Integrand size = 20, antiderivative size = 162 \[ \int \frac {\left (a+b x^3+c x^6\right )^{3/2}}{x^{16}} \, dx=-\frac {b \left (b^2-4 a c\right ) \left (2 a+b x^3\right ) \sqrt {a+b x^3+c x^6}}{128 a^3 x^6}+\frac {b \left (2 a+b x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{48 a^2 x^{12}}-\frac {\left (a+b x^3+c x^6\right )^{5/2}}{15 a x^{15}}+\frac {b \left (b^2-4 a c\right )^2 \text {arctanh}\left (\frac {2 a+b x^3}{2 \sqrt {a} \sqrt {a+b x^3+c x^6}}\right )}{256 a^{7/2}} \]
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Time = 0.09 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1371, 744, 734, 738, 212} \[ \int \frac {\left (a+b x^3+c x^6\right )^{3/2}}{x^{16}} \, dx=\frac {b \left (b^2-4 a c\right )^2 \text {arctanh}\left (\frac {2 a+b x^3}{2 \sqrt {a} \sqrt {a+b x^3+c x^6}}\right )}{256 a^{7/2}}-\frac {b \left (b^2-4 a c\right ) \left (2 a+b x^3\right ) \sqrt {a+b x^3+c x^6}}{128 a^3 x^6}+\frac {b \left (2 a+b x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{48 a^2 x^{12}}-\frac {\left (a+b x^3+c x^6\right )^{5/2}}{15 a x^{15}} \]
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Rule 212
Rule 734
Rule 738
Rule 744
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^6} \, dx,x,x^3\right ) \\ & = -\frac {\left (a+b x^3+c x^6\right )^{5/2}}{15 a x^{15}}-\frac {b \text {Subst}\left (\int \frac {\left (a+b x+c x^2\right )^{3/2}}{x^5} \, dx,x,x^3\right )}{6 a} \\ & = \frac {b \left (2 a+b x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{48 a^2 x^{12}}-\frac {\left (a+b x^3+c x^6\right )^{5/2}}{15 a x^{15}}+\frac {\left (b \left (b^2-4 a c\right )\right ) \text {Subst}\left (\int \frac {\sqrt {a+b x+c x^2}}{x^3} \, dx,x,x^3\right )}{32 a^2} \\ & = -\frac {b \left (b^2-4 a c\right ) \left (2 a+b x^3\right ) \sqrt {a+b x^3+c x^6}}{128 a^3 x^6}+\frac {b \left (2 a+b x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{48 a^2 x^{12}}-\frac {\left (a+b x^3+c x^6\right )^{5/2}}{15 a x^{15}}-\frac {\left (b \left (b^2-4 a c\right )^2\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x+c x^2}} \, dx,x,x^3\right )}{256 a^3} \\ & = -\frac {b \left (b^2-4 a c\right ) \left (2 a+b x^3\right ) \sqrt {a+b x^3+c x^6}}{128 a^3 x^6}+\frac {b \left (2 a+b x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{48 a^2 x^{12}}-\frac {\left (a+b x^3+c x^6\right )^{5/2}}{15 a x^{15}}+\frac {\left (b \left (b^2-4 a c\right )^2\right ) \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 )}{128 a^3} \\ & = -\frac {b \left (b^2-4 a c\right ) \left (2 a+b x^3\right ) \sqrt {a+b x^3+c x^6}}{128 a^3 x^6}+\frac {b \left (2 a+b x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{48 a^2 x^{12}}-\frac {\left (a+b x^3+c x^6\right )^{5/2}}{15 a x^{15}}+\frac {b \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 )}{256 a^{7/2}} \\ \end{align*}
Time = 0.97 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.99 \[ \int \frac {\left (a+b x^3+c x^6\right )^{3/2}}{x^{16}} \, dx=\frac {-\frac {\sqrt {a} \sqrt {a+b x^3+c x^6} \left (128 a^4+15 b^4 x^{12}-10 a b^2 x^9 \left (b+10 c x^3\right )+16 a^3 \left (11 b x^3+16 c x^6\right )+8 a^2 x^6 \left (b^2+7 b c x^3+16 c^2 x^6\right )\right )}{x^{15}}-15 b \left (b^2-4 a c\right )^2 \text {arctanh}\left (\frac {\sqrt {c} x^3-\sqrt {a+b x^3+c x^6}}{\sqrt {a}}\right )}{1920 a^{7/2}} \]
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\[\int \frac {\left (c \,x^{6}+b \,x^{3}+a \right )^{\frac {3}{2}}}{x^{16}}d x\]
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none
Time = 0.35 (sec) , antiderivative size = 383, normalized size of antiderivative = 2.36 \[ \int \frac {\left (a+b x^3+c x^6\right )^{3/2}}{x^{16}} \, dx=\left [\frac {15 \, {\left (b^{5} - 8 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} \sqrt {a} x^{15} \log \left (-\frac {{\left (b^{2} + 4 \, a c\right )} x^{6} + 8 \, a b x^{3} + 4 \, \sqrt {c x^{6} + b x^{3} + a} {\left (b x^{3} + 2 \, a\right )} \sqrt {a} + 8 \, a^{2}}{x^{6}}\right ) - 4 \, {\left ({\left (15 \, a b^{4} - 100 \, a^{2} b^{2} c + 128 \, a^{3} c^{2}\right )} x^{12} - 2 \, {\left (5 \, a^{2} b^{3} - 28 \, a^{3} b c\right )} x^{9} + 176 \, a^{4} b x^{3} + 8 \, {\left (a^{3} b^{2} + 32 \, a^{4} c\right )} x^{6} + 128 \, a^{5}\right )} \sqrt {c x^{6} + b x^{3} + a}}{7680 \, a^{4} x^{15}}, -\frac {15 \, {\left (b^{5} - 8 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} \sqrt {-a} x^{15} \arctan \left (\frac {\sqrt {c x^{6} + b x^{3} + a} {\left (b x^{3} + 2 \, a\right )} \sqrt {-a}}{2 \, {\left (a c x^{6} + a b x^{3} + a^{2}\right )}}\right ) + 2 \, {\left ({\left (15 \, a b^{4} - 100 \, a^{2} b^{2} c + 128 \, a^{3} c^{2}\right )} x^{12} - 2 \, {\left (5 \, a^{2} b^{3} - 28 \, a^{3} b c\right )} x^{9} + 176 \, a^{4} b x^{3} + 8 \, {\left (a^{3} b^{2} + 32 \, a^{4} c\right )} x^{6} + 128 \, a^{5}\right )} \sqrt {c x^{6} + b x^{3} + a}}{3840 \, a^{4} x^{15}}\right ] \]
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\[ \int \frac {\left (a+b x^3+c x^6\right )^{3/2}}{x^{16}} \, dx=\int \frac {\left (a + b x^{3} + c x^{6}\right )^{\frac {3}{2}}}{x^{16}}\, dx \]
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Exception generated. \[ \int \frac {\left (a+b x^3+c x^6\right )^{3/2}}{x^{16}} \, dx=\text {Exception raised: ValueError} \]
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\[ \int \frac {\left (a+b x^3+c x^6\right )^{3/2}}{x^{16}} \, dx=\int { \frac {{\left (c x^{6} + b x^{3} + a\right )}^{\frac {3}{2}}}{x^{16}} \,d x } \]
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Timed out. \[ \int \frac {\left (a+b x^3+c x^6\right )^{3/2}}{x^{16}} \, dx=\int \frac {{\left (c\,x^6+b\,x^3+a\right )}^{3/2}}{x^{16}} \,d x \]
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