Integrand size = 20, antiderivative size = 204 \[ \int x^8 \left (a+b x^3+c x^6\right )^{3/2} \, dx=-\frac {\left (b^2-4 a c\right ) \left (7 b^2-4 a c\right ) \left (b+2 c x^3\right ) \sqrt {a+b x^3+c x^6}}{1536 c^4}+\frac {\left (7 b^2-4 a c\right ) \left (b+2 c x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{576 c^3}-\frac {7 b \left (a+b x^3+c x^6\right )^{5/2}}{180 c^2}+\frac {x^3 \left (a+b x^3+c x^6\right )^{5/2}}{18 c}+\frac {\left (b^2-4 a c\right )^2 \left (7 b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c x^3}{2 \sqrt {c} \sqrt {a+b x^3+c x^6}}\right )}{3072 c^{9/2}} \]
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Time = 0.12 (sec) , antiderivative size = 204, 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, 756, 654, 626, 635, 212} \[ \int x^8 \left (a+b x^3+c x^6\right )^{3/2} \, dx=\frac {\left (b^2-4 a c\right )^2 \left (7 b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c x^3}{2 \sqrt {c} \sqrt {a+b x^3+c x^6}}\right )}{3072 c^{9/2}}-\frac {\left (b^2-4 a c\right ) \left (7 b^2-4 a c\right ) \left (b+2 c x^3\right ) \sqrt {a+b x^3+c x^6}}{1536 c^4}+\frac {\left (7 b^2-4 a c\right ) \left (b+2 c x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{576 c^3}-\frac {7 b \left (a+b x^3+c x^6\right )^{5/2}}{180 c^2}+\frac {x^3 \left (a+b x^3+c x^6\right )^{5/2}}{18 c} \]
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Rule 212
Rule 626
Rule 635
Rule 654
Rule 756
Rule 1371
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \text {Subst}\left (\int x^2 \left (a+b x+c x^2\right )^{3/2} \, dx,x,x^3\right ) \\ & = \frac {x^3 \left (a+b x^3+c x^6\right )^{5/2}}{18 c}+\frac {\text {Subst}\left (\int \left (-a-\frac {7 b x}{2}\right ) \left (a+b x+c x^2\right )^{3/2} \, dx,x,x^3\right )}{18 c} \\ & = -\frac {7 b \left (a+b x^3+c x^6\right )^{5/2}}{180 c^2}+\frac {x^3 \left (a+b x^3+c x^6\right )^{5/2}}{18 c}+\frac {\left (7 b^2-4 a c\right ) \text {Subst}\left (\int \left (a+b x+c x^2\right )^{3/2} \, dx,x,x^3\right )}{72 c^2} \\ & = \frac {\left (7 b^2-4 a c\right ) \left (b+2 c x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{576 c^3}-\frac {7 b \left (a+b x^3+c x^6\right )^{5/2}}{180 c^2}+\frac {x^3 \left (a+b x^3+c x^6\right )^{5/2}}{18 c}-\frac {\left (\left (b^2-4 a c\right ) \left (7 b^2-4 a c\right )\right ) \text {Subst}\left (\int \sqrt {a+b x+c x^2} \, dx,x,x^3\right )}{384 c^3} \\ & = -\frac {\left (b^2-4 a c\right ) \left (7 b^2-4 a c\right ) \left (b+2 c x^3\right ) \sqrt {a+b x^3+c x^6}}{1536 c^4}+\frac {\left (7 b^2-4 a c\right ) \left (b+2 c x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{576 c^3}-\frac {7 b \left (a+b x^3+c x^6\right )^{5/2}}{180 c^2}+\frac {x^3 \left (a+b x^3+c x^6\right )^{5/2}}{18 c}+\frac {\left (\left (b^2-4 a c\right )^2 \left (7 b^2-4 a c\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x+c x^2}} \, dx,x,x^3\right )}{3072 c^4} \\ & = -\frac {\left (b^2-4 a c\right ) \left (7 b^2-4 a c\right ) \left (b+2 c x^3\right ) \sqrt {a+b x^3+c x^6}}{1536 c^4}+\frac {\left (7 b^2-4 a c\right ) \left (b+2 c x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{576 c^3}-\frac {7 b \left (a+b x^3+c x^6\right )^{5/2}}{180 c^2}+\frac {x^3 \left (a+b x^3+c x^6\right )^{5/2}}{18 c}+\frac {\left (\left (b^2-4 a c\right )^2 \left (7 b^2-4 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 )}{1536 c^4} \\ & = -\frac {\left (b^2-4 a c\right ) \left (7 b^2-4 a c\right ) \left (b+2 c x^3\right ) \sqrt {a+b x^3+c x^6}}{1536 c^4}+\frac {\left (7 b^2-4 a c\right ) \left (b+2 c x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{576 c^3}-\frac {7 b \left (a+b x^3+c x^6\right )^{5/2}}{180 c^2}+\frac {x^3 \left (a+b x^3+c x^6\right )^{5/2}}{18 c}+\frac {\left (b^2-4 a c\right )^2 \left (7 b^2-4 a c\right ) \tanh ^{-1}\left (\frac {b+2 c x^3}{2 \sqrt {c} \sqrt {a+b x^3+c x^6}}\right )}{3072 c^{9/2}} \\ \end{align*}
