Integrand size = 20, antiderivative size = 256 \[ \int \frac {1}{x^{10} \left (a+b x^3+c x^6\right )^{3/2}} \, dx=\frac {2 \left (b^2-2 a c+b c x^3\right )}{3 a \left (b^2-4 a c\right ) x^9 \sqrt {a+b x^3+c x^6}}-\frac {\left (7 b^2-16 a c\right ) \sqrt {a+b x^3+c x^6}}{9 a^2 \left (b^2-4 a c\right ) x^9}+\frac {b \left (35 b^2-116 a c\right ) \sqrt {a+b x^3+c x^6}}{36 a^3 \left (b^2-4 a c\right ) x^6}-\frac {\left (105 b^4-460 a b^2 c+256 a^2 c^2\right ) \sqrt {a+b x^3+c x^6}}{72 a^4 \left (b^2-4 a c\right ) x^3}+\frac {5 b \left (7 b^2-12 a c\right ) \text {arctanh}\left (\frac {2 a+b x^3}{2 \sqrt {a} \sqrt {a+b x^3+c x^6}}\right )}{48 a^{9/2}} \]
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Time = 0.19 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {1371, 754, 848, 820, 738, 212} \[ \int \frac {1}{x^{10} \left (a+b x^3+c x^6\right )^{3/2}} \, dx=\frac {5 b \left (7 b^2-12 a c\right ) \text {arctanh}\left (\frac {2 a+b x^3}{2 \sqrt {a} \sqrt {a+b x^3+c x^6}}\right )}{48 a^{9/2}}+\frac {b \left (35 b^2-116 a c\right ) \sqrt {a+b x^3+c x^6}}{36 a^3 x^6 \left (b^2-4 a c\right )}-\frac {\left (7 b^2-16 a c\right ) \sqrt {a+b x^3+c x^6}}{9 a^2 x^9 \left (b^2-4 a c\right )}-\frac {\left (256 a^2 c^2-460 a b^2 c+105 b^4\right ) \sqrt {a+b x^3+c x^6}}{72 a^4 x^3 \left (b^2-4 a c\right )}+\frac {2 \left (-2 a c+b^2+b c x^3\right )}{3 a x^9 \left (b^2-4 a c\right ) \sqrt {a+b x^3+c x^6}} \]
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Rule 212
Rule 738
Rule 754
Rule 820
Rule 848
Rule 1371
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \text {Subst}\left (\int \frac {1}{x^4 \left (a+b x+c x^2\right )^{3/2}} \, dx,x,x^3\right ) \\ & = \frac {2 \left (b^2-2 a c+b c x^3\right )}{3 a \left (b^2-4 a c\right ) x^9 \sqrt {a+b x^3+c x^6}}-\frac {2 \text {Subst}\left (\int \frac {\frac {1}{2} \left (-7 b^2+16 a c\right )-3 b c x}{x^4 \sqrt {a+b x+c x^2}} \, dx,x,x^3\right )}{3 a \left (b^2-4 a c\right )} \\ & = \frac {2 \left (b^2-2 a c+b c x^3\right )}{3 a \left (b^2-4 a c\right ) x^9 \sqrt {a+b x^3+c x^6}}-\frac {\left (7 b^2-16 a c\right ) \sqrt {a+b x^3+c x^6}}{9 a^2 \left (b^2-4 a c\right ) x^9}+\frac {2 \text {Subst}\left (\int \frac {-\frac {1}{4} b \left (35 b^2-116 a c\right )-c \left (7 b^2-16 a c\right ) x}{x^3 \sqrt {a+b x+c x^2}} \, dx,x,x^3\right )}{9 a^2 \left (b^2-4 a c\right )} \\ & = \frac {2 \left (b^2-2 a c+b c x^3\right )}{3 a \left (b^2-4 a c\right ) x^9 \sqrt {a+b x^3+c x^6}}-\frac {\left (7 b^2-16 a c\right ) \sqrt {a+b x^3+c x^6}}{9 a^2 \left (b^2-4 a c\right ) x^9}+\frac {b \left (35 b^2-116 a c\right ) \sqrt {a+b x^3+c x^6}}{36 a^3 \left (b^2-4 a c\right ) x^6}-\frac {\text {Subst}\left (\int \frac {\frac {1}{8} \left (-105 b^4+460 a b^2 c-256 a^2 c^2\right )-\frac {1}{4} b c \left (35 b^2-116 a c\right ) x}{x^2 \sqrt {a+b x+c x^2}} \, dx,x,x^3\right )}{9 a^3 \left (b^2-4 a c\right )} \\ & = \frac {2 \left (b^2-2 a c+b c x^3\right )}{3 a \left (b^2-4 a c\right ) x^9 \sqrt {a+b x^3+c x^6}}-\frac {\left (7 b^2-16 a c\right ) \sqrt {a+b x^3+c x^6}}{9 a^2 \left (b^2-4 a c\right ) x^9}+\frac {b \left (35 b^2-116 a c\right ) \sqrt {a+b x^3+c x^6}}{36 a^3 \left (b^2-4 a c\right ) x^6}-\frac {\left (105 b^4-460 a b^2 c+256 a^2 c^2\right ) \sqrt {a+b x^3+c x^6}}{72 a^4 \left (b^2-4 a c\right ) x^3}-\frac {\left (5 b \left (7 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^4} \\ & = \frac {2 \left (b^2-2 a c+b c x^3\right )}{3 a \left (b^2-4 a c\right ) x^9 \sqrt {a+b x^3+c x^6}}-\frac {\left (7 b^2-16 a c\right ) \sqrt {a+b x^3+c x^6}}{9 a^2 \left (b^2-4 a c\right ) x^9}+\frac {b \left (35 b^2-116 a c\right ) \sqrt {a+b x^3+c x^6}}{36 a^3 \left (b^2-4 a c\right ) x^6}-\frac {\left (105 b^4-460 a b^2 c+256 a^2 c^2\right ) \sqrt {a+b x^3+c x^6}}{72 a^4 \left (b^2-4 a c\right ) x^3}+\frac {\left (5 b \left (7 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^4} \\ & = \frac {2 \left (b^2-2 a c+b c x^3\right )}{3 a \left (b^2-4 a c\right ) x^9 \sqrt {a+b x^3+c x^6}}-\frac {\left (7 b^2-16 a c\right ) \sqrt {a+b x^3+c x^6}}{9 a^2 \left (b^2-4 a c\right ) x^9}+\frac {b \left (35 b^2-116 a c\right ) \sqrt {a+b x^3+c x^6}}{36 a^3 \left (b^2-4 a c\right ) x^6}-\frac {\left (105 b^4-460 a b^2 c+256 a^2 c^2\right ) \sqrt {a+b x^3+c x^6}}{72 a^4 \left (b^2-4 a c\right ) x^3}+\frac {5 b \left (7 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^{9/2}} \\ \end{align*}
Time = 0.92 (sec) , antiderivative size = 210, normalized size of antiderivative = 0.82 \[ \int \frac {1}{x^{10} \left (a+b x^3+c x^6\right )^{3/2}} \, dx=\frac {-32 a^4 c+105 b^4 x^9 \left (b+c x^3\right )+5 a b^2 x^6 \left (7 b^2-106 b c x^3-92 c^2 x^6\right )+8 a^3 \left (b^2+7 b c x^3+16 c^2 x^6\right )+2 a^2 x^3 \left (-7 b^3-86 b^2 c x^3+244 b c^2 x^6+128 c^3 x^9\right )}{72 a^4 \left (-b^2+4 a c\right ) x^9 \sqrt {a+b x^3+c x^6}}+\frac {5 b \left (-7 b^2+12 a c\right ) \text {arctanh}\left (\frac {\sqrt {c} x^3-\sqrt {a+b x^3+c x^6}}{\sqrt {a}}\right )}{24 a^{9/2}} \]
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\[\int \frac {1}{x^{10} \left (c \,x^{6}+b \,x^{3}+a \right )^{\frac {3}{2}}}d x\]
