.94
method | result | size |
default | \(\frac {\ln \left (c \,x^{6}+b \,x^{3}+a \right )}{3}\) | \(16\) |
norman | \(\frac {\ln \left (c \,x^{6}+b \,x^{3}+a \right )}{3}\) | \(16\) |
risch | \(\frac {\ln \left (c \,x^{6}+b \,x^{3}+a \right )}{3}\) | \(16\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 15, normalized size = 0.88 \begin {gather*} \frac {1}{3} \, \log \left (c x^{6} + b x^{3} + a\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.40, size = 15, normalized size = 0.88 \begin {gather*} \frac {1}{3} \, \log \left (c x^{6} + b x^{3} + a\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.24, size = 14, normalized size = 0.82 \begin {gather*} \frac {\log {\left (a + b x^{3} + c x^{6} \right )}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.44, size = 16, normalized size = 0.94 \begin {gather*} \frac {1}{3} \, \log \left ({\left | c x^{6} + b x^{3} + a \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.05, size = 15, normalized size = 0.88 \begin {gather*} \frac {\ln \left (c\,x^6+b\,x^3+a\right )}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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