Integrand size = 20, antiderivative size = 133 \[ \int \frac {\left (a+b x^3+c x^6\right )^{3/2}}{x^{13}} \, dx=\frac {\left (b^2-4 a c\right ) \left (2 a+b x^3\right ) \sqrt {a+b x^3+c x^6}}{64 a^2 x^6}-\frac {\left (2 a+b x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{24 a x^{12}}-\frac {\left (b^2-4 a c\right )^2 \text {arctanh}\left (\frac {2 a+b x^3}{2 \sqrt {a} \sqrt {a+b x^3+c x^6}}\right )}{128 a^{5/2}} \]
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Time = 0.08 (sec) , antiderivative size = 133, 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 = {1371, 734, 738, 212} \[ \int \frac {\left (a+b x^3+c x^6\right )^{3/2}}{x^{13}} \, dx=-\frac {\left (b^2-4 a c\right )^2 \text {arctanh}\left (\frac {2 a+b x^3}{2 \sqrt {a} \sqrt {a+b x^3+c x^6}}\right )}{128 a^{5/2}}+\frac {\left (b^2-4 a c\right ) \left (2 a+b x^3\right ) \sqrt {a+b x^3+c x^6}}{64 a^2 x^6}-\frac {\left (2 a+b x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{24 a x^{12}} \]
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Rule 212
Rule 734
Rule 738
Rule 1371
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \text {Subst}\left (\int \frac {\left (a+b x+c x^2\right )^{3/2}}{x^5} \, dx,x,x^3\right ) \\ & = -\frac {\left (2 a+b x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{24 a x^{12}}-\frac {\left (b^2-4 a c\right ) \text {Subst}\left (\int \frac {\sqrt {a+b x+c x^2}}{x^3} \, dx,x,x^3\right )}{16 a} \\ & = \frac {\left (b^2-4 a c\right ) \left (2 a+b x^3\right ) \sqrt {a+b x^3+c x^6}}{64 a^2 x^6}-\frac {\left (2 a+b x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{24 a x^{12}}+\frac {\left (b^2-4 a c\right )^2 \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x+c x^2}} \, dx,x,x^3\right )}{128 a^2} \\ & = \frac {\left (b^2-4 a c\right ) \left (2 a+b x^3\right ) \sqrt {a+b x^3+c x^6}}{64 a^2 x^6}-\frac {\left (2 a+b x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{24 a x^{12}}-\frac {\left (b^2-4 a c\right )^2 \text {Subst}\left (\int \frac {1}{4 a-x^2} \, dx,x,\frac {2 a+b x^3}{\sqrt {a+b x^3+c x^6}}\right )}{64 a^2} \\ & = \frac {\left (b^2-4 a c\right ) \left (2 a+b x^3\right ) \sqrt {a+b x^3+c x^6}}{64 a^2 x^6}-\frac {\left (2 a+b x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{24 a x^{12}}-\frac {\left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac {2 a+b x^3}{2 \sqrt {a} \sqrt {a+b x^3+c x^6}}\right )}{128 a^{5/2}} \\ \end{align*}
Time = 0.64 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.89 \[ \int \frac {\left (a+b x^3+c x^6\right )^{3/2}}{x^{13}} \, dx=-\frac {\left (2 a+b x^3\right ) \sqrt {a+b x^3+c x^6} \left (8 a^2+8 a b x^3-3 b^2 x^6+20 a c x^6\right )}{192 a^2 x^{12}}+\frac {\left (b^2-4 a c\right )^2 \text {arctanh}\left (\frac {\sqrt {c} x^3-\sqrt {a+b x^3+c x^6}}{\sqrt {a}}\right )}{64 a^{5/2}} \]
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\[\int \frac {\left (c \,x^{6}+b \,x^{3}+a \right )^{\frac {3}{2}}}{x^{13}}d x\]
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none
Time = 0.30 (sec) , antiderivative size = 319, normalized size of antiderivative = 2.40 \[ \int \frac {\left (a+b x^3+c x^6\right )^{3/2}}{x^{13}} \, dx=\left [\frac {3 \, {\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \sqrt {a} x^{12} \log \left (-\frac {{\left (b^{2} + 4 \, a c\right )} x^{6} + 8 \, a b x^{3} - 4 \, \sqrt {c x^{6} + b x^{3} + a} {\left (b x^{3} + 2 \, a\right )} \sqrt {a} + 8 \, a^{2}}{x^{6}}\right ) + 4 \, {\left ({\left (3 \, a b^{3} - 20 \, a^{2} b c\right )} x^{9} - 24 \, a^{3} b x^{3} - 2 \, {\left (a^{2} b^{2} + 20 \, a^{3} c\right )} x^{6} - 16 \, a^{4}\right )} \sqrt {c x^{6} + b x^{3} + a}}{768 \, a^{3} x^{12}}, \frac {3 \, {\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \sqrt {-a} x^{12} \arctan \left (\frac {\sqrt {c x^{6} + b x^{3} + a} {\left (b x^{3} + 2 \, a\right )} \sqrt {-a}}{2 \, {\left (a c x^{6} + a b x^{3} + a^{2}\right )}}\right ) + 2 \, {\left ({\left (3 \, a b^{3} - 20 \, a^{2} b c\right )} x^{9} - 24 \, a^{3} b x^{3} - 2 \, {\left (a^{2} b^{2} + 20 \, a^{3} c\right )} x^{6} - 16 \, a^{4}\right )} \sqrt {c x^{6} + b x^{3} + a}}{384 \, a^{3} x^{12}}\right ] \]
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\[ \int \frac {\left (a+b x^3+c x^6\right )^{3/2}}{x^{13}} \, dx=\int \frac {\left (a + b x^{3} + c x^{6}\right )^{\frac {3}{2}}}{x^{13}}\, dx \]
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Exception generated. \[ \int \frac {\left (a+b x^3+c x^6\right )^{3/2}}{x^{13}} \, dx=\text {Exception raised: ValueError} \]
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\[ \int \frac {\left (a+b x^3+c x^6\right )^{3/2}}{x^{13}} \, dx=\int { \frac {{\left (c x^{6} + b x^{3} + a\right )}^{\frac {3}{2}}}{x^{13}} \,d x } \]
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Timed out. \[ \int \frac {\left (a+b x^3+c x^6\right )^{3/2}}{x^{13}} \, dx=\int \frac {{\left (c\,x^6+b\,x^3+a\right )}^{3/2}}{x^{13}} \,d x \]
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