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Time = 0.41 (sec) , antiderivative size = 705, normalized size of antiderivative = 2.75 \[ \int \frac {1}{x^{10} \left (a+b x^3+c x^6\right )^{3/2}} \, dx=\left [-\frac {15 \, {\left ({\left (7 \, b^{5} c - 40 \, a b^{3} c^{2} + 48 \, a^{2} b c^{3}\right )} x^{15} + {\left (7 \, b^{6} - 40 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2}\right )} x^{12} + {\left (7 \, a b^{5} - 40 \, a^{2} b^{3} c + 48 \, a^{3} b c^{2}\right )} x^{9}\right )} \sqrt {a} \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 (105 \, a b^{4} c - 460 \, a^{2} b^{2} c^{2} + 256 \, a^{3} c^{3}\right )} x^{12} + {\left (105 \, a b^{5} - 530 \, a^{2} b^{3} c + 488 \, a^{3} b c^{2}\right )} x^{9} + {\left (35 \, a^{2} b^{4} - 172 \, a^{3} b^{2} c + 128 \, a^{4} c^{2}\right )} x^{6} + 8 \, a^{4} b^{2} - 32 \, a^{5} c - 14 \, {\left (a^{3} b^{3} - 4 \, a^{4} b c\right )} x^{3}\right )} \sqrt {c x^{6} + b x^{3} + a}}{288 \, {\left ({\left (a^{5} b^{2} c - 4 \, a^{6} c^{2}\right )} x^{15} + {\left (a^{5} b^{3} - 4 \, a^{6} b c\right )} x^{12} + {\left (a^{6} b^{2} - 4 \, a^{7} c\right )} x^{9}\right )}}, -\frac {15 \, {\left ({\left (7 \, b^{5} c - 40 \, a b^{3} c^{2} + 48 \, a^{2} b c^{3}\right )} x^{15} + {\left (7 \, b^{6} - 40 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2}\right )} x^{12} + {\left (7 \, a b^{5} - 40 \, a^{2} b^{3} c + 48 \, a^{3} b c^{2}\right )} x^{9}\right )} \sqrt {-a} \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 (105 \, a b^{4} c - 460 \, a^{2} b^{2} c^{2} + 256 \, a^{3} c^{3}\right )} x^{12} + {\left (105 \, a b^{5} - 530 \, a^{2} b^{3} c + 488 \, a^{3} b c^{2}\right )} x^{9} + {\left (35 \, a^{2} b^{4} - 172 \, a^{3} b^{2} c + 128 \, a^{4} c^{2}\right )} x^{6} + 8 \, a^{4} b^{2} - 32 \, a^{5} c - 14 \, {\left (a^{3} b^{3} - 4 \, a^{4} b c\right )} x^{3}\right )} \sqrt {c x^{6} + b x^{3} + a}}{144 \, {\left ({\left (a^{5} b^{2} c - 4 \, a^{6} c^{2}\right )} x^{15} + {\left (a^{5} b^{3} - 4 \, a^{6} b c\right )} x^{12} + {\left (a^{6} b^{2} - 4 \, a^{7} c\right )} x^{9}\right )}}\right ] \]
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\[ \int \frac {1}{x^{10} \left (a+b x^3+c x^6\right )^{3/2}} \, dx=\int \frac {1}{x^{10} \left (a + b x^{3} + c x^{6}\right )^{\frac {3}{2}}}\, dx \]
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Exception generated. \[ \int \frac {1}{x^{10} \left (a+b x^3+c x^6\right )^{3/2}} \, dx=\text {Exception raised: ValueError} \]
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\[ \int \frac {1}{x^{10} \left (a+b x^3+c x^6\right )^{3/2}} \, dx=\int { \frac {1}{{\left (c x^{6} + b x^{3} + a\right )}^{\frac {3}{2}} x^{10}} \,d x } \]
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Timed out. \[ \int \frac {1}{x^{10} \left (a+b x^3+c x^6\right )^{3/2}} \, dx=\int \frac {1}{x^{10}\,{\left (c\,x^6+b\,x^3+a\right )}^{3/2}} \,d x \]
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