Integrand size = 52, antiderivative size = 207 \[ \int \frac {\sqrt {b^2+a^2 x^3} \left (2 b^2+c x^3+a^2 x^6\right )}{x^7 \left (-b^2+a^2 x^6\right )} \, dx=\frac {\sqrt {b^2+a^2 x^3} \left (2 b^2+a^2 x^3+2 c x^3\right )}{6 b^2 x^6}+\frac {\left (-a^4+12 a^2 b^2+2 a^2 c\right ) \text {arctanh}\left (\frac {\sqrt {b^2+a^2 x^3}}{b}\right )}{6 b^3}-\frac {\sqrt {-a+b} \left (3 a^2 b-a c\right ) \text {arctanh}\left (\frac {\sqrt {b^2+a^2 x^3}}{\sqrt {b} \sqrt {-a+b}}\right )}{3 b^{5/2}}-\frac {\sqrt {a+b} \left (3 a^2 b+a c\right ) \text {arctanh}\left (\frac {\sqrt {b^2+a^2 x^3}}{\sqrt {b} \sqrt {a+b}}\right )}{3 b^{5/2}} \]
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Time = 0.60 (sec) , antiderivative size = 359, normalized size of antiderivative = 1.73, number of steps used = 23, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.173, Rules used = {6857, 272, 43, 44, 65, 214, 52, 455, 211} \[ \int \frac {\sqrt {b^2+a^2 x^3} \left (2 b^2+c x^3+a^2 x^6\right )}{x^7 \left (-b^2+a^2 x^6\right )} \, dx=-\frac {a \sqrt {a-b} (3 a b-c) \arctan \left (\frac {\sqrt {a^2 x^3+b^2}}{\sqrt {b} \sqrt {a-b}}\right )}{3 b^{5/2}}+\frac {2 a^2 \text {arctanh}\left (\frac {\sqrt {a^2 x^3+b^2}}{b}\right )}{b}-\frac {a \sqrt {a+b} (3 a b+c) \text {arctanh}\left (\frac {\sqrt {a^2 x^3+b^2}}{\sqrt {b} \sqrt {a+b}}\right )}{3 b^{5/2}}+\frac {a^2 c \text {arctanh}\left (\frac {\sqrt {a^2 x^3+b^2}}{b}\right )}{3 b^3}+\frac {c \sqrt {a^2 x^3+b^2}}{3 b^2 x^3}-\frac {2 a^2 \sqrt {a^2 x^3+b^2}}{b^2}+\frac {a^2 \sqrt {a^2 x^3+b^2}}{6 b^2 x^3}+\frac {\sqrt {a^2 x^3+b^2}}{3 x^6}+\frac {a (3 a b-c) \sqrt {a^2 x^3+b^2}}{3 b^3}+\frac {a (3 a b+c) \sqrt {a^2 x^3+b^2}}{3 b^3}-\frac {a^4 \text {arctanh}\left (\frac {\sqrt {a^2 x^3+b^2}}{b}\right )}{6 b^3} \]
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Rule 43
Rule 44
Rule 52
Rule 65
Rule 211
Rule 214
Rule 272
Rule 455
Rule 6857
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {2 \sqrt {b^2+a^2 x^3}}{x^7}-\frac {c \sqrt {b^2+a^2 x^3}}{b^2 x^4}-\frac {3 a^2 \sqrt {b^2+a^2 x^3}}{b^2 x}-\frac {a^2 (3 a b+c) x^2 \sqrt {b^2+a^2 x^3}}{2 b^3 \left (b-a x^3\right )}+\frac {a^2 (3 a b-c) x^2 \sqrt {b^2+a^2 x^3}}{2 b^3 \left (b+a x^3\right )}\right ) \, dx \\ & = -\left (2 \int \frac {\sqrt {b^2+a^2 