Integrand size = 20, antiderivative size = 145 \[ \int \frac {1}{x^{10} \sqrt {a+b x^3+c x^6}} \, dx=-\frac {\sqrt {a+b x^3+c x^6}}{9 a x^9}+\frac {5 b \sqrt {a+b x^3+c x^6}}{36 a^2 x^6}-\frac {\left (15 b^2-16 a c\right ) \sqrt {a+b x^3+c x^6}}{72 a^3 x^3}+\frac {b \left (5 b^2-12 a c\right ) \text {arctanh}\left (\frac {2 a+b x^3}{2 \sqrt {a} \sqrt {a+b x^3+c x^6}}\right )}{48 a^{7/2}} \]
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Time = 0.11 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {1371, 758, 848, 820, 738, 212} \[ \int \frac {1}{x^{10} \sqrt {a+b x^3+c x^6}} \, dx=\frac {b \left (5 b^2-12 a c\right ) \text {arctanh}\left (\frac {2 a+b x^3}{2 \sqrt {a} \sqrt {a+b x^3+c x^6}}\right )}{48 a^{7/2}}-\frac {\left (15 b^2-16 a c\right ) \sqrt {a+b x^3+c x^6}}{72 a^3 x^3}+\frac {5 b \sqrt {a+b x^3+c x^6}}{36 a^2 x^6}-\frac {\sqrt {a+b x^3+c x^6}}{9 a x^9} \]
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Rule 212
Rule 738
Rule 758
Rule 820
Rule 848
Rule 1371
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \text {Subst}\left (\int \frac {1}{x^4 \sqrt {a+b x+c x^2}} \, dx,x,x^3\right ) \\ & = -\frac {\sqrt {a+b x^3+c x^6}}{9 a x^9}-\frac {\text {Subst}\left (\int \frac {\frac {5 b}{2}+2 c x}{x^3 \sqrt {a+b x+c x^2}} \, dx,x,x^3\right )}{9 a} \\ & = -\frac {\sqrt {a+b x^3+c x^6}}{9 a x^9}+\frac {5 b \sqrt {a+b x^3+c x^6}}{36 a^2 x^6}+\frac {\text {Subst}\left (\int \frac {\frac {1}{4} \left (15 b^2-16 a c\right )+\frac {5 b c x}{2}}{x^2 \sqrt {a+b x+c x^2}} \, dx,x,x^3\right )}{18 a^2} \\ & = -\frac {\sqrt {a+b x^3+c x^6}}{9 a x^9}+\frac {5 b \sqrt {a+b x^3+c x^6}}{36 a^2 x^6}-\frac {\left (15 b^2-16 a c\right ) \sqrt {a+b x^3+c x^6}}{72 a^3 x^3}-\frac {\left (b \left (5 b^2-12 a c\right )\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x+c x^2}} \, dx,x,x^3\right )}{48 a^3} \\ & = -\frac {\sqrt {a+b x^3+c x^6}}{9 a x^9}+\frac {5 b \sqrt {a+b x^3+c x^6}}{36 a^2 x^6}-\frac {\left (15 b^2-16 a c\right ) \sqrt {a+b x^3+c x^6}}{72 a^3 x^3}+\frac {\left (b \left (5 b^2-12 a c\right )\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 )}{24 a^3} \\ & = -\frac {\sqrt {a+b x^3+c x^6}}{9 a x^9}+\frac {5 b \sqrt {a+b x^3+c x^6}}{36 a^2 x^6}-\frac {\left (15 b^2-16 a c\right ) \sqrt {a+b x^3+c x^6}}{72 a^3 x^3}+\frac {b \left (5 b^2-12 a c\right ) \tanh ^{-1}\left (\frac {2 a+b x^3}{2 \sqrt {a} \sqrt {a+b x^3+c x^6}}\right )}{48 a^{7/2}} \\ \end{align*}
Time = 0.37 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.76 \[ \int \frac {1}{x^{10} \sqrt {a+b x^3+c x^6}} \, dx=\frac {\sqrt {a+b x^3+c x^6} \left (-8 a^2+10 a b x^3-15 b^2 x^6+16 a c x^6\right )}{72 a^3 x^9}+\frac {\left (-5 b^3+12 a b c\right ) \text {arctanh}\left (\frac {\sqrt {c} x^3-\sqrt {a+b x^3+c x^6}}{\sqrt {a}}\right )}{24 a^{7/2}} \]
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\[\int \frac {1}{x^{10} \sqrt {c \,x^{6}+b \,x^{3}+a}}d x\]
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Time = 0.28 (sec) , antiderivative size = 263, normalized size of antiderivative = 1.81 \[ \int \frac {1}{x^{10} \sqrt {a+b x^3+c x^6}} \, dx=\left [-\frac {3 \, {\left (5 \, b^{3} - 12 \, a b c\right )} \sqrt {a} x^{9} \log \left (-\frac {{\left (b^{2} + 4 \, a c\right )} x^{6} + 8 \, a b x^{3} - 4 \, \sqrt {c x^{6} + b x^{3} + a} {\left (b x^{3} + 2 \, a\right )} \sqrt {a} + 8 \, a^{2}}{x^{6}}\right ) + 4 \, {\left ({\left (15 \, a b^{2} - 16 \, a^{2} c\right )} x^{6} - 10 \, a^{2} b x^{3} + 8 \, a^{3}\right )} \sqrt {c x^{6} + b x^{3} + a}}{288 \, a^{4} x^{9}}, -\frac {3 \, {\left (5 \, b^{3} - 12 \, a b c\right )} \sqrt {-a} x^{9} \arctan \left (\frac {\sqrt {c x^{6} + b x^{3} + a} {\left (b x^{3} + 2 \, a\right )} \sqrt {-a}}{2 \, {\left (a c x^{6} + a b x^{3} + a^{2}\right )}}\right ) + 2 \, {\left ({\left (15 \, a b^{2} - 16 \, a^{2} c\right )} x^{6} - 10 \, a^{2} b x^{3} + 8 \, a^{3}\right )} \sqrt {c x^{6} + b x^{3} + a}}{144 \, a^{4} x^{9}}\right ] \]
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\[ \int \frac {1}{x^{10} \sqrt {a+b x^3+c x^6}} \, dx=\int \frac {1}{x^{10} \sqrt {a + b x^{3} + c x^{6}}}\, dx \]
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Exception generated. \[ \int \frac {1}{x^{10} \sqrt {a+b x^3+c x^6}} \, dx=\text {Exception raised: ValueError} \]
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\[ \int \frac {1}{x^{10} \sqrt {a+b x^3+c x^6}} \, dx=\int { \frac {1}{\sqrt {c x^{6} + b x^{3} + a} x^{10}} \,d x } \]
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Timed out. \[ \int \frac {1}{x^{10} \sqrt {a+b x^3+c x^6}} \, dx=\int \frac {1}{x^{10}\,\sqrt {c\,x^6+b\,x^3+a}} \,d x \]
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