Integrand size = 20, antiderivative size = 38 \[ \int \frac {x^2}{\left (a+b x^3+c x^6\right )^{3/2}} \, dx=-\frac {2 \left (b+2 c x^3\right )}{3 \left (b^2-4 a c\right ) \sqrt {a+b x^3+c x^6}} \]
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Time = 0.02 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {1366, 627} \[ \int \frac {x^2}{\left (a+b x^3+c x^6\right )^{3/2}} \, dx=-\frac {2 \left (b+2 c x^3\right )}{3 \left (b^2-4 a c\right ) \sqrt {a+b x^3+c x^6}} \]
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Rule 627
Rule 1366
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \text {Subst}\left (\int \frac {1}{\left (a+b x+c x^2\right )^{3/2}} \, dx,x,x^3\right ) \\ & = -\frac {2 \left (b+2 c x^3\right )}{3 \left (b^2-4 a c\right ) \sqrt {a+b x^3+c x^6}} \\ \end{align*}
Time = 0.26 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00 \[ \int \frac {x^2}{\left (a+b x^3+c x^6\right )^{3/2}} \, dx=-\frac {2 \left (b+2 c x^3\right )}{3 \left (b^2-4 a c\right ) \sqrt {a+b x^3+c x^6}} \]
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Time = 2.47 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.97
method | result | size |
gosper | \(\frac {\frac {4 c \,x^{3}}{3}+\frac {2 b}{3}}{\sqrt {c \,x^{6}+b \,x^{3}+a}\, \left (4 a c -b^{2}\right )}\) | \(37\) |
trager | \(\frac {\frac {4 c \,x^{3}}{3}+\frac {2 b}{3}}{\sqrt {c \,x^{6}+b \,x^{3}+a}\, \left (4 a c -b^{2}\right )}\) | \(37\) |
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none
Time = 0.27 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.76 \[ \int \frac {x^2}{\left (a+b x^3+c x^6\right )^{3/2}} \, dx=-\frac {2 \, \sqrt {c x^{6} + b x^{3} + a} {\left (2 \, c x^{3} + b\right )}}{3 \, {\left ({\left (b^{2} c - 4 \, a c^{2}\right )} x^{6} + {\left (b^{3} - 4 \, a b c\right )} x^{3} + a b^{2} - 4 \, a^{2} c\right )}} \]
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\[ \int \frac {x^2}{\left (a+b x^3+c x^6\right )^{3/2}} \, dx=\int \frac {x^{2}}{\left (a + b x^{3} + c x^{6}\right )^{\frac {3}{2}}}\, dx \]
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Exception generated. \[ \int \frac {x^2}{\left (a+b x^3+c x^6\right )^{3/2}} \, dx=\text {Exception raised: ValueError} \]
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none
Time = 0.40 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.18 \[ \int \frac {x^2}{\left (a+b x^3+c x^6\right )^{3/2}} \, dx=-\frac {2 \, {\left (\frac {2 \, c x^{3}}{b^{2} - 4 \, a c} + \frac {b}{b^{2} - 4 \, a c}\right )}}{3 \, \sqrt {c x^{6} + b x^{3} + a}} \]
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Time = 8.68 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.97 \[ \int \frac {x^2}{\left (a+b x^3+c x^6\right )^{3/2}} \, dx=\frac {4\,c\,x^3+2\,b}{\left (12\,a\,c-3\,b^2\right )\,\sqrt {c\,x^6+b\,x^3+a}} \]
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