Integrand size = 20, antiderivative size = 121 \[ \int \frac {x^{11}}{\sqrt {a+b x^3+c x^6}} \, dx=\frac {x^6 \sqrt {a+b x^3+c x^6}}{9 c}+\frac {\left (15 b^2-16 a c-10 b c x^3\right ) \sqrt {a+b x^3+c x^6}}{72 c^3}-\frac {b \left (5 b^2-12 a c\right ) \text {arctanh}\left (\frac {b+2 c x^3}{2 \sqrt {c} \sqrt {a+b x^3+c x^6}}\right )}{48 c^{7/2}} \]
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Time = 0.07 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1371, 756, 793, 635, 212} \[ \int \frac {x^{11}}{\sqrt {a+b x^3+c x^6}} \, dx=-\frac {b \left (5 b^2-12 a c\right ) \text {arctanh}\left (\frac {b+2 c x^3}{2 \sqrt {c} \sqrt {a+b x^3+c x^6}}\right )}{48 c^{7/2}}+\frac {\left (-16 a c+15 b^2-10 b c x^3\right ) \sqrt {a+b x^3+c x^6}}{72 c^3}+\frac {x^6 \sqrt {a+b x^3+c x^6}}{9 c} \]
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Rule 212
Rule 635
Rule 756
Rule 793
Rule 1371
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \text {Subst}\left (\int \frac {x^3}{\sqrt {a+b x+c x^2}} \, dx,x,x^3\right ) \\ & = \frac {x^6 \sqrt {a+b x^3+c x^6}}{9 c}+\frac {\text {Subst}\left (\int \frac {x \left (-2 a-\frac {5 b x}{2}\right )}{\sqrt {a+b x+c x^2}} \, dx,x,x^3\right )}{9 c} \\ & = \frac {x^6 \sqrt {a+b x^3+c x^6}}{9 c}+\frac {\left (15 b^2-16 a c-10 b c x^3\right ) \sqrt {a+b x^3+c x^6}}{72 c^3}-\frac {\left (b \left (5 b^2-12 a c\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x+c x^2}} \, dx,x,x^3\right )}{48 c^3} \\ & = \frac {x^6 \sqrt {a+b x^3+c x^6}}{9 c}+\frac {\left (15 b^2-16 a c-10 b c x^3\right ) \sqrt {a+b x^3+c x^6}}{72 c^3}-\frac {\left (b \left (5 b^2-12 a c\right )\right ) \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 )}{24 c^3} \\ & = \frac {x^6 \sqrt {a+b x^3+c x^6}}{9 c}+\frac {\left (15 b^2-16 a c-10 b c x^3\right ) \sqrt {a+b x^3+c x^6}}{72 c^3}-\frac {b \left (5 b^2-12 a c\right ) \tanh ^{-1}\left (\frac {b+2 c x^3}{2 \sqrt {c} \sqrt {a+b x^3+c x^6}}\right )}{48 c^{7/2}} \\ \end{align*}
Time = 0.23 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.83 \[ \int \frac {x^{11}}{\sqrt {a+b x^3+c x^6}} \, dx=\frac {\sqrt {a+b x^3+c x^6} \left (15 b^2-16 a c-10 b c x^3+8 c^2 x^6\right )}{72 c^3}+\frac {\left (5 b^3-12 a b c\right ) \log \left (b+2 c x^3-2 \sqrt {c} \sqrt {a+b x^3+c x^6}\right )}{48 c^{7/2}} \]
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\[\int \frac {x^{11}}{\sqrt {c \,x^{6}+b \,x^{3}+a}}d x\]
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none
Time = 0.27 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.99 \[ \int \frac {x^{11}}{\sqrt {a+b x^3+c x^6}} \, dx=\left [-\frac {3 \, {\left (5 \, b^{3} - 12 \, a b c\right )} \sqrt {c} \log \left (-8 \, c^{2} x^{6} - 8 \, b c x^{3} - b^{2} - 4 \, \sqrt {c x^{6} + b x^{3} + a} {\left (2 \, c x^{3} + b\right )} \sqrt {c} - 4 \, a c\right ) - 4 \, {\left (8 \, c^{3} x^{6} - 10 \, b c^{2} x^{3} + 15 \, b^{2} c - 16 \, a c^{2}\right )} \sqrt {c x^{6} + b x^{3} + a}}{288 \, c^{4}}, \frac {3 \, {\left (5 \, b^{3} - 12 \, a b c\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{6} + b x^{3} + a} {\left (2 \, c x^{3} + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{6} + b c x^{3} + a c\right )}}\right ) + 2 \, {\left (8 \, c^{3} x^{6} - 10 \, b c^{2} x^{3} + 15 \, b^{2} c - 16 \, a c^{2}\right )} \sqrt {c x^{6} + b x^{3} + a}}{144 \, c^{4}}\right ] \]
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\[ \int \frac {x^{11}}{\sqrt {a+b x^3+c x^6}} \, dx=\int \frac {x^{11}}{\sqrt {a + b x^{3} + c x^{6}}}\, dx \]
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Exception generated. \[ \int \frac {x^{11}}{\sqrt {a+b x^3+c x^6}} \, dx=\text {Exception raised: ValueError} \]
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\[ \int \frac {x^{11}}{\sqrt {a+b x^3+c x^6}} \, dx=\int { \frac {x^{11}}{\sqrt {c x^{6} + b x^{3} + a}} \,d x } \]
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Timed out. \[ \int \frac {x^{11}}{\sqrt {a+b x^3+c x^6}} \, dx=\int \frac {x^{11}}{\sqrt {c\,x^6+b\,x^3+a}} \,d x \]
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