x^3}}{x^7} \, dx\right )-\frac {\left (3 a^2\right ) \int \frac {\sqrt {b^2+a^2 x^3}}{x} \, dx}{b^2}+\frac {\left (a^2 (3 a b-c)\right ) \int \frac {x^2 \sqrt {b^2+a^2 x^3}}{b+a x^3} \, dx}{2 b^3}-\frac {c \int \frac {\sqrt {b^2+a^2 x^3}}{x^4} \, dx}{b^2}-\frac {\left (a^2 (3 a b+c)\right ) \int \frac {x^2 \sqrt {b^2+a^2 x^3}}{b-a x^3} \, dx}{2 b^3} \\ & = -\left (\frac {2}{3} \text {Subst}\left (\int \frac {\sqrt {b^2+a^2 x}}{x^3} \, dx,x,x^3\right )\right )-\frac {a^2 \text {Subst}\left (\int \frac {\sqrt {b^2+a^2 x}}{x} \, dx,x,x^3\right )}{b^2}+\frac {\left (a^2 (3 a b-c)\right ) \text {Subst}\left (\int \frac {\sqrt {b^2+a^2 x}}{b+a x} \, dx,x,x^3\right )}{6 b^3}-\frac {c \text {Subst}\left (\int \frac {\sqrt {b^2+a^2 x}}{x^2} \, dx,x,x^3\right )}{3 b^2}-\frac {\left (a^2 (3 a b+c)\right ) \text {Subst}\left (\int \frac {\sqrt {b^2+a^2 x}}{b-a x} \, dx,x,x^3\right )}{6 b^3} \\ & = -\frac {2 a^2 \sqrt {b^2+a^2 x^3}}{b^2}+\frac {a (3 a b-c) \sqrt {b^2+a^2 x^3}}{3 b^3}+\frac {a (3 a b+c) \sqrt {b^2+a^2 x^3}}{3 b^3}+\frac {\sqrt {b^2+a^2 x^3}}{3 x^6}+\frac {c \sqrt {b^2+a^2 x^3}}{3 b^2 x^3}-\frac {1}{6} a^2 \text {Subst}\left (\int \frac {1}{x^2 \sqrt {b^2+a^2 x}} \, dx,x,x^3\right )-a^2 \text {Subst}\left (\int \frac {1}{x \sqrt {b^2+a^2 x}} \, dx,x,x^3\right )-\frac {\left (a^2 (a-b) (3 a b-c)\right ) \text {Subst}\left (\int \frac {1}{(b+a x) \sqrt {b^2+a^2 x}} \, dx,x,x^3\right )}{6 b^2}-\frac {\left (a^2 c\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {b^2+a^2 x}} \, dx,x,x^3\right )}{6 b^2}-\frac {\left (a^2 (a+b) (3 a b+c)\right ) \text {Subst}\left (\int \frac {1}{(b-a x) \sqrt {b^2+a^2 x}} \, dx,x,x^3\right )}{6 b^2} \\ & = -\frac {2 a^2 \sqrt {b^2+a^2 x^3}}{b^2}+\frac {a (3 a b-c) \sqrt {b^2+a^2 x^3}}{3 b^3}+\frac {a (3 a b+c) \sqrt {b^2+a^2 x^3}}{3 b^3}+\frac {\sqrt {b^2+a^2 x^3}}{3 x^6}+\frac {a^2 \sqrt {b^2+a^2 x^3}}{6 b^2 x^3}+\frac {c \sqrt {b^2+a^2 x^3}}{3 b^2 x^3}-2 \text {Subst}\left (\int \frac {1}{-\frac {b^2}{a^2}+\frac {x^2}{a^2}} \, dx,x,\sqrt {b^2+a^2 x^3}\right )+\frac {a^4 \text {Subst}\left (\int \frac {1}{x \sqrt {b^2+a^2 x}} \, dx,x,x^3\right )}{12 b^2}-\frac {((a-b) (3 a b-c)) \text {Subst}\left (\int \frac {1}{b-\frac {b^2}{a}+\frac {x^2}{a}} \, dx,x,\sqrt {b^2+a^2 x^3}\right )}{3 b^2}-\frac {c \text {Subst}\left (\int \frac {1}{-\frac {b^2}{a^2}+\frac {x^2}{a^2}} \, dx,x,\sqrt {b^2+a^2 x^3}\right )}{3 b^2}-\frac {((a+b) (3 a b+c)) \text {Subst}\left (\int \frac {1}{b+\frac {b^2}{a}-\frac {x^2}{a}} \, dx,x,\sqrt {b^2+a^2 x^3}\right )}{3 b^2} \\ & = -\frac {2 a^2 \sqrt {b^2+a^2 x^3}}{b^2}+\frac {a (3 a b-c) \sqrt {b^2+a^2 x^3}}{3 b^3}+\frac {a (3 a b+c) \sqrt {b^2+a^2 x^3}}{3 b^3}+\frac {\sqrt {b^2+a^2 x^3}}{3 x^6}+\frac {a^2 \sqrt {b^2+a^2 x^3}}{6 b^2 x^3}+\frac {c \sqrt {b^2+a^2 x^3}}{3 b^2 x^3}-\frac {a \sqrt {a-b} (3 a b-c) \arctan \left (\frac {\sqrt {b^2+a^2 x^3}}{\sqrt {a-b} \sqrt {b}}\right )}{3 b^{5/2}}+\frac {2 a^2 \text {arctanh}\left (\frac {\sqrt {b^2+a^2 x^3}}{b}\right )}{b}+\frac {a^2 c \text {arctanh}\left (\frac {\sqrt {b^2+a^2 x^3}}{b}\right )}{3 b^3}-\frac {a \sqrt {a+b} (3 a b+c) \text {arctanh}\left (\frac {\sqrt {b^2+a^2 x^3}}{\sqrt {b} \sqrt {a+b}}\right )}{3 b^{5/2}}+\frac {a^2 \text {Subst}\left (\int \frac {1}{-\frac {b^2}{a^2}+\frac {x^2}{a^2}} \, dx,x,\sqrt {b^2+a^2 x^3}\right )}{6 b^2} \\ & = -\frac {2 a^2 \sqrt {b^2+a^2 x^3}}{b^2}+\frac {a (3 a b-c) \sqrt {b^2+a^2 x^3}}{3 b^3}+\frac {a (3 a b+c) \sqrt {b^2+a^2 x^3}}{3 b^3}+\frac {\sqrt {b^2+a^2 x^3}}{3 x^6}+\frac {a^2 \sqrt {b^2+a^2 x^3}}{6 b^2 x^3}+\frac {c \sqrt {b^2+a^2 x^3}}{3 b^2 x^3}-\frac {a \sqrt {a-b} (3 a b-c) \arctan \left (\frac {\sqrt {b^2+a^2 x^3}}{\sqrt {a-b} \sqrt {b}}\right )}{3 b^{5/2}}-\frac {a^4 \text {arctanh}\left (\frac {\sqrt {b^2+a^2 x^3}}{b}\right )}{6 b^3}+\frac {2 a^2 \text {arctanh}\left (\frac {\sqrt {b^2+a^2 x^3}}{b}\right )}{b}+\frac {a^2 c \text {arctanh}\left (\frac {\sqrt {b^2+a^2 x^3}}{b}\right )}{3 b^3}-\frac {a \sqrt {a+b} (3 a b+c) \text {arctanh}\left (\frac {\sqrt {b^2+a^2 x^3}}{\sqrt {b} \sqrt {a+b}}\right )}{3 b^{5/2}} \\ \end{align*}
Time = 0.69 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.91 \[ \int \frac {\sqrt {b^2+a^2 x^3} \left (2 b^2+c x^3+a^2 x^6\right )}{x^7 \left (-b^2+a^2 x^6\right )} \, dx=-\frac {-\frac {b \sqrt {b^2+a^2 x^3} \left (2 b^2+\left (a^2+2 c\right ) x^3\right )}{x^6}+2 a \sqrt {a-b} \sqrt {b} (3 a b-c) \arctan \left (\frac {\sqrt {b^2+a^2 x^3}}{\sqrt {a-b} \sqrt {b}}\right )+a^2 \left (a^2-2 \left (6 b^2+c\right )\right ) \text {arctanh}\left (\frac {\sqrt {b^2+a^2 x^3}}{b}\right )+2 a \sqrt {b} \sqrt {a+b} (3 a b+c) \text {arctanh}\left (\frac {\sqrt {b^2+a^2 x^3}}{\sqrt {b} \sqrt {a+b}}\right )}{6 b^3} \]
