Optimal. Leaf size=85 \[ \frac {3}{2} a^2 b x^2+\frac {3}{4} a \left (b^2+a c\right ) x^4+\frac {1}{6} b \left (b^2+6 a c\right ) x^6+\frac {3}{8} c \left (b^2+a c\right ) x^8+\frac {3}{10} b c^2 x^{10}+\frac {c^3 x^{12}}{12}+a^3 \log (x) \]
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Rubi [A]
time = 0.05, antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {1128, 712}
\begin {gather*} a^3 \log (x)+\frac {3}{2} a^2 b x^2+\frac {3}{8} c x^8 \left (a c+b^2\right )+\frac {1}{6} b x^6 \left (6 a c+b^2\right )+\frac {3}{4} a x^4 \left (a c+b^2\right )+\frac {3}{10} b c^2 x^{10}+\frac {c^3 x^{12}}{12} \end {gather*}
Antiderivative was successfully verified.
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Rule 712
Rule 1128
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2+c x^4\right )^3}{x} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {\left (a+b x+c x^2\right )^3}{x} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (3 a^2 b+\frac {a^3}{x}+3 a \left (b^2+a c\right ) x+b \left (b^2+6 a c\right ) x^2+3 c \left (b^2+a c\right ) x^3+3 b c^2 x^4+c^3 x^5\right ) \, dx,x,x^2\right )\\ &=\frac {3}{2} a^2 b x^2+\frac {3}{4} a \left (b^2+a c\right ) x^4+\frac {1}{6} b \left (b^2+6 a c\right ) x^6+\frac {3}{8} c \left (b^2+a c\right ) x^8+\frac {3}{10} b c^2 x^{10}+\frac {c^3 x^{12}}{12}+a^3 \log (x)\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 85, normalized size = 1.00 \begin {gather*} \frac {3}{2} a^2 b x^2+\frac {3}{4} a \left (b^2+a c\right ) x^4+\frac {1}{6} b \left (b^2+6 a c\right ) x^6+\frac {3}{8} c \left (b^2+a c\right ) x^8+\frac {3}{10} b c^2 x^{10}+\frac {c^3 x^{12}}{12}+a^3 \log (x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 85, normalized size = 1.00
method | result | size |
norman | \(\left (\frac {3}{4} a^{2} c +\frac {3}{4} a \,b^{2}\right ) x^{4}+\left (\frac {3}{8} c^{2} a +\frac {3}{8} b^{2} c \right ) x^{8}+\left (a b c +\frac {1}{6} b^{3}\right ) x^{6}+\frac {c^{3} x^{12}}{12}+\frac {3 a^{2} b \,x^{2}}{2}+\frac {3 b \,c^{2} x^{10}}{10}+a^{3} \ln \left (x \right )\) | \(82\) |
default | \(\frac {c^{3} x^{12}}{12}+\frac {3 b \,c^{2} x^{10}}{10}+\frac {3 a \,c^{2} x^{8}}{8}+\frac {3 b^{2} c \,x^{8}}{8}+a b c \,x^{6}+\frac {b^{3} x^{6}}{6}+\frac {3 a^{2} c \,x^{4}}{4}+\frac {3 a \,b^{2} x^{4}}{4}+\frac {3 a^{2} b \,x^{2}}{2}+a^{3} \ln \left (x \right )\) | \(85\) |
risch | \(\frac {3 a^{2} b \,x^{2}}{2}+a b c \,x^{6}+\frac {3 a \,b^{2} x^{4}}{4}+\frac {3 a \,c^{2} x^{8}}{8}-\frac {a \,b^{4}}{8 c^{2}}+\frac {b^{6}}{120 c^{3}}+\frac {c^{3} x^{12}}{12}+\frac {3 a^{2} c \,x^{4}}{4}+\frac {3 a^{2} b^{2}}{4 c}+\frac {3 b \,c^{2} x^{10}}{10}+\frac {3 b^{2} c \,x^{8}}{8}+\frac {b^{3} x^{6}}{6}+a^{3} \ln \left (x \right )\) | \(113\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 82, normalized size = 0.96 \begin {gather*} \frac {1}{12} \, c^{3} x^{12} + \frac {3}{10} \, b c^{2} x^{10} + \frac {3}{8} \, {\left (b^{2} c + a c^{2}\right )} x^{8} + \frac {1}{6} \, {\left (b^{3} + 6 \, a b c\right )} x^{6} + \frac {3}{2} \, a^{2} b x^{2} + \frac {3}{4} \, {\left (a b^{2} + a^{2} c\right )} x^{4} + \frac {1}{2} \, a^{3} \log \left (x^{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.33, size = 79, normalized size = 0.93 \begin {gather*} \frac {1}{12} \, c^{3} x^{12} + \frac {3}{10} \, b c^{2} x^{10} + \frac {3}{8} \, {\left (b^{2} c + a c^{2}\right )} x^{8} + \frac {1}{6} \, {\left (b^{3} + 6 \, a b c\right )} x^{6} + \frac {3}{2} \, a^{2} b x^{2} + \frac {3}{4} \, {\left (a b^{2} + a^{2} c\right )} x^{4} + a^{3} \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.06, size = 92, normalized size = 1.08 \begin {gather*} a^{3} \log {\left (x \right )} + \frac {3 a^{2} b x^{2}}{2} + \frac {3 b c^{2} x^{10}}{10} + \frac {c^{3} x^{12}}{12} + x^{8} \cdot \left (\frac {3 a c^{2}}{8} + \frac {3 b^{2} c}{8}\right ) + x^{6} \left (a b c + \frac {b^{3}}{6}\right ) + x^{4} \cdot \left (\frac {3 a^{2} c}{4} + \frac {3 a b^{2}}{4}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 5.01, size = 87, normalized size = 1.02 \begin {gather*} \frac {1}{12} \, c^{3} x^{12} + \frac {3}{10} \, b c^{2} x^{10} + \frac {3}{8} \, b^{2} c x^{8} + \frac {3}{8} \, a c^{2} x^{8} + \frac {1}{6} \, b^{3} x^{6} + a b c x^{6} + \frac {3}{4} \, a b^{2} x^{4} + \frac {3}{4} \, a^{2} c x^{4} + \frac {3}{2} \, a^{2} b x^{2} + \frac {1}{2} \, a^{3} \log \left (x^{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.03, size = 73, normalized size = 0.86 \begin {gather*} a^3\,\ln \left (x\right )+x^6\,\left (\frac {b^3}{6}+a\,c\,b\right )+\frac {c^3\,x^{12}}{12}+\frac {3\,a^2\,b\,x^2}{2}+\frac {3\,b\,c^2\,x^{10}}{10}+\frac {3\,a\,x^4\,\left (b^2+a\,c\right )}{4}+\frac {3\,c\,x^8\,\left (b^2+a\,c\right )}{8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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