Integrand size = 35, antiderivative size = 60 \[ \int \frac {\left (-2 c+a x^3\right ) \sqrt {c+b x^2+a x^3}}{\left (c+a x^3\right )^2} \, dx=-\frac {x \sqrt {c+b x^2+a x^3}}{c+a x^3}-\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {c+b x^2+a x^3}}\right )}{\sqrt {b}} \]
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Timed out. \[ \int \frac {\left (-2 c+a x^3\right ) \sqrt {c+b x^2+a x^3}}{\left (c+a x^3\right )^2} \, dx=\text {\$Aborted} \]
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Rubi steps Aborted
Time = 0.47 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.97 \[ \int \frac {\left (-2 c+a x^3\right ) \sqrt {c+b x^2+a x^3}}{\left (c+a x^3\right )^2} \, dx=-\frac {x \sqrt {c+x^2 (b+a x)}}{c+a x^3}-\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {c+x^2 (b+a x)}}\right )}{\sqrt {b}} \]
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Time = 7.47 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.92
method | result | size |
default | \(-\frac {x \sqrt {a \,x^{3}+b \,x^{2}+c}}{a \,x^{3}+c}-\frac {\operatorname {arctanh}\left (\frac {\sqrt {a \,x^{3}+b \,x^{2}+c}}{x \sqrt {b}}\right )}{\sqrt {b}}\) | \(55\) |
pseudoelliptic | \(-\frac {x \sqrt {a \,x^{3}+b \,x^{2}+c}}{a \,x^{3}+c}-\frac {\operatorname {arctanh}\left (\frac {\sqrt {a \,x^{3}+b \,x^{2}+c}}{x \sqrt {b}}\right )}{\sqrt {b}}\) | \(55\) |
elliptic | \(\text {Expression too large to display}\) | \(5094\) |
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none
Time = 0.28 (sec) , antiderivative size = 244, normalized size of antiderivative = 4.07 \[ \int \frac {\left (-2 c+a x^3\right ) \sqrt {c+b x^2+a x^3}}{\left (c+a x^3\right )^2} \, dx=\left [-\frac {4 \, \sqrt {a x^{3} + b x^{2} + c} b x - {\left (a x^{3} + c\right )} \sqrt {b} \log \left (\frac {a^{2} x^{6} + 8 \, a b x^{5} + 8 \, b^{2} x^{4} + 2 \, a c x^{3} + 8 \, b c x^{2} - 4 \, {\left (a x^{4} + 2 \, b x^{3} + c x\right )} \sqrt {a x^{3} + b x^{2} + c} \sqrt {b} + c^{2}}{a^{2} x^{6} + 2 \, a c x^{3} + c^{2}}\right )}{4 \, {\left (a b x^{3} + b c\right )}}, -\frac {2 \, \sqrt {a x^{3} + b x^{2} + c} b x - {\left (a x^{3} + c\right )} \sqrt {-b} \arctan \left (\frac {{\left (a x^{3} + 2 \, b x^{2} + c\right )} \sqrt {a x^{3} + b x^{2} + c} \sqrt {-b}}{2 \, {\left (a b x^{4} + b^{2} x^{3} + b c x\right )}}\right )}{2 \, {\left (a b x^{3} + b c\right )}}\right ] \]
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Timed out. \[ \int \frac {\left (-2 c+a x^3\right ) \sqrt {c+b x^2+a x^3}}{\left (c+a x^3\right )^2} \, dx=\text {Timed out} \]
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\[ \int \frac {\left (-2 c+a x^3\right ) \sqrt {c+b x^2+a x^3}}{\left (c+a x^3\right )^2} \, dx=\int { \frac {\sqrt {a x^{3} + b x^{2} + c} {\left (a x^{3} - 2 \, c\right )}}{{\left (a x^{3} + c\right )}^{2}} \,d x } \]
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\[ \int \frac {\left (-2 c+a x^3\right ) \sqrt {c+b x^2+a x^3}}{\left (c+a x^3\right )^2} \, dx=\int { \frac {\sqrt {a x^{3} + b x^{2} + c} {\left (a x^{3} - 2 \, c\right )}}{{\left (a x^{3} + c\right )}^{2}} \,d x } \]
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Time = 6.69 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.32 \[ \int \frac {\left (-2 c+a x^3\right ) \sqrt {c+b x^2+a x^3}}{\left (c+a x^3\right )^2} \, dx=\frac {\ln \left (\frac {c+a\,x^3+2\,b\,x^2-2\,\sqrt {b}\,x\,\sqrt {a\,x^3+b\,x^2+c}}{c^2+a\,c\,x^3}\right )}{2\,\sqrt {b}}-\frac {x\,\sqrt {a\,x^3+b\,x^2+c}}{a\,x^3+c} \]
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