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Time = 1.24 (sec) , antiderivative size = 204, normalized size of antiderivative = 0.99
method | result | size |
risch | \(\frac {\sqrt {a^{2} x^{3}+b^{2}}\, \left (a^{2} x^{3}+2 c \,x^{3}+2 b^{2}\right )}{6 b^{2} x^{6}}+\frac {a^{2} \left (-\frac {2 \left (a^{2}-12 b^{2}-2 c \right ) \operatorname {arctanh}\left (\frac {\sqrt {a^{2} x^{3}+b^{2}}}{\sqrt {b^{2}}}\right )}{3 \sqrt {b^{2}}}+\frac {2 \left (-6 a^{2} b +6 a \,b^{2}+2 a c -2 b c \right ) \arctan \left (\frac {\sqrt {a^{2} x^{3}+b^{2}}}{\sqrt {b \left (a -b \right )}}\right )}{3 \sqrt {b \left (a -b \right )}\, a}-\frac {2 \left (6 a^{2} b +6 a \,b^{2}+2 a c +2 b c \right ) \operatorname {arctanh}\left (\frac {\sqrt {a^{2} x^{3}+b^{2}}}{\sqrt {b \left (a +b \right )}}\right )}{3 \sqrt {b \left (a +b \right )}\, a}\right )}{4 b^{2}}\) | \(204\) |
pseudoelliptic | \(\frac {a^{4} \left (-12 \left (a -b \right ) x^{6} \left (a b -\frac {c}{3}\right ) a \sqrt {b \left (a +b \right )}\, b \arctan \left (\frac {\sqrt {a^{2} x^{3}+b^{2}}}{\sqrt {b \left (a -b \right )}}\right )+\left (-12 x^{6} a \left (a +b \right ) \left (a b +\frac {c}{3}\right ) b \,\operatorname {arctanh}\left (\frac {\sqrt {a^{2} x^{3}+b^{2}}}{\sqrt {b \left (a +b \right )}}\right )+\left (a^{2} x^{6} \left (a^{2}-12 b^{2}-2 c \right ) \ln \left (-b +\sqrt {a^{2} x^{3}+b^{2}}\right )-a^{2} x^{6} \left (a^{2}-12 b^{2}-2 c \right ) \ln \left (\sqrt {a^{2} x^{3}+b^{2}}+b \right )+2 b \sqrt {a^{2} x^{3}+b^{2}}\, \left (a^{2} x^{3}+2 c \,x^{3}+2 b^{2}\right )\right ) \sqrt {b \left (a +b \right )}\right ) \sqrt {b \left (a -b \right )}\right )}{12 \sqrt {b \left (a +b \right )}\, \sqrt {b \left (a -b \right )}\, b^{3} \left (\sqrt {a^{2} x^{3}+b^{2}}+b \right )^{2} \left (b -\sqrt {a^{2} x^{3}+b^{2}}\right )^{2}}\) | \(279\) |
