Integrand size = 17, antiderivative size = 113 \[ \int \frac {\sqrt {a x+b x^3}}{x} \, dx=\frac {2}{3} \sqrt {a x+b x^3}+\frac {2 a^{3/4} \sqrt {x} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{3 \sqrt [4]{b} \sqrt {a x+b x^3}} \]
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Time = 0.05 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {2046, 2036, 335, 226} \[ \int \frac {\sqrt {a x+b x^3}}{x} \, dx=\frac {2 a^{3/4} \sqrt {x} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{3 \sqrt [4]{b} \sqrt {a x+b x^3}}+\frac {2}{3} \sqrt {a x+b x^3} \]
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Rule 226
Rule 335
Rule 2036
Rule 2046
Rubi steps \begin{align*} \text {integral}& = \frac {2}{3} \sqrt {a x+b x^3}+\frac {1}{3} (2 a) \int \frac {1}{\sqrt {a x+b x^3}} \, dx \\ & = \frac {2}{3} \sqrt {a x+b x^3}+\frac {\left (2 a \sqrt {x} \sqrt {a+b x^2}\right ) \int \frac {1}{\sqrt {x} \sqrt {a+b x^2}} \, dx}{3 \sqrt {a x+b x^3}} \\ & = \frac {2}{3} \sqrt {a x+b x^3}+\frac {\left (4 a \sqrt {x} \sqrt {a+b x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^4}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {a x+b x^3}} \\ & = \frac {2}{3} \sqrt {a x+b x^3}+\frac {2 a^{3/4} \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 )}{3 \sqrt [4]{b} \sqrt {a x+b x^3}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.01 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.42 \[ \int \frac {\sqrt {a x+b x^3}}{x} \, dx=\frac {2 \sqrt {x \left (a+b x^2\right )} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{4},\frac {5}{4},-\frac {b x^2}{a}\right )}{\sqrt {1+\frac {b x^2}{a}}} \]
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Time = 2.18 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.10
method | result | size |
default | \(\frac {2 \sqrt {b \,x^{3}+a x}}{3}+\frac {2 a \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{3 b \sqrt {b \,x^{3}+a x}}\) | \(124\) |
elliptic | \(\frac {2 \sqrt {b \,x^{3}+a x}}{3}+\frac {2 a \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{3 b \sqrt {b \,x^{3}+a x}}\) | \(124\) |
risch | \(\frac {2 x \left (b \,x^{2}+a \right )}{3 \sqrt {x \left (b \,x^{2}+a \right )}}+\frac {2 a \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{3 b \sqrt {b \,x^{3}+a x}}\) | \(132\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.30 \[ \int \frac {\sqrt {a x+b x^3}}{x} \, dx=\frac {2 \, {\left (2 \, a \sqrt {b} {\rm weierstrassPInverse}\left (-\frac {4 \, a}{b}, 0, x\right ) + \sqrt {b x^{3} + a x} b\right )}}{3 \, b} \]
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\[ \int \frac {\sqrt {a x+b x^3}}{x} \, dx=\int \frac {\sqrt {x \left (a + b x^{2}\right )}}{x}\, dx \]
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\[ \int \frac {\sqrt {a x+b x^3}}{x} \, dx=\int { \frac {\sqrt {b x^{3} + a x}}{x} \,d x } \]
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\[ \int \frac {\sqrt {a x+b x^3}}{x} \, dx=\int { \frac {\sqrt {b x^{3} + a x}}{x} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {a x+b x^3}}{x} \, dx=\int \frac {\sqrt {b\,x^3+a\,x}}{x} \,d x \]
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