Integrand size = 20, antiderivative size = 124 \[ \int x^2 \left (a+b x^3+c x^6\right )^{3/2} \, dx=-\frac {\left (b^2-4 a c\right ) \left (b+2 c x^3\right ) \sqrt {a+b x^3+c x^6}}{64 c^2}+\frac {\left (b+2 c x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{24 c}+\frac {\left (b^2-4 a c\right )^2 \text {arctanh}\left (\frac {b+2 c x^3}{2 \sqrt {c} \sqrt {a+b x^3+c x^6}}\right )}{128 c^{5/2}} \]
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Time = 0.06 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1366, 626, 635, 212} \[ \int x^2 \left (a+b x^3+c x^6\right )^{3/2} \, dx=\frac {\left (b^2-4 a c\right )^2 \text {arctanh}\left (\frac {b+2 c x^3}{2 \sqrt {c} \sqrt {a+b x^3+c x^6}}\right )}{128 c^{5/2}}-\frac {\left (b^2-4 a c\right ) \left (b+2 c x^3\right ) \sqrt {a+b x^3+c x^6}}{64 c^2}+\frac {\left (b+2 c x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{24 c} \]
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Rule 212
Rule 626
Rule 635
Rule 1366
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \text {Subst}\left (\int \left (a+b x+c x^2\right )^{3/2} \, dx,x,x^3\right ) \\ & = \frac {\left (b+2 c x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{24 c}-\frac {\left (b^2-4 a c\right ) \text {Subst}\left (\int \sqrt {a+b x+c x^2} \, dx,x,x^3\right )}{16 c} \\ & = -\frac {\left (b^2-4 a c\right ) \left (b+2 c x^3\right ) \sqrt {a+b x^3+c x^6}}{64 c^2}+\frac {\left (b+2 c x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{24 c}+\frac {\left (b^2-4 a c\right )^2 \text {Subst}\left (\int \frac {1}{\sqrt {a+b x+c x^2}} \, dx,x,x^3\right )}{128 c^2} \\ & = -\frac {\left (b^2-4 a c\right ) \left (b+2 c x^3\right ) \sqrt {a+b x^3+c x^6}}{64 c^2}+\frac {\left (b+2 c x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{24 c}+\frac {\left (b^2-4 a c\right )^2 \text {Subst}\left (\int \frac {1}{4 c-x^2} \, dx,x,\frac {b+2 c x^3}{\sqrt {a+b x^3+c x^6}}\right )}{64 c^2} \\ & = -\frac {\left (b^2-4 a c\right ) \left (b+2 c x^3\right ) \sqrt {a+b x^3+c x^6}}{64 c^2}+\frac {\left (b+2 c x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{24 c}+\frac {\left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac {b+2 c x^3}{2 \sqrt {c} \sqrt {a+b x^3+c x^6}}\right )}{128 c^{5/2}} \\ \end{align*}
Time = 0.64 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.92 \[ \int x^2 \left (a+b x^3+c x^6\right )^{3/2} \, dx=\frac {\left (b+2 c x^3\right ) \sqrt {a+b x^3+c x^6} \left (-3 b^2+20 a c+8 b c x^3+8 c^2 x^6\right )}{192 c^2}+\frac {\left (-b^2+4 a c\right )^2 \text {arctanh}\left (\frac {\sqrt {c} x^3}{-\sqrt {a}+\sqrt {a+b x^3+c x^6}}\right )}{64 c^{5/2}} \]
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\[\int x^{2} \left (c \,x^{6}+b \,x^{3}+a \right )^{\frac {3}{2}}d x\]
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none
Time = 0.26 (sec) , antiderivative size = 297, normalized size of antiderivative = 2.40 \[ \int x^2 \left (a+b x^3+c x^6\right )^{3/2} \, dx=\left [\frac {3 \, {\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \sqrt {c} \log \left (-8 \, c^{2} x^{6} - 8 \, b c x^{3} - b^{2} - 4 \, \sqrt {c x^{6} + b x^{3} + a} {\left (2 \, c x^{3} + b\right )} \sqrt {c} - 4 \, a c\right ) + 4 \, {\left (16 \, c^{4} x^{9} + 24 \, b c^{3} x^{6} - 3 \, b^{3} c + 20 \, a b c^{2} + 2 \, {\left (b^{2} c^{2} + 20 \, a c^{3}\right )} x^{3}\right )} \sqrt {c x^{6} + b x^{3} + a}}{768 \, c^{3}}, -\frac {3 \, {\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{6} + b x^{3} + a} {\left (2 \, c x^{3} + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{6} + b c x^{3} + a c\right )}}\right ) - 2 \, {\left (16 \, c^{4} x^{9} + 24 \, b c^{3} x^{6} - 3 \, b^{3} c + 20 \, a b c^{2} + 2 \, {\left (b^{2} c^{2} + 20 \, a c^{3}\right )} x^{3}\right )} \sqrt {c x^{6} + b x^{3} + a}}{384 \, c^{3}}\right ] \]
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\[ \int x^2 \left (a+b x^3+c x^6\right )^{3/2} \, dx=\int x^{2} \left (a + b x^{3} + c x^{6}\right )^{\frac {3}{2}}\, dx \]
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Exception generated. \[ \int x^2 \left (a+b x^3+c x^6\right )^{3/2} \, dx=\text {Exception raised: ValueError} \]
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none
Time = 0.35 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.07 \[ \int x^2 \left (a+b x^3+c x^6\right )^{3/2} \, dx=\frac {1}{192} \, \sqrt {c x^{6} + b x^{3} + a} {\left (2 \, {\left (4 \, {\left (2 \, c x^{3} + 3 \, b\right )} x^{3} + \frac {b^{2} c^{2} + 20 \, a c^{3}}{c^{3}}\right )} x^{3} - \frac {3 \, b^{3} c - 20 \, a b c^{2}}{c^{3}}\right )} - \frac {{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \log \left ({\left | 2 \, {\left (\sqrt {c} x^{3} - \sqrt {c x^{6} + b x^{3} + a}\right )} \sqrt {c} + b \right |}\right )}{128 \, c^{\frac {5}{2}}} \]
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Time = 8.53 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.93 \[ \int x^2 \left (a+b x^3+c x^6\right )^{3/2} \, dx=\frac {\left (c\,x^3+\frac {b}{2}\right )\,{\left (c\,x^6+b\,x^3+a\right )}^{3/2}}{12\,c}+\frac {\left (3\,a\,c-\frac {3\,b^2}{4}\right )\,\left (\left (\frac {b}{4\,c}+\frac {x^3}{2}\right )\,\sqrt {c\,x^6+b\,x^3+a}+\frac {\ln \left (\sqrt {c\,x^6+b\,x^3+a}+\frac {c\,x^3+\frac {b}{2}}{\sqrt {c}}\right )\,\left (a\,c-\frac {b^2}{4}\right )}{2\,c^{3/2}}\right )}{12\,c} \]
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