Integrand size = 20, antiderivative size = 108 \[ \int \frac {1}{x^7 \sqrt {a+b x^3+c x^6}} \, dx=-\frac {\sqrt {a+b x^3+c x^6}}{6 a x^6}+\frac {b \sqrt {a+b x^3+c x^6}}{4 a^2 x^3}-\frac {\left (3 b^2-4 a c\right ) \text {arctanh}\left (\frac {2 a+b x^3}{2 \sqrt {a} \sqrt {a+b x^3+c x^6}}\right )}{24 a^{5/2}} \]
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Time = 0.07 (sec) , antiderivative size = 108, 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, 758, 820, 738, 212} \[ \int \frac {1}{x^7 \sqrt {a+b x^3+c x^6}} \, dx=-\frac {\left (3 b^2-4 a c\right ) \text {arctanh}\left (\frac {2 a+b x^3}{2 \sqrt {a} \sqrt {a+b x^3+c x^6}}\right )}{24 a^{5/2}}+\frac {b \sqrt {a+b x^3+c x^6}}{4 a^2 x^3}-\frac {\sqrt {a+b x^3+c x^6}}{6 a x^6} \]
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Rule 212
Rule 738
Rule 758
Rule 820
Rule 1371
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \text {Subst}\left (\int \frac {1}{x^3 \sqrt {a+b x+c x^2}} \, dx,x,x^3\right ) \\ & = -\frac {\sqrt {a+b x^3+c x^6}}{6 a x^6}-\frac {\text {Subst}\left (\int \frac {\frac {3 b}{2}+c x}{x^2 \sqrt {a+b x+c x^2}} \, dx,x,x^3\right )}{6 a} \\ & = -\frac {\sqrt {a+b x^3+c x^6}}{6 a x^6}+\frac {b \sqrt {a+b x^3+c x^6}}{4 a^2 x^3}+\frac {\left (3 b^2-4 a c\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x+c x^2}} \, dx,x,x^3\right )}{24 a^2} \\ & = -\frac {\sqrt {a+b x^3+c x^6}}{6 a x^6}+\frac {b \sqrt {a+b x^3+c x^6}}{4 a^2 x^3}-\frac {\left (3 b^2-4 a c\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 )}{12 a^2} \\ & = -\frac {\sqrt {a+b x^3+c x^6}}{6 a x^6}+\frac {b \sqrt {a+b x^3+c x^6}}{4 a^2 x^3}-\frac {\left (3 b^2-4 a c\right ) \tanh ^{-1}\left (\frac {2 a+b x^3}{2 \sqrt {a} \sqrt {a+b x^3+c x^6}}\right )}{24 a^{5/2}} \\ \end{align*}
Time = 0.23 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.84 \[ \int \frac {1}{x^7 \sqrt {a+b x^3+c x^6}} \, dx=\frac {\left (-2 a+3 b x^3\right ) \sqrt {a+b x^3+c x^6}}{12 a^2 x^6}+\frac {\left (3 b^2-4 a c\right ) \text {arctanh}\left (\frac {\sqrt {c} x^3-\sqrt {a+b x^3+c x^6}}{\sqrt {a}}\right )}{12 a^{5/2}} \]
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\[\int \frac {1}{x^{7} \sqrt {c \,x^{6}+b \,x^{3}+a}}d x\]
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none
Time = 0.27 (sec) , antiderivative size = 221, normalized size of antiderivative = 2.05 \[ \int \frac {1}{x^7 \sqrt {a+b x^3+c x^6}} \, dx=\left [-\frac {{\left (3 \, b^{2} - 4 \, a c\right )} \sqrt {a} x^{6} \log \left (-\frac {{\left (b^{2} + 4 \, a c\right )} x^{6} + 8 \, a b x^{3} + 4 \, \sqrt {c x^{6} + b x^{3} + a} {\left (b x^{3} + 2 \, a\right )} \sqrt {a} + 8 \, a^{2}}{x^{6}}\right ) - 4 \, \sqrt {c x^{6} + b x^{3} + a} {\left (3 \, a b x^{3} - 2 \, a^{2}\right )}}{48 \, a^{3} x^{6}}, \frac {{\left (3 \, b^{2} - 4 \, a c\right )} \sqrt {-a} x^{6} \arctan \left (\frac {\sqrt {c x^{6} + b x^{3} + a} {\left (b x^{3} + 2 \, a\right )} \sqrt {-a}}{2 \, {\left (a c x^{6} + a b x^{3} + a^{2}\right )}}\right ) + 2 \, \sqrt {c x^{6} + b x^{3} + a} {\left (3 \, a b x^{3} - 2 \, a^{2}\right )}}{24 \, a^{3} x^{6}}\right ] \]
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\[ \int \frac {1}{x^7 \sqrt {a+b x^3+c x^6}} \, dx=\int \frac {1}{x^{7} \sqrt {a + b x^{3} + c x^{6}}}\, dx \]
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Exception generated. \[ \int \frac {1}{x^7 \sqrt {a+b x^3+c x^6}} \, dx=\text {Exception raised: ValueError} \]
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\[ \int \frac {1}{x^7 \sqrt {a+b x^3+c x^6}} \, dx=\int { \frac {1}{\sqrt {c x^{6} + b x^{3} + a} x^{7}} \,d x } \]
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Timed out. \[ \int \frac {1}{x^7 \sqrt {a+b x^3+c x^6}} \, dx=\int \frac {1}{x^7\,\sqrt {c\,x^6+b\,x^3+a}} \,d x \]
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