default | \(\frac {\sqrt {a^{2} x^{3}+b^{2}}}{3 x^{6}}+\frac {a^{2} \sqrt {a^{2} x^{3}+b^{2}}}{6 b^{2} x^{3}}-\frac {a^{4} \operatorname {arctanh}\left (\frac {\sqrt {a^{2} x^{3}+b^{2}}}{\sqrt {b^{2}}}\right )}{6 b^{2} \sqrt {b^{2}}}-\frac {c \left (-\frac {\sqrt {a^{2} x^{3}+b^{2}}}{3 x^{3}}-\frac {a^{2} \operatorname {arctanh}\left (\frac {\sqrt {a^{2} x^{3}+b^{2}}}{\sqrt {b^{2}}}\right )}{3 \sqrt {b^{2}}}\right )}{b^{2}}-\frac {3 a^{2} \left (\frac {2 \sqrt {a^{2} x^{3}+b^{2}}}{3}-\frac {2 b^{2} \operatorname {arctanh}\left (\frac {\sqrt {a^{2} x^{3}+b^{2}}}{\sqrt {b^{2}}}\right )}{3 \sqrt {b^{2}}}\right )}{b^{2}}+\frac {a \left (3 a b +c \right ) \left (\sqrt {a^{2} x^{3}+b^{2}}-\frac {b \left (a +b \right ) \operatorname {arctanh}\left (\frac {\sqrt {a^{2} x^{3}+b^{2}}}{\sqrt {b \left (a +b \right )}}\right )}{\sqrt {b \left (a +b \right )}}\right )}{3 b^{3}}+\frac {a \left (3 a b -c \right ) \left (\sqrt {a^{2} x^{3}+b^{2}}-\frac {b \left (a -b \right ) \arctan \left (\frac {\sqrt {a^{2} x^{3}+b^{2}}}{\sqrt {b \left (a -b \right )}}\right )}{\sqrt {b \left (a -b \right )}}\right )}{3 b^{3}}\) | \(316\) |
elliptic | \(\text {Expression too large to display}\) | \(3271\) |
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Time = 0.77 (sec) , antiderivative size = 1051, normalized size of antiderivative = 5.08 \[ \int \frac {\sqrt {b^2+a^2 x^3} \left (2 b^2+c x^3+a^2 x^6\right )}{x^7 \left (-b^2+a^2 x^6\right )} \, dx=\left [-\frac {2 \, {\left (3 \, a^{2} b^{2} - a b c\right )} x^{6} \sqrt {-\frac {a - b}{b}} \log \left (\frac {a^{2} x^{3} - a b + 2 \, b^{2} + 2 \, \sqrt {a^{2} x^{3} + b^{2}} b \sqrt {-\frac {a - b}{b}}}{a x^{3} + b}\right ) - 2 \, {\left (3 \, a^{2} b^{2} + a b c\right )} x^{6} \sqrt {\frac {a + b}{b}} \log \left (\frac {a^{2} x^{3} + a b + 2 \, b^{2} - 2 \, \sqrt {a^{2} x^{3} + b^{2}} b \sqrt {\frac {a + b}{b}}}{a x^{3} - b}\right ) + {\left (a^{4} - 12 \, a^{2} b^{2} - 2 \, a^{2} c\right )} x^{6} \log \left (b + \sqrt {a^{2} x^{3} + b^{2}}\right ) - {\left (a^{4} - 12 \, a^{2} b^{2} - 2 \, a^{2} c\right )} x^{6} \log \left (-b + \sqrt {a^{2} x^{3} + b^{2}}\right ) - 2 \, \sqrt {a^{2} x^{3} + b^{2}} {\left ({\left (a^{2} b + 2 \, b c\right )} x^{3} + 2 \, b^{3}\right )}}{12 \, b^{3} x^{6}}, \frac {4 \, {\left (3 \, a^{2} b^{2} - a b c\right )} x^{6} \sqrt {\frac {a - b}{b}} \arctan \left (\frac {b \sqrt {\frac {a - b}{b}}}{\sqrt {a^{2} x^{3} + b^{2}}}\right ) + 2 \, {\left (3 \, a^{2} b^{2} + a b c\right )} x^{6} \sqrt {\frac {a + b}{b}} \log \left (\frac {a^{2} x^{3} + a