Integrand size = 20, antiderivative size = 155 \[ \int \frac {\left (a+b x^3+c x^6\right )^{3/2}}{x} \, dx=\frac {\left (b^2+8 a c+2 b c x^3\right ) \sqrt {a+b x^3+c x^6}}{24 c}+\frac {1}{9} \left (a+b x^3+c x^6\right )^{3/2}-\frac {1}{3} a^{3/2} \text {arctanh}\left (\frac {2 a+b x^3}{2 \sqrt {a} \sqrt {a+b x^3+c x^6}}\right )-\frac {b \left (b^2-12 a c\right ) \text {arctanh}\left (\frac {b+2 c x^3}{2 \sqrt {c} \sqrt {a+b x^3+c x^6}}\right )}{48 c^{3/2}} \]
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Time = 0.12 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {1371, 748, 828, 857, 635, 212, 738} \[ \int \frac {\left (a+b x^3+c x^6\right )^{3/2}}{x} \, dx=-\frac {1}{3} a^{3/2} \text {arctanh}\left (\frac {2 a+b x^3}{2 \sqrt {a} \sqrt {a+b x^3+c x^6}}\right )-\frac {b \left (b^2-12 a c\right ) \text {arctanh}\left (\frac {b+2 c x^3}{2 \sqrt {c} \sqrt {a+b x^3+c x^6}}\right )}{48 c^{3/2}}+\frac {\left (8 a c+b^2+2 b c x^3\right ) \sqrt {a+b x^3+c x^6}}{24 c}+\frac {1}{9} \left (a+b x^3+c x^6\right )^{3/2} \]
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Rule 212
Rule 635
Rule 738
Rule 748
Rule 828
Rule 857
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} \, dx,x,x^3\right ) \\ & = \frac {1}{9} \left (a+b x^3+c x^6\right )^{3/2}-\frac {1}{6} \text {Subst}\left (\int \frac {(-2 a-b x) \sqrt {a+b x+c x^2}}{x} \, dx,x,x^3\right ) \\ & = \frac {\left (b^2+8 a c+2 b c x^3\right ) \sqrt {a+b x^3+c x^6}}{24 c}+\frac {1}{9} \left (a+b x^3+c x^6\right )^{3/2}+\frac {\text {Subst}\left (\int \frac {8 a^2 c-\frac {1}{2} b \left (b^2-12 a c\right ) x}{x \sqrt {a+b x+c x^2}} \, dx,x,x^3\right )}{24 c} \\ & = \frac {\left (b^2+8 a c+2 b c x^3\right ) \sqrt {a+b x^3+c x^6}}{24 c}+\frac {1}{9} \left (a+b x^3+c x^6\right )^{3/2}+\frac {1}{3} a^2 \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x+c x^2}} \, dx,x,x^3\right )-\frac {\left (b \left (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} \\ & = \frac {\left (b^2+8 a c+2 b c x^3\right ) \sqrt {a+b x^3+c x^6}}{24 c}+\frac {1}{9} \left (a+b x^3+c x^6\right )^{3/2}-\frac {1}{3} \left (2 a^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 )-\frac {\left (b \left (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} \\ & = \frac {\left (b^2+8 a c+2 b c x^3\right ) \sqrt {a+b x^3+c x^6}}{24 c}+\frac {1}{9} \left (a+b x^3+c x^6\right )^{3/2}-\frac {1}{3} a^{3/2} \tanh ^{-1}\left (\frac {2 a+b x^3}{2 \sqrt {a} \sqrt {a+b x^3+c x^6}}\right )-\frac {b \left (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^{3/2}} \\ \end{align*}
Time = 0.51 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.92 \[ \int \frac {\left (a+b x^3+c x^6\right )^{3/2}}{x} \, dx=\frac {1}{144} \left (\frac {2 \sqrt {a+b x^3+c x^6} \left (3 b^2+14 b c x^3+8 c \left (4 a+c x^6\right )\right )}{c}-\frac {3 \left (b^3-12 a b c\right ) \text {arctanh}\left (\frac {b+2 c x^3}{2 \sqrt {c} \sqrt {a+b x^3+c x^6}}\right )}{c^{3/2}}+96 a^{3/2} \text {arctanh}\left (\frac {\sqrt {c} x^3-\sqrt {a+b x^3+c x^6}}{\sqrt {a}}\right )\right ) \]
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\[\int \frac {\left (c \,x^{6}+b \,x^{3}+a \right )^{\frac {3}{2}}}{x}d x\]
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none
Time = 0.35 (sec) , antiderivative size = 727, normalized size of antiderivative = 4.69 \[ \int \frac {\left (a+b x^3+c x^6\right )^{3/2}}{x} \, dx=\left [\frac {48 \, a^{\frac {3}{2}} c^{2} \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 ) - 3 \, {\left (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} + 14 \, b c^{2} x^{3} + 3 \, b^{2} c + 32 \, a c^{2}\right )} \sqrt {c x^{6} + b x^{3} + a}}{288 \, c^{2}}, \frac {24 \, a^{\frac {3}{2}} c^{2} \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 ) + 3 \, {\left (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} + 14 \, b c^{2} x^{3} + 3 \, b^{2} c + 32 \, a c^{2}\right )} \sqrt {c x^{6} + b x^{3} + a}}{144 \, c^{2}}, \frac {96 \, \sqrt {-a} a c^{2} \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 ) - 3 \, {\left (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} + 14 \, b c^{2} x^{3} + 3 \, b^{2} c + 32 \, a c^{2}\right )} \sqrt {c x^{6} + b x^{3} + a}}{288 \, c^{2}}, \frac {48 \, \sqrt {-a} a c^{2} \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 ) + 3 \, {\left (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} + 14 \, b c^{2} x^{3} + 3 \, b^{2} c + 32 \, a c^{2}\right )} \sqrt {c x^{6} + b x^{3} + a}}{144 \, c^{2}}\right ] \]
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\[ \int \frac {\left (a+b x^3+c x^6\right )^{3/2}}{x} \, dx=\int \frac {\left (a + b x^{3} + c x^{6}\right )^{\frac {3}{2}}}{x}\, dx \]
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Exception generated. \[ \int \frac {\left (a+b x^3+c x^6\right )^{3/2}}{x} \, dx=\text {Exception raised: ValueError} \]
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\[ \int \frac {\left (a+b x^3+c x^6\right )^{3/2}}{x} \, dx=\int { \frac {{\left (c x^{6} + b x^{3} + a\right )}^{\frac {3}{2}}}{x} \,d x } \]
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Timed out. \[ \int \frac {\left (a+b x^3+c x^6\right )^{3/2}}{x} \, dx=\int \frac {{\left (c\,x^6+b\,x^3+a\right )}^{3/2}}{x} \,d x \]
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