Time = 0.51 (sec) , antiderivative size = 194, normalized size of antiderivative = 0.95 \[ \int x^8 \left (a+b x^3+c x^6\right )^{3/2} \, dx=\frac {2 \sqrt {c} \sqrt {a+b x^3+c x^6} \left (-105 b^5+70 b^4 c x^3+8 b^3 c \left (95 a-7 c x^6\right )+48 b^2 c^2 x^3 \left (-9 a+c x^6\right )+160 c^3 x^3 \left (3 a^2+14 a c x^6+8 c^2 x^{12}\right )+16 b c^2 \left (-81 a^2+18 a c x^6+104 c^2 x^{12}\right )\right )-15 \left (b^2-4 a c\right )^2 \left (7 b^2-4 a c\right ) \log \left (b+2 c x^3-2 \sqrt {c} \sqrt {a+b x^3+c x^6}\right )}{46080 c^{9/2}} \]
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\[\int x^{8} \left (c \,x^{6}+b \,x^{3}+a \right )^{\frac {3}{2}}d x\]
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Time = 0.28 (sec) , antiderivative size = 451, normalized size of antiderivative = 2.21 \[ \int x^8 \left (a+b x^3+c x^6\right )^{3/2} \, dx=\left [-\frac {15 \, {\left (7 \, b^{6} - 60 \, a b^{4} c + 144 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} \sqrt {c} \log \left (-8 \, c^{2} x^{6} - 8 \, b c x^{3} - b^{2} + 4 \, \sqrt {c x^{6} + b x^{3} + a} {\left (2 \, c x^{3} + b\right )} \sqrt {c} - 4 \, a c\right ) - 4 \, {\left (1280 \, c^{6} x^{15} + 1664 \, b c^{5} x^{12} + 16 \, {\left (3 \, b^{2} c^{4} + 140 \, a c^{5}\right )} x^{9} - 8 \, {\left (7 \, b^{3} c^{3} - 36 \, a b c^{4}\right )} x^{6} - 105 \, b^{5} c + 760 \, a b^{3} c^{2} - 1296 \, a^{2} b c^{3} + 2 \, {\left (35 \, b^{4} c^{2} - 216 \, a b^{2} c^{3} + 240 \, a^{2} c^{4}\right )} x^{3}\right )} \sqrt {c x^{6} + b x^{3} + a}}{92160 \, c^{5}}, -\frac {15 \, {\left (7 \, b^{6} - 60 \, a b^{4} c + 144 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{6} + b x^{3} + a} {\left (2 \, c x^{3} + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{6} + b c x^{3} + a c\right )}}\right ) - 2 \, {\left (1280 \, c^{6} x^{15} + 1664 \, b c^{5} x^{12} + 16 \, {\left (3 \, b^{2} c^{4} + 140 \, a c^{5}\right )} x^{9} - 8 \, {\left (7 \, b^{3} c^{3} - 36 \, a b c^{4}\right )} x^{6} - 105 \, b^{5} c + 760 \, a b^{3} c^{2} - 1296 \, a^{2} b c^{3} + 2 \, {\left (35 \, b^{4} c^{2} - 216 \, a b^{2} c^{3} + 240 \, a^{2} c^{4}\right )} x^{3}\right )} \sqrt {c x^{6} + b x^{3} + a}}{46080 \, c^{5}}\right ] \]
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\[ \int x^8 \left (a+b x^3+c x^6\right )^{3/2} \, dx=\int x^{8} \left (a + b x^{3} + c x^{6}\right )^{\frac {3}{2}}\, dx \]
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Exception generated. \[ \int x^8 \left (a+b x^3+c x^6\right )^{3/2} \, dx=\text {Exception raised: ValueError} \]
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\[ \int x^8 \left (a+b x^3+c x^6\right )^{3/2} \, dx=\int { {\left (c x^{6} + b x^{3} + a\right )}^{\frac {3}{2}} x^{8} \,d x } \]
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Timed out. \[ \int x^8 \left (a+b x^3+c x^6\right )^{3/2} \, dx=\int x^8\,{\left (c\,x^6+b\,x^3+a\right )}^{3/2} \,d x \]
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