b + 2 \, b^{2} - 2 \, \sqrt {a^{2} x^{3} + b^{2}} b \sqrt {\frac {a + b}{b}}}{a x^{3} - b}\right ) - {\left (a^{4} - 12 \, a^{2} b^{2} - 2 \, a^{2} c\right )} x^{6} \log \left (b + \sqrt {a^{2} x^{3} + b^{2}}\right ) + {\left (a^{4} - 12 \, a^{2} b^{2} - 2 \, a^{2} c\right )} x^{6} \log \left (-b + \sqrt {a^{2} x^{3} + b^{2}}\right ) + 2 \, \sqrt {a^{2} x^{3} + b^{2}} {\left ({\left (a^{2} b + 2 \, b c\right )} x^{3} + 2 \, b^{3}\right )}}{12 \, b^{3} x^{6}}, \frac {4 \, {\left (3 \, a^{2} b^{2} + a b c\right )} x^{6} \sqrt {-\frac {a + b}{b}} \arctan \left (\frac {b \sqrt {-\frac {a + b}{b}}}{\sqrt {a^{2} x^{3} + b^{2}}}\right ) - 2 \, {\left (3 \, a^{2} b^{2} - a b c\right )} x^{6} \sqrt {-\frac {a - b}{b}} \log \left (\frac {a^{2} x^{3} - a b + 2 \, b^{2} + 2 \, \sqrt {a^{2} x^{3} + b^{2}} b \sqrt {-\frac {a - b}{b}}}{a x^{3} + b}\right ) - {\left (a^{4} - 12 \, a^{2} b^{2} - 2 \, a^{2} c\right )} x^{6} \log \left (b + \sqrt {a^{2} x^{3} + b^{2}}\right ) + {\left (a^{4} - 12 \, a^{2} b^{2} - 2 \, a^{2} c\right )} x^{6} \log \left (-b + \sqrt {a^{2} x^{3} + b^{2}}\right ) + 2 \, \sqrt {a^{2} x^{3} + b^{2}} {\left ({\left (a^{2} b + 2 \, b c\right )} x^{3} + 2 \, b^{3}\right )}}{12 \, b^{3} x^{6}}, \frac {4 \, {\left (3 \, a^{2} b^{2} + a b c\right )} x^{6} \sqrt {-\frac {a + b}{b}} \arctan \left (\frac {b \sqrt {-\frac {a + b}{b}}}{\sqrt {a^{2} x^{3} + b^{2}}}\right ) + 4 \, {\left (3 \, a^{2} b^{2} - a b c\right )} x^{6} \sqrt {\frac {a - b}{b}} \arctan \left (\frac {b \sqrt {\frac {a - b}{b}}}{\sqrt {a^{2} x^{3} + b^{2}}}\right ) - {\left (a^{4} - 12 \, a^{2} b^{2} - 2 \, a^{2} c\right )} x^{6} \log \left (b + \sqrt {a^{2} x^{3} + b^{2}}\right ) + {\left (a^{4} - 12 \, a^{2} b^{2} - 2 \, a^{2} c\right )} x^{6} \log \left (-b + \sqrt {a^{2} x^{3} + b^{2}}\right ) + 2 \, \sqrt {a^{2} x^{3} + b^{2}} {\left ({\left (a^{2} b + 2 \, b c\right )} x^{3} + 2 \, b^{3}\right )}}{12 \, b^{3} x^{6}}\right ] \]
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Timed out. \[ \int \frac {\sqrt {b^2+a^2 x^3} \left (2 b^2+c x^3+a^2 x^6\right )}{x^7 \left (-b^2+a^2 x^6\right )} \, dx=\text {Timed out} \]
