Integrand size = 17, antiderivative size = 248 \[ \int \frac {\sqrt {a x+b x^3}}{x^2} \, dx=\frac {4 \sqrt {b} x \left (a+b x^2\right )}{\left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {a x+b x^3}}-\frac {2 \sqrt {a x+b x^3}}{x}-\frac {4 \sqrt [4]{a} \sqrt [4]{b} \sqrt {x} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{\sqrt {a x+b x^3}}+\frac {2 \sqrt [4]{a} \sqrt [4]{b} \sqrt {x} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{\sqrt {a x+b x^3}} \]
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Time = 0.13 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {2045, 2057, 335, 311, 226, 1210} \[ \int \frac {\sqrt {a x+b x^3}}{x^2} \, dx=\frac {2 \sqrt [4]{a} \sqrt [4]{b} \sqrt {x} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{\sqrt {a x+b x^3}}-\frac {4 \sqrt [4]{a} \sqrt [4]{b} \sqrt {x} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{\sqrt {a x+b x^3}}-\frac {2 \sqrt {a x+b x^3}}{x}+\frac {4 \sqrt {b} x \left (a+b x^2\right )}{\left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {a x+b x^3}} \]
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Rule 226
Rule 311
Rule 335
Rule 1210
Rule 2045
Rule 2057
Rubi steps \begin{align*} \text {integral}& = -\frac {2 \sqrt {a x+b x^3}}{x}+(2 b) \int \frac {x}{\sqrt {a x+b x^3}} \, dx \\ & = -\frac {2 \sqrt {a x+b x^3}}{x}+\frac {\left (2 b \sqrt {x} \sqrt {a+b x^2}\right ) \int \frac {\sqrt {x}}{\sqrt {a+b x^2}} \, dx}{\sqrt {a x+b x^3}} \\ & = -\frac {2 \sqrt {a x+b x^3}}{x}+\frac {\left (4 b \sqrt {x} \sqrt {a+b x^2}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {a+b x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {a x+b x^3}} \\ & = -\frac {2 \sqrt {a x+b x^3}}{x}+\frac {\left (4 \sqrt {a} \sqrt {b} \sqrt {x} \sqrt {a+b x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {a x+b x^3}}-\frac {\left (4 \sqrt {a} \sqrt {b} \sqrt {x} \sqrt {a+b x^2}\right ) \text {Subst}\left (\int \frac {1-\frac {\sqrt {b} x^2}{\sqrt {a}}}{\sqrt {a+b x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {a x+b x^3}} \\ & = \frac {4 \sqrt {b} x \left (a+b x^2\right )}{\left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {a x+b x^3}}-\frac {2 \sqrt {a x+b x^3}}{x}-\frac {4 \sqrt [4]{a} \sqrt [4]{b} \sqrt {x} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{\sqrt {a x+b x^3}}+\frac {2 \sqrt [4]{a} \sqrt [4]{b} \sqrt {x} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{\sqrt {a x+b x^3}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.02 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.21 \[ \int \frac {\sqrt {a x+b x^3}}{x^2} \, dx=-\frac {2 \sqrt {x \left (a+b x^2\right )} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},-\frac {1}{4},\frac {3}{4},-\frac {b x^2}{a}\right )}{x \sqrt {1+\frac {b x^2}{a}}} \]
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Time = 2.13 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.71
method | result | size |
default | \(-\frac {2 \left (b \,x^{2}+a \right )}{\sqrt {x \left (b \,x^{2}+a \right )}}+\frac {2 \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {-a b}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{\sqrt {b \,x^{3}+a x}}\) | \(177\) |
risch | \(-\frac {2 \left (b \,x^{2}+a \right )}{\sqrt {x \left (b \,x^{2}+a \right )}}+\frac {2 \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {-a b}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{\sqrt {b \,x^{3}+a x}}\) | \(177\) |
elliptic | \(-\frac {2 \left (b \,x^{2}+a \right )}{\sqrt {x \left (b \,x^{2}+a \right )}}+\frac {2 \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {-a b}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{\sqrt {b \,x^{3}+a x}}\) | \(177\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.08 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.16 \[ \int \frac {\sqrt {a x+b x^3}}{x^2} \, dx=-\frac {2 \, {\left (2 \, \sqrt {b} x {\rm weierstrassZeta}\left (-\frac {4 \, a}{b}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, a}{b}, 0, x\right )\right ) + \sqrt {b x^{3} + a x}\right )}}{x} \]
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\[ \int \frac {\sqrt {a x+b x^3}}{x^2} \, dx=\int \frac {\sqrt {x \left (a + b x^{2}\right )}}{x^{2}}\, dx \]
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\[ \int \frac {\sqrt {a x+b x^3}}{x^2} \, dx=\int { \frac {\sqrt {b x^{3} + a x}}{x^{2}} \,d x } \]
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\[ \int \frac {\sqrt {a x+b x^3}}{x^2} \, dx=\int { \frac {\sqrt {b x^{3} + a x}}{x^{2}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {a x+b x^3}}{x^2} \, dx=\int \frac {\sqrt {b\,x^3+a\,x}}{x^2} \,d x \]
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