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\[ \int \frac {\sqrt {b^2+a^2 x^3} \left (2 b^2+c x^3+a^2 x^6\right )}{x^7 \left (-b^2+a^2 x^6\right )} \, dx=\int { \frac {{\left (a^{2} x^{6} + c x^{3} + 2 \, b^{2}\right )} \sqrt {a^{2} x^{3} + b^{2}}}{{\left (a^{2} x^{6} - b^{2}\right )} x^{7}} \,d x } \]
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Time = 0.28 (sec) , antiderivative size = 308, normalized size of antiderivative = 1.49 \[ \int \frac {\sqrt {b^2+a^2 x^3} \left (2 b^2+c x^3+a^2 x^6\right )}{x^7 \left (-b^2+a^2 x^6\right )} \, dx=-\frac {{\left (3 \, a^{3} b - 3 \, a^{2} b^{2} - a^{2} c + a b c\right )} \arctan \left (\frac {\sqrt {a^{2} x^{3} + b^{2}}}{\sqrt {a b - b^{2}}}\right )}{3 \, \sqrt {a b - b^{2}} b^{2}} + \frac {{\left (3 \, a^{3} b + 3 \, a^{2} b^{2} + a^{2} c + a b c\right )} \arctan \left (\frac {\sqrt {a^{2} x^{3} + b^{2}}}{\sqrt {-a b - b^{2}}}\right )}{3 \, \sqrt {-a b - b^{2}} b^{2}} - \frac {{\left (a^{4} - 12 \, a^{2} b^{2} - 2 \, a^{2} c\right )} \log \left ({\left | b + \sqrt {a^{2} x^{3} + b^{2}} \right |}\right )}{12 \, b^{3}} + \frac {{\left (a^{4} - 12 \, a^{2} b^{2} - 2 \, a^{2} c\right )} \log \left ({\left | -b + \sqrt {a^{2} x^{3} + b^{2}} \right |}\right )}{12 \, b^{3}} + \frac {\sqrt {a^{2} x^{3} + b^{2}} a^{4} b^{2} + {\left (a^{2} x^{3} + b^{2}\right )}^{\frac {3}{2}} a^{4} - 2 \, \sqrt {a^{2} x^{3} + b^{2}} a^{2} b^{2} c + 2 \, {\left (a^{2} x^{3} + b^{2}\right )}^{\frac {3}{2}} a^{2} c}{6 \, a^{4} b^{2} x^{6}} \]
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Time = 15.00 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.20 \[ \int \frac {\sqrt {b^2+a^2 x^3} \left (2 b^2+c x^3+a^2 x^6\right )}{x^7 \left (-b^2+a^2 x^6\right )} \, dx=\frac {\sqrt {a^2\,x^3+b^2}}{3\,x^6}+\frac {a^2\,\ln \left (\frac {{\left (b+\sqrt {a^2\,x^3+b^2}\right )}^3\,\left (b-\sqrt {a^2\,x^3+b^2}\right )}{x^6}\right )\,\left (-a^2+12\,b^2+2\,c\right )}{12\,b^3}+\frac {\sqrt {a^2\,x^3+b^2}\,\left (a^2+2\,c\right )}{6\,b^2\,x^3}+\frac {a\,\ln \left (\frac {a\,b+2\,b^2+a^2\,x^3-2\,\sqrt {b}\,\sqrt {a^2\,x^3+b^2}\,\sqrt {a+b}}{b-a\,x^3}\right )\,\sqrt {a+b}\,\left (c+3\,a\,b\right )}{6\,b^{5/2}}+\frac {a\,\ln \left (\frac {2\,b^2-a\,b+a^2\,x^3+2\,\sqrt {b}\,\sqrt {a^2\,x^3+b^2}\,\sqrt {b-a}}{a\,x^3+b}\right )\,\sqrt {b-a}\,\left (c-3\,a\,b\right )}{6\,b^{5/